Related papers: Duality and tangles of set separations
Properties and examples of the dual transformation between two planes, which is such that the coordinates of a point in the original plane give the coefficients of a line in the dual plane and the coefficients of a line in the original…
We introduce and discuss the concept of modular entanglement. This is the entanglement that is established between the end points of modular systems composed by sets of interacting moduli of arbitrarily fixed size. We show that end-to-end…
A convexity space is a set X with a chosen family of subsets (called convex subsets) that is closed under arbitrary intersections and directed unions. There is a lot of interest in spaces that have both a convexity space and a topological…
Owing to their versatility, graph structures admit representations of intricate relationships between the separate entities comprising the data. We formalise the notion of connection between two vertex sets in terms of edge and vertex…
Syntax connects words to each other in very specific ways. Two words are syntactically connected if they depend directly on each other. Syntactic connections usually happen within a sentence. Gathering all those connection across several…
A tangle of order $k$ in a matroid or graph may be thought of as a "$k$-connected component". For a tangle of order $k$ in a matroid or graph that satisfies a certain robustness condition, we describe a tree decomposition of the matroid or…
Well-graded families, extremal systems and maximum systems (the last two in the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three important classes of set systems. This paper aims to study the notion of duality in the…
Connections are an important tool of differential geometry. This paper investigates their definition and structure in the abstract setting of tangent categories. At this level of abstraction we derive several classically important results…
We first recall the basic theory of double vector bundles and the canonical pairing of their duals introduced by the author and by Konieczna and Urbanski. We then show that the relationship between a double vector bundle and its two duals…
In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially…
We investigate entanglement dynamics in multipartite systems, establishing a quantitative concept of entanglement flow: both flow through individual particles, and flow along general networks of interacting particles. In the former case,…
Homomorphism duality pairs play crucial role in the theory of relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be…
We construct several examples where duality transformation commutes with the orbifolding procedure even when the orbifolding group does not act freely, and there are massless states from the twisted sector at a generic point in the moduli…
Great part of the interest in complex networks has been motivated by the presence of structured, frequently non-uniform, connectivity. Because diverse connectivity patterns tend to result in distinct network dynamics, and also because they…
We call a function $f: X\to Y$ $P$-preserving if, for every subspace $A \subset X$ with property $P$, its image $f(A)$ also has property $P$. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural…
We describe the duality between different geometries which arises by considering the classical and quantum harmonic map problem. To appear in ``Essays on Mirror Manifolds II''.
We investigate the entanglement properties of multiparticle systems, concentrating on the case where the entanglement is robust against disposal of particles. Two qubits -belonging to a multipartite system- are entangled in this sense iff…
Graphs are called navigable if one can find short paths through them using only local knowledge. It has been shown that for a graph to be navigable, its construction needs to meet strict criteria. Since such graphs nevertheless seem to…
In this paper, we consider to what degree the structure of a linear system is determined by the system's input/output behavior. The structure of a linear system is a directed graph where the vertices represent the variables in the system…
The open-system dynamics of entanglement plays an important role in the assessment of the robustness of quantum information processes and also in the investigation of the classical limit of quantum mechanics. Here we show that, subjacent to…