Solution of a problem of Skolem

T. Skolem shows that there are at most six integer solutions to the Diophantine equation x5 + 2y5 + 4z5 - 10xy3z + 10x2yz2 = 1. The author shows here that there are precisely three integer solutions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23064/1/0000636.pd

Tapping network traffic in Kubernetes

The rapid increase in cloud usage among organizations has led to a shift in the cybersecurity industry. Whereas before, organizations wanted traditional security monitoring using statically placed IDS sensors within their data centers and networks, they now want dynamic security monitoring of their cloud solutions. As more and more organizations move their infrastructure and applications to the cloud the need for cybersecurity solutions that can adapt and transform to meet this new demand is increasing. Although many cloud providers, provide integrated security solutions, these are dependent on the correct configuration from the customers, which may rather want to pay a security firm instead. Telenor Security Operation Center is a long contender in the traditional cybersecurity firm space and is looking to move into IDS monitoring of cloud solutions, more specifically providing network IDS monitoring of traffic within managed Kubernetes clusters at cloud providers, such as Amazon Web Services Elastic Kubernetes Service. This is to be accomplished by providing all the desired pods within a cluster their own sidecar container, which acts as a network sniffer that sends the recorded traffic through vxlan to an external sensor also operating in the cloud. By doing this, traditional IDS monitoring suddenly becomes available in the cloud, and is covering a part that is often neglected in cloud environments, and that is monitoring the internal Kubernetes cluster traffic. AWS EKS was used as a testing ground for a simulated Kubernetes cluster running sample applications monitored by the sidecar container. Which is essentially a Python script sniffing the localhost traffic of the shared network namespace of a Kubernetes pod. This infrastructure will be generated by a set of Terraform files for automated setup and reproducibility, as well as making use of the gitops tool Fluxcd for syncing Kubernetes manifests. The solution will also be monitored by a complete monitoring solution in the form of kube-prometheus-stack which will provide complete insight into performance metrics down at the container level, through Prometheus and Grafana. Finally, a series of performance tests will be conducted, using k6s and iperf, automated by Ansible, to gather the performance impact of the sidecar container. A series of iperf and k6s tests were conducted against the sidecar container. The k6s test was run at a data rate of 3 Mb/s and showed that the data rate needed to be higher to gather useful performance metrics. This is where iperf took over and tested the sidecar container at data rates of 50,100,250 and 500 Mb/s using a server at the University of Agder as base. These initial raw performance results showed a max CPU usage of 11.8% of the Kubernetes node’s 2 vCPU’s. Together with a max memory usage of 14 MB this showed that the sidecar container does not consume a vast amount of resources. And has the potential as a scalable and efficient network tapping method in Kubernetes. However, some anomalies were discovered during the performance testing that revealed undiscovered issues with the method. One of which was packet anomalies between the number of packets at the sensor and the number of packets observed by the iperf server at the University of Agder. Due to the many layers involved in the networking stack for this method, there needs to be conducted additional research into how these anomalies arise. While also considering alternative transport methods to vxlan

Level Eulerian Posets

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem--Mahler--Lech theorem, the ${\bf ab}$-series of a level poset is shown to be a rational generating function in the non-commutative variables ${\bf a}$ and ${\bf b}$. In the case the poset is also Eulerian, the analogous result holds for the ${\bf cd}$-series. Using coalgebraic techniques a method is developed to recognize the ${\bf cd}$-series matrix of a level Eulerian poset

Finitely generated ideals of the ring of integer-valued polynomials

Throughout this paper, Z denoes the integers, Q the rational numbers, and D the collection of polynomials over Q having the property thatf(a) E Z for every a in Z. After first being studied by Polya [2 1 ] and Skolem [23], the domain D has been the subject of several more recent papers [2-14, 16, 17 ]. In particular, Brizolis established in [4] that D is a Priifer domain with each finitely generated ideal I determined by the values at integers of the polynomials in I. Specifically, he showed that if I= (f,(r),...,&(f))D, then g(t) E I if and only if g(u) E (f,(a),...,fJ(u))Z for every a E Z. In this paper we continue the study of the finitely generated ideals of D. While our initial efforts were directed toward answering a question of Brizolis [4] as to whether or not each finitely generated ideal of D can be gnerated by two elements, in time we became interested in giving a more explicit description of finite generating sets for ideals of D. Our methods are constructive, and we feel that we have had some success in accomplising this goal

A Cauchy-Dirac delta function

The Dirac delta function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his infinitesimal definition of delta.Comment: 24 pages, 2 figures; Foundations of Science, 201

Stevin numbers and reality

We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with arXiv:1104.0375, arXiv:1108.2885, arXiv:1108.420