Search Results (44)

We introduce a novel family of compactly supported basis functions, termed Legendre Delta-Shaped Functions (LDSFs), constructed using the eigenfunctions of the Legendre differential equation. We begin by proving that LDSFs form a basis for a Haar space. We then demonstrate that interpolation using classical Legendre polynomials is a special case of interpolation with the proposed Legendre Delta-Shaped Basis Functions (LDSBFs). To illustrate the potential of LDSBFs, we apply the corresponding series to approximate a rectangular pulse. The results reveal that Gibbs oscillations decay rapidly, resulting in significantly improved accuracy across smooth regions. This example underscores the effectiveness and novelty of our approach. Furthermore, LDSBFs are employed within the collocation framework to solve Poisson-type equations and systems of nonlinear differential equations arising in energy transfer problems. We also derive new error bounds for interpolation polynomials in a special case, expressed in both the discrete ${(L}_2)$ norm and the Sobolev $\left(H_p\right)$ norm. To validate the proposed method, we compare our results with those obtained using the Legendre pseudospectral method. Numerical experiments confirm that our approach is accurate, efficient, and highly competitive with existing techniques. Full article

The analysis of rockfall events provides critical insights for deciphering planetary geological processes and reconstructing environmental evolutionary timelines. Conventional visual interpretation methods that rely on orbiter imagery can be inefficient due to their massive datasets and subtle morphological signatures. While deep learning technologies, particularly object detection models, demonstrate transformative potential, they require specific adaptation to planetary imaging constraints, including low contrast, grayscale inputs, and small-target detection. Our coordinated optimization strategy integrates dynamic cropping optimization with architectural innovations: Kolmogorov–Arnold Network based C3 module (KANC3) replaces RepC3 through Legendre polynomial decomposition to strengthen feature representation, while our dynamic cropping strategy significantly improves small-target detection in low-contrast grayscale imagery by mitigating background and target imbalance. Experimental validation on the optimized RMaM-2020 dataset demonstrates that Real-Time Detection Transformer with a ResNet-18 backbone and Kolmogorov–Arnold Network based C3 module (RT-DETR-R18-KANC3) achieves 0.982 precision, 0.955 recall, and 0.964 mAP50 under low-contrast conditions, representing a 1% improvement over the baseline model and exceeding YOLO-series models by >40% in relative performance metrics. Full article

The G-modified Helmholtz equation is a partial differential equation that allows gravity intensity g to be expressed as a series of spherical harmonics, with the radial distance r raised to irrational powers. In this study, we consider a non-rotating triaxial ellipsoid parameterized by the geodetic latitude φ and geodetic longitude λ, and eccentricities e_e, e_x, e_y. On its surface, the value of gravity potential has a constant value, defining a level triaxial ellipsoid. In addition, the gravity intensity is known on the surface, which allows us to formulate a Dirichlet boundary value problem for determining the gravity intensity as a series of spherical harmonics. This expression for gravity intensity is presented here for the first time, filling a gap in the study of triaxial ellipsoids and spheroids. Given that the triaxial ellipsoid has very small eccentricities, a first order approximation can be made by retaining only the terms containing e_e² and e_x². The resulting expression in spherical harmonics contains even degree and even order harmonic coefficients, along with the associated Legendre functions. The maximum degree and order that occurs is four. Finally, as a special case, we present the geometrical degeneration of an oblate spheroid. Full article

Lewis’ theory of multiple scattering has been modeled as a random walk on a unit sphere for calculating the multiple scattering angular distribution of charged particles, which is more intuitive and mathematically simpler. This formalism can lead to the Goudsmit–Saunderson theory and the Lewis theory of multiple scattering angular distribution, thus providing an easier-to-understand framework to unify both the Goudsmit–Saunderson and the Lewis theories. This new random walk method eliminates the need for integro-differential expansions in Lewis theory and is faster at calculating multiple scattering angular distributions, reducing the required Legendre series terms by 80% at small step (path) length (<20) and providing much greater calculation efficiency. Crucially, the random walk formalism explicitly preserves spherical symmetry by treating angular deflections as steps on a unit sphere, enabling the efficient sampling of scattering events while maintaining accuracy. Further, a robust algorithm for numerically calculating multiple scattering angular distributions of electrons based on the Goudsmit–Saunderson and Lewis theories has been developed. Partial wave elastic scattering differential cross-sections, generated with the program ELSEPA, have been used in the calculations. A two-point Gauss–Legendre quadrature method is used to calculate the Legendre coefficients (multiple scattering moments). Full article

This paper presents an improved methodology for optimizing the fed-batch fermentation process of xylitol production, aiming to maximize the final concentration in a bioreactor co-fed with xylose and glucose. Xylitol is a valuable sugar alcohol widely used in the food and pharmaceutical industries, and its microbial production requires precise control over substrate feeding strategies. The proposed technique employs Legendre polynomials to parameterize two control actions (the feeding rates of glucose and xylose), and it uses a hybrid optimization algorithm combining Monte Carlo sampling with genetic algorithms for coefficient selection. Unlike traditional optimization approaches based on piecewise parameterization, which produce discontinuous control profiles and require post-processing, this method generates smooth profiles directly applicable to real systems. Additionally, it significantly reduces mathematical complexity compared to strategies that combine Fourier series with orthonormal polynomials while maintaining similar optimization results. The methodology achieves good results in xylitol production using only eight parameters, compared to at least twenty in other approaches. This dimensionality reduction improves the robustness of the optimization by decreasing the likelihood of convergence to local optima while also reducing the computational cost and enhancing feasibility for implementation. The results highlight the potential of this strategy as a practical and efficient tool for optimizing nonlinear multivariable bioprocesses. Full article

A multifractal formalism relates multiscale quantities to the multifractal spectrum. The multifractal framework provides significant analytical advantages by incorporating a wide range of statistical moment orders $\left(q\right)$, thereby enabling a more comprehensive characterization of the intrinsic structural variability embedded in the dataset. The scaling properties of the analyzed rainfall time series was studied using Legendre transformation. This tool is effective for detecting multifractality in the time series of interest and for extracting information on scaling behavior. The obtained parameters may ultimately aid in performing multifractal modeling. The 50-year-long daily rainfall time series shows multifractal properties. The analysis of the generalized Hurst exponent $h(q$) enabled the classification of time series’ temporal dynamics, distinguishing between persistent, anti-persistent, and uncorrelated behavior. The multifractal analysis proves to be an effective and robust tool to characterize precipitation time series in the context of climate change research. Ultimately, the parameters and features derived from the multifractal spectrum—such as singularity strengths and spectrum width—serve as both quantitative and qualitative metrics for characterizing the spatiotemporal dynamics of rainfall in the semi-arid region of the Central Mexican Plateau. Full article

By examining Fourier–Legendre series pairs $\mathcal{F}\left(y\right)$ and $\mathcal{G}\left(y\right)$, fifteen classes of symmetric and asymmetric bivariate series will systematically be investigated. A compendium of 360 remarkable summation formulae will be established. They may serve as a complementary work to the paper by Chu and Campbell (The Ramanujan Journal (2023)). Full article

In this paper, by using the zeroth-order q-Tricomi functions, the theory of three-variable q-Legendre-based Appell polynomials is introduced. These polynomials are studied by means of generating functions, series expansions, and determinant representation. Further, by utilizing the concepts of q-quasi-monomiality, these polynomials are examined as several q-quasi-monomial and operational representations; the q-differential equations for the three-variable q-Legendre-based Appell polynomials were obtained. In addition, we established a new generalization of three-variable q-Legendre-Hermite-Appell polynomials, and we derive series expansion, determinant representation, and q-quasi-monomial and q-differential equations. Some examples are framed to better illustrate the theory of three-variable q-Legendre-based Appell polynomials, and this is characterized by the above properties. Full article

This paper shows that certain ${}_{3}F_4$ hypergeometric functions can be expanded in sums of pair products of ${}_{1}F_2$ functions. In special cases, the ${}_{3}F_4$ hypergeometric functions reduce to ${}_{2}F_3$ functions. Further special cases allow one to reduce the ${}_{2}F_3$ functions to ${}_{1}F_2$ functions, and the sums to products of ${}_{0}F_1$ (Bessel) and ${}_{1}F_2$ functions. The class of hypergeometric functions with summation theorems are thereby expanded beyond those expressible as pair-products of ${}_{2}F_1$ functions, ${}_{3}F_2$ functions, and generalized Whittaker functions, into the realm of ${}_{p}F_q$ functions where $p<q$ for both the summand and terms in the series. Full article

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind $J_N\left(kx\right)$ and modified Bessel functions of the first kind $I_N\left(kx\right)$ lead to an infinite set of series involving ${}_{1}F_2$ hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving ${}_{1}F_2$ hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving ${}_{1}F_2$ hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution. Full article

In a recently published work, we introduced local Legendre polynomial fitting-based permutation entropy (LPPE) as a new complexity measure for quantifying disorder or randomness in time series. LPPE benefits from the ordinal pattern (OP) concept and incorporates a natural, aliasing-free multiscaling effect by design. The current work extends our previous study by investigating LPPE’s capability to assess fatigue levels using both synthetic and real surface electromyography (sEMG) signals. Real sEMG signals were recorded during biceps brachii fatiguing exercise maintained at 70% of maximal voluntary contraction (MVC) until exhaustion and were divided into four consecutive temporal segments reflecting sequential stages of exhaustion. As fatigue levels rise, LPPE values can increase or decrease significantly depending on the selection of embedding dimensions. Our analysis reveals two key insights. First, using LPPE with limited embedding dimensions shows consistency with the literature. Specifically, fatigue induces a decrease in sEMG complexity measures. This observation is supported by a comparison with the existing multiscale permutation entropy (MPE) variant, that is, the refined composite downsampling (rcDPE). Second, given a fixed OP length, higher embedding dimensions increase LPPE’s sensitivity to low-frequency components, which are notably present under fatigue conditions. Consequently, specific higher embedding dimensions appear to enhance the discrimination of fatigue levels. Thus, LPPE, as the only MPE variant that allows a practical exploration of higher embedding dimensions, offers a new perspective on fatigue’s impact on sEMG complexity, complementing existing MPE approaches. Full article

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solutions, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively. Full article

The principal aim of this paper is to introduce the extended Voigt-type function ${\mathbb{V}}_{\mu ,\nu }(x,y)$ and its counterpart extension ${\mathbb{W}}_{\mu ,\nu }(x,y)$, involving the Neumann function $Y_{\nu }$ in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions $K(x,y)$ and $L(x,y)$, and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions ${\mathbb{V}}_{\mu ,\nu }^{\left(r\right)}(x,y),{\mathbb{W}}_{\mu ,\nu }^{\left(r\right)}(x,y)$ and the r positive integer of the initial extensions ${\mathbb{V}}_{\mu ,\nu }(x,y),{\mathbb{W}}_{\mu ,\nu }(x,y)$. Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions ${\Psi }_2^{\left(r\right)}$. Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind $Q_{\eta }^{-\nu }$ in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for ${\mathbb{V}}_{\mu ,\nu }(x,y)$ and ${\mathbb{W}}_{\mu ,\nu }(x,y)$. Particularly interesting results are presented for the Neumann function $Y_{\nu }$ and for the Struve ${\mathbf{H}}_{\nu }$ function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks. Full article

In this article, new results are investigated in the context of the recently introduced Abu-Shady–Kaabar fractional derivative. First, we solve the generalized Legendre fractional differential equation. As in the classical case, the generalized Legendre polynomials constitute notable solutions to the aforementioned fractional differential equation. In the sense of the fractional derivative of Abu-Shady–Kaabar, we establish important properties of the generalized Legendre polynomials such as Rodrigues formula and recurrence relations. Special attention is also devoted to another very important property of Legendre polynomials and their orthogonal character. Finally, the representation of a function $f\in L_{\alpha }^2\left(\left[-1,1\right]\right)$ in a series of generalized Legendre polynomials is addressed. Full article

Tomographic Synthetic Aperture Radar (SAR) allows the reconstruction of the 3D radar reflectivity of forests from a large(r) number of multi-angular acquisitions. However, in most practical implementations it suffers from limited vertical resolution and/or reconstruction artefacts as the result of non-ideal acquisition setups. Polarisation Coherence Tomography (PCT) offers an alternative to traditional tomographic techniques that allow the reconstruction of the low-frequency 3D radar reflectivity components from a small(er) number of multi-angular SAR acquisitions. PCT formulates the tomographic reconstruction problem as a series expansion on a given function basis. The expansion coefficients are estimated from interferometric coherence measurements between acquisitions. In its original form, PCT uses the Legendre polynomial basis for the reconstruction of the 3D radar reflectivity. This paper investigates the use of new basis functions for the reconstruction of X-band 3D radar reflectivity of forests derived from available lidar waveforms. This approach enables an improved 3D radar reflectivity reconstruction with enhanced vertical resolution, tailored to individual forest conditions. It also allows the translation from sparse lidar waveform vertical reflectivity information into continuous vertical reflectivity estimates when combined with interferometric SAR measurements. This is especially relevant for exploring the synergy of actual missions such as GEDI and TanDEM-X. The quality of the reconstructed 3D radar reflectivity is assessed by comparing simulated InSAR coherences derived from the reconstructed 3D radar reflectivity against measured coherences at different spatial baselines. The assessment is performed and discussed for interferometric TanDEM-X acquisitions performed over two tropical Gabonese rainforest sites: Mondah and Lopé. The results demonstrate that the lidar-derived basis provides more physically realistic vertical reflectivity profiles, which also produce a smaller bias in the simulated coherence validation, compared to the conventional Legendre polynomial basis. Full article