A graph partition problem

Given a graph $G$ on $n$ vertices, for which $m$ is it possible to partition the edge set of the $m$-fold complete graph $mK_n$ into copies of $G$? We show that there is an integer $m_0$, which we call the \emph{partition modulus of $G$}, such that the set $M(G)$ of values of $m$ for which such a partition exists consists of all but finitely many multiples of $m_0$. Trivial divisibility conditions derived from $G$ give an integer $m_1$ which divides $m_0$; we call the quotient $m_0/m_1$ the \emph{partition index of $G$}. It seems that most graphs $G$ have partition index equal to $1$, but we give two infinite families of graphs for which this is not true. We also compute $M(G)$ for various graphs, and outline some connections between our problem and the existence of designs of various types

Big Data Now, 2015 Edition

Now in its fifth year, O’Reilly’s annual Big Data Now report recaps the trends, tools, applications, and forecasts we’ve talked about over the past year. For 2015, we’ve included a collection of blog posts, authored by leading thinkers and experts in the field, that reflect a unique set of themes we’ve identified as gaining significant attention and traction. Our list of 2015 topics include: Data-driven cultures Data science Data pipelines Big data architecture and infrastructure The Internet of Things and real time Applications of big data Security, ethics, and governance Is your organization on the right track? Get a hold of this free report now and stay in tune with the latest significant developments in big data

Multipartite graph decomposition: cycles and closed trails

This paper surveys results on cycle decompositions of complete multipartite graphs (where the parts are not all of size 1, so the graph is not K_n ), in the case that the cycle lengths are “small”. Cycles up to length n are considered, when the complete multipartite graph has n parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned

Multipartite graph decomposition: cycles and closed trails

This paper surveys results on cycle decompositions of complete multipartite graphs (where the parts are not all of size 1, so the graph is not <em>K</em>_<em>n</em> ), in the case that the cycle lengths are “small”. Cycles up to length <em>n</em> are considered, when the complete multipartite graph has <em>n</em> parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned.<br /

PiCo: A Domain-Specific Language for Data Analytics Pipelines

In the world of Big Data analytics, there is a series of tools aiming at simplifying programming applications to be executed on clusters. Although each tool claims to provide better programming, data and execution models—for which only informal (and often confusing) semantics is generally provided—all share a common under- lying model, namely, the Dataflow model. Using this model as a starting point, it is possible to categorize and analyze almost all aspects about Big Data analytics tools from a high level perspective. This analysis can be considered as a first step toward a formal model to be exploited in the design of a (new) framework for Big Data analytics. By putting clear separations between all levels of abstraction (i.e., from the runtime to the user API), it is easier for a programmer or software designer to avoid mixing low level with high level aspects, as we are often used to see in state-of-the-art Big Data analytics frameworks. From the user-level perspective, we think that a clearer and simple semantics is preferable, together with a strong separation of concerns. For this reason, we use the Dataflow model as a starting point to build a programming environment with a simplified programming model implemented as a Domain-Specific Language, that is on top of a stack of layers that build a prototypical framework for Big Data analytics. The contribution of this thesis is twofold: first, we show that the proposed model is (at least) as general as existing batch and streaming frameworks (e.g., Spark, Flink, Storm, Google Dataflow), thus making it easier to understand high-level data-processing applications written in such frameworks. As result of this analysis, we provide a layered model that can represent tools and applications following the Dataflow paradigm and we show how the analyzed tools fit in each level. Second, we propose a programming environment based on such layered model in the form of a Domain-Specific Language (DSL) for processing data collections, called PiCo (Pipeline Composition). The main entity of this programming model is the Pipeline, basically a DAG-composition of processing elements. This model is intended to give the user an unique interface for both stream and batch processing, hiding completely data management and focusing only on operations, which are represented by Pipeline stages. Our DSL will be built on top of the FastFlow library, exploiting both shared and distributed parallelism, and implemented in C++11/14 with the aim of porting C++ into the Big Data world

Renormalization: an advanced overview

We present several approaches to renormalization in QFT: the multi-scale analysis in perturbative renormalization, the functional methods \`a la Wetterich equation, and the loop-vertex expansion in non-perturbative renormalization. While each of these is quite well-established, they go beyond standard QFT textbook material, and may be little-known to specialists of each other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added, minor improvements; v3: some changes to sect. 5, refs. adde