Search Results (48)

This paper undertakes an in-depth exploration into the issue of quantization errors that occur during color gamut conversion within LED full-color display systems. To commence, a CIE-xyY colorimetric framework, which is customized to the unique characteristics of LED, is constructed. This framework serves as the bedrock for formulating the principles governing the operation of LED color gamuts. Subsequently, the conversions among diverse color spaces are scrutinized with great meticulousness. The core emphasis then shifts to dissecting how discrete control systems, in conjunction with quantization errors at low grayscale levels, precipitate the distortion of color gamut boundaries during the conversion process. The Laplacian operator is deployed to furnish a geometric comprehension of the distortion points, thereby delineating the topological discrepancies between the target and actual points. The quantitative analysis precisely delineates the correlation between quantization precision and the quantity of distortion points. The research endeavors to disclose the intricate relationships among quantization, color spaces, and colorimetric fidelity. This paper is conducive to the prospective calibration and rectification of LED display systems, furnishing a theoretical underpinning for the further enhancement of color reproduction in LED displays. Consequently, LED monitors can be rendered capable of satisfying the stringent accuracy requisites of advanced imaging and media. Full article

We consider first-order impulses for impulsive stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $H\in ({\displaystyle \frac{1}{2}},1)$ involving a nonlinear $\varphi$-Laplacian operator. The system incorporates both state and derivative impulses at fixed time instants. First, we establish the existence of at least one mild solution under appropriate conditions in terms of nonlinearities, impulses, and diffusion coefficients. We achieve this by applying a nonlinear alternative of the Leray–Schauder fixed-point theorem in a generalized Banach space setting. The topological structure of the solution set is established, showing that the set of all solutions is compact, closed, and convex in the function space considered. Our results extend existing impulsive differential equation frameworks to include fractional stochastic perturbations (via fBm) and general $\varphi$-Laplacian dynamics, which have not been addressed previously in tandem. These contributions provide a new existence framework for impulsive systems with memory and hereditary properties, modeled in stochastic environments with long-range dependence. Full article

Manifold learning is a significant computer vision task used to describe high-dimensional visual data in lower-dimensional manifolds without sacrificing the intrinsic structural properties required for 3D reconstruction. Isomap, Locally Linear Embedding (LLE), Laplacian Eigenmaps, and t-SNE are helpful in data topology preservation but are typically indifferent to the intrinsic differential geometric characteristics of the manifolds, thus leading to deformation of spatial relations and reconstruction accuracy loss. This research proposes an Optimization of Manifold Learning using Differential Geometry Framework (OML-DGF) to overcome the drawbacks of current manifold learning techniques in 3D reconstruction. The framework employs intrinsic geometric properties—like curvature preservation, geodesic coherence, and local–global structure correspondence—to produce structurally correct and topologically consistent low-dimensional embeddings. The model utilizes a Riemannian metric-based neighborhood graph, approximations of geodesic distances with shortest path algorithms, and curvature-sensitive embedding from second-order derivatives in local tangent spaces. A curvature-regularized objective function is derived to steer the embedding toward facilitating improved geometric coherence. Principal Component Analysis (PCA) reduces initial dimensionality and modifies LLE with curvature weighting. Experiments on the ModelNet40 dataset show an impressive improvement in reconstruction quality, with accuracy gains of up to 17% and better structure preservation than traditional methods. These findings confirm the advantage of employing intrinsic geometry as an embedding to improve the accuracy of 3D reconstruction. The suggested approach is computationally light and scalable and can be utilized in real-time contexts such as robotic navigation, medical image diagnosis, digital heritage reconstruction, and augmented/virtual reality systems in which strong 3D modeling is a critical need. Full article

While tank vehicles are in the process of transporting liquid chemicals, the transportation task may be delayed or terminated due to disruptions in the transportation network. To reduce transportation losses, it may be necessary to evaluate network performance to cope with emergencies in the transportation network. This study investigates the vulnerability of the road network for liquid chemical transportation (RNLCT) under both intentional and random attacks. We selected six network indices, including average degree, average path length, average betweenness of node and edge, average clustering coefficient, and network efficiency to analyze the topological structure of the network. A reverse greedy algorithm is applied to identify 40 key nodes for simulated intentional attacks. The results show that network efficiency drops by 57.3% under targeted attacks, compared to 37.8% under random attacks. Moreover, the proposed method outperforms the distance laplacian centrality, which leads to a 43.3% reduction, demonstrating its effectiveness in identifying key nodes and enhancing network robustness. Full article

In this paper, we investigate the geometry and topology of compact warped product minimal submanifolds of arbitrary codimension immersed in a sphere. These submanifolds satisfy a specific pinching condition relating the length and Laplacian of the warping function to the dimensions of the warped product. Our results extend previous work on minimal immersions into the sphere. Full article

This paper studies the robust $H_{\infty }$ time-varying formation tracking (TVFT) problem for heterogeneous nonlinear multi-agent systems (MASs) with parameter uncertainties, external disturbances, and unknown leader inputs. The objective is to ensure that follower agents track the leader’s trajectory while achieving a desired time-varying formation, even under unmodeled dynamics and disturbances. Unlike existing methods that rely on global topology information or homogeneous system assumptions, an adaptive control protocol is proposed in full distribution, requiring no global topology information, and integrates nonlinear compensation terms to handle unknown leader inputs and parameter uncertainties. Based on the Lyapunov theory and laplacian matrix, a robust $H_{\infty }$ TVFT criterion is developed. Finally, a numerical example is given to verify the theory. Full article

Based on Graph Balancing Theory, this study proposes an anomaly detection algorithm, the Supply Chain Proof of Relation (PoR), applied to enterprise procurement networks formalized as weighted directed graphs. A mathematical framework is constructed by integrating Laplacian flow conservation and the Sheaf topological coherence principle to identify anomalous nodes whose local characteristics deviate significantly from the global features of the supply network. PoR was empirically implemented on a dataset comprising 856 Taiwanese enterprises, successfully detecting 56 entities exhibiting abnormal behavior. Anomaly intensity was visualized through trend plots, revealing nodes with rapidly increasing deviations. To validate the effectiveness of this detection, the study further analyzed the correlation between internal and external performance metrics. The results demonstrate that anomalous nodes exhibit near-zero correlations, in contrast to the significant correlations observed in normal nodes—indicating a disruption of information consistency. This research establishes a graph-theoretic framework for anomaly detection, presents a mathematical model independent of training data, and highlights the linkage between structural deviations and informational distortions. By incorporating Sheaf Theory, the study enhances the analytical depth of topological consistency. Moreover, this work demonstrates the observability of flow conservation violations within a highly complex, non-physical system such as the supply chain. It completes a logical integration of Sheaf Coherence, Graph Balancing, and High-Dimensional Anomaly Projection, and achieves a cross-mapping between Graph Structural Deviations and Statistical Inconsistencies in weighted directed graphs. This contribution advances the field of graph topology-based statistical anomaly detection, opening new avenues for the methodological integration between physical systems and economic networks. Full article

Laplacian controllability and observability of a consensus network is a widely considered topic in the area of multi-agent systems, complex networks, and large-scale systems. In this paper, this problem is addressed when the communication among nodes is described through a starlike tree topology. After a brief description of the mathematical setting of the problem adopted in a wide number of multi-agent systems engineering applications, some novel results are drawn based on node positions within the network only. The resulting methods are graphical and thus effective and exempt from numerical errors, and the final algorithm is provided to perform the analysis by machine computation. Several examples are provided to show the effectiveness of the algorithm proposed. Full article

This paper analyzes two approximation methods for the Laplacian eigenvectors of the Kronecker product, as recently presented in the literature. We enhance the approximations by comparing the correlation coefficients of the eigenvectors, which indicate how well an arbitrary vector approximates a matrix’s eigenvector. In the first method, some correlation coefficients are explicitly calculable, while others are not. In the second method, only certain coefficients can be estimated with good accuracy, as supported by empirical and theoretical evidence, with the rest remaining incalculable. The primary objective is to evaluate the accuracy of the approximation methods by analyzing and comparing limited sets of coefficients on one hand and the estimation on the other. Therefore, we compute the extreme values of the mentioned sets and theoretically compare them. Our observations indicate that, in most cases, the relationship between the majority of the values in the first set and those in the second set reflects the relationship between the remaining coefficients of both approximations. Moreover, it can be observed that each of the sets generally contains smaller values compared to the values found among the remaining correlation coefficients. Finally, we find that the performance of the two approximation methods is significantly influenced by imbalanced graph structures, exemplified by a class of almost regular graphs discussed in the paper. Full article

Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and N-chain complexes. Full article

In recent years, the application of free-form surface spatial grid structures in large public buildings has become increasingly common. The layouts of grids are important factors that affect both the mechanical performance and aesthetic appeal of such structures. To achieve a triangular grid with good mechanical performance and uniformity on free-form surfaces, this study proposes a new method called the “strain energy gradient optimization method”. The grid topology is optimized to maximize the overall stiffness, by analyzing the sensitivity of nodal coordinates to the overall strain energy. The results indicate that the overall strain energy of the optimized grid has decreased, indicating an improvement in the structural stiffness. Specifically, compared to the initial grid, the optimized grid has a 30% decrease in strain energy and a 43.3% decrease in maximum nodal displacement. To optimize the smoothness of the grid, the study further applies the Laplacian grid smoothing method. Compared to the mechanically adjusted grid, the structural mechanical performance does not significantly change after smoothing, while the geometric indicators are noticeably improved, with smoother lines and regular shapes. On the other hand, compared to the initial grid, the smoothed grid has a 21.4% decrease in strain energy and a 28.3% decrease in maximum nodal displacement. Full article

The use of graph theory for solving labyrinths and mazes is well known, understanding the possible paths as the connections between the nodes that represent the corners or bifurcations. This work presents a new idea: minimizing the length of the optimal path formulated as a topology optimization problem. The maze is mapped with finite elements, and then, through the eigenvalues of the Laplacian matrix of the graph, a constraint is imposed over the connectivity between the input and the output. Several 2D examples are provided to support this approach, allowing for unequivocally finding the shortest path in mazes with multiple connections between entrance and exit, resulting in an nonheuristic algorithm. Full article

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation ${(-\Delta )}^{s}u+V\left(ϵx\right)u=\lambda u+ulog{u}^2\mathrm{in}{\mathbb{R}}^N,$ under the mass constraint ${\int }_{{\mathbb{R}}^N}{\left|u\right|}^{2}dx=a.$ Here, $N\ge 2$, $a,ϵ>0$, $\lambda \in \mathbb{R}$ is an unknown parameter, ${(-\Delta )}^s$ is the fractional Laplacian and $s\in (0,1)$. We introduce a function space where the energy functional associated with the problem is of class ${\mathcal{C}}^1$. Then, under some assumptions on the potential V and using the Lusternik–Schnirelmann category, we show that the number of normalized solutions depends on the topology of the set for which the potential V reaches its minimum. Full article

This paper investigates the long-time behavior of fractional-order complex memristive neural networks in order to analyze the synchronization of both anatomical and functional brain networks, for predicting therapy response, and ensuring safe diagnostic and treatments of neurological disorder (such as epilepsy, Alzheimer’s disease, or Parkinson’s disease). A new mathematical brain connectivity model, taking into account the memory characteristics of neurons and their past history, the heterogeneity of brain tissue, and the local anisotropy of cell diffusion, is proposed. This developed model, which depends on topology, interactions, and local dynamics, is a set of coupled nonlinear Caputo fractional reaction–diffusion equations, in the shape of a fractional-order ODE coupled with a set of time fractional-order PDEs, interacting via an asymmetric complex network. In order to introduce into the model the connection structure between neurons (or brain regions), the graph theory, in which the discrete Laplacian matrix of the communication graph plays a fundamental role, is considered. The existence of an absorbing set in state spaces for system is discussed, and then the dissipative dynamics result, with absorbing sets, is proved. Finally, some Mittag–Leffler synchronization results are established for this complex memristive neural network under certain threshold values of coupling forces, memristive weight coefficients, and diffusion coefficients. Full article

With the advancement of information technology, social media has become increasingly prevalent. The complex networks of social relationships among decision-makers (DMs) have given rise to the problem of social network group decision-making (SNGDM), which has garnered considerable attention in recent years. However, most existing consensus-reaching methods in SNGDM only consider local network information when determining the influence of DMs within the social network. This approach fails to adequately reflect the crucial role of key DMs in regulating information propagation during the consensus-reaching process. Additionally, the partial absence of linguistic evaluations in the decision-making problems also poses obstacles to identifying the optimal alternative. Therefore, this paper proposes an improved Laplacian gravity centrality-based consensus method that can effectively handle incomplete decision information in social network environments. First, the extended comparative linguistic expressions with symbolic translation (ELICIT) are utilized to describe DMs’ linguistic evaluations and construct the incomplete decision matrix. Second, the improved Laplacian gravity centrality (ILGC) is proposed to quantify the influence of DMs in the social network by considering local and global topological structures. Based on the ILGC measure, we develop a trust-driven consensus-reaching model to enhance group consensus, which can better simulate opinion interactions in real-world situations. Lastly, we apply the proposed method to a smart city evaluation problem. The results show that our method can more reasonably handle incomplete linguistic evaluations, more comprehensively capture the influence of DMs, and more effectively improve group consensus. Full article