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In this paper, a fractional differential equation with integral conditions is studied. The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. It is shown that the scheme is second-order convergent, which improves the results on the uniform mesh. [ABSTRACT FROM AUTHOR]

The ArtGallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of ArtGallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of ArtGallery are all W[1]-hard with respect to k [Bonnet and Miltzow in 24th Annual European Symposium on Algorithms (Aarhus 2016)], the classic variant has an O (log k) -approximation algorithm [Bonnet and Miltzow in 33rd International Symposium on Computational Geometry (Brisbane 2017)], and it may require irrational guards [Abrahamsen et al. in 33rd International Symposium on Computational Geometry (Brisbane 2017)]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be ∃ R -complete [Abrahamsen et al. in 50th Annual ACM SIGACT Symposium on Theory of Computing (Los Angeles 2018)]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Workshop on Fixed-Parameter Computational Geometry (Leiden 2016)]: Is ArtGallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of ArtGallery, this case can also be viewed as the classic DominatingSet problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-VertexArtGallery is solvable in time r O (r 2) · n O (1) . Furthermore, our approach extends to assert that Vertex-BoundaryArtGallery and Boundary-VertexArtGallery are both FPT as well. To this end, we utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-VertexArtGallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions. [ABSTRACT FROM AUTHOR]

The popularity of Internet of Things (IoT) devices in smart homes has raised significant concerns regarding data security and privacy. Traditional machine learning (ML) methods for anomaly detection often require sharing sensitive IoT data with a central server, posing security and efficiency challenges. In response, this paper introduces FedGroup, a novel Federated Learning (FL) method inspired by FedAvg. FedGroup revolutionizes the central model’s learning process by updating it based on the learning patterns of distinct groups of IoT devices. Our experimental results demonstrate that FedGroup consistently achieves comparable or superior accuracy in anomaly detection when compared to both federated and non-federated learning methods. Additionally, Ensemble Learning (EL) collects intelligence from numerous contributing models, leading to enhanced prediction performance. Furthermore, FedGroup significantly improves the detection of attack types and their details, contributing to a more robust security framework for smart homes. Our approach demonstrates exceptional performance, achieving an accuracy rate of 99.64% with a minimal false positive rate (FPR) of 0.02% in attack type detection, and an impressive 99.89% accuracy in attack type detail detection. [ABSTRACT FROM AUTHOR]

As users have higher and higher requirements for the quality of experience, traditional cloud computing is gradually unable to meet the needs of user equipments. Hence mobile edge computing networks mounted by unmanned aerial vehicles are introduced to improve user experience and reduce energy consumption. However, most current work is based on neural networks, which require large amounts of labeled data or long training times. Given these challenges, this paper proposes an energy-efficient multi-stage alternating optimization scheme to reduce the weighted energy consumption of the entire network. We analyze the energy consumption of each device and formulate a non-convex optimization problem. Considering the impact of task offloading, resource allocation, and path planning on network energy consumption, we transform the energy consumption problem into three subproblems. And use the coordinate descent algorithm, interior point method, and successive convex approximation method to optimize them alternately. The simulation results show that the proposed optimization scheme can significantly reduce the network's total energy consumption. [ABSTRACT FROM AUTHOR]

The numerical solution of a class of second-order singularly perturbed three-point boundary value problems (BVPs) in 1D is achieved using a uniformly convergent, stable, and efficient difference method on a piecewise-uniform mesh. The presence of a boundary layer(s) on one (or both) of the interval’s endpoints is caused by the presence of the tiny parameter in the highest order derivative. As the perturbation parameter approaches 0, traditional numerical techniques on the uniform mesh become insufficient, resulting in poor accuracy and large blows without the use of an excessive number of points. Specially customised techniques, such as fitted operator methods or methods linked to adapted or fitted meshes that solve essential characteristics such as boundary and/or inner layers, are necessary to overcome this drawback. We developed a fitted-mesh technique in this paper that works for all perturbation parameter values. The monotone hybrid technique, which includes midway upwinding in the outer area and centre differencing in the layer region on a fitted-mesh condensing in the border layer region, is the basis for our difference scheme. In a discrete L∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^\infty $$\end{document} norm, uniform error estimates are constructed, and the technique is demonstrated to be parameter-uniform convergent of order two (up to a logarithmic factor). To show the effectiveness of the recommended technique and to corroborate the theoretical findings, a numerical example is presented. In practise, the convergence obtained matches the theoretical expectations. [ABSTRACT FROM AUTHOR]