Linear-width is a well-regarded width parameter in graph theory. The constructs of linear loose tangle and linear tangle present obstacles to attaining linear-width. In this succinct paper, our primary focus will be the exploration of maximal linear loose tangles.

Loose Tangle is a concept in graph theory that has a dual relationship with branch-width which is well-known graph width parameter. Ultrafilter, a fundamental notion in mathematics, is similarly known to have a dual relationship with branch-width when extended to a connectivity system (X, f). In this compact paper, we revisit and contemplate the interplay between Loose Tangle and Filter through the lens of a submodular partition function.

: In the realm of data management, data preservation stands as a critical undertaking aimed at preserving and upholding the integrity of data. Regardless of whether it concerns personal or enterprise data, the detrimental effects of malicious alterations implemented by attackers cannot be overlooked. Particularly in conventional industrial control environments, the prevalent practice involves the transmission of data from sensors to databases for storage purposes. However, it is essential to recognize that this process exposes the data to various vulnerabilities. Thus, to ensure the long-term security and reliability of the data, it becomes imperative to implement robust data preservation strategies within these industrial control systems. However, the reliance of these databases on physical hard disks introduces inherent vulnerabilities, including the potential for data loss due to disk damage or targeted malicious attacks. Consequently, it becomes imperative to prioritize the implementation of robust data preservation measures. These measures are crucial in mitigating the risk of disruptions and protecting critical data from compromise. By establishing effective data backup systems, employing advanced security protocols, and implementing proactive monitoring mechanisms, organizations can bolster their data preservation capabilities and safeguard against potential threats to data integrity and availability. As a result, many enterprises opt to store their data with third-party providers to ensure data integrity. However, this approach carries inherent risks. If the third-party service experiences an attack or if the data is tampered with, it becomes challenging to verify the integrity of the data. To address these concerns and ensure data preservation within the context of the Internet of Things (IoT), a growing number of individuals are integrating IoT with Distributed Ledger Technology (DLT). By leveraging DLT, the integrity of data can be ensured, reducing reliance on centralized third-party storage and enhancing security in the IoT ecosystem. In this article, IOTA is the DLT, which employs Directed Acyclic Graph (DAG) to store transaction information. Compared to Ethereum or other blockchain technologies, IOTA offers notable advantages in terms of transaction verification speed, making it highly suitable for real-time IoT environments. However, the conventional transmission path from sensors to IOTA nodes entails a complex route, involving multiple hardware devices before reaching the intended destination. This complexity poses challenges in ensuring data integrity during transmission and introduces vulnerabilities such as man-in-the-middle attacks or SQL injection attacks. To address these issues, we propose a method to streamline the transmission path between sensors and IOTA, specifically tailored for industrial fields with numerous IoT devices. Our approach involves preprocessing the data stored on the server using our method before uploading, ensuring data confidentiality, and leveraging IOTA to guarantee data integrity. To achieve the shortest path between IoT and DLT nodes, it becomes necessary to establish IOTA nodes on lower-level devices, such as Raspberry Pi or IoT controllers. By simplifying the transmission path, we can reduce the potential for tampering and enhance overall data security. Implementing our proposed method enables the assurance of data confidentiality and integrity during both transmission and storage on the server, strengthening the trustworthiness of the IoT, and IOTA integration.

A geometric Planck-scale model of quantum electrodynamics is tested against observations. Based on Dirac’s proposal to describe spin 1/2 particles as tethered objects, elementary fermions are conjectured to be fluctuating rational tangles with unobservable tethers. As Battey-Pratt and Racey showed, when such tangles propagate, they obey the free Dirac equation. Classifying rational tangles yields the observed spectrum of elementary fermions. Classifying deformations of tangle cores yields exactly three types of gauge interactions. They exchange three types of elementary gauge bosons and have the symmetry groups U(1), broken SU(2) and SU(3). The possible rational tangles for fermions, Higgs and gauge bosons allow only the observed Feynman diagrams. The complete Lagrangian of the standard model arises, including the Lagrangian of quantum electrodynamics. Over 50 tests of the tangle model are deduced. They include details on the perturbation expansion of the g-factor and the existence of limit values for electric and magnetic fields. Measurements agree with most tests, though this could change in the future. Some tests are genuine predictions that still need to be checked. In particular, the geometry of the process occurring at QED interaction vertices suggests an ab-initio estimate for the fine structure constant. In addition, the average geometric shapes of the elementary particle tangles suggest ab-initio lower and upper limits for the mass values of the electron and the other leptons.