arXiv research

In this paper we define $n$th order Hessian structures on manifolds and study them. In particular, when $n = 3$, we make a detailed study and establish a one-to-one correspondence between {\it third-order Hessian structures} and a {\it certain class of connections} on the second-order tangent bundle of a manifold. Furt…

New methods using natural gradient for structured optimization.

Using a model for the bundle $\hat{\mathcal F}^2M$ of semi-holonomic second order frames of a manifold $M$ as an extension of the bundle ${\mathcal F}^2M$ of holonomic second order frames of $M$, we introduce in $\hat{\mathcal F}^2M$ a principal bundle structure over ${\mathcal F}^2M$, the structure group being the add…

A dynamical system on the total space of the fibre bundle of second order accelerations, $T^2M$, is defined as a third order vector field $S$ on $T^2M$, called semispray, which is mapped by the second order tangent structure into one of the Liouville vector field. For a regular Lagrangian of second order we prove that …

New superintegrable systems derived from Frobenius structures.

Paper generalizes connections between Lie groups and affine connections.

Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.

Superintegrable systems on curved manifolds found to have Hessian structures.

Paper uses second-order differential geometry to study stochastic mechanics.

Superintegrable systems on surfaces are classified geometrically.

Clustering is fundamental for gaining insights from complex networks, and spectral clustering (SC) is a popular approach. Conventional SC focuses on second-order structures (e.g., edges connecting two nodes) without direct consideration of higher-order structures (e.g., triangles and cliques). This has motivated SC ext…

Study second-order obstruction to nearly $G_2$ structure deformations.

We study second-order PDEs in 4D for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge-Ampere property. Since half-flatness of the conformal structure is equivalent to the exi…

Equivalence of second order differential operators in vector bundles studied.

In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the topology of fibers of the rank two prolongations. Moreover, by using properties of …

Develops second order infinitesimal structures on Teichmüller space.

SOLBP extends efficient inference to uncertain Bayesian networks.

Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.

Study reveals geometric context of second-order superintegrable systems.

New insights into 3D PDEs via Einstein-Weyl geometry.

New pruning method captures global correlations for efficient neural network inference.

The paper studies connections in superintegrable systems, revealing geometric insights.

Proves and tests methods for learning time-series with breaks.

This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote the configuration fiber bundle, we show that both the multisymplectic structure on $J^3Y$ as well…

In this paper we investigate the relations between semispray, nonlinear connection, dynamical covariant derivative and Jacobi endomorphism on Lie algebroids. Using these geometric structures, we study the symmetries of second order differential equations in the general framework of Lie algebroids.

We give a summary of recent results on the explicit local form of the second-order symmetric Lorentzian manifolds in arbitrary dimension, and its global version. These spacetimes turn out to be essentially a specific subclass of plane waves.

Global invariant for path structures and differential equations defined on torus.

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well de…

A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-rec…

We study the fillability (or embeddability) of 3-dimensional $CR$ structures under the geometric flows. Suppose we can solve a certain second order equation for the geometric quantity associated to the flow. Then we prove that if the initial $CR$ structure is fillable, then it keeps having the same property as long as …

We give the definition of angles on a Gromov-Hausdorff limit space of a sequence of complete n-dimensional Riemannian manifolds with a lower Ricci curvature bound. We apply this to prove there is a weakly second order differential structure on these spaces and prove there is a unique Levi-Civita connection allowing us …

For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet space) and are defined by non-degenerate scalar second-order-only (Hessian) PDEs in…

A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geome…

A new method for faster optimization on statistical manifolds.

Smooth manifolds have been always understood intuitively as spaces with an affine geometry on the infinitesimal scale. In Synthetic Differential Geometry this can be made precise by showing that a smooth manifold carries a natural structure of an infinitesimally affine space. This structure is comprised of two pieces o…

We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We use Vessiot theory and solvable structures to find the largest integrable distributions contained in the Vessiot distribution assoc…

Study second order integrability of Einstein deformations on Riemannian and Kähler manifolds.

New theory allows ICA without assuming non-Gaussian sources.

A Lie groupoid, called \textit{second-order non-holonomic material Lie groupoid}, is associated in a natural way to any Cosserat media. This groupoid is used to give a new definition of homogeneity which does not depend on a reference crystal. The corresponding Lie algebroid, called \textit{second-order non-holonomic m…

These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting f…

We show that Plebanski's second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magri's theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.

TSCD is an algorithm for causal discovery using second-order statistics.

The class of second order ODE's cubic with respect to the first order derivative is considered. Using geometric structures associated with these equations, the subclasses of umbilical equations, zero mean curvature equations, and zero Gaussian curvature equations are defined. Zero mean curvature equations are studied w…

The paper classifies second-order superintegrable systems with torsion and semi-degeneracy.

Finite-sum optimization problems are ubiquitous in machine learning, and are commonly solved using first-order methods which rely on gradient computations. Recently, there has been growing interest in \emph{second-order} methods, which rely on both gradients and Hessians. In principle, second-order methods can require …

We study locally conformal symplectic (LCS) structures of the second kind on a Lie algebra. We show a method to build new examples of Lie algebras admitting LCS structures of the second kind starting with a lower dimensional Lie algebra endowed with a LCS structure and a suitable representation. Moreover, we characteri…

Study shows how macroeconomic news affects intraday price and order flow dynamics.

PCA outperforms random projections in retaining second order signals from latent groups.