A new model for curves on manifolds using rolling operations.
problem Modeling curves on manifolds without explicit parametrization.
method Using rolling operations to construct Gaussian processes on manifolds.
result Conditions for the rolling of mean to equal Fréchet mean and estimators of parameters.
New kinematic model for a spin-rolling sphere using Darboux frame.
problem Control and planning of an underactuated sphere on a plane.
method Darboux frame transformation to fully-actuated model.
result Underactuated sphere model transformed into fully-actuated model.
New geometric interpretation of discrete Willmore energy using rolling spheres connection.
problem Discrete formulation of Willmore energy for simplicial surfaces.
method Geometric interpretation of Möbius invariant discrete Willmore energy using rolling spheres connection.
result Clear geometric interpretations of discrete Willmore energy with manifest Möbius invariance.
Gluck twisting certain knots results in standard 4-spheres.
problem Understanding when Gluck twists yield standard 4-spheres.
method Analyzing smooth homotopy 4-spheres and diffeomorphisms.
result Infinite collection of twisted doubles of corks are standard.
New geometric approach controls motion of a spinning sphere on a plane.
problem Controlling the motion of a spinning sphere on a straight path.
method Geometric-based planning approach using Darboux frame kinematics.
result Iterative algorithm tunes inputs for desired configurations.
We study the rolling of the Chaplygin ball in Rn 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…
We study the control system of a Riemannian manifold M of dimension n rolling on the sphere Sn. 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…
In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds (M,g) and (M^,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 …
The paper characterizes surfaces where the speed of a ball is constant.
problem Understanding surfaces where the speed of a ball is constant.
method Analyzing the relationship between the speed of a ball and the mean curvature of the surface.
result The only surfaces where the speed is constant are subsets of planes, circular cylinders, and spheres.
The paper proves properties of branched covers of specific knots and tori.
problem Investigating the smoothness and diffeomorphism of specific 4-manifolds.
method Analyzing double branched covers of twist-roll spun knots and turned twisted tori, applying techniques to show diffeomorphism.
result Proves that certain 4-manifolds are diffeomorphic to standard manifolds.
We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and…
Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.
problem Constructing exotic embeddings of RP^2 and homotopy spheres.
method Using Montesinos knots and roll-spun knots to prove the existence of homotopy spheres.
result An infinite family of homotopy spheres and homotopy CP^2s are produced.
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 2n(n+3)−dimensional manifold. The conditions of no-twisting and no-slipping are decoded by means of …
We present the classical Wagner construction from 1935 of the curvature tensor for completely nonholonomic manifolds in both invariant and coordinate way. The starting point is the Shouten curvature tensor for nonholonomic connection introduced by Vranceanu and Shouten. We illustrate the construction on two mechanical …
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…
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 …
Louis Poinsot has shown in 1854 that the motion of a rigid body, with one of its points fixed, can be described as the rolling without slipping of one cone, the 'body cone', along another, the 'space cone', with their common vertex at the fixed point. This description has been further refined by the second author in 19…
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≥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.
problem Clarifying the difference between two types of rolling.
method Detailed explanation and illustrative examples.
result Theoretical results complemented with examples.
Study on rolling Stiefel manifolds with specific metrics.
problem Intrinsic and extrinsic rolling of Stiefel manifolds with α-metrics. method Investigation of intrinsic rolling of normal naturally reductive homogeneous spaces, derivation of ODEs for rolling, and explicit solutions.
result Explicit solutions for intrinsic and extrinsic rolling of Stiefel manifolds.
Study on rolling of 2D and 3D manifolds, identifying orbit dimensions.
problem Understanding rolling dynamics of 2D and 3D manifolds with constraints.
method Modeling rolling as a control affine system on a fibered space Q, analyzing reachable sets.
result Identified possible dimensions of non-open rolling orbits: 2, 5, 6, 7.
The Blaschke rolling disk theorem is extended to non-convex domains.
problem Classical inclusion principle for non-convex domains.
method Geometric conditions based on curvature, algorithm for decomposition.
result Necessary and sufficient conditions for rolling disks in non-convex domains.
Local equivalence found between maximally symmetric rolling and flat Cartan distributions.
problem Establishing local equivalence between maximally symmetric rolling and flat Cartan distributions.
method Using complex parametrisation of su(2), a change of coordinates maps the maximally symmetric rolling (2,3,5)-distribution to the flat Cartan distribution. result Local equivalence between maximally symmetric rolling and flat Cartan distributions established.
Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
Rolling two hyperboloid surfaces is described using a Monge normal form.
problem Describing the rolling motion of hyperboloid surfaces.
method Parametrization of sl2 using unimodular fractional linear transformations. result Found a Monge normal form for the rolling of hyperboloid surfaces.
Shapes can roll downhill following any curve, but often return to initial orientation after crossing multiple copies.
problem How to design shapes that roll downhill along a given curve and its translations.
method Analyzing the geometric properties and motion of shapes on inclined planes.
result Most curves allow shapes to roll downhill following them and their translations, but some require crossing multiple copies.
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…
Study rolling dynamics with random slipping and twisting using large deviation principles.
problem Analyzing the stability of a rolling model with random slipping and twisting.
method Modelled as a stochastic differential equation on the orthonormal frame bundle, examined via large deviations.
result Proved large deviation principles for projection curves and their horizontal lifts on the base manifold.
Rollings of reductive homogeneous spaces are studied using intrinsic curves.
problem Investigate rollings of reductive homogeneous spaces without slip and twist.
method An intrinsic point of view, considering rollings as curves in the configuration space Q tangent to a certain distribution. result Explicit solutions for rollings of m over G/H are obtained for specific cases. New method for rolling bodies on inclined planes, with applications to rescue operations.
problem Constructing solid bodies rolling along curves on inclined planes.
method Rigorous existence theorems and connections to maritime rescue operations.
result Comprehensive existence theorems for rolling bodies on inclined planes.
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
Study rolling control of Lorentzian manifolds on flat space.
problem Complete controllability of rolling Lorentzian manifolds.
method Examining the holonomy group of the distribution encoding rolling constraints.
result Rolling problem is completely controllable if and only if the holonomy group equals SO0(n,1). 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…
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}…
Dancing polygons and rolling balls linked via a special geometric distribution.
problem Understanding the geometric and mechanical relationship between dancing polygons and rolling balls.
method Mapping dancing polygons to trajectories of a rolling ball on a 3D surface, both described by a specific geometric distribution.
result Non-degenerate dancing pairs of polygons exist for all n≥6 and correspond to rolling ball trajectories. Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they …
ERDM integrates rolling forecasts with diffusion models for complex dynamics.
problem Forecasting complex dynamics with rolling forecasts and diffusion models.
method Adapting EDM components for rolling forecasts, introducing novel loss weighting, efficient initialization, and hybrid architecture.
result ERDM outperforms diffusion-based baselines in 2D Navier-Stokes simulations and ERA5 weather forecasting.
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…
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.
New definition of Bäcklund transformation for surface isometric deformation.
problem Defining Bäcklund transformation in surface isometric deformation.
method Proving generic 4D integrable rolling distribution splits into 1D family of 3D distributions.
result Introducing new definition of Bäcklund transformation.
Study maximally symmetric distribution of An-Nurowski surface rolling on a plane.
problem Maximally symmetric (2,3,5)-distribution of An-Nurowski surface rolling without slipping or twisting. method Calculated vector fields defining a split g2 Lie algebra and projected to an action of SL(3,R). result Obtained an action of SL(3,R) on the configuration space without a surface. Rolling systems limit to billiard models with no-slip collisions.
problem Understanding how rolling systems behave as billiard models with no-slip collisions.
method Showed that no-slip billiards arise as limits of non-holonomic rolling systems.
result Rolling systems limit to billiard models with no-slip collisions.
Rolling Diffusion improves video prediction by progressively corrupting frames based on their temporal position.
problem Improving video prediction accuracy by accounting for temporal dynamics.
method A sliding window denoising process that assigns more noise to frames that appear later in a sequence.
result Rolling Diffusion outperforms standard diffusion models in tasks with complex temporal dynamics.
The paper uses stochastic control to analyze interest rate markets with roll-over risk.
problem Analyzing interest rate markets with roll-over risk without classical arbitrage assumptions.
method Stochastic optimal control problems with power-type objective functionals.
result Endogenously determined funding-liquidity spread.
New proof confirms rolling objects can follow any path.
problem Existence of rolling objects following any path.
method Geometric proof for period-n trajectoids.
result Existence of period-n trajectoids for any smooth curve.
Robotics: Rolling robots on a moving platform can be controlled.
problem Controlling the motion of rolling robots atop a moving platform.
method Developed a mathematical model and demonstrated simulations.
result Platform acceleration can control robot's heading and motion.
Khovanov homology fails to differentiate certain slice disks.
problem Differentiating roll-spun slice disks from trivial ones.
method Using Khovanov homology and Morse theory.
result Khovanov homology cannot distinguish roll-spun slice disks from trivial ones.