Related papers: Equivariant topological complexities
We explore the topology of configuration spaces of hard disks experimentally, and show that several changes in the topology can already be observed with a small number of particles. The results illustrate a theorem of Baryshnikov, Bubenik,…
Central configurations have been of great interest over many years, with the earliest examples due to Euler and Lagrange. There are numerous results in the literature demonstrating the existence of central configurations with specific…
Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called {\it…
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…
We establish sharp upper bounds for the topological complexity of motion planning problem in spaces with small fundamental group, i.e. when it is finite of small order or has small cohomological dimension.
We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On…
In this paper we study symmetric motion planning algorithms, i.e. such that the motion from one state A to another B, prescribed by the algorithm, is the time reverse of the motion from B to A. We experiment with several different notions…
The configuration space of a mechanical linkage, consisting of rigid bodies moving in space constrained by joints, is defined by algebraic conditions. If these equations do not define a complete intersection, then the dimension of the…
To determine the relative position of any two surfaces in a system, one approach is to useoperations (Minkowski sum and intersection) on sets of constraints. These constraints aremade compliant with half-spaces of R^n where each set of…
We study the topological complexity of work maps with respect to some subspaces of the configuration space and a workspace considered as the target set of the motion of robots. The motivation is to optimize and reduce the number of motion…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
The concept of complexity appears in virtually all areas of knowledge. Its intuitive meaning shares similarities across fields, but disagreements between its details hinders a general definition, leading to a plethora of proposed…
Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version…
We study certain topological problems that are inspired by applications to autonomous robot manipulation. Consider a continuous map $f\colon X\to Y$, where $f$ can be a kinematic map from the configuration space $X$ to the working space $Y$…
Unlike conventional mechanisms, compliant mechanisms produce the desired deformations by exploiting elastic strain and do not need, therefore, moving parts. The number of degrees of freedom of a conventional mechanism, also called mobility,…
Bielavsky introduced and investigated the class of symmetric symplectic spaces, that is, symmetric spaces endowed with a symplectic form invariant with respect to symmetries. Since the theory of symmetric spaces has generalizations, we ask…
Statistical models that possess symmetry arise in diverse settings such as random fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the…
Function space topologies are developed for EC(Y,Z), the class of equi-continuous mappings from a topological space Y to a uniform space Z. Properties such as splittingness, admissibility etc. are defined for such spaces. The net theoretic…
The Topological complexity a la Farber $\text{TC}(-)$ is a homotopy invariant which have interesting applications in Robotics, specifically, in the robot motion planning problem. In this work we calculate the topological complexity of the…
J. Milnor introduced a specific class of codimension-$1$ submanifolds in the product of projective spaces, known as Milnor manifolds. This paper establishes precise bounds on the higher topological complexity of these manifolds and provides…