Related papers: Towards Directed Collapsibility
Coordination geometries describe how the neighbours of a central particle are arranged around it. Such geometries can be thought to lie in an abstract topological space; a model of this space could provide a mathematical basis for…
A salient feature of topological phases are surface states and many of the widely studied physical properties are directly tied to their existence. Although less explored, a variety of topological phases can however similarly be…
We introduce the weighted path homology on the category of weigh\-ted directed hypergraphs and describe conditions of homotopy invariance of weighted path homology groups. We give several examples that explain the nontriviality of the…
In this paper we explore mathematical tools that can be used to relate directed and undirected random graph models to each other. We identify probability spaces on which a directed and an undirected graph model are equivalent, and…
We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common…
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial…
Vector field guided path following (VF-PF) algorithms are fundamental in robot navigation tasks, but may not deliver the desirable performance when robots encounter singular points where the vector field becomes zero. The existence of…
This thesis captures the ongoing development of twisted cubes, which is a modification of cubes (in a topological sense) where its homotopy type theory does not require paths or higher paths to be invertible. My original motivation to…
Globular complexes were introduced by E. Goubault and the author in arXiv:math/0107060 to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of…
Many networks describing complex systems are directed: the interactions between elements are not symmetric. Recent work has shown that these networks can display properties such as trophic coherence or non-normality, which in turn affect…
We consider the problem of classifying trajectories on a discrete or discretised 2-dimensional manifold modelled by a simplicial complex. Previous works have proposed to project the trajectories into the harmonic eigenspace of the Hodge…
We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for spaces X whose higher topological complexity TC_n(X) is…
We define a class of probability distributions that we call simplicial mixture models, inspired by simplicial complexes from algebraic topology. The parameters of these distributions represent their topology and we show that it is possible…
Path sets are spaces of one-sided infinite symbol sequences associated to pointed graphs (G_v_0), which are edge-labeled directed graphs G with a distinguished vertex v_0. Such sets arise naturally as address labels in geometric fractal…
We show that in codimension at least 3, spaces of locally flat topological embeddings of manifolds are correctly modelled by derived spaces of maps between their configuration categories (under mild smoothability conditions). That general…
Closure spaces are a generalisation of topological spaces obtained by removing the idempotence requirement on the closure operator. We adapt the standard notion of bisimilarity for topological models, namely Topo-bisimilarity, to closure…
A new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties…
The category of monotone determined spaces is an extended topological framework for dcpos in domain theory. We first show that monotone determined spaces are exactly the spaces generated by one-point convergence spaces, and then naturally…
A rotation in a Euclidean space V is an orthogonal map on V which acts locally as a plane rotation with some fixed angle. We give a classification of all pairs of rotations in finite-dimensional Euclidean space, up to simultaneous…
We study infinite Euclidean distance discriminants of algebraic varieties, defined as the loci of data points whose fibers under the second projection from the Euclidean distance correspondence are positive-dimensional. In particular, these…