Equations In Plane Research Articles

A novel method: YOLO-CE and 3D point cloud-based feature extraction for welding seams of tower bases

Abstract Robotic automated welding of non-standard steel structures presents significant challenges, particularly for electric power tower bases. This study introduces a novel approach that integrates the You Only Look Once—Compact Invert Block and Efficient Local Attention (YOLO-CE) model, an enhanced version of YOLOV8 for 2D image segmentation, with 3D point cloud technology. The YOLO-CE model is used to accurately extract point cloud data from the target area, which is then processed using the MSAC algorithm for efficient plane segmentation. Weld lines are identified through plane equations, allowing for initial weld point cloud extraction. To further refine accuracy, an optimized evaluation equation is developed that accounts for both the distance between the weld point cloud and the fitted plane, and the angle between their normal vectors. This enables precise classification of the weld point cloud. From this classification, key weld feature points are identified, and their exact positions are determined by calculating the distances between these points and their intersections with three planes. The reliability of the proposed method was validated using a robot for precise measurements, with a total error margin of less than 1.5084 mm, demonstrating high accuracy and stability. Post-operation inspections confirmed that the welds were filled and free from defects, meeting all process requirements. The YOLO-CE model achieved a mIoU of 96.38% and a precision of 99.8%, highlighting its effectiveness. This method provides an efficient and precise solution for the automated welding of non-standard steel structural components and has promising application potential.

Review of Some Modified Generalized Korteweg–de Vries–Kuramoto–Sivashinsky Equations (Part II)

In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form N(ux), where ux denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto–Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in Rn. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case.

Review of Some Modified Generalized Korteweg–De Vries–Kuramoto–Sivashinsky (mgKdV-KS) Equations

This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg–De Vries equation, the Kuramoto–Sivashinsky equation, the generalized Korteweg–De Vries–Kuramoto–Sivashinsky equation and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane.

Devising an analytical method for solving the eighth-order Kolmogorov equations for an asymmetric Markov chain

The object of research is a complex system of three subsystems, which function independently of each other and are in a working or failed state. There is a need to analytically model and manage the Markov random process in the system, varying the intensity of their development-restoration and degradation-destruction flows. In the study, an analytical method for solving Kolmogorov equations of the eighth order for an asymmetric Markov chain was devised. The corresponding Kolmogorov equations of the eighth order have an ordered transition probability matrix. The distribution of the eight roots of this equation in the complex plane has central symmetry. The results are analytical solutions for the probabilities of the eight states of the Markov chain in time in the form of ordered determinants with respect to the indices of the eight roots and the indices of the eight states, including the column vector of the initial conditions. Symmetry has been established in the distribution on the complex plane of eight real, negative roots of the characteristic Kolmogorov equation centered at the point defined as Re ϑ = –a7/8, where a7 is the coefficient of the characteristic equation of the eighth degree at the seventh power. Formulas expressing eight roots of the characteristic Kolmogorov equation have been heuristically derived, one of which is zero, due to the intensities of failures and recovery of three subsystems, the eight states of which in general make up an asymmetric Markov chain. For structures consisting of three independently functioning processes, the random process of the transition of the structure through eight possible states with a known initial state is determined in time. An analytical solution to Kolmogorov differential equations of the eighth order for an asymmetric state graph is proposed in harmonic form for the purpose of analysis and synthesis of a random Markov process in a triple system.

Multi-line laser 3D reconstruction method based on spatial quadric surface and geometric estimation

The traditional multi-line laser 3D reconstruction is based on binocular epipolar constraints and the equation of the laser optical plane. Firstly, matching points are identified based on the epipolar constraints. Then, the correct matching points are selected based on the optical plane equation of the multi-line laser. Finally, 3D reconstruction is performed using the selected matching points. In actual production processes involving multi-line lasers, noticeable distortion occurs due to the projection of Diffractive Optical Elements (DOE) at a large angle. This results in curved laser light planes and curved reflections on the measured objects. Under such circumstances, efficiently screening matching points using the traditional method based on the laser plane equation becomes challenging. Additionally, inherent noise in multi-line laser systems introduces errors in extracted laser center coordinates, making it impossible to directly obtain high-precision 3D data. To address these issues, this paper proposes a method that utilizes spatial quadric surfaces and geometric estimation for completing multi-line laser 3D reconstructions. By analyzing the distortion principle of DOE and the positional relationship after optical diffraction, the multi-line laser manifests as a quadric surface on the optical output plane. Consequently, by calibrating the equations of the quadric surface and applying binocular polar constraints, suitable matching points for the multi-line laser can be chosen. Once an accurate matching point is identified, a minimum geometric distance estimation can be established based on the distance constraint between the point and its corresponding epipolar line. This distance represents the separation between the left and right camera’s laser center points and their respective epipolar lines. Utilizing this estimate of the minimum geometric distance, a refined calculation can be performed to better satisfy epipolar constraints and obtain new matching points. Ultimately, utilizing these new matching points enables the completion of 3D reconstruction for multi-line lasers. In comparison with methods relying on spatial plane and epipolar constraints, our algorithm exhibits a superior degree of matchability and accuracy.According to multiple groups of test data, the average error before optimization is 0.3126 mm, and can be increased to 0.0336 mm after optimization.

On a planar equation involving [formula omitted]-Laplacian with zero mass and Trudinger–Moser nonlinearity

In this work, we study existence of positive solutions to a class of (2,q)-equations in the zero mass case in R2. We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.

Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation

Abstract Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on tracking the dynamics of the singularities in the complexified space domain all the way from the initial time until the blow-up time, which occurs when the singularities reach the real axis. This widely applicable approach gives forewarning of the possibility of blow up and an understanding of the influence of singularities on the solution behaviour on the real axis, aiding the (perhaps surprisingly involved) asymptotic analysis of the real-line behaviour. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales over which blow up is approached. The solution to the nonlinear heat equation in the complex spatial plane is shown to be related asymptotically to a nonlinear ordinary differential equation. This latter equation is studied in detail, including its computation on multiple Riemann sheets, providing further insight into the singularities of blow-up solutions of the nonlinear heat equation when viewed as multivalued functions in the complex space domain and illustrating the potential intricacy of singularity dynamics in such (non-integrable) nonlinear contexts.

Vanishing shear viscosity limit for the compressible planar MHD system with boundary layer

This paper is devoted to the study of the vanishing shear viscosity limit and strong boundary layer problem for the compressible, viscous, and heat-conducting planar MHD equations. The main aim is to obtain a sharp convergence rate which is usually connected to the boundary layer thickness. However, The convergence rate would be possibly slowed down due to the presence of the strong boundary layer effect and the interactions among the magnetic field, temperature, and fluids through not only the velocity equations but also the strongly nonlinear terms in the temperature equation. Our main strategy is to construct some new functions via asymptotic matching method which can cancel some quantities decaying in a lower speed. It leads to a sharp L∞ convergence rate as the shear viscosity vanishes for global-in-time solution with arbitrarily large initial data.

Flexible technique for global calibration of multi-view stereo vision with a non-overlapping FOV using linear structured light projector

A flexible global calibration method for multi-view stereo vision with non-overlapping field of view (FOV) using linear structured light projector is presented in this paper. At least three groups of linear structured light plane equations are utilized to calculate the rotation between any pair of cameras with non-overlapping FOV, and then the closed solution of the translation is determined based on the constraint that the normal vector is perpendicular to the lines on linear structured light plane. Once the rotation and translation between all pairs of cameras with non-overlapping FOV are obtained, the rotation and translation of each camera relative to the reference position is eventually computed through global optimization. The experimental data reflects that the calibration accuracy is improved by 35.2% after global optimization, and the global calibration results of the proposed method deviates less from that of Zhang’s method.

Dirichlet Problem for Nonlinear Higher-Order Equations in Upper Half Plane

In this article, we consider the Dirichlet problem for nonlinear higher-order equations in upper half plane. Firstly we introduce the solutions of inhomogeneous polyanalytic equation in upper half plane. Then we investigate the properties of relevant integral operators. Lastly we transform the Dirichlet problem for nonlinear higher-order equations in upper half plane into the system of integro-differential equations and we obtain the existence of unique solution using Banach fixed point theorem.

A geometric solution to microphone array position calibration and its confidence interval analysis

Wireless acoustic sensor networks (WASNs) comprised of spatially distributed acoustic nodes have been increasingly popular in many areas such as speech enhancement, blind source separation, and audio surveillance, where node position information is generally required to implement relevant processing algorithms. In this article, we focus on the node position calibration only using direction-of-arrival (DOA) information with each node consisting of a microphone array. State-of-the-art techniques translate the position calibration into optimization of DOA-based cost functions, and have shown notable success. However, a compromise between position calibration accuracy and computational load still exists. Furthermore, none of the existing studies can offer a quantitative evaluation of calibration performance before the optimization procedure is finished. To address these problems, a geometric solution to array node position calibration is proposed using DOA measurements. For a given array node, DOAs of three sources are first utilized to construct cylinder, plane, and cone equations. Next, the two-dimensional projection of intersections among these equations is calculated using Sylvesterresultantmatrix. Finally, a closed-form solution of node position in three-dimensional space is established. Moreover, explicit mathematical derivation of the confidence interval analysis on the calibration results under given node angle estimation error condition is presented, indicating that the proposed method can provide predictable performance with corresponding probabilities. The Cramér-Rao bound (CRB) is further investigated. Simulation results reveal that as compared with existing methods our approach achieves comparable calibration accuracy with much less computational burden. Real-world experiments also verify its effectiveness.

Multi-method examination of contact mechanics in orthotropic layers under gravity

Analyzing the behavior of intelligent unconventional materials under diverse contact scenarios, in comparison to conventional materials, is a critical step in the initial design of engineering systems. This paper presents the development of analytical and numerical approaches for analyzing contact mechanics in a system comprising an orthotropic layer resting on a rigid foundation. Parametric analyses include the consideration of cylindrical and flat punch profiles. Analytical formulation utilizes integral transform methods to convert planar elasticity equations into a Cauchy-type singular integral equation of the second kind. A detailed description of the solution technique for the integral equation is provided, encompassing both the analytical formulation and the required discretization for obtaining the solution. Subsequently, a finite element approach (FEA) is employed to approximate contact bodies using a collection of finite elements, while contact boundaries are approximated by utilizing a set of polygons. Alternatively, the problem is addressed using the Multilayer Perceptron approach, a form of artificial neural network frequently applied in diverse machine learning applications, including scenarios involving contact problems. Finally, the resolution to the problem is achieved by employing the multilayer perceptron (MLP) method. The study yields determinations of contact stresses under the punch, the critical separation load resulting in the detachment of the orthotropic layer from the rigid foundation, and the corresponding separation distance. This analysis considers a range of dimensionless parameters and explored the behavior of diverse orthotropic materials. Comparisons between the results of the analytical method and computations from FEA and MLP reveal the exceptional accuracy attained by all three approaches. As the pioneer study employing three distinct approaches to examine continuous contact mechanics within an orthotropic layer under the influence of gravity, this research constitutes a valuable point of reference for upcoming scholars in the field.

On smoothness conditions and selection of the edge of a scattering surface in one class of 3D time-optimal problems

A class of time-optimal problems in three-dimensional space with a spherical velocity vectogram is considered. The target set is a parametrically defined smooth curve. Numerical and analytical approaches to constructing the bisector of the target set — the scattering surface — in the time-optimal problem are proposed. The algorithms are based on formulas for the edge points of the scattering surface, written in terms of curve invariants. It is shown that these points form the edge of the bisector and lie at the centers of the touching spheres to the curve. A theorem on sufficient conditions for the smoothness of a scattering surface is proven. The equations of the tangent plane to the bisector are found for those points from which exactly two optimal trajectories emerge. An example of solving a time-optimal problem in the form of a set of level surfaces of the optimal result function is given, highlighting the surface of their non-smoothness.

Analytical modelling and design of linear controlled dynamic systems

A generalized canonical mathematical model of multidimensional controlled mining transportation complexes and other technical systems in form by A.M. Letov is considered. A principle of symmetry and an algebraic concept are in the basis of the developed analytical methods of design. A principle of symmetry is realized on a set of indexes of roots of a characteristic equation of the system and on a set of indexes of phase coordinates of the mathematical model. The problem of quality of dynamic processes in time is reduced to an algebraic problem of distribution of roots of a corresponding characteristic equation in a complex plane. An analogy in a procedure of transformation of a characteristic determinant into a polynomial and a structure of elementary symmetrical polynomials of roots is established. A new formulation of an analytical representation of change of phase coordinates in time in a form of ordered determinants with respect to indexes of roots and indexes of phase coordinates is obtained based on residue theorem. An illustration of the developed analytical method of design is performed on a special case of a well-known controlled technical system of the fourth order.

Growth of meromorphic solution related to linear difference equations and value sharing

In this paper, we consider the growth of meromorphic solutions of complex linear difference equations of certain types and derive some results, which improve the results of Chiang and Feng [On the Nevanlinna characteristic of f ( z + η ) and difference equations in the complex plane. Ramanujan J. 2008;16(1):105–129; On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans Am Math Soc. 2009;361:3767–3791] and Chen [Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl. 2011;373:235–241]. Also in this paper, we consider the sharing value problems of entire functions with their difference operators and obtain a result which improves the result of Li, Yang and Yi [Entire functions sharing an entire function of smaller order with their shifts. Proc Jpn Acad. 89 (A) (2013), 34–39]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.

A logarithmic Choquard equation with critical exponential growth in ℝ2

ABSTRACT In this paper, we are paid attention to the following logarithmic Choquard equation − Δ u + u = [ I 2 ∗ F ( u ) ] f ( u ) , x ∈ R 2 , where I 2 = 1 2 π ln ⁡ 1 | x | is the Newtonian kernel in dimension 2, F ( u ) = ∫ 0 u f ( t ) d t and f ∈ C 1 ( R , R ) is critical exponential growth with respect to Trudinger–Moser. Based on a variational framework proposed in Liu et al. (J Differ Equ 2022;328), we obtain a positive finite energy solution of the above logarithmic Choquard equation via the variational methods. Furthermore, we use another approach to investigate the planar logarithmic Choquard equations involving critical exponential growth.

Navier–Stokes Equations in the Half Space with Non Compatible Data

This paper considers the Navier–Stokes equations in the half plane with Euler-type initial conditions, i.e., initial conditions with a non-zero tangential component at the boundary. Under analyticity assumptions for the data, we prove that the solution exists for a short time independent of the viscosity. We construct the Navier–Stokes solution through a composite asymptotic expansion involving solutions of the Euler and Prandtl equations plus an error term. The norm of the error goes to zero with the square root of the viscosity. The Prandtl solution contains a singular term, which influences the regularity of the error. The error term is the sum of a first-order Euler correction, a first-order Prandtl correction, and a further error term. The use of an analytic setting is mainly due to the Prandtl equation.

Investigating the fractional wave function and the impact of topological defects with anisotropic plasma on the dissociation of bottomonium in the fractional non-relativistic quark model

Incorporating a topological defect and anisotropic plasma, this work used the generalized fractional of the Nikiforov–Uvarov technique to solve the fractional-radial Schrödinger equation in the longitudinal-transverse plane. The study produced wave functions and energy eigenvalues in their fractional forms. The results showed that the presence of an anisotropic plasma and a topological defect increases the dissociation energy of bottomonium. Furthermore, regardless of whether the fractional or classical models are taken into account, it was shown that the effect of temperature on the dissociation energy is stronger than the effect of baryonic chemical potential. In addition, the dissociation energy of bottomonium is significantly larger at lower chemical potential levels. Last but not least, the energy of bottomonium is only little influenced by magnetic auxiliaries.

Spatial quadric calibration method for multi-line laser based on diffractive optical element

The traditional multi-line laser calibration method relies primarily on calibrating the spatial plane equations of multiple laser lines and reconstructing the multi-line laser in three dimensions through the plane equation and camera parameters. In traditional methods, the multi-line laser is projected as a straight line by default, but in reality, the laser line projected by the multi-line laser is not a plane due to the Diffractive Optical Element (DOE). Instead, it is a surface with a certain curvature in space. During the production process, the multi-line laser experiences significant distortion because of the DOE projecting at a large angle. Consequently, if traditional methods are used, accurate calibration of the multi-line laser cannot be achieved. To address this issue, this paper introduces a new calibration algorithm for multi-line laser based on a spatial quadric equation. By calibrating the spatial quadric equation of the multi-line laser, the spatial characteristics of the laser can be accurately captured. The algorithm involves projecting a crossed 7-line laser onto a calibration plate and changing the position of the calibration plate while capturing images. This allows for obtaining the spatial three-dimensional coordinates of each laser line in the camera coordinate system. Next, spatial quadric equations are fitted for each laser line based on the three-dimensional coordinates. Finally, 3D reconstruction is performed using the quadratic surface parameters of the multi-line laser and camera parameters. The results from 3D reconstruction demonstrate that, compared to traditional laser calibration methods based on planes, the proposed algorithm achieves higher calibration accuracy. It also improves the number of successful binocular multi-line laser matches and overall accuracy.

Temporal evolution of sound pulse in shallow water under conditions of horizontal refraction. Vertical modes and space-time horizontal rays

As is known, to describe horizontal refraction in a shallow water waveguide, the “vertical modes and horizontal rays” approach (Burridge & Weinberg, 1974) is used, within which the two-dimensional eikonal equation for the modal amplitude includes the refractive index determined by the eigenvalues depending on the horizontal coordinates (x,y) and frequency (f). In other words, we have an effective two-dimensional inhomogeneous dispersive medium, to describe the propagation of signals in which, the authors propose to use the so-called space-time rays (Rytov, 1940, Connor & Felsen, 1974, Hillion, 1993) applied for the horizontal plane (STHR). The set of STHRs in a given situation is constructed for each mode. Within the framework of this approach, it is possible to obtain the evolution of the pulses of each mode from the solution of the nonstationary eikonal equation in the horizontal plane or from the corresponding Hamilton-Jacobi equations, obtained using non-local integral wave equation. Examples of constructing STHR in some typical situations (in particular in a wedge) are considered, some specific effects during signal propagation are considered, in particular, the so-called the pulse front tilt, previously discovered in laser optics (Bor & Rasz, 1985). Work was supported by ISF, grant 946/20.