arXiv research

Geometric structures on surfaces relate to 2-plane distributions in 5D.

Geometric approach solves maximum likelihood for Cauchy-like distributions.

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

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic point…

We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold $M$ there is a metric, such that th…

By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i…

Study on maps with horizontal $α$-harmonic properties in 1D and 2D.

The equitangent locus of a convex plane curve consists of the points from which the two tangent segments to the curve have equal length. The equitangent problem concerns the relation between the curve and its equitangent locus. An equitangent n-gon of a convex curve is a circumscribed n-gon whose vertices belong to the…

Generalizes Kauffman's clock theorem to surfaces.

Proposes a general model for plane-based clustering with a new loss function.

Random translation surfaces converge to a Poisson plane as genus grows.

Constructs polyhedral chains with prescribed tangent plane distributions.

Study the geometry and holonomy of indecomposable cones.

Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.

A beta function for double layers is defined and analyzed.

New findings on currents and Frobenius theorem properties.

Study curvature of orthogonal distributions on manifolds.

New numerical methods for evolving curves on curved spaces.

The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.

The study finds many tight contact structures on hyperbolic 3-spheres.

Two flat sub-Lorentzian problems on Martinet distribution differ in attainable set intersections.

As was shown recently by P. Nurowski, to any rank 2 maximally nonholonomic vector distribution on a 5-dimensional manifold M one can assign the canonical conformal structure of signature (3,2). His construction is based on the properties of the special 12-dimensional coframe bundle over M, which was distinguished by E.…

A new probabilistic polygonal curve representation using Gaussian Mixture Models.

We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+ε)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $ε\in {0,1}$ and $m \geq 2$. In particular, given such a distribution $\mathcal{N}$, say, we obtain algebraic conditions on the Weyl tensor and t…

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…

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…

Proposes a new complex Gaussian distribution for better modeling of complex-valued signals.

In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parametrization that characterizes any collection of non-crossing quantile planes over arbitrarily shaped convex predictor domains in a…

Study on 5-manifolds with specific geometric structures.

In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection $\bar{\nabla}$ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…

In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or $(2,3,5)$-distributions determined by a single function of the form $F(q)$, the vanishing condition for the curvature invar…

Invariants for 3D manifolds with plane fields defined by 1-forms and vector fields.

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…

We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.

Extends results on automorphism groups of flat Minkowski planes to toroidal circle planes.

Stable planes are locally isomorphic to classical projective planes.

Only vertical planes are asymptotic to other planes in 3D space.

This paper continues a series of studies devoted to analysis of the bivariate probability distribution P(x,y) of two consecutive price increments x (push) and y (response) at intraday timescales for a group of stocks. Besides the asymmetry properties of P(x,y) such as Market Mill dependence patterns described in preced…

We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensor…

Study of elementary planes in Apollonian orbifold with unusual equidistribution.

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed fo…

This paper explores VAEs in Fisher-Shannon plane, revealing the relationship between Fisher information and Shannon entropy.

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…

Criteria for sharksfin and deltoid singularities from plane to plane, with applications.

Crooked planes in 3D Minkowski space can be foliated.

Analytic torsion defined for rank 2 distributions on 5-manifolds.

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…