Related papers: Geodesic complexity of a cube
A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological…
The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of…
We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity…
We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and non-orientable). We also give improved bounds for the topological complexity of unordered configuration spaces…
We determine lower bounds for the topological complexity of many planar polygon spaces mod isometry. With very few exceptions, the upper and lower bounds given by dimension and cohomology considerations differ by 1. This is true for 130 of…
Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than…
In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, $U_{max}$ of geodesic curvature of its path; this constrains the…
A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. For a probability distribution, the length along a path between two points can be defined as the amount of probability mass accumulated…
A fundamental characteristic of computer networks is their topological structure. The question of the description of the structural characteristics of computer networks represents a problem that is not completely solved. Search methods for…
``One could imagine that as a result of enormously extended astronomical experience, the entire universe consists of countless identical copies of our Milky Way, that the infinite space can be partitioned into cubes each containing an…
Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a…
By a formula of Farber the topological complexity TC(X) of a (p-1)-connected, m-dimensional CW-complex X is bounded above by (2m+1)/p+1. There are also various lower estimates for TC(X) such as the nilpotency of the ring $H^*(X\times…
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
In this paper we prove the probabilistic continuous complexity conjecture. In continuous complexity theory, this states that the complexity of solving a continuous problem with probability approaching 1 converges (in this limit) to the…
We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex…
It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.
The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties.…
We explore several problems related to ruled polygons. Given a ruling of a polygon $P$, we consider the Reeb graph of $P$ induced by the ruling. We define the Reeb complexity of $P$, which roughly equates to the minimum number of points…
For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an $\epsilon$-splitting map on a concentric geodesic ball with uniformly small radius. There are two new…