Distance functional dependencies in the presence of complex values

Distance functional dependencies (dFDs) have been introduced in the context of the relational data model as a generalisation of error-robust functional dependencies (erFDs). An erFD is a dependency that still holds, if errors are introduced into a relation, which cause the violation of an original functional dependency. A dFD with a distance d=2e+1 corresponds to an erFD with at most e errors in each tuple. Recently, an axiomatisation of dFDs has been obtained. Database theory, however, does no longer deal only with flat relations. Modern data models such as the higher-order Entity-Relationship model (HERM), object oriented datamodels (OODM), or the eXtensible Meakup Language (XML) provide constructors for complex values such as finite sets, multisets and lists. In this article, dFDs with complex values are investigated. Based on a generalisation of the HAmming distance for tuples to complex values, which exploits a lattice structure on subattributes, the major achievement is a finite axiomatisation of the new class of dependencies

Topological dualities in the Ising model

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension

Topological dualities in the Ising model

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.Comment: 62 pages, 22 figures; v2 adds important reference [S]; v2 has reworked introduction, additional reference [KS], and minor changes; v4 for publication in Geometry and Topology has all new figures and a few minor changes and additional reference

Using FCA to Suggest Refactorings to Correct Design Defects

Design defects are poor design choices resulting in a hard-to- maintain software, hence their detection and correction are key steps of a\ud disciplined software process aimed at yielding high-quality software\ud artifacts. While modern structure- and metric-based techniques enable\ud precise detection of design defects, the correction of the discovered\ud defects, e.g., by means of refactorings, remains a manual, hence\ud error-prone, activity. As many of the refactorings amount to re-distributing\ud class members over a (possibly extended) set of classes, formal concept\ud analysis (FCA) has been successfully applied in the past as a formal\ud framework for refactoring exploration. Here we propose a novel approach\ud for defect removal in object-oriented programs that combines the\ud effectiveness of metrics with the theoretical strength of FCA. A\ud case study of a specific defect, the Blob, drawn from the\ud Azureus project illustrates our approach

Refactorings of Design Defects using Relational Concept Analysis

Software engineers often need to identify and correct design defects, ıe} recurring design problems that hinder development and maintenance\ud by making programs harder to comprehend and--or evolve. While detection\ud of design defects is an actively researched area, their correction---mainly\ud a manual and time-consuming activity --- is yet to be extensively\ud investigated for automation. In this paper, we propose an automated\ud approach for suggesting defect-correcting refactorings using relational\ud concept analysis (RCA). The added value of RCA consists in exploiting\ud the links between formal objects which abound in a software re-engineering\ud context. We validated our approach on instances of the <span class='textit'></span>Blob\ud design defect taken from four different open-source programs