Related papers: Reparametrizations of Continuous Paths
In order to obtain a well-defined path integral one often employs discretizations. In the case of General Relativity these generically break diffeomorphism symmetry, which has severe consequences since these symmetries determine the…
We consider the following problem for a fixed graph H: given a graph G and two H-colorings of G, i.e. homomorphisms from G to H, can one be transformed (reconfigured) into the other by changing one color at a time, maintaining an H-coloring…
In this short note, we argue that directed homotopy can be given the structure of generalized modules, over particular monoids. This is part of a general attempt for refoundation of directed topology.
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
Persistent homology has undergone significant development in recent years. However, one outstanding challenge is to build a coherent statistical inference procedure on persistent diagrams. In this paper, we first present a new lattice path…
This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their…
We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We…
We give the detale description from various points of view of Pascal automorphism,--- a natural transformation of the space of paths in the Pascal graph (= infinite Pascal triangle), and describetha plan of the proof of continuiuty of its…
Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression…
Cardiac parametric mapping is useful for evaluating cardiac fibrosis and edema. Parametric mapping relies on single-shot heartbeat-by-heartbeat imaging, which is susceptible to intra-shot motion during the imaging window. However, reducing…
Despite the attention that the problem of path planning for tethered robots has garnered in the past few decades, the approaches proposed to solve it typically rely on a discrete representation of the configuration space and do not exploit…
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or,…
We want to investigate 'spaces' where paths have a 'weight', or 'cost', expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to path-reversion. Thus, 'weighted algebraic topology' can be…
Multiparameter persistence module can capture more topological differences across data instances compared to using a single parameter, where the well-studied matching distance investigates the distance along a straight line in the…
A precise meaning is given to the notion of continuous iteration of a mapping. Usual discrete iterations are extended into a dynamical flow which is a homotopy of them all. The continuous iterate reveals that a dynamical map is formend by…
For Gamma a finite, connected metric graph, we consider the space of configurations of n points in Gamma with a restraint parameter r dictating the minimum distance allowed between each pair of points. These restricted configuration spaces…
We survey research on the homotopy theory of the space map(X, Y) consisting of all continuous functions between two topological spaces. We summarize progress on various classification problems for the homotopy types represented by the…
This note presents an approach to studying the iterates of a mapping whose restriction to the complement of a finite set is continuous and open. The main examples to which the approach can be applied are piecewise monotone mappings defined…
In this paper we focus on the map matching problem where the goal is to find a path through a planar graph such that the path through the vertices closely matches a given polygonal curve. The map matching problem is usually approached with…
Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case…