Related papers: Topological complexity of a map
We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the earlier theoretical work with explicit algorithms and experiments with an implementation. More formally, the robustness…
In this paper, we study three relative LS categories of a map and study some of their properties. Then we introduce the `higher topological complexity' and `weak higher topological complexity' of a map. Each of them are homotopy invariants.…
A system of differential forms will establish a topology and a topological structure on a domain of independent variables such that is possible to determine which maps or processes acting on the system are continuous. Perhaps the most…
Given a set R of robots, each one located at different vertices of an infinite regular tessellation graph, we aim to explore the Arbitrary Pattern Formation (APF) problem. Given a multiset F of grid vertices such that |R|=|F|, APF asks for…
We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant,…
Transitivity, the existence of periodic points and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that for graphs that are not trees, for every $\varepsilon>0,$ there exist (complicate)…
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.…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
In this paper we analyze some relationships between the topological complexity of a space $X$ and the category of $C_{\Delta_X},$ the homotopy cofibre of the diagonal map $\Delta_X:X\rightarrow X\times X.$ We establish the equality of the…
Mobile robots with complex morphology are essential for traversing rough terrains in Urban Search & Rescue missions (USAR). Since teleoperation of the complex morphology causes high cognitive load of the operator, the morphology is…
The unordered configuration space of $n$ points on a graph $\Gamma,$ denoted here by $UC^n(\Gamma),$ can be viewed as the space of all configurations of $n$ unlabeled robots on a system of one-dimensional tracks, which is interpreted as a…
In a topological dynamical system the complexity of an orbit is a measure of the amount of information (algorithmic information content) that is necessary to describe the orbit. This indicator is invariant up to topological conjugation. We…
In this paper, we present a topology optimization (TO) framework to enable automated design of mechanical components while ensuring the result can be manufactured using multi-axis machining. Although TO improves the part's performance, the…
We use the complexity function of an invariant, not necessary closed, subset of a two-sided shift space to compute the polynomial entropy of the induced dynamics on the hyperspace of continua for certain one-dimensional dynamical systems.…
We define a simpler notion of symmetric topological complexity more ad hoc to the motion planning problem which was the original motivation for the definition of topological complexity. This is a homotopy invariant that we call…
In this paper we study paramertized motion planning algorithms which provide universal and flexible solutions to diverse motion planning problems. Such algorithms are intended to function under a variety of external conditions which are…
Autonomous motion of a system (robot) is controlled by a motion planning algorithm. A sequential parametrized motion planning algorithm \cite{FP22} works under variable external conditions and generates continuous motions of the system to…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
The topological complexity of a path-connected space $X,$ denoted $TC(X),$ can be thought of as the minimum number of continuous rules needed to describe how to move from one point in $X$ to another. The space $X$ is often interpreted as a…
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…