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We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric…
We set the foundations for a new approach to Topological Data Analysis (TDA) based on homotopical methods at chain complexes level. We present the category of tame parametrised chain complexes as a comprehensive environment that includes…
The limited penetrable horizontal visibility graph algorithm was recently introduced to map time series in complex networks. We extend this visibility graph and create a directed limited penetrable horizontal visibility graph and an image…
The neighborhood complex of a graph was introduced by Lov\'asz to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such `Hom complexes' are also…
Our main result states that for each finite complex L the category ${\bf TOP}$ of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all…
This paper revisits the robustness analysis and distributed ${ H}_\infty$ controller design for the platooning of connected vehicles. Recently, the relevant result subjected to the undirected topology has been studied, in the light of the…
We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the other one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal…
We bound the complexity of the fibers of the generic linear projection of a smooth variety in terms of a new family of invariants. These invariants are closely related to ideas of John Mather, and we give a simple proof of his bound on the…
We construct "higher" motion planners for automated systems whose space of states are homotopy equivalent to a polyhedral product space $Z(K,\{(S^{k_i},\star)\})$, e.g. robot arms with restrictions on the possible combinations of…
We present a machine learning approach that leverages persistent homology to classify bacterial flagellar motors into two functional states: rotated and stalled. By embedding protein structural data into a topological framework, we extract…
Although deep learning approaches have had tremendous success in image, video and audio processing, computer vision, and speech recognition, their applications to three-dimensional (3D) biomolecular structural data sets have been hindered…
We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional…
For a smooth, closed $n$-manifold $M$, we define an upper semi-continuous integer-valued complexity function on $H^1(M;{\mathbb R})$ using Morse theory. This measures how far an integral class is from being a fiber of a fibration. The fact…
Contact manifolds are odd-dimensional smooth manifolds endowed with a maximally non-integrable field of hyperplanes. They are intimately related to symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a closed…
Motion path planning is an intrinsically geometric problem which is central for design of robot systems. Since the early years of AI, robotics together with computer vision have been the areas of computer science that drove its development.…
We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in…
Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of…
Symmetric transverse sets were introduced to make the construction of the parallel product with synchronization for process algebras functorial. It is proved that one can do directed homotopy on symmetric transverse sets in the following…
The concept of relative sectional category expands upon classical sectional category theory by incorporating the pullback of a fibration along a map. Our paper aims not only to explore this extension but also to thoroughly investigate its…
This work presents an extended framework for learning-based bipedal locomotion that incorporates a heuristic step-planning strategy guided by desired torso velocity tracking. The framework enables precise interaction between a humanoid…