arXiv research

Local equivalence found between maximally symmetric rolling and flat Cartan distributions.

Dancing polygons and rolling balls linked via a special geometric distribution.

Rolling two hyperboloid surfaces is described using a Monge normal form.

Study maximally symmetric distribution of An-Nurowski surface rolling on a plane.

Study rolling control of Lorentzian manifolds on flat space.

New definition of Bäcklund transformation for surface isometric deformation.

In the present work we define the rolling of one pseudo-Riemannian manifold over another without slipping and twisting. We compare the definition of the rolling without slipping and twisting of two manifolds isometrically embedded into a pseudo-Euclidean space with the rolling defined only by the intrinsic data, namely…

We give a description of Nurowski's conformal structure for some examples of bracket-generating rank 2 distributions in dimension 5, aka $(2,3,5)$-distributions, namely the An-Nurowski circle twistor distribution for pairs of surfaces of constant Gauss curvature rolling without slipping or twisting over each other. In …

In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds $(M,g)$ and $(\hat M,\hat g)$ rolling without twisting or slipping. We show that, under certain genericity hypotheses, the natural bundle projection from the state space $Q$ of the rolling model onto $M$ is a principal …

Rollings of reductive homogeneous spaces are studied using intrinsic curves.

New classification of conformal structures with maximal $G_2$ symmetry.

We present an intrinsic formulation of the kinematic problem of two $n-$dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an $\frac{n(n+3)}2-$dimensional manifold. The conditions of no-twisting and no-slipping are decoded by means of …

On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g i…

In this paper, we consider two cases of rolling of one smooth connected complete Riemannian manifold $(M,g)$ onto another one $(\hM,\hg)$ of equal dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation where there is no relative spin (or twist) of one manifold with respect to the other one. As for…

Explains rolling of symmetric spaces on flat spaces.

Study on rolling Stiefel manifolds with specific metrics.

In this semi-expository paper we disclose hidden symmetries of a classical nonholonomic kinematic model and try to explain geometric meaning of basic invariants of vector distributions.

A new model for curves on manifolds using rolling operations.

The Blaschke rolling disk theorem is extended to non-convex domains.

Generalized Blaschke rolling theorem for curved spaces.

Shapes can roll downhill following any curve, but often return to initial orientation after crossing multiple copies.

Associated to the problem of rolling one surface along another there is a five-manifold M with a rank two distribution. If the two surfaces are spheres then M is the product of the rotation group SO_3 with the two-sphere and its distribution enjoys an obvious symmetry group; the product of two SO_3's, one for each sphe…

New method for rolling bodies on inclined planes, with applications to rescue operations.

In this article, we consider the rolling (or development) of two Riemannian connected manifolds $(M,g)$ and $(\hat{M},\hat{g})$ of dimensions $2$ and $3$ respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of \cite{MortadaKokkonenChitour}, which modelled the general set…

We prove that for a generic $3$-dimensional integrable rolling distribution of contact elements (excluding developable seed and isotropic developable leaves) isometric correspondence of leaves of a general nature (independent of the shape of the seed) requires the Bäcklund transformation.

We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each cu…

This paper tackles belief-state selection in simulators with latent states.

Given any smooth plane curve α(s)representing a mirror that reflects light the usual way and any radiant light source at a point in the plane, the reflected light will produce a caustic envelope. For such an envelope, we show that there is an associated curve \b{eta}(s) and a family of circles C(s) that roll on \b{eta}…

We study a rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against Euclidean space, where the rolling is coupled with random slipping and twisting. The system is modelled by a stochastic differential equation of Stratonovich-type driven by semimartingales, on…

ERDM integrates rolling forecasts with diffusion models for complex dynamics.

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the d…

We study the dynamics of the linear and non-linear serial dependencies in financial time series in a rolling window framework. In particular, we focus on the detection of episodes of statistically significant two- and three-point correlations in the returns of several leading currency exchange rates that could offer so…

In this paper we bring to bear some new tools from statistical learning on the analysis of roll call data. We present a new data-driven model for roll call voting that is geometric in nature. We construct the model by adapting the "Partition Decoupling Method," an unsupervised learning technique originally developed fo…

We use the methods of geometric control theory to study extremal trajectories of vertical rolling disk. We focus on the role of symmetries of the underlying geometric structures. We demonstrate the computations in the CAS Maple package DifferentialGeometry.

We study the rolling of the Chaplygin ball in $\mathbb R^n$ over a fixed $(n-1)$--dimensional sphere without slipping and without slipping and twisting. The problems can be naturally considered within a framework of appropriate modifications of the L+R and LR systems -- well known systems on Lie groups groups with an i…

New kinematic model for a spin-rolling sphere using Darboux frame.

Rolling systems limit to billiard models with no-slip collisions.

Rolling Diffusion improves video prediction by progressively corrupting frames based on their temporal position.

The paper uses stochastic control to analyze interest rate markets with roll-over risk.

New proof confirms rolling objects can follow any path.

Robotics: Rolling robots on a moving platform can be controlled.

Study reveals investor heterogeneity in Korean equity market cash flows.

Khovanov homology fails to differentiate certain slice disks.

The paper proves properties of branched covers of specific knots and tori.

New geometric interpretation of discrete Willmore energy using rolling spheres connection.

We study the control system of a Riemannian manifold $M$ of dimension $n$ rolling on the sphere $S^n$. The controllability of this system is described in terms of the holonomy of a vector bundle connection which, we prove, is isomorphic to the Riemannian holonomy group of the cone $C(M)$ of $M$. Using Berger's list, we…

The paper characterizes surfaces where the speed of a ball is constant.

A well-known and interesting family of sub-Riemannian space are the systems involving two balls rolling against each other without slipping or twisting. In this note, we show how the sub-Riemannian geodesics of these space, when the two balls are embedded in $\mathbb{R}^3 \times \mathbb{R}^3$, are horizontal curves on …