Study on planar graphs in Poincare model of hyperbolic geometry.
problem Investigating Morse flows on a 2-disk using planar graphs.
method Using planar graphs and spherical graphs to describe topological structures.
result Listed all planar graphs with at least 3 edges and described those with 4 edges.
The paper examines the geometry of a curve's centre symmetry set.
problem Global geometrical properties of a curve's centre symmetry set.
method Study of the envelope of affine chords.
result Number of singularities and asymptotes of the centre symmetry set.
New method for sensing non-planar surfaces using ERT.
problem Limited computational techniques for planar surfaces in ERT-based sensing skins.
method Generalized ERT to non-planar surfaces using Riemannian geometry.
result Feasibility and applicability of ERT-based sensing skins for non-planar geometries demonstrated.
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.
problem Estimating spanning tree entropy in planar lattice graphs.
method Using hyperbolic geometry and polyhedra volumes.
result Proved bounds are easy to compute and provide excellent estimates.
Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
3D analog of Whitney's planarity criterion for 2-complexes.
problem Embedding 2-complexes in 3-space.
method Proving dual matroids of 2-complexes are graphic if and only if the complexes embed in 3-space.
result Simply connected and locally 2-dimensional complexes embed in 3-space if and only if their dual matroids are graphic.
Planar multilinks prove rational singularities in surface geometry.
problem Characterizing surface singularities using planar multilinks.
method Combining topological and combinatorial approaches, including Min--Roy--Wang's work.
result Planar multilinks imply rational singularities and sandwiched singularities.
Approximates smooth surfaces using Laguerre geometry meshes.
problem Approximating smooth surfaces in Laguerre geometry.
method Using Laguerre meshes composed of quadrilaterals, cones, and spherical faces.
result Laguerre conjugate nets and directions for surface approximation.
Approximates surfaces using Laguerre geometry with spherical faces.
problem Approximating smooth surfaces using Laguerre geometry.
method Using Laguerre conjugate nets and spherical faces to approximate surfaces.
result Laguerre conjugate nets provide a method for surface approximation.
New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.
problem Designing aesthetic shapes in equiaffine geometry.
method Introducing a new symmetry (ESA) to characterize planar curves.
result The new class of curves includes the quadratic curve and logarithmic spiral.
Paper estimates diameter for Minkowski problem solutions.
problem Estimating diameter of solutions to Minkowski problem.
method Uniform diameter estimate for Lp dual Minkowski problem. result Uniform diameter estimate for planar Lp dual Minkowski problem. 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.
problem Understanding the properties of p-elasticae in the plane.
method Introducing new types of p-elliptic functions to classify all planar p-elasticae.
result Explicit formulae and optimal regularity of all planar p-elasticae are obtained.
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar 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.
problem Geometric properties of interception curves defined by differential equations.
method Parametric representation and spherical curve defined by Gudermannian function.
result Symmetry/asymmetry between spherical and planar cases, connections to lemniscate constants.
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…
The paper characterizes measures preserving independence through planar web geometry.
problem Characterizing measures with preserved independence.
method Planar web geometry and inhomogeneous Abelian functional equations.
result The independence-preserving property is preserved by coordinatewise reparametrizations and forms a natural invariant.
The paper extends curve curvature types from normed to gauge planes.
problem Extending curve curvature classification to gauge planes.
method Using gauge analogue of Birkhoff orthogonality and differential geometry.
result Four curvature types (Minkowski, normal, circular, arc-length) are identified.
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.
problem Understanding the kinematics and geometry of a snake robot.
method Analysis of (2,3,5) distributions and solving Cartan equivalence problem.
result Discovery of a CR structure with CR dimension 1 and real codimension 3.
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.
problem Realizing planar graphs as Reeb graphs of algebraic functions.
method Generic embedding and elementary procedures.
result Generically embedded planar graphs are homeomorphic to Reeb graphs of algebraic functions.
Computes elastic grids that approximate 3D surfaces without physical simulations.
problem Creating planar grids that fit complex 3D surfaces efficiently.
method Uses differential geometry to minimize bending energy and nestle to the surface.
result Elastic grids can approximate 3D surfaces without physical simulations.
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.
Study of minimal surfaces in a specific symmetric space with polynomial growth.
problem Asymptotic geometry of minimal surfaces in a symmetric space.
method Homeomorphism between Hitchin components and maximal surfaces, identification of convex embeddings, local limits of equivariant surfaces.
result Identification of planar maximal surfaces as local limits of equivariant surfaces.
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.
problem Understanding the formation of shapes resembling honeycombs in low-energy configurations.
method Detailed quantitative estimates and a revision of the global isoperimetric principle for honeycomb clusters.
result The majority of chambers in low-energy planar clusters are generalized hexagons, closely resembling regular hexagons.
Tropical curves match to special Lagrangian shapes.
problem Connecting tropical geometry to special Lagrangian shapes.
method Gluing construction that matches tropical local models to Lagrangian shapes.
result Locally planar tropical curves can be realized as special Lagrangian limits.
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.
problem Characterizing para-Kaehler metrics with specific properties.
method Developed c-projective geometry for para-Kaehler metrics, focusing on 4D case.
result Local description and characterization of 4D pc-projectively equivalent metrics, including Einstein type.
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.
problem Finding the shortest essential curve on arithmetic punctured spheres.
method Correspondence between surfaces and planar triangulations to bound systole length.
result Arithmetic surfaces do not achieve maximal systole length for n=7,10,11. 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 …
The study finds lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.
problem Finding lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.
method Analyzing the spectrum of the Laplacian with magnetic Neumann boundary conditions, focusing on multiply connected domains with convex curves. Lower bounds are derived based on geometric invariants such as area, perimeter, diameter, and fluxes around inner holes.
result Sharp lower bounds for the first eigenvalue are derived for doubly connected domains and domains with an arbitrary number of holes, and a lower bound is obtained for Aharonov-Bohm operators with an arbitrary number of poles when holes shrink to points.
Study local invariants of divergence-free webs in geometry.
problem Characterize triviality of divergence-free webs.
method Introduce two local invariants: differential and geometric.
result Triviality of either invariant characterizes trivial divergence-free web-germs.
Study of group boundaries and subgroup properties.
problem Characterizing boundaries of relatively hyperbolic group pairs.
method Analyzing Bowditch boundaries and convergence group actions.
result Rigidity of group pairs leads to specific subgroup properties.
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…
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.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of 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.
problem Finding non-planar minimal spheres in ellipsoids.
method Quantitative proof of minimal spheres existence with constraints.
result Ellipsoids with large axes contain at least three non-planar minimal spheres.
Paper reveals how minimal surfaces' volumes can deduce their Riemannian structure.
problem Determining the Riemannian structure of minimal surfaces from their volumes.
method Analysis of Dirichlet-Neumann map and Carleman estimates.
result Volumes of minimal surfaces determine their Riemannian structure.
We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.