arXiv research

The H-type deviation measures how close step two Carnot groups are to H-type groups.

Characterizes GM-groups via sub-Riemannian geometry properties.

Solves a long-standing problem on step-two groups with exact formulas.

Study shows a specific Carnot group violates a curvature exponent bound.

New bounds on geodesic dimension and curvature exponent in Carnot groups.

New stability theorems for H-type Carnot groups established.

A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step t…

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics $σ_\e$ collapsing to a subRiemannian metric $σ_0$ as $\e\to 0$. We establish $C^{k,α}$ estimates for this…

In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisation of a previous analogous result for step 2 free nilpotent groups.

In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particu…

Study compares two subriemannian structures on S7, finding non-isometric and non-isospectral properties.

In the present paper we give a proof of the fact that the sub-Riemannian cut locus of a wide class of nilpotent groups of step two, called $H$-type groups, starting from the origin corresponds to the center of the group. We obtain this result by completely describing the sub-Riemannian geodesics in the group, and using…

In this note, we study the cut locus of the free, step two Carnot groups $\mathbb{G}_k$ with $k$ generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Myasnichenko - 2002] and [Montanari, Morbidelli - 2016], by exh…

By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and…

We give the complete classification of all sub-Riemannian model spaces with both step and rank three. They will be divided into three families based on their nilpotentization. Each family will depend on a different number of parameters, making the result crucially different from the known case of step two model spaces.…

The paper constructs optimal sub-Riemannian geodesics in specific Carnot groups.

We study a new class of rank two sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds, and graded nilpotent Lie groups of step two. These manifolds admit a canonical horizontal connection and a canonical sub-Laplaci…

Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.

In this paper, we consider a smooth connected finite-dimensional manifold $M$, an affine connection $\nabla$ with holonomy group $H^{\nabla}$ and $Δ$ a smooth completely non integrable distribution. We define the $Δ$-horizontal holonomy group $H^{\;\nabla}_Δ$ as the subgroup of $H^{\nabla}$ obtained by $\nabla$-paralle…

Let $\M$ be a smooth connected non-compact manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter $ρ_1$, the generalized curvature inequality …

Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced…

Brokerage algorithm learns from context to minimize trading regret.

The paper studies algebraic relations of first integrals on specific Lie groups.

In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian information of the smooth part of the objective function is available via…

New sub-Riemannian structures fail synthetic curvature bounds.

Study uses stacked generalization to improve fraud detection algorithms.

De-identification of electronic health records (EHR) is a vital step towards advancing health informatics research and maximising the use of available data. It is a two-step process where step one is the identification of protected health information (PHI), and step two is replacing such PHI with surrogates. Despite th…

We study the topology of admissible-loop spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical points of the 'Energy' functional, defined on the loop space). These geodesics t…

Let $\M$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. Associated with $L$ one has \textit{le carré du champ} $Γ$ and a canonical distance $d$, with respect to which we suppose that …

This paper describes a practical system for Multi Touch Attribution (MTA) for use by a publisher of digital ads. We developed this system for JD.com, an eCommerce company, which is also a publisher of digital ads in China. The approach has two steps. The first step ('response modeling') fits a user-level model for purc…