arXiv research

The paper finds bounds on shortest dense curves on surfaces.

Explicit bounds found for shortest orthogeodesics and volumes of hyperbolic manifolds.

Short proof for ideal polygons with near optimal orthogeodesic decomposition.

The paper explores orthogeodesics on hyperbolic surfaces and their integer traces.

Uniform counting formulas for orthogeodesics in Kleinian groups converge.

Proves identities linking curve lengths and orthogeodesics on hyperbolic surfaces.

This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.

Graded identities for hyperbolic surfaces with cusps and cone points.

For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian's identity from Teichmueller theory. Any equality characterizes diagonal embeddings.

Study geodesics on flat tori, focusing on convex bodies.

The Poincaré series for surfaces with boundary extends to the complex plane.

Algorithms find second and third shortest non-trivial closed walks on surfaces.

Study geodesics and shortest arcs on Lie groups with specific metrics.

Shortest geodesic on curved spheres is no longer than 3 times the diameter.

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

Study geodesics and shortest arcs on Lie groups with specific metrics.

Generative Flow Networks solve shortest path problems in graphs.

Deep learning approximates shortest path distances in large graphs.

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

The paper finds shortest geodesic bounds on orbifolds with diameter limits.

Study on shortest arcs on hyperbolic surfaces with boundary.

Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.

Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.

Shortest non-simple closed geodesics on hyperbolic surfaces found.

Sharp bounds found on shortest geodesic on punctured spheres.

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest pa…

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.

Study shows shortest periodic geodesic on hyperbolic orbisphere complements figure-eight knot.

The paper improves bounds on the shortest closed geodesic length on surfaces.

Continuity of earthquake flow map transfers Teichmüller dynamics results.

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.

Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R^d. We study the convergence of the shortest path distance in such graphs as the sample size tends to infinity. We prove that for unweighted kNN graphs, this distance converges …

Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.

Infinite fractal tree solves shortest connection problem.

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.

Study shortest geodesics on flat cone spheres with conical singularities.

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into ac…

Algorithm reduces regret in SSP problems with LFA.

Study on OI surfaces with unique geometric properties.

Study bounds the length of shortest periodic geodesics on certain curved spaces.

In every conformal class of Finsler (or Riemannian) metrics on a closed manifold there exists a residual subset of Finsler metrics, such that, with respect to the residual Finsler metrics, in any non-trivial homotopy class of free loops there is precisely one shortest geodesic loop.

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

The authors find geodesics, shortest arcs, diameter, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO(3), under condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO(3)$.

New bounds on shortest geodesic loops on a sphere.

Landmark-based node embeddings approximate shortest path distances in random graphs.

There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as…

In this paper, we prove some convergence results of a special case of optimistic policy iteration algorithm for stochastic shortest path problem. We consider both Monte Carlo and $TD(λ)$ methods for the policy evaluation step under the condition that the termination state will eventually be reached almost surely.