arXiv research

Study on planar graphs in Poincare model of hyperbolic geometry.

The paper examines the geometry of a curve's centre symmetry set.

New method for sensing non-planar surfaces using ERT.

Motivated by the analogies between the projective and the almost quaternionic geometries, we study the generalized planar curves and mappings. We follow, recover, and extend the classical approach as developed by Mikes and Sinyukov. Then we exploit the impact of the general results in the almost quaternionic geometry. …

Sharp bounds for spanning tree entropy in planar lattices.

Unified description of aesthetic curves through self-affinities.

3D analog of Whitney's planarity criterion for 2-complexes.

Planar multilinks prove rational singularities in surface geometry.

Approximates smooth surfaces using Laguerre geometry meshes.

Approximates surfaces using Laguerre geometry with spherical faces.

New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.

Paper estimates diameter for Minkowski problem solutions.

In the complex setting, let $F(x,y,y')=0$ be an analytic or algebraic differential equation with $y'$-degree $d$. We deal with the qualitative study of such equations through the geometry of the planar $d$-web generated by the generic family of integral curves. Infinitesimal symmetries of these configurations are discu…

The secant caustic of a planar curve $M$ is the image of the singular set of the secant map of $M$. We analyse the geometrical properties of the secant caustic of a planar curve, i.e. the number of branches of the secant caustic, the parity of the number of cusps and the number of inflexion points in each branch of thi…

This paper classifies all planar p-elasticae and their properties.

The paper finds new inequalities for convex polygons.

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …

The paper explores geometric properties of interception curves on planes and spheres.

The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill's region still has three connected components. The appro…

This article introduces planar shape signatures derived from homology nerves, which are intersecting 1-cycles in a collection of homology groups endowed with a proximal relator (set of nearness relations) that includes a descriptive proximity. A 1-cycle is a closed, connected path with a zero boundary in a simplicial c…

Study reveals CR structure of snake robot's geometry.

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H^2 we get a formula that appears to be new. We show that it implies the standard Cauchy-Santalo formula involving a central ang…

Geometric reduction of the Newtonian planar three-body problem is investigated in the framework of equivariant Riemannian geometry, which reduces the study of trajectories of three-body motions to the study of their moduli curves, that is, curves which record the change of size and shape, in the moduli space of oriente…

New method realizes planar graphs as Reeb graphs of algebraic functions.

Computes elastic grids that approximate 3D surfaces without physical simulations.

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

Study of minimal surfaces in a specific symmetric space with polynomial growth.

In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this re…

The paper analyzes the emergence of almost-honeycomb structures in low-energy planar clusters.

Tropical curves match to special Lagrangian shapes.

The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a topological, not a metric concept. That leads quickly to the notion of connectedness with…

The paper studies projectively equivalent para-Kaehler metrics in 4D.

In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems. In particular, we generalize the cosymplectic structures to time-dependent Nambu-Poisson Hamiltonian systems and corresponding Jacobi's last multiplier for 3D systems. We illustrate our cons…

We show that the planar circular restricted three body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of Contact Topology to understand this particular dy…

Study finds bounds for systole length on arithmetic punctured spheres.

We develop a new method to construct explicit, regular minimal surfaces in Euclidean space that are defined on the entire complex plane with controlled geometry. More precisely we show that for a large class of planar curves $(x(t), y(t))$ one can find a third coordinate $z(t)$ and normal fields $n(t)$ along the space …

Kernel methods have had great success in Statistics and Machine Learning. Despite their growing popularity, however, less effort has been drawn towards developing kernel based classification methods on Riemannian manifolds due to difficulty in dealing with non-Euclidean geometry. In this paper, motivated by the extrins…

The study finds lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.

In this communication, complex systems with a near trivial dynamics are addressed. First, under the hypothesis of equiprobability in the asymptotic equilibrium, it is shown that the (hyper) planar geometry of an $N$-dimensional multi-agent economic system implies the exponential (Boltzmann-Gibss) wealth distribution an…

Study local invariants of divergence-free webs in geometry.

Study of group boundaries and subgroup properties.

We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic…

Characterizes minor-minimal separating projective planar graphs and their generalizations.

If a hyperbolic 3-manifold admits an exceptional Dehn filling, then the length of the slope of that Dehn filling is known to be at most six. However, the bound of six appears to be sharp only in the toroidal case. In this paper, we investigate slope lengths of other exceptional fillings. We construct hyperbolic 3-manif…

Minimal spheres found in ellipsoids with large axes.

We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.

Paper reveals how minimal surfaces' volumes can deduce their Riemannian structure.

Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.