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The Industrial Internet of Things (IIoT) refers to the use of interconnected smart devices, sensors, and other technologies to create a network of intelligent systems that can monitor and manage industrial processes. 6TiSCH (IPv6 over the Time Slotted Channel Hopping mode of IEEE 802.15.4e) as an enabling technology facilitates low-power and low-latency communication between IoT devices in industrial environments. The Routing Protocol for Low power and lossy networks (RPL), which is used as the de-facto routing protocol for 6TiSCH networks is observed to suffer from several limitations, especially during congestion in the network. Therefore, there is an immediate need for some modifications to the RPL to deal with this problem. Under traffic load which keeps on changing continuously at different instants of time, the proposed mechanism aims at finding the appropriate parent for a node that can forward the packet to the destination through the least congested path with minimal packet loss. This facilitates congestion management under dynamic traffic loads. For this, a new metric for routing using the concept of exponential weighting has been proposed, which takes the number of packets present in the queue of the node into account when choosing the parent at a particular instance of time. Additionally, the paper proposes a parent selection and swapping mechanism for congested networks. Performance evaluations are carried out in order to validate the proposed work. The results show an improvement in the performance of RPL under heavy and dynamic traffic loads., Comment: The paper has been submitted, accepted, and presented at the 2023 IEEE Global Communications Conference: Next-Generation Networking and Internet, with plans for publication. It was delivered during the IEEE Global Communications Conference held on December 6th, 2023, in Kuala Lumpur, Malaysia

Here we show that for $k\in \mathbb N,$ the closure of the $k$-rank numerical range of a contraction $A$ acting on an infinite-dimensional Hilbert space $\mathcal{H}$ is the intersection of the closure of the $k$-rank numerical ranges of all unitary dilations of $A$ to $\mathcal{H}\oplus\mathcal{H}.$ The same is true for $k=\infty$ provided the $\infty$-rank numerical range of $A$ is non-empty. These generalize a finite dimensional result of Gau, Li and Wu. We also show that when both defect numbers of a contraction are equal and finite ($=N$), one may restrict the intersection to a smaller family consisting of all unitary $N$-dilations. A result of {Bercovici and Timotin} on unitary $N$-dilations is used to prove it. Finally, we have investigated the same problem for the $C$-numerical range and obtained the answer in negative., Comment: 15 pages; Published in Adv. Oper. Theory 7, 56 (2022). Title is changed to "Generalized Halmos Conjectures and Constrained Unitary Dilations". In this published version, section 5 is the new addition. Based on the referee report, more examples are given in section 6

The advantage of computational resources in edge computing near the data source has kindled growing interest in delay-sensitive Internet of Things (IoT) applications. However, the benefit of the edge server is limited by the uploading and downloading links between end-users and edge servers when these end-users seek computational resources from edge servers. The scenario becomes more severe when the user-end's devices are in the shaded region resulting in low uplink/downlink quality. In this paper, we consider a reconfigurable intelligent surface (RIS)-assisted edge computing system, where the benefits of RIS are exploited to improve the uploading transmission rate. We further aim to minimize the delay of worst-case in the network when the end-users either compute task data in their local CPU or offload task data to the edge server. Next, we optimize the uploading bandwidth allocation for every end-user's task data to minimize the maximum delay in the network. The above optimization problem is formulated as quadratically constrained quadratic programming. Afterward, we solve this problem by semidefinite relaxation. Finally, the simulation results demonstrate that the proposed strategy is scalable under various network settings., Comment: 6 pages, 3 figures, accepted for publication in IEEE GLOBECOM 2021

Multiple-input multiple-output non-orthogonal multiple access (MIMO-NOMA) cellular network is promising for supporting massive connectivity. This paper exploits low-latency machine learning in the MIMO-NOMA uplink transmission environment, where a substantial amount of data must be uploaded from multiple data sources to a one-hop away edge server for machine learning. A delay-aware edge learning framework with the collaboration of data sources, the edge server, and the base station, referred to as DACEL, is proposed. Based on the delay analysis of DACEL, a NOMA channel allocation algorithm is further designed to minimize the learning delay. The simulation results show that the proposed algorithm outperforms the baseline schemes in terms of learning delay reduction., Comment: 5 pages, 3 figures, accepted for publication in IEEE/CIC ICCC 2021

We describe here the higher rank numerical range, as defined by Choi, Kribs and Zyczkowski, of a normal operator on an infinite dimensional Hilbert space in terms of its spectral measure. This generalizes a result of Avendano for self-adjoint operators. An analogous description of the numerical range of a normal operator by Durszt is derived for the higher rank numerical range as an immediate consequence. It has several interesting applications. We show using Durszt's example that there exists a normal contraction $T$ for which the intersection of the higher rank numerical ranges of all unitary dilations of $T$ contains the higher rank numerical range of $T$ as a proper subset. Finally, we strengthen and generalize a result of Wu by providing a necessary and sufficient condition for the higher rank numerical range of a normal contraction being equal to the intersection of the higher rank numerical ranges of all possible unitary dilations of it., Comment: 21 pages

Reconfigurable intelligent surface (RIS)-empowered communication is being considered as an enabling technology for sixth generation (6G) wireless networks. The key idea of RIS-assisted communication is to enhance the capacity, coverage, energy efficiency, physical layer security, and many other aspects of modern wireless networks. At the same time, mobile edge computing (MEC) has already shown its huge potential by extending the computation, communication, and caching capabilities of a standalone cloud server to the network edge. In this article, we first provide an overview of how MEC and RIS can benefit each other. We envision that the integration of MEC and RIS will bring an unprecedented transformation to the future evolution of wireless networks. We provide a system-level perspective on the MEC-aided RIS (and RIS-assisted MEC) that will evolve wireless network towards 6G. We also outline some of the fundamental challenges that pertain to the implementation of MEC-aided RIS (and RIS-assisted MEC) networks. Finally, the key research trends in the RIS-assisted MEC are discussed.

As Critical National Infrastructures are becoming more vulnerable to cyber attacks, their protection becomes a significant issue for any organization as well as a nation. Moreover, the ability to attribute is a vital element of avoiding impunity in cyberspace. In this article, we present main threats to critical infrastructures along with protective measures that one nation can take, and which are classified according to legal, technical, organizational, capacity building, and cooperation aspects. Finally we provide an overview of current methods and practices regarding cyber attribution and cyber peace keeping

This paper presents a comprehensive survey of the existing blockchain protocols for the Internet of Things (IoT) networks. We start by describing the blockchains and summarizing the existing surveys that deal with blockchain technologies. Then, we provide an overview of the application domains of blockchain technologies in IoT, e.g, Internet of Vehicles, Internet of Energy, Internet of Cloud, Fog computing, etc. Moreover, we provide a classification of threat models, which are considered by blockchain protocols in IoT networks, into five main categories, namely, identity-based attacks, manipulation-based attacks, cryptanalytic attacks, reputation-based attacks, and service-based attacks. In addition, we provide a taxonomy and a side-by-side comparison of the state-of-the-art methods towards secure and privacy-preserving blockchain technologies with respect to the blockchain model, specific security goals, performance, limitations, computation complexity, and communication overhead. Based on the current survey, we highlight open research challenges and discuss possible future research directions in the blockchain technologies for IoT., Comment: 14 pages, 5 figures

In this paper, we address the symbol synchronization issue in molecular communication via diffusion (MCvD). Symbol synchronization among chemical sensors and nanomachines is one of the critical challenges to manage complex tasks in the nanonetworks with molecular communication (MC). As in diffusion-based MC, most of the molecules arrive at the receptor closer to the start of the symbol duration, the wrong estimation of the start of the symbol interval leads to a high symbol detection error. By utilizing two types of molecules with different diffusion coefficients we propose a synchronization technique for MCvD. Moreover, we evaluate the symbol-error-rate performance under the proposed symbol synchronization scheme for equal and non-equal symbol duration in MCvD systems., Comment: Changed title, revised text, IEEE International Conference on Communications (ICC), Shanghai, China, May, 2019

D. Bures defined a metric $\beta $ on states of a $C^*$-algebra and this concept has been generalized to unital completely positive maps $\phi : \mathcal A \to \mathcal B$, where $\mathcal B$ is either an injective $C^*$-algebra or a von Neumann algebra. We introduce a new distance $\gamma $ for the same classes of unital completely positive maps. We use in our definition the distance between representations on the same Hilbert $C^*$-module in contrast to the Bures metric which uses one representation and distinct vectors. This metric can be expressed in terms of a class of completely positive maps on free products of $C^*$-algebras and in this setting $\gamma $ looks like Wasserstein metric on probability measures. Surprisingly, when the range algebra $\mathcal B$ is injective, $\gamma $ and $\beta $ are related by the following explicit formula: $\beta ^2= 2-\sqrt{4- \gamma ^2} .$ A deep result of Choi and Li on constrained dilation is the main tool in proving this formula., Comment: 28 pages, The abstract and some statements revised based on a referee report. To appear in the Houston Journal of Mathematics

We introduce the notion of additive units, or `addits', of a pointed Arveson system, and demonstrate their usefulness through several applications. By a pointed Arveson system we mean a spatial Arveson system with a fixed normalised reference unit. We show that the addits form a Hilbert space whose codimension-one subspace of `roots' is isomorphic to the index space of the Arveson system, and that the addits generate the type I part of the Arveson system. Consequently the isomorphism class of the Hilbert space of addits is independent of the reference unit. The addits of a pointed inclusion system are shown to be in natural correspondence with the addits of the generated pointed product system. The theory of amalgamated products is developed using addits and roots, and an explicit formula for the amalgamation of pointed Arveson systems is given, providing a new proof of its independence of the particular reference units. (This independence justifies the terminology `spatial product' of spatial Arveson systems). Finally a cluster construction for inclusion subsystems of an Arveson system is introduced and we demonstrate its correspondence with the action of the Cantor--Bendixson derivative in the context of the random closed set approach to product systems due to Tsirelson and Liebscher., Comment: 26 pages. Note added in proof. To appear in the Transactions of the American Mathematical Society

It is known that the spatial product of two product systems is intrinsic. Here we extend this result by analyzing subsystems of the tensor product of product systems. A relation with cluster systems is established. In a special case, we show that the amalgamated product of product systems through strictly contractive units is independent of the choices of the units. The amalgamated product in this case is isomorphic to the tensor product of the spatial product of the two and the type I product system of index one., Comment: 9 pages

We introduce the notion of additive units and roots of a unit in a spatial product system. The set of all roots of any unit forms a Hilbert space and its dimension is the same as the index of the product system. We show that a unit and all of its roots generate the type I part of the product system. Using properties of roots, we also provide an alternative proof of the Powers' problem that the cocycle conjugacy class of Powers sum is independent of the choice of intertwining isometries. In the last section, we introduce the notion of cluster of a product subsystem and establish its connection with random sets in the sense of Tsirelson and Liebscher., Comment: 20 pages

We prove that the spatial product of two spatial Arveson systems is independent of the choice of the reference units. This also answers the same question for the minimal dilation the Powers sum of two spatial CP-semigroups: It is independent up to cocycle conjugacy.

Here we generalize the concept of spatial tensor product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated tensor product of product systems, and now the amalgamation can be done using a contractive morphism. Index of amalgamation product (when done through units) adds up for normalized units but for non-normalized units, the index is one more than the sum. We define inclusion systems and use it as a tool for index computations. It is expected that this notion will have other uses., Comment: 13 pages, Improved version.