Search Results (9)

Accurate and reliable positioning is essential for unmanned marine vehicles (UMVs), especially in complex maritime environments. Existing algorithms often underutilize historical information, struggle with nonlinear dynamics, and lack adaptability in the GNSS Measurement Noise Covariance, leading to degraded performance. This study proposes an enhanced Factor Graph Optimization (FGO) method integrated with an adaptive neuro-fuzzy inference system (ANFIS) to overcome these challenges. First, an improved GNSS/Dead Reckoning (DR) factor graph is built using refined error models to enhance baseline accuracy. Second, a marginalization factor is introduced utilizing a sliding window and the Schur complement method to retain informative historical data while reducing computational load, thereby improving stability and field performance. Third, an ANFIS-based adaptive GNSS factor dynamically updates the GNSS Measurement Noise Covariance Matrix (GMNCM) to strengthen robustness under variable maritime conditions. Simulation and field tests demonstrate significant improvements: the proposed method achieves 29.1%, 26.5%, and 9.9% higher accuracy than EKF, UKF, and conventional FGO, respctively. Under GNSS interruptions, EKF and UKF diverge with errors exceeding 500 m, while FGO limits drift to 20 m. The proposed ANFIS–FGO shows the smallest fluctuations and fastest recovery, confirming its strong resilience and practical applicability for UMV navigation. Full article

In the digital era, protecting the integrity and ownership of digital content is increasingly crucial, particularly against unauthorized copying and tampering. Traditional watermarking techniques often struggle to remain robust under various image manipulations, leading to a need for more resilient methods. To address this challenge, we propose a novel watermarking technique that integrates the Discrete Wavelet Transform (DWT), Singular Value Decomposition (SVD), and Schur matrix factorization to embed a QR code as a watermark into digital images. Our method was rigorously tested across a range of common image attacks, including histogram equalization, salt-and-pepper noise, ripple distortions, smoothing, and extensive cropping. The results demonstrate that our approach significantly outperforms existing methods, achieving high normalized correlation (NC) values such as 0.9949 for histogram equalization, 0.9846 for salt-and-pepper noise (2%), 0.96063 for ripple distortion, 0.9670 for smoothing, and up to 0.9995 under 50% cropping. The watermark consistently maintained its integrity and scannability under all tested conditions, making our method a reliable solution for enhancing digital copyright protection. Full article

The concept of fractional powers of operators was initially developed within the framework of functional analysis and has since played a significant role in the study of evolution equations, abstract differential equations, and complex dynamical systems. Meanwhile, the operator matrix approach has emerged as a powerful and widely used tool for analyzing the structural properties of operators. Motivated by these developments, this paper focuses on the explicit computation of fractional powers of anti-triangular operator matrices. Specifically, we first derive an explicit formula for the fractional powers of off-diagonal operator matrices by employing the formal determinant of block operator matrices. Then, based on the Schur factorization, we obtain a representation for the fractional powers of the general anti-triangular case. As applications, the obtained results are further applied to certain differential equations. Full article

In this paper, we investigate the geometric realization of $\mathrm{Spin}\left(8\right)$ triality through vector fields on the octonionic algebra $\mathbb{O}$. The triality automorphism group of $\mathrm{Spin}\left(8\right)$, isomorphic to ${\mathfrak{S}}_3$, cyclically permutes the three inequivalent 8-dimensional representations: the vector representation V and the spinor representations $S_{+}$ and $S_{-}$. While triality automorphisms are well known through representation theory, their concrete geometric realization as flows on octonionic space has remained unexplored. We construct explicit smooth vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ on $\mathbb{O}\cong {\mathbb{R}}^8$ whose flows generate infinitesimal triality transformations. These vector fields have a linear structure arising from skew-symmetric matrices that implement simultaneous rotations in three orthogonal coordinate planes, providing the first elementary geometric description of triality symmetry. The main results establish that these vector fields preserve the octonionic multiplication structure up to automorphisms in $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$, demonstrating fundamental compatibility between geometric flows and octonionic algebra. We prove that the standard Euclidean metric on $\mathbb{O}$ is triality-invariant and classify all triality-invariant Riemannian metrics as conformal to the Euclidean metric with a conformal factor depending only on the isotonic norm. This classification employs Schur’s lemma applied to the irreducible $\mathrm{Spin}\left(8\right)$ action, revealing the rigidity imposed by triality symmetry. We provide a complete classification of triality-symmetric minimal surfaces, showing they are locally isometric to totally geodesic planes, surfaces of revolution about triality-fixed axes, or surfaces generated by triality orbits of geodesic curves. This trichotomy reflects the threefold triality symmetry and establishes correspondence between representation-theoretic decomposition and geometric surface types. For complete surfaces with finite total curvature, we establish global classification and develop explicit Weierstrass-type representations adapted to triality symmetry. Full article

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem $Ly=\lambda y$, for an algebraic parameter $\lambda$ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve $\Gamma$. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve $\Gamma$, defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, $\Gamma$ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of $L-\lambda$ becomes explicit over a new coefficient field containing $\Gamma$. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems. Full article

Volume increment is one of the main concerns in forestry practice. The aim of our study was to examine the impact of factors influencing the periodic annual increment of oak. To meet our objective, we used measurement data from the national forest inventory in Poland from 2005 to 2019 for oak-dominated stands. Our study used data of 1464 sample plots with dominant oak species (Quercus sessilis Ehrh. ex Schur and Quercus robur L.) measured within the national forest inventory in Poland. We developed models explaining the dependence of the periodic annual volume increment on stand characteristics using the generalized additive model. The generalized additive model allows us to analyze each variable’s effect on the dependent variable, with all other variables fixed. We documented the effect of age, height, basal area, and relative spacing index (RSI) on the periodic annual volume increment (PAIv) of oaks in Poland. The PAIv of oaks decreased gradually as the tree aged. The dependence of the PAIv on stand density was shown through its relationship with the basal area and RSI. The developed model explained about 64.6% of the periodic annual volume increment variance. Full article

In ensemble data assimilation systems, the impracticalities of full sampling and systematic error often lead to spurious correlations between two variables with low actual correlations. To solve these problems, researchers have previously proposed a covariance localization (CL) method, which mainly involves the Schur product between a state error covariance matrix and a distance-based correlation matrix. Although this CL method can reduce spurious correlations to a certain extent, observational data remain difficult to be used effectively, which results in unreasonable assimilation. In this study, we develop a new CL method coupled with a fuzzy logic control algorithm, which we call the covariance fuzzy (CF) method. The proposed CF method is a distance-based localization method with “fuzzy” vanishing correlations in data assimilation (DA) systems. To verify the effectiveness of the new algorithm, we conducted a set of experiments using an ensemble Kalman filter (EnKF) that combines the nonlinear Lorenz-96 model or the quasi-geostrophic (QG) models. First, the performances of the CL and CF methods are discussed with respect to different strength forcings, ensemble sizes, and covariance inflation factors. The experimental results show that the proposed CF method can obtain a more effective observation weight than the CL method and can reduce the errors caused by spurious correlations. Additionally, using power spectral density (PSD) as a performance evaluation index, the robustness of the proposed fuzzy logic localization method is demonstrated. However, the application of the fuzzy logic-based localization methodology to a real atmospheric model remains to be tested. Full article

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the $QR$ and $LU$ factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the $QR$ factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices. Full article

This paper introduces a general solution of singular fractional-order linear-time invariant (FoLTI) continuous systems using the Adomian Decomposition Method (ADM) based on the Caputo's definition of the fractional-order derivative. The complexity of their entropy lies in defining the complete solution of such systems, which depends on introducing a method of decomposing their dynamic states from their static states. The solution is formulated by converting the singular system of regular pencils into a recursive form using the sequence of transformations, which separates the dynamic variables from the algebraic variables. The main idea of this work is demonstrated via numerical examples. Full article