Online Automated Synthesis of Compact Normative Systems

Peer reviewedPostprin

Regularity lemmas in a Banach space setting

Szemer\'edi's regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lov\'asz and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst, Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space interpretation of the lemma and an interpretation in terms of compact- ness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lov\'asz and Szegedy as well as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H. Cohn and Y. Zhao: An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions, arXiv preprint arXiv:1401.2906 (2014)].Comment: 15 pages. The topological part has been substantially improved based on referees comments. To appear in European Journal of Combinatoric

The Theory of Quasiconformal Mappings in Higher Dimensions, I

We present a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts

Revisiting parton evolution and the large-x limit

This remark is part of an ongoing project to simplify the structure of the multi-loop anomalous dimensions for parton distributions and fragmentation functions. It answers the call for a "structural explanation" of a "very suggestive" relation found by Moch, Vermaseren and Vogt in the context of the x->1 behaviour of three-loop DIS anomalous dimensions. It also highlights further structure that remains to be fully explained.Comment: 6 pages, v2 corrects misprints and contains an additional referenc

Model theoretic stability and definability of types, after A. Grothendieck

We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Crit{\`e}res de compacit{\'e}" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces

Lattice structures for bisimilar Probabilistic Automata

The paper shows that there is a deep structure on certain sets of bisimilar Probabilistic Automata (PA). The key prerequisite for these structures is a notion of compactness of PA. It is shown that compact bisimilar PA form lattices. These results are then used in order to establish normal forms not only for finite automata, but also for infinite automata, as long as they are compact.Comment: In Proceedings INFINITY 2013, arXiv:1402.661

Two topologies are better than one

Partially ordered sets and metric spaces are used in studying semantics in Computer Science. Sets with both these structures are hence of particular interest. The partial metric spaces introduced by Matthews are an attempt to bring these ideas together in a single axiomatic framework. We consider an appropriate context in which to consider these spaces is as a bitopological space, i.e. a space with two (related) topologies. From this starting point, we cover the groundwork for a theory of partial metric spaces by generalising ideas from topology and metric spaces. For intuition we repeatedly refer to the real line with the usual ordering and metric as a natural example. We also examine in detail some other examples of more relevance to Computer Science