TY - JOUR AU - Pach, János ID - 8323 JF - Discrete and Computational Geometry SN - 01795376 TI - A farewell to Ricky Pollack VL - 64 ER - TY - CONF AB - We evaluate the usefulness of persistent homology in the analysis of heart rate variability. In our approach we extract several topological descriptors characterising datasets of RR-intervals, which are later used in classical machine learning algorithms. By this method we are able to differentiate the group of patients with the history of transient ischemic attack and the group of hypertensive patients. AU - Graff, Grzegorz AU - Graff, Beata AU - Jablonski, Grzegorz AU - Narkiewicz, Krzysztof ID - 8580 SN - 9781728157511 T2 - 11th Conference of the European Study Group on Cardiovascular Oscillations: Computation and Modelling in Physiology: New Challenges and Opportunities, TI - The application of persistent homology in the analysis of heart rate variability ER - TY - JOUR AB - In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces. AU - Akopyan, Arseniy AU - Karasev, Roman ID - 10867 IS - 3 JF - International Mathematics Research Notices KW - General Mathematics SN - 1073-7928 TI - Waist of balls in hyperbolic and spherical spaces VL - 2020 ER - TY - THES AB - Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications. For the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries. AU - Ölsböck, Katharina ID - 7460 KW - shape reconstruction KW - hole manipulation KW - ordered complexes KW - Alpha complex KW - Wrap complex KW - computational topology KW - Bregman geometry SN - 2663-337X TI - The hole system of triangulated shapes ER - TY - THES AB - This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars. AU - Masárová, Zuzana ID - 7944 KW - reconfiguration KW - reconfiguration graph KW - triangulations KW - flip KW - constrained triangulations KW - shellability KW - piecewise-linear balls KW - token swapping KW - trees KW - coloured weighted token swapping SN - 2663-337X TI - Reconfiguration problems ER -