Structured singular values and their application in computing eigenvalue backward errors of the Rosenbrock system matrix

The structured singular values (aka the {\mu}-values) are essential in analyzing the stability of control systems and in the structured eigenvalue perturbation theory of matrices and matrix polynomials. In this paper, we study the {\mu}-value of a matrix under block-diagonal structured perturbations (full blocks but possibly rectangular). We provide an explicit expression for the {\mu}-value and also obtain a computable upper bound in terms of minimizing the largest singular value of a parameter-dependent matrix. This upper bound equals the {\mu}-value when the perturbation matrix has no more than three blocks on the diagonal. We then apply the {\mu}-value results in computing eigenvalue backward errors of a Rosenbrock system matrix corresponding to a rational matrix function when some or all blocks of the Rosenbrock system matrix are subject to perturbation. The results are illustrated through numerical experiments.Comment: 23 page

Comparison between Tubular Discectomy and Open Microdiscectomy surgery for Symptomatic Lumbar Disk Herniation

Objectives: There are many different minimally invasive procedures that can be used to treat lumbar disc herniation. In the management of lumbar disc herniation, Open microdiscectomy MD is the gold standard and is used as a benchmark for comparison with more recent techniques like tubular discectomy. The purpose of this study was to assess the complication rates of tubular discectomy and to compare the postoperative outcomes of individuals undergoing tubular discectomy with those undergoing MD. Methods: A retrospective analysis of 250 patients who underwent single-level lumbar discectomy either by tubular TD or MD between JULY 2017- JUNE 2022 was performed. The differences in the 2 groups' demographics, surgical length, intraoperative blood loss, overall hospital stay, pain score on the visual analogue scale (VAS), Oswestry Disability Index (ODI) score both before and after the procedure, and complications were assessed. Results: Out of the 250 patients, 130 patients were treated with MD and 120 by tubular disectomy TD. The mean age in MD and tubular TD group was 45.5 and 45.8 years. There was a improvement in VAS and ODI scores at 4 weeks in both the groups. There was a greater reduction of back pain in the TD group at 2 weeks compared to MD group but at 1 month both are comparable. Average surgical time was shorter in MD (71.6 min) as compared to TD group (80.2 min). the Average blood loss was higher (90.2 mL) in MD group as compared to TD group (35.8 mL) (Table 3 ). Length of incision as measured from the surgical scar was 1.7 cm in TD while it was 3.0 cm in MD group. Average hospital length of stay in TD (1.2 days) which was less than MD group (2.1 days). Likewise, MD group patients took 5.9 weeks to return to activity which was higher than the TD group (4.1 weeks). There were 9 cases (6.9%) of dural tear in MD and 10 cases (8.3%) in TD group. Conclusion: Sciatica caused by disc herniation can be effectively and safely treated with lumbar discectomy, whether it is TD or MD. In comparison to MD, the TD method for treating symptomatic lumbar radiculopathy is superior in that it causes less postoperative back pain, less blood loss, a shorter hospital stay, and a quicker return to work. Despite the learning curve, TD has been demonstrated to be a viable choice for treating lumbar disc herniation in patients who are carefully chosen

Eigenvalue backward errors of Rosenbrock systems and optimization of sums of Rayleigh quotient

We address the problem of computing the eigenvalue backward error of the Rosenbrock system matrix under various types of block perturbations. We establish computable formulas for these backward errors using a class of minimization problems involving the Sum of Two generalized Rayleigh Quotients (SRQ2). For computational purposes and analysis, we reformulate such optimization problems as minimization of a rational function over the joint numerical range of three Hermitian matrices. This reformulation eliminates certain local minimizers of the original SRQ2 minimization and allows for convenient visualization of the solution. Furthermore, by exploiting the convexity within the joint numerical range, we derive a characterization of the optimal solution using a Nonlinear Eigenvalue Problem with Eigenvector dependency (NEPv). The NEPv characterization enables a more efficient solution of the SRQ2 minimization compared to traditional optimization techniques. Our numerical experiments demonstrate the benefits and effectiveness of the NEPv approach for SRQ2 minimization in computing eigenvalue backward errors of Rosenbrock systems

Implementing Security in IoT Ecosystem Using 5G Network Slicing and Pattern Matched Intrusion Detection System: A Simulation Study

Aim/Purpose: 5G and IoT are two path-breaking technologies, and they are like wall and climbers, where IoT as a climber is growing tremendously, taking the support of 5G as a wall. The main challenge that emerges here is to secure the ecosystem created by the collaboration of 5G and IoT, which consists of a network, users, endpoints, devices, and data. Other than underlying and hereditary security issues, they bring many Zero-day vulnerabilities, which always pose a risk. This paper proposes a security solution using network slicing, where each slice serves customers with different problems. Background: 5G and IoT are a combination of technology that will enhance the user experience and add many security issues to existing ones like DDoS, DoS. This paper aims to solve some of these problems by using network slicing and implementing an Intrusion Detection System to identify and isolate the compromised resources. Methodology: This paper proposes a 5G-IoT architecture using network slicing. Research here is an advancement to our previous implementation, a Python-based software divided into five different modules. This paper’s amplification includes induction of security using pattern matching intrusion detection methods and conducting tests in five different scenarios, with 1000 up to 5000 devices in different security modes. This enhancement in security helps differentiate and isolate attacks on IoT endpoints, base stations, and slices. Contribution: Network slicing is a known security technique; we have used it as a platform and developed a solution to host IoT devices with peculiar requirements and enhance their security by identifying intruders. This paper gives a different solution for implementing security while using slicing technology. Findings: The study entails and simulates how the IoT ecosystem can be variedly deployed on 5G networks using network slicing for different types of IoT devices and users. Simulation done in this research proves that the suggested architecture can be successfully implemented on IoT users with peculiar requirements in a network slicing environment. Recommendations for Practitioners: Practitioners can implement this solution in any live or production IoT environment to enhance security. This solution helps them get a cost-effective method for deploying IoT devices on a 5G network, which would otherwise have been an expensive technology to implement. Recommendation for Researchers: Researchers can enhance the simulations by amplifying the different types of IoT devices on varied hardware. They can even perform the simulation on a real network to unearth the actual impact. Impact on Society: This research provides an affordable and modest solution for securing the IoT ecosystem on a 5G network using network slicing technology, which will eventually benefit society as an end-user. This research can be of great assistance to all those working towards implementing security in IoT ecosystems. Future Research: All the configuration and slicing resources allocation done in this research was performed manually; it can be automated to improve accuracy and results. Our future direction will include machine learning techniques to make this application and intrusion detection more intelligent and advanced. This simulation can be combined and performed with smart network devices to obtain more varied results. A proof-of-concept system can be implemented on a real 5G network to amplify the concept further. </jats:p