Search Results (65)

In this paper, we extend some differential and structural results for monic Jacobi–Sobolev orthogonal polynomials, associated with a general discrete Sobolev inner product, with a Jacobi continuous part. We consider finitely many exterior mass points and a positive semidefinite Sobolev product matrix. Using Christoffel–Darboux kernels, we derive several structure and connection formulas involving two consecutive Jacobi and Jacobi–Sobolev polynomials. This representation leads to lowering and raising operators with rational coefficients; a second-order ordinary differential equation and a three-term recurrence relation, with polynomial coefficients. These coefficients depend on n. These results extend several classical structural properties of Jacobi polynomials to a general discrete Sobolev setting. Full article

We extend the phase-tagged fluctuation framework for cumulative shock models from a fixed failure threshold to a linearly moving boundary $u_0+c{\tau }_n$, the premium drift regime that underlies insurance ruin theory. The moving boundary turns the first-passage problem from a fixed-level crossing into a crossing of a drifting walk, for which the partial sum truncation of the fixed-threshold theory no longer applies. We resolve this with a drift-aware ladder apparatus: a rank-one drift ladder matrix${\mathit{G}}^{\uparrow }\left(c\right)=g\left(c\right)\mathbf{e}\mathit{\alpha }$ whose scalar $g\left(c\right)$ solves a discrete Lundberg equation $a\left(z\right)=z^c$, a closed-form ascending ladder-height law obtained from the roots of the Lundberg polynomial, and a compound-geometric (Pollaczek–Khinchine) representation of the ruin probability. Building on these, we derive in closed form the full five-variable moving-boundary reliability functional ${\mathbf{\Phi }}_{\nu }^{\mathrm{ruin}}(\xi ,u,v,\vartheta ,\theta )$, the drift analogue of the fixed-threshold phase-tagged functional, and show it lies in the same two-dimensional matrix subspace $\mathrm{span}\left\{H\right(\theta ),H(\omega \left)\right\}$: the premium drift deforms the scalar coefficients through a first-passage transform while leaving the matrix structure invariant. As the principal application, we obtain the phase-resolved Gerber–Shiu expected-discounted-penalty function, with the joint transform of the time of ruin, the deficit at ruin, and the surplus prior to ruin. The classical scalar Gerber–Shiu function and the fixed-threshold functional are recovered as projections and as the $c\to 0$ limit, respectively. All closed forms reduce to root-finding on a single polynomial for rational model primitives, and every structural result is verified against Monte Carlo simulation and exact recursion. Full article

The paper investigates the Lie symmetries of a generalized nonlinear $(2+1)$-dimensional reaction–diffusion equation with time-dependent diffusion and reaction coefficients. The main contributions consist in the derivation of compatibility conditions that is the values of these coefficients for which the equation can be integrated. We consider the cases where the diffusion is linear in the main variable and the reaction term is a polynomial up to the fourth order, choices covering almost all models of practical interest Associated classes of self-similar solutions for these models selected via Lie compatibility conditions are highlighted. In particular, exact solutions of rational and exponential types, associated with linear, quadratic and cubic reaction terms, are mentioned. Full article

This research focuses on the following equation: $(\overline{g}g-1)(g\underline{g}-1)+B\left(z\right){\displaystyle \frac{g^{\prime }\left(z\right)}{g\left(z\right)}}=R(z,g)={\displaystyle \frac{P(z,g)}{Q(z,g)}},$ where $P(z,g)$ and $Q(z,g)$ represent polynomials in g, possessing rational coefficients in z and lacking shared roots, $B\left(z\right)$ is rational, and $\overline{g}\equiv g(z+1),\underline{g}\equiv g(z-1)$. Reduced forms arise when this equation permits an entire solution with a hyper-order strictly below one. This paper investigates the forms and value distribution of entire solutions to the reduced Painlevé-type differential-difference equation. Furthermore, the existence of entire solutions to the above equation is examined when ${deg}_g\left(P\right)\ge 5$ and ${deg}_g\left(Q\right)=0$ or ${deg}_g\left(Q\right)\ge 1$ and $Q(z,0)\not\equiv 0$. Full article

Accurate 3D terrain reconstruction from multi-view whisk-broom thermal infrared imagery with small intersection angles remains challenging because stereo geometry is weak and height sensitivity is limited. To address this challenge, we develop an affine-initialized rational polynomial coefficient (RPC) reconstruction framework for 3D positioning under weak geometric conditions. An affine model is first used to estimate initial 3D coordinates from image tie points, which are then used to initialize RPC-based refinement. The refinement adopts an iterative scheme with hierarchical updates, where longitude and latitude are optimized before altitude to mitigate error propagation when height observability is low. The method is evaluated using multi-view data acquired by the SDGSAT-1 Thermal Infrared Spectrometer (TIS) over plain, hilly, and mountainous terrains, with intersection angles ranging from 0.57° to 6.5°. The results show that approximately 80% of the reconstruction errors fall within 2 pixels and more than 90% fall within 3 pixels, corresponding to 60 m and 90 m at the image resolution used in this study. The root-mean-square error (RMSE) remains below 0.3 pixels in plains, 1.3 pixels in hilly areas, and 1.8 pixels in mountainous areas. Overall, the proposed framework facilitates stable 3D terrain reconstruction from whisk-broom thermal infrared imagery and reduces reliance on confidential rigorous sensor models. Full article

We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input–output pairs of an unknown function defined on a tropical semifield, the problem is to determine an approximating rational function formed by two Puiseux polynomials as numerator and denominator. With specified numbers of monomials in both polynomials, the approximation aims at evaluating the exponent and coefficient for each monomial in the polynomials to fit the rational function to the data in the sense of a tropical distance function. To solve the problem, we transform it into an approximation of a vector equation with unknown vectors on both sides, where one side corresponds to the numerator polynomial and the other side to the denominator. Each side involves a matrix with entries dependent on the unknown exponents, multiplied by the vector of unknown coefficients of monomials. We propose an algorithm that constructs a series of approximate solutions by alternately fixing one side of the equation to an already-found result and leaving the other side intact. Each equation obtained is approximated with respect to the vector of coefficients, which yields this vector and approximation error, both parameterized by exponents. The exponents are found by minimizing the error with an optimization procedure based on an agglomerative clustering technique. To illustrate, we present results for an approximation problem in terms of max-plus algebra (a real semifield with addition defined as maximum and multiplication as arithmetic addition), which corresponds to an ordinary problem of piecewise linear approximation of real functions. As our numerical experience shows, the proposed algorithm converges in a finite number of steps and provides a reasonably accurate solution to the problems considered. Full article

This study investigates the dynamics of optical solitons for the variable-coefficient coupled higher-order nonlinear Schrödinger equation (VCHNLSE) enriched with $\beta$-derivatives. By employing an extended direct algebraic method (EDAM), we successfully derive explicit soliton solutions that illustrate the intricate interplay between nonlinearities and variable coefficients. Our approach facilitates the transformation of the complex NLS into a more manageable form, allowing for the systematic exploration of diverse solitonic structures, including bright, dark, and singular solitons, as well as exponential, polynomial, hyperbolic, rational, and Jacobi elliptic solutions. This diverse family of solutions substantially expands beyond the limited soliton interactions studied in conventional approaches, demonstrating the superior capability of our method in unraveling new wave phenomena. Furthermore, we rigorously demonstrate the robustness of these soliton solutions against various perturbations through comprehensive stability analysis and numerical simulations under parameter variations. The practical significance of this work lies in its potential applications in advanced optical communication systems. The derived soliton solutions and the analysis of their dynamics provide crucial insights for designing robust signal carriers in nonlinear optical media. Specifically, the management of variable coefficients and fractional-order effects can be leveraged to model and engineer sophisticated dispersion-managed optical fibers, tunable photonic devices, and ultrafast laser systems, where controlling pulse propagation and stability is paramount. The presence of $\beta$-fractional derivatives introduces additional complexity to the wave propagation behaviors, leading to novel dynamics that we analyze through numerical simulations and graphical representations. The findings highlight the potential of the proposed methodology to uncover rich patterns in soliton dynamics, offering insights into their robustness and stability under varying conditions. This work not only contributes to the theoretical foundation of nonlinear optics but also provides a framework for practical applications in optical fiber communications and other fields involving nonlinear wave phenomena. Full article

High-resolution pushbroom satellites routinely acquire multi-tenskilometer-scale strips whose vendors’ rational polynomial coefficients (RPCs) exhibit systematic, direction-dependent biases that accumulate downstream when ground control is sparse. This study presents a physically interpretable stripwise extrapolation framework that predicts along- and across-track RPC correlation coefficients for inaccessible segments from an upstream calibration subset. Terrain-independent RPCs were regenerated and residual image-space errors were modeled with weighted least squares using elapsed time, off-nadir evolution, and morphometric descriptors of the target terrain. Gaussian kernel weights favor calibration scenes with a Jarque–Bera-indexed relief similar to the target. When applied to three KOMPSAT-3A panchromatic strips, the approach preserves native scene geometry while transporting calibrated coefficients downstream, reducing positional errors in two strips to <2.8 pixels (~2.0 m at 0.710 m Ground Sample Distance, GSD). The first strip with a stronger attitude drift retains 4.589 pixel along-track errors, indicating the need for wider predictor coverage under aggressive maneuvers. The results clarify the directional error structure with a near-constant across-track bias and low-frequency along-track drift and show that a compact predictor set can stabilize extrapolation without full-block adjustment or dense tie networks. This provides a GCP-efficient alternative to full-block adjustment and enables accurate georeferencing in controlled environments. Full article

We study a class of continued fraction transformations where the partial numerators are quadratic polynomials and the denominators are linear or constant. Using the Bauer–Muir transform, we establish two theorems that yield structurally distinct but equivalent continued fractions—one with rational coefficients and another with alternating forms. These transformations provide a unified framework for evaluating and simplifying continued fractions, including classical identities such as one of Euler, a recent result by Campbell and Chen, and several conjectures from the Ramanujan Machine involving $\pi$ and $\log 2.$ We conclude by discussing the potential extension of our methods to more general polynomial cases. Full article

In the era of big data, the security of information encryption systems has garnered extensive attention, particularly in critical domains such as financial transactions and medical data management. While traditional Shamir’s Secret Sharing (SSS) ensures secure integer sharing through threshold cryptography, it exhibits inherent limitations when applied to floating-point domains and high-precision numerical scenarios. To address these issues, this paper proposes an innovative algorithm to optimize SSS via type-specific coding for real numbers. By categorizing real numbers into four types—rational numbers, special irrationals, common irrationals, and general irrationals—our approach achieves lossless transmission for rational numbers, special irrationals, and common irrationals, while enabling low-loss recovery for general irrationals. The scheme leverages a type-coding system to embed data category identifiers in polynomial coefficients, combined with Bernoulli-distributed random bit injection to enhance security. The experimental results validate its effectiveness in balancing precision and security across various real-number types. 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

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

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

In this paper, we propose a 3D Digital Surface Model (DSM) reconstruction method from uncalibrated Multi-view Satellite Stereo (MVSS) images, where Rational Polynomial Coefficient (RPC) sensor parameters are not available. While recent investigations have introduced several techniques to reconstruct high-precision and high-density DSMs from MVSS images, they inherently depend on the use of geo-corrected RPC sensor parameters. However, RPC parameters from satellite sensors are subject to being erroneous due to inaccurate sensor data. In addition, due to the increasing data availability from the internet, uncalibrated satellite images can be easily obtained without RPC parameters. This study proposes a novel method to reconstruct a 3D DSM from uncalibrated MVSS images by estimating and integrating RPC parameters. To do this, we first employ a structure from motion (SfM) and 3D homography-based geo-referencing method to reconstruct an initial DSM. Second, we sample 3D points from the initial DSM as references and reproject them to the 2D image space to determine 3D–2D correspondences. Using the correspondences, we directly calculate all RPC parameters. To overcome the memory shortage problem while running the large size of satellite images, we also propose an RPC integration method. Image space is partitioned to multiple tiles, and RPC estimation is performed independently in each tile. Then, all tiles’ RPCs are integrated into the final RPC to represent the geometry of the whole image space. Finally, the integrated RPC is used to run a true MVSS pipeline to obtain the 3D DSM. The experimental results show that the proposed method can achieve 1.455 m Mean Absolute Error (MAE) in the height map reconstruction from multi-view satellite benchmark datasets. We also show that the proposed method can be used to reconstruct a geo-referenced 3D DSM from uncalibrated and freely available Google Earth imagery. Full article

The Rational Function Model (RFM) is composed of numerous highly correlated Rational Polynomial Coefficients (RPCs), establishing a mathematical relationship between two-dimensional images and three-dimensional spatial coordinates. Due to the existence of ill-posedness and overparameterization, the estimated RPCs are sensitive to any slight perturbations in the observation data, particularly when handling a limited number of Ground Control Points (GCPs). Recently, Principal Component Analysis (PCA) has demonstrated significant performance improvements in the RFM optimization problem. In the PCA-based RFM, each Principal Component (PC) is a linear combination of all variables in the design matrix. However, some original variables are noise related and have very small or almost zero contributions to the construction of PCs, which leads to the overparameterization problem and makes the RPC estimation process ill posed. To address this problem, in this paper, we propose an Adaptive Sparse Principal Component Analysis-based RFM method (ASPCA-RFM) for RPC estimation. In this method, the Elastic Net sparsity constraint is introduced to ensure that each PC contains only a small number of original variables, which automatically eliminates unnecessary variables during PC computation. Since the optimal regularization parameters of the Elastic Net vary significantly in different scenarios, an adaptive regularization parameter approach is proposed to dynamically adjust the regularization parameters according to the explained variance of PCs and degrees of freedom. By adopting the proposed method, the noise and error in the design matrix can be reduced, and the ill-posedness and overparameterization of the RPC estimation can be significantly mitigated. Additionally, we conduct extensive experiments to validate the effectiveness of our method. Compared to existing state-of-the-art methods, the proposed method yields markedly improved or competitive performance. Full article