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This paper studies the robustness of large-scale interconnected systems with respect to external disturbances, focussing on their scalability properties. Specifically, a notion of scalability is introduced that asks for these robustness…
It is shown that if a non-invertible area preserving local homeomorphism on $\mathbb{T}^2$ is homotopic to a linear expanding or hyperbolic endomorphism, then it must be topologically transitive. This gives a complete characterization, in…
We introduce the concept of deck transformations within the category of developable complexes of groups. Drawing inspiration from classical covering theory for topological spaces, we propose an alternative construction of the universal…
The classical notions of structural controllability and structural observability are receiving increasing attention in Network Science, since they provide a mathematical basis to answer how the network structure of a dynamic system affects…
The question whether an ontology can safely be replaced by another, possibly simpler, one is fundamental for many ontology engineering and maintenance tasks. It underpins, for example, ontology versioning, ontology modularization,…
The usual concept of shape invariance is discussed and one extension of this concept is suggested.
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
In this chapter, we discuss urban mobility from a complexity science perspective. First, we give an overview of the datasets that enable this approach, such as mobile phone records, location-based social network traces, or GPS trajectories…
Flexible mechanical metamaterials possess repeating structural motifs that imbue them with novel, exciting properties including programmability, anomalous elastic moduli and nonlinear and robust response. We address such structures via…
This paper considers systems subject to nonholonomic constraints which are not uniform on the whole configuration manifold. When the constraints change, the system undergoes a transition in order to comply with the new imposed conditions.…
We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex by means of the concept of contiguous simplicial maps. We study the links of this new…
In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.
Inversion of various inclusions, that characterize continuity in topological spaces, results in numerous variants of quotient and perfect maps. In the framework of convergences, the said inclusions are no longer equivalent, and each of them…
The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of…
The topic of the glass transition gives rise to a a wide diversity of views. It is, accordingly, characterized by a lack of agreement on which would be the most profitable theoretical perspective. In this chapter, I provide some elements…
We define and study notions of amenability and skew-amenability of continuous actions of topological groups on compact topological spaces. Our main motivation is the question under what conditions amenability of a topological group passes…
This paper presents reflections on the validity of a series of mathematical methods and technical assumptions that are encrusted in macrophysics (related to gravitational interaction), that seem to have little or no physical significance.…
Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology.…
The dual concepts of `universality' and `hypercyclicity' are better understood and studied as `topological transitivity'. In this article we consider transitivity properties of skew products, essentially with non-compact fibers. We study…
In this paper we intend to study implications in their most general form, generalizing different classes of implications including the Heyting implication, sub-structural implications and weak strict implications. Following the topological…