The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Automatic enumeration of regular objects

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer Sequence

Content-Aware DataGuides for Indexing Large Collections of XML Documents

XML is well-suited for modelling structured data with textual content. However, most indexing approaches perform structure and content matching independently, combining the retrieved path and keyword occurrences in a third step. This paper shows that retrieval in XML documents can be accelerated significantly by processing text and structure simultaneously during all retrieval phases. To this end, the Content-Aware DataGuide (CADG) enhances the wellknown DataGuide with (1) simultaneous keyword and path matching and (2) a precomputed content/structure join. Extensive experiments prove the CADG to be 50-90% faster than the DataGuide for various sorts of query and document, including difficult cases such as poorly structured queries and recursive document paths. A new query classification scheme identifies precise query characteristics with a predominant influence on the performance of the individual indices. The experiments show that the CADG is applicable to many real-world applications, in particular large collections of heterogeneously structured XML documents

Real structures on rational surfaces and automorphisms acting trivially on Picard groups

In this article, we prove that any complex smooth rational surface $X$ which has no automorphism of positive entropy has a finite number of real forms (this is especially the case if $X$ cannot be obtained by blowing up $\mathbb P^2_{\mathbb C}$ at $r\geq 10$ points). In particular, we prove that the group $\mathrm{Aut}^{\#}X$ of complex automorphisms of $X$ which act trivially on the Picard group of $X$ is a linear algebraic group defined over $\mathbb R$.Comment: Minor changes. To appear in Mathematische Zeitschrif

Specifying Reusable Components

Reusable software components need expressive specifications. This paper outlines a rigorous foundation to model-based contracts, a method to equip classes with strong contracts that support accurate design, implementation, and formal verification of reusable components. Model-based contracts conservatively extend the classic Design by Contract with a notion of model, which underpins the precise definitions of such concepts as abstract equivalence and specification completeness. Experiments applying model-based contracts to libraries of data structures suggest that the method enables accurate specification of practical software