F-threshold functions: syzygy gap fractals and the two-variable homogeneous case

In this article we study F-pure thresholds (and, more generally, F-thresholds) of homogeneous polynomials in two variables over a field of characteristic p>0. Passing to a field extension, we factor such a polynomial into a product of powers of pairwise prime linear forms, and to this collection of linear forms we associate a special type of function called a syzygy gap fractal. We use this syzygy gap fractal to study, at once, the collection of all F-pure thresholds of all polynomials constructed with the same fixed linear forms. This allows us to describe the structure of the denominator of such an F-pure threshold, showing in particular that whenever the F-pure threshold differs from its expected value its denominator is a multiple of p. This answers a question of Schwede in the two-variable homogeneous case. In addition, our methods give an algorithm to compute F-pure thresholds of homogenous polynomials in two variables.Comment: 42 pages; 6 figures. Section 6 was mostly rewritten; a new appendix was included; other smaller changes throughout. Comments welcom

Existence of an unbounded vacant set for subcritical continuum percolation

We consider the Poisson Boolean percolation model in $\mathbb{R}^2$, where the radii of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in $\mathbb{R}^d$, for any $d\ge2$, finite moment of order $d$ is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.Comment: 9 page

Multitype Contact Process on Z\Z: Extinction and Interface

We consider a two-type contact process on $\Z$ in which both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval $[-L,L]$ and the other type occupies infinitely many sites both in $(-\infty, L)$ and $(L, \infty)$. We also show that, starting from the configuration in which all sites in $(-\infty, 0]$ are occupied by type 1 particles and all sites in $(0, \infty)$ are occupied by type 2 particles, the process $\rho_t$ defined by the size of the interface area between the two types at time $t$ is tight

Maturity model for DevOps

Businesses today need to respond to customer needs at unprecedented speed. Driven by this need for speed, many companies are rushing to the DevOps movement. DevOps, the combination of Development and Operations, is a new way of thinking in the software engineering domain that recently received much attention. Since DevOps has recently been introduced as a new term and novel concept, no common understanding of what it means has yet been achieved. Therefore, the definitions of DevOps often are only a part relevant to the concept. When further observing DevOps, it could be seen as a movement, but is still young and not yet formally defined. Also, no adoption models or fine-grained maturity models showing what to consider to adopt DevOps and how to mature it were identified. As a consequence, this research attempted to fill these gaps and consequently brought forward a Systematic Literature Review to identify the determining factors contributing to the implementation of DevOps, including the main capabilities and areas with which it evolves. This resulted in a list of practices per area and capability that was used in the interviews with DevOps practitioners that, with their experience, contributed to define the maturity of those DevOps practices. This combination of factors was used to construct a DevOps maturity model showing the areas and capabilities to be taken into account in the adoption and maturation of DevOps.Hoje em dia, as empresas precisam de responder às necessidades dos clientes a uma velocidade sem precedentes. Impulsionadas por esta necessidade de velocidade, muitas empresas apressam-se para o movimento DevOps. O DevOps, a combinação de Desenvolvimento e Operações, é uma nova maneira de pensar no domínio da engenharia de software que recentemente recebeu muita atenção. Desde que o DevOps foi introduzido como um novo termo e um novo conceito, ainda não foi alcançado um entendimento comum do que significa. Portanto, as definições do DevOps geralmente são apenas uma parte relevante para o conceito. Ao observar o DevOps, o fenómeno aborda questões culturais e técnicas para obter uma produção mais rápida de software, tem um âmbito amplo e pode ser visto como um movimento, mas ainda é jovem e ainda não está formalmente definido. Além disso, não foram identificados modelos de adoção ou modelos de maturidade refinados que mostrem o que considerar para adotar o DevOps e como fazê-lo crescer. Como consequência, esta pesquisa tentou preencher essas lacunas e, consequentemente, apresentou uma Revisão sistemática da literatura para identificar os fatores determinantes que contribuem para a implementação de DevOps, incluindo os principais recursos e áreas com os quais ele evolui. Isto resultou numa lista de práticas por área e por capacidade, que foi utilizado como base nas entrevistas realizadas com peritos em DevOps que, com a sua experiência, ajudaram a atribuir níveis de maturidade a cada prática. Esta combinação de fatores foi usada para construir um modelo de maturidade de DevOps mostrando as áreas e as capacidades a serem levados em consideração na sua adoção e maturação