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The forest fire burned area is one of the most basic factors used to describe forest fires and plays a vital role in damage assessment. The development of the NSSI-NDVI vegetation index triangular space method enables simultaneous calculation of the flammable non-photosynthetic vegetation (NPV), combustible photosynthetic vegetation (PV), and incombustible bare soil (BS) fractional cover in forest areas. This can be used to compensate for the calculation method that was based on NDVI vegetation index only by comparing vegetation cover before and after forest fires, with the omission of the NPV burned area. To this end, the NSSI-NDVI triangular space shape consistency before and after forest fires was elucidated through combustion and ash wetting experiments. In addition, the feasibility of the NSSI-NDVI triangular space method for the accurate calculation of the post-fire vegetation damage area was verified. Finally, the applicability and accuracy of this research method were verified based on 10 m spatial resolution satellite hyperspectral images from before and after the forest fire in Lushan, Sichuan Province, China. The NSSI-NDVI triangular space method was used to calculate the PV, NPV, and BS coverage simultaneously, and component transformation was used to calculate the burned area and burned site separately.

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly reducing the computational complexity of nonlinear terms. Temporal discretization is achieved using a second-order difference method, while spatial discretization utilizes a simple and easily implementable discrete approximation for the fractional Laplacian operator. The boundedness and convergence of the proposed numerical scheme under the maximum norm are rigorously analyzed, demonstrating its adherence to discrete energy conservation or dissipation laws. Numerical experiments validate the scheme’s effectiveness, structure-preserving properties, and capability for long-time simulations for both one- and two-dimensional problems. Additionally, the impact of the parameter ε on error dynamics is investigated.

We develop a new approach to construct finite element methods to solve the Poisson equation. The idea is to use the pointwise Laplacian as a degree of freedom followed by interpolating the solution at the degree of freedom by the given right-hand side function in the partial differential equation. The finite element solution is then the Galerkin projection in a smaller vector space. This idea is similar to that of interpolating the boundary condition in the standard finite element method. Our approach results in a smaller system of equations and of a better condition number. The number of unknowns on each element is reduced significantly from ( k 2 + 3 k + 2 ) / 2 to 3 k for the P k ( k ≥ 3 ) finite element. We construct bivariate P 2 conforming and nonconforming, and P k ( k ≥ 3 ) conforming interpolated Galerkin finite elements on triangular grids; prove their optimal order of convergence; and confirm our findings by numerical tests.

Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian elimination requires O NM 1 M 2 M 1 2 + M 2 2 + N M 1 M 2 complexity and O N M 1 2 M 2 2 storage, where N is the number of temporal unknown and M 1 , M 2 are the numbers of spatial unknown in x , y directions respectively. By exploring the structure of the coefficient matrix in fully coupled form, it possesses block lower-triangular Toeplitz structure and its blocks are block-dense Toeplitz matrices with dense-Toeplitz blocks. Based on this special structure and cooperating with time-marching or divide-and-conquer technique, two fast solvers with storage O NM 1 M 2 are developed. The complexity for the fast solver via time-marching is O NM 1 M 2 N + log M 1 M 2 and the one via divide-and-conquer technique is O NM 1 M 2 log 2 N + log M 1 M 2 . It is worth to remark that the proposed solvers are not lossy. Some discussions on achieving convergence rate for smooth and non-smooth solutions are given. Numerical results show the high efficiency of the proposed fast solvers.

Based on the circulant-and-skew-circulant representation of Toeplitz matrix inversion and the divide-and-conquer technique, a fast numerical method is developed for solving N -by- N block lower triangular Toeplitz with M -by- M dense Toeplitz blocks system with 𝓞 ( M N log N ( log N + log M ) ) complexity and 𝓞 ( NM ) storage. Moreover, the method is employed for solving the linear system that arises from compact finite difference scheme for time-space fractional diffusion equations with significant speedup. Numerical examples are given to show the efficiency of the proposed method.

Localization of mobile robots is crucial for deploying robots in real-world applications such as search and rescue missions. This work aims to develop an accurate localization system applicable to swarm robots equipped only with low-cost monocular vision sensors and visual markers. The system is designed to operate in fully open spaces, without landmarks or support from positioning infrastructures. To achieve this, we propose a localization method based on equilateral triangular formations. By leveraging the geometric properties of equilateral triangles, the accurate two-dimensional position of each participating robot is estimated using one-dimensional lateral distance information between robots, which can be reliably and accurately obtained with a low-cost monocular vision sensor. Experimental and simulation results demonstrate that, as travel time increases, the positioning error of the proposed method becomes significantly smaller than that of a conventional dead-reckoning system, another low-cost localization approach applicable to open environments.

Typically, the upper part of the roof a gymnasium building is a radial inverted triangular truss structure, and the lower part is a cable structure. They are connected by vertical braces to form a self-balancing structural system. The whole roof is supported by a complex, spatial, prestressed structure comprising tilted Y-shaped laced columns. Such structures rely on the integrity of the form and the application of prestress to achieve the best performance; it is in an extremely unstable state during construction. In order to study the mechanical behavior of the structure in this process, finite element software was used to analyze the cumulative slip of the structure and the construction process of cable tension, and the simulation values were compared to the actual monitoring values. The stress and deformation of the structure in different construction stages were investigated, and a reasonable structural unloading scheme was put forward. The study results showed that the stiffness of the long-span space truss suspended dome structure gradually increased with the structural integrity during construction, and the vertical deformation decreased from 25.4 mm to 19.26 mm with the construction process. The location and magnitude of the structure’s maximum internal force and maximum stress varied greatly compared to the static analysis when considering the construction process effects. Hence, conducting a construction process analysis is necessary. The construction technology of symmetrical rotating cumulative slip proposed in this paper has the advantages of a short construction duration, safe and stable construction process, etc., providing technical references for similar engineering constructions.

Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics. Previous methods of constructing such Voronoi diagrams generally depended on having an exact geodesic metric. However, exact geodesic computation is time-consuming and has high memory usage, limiting wider application of geodesic Voronoi diagrams (GVDs). In order to overcome this issue, instead of using exact methods, we reformulate a graph method based on Steiner point insertion, as an effective way to obtain geodesic distances. Further, since a bisector comprises hyperbolic and line segments, we utilize Apollonius diagrams to encode complicated structures, enabling Voronoi diagrams to encode a medial-axis surface for a dense set of boundary samples. Based on these strategies, we present an approximation algorithm for efficient Voronoi diagram construction on triangulated surfaces. We also suggest a measure for evaluating similarity of our results to the exact GVD. Although our GVD results are constructed using approximate geodesic distances, we can get GVD results similar to exact results by inserting Steiner points on triangle edges. Experimental results on many 3D models indicate the improved speed and memory requirements compared to previous leading methods.

Recommendation system technologies predominantly focus on user-item interaction data, which are mapped into shared vector spaces for digital representation. These representations are then analyzed to uncover the relationships between users and items. As recommendation technologies have seen widespread adoption, a novel challenge has emerged in supply-demand matching contexts: the dual-triangular recommendation problem, involving four key entities, i.e., users with their demands, and suppliers with their offered items, forming a heterogeneous information network. In this work, we introduce the concept of dual-triangular recommendation and formally define this scientific problem. We propose a dual-triangular recommendation algorithm, enhanced by large language models, which utilizes knowledge graph encoder and LLM-augmented encoder to generate embedding representations for the four entities. A multi-task framework is employed to enable the sharing of underlying parameters across multiple recommendation tasks within the dual-triangular context. Through extensive experiments conducted on a real-world technology commercialization platform dataset, patent transfer dataset, and talent recruitment dataset, we demonstrate the effectiveness of our approach, offering a feasible and scalable solution to the dual-triangular recommendation problem.