On computing linearizing coordinates from the symmetry algebra

A characterization of the symmetry algebra of the N th-order ordinary differential equations (ODEs) with maximal symmetry and all third-order linearizable ODEs is given. This is used to show that such an algebra g determines - up to a point transformation - only one linear equation whose symmetry algebra is g and an algorithmic procedure is given to find the linearizing coordinates. This procedure is applied to several examples from the literature

A Note on the Integration of Scalar Fourth-Order Ordinary Differential Equations with Four-Dimensional Symmetry Algebras

The complete integration of scalar fourth-order ODEs with four-dimensional symmetry algebras is performed by utilizing Lie’s method which was invoked to integrate scalar second-order ODEs admitting two-dimensional symmetry algebras. We obtain a complete integration of all scalar fourth-order ODEs that possess four Lie point symmetries

Proactive Caching at the Edge Leveraging Influential User Detection in Cellular D2D Networks

Caching close to users in a radio access network (RAN) has been identified as a promising method to reduce a backhaul traffic load and minimize latency in 5G and beyond. In this paper, we investigate a novel community detection inspired by a proactive caching scheme for device-to-device (D2D) enabled networks. The proposed scheme builds on the idea that content generated/accessed by influential users is more probable to become popular and thus can be exploited for pro-caching. We use a Clustering Coefficient based Genetic Algorithm (CC-GA) for community detection to discover a group of cellular users present in close vicinity. We then use an Eigenvector Centrality measure to identify the influential users with respect to the community structure, and the content associated to it is then used for pro-active caching using D2D communications. The numerical results show that, compared to reactive caching, where historically popular content is cached, depending on cache size, load and number of requests, up to 30% more users can be satisfied using a proposed scheme while achieving significant reduction in backhaul traffic load

Recent advancements in metal oxides for energy storage materials: design, classification, and electrodes configuration of supercapacitor

Among different energy storage devices, supercapacitors have garnered the attention due to their higher charge storage capacity, superior charging-discharging performance, higher power density, and long cycle life. Subsequently, introducing low-cost and highly-efficient supercapacitors is a hot topic in the industrial and scientific realms. Metal oxides are considered as the most suitable electrode materials due to their intrinsic properties, economic attractiveness, environmental friendliness, and abundant availability. However, there is still a salient need to have e comprehensive study of the significance of metal oxide nanomaterials for supercapacitors in relation to their sustainable synthesis approaches. Thus, this review delineated the different sustainable synthesis approaches for metal oxide nanomaterials following their morphological, compositional, and supercapacitive properties. Initially, the review covers designs or configurations of supercapacitors followed by the types of supercapacitors based on electrode materials. Further, metal oxides-based pseudocapacitive-/battery-type electrodes have been articulated comprehensively. Moreover, this review aims to provide sustainable fabrication of metal oxide nanomaterial-based supercapacitors with a superior interpretation of the design and functioning of the device that could serve as guidelines for new synthesis and fabrication approaches to develop sustainable supercapacitors for practical applications