On a class of vector fields with discontinuity of divide-by-zero type and its applications

We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces

How Much is the Whole Really More than the Sum of its Parts? 1 + 1 = 2.5: Superlinear Productivity in Collective Group Actions

In a variety of open source software projects, we document a superlinear growth of production ($R \sim c^\beta$) as a function of the number of active developers $c$, with $\beta \simeq 4/3$ with large dispersions. For a typical project in this class, doubling of the group size multiplies typically the output by a factor $2^\beta=2.5$, explaining the title. This superlinear law is found to hold for group sizes ranging from 5 to a few hundred developers. We propose two classes of mechanisms, {\it interaction-based} and {\it large deviation}, along with a cascade model of productive activity, which unifies them. In this common framework, superlinear productivity requires that the involved social groups function at or close to criticality, in the sense of a subtle balance between order and disorder. We report the first empirical test of the renormalization of the exponent of the distribution of the sizes of first generation events into the renormalized exponent of the distribution of clusters resulting from the cascade of triggering over all generation in a critical branching process in the non-meanfield regime. Finally, we document a size effect in the strength and variability of the superlinear effect, with smaller groups exhibiting widely distributed superlinear exponents, some of them characterizing highly productive teams. In contrast, large groups tend to have a smaller superlinearity and less variability.Comment: 29 pages, 8 figure

Discrete-time dynamic modeling for software and services composition as an extension of the Markov chain approach

Discrete Time Markov Chains (DTMCs) and Continuous Time Markov Chains (CTMCs) are often used to model various types of phenomena, such as, for example, the behavior of software products. In that case, Markov chains are widely used to describe possible time-varying behavior of “self-adaptive” software systems, where the transition from one state to another represents alternative choices at the software code level, taken according to a certain probability distribution. From a control-theoretical standpoint, some of these probabilities can be interpreted as control signals and others can just be observed. However, the translation between a DTMC or CTMC model and a corresponding first principle model, that can be used to design a control system is not immediate. This paper investigates a possible solution for translating a CTMC model into a dynamic system, with focus on the control of computing systems components. Notice that DTMC models can be translated as well, providing additional information