Search Results (177)

This paper extends Woźny and Chudy’s linear-complexity Bézier evaluation algorithm (2020) to all parametric curves/surfaces with normalized basis functions via a novel basis function matrix decomposition. The unified framework covers the following: (i) B-spline/NURBS models; (ii) Bézier-type surfaces (tensor-product, rational, and triangular); (iii) enhanced models with shape parameters or non-polynomial basis spaces. For curves, we propose sequential and reverse corner-cutting modes. Surface evaluation adapts to type: non-tensor-product surfaces are processed through index-linearization to match the curve format, while tensor-product surfaces utilize nested curve evaluation. This approach reduces computational complexity, resolves cross-model compatibility issues, and establishes an efficient evaluation framework for diverse parametric geometries. Full article

Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of applied mathematics and mathematical physics. Here, we describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii–Vorob’ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely, a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article

The present review discusses the solution of the Heun, confluent, biconfluent, double confluent, and triconfluent equations in terms of polynomial expansions, and applies the results to generate exactly solvable Schrödinger potentials. Although there are more general approaches to solve these differential equations in terms of the expansions of certain special functions, the importance of polynomial solutions is unquestionable, as most of the known potentials are solvable in terms of the hypergeometric and confluent hypergeometric functions; i.e., Natanzon-class potentials possess bound-state solutions in terms of classical orthogonal polynomials, to which the (confluent) hypergeometric functions can be reduced. Since some of the Heun-type equations contain the hypergeometric and/or confluent hypergeometric differential equations as special limits, the potentials generated from them may also contain Natanzon-class potentials as special cases. A power series expansion is assumed around one of the singular points of each differential equation, and recurrence relations are obtained for the expansion coefficients. With the exception of the triconfluent Heun equations, these are three-term recurrence relations, the termination of which is achieved by prescribing certain conditions. In the case of the biconfluent and double confluent Heun equations, the expansion coefficients can be obtained in the standard way, i.e., after finding the roots of an (N + 1)th-order polynomial in one of the parameters, which, in turn, follows from requiring the vanishing of an (N + 1) × (N + 1) determinant. However, in the case of the Heun and confluent Heun equations, the recurrence relation can be solved directly, and the solutions are obtained in terms of rationally extended X₁-type Jacobi and Laguerre polynomials, respectively. Examples for solvable potentials are presented for the Heun, confluent, biconfluent, and double confluent Heun equations, and alternative methods for obtaining the same potentials are also discussed. These are the schemes based on the rational extension of Bochner-type differential equations (for the Heun and confluent Heun equation) and solutions based on quasi-exact solvability (QES) and on continued fractions (for the biconfluent and double confluent equation). Possible further lines of investigations are also outlined concerning physical problems that require the solution of second-order differential equations, i.e., the Schrödinger equation with position-dependent mass and relativistic wave equations. Full article

Transfinite interpolation, originally proposed in the early 1970s as a global interpolation method, was first implemented using Lagrange polynomials and cubic Hermite splines. While initially developed for computer-aided geometric design (CAGD), the method also found application in global finite element analysis. With the advent of isogeometric analysis (IGA), Bernstein–Bézier polynomials have increasingly replaced Lagrange polynomials, particularly in conjunction with tensor product B-splines and non-uniform rational B-splines (NURBSs). Despite its early promise, transfinite interpolation has seen limited adoption in modern CAD/CAE workflows, primarily due to its mathematical complexity—especially when blending polynomials of different degrees. In this context, the present study revisits transfinite interpolation and demonstrates that, in four broad classes, Lagrange polynomials can be systematically replaced by Bernstein polynomials in a one-to-one manner, thus giving the same accuracy. In a fifth class, this replacement yields a robust dual set of basis functions with improved numerical properties. A key advantage of Bernstein polynomials lies in their natural compatibility with weighted formulations, enabling the accurate representation of conic sections and quadrics—scenarios where IGA methods are particularly effective. The proposed methodology is validated through its application to a boundary-value problem governed by the Laplace equation, as well as to the eigenvalue analysis of an acoustic cavity, thereby confirming its feasibility and accuracy. Full article

A common and simple estimate of variability is the sample range, which is the difference between the maximum and minimum values in the sample. While other measures of variability are preferred in most instances, process owners and operators regularly use range (R) control charts to monitor process variability. The center line and limits of the R charts use constants that are based on the first two moments (mean and variance) of the distribution of the range of normal random variables. Historically, the computation of moments requires the use of tabulated constants approximated using numerical integration. We provide exact results for the moments for sample sizes 2 through 5. For sample sizes from 6 to 1000, we used the differential correction method to find Chebyshev minimax rational-function approximations of the moments. The rational function we recommend for the mean (R-chart constant d₂) has a polynomial of order two in the numerator and six in the denominator and achieves a maximum error of 4.4 × 10⁻⁶. The function for the standard deviation (R-chart constant d₃) has a polynomial of order two in the numerator and seven in the denominator and achieves a maximum error of 1.5 × 10⁻⁵. The exact and approximate expressions eliminate the need for table lookup in the control chart design phase. Full article

The paper advances a new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the indexes, we generate the Darboux–Crum nets of the rational canonical Sturm–Liouville equations (RCSLEs). It is shown that each RCSLE in question has four infinite sequences of quasi-rational solutions (q-RSs) such that their polynomial components from each sequence form a X-Jacobi DPS composed of simple pseudo-Wronskian polynomials (p-WPs). For each p-th order rational Darboux Crum transform of the Jacobi-reference (JRef) CSLE, used as the starting point, we formulate two rational Sturm–Liouville problems (RSLPs) by imposing the Dirichlet boundary conditions on the solutions of the so-called ‘prime’ SLE (p-SLE) at the ends of the intervals (−1, +1) or (+1, ∞). Finally, we demonstrate that the polynomial components of the q-RSs representing the eigenfunctions of these two problems have the form of simple p-WPs composed of p Romanovski–Jacobi (R-Jacobi) polynomials with the same pair of indexes and a single classical Jacobi polynomial, or, accordingly, p classical Jacobi polynomials with the same pair of positive indexes and a single R-Jacobi polynomial. The common, fundamentally important feature of all the simple p-WPs involved is that they do not vanish at the finite singular endpoints—the main reason why they were selected for the current analysis in the first place. The discussion is accompanied by a sketch of the one-dimensional quantum-mechanical problems exactly solvable by the aforementioned infinite and finite EOP sequences. Full article

Satellite imagery is a widely used source of spatial information in many applications, such as land use/land cover, object detection, agricultural monitoring, and urban area monitoring. Numerous factors, including projection, tilt angle, scanner, atmospheric conditions, terrain curvature, and fluctuations, can cause satellite images to become distorted. Eliminating systematic errors caused by the sensor and platform is a crucial step to obtaining reliable information from satellite images. To utilize satellite images directly in applications requiring high accuracy, the errors in the images should be removed by geometric correction. In this study, geometric correction was applied to the Pléiades 1A (PHR) image using non-parametric methods, and the effects of different transformation models and digital elevation models (DEMs) were investigated. Ground control points (GCPs) were obtained from orthophotos created by the photogrammetric method using precise positioning. The effect of photogrammetric DEMs with various spatial resolutions on geometric correction was investigated. Additionally, the effect of DEMs obtained using the photogrammetric method was compared with those from open-source DEMs, including SRTM, ASTER GDEM, COP30, AW3D30, and NASADEM. Two-dimensional polynomial transformation, the thin plate spline (TPS), and the rational function model (RFM) were applied as transformation methods. Our results showed that a higher-accuracy geometric correction process could be achieved with orthophotos and DEMs created using precise positioning techniques such as RTK and PPK. According to the results obtained, an RMSE of 0.633 m was achieved with RFM using RTK-DEM, while an RMSE of 0.615 m was achieved with RFM using PPK-DEM. Full article

With the development of pulse technology, reinforced concrete buildings are exposed to increasingly complex high-power electromagnetic pulse (EMP) environments, posing risks of functional degradation or destruction of indoor electronic equipment and systems. Therefore, it is imperative to assess the internal fields of buildings under EMP irradiation. The challenge lies in the multi-scale characteristics of reinforced concrete buildings, where fine grids are required for the accurate modelling of rebar, thereby consuming substantial computing resources. To address this challenge, this paper proposes a fast calculation method of the time–domain coupling characteristics between buildings and EMPs based on the electromagnetic parameter equivalence of reinforced concrete walls. The method first calculates the equivalent electromagnetic parameters from the S-parameters of the walls, which are then fitted into polynomial rational functions. Then, the auxiliary differential equation finite-difference time–domain (ADE-FDTD) method is used to analyze the time–domain coupling characteristics of reinforced concrete walls and buildings under EMP irradiation. The results show that the proposed method significantly enhances computational efficiency while maintaining high accuracy. Specifically, for a large two-story reinforced concrete building, the method achieves a 3.2-fold increase in computational speed and a 4.3-fold reduction in memory usage compared to conventional commercial software (CST Studio Suite 2022). This approach provides an effective solution for simulating the coupling characteristics between large reinforced concrete buildings and external EMPs. Full article

The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. The first three X-DPSs formed by pseudo-Wronskians of two Jacobi polynomials contain both exceptional orthogonal polynomial systems (X-Jacobi OPSs) on the interval (−1, +1) and the finite EOP sequences on the positive interval (1, ∞). On the contrary, the X-DPS of series W formed by Wronskians of two Jacobi polynomials contains only (infinitely many) finite EOP sequences on the interval (1, ∞). In addition, the paper rigorously examines the three isospectral families of the associated Liouville potentials (rationally extended hyperbolic Pöschl-Teller potentials of types a, b, and a′) exactly quantized by the EOPs in question. Full article

Safe driving and effective collision avoidance are critical challenges in the development of autonomous driving technology. As the dynamic interactions between vehicles and pedestrians become increasingly complex, making rational decisions and accurately executing planning and control in emergency situations has become a core issue for sustainable development relating to traffic mobility and safety. This paper proposes an active collision-avoidance control strategy based on emergency decisions and planning in the context of vehicle–pedestrian interactions. A safety-distance model is developed with consideration given to the dynamic interactions between these two entities, and an emergency-decision mechanism is designed using the integration of priority rules. To generate smooth collision-avoidance trajectories, a quintic polynomial method is employed to construct trajectory clusters that meet the desired specifications. Moreover, a multi-objective optimization value function which considers multiple factors comprehensively is used to select the optimal path. To enhance collision-avoidance control accuracy, an RBF (radial basis function)–optimized SMC (sliding mode control) algorithm is introduced. Additionally, an FD-SF (force demand–based speed feedback) algorithm is designed to accurately track the longitudinal braking path. The results indicate that the proposed strategy can generate efficient, comfortable, and smooth optimal collision-avoidance paths, significantly improving vehicle response speed and control accuracy. Full article

This study addresses the issue of atmospheric delay correction for the rational polynomial coefficient (RPC) model associated with spaceborne synthetic aperture radar (SAR) imagery under conditions lacking ephemeris data, proposing a novel approach to enhance the geometric positioning accuracy of RPC models. A satellite position inversion method based on the vector-autonomous intersection technique was developed, incorporating ionospheric delay and neutral atmospheric delay models to derive atmospheric delay errors. Additionally, an RPC model reconstruction approach, which integrates atmospheric correction, is proposed. Validation experiments using GF-3 satellite imagery demonstrated that the atmospheric delay values obtained by this method differed by only 0.0001 m from those derived using the traditional ephemeris-based approach, a negligible difference. The method also exhibited high robustness in long-strip imagery. The reconstructed RPC parameters improved image-space accuracy by 18–44% and object-space accuracy by 19–32%. The results indicate that this approach can fully replace traditional ephemeris-based methods for atmospheric delay extraction under ephemeris-free conditions, significantly enhancing the geometric positioning accuracy of SAR imagery RPC models, with substantial application value and development potential. Full article

We give some of our results over the past few years about rogue waves concerning some partial differential equations, such as the focusing nonlinear Schrödinger equation (NLS), the Kadomtsev–Petviashvili equation (KPI), the Lakshmanan–Porsezian–Daniel equation (LPD) and the Hirota equation (H). For the NLS and KP equations, we give different types of representations of the solutions, in terms of Fredholm determinants, Wronskians and degenerate determinants of order $2N$. These solutions are called solutions of order N. In the case of the NLS equation, the solutions, explicitly constructed, appear as deformations of the Peregrine breathers $P_N$ as the last one can be obtained when all parameters are equal to zero. At order N, these solutions are the product of a ratio of two polynomials of degree $N(N+1)$ in x and t by an exponential depending on time t and depending on $2N-2$ real parameters: they are called quasi-rational solutions. For the KPI equation, we explicitly obtain solutions at order N depending on $2N-2$ real parameters. We present different examples of rogue waves for the LPD and Hirota equations. Full article

An algorithmic method is proposed to generate all cubic polynomials with a critical orbit relation. We generate curves (polynomials of parameters) that correspond to those functions with critical orbit relations. The irreducibility of the polynomials obtained is left as an open problem. Our approach also works to generate critical orbit relations in all families of rational functions with active critical points. Full article

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X₁-Jacobi DPSs composed of Heun polynomials contains both the X₁-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way. Full article