Hexnet framework enhances image processing with hexagonal structures.
problem Improve image processing systems with biological-inspired hexagonal structures.
method Develops a hexagonal deep learning framework (Hexnet) for image processing.
result Hexnet surpasses current hexagonal image processing systems and artificial neural networks.
New hexagonal circular 3-webs with reducible curves classified.
problem Classifying hexagonal circular 3-webs with reducible polar curves of degree 3.
method New examples and classifications presented.
result Classification of hexagonal circular 3-webs with reducible polar curves of degree 3.
Classifies hexagonal circular 3-webs with cubic polar curves.
problem Classifying hexagonal circular 3-webs with algebraic polar curves of degree three.
method Analyzes hexagonal circular 3-webs on unit sphere with polar points on a twisted cubic.
result Completes the classification of hexagonal circular 3-webs with algebraic polar curves of degree three.
Hexagonal triangulation remains rigid under certain conditions.
problem Maintaining the rigidity of hexagonal triangulation under specific transformations.
method PL conformality for hexagonal Delaunay triangulation.
result Rigidity theorem for convex ideal hyperbolic polyhedra.
Study of graphs from hexagon decompositions of surfaces.
problem Understanding geometric properties of hexagon decompositions.
method Define and analyze graphs associated with hexagon decompositions of surfaces.
result Quasi-isometric relationships between studied graphs and known groups.
In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles…
Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.
problem Proving Doyle conjecture for hexagonal lattice circle packings.
method Using Liouville theorem of discrete harmonic functions based on logarithmic radii ratio observation.
result Proves rigidity of Doyle spirals in hexagonal lattice circle packings with bounded radii ratios.
New bounds on knotting probability of equilateral hexagons found.
problem Determining the knotting probability of equilateral hexagons.
method Symplectic geometry techniques to parametrize and analyze the space of equilateral hexagons.
result New bounds on the knotting probability of equilateral hexagons.
We study the geometry of oriented right-angled hexagons in H^4, the hyperbolic 4-space, via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon so that for the non-augmented sides, we can define quaternion half side-lengths whose angular parts are obtained from half the Euler angle…
Study hexagonal network evolution under curvature flow.
problem Understanding hexagonal network evolution under curvature flow.
method Proved local existence of classical solutions and classified homothetically shrinking solutions.
result Provided an example of network shrinking to a segment with multiplicity two.
The study examines polyhedra with hexagonal and triangular faces, focusing on their 3-regular planar graphs.
problem Analyzing polyhedra with hexagonal and triangular faces and three faces around each vertex.
method Representing polyhedra as quotients of hexagonal tilings under isometries, using signatures to describe the arrangement of rotations, and establishing a bijection between trihexes and equivalence classes of signatures.
result A bijection between trihexes and equivalence classes of signatures, allowing bounds on the number of trihexes for a given number of vertices.
Hexagonal norm double bubble problem solved with minimal configurations.
problem Finding the optimal shapes for minimizing perimeter in hexagonal geometry.
method Elementary proof and geometric exclusions to simplify minimizer search.
result Existence of minimizing sets for volume ratio parameter α in (0,1].
The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elem…
Improved bounds for knot crossings in different mosaic patterns.
problem Finding tighter bounds for knot crossings in rectangular and hexagonal mosaics.
method Extended Howard and Kobin's proof to hexagonal mosaics and shortened the rectangular proof.
result New bounds for hexagonal mosaics with improved efficiency in rectangular mosaics.
We provide a complete classification of hexagonal singular 3-web germs in the complex plane, satisfying the following two conditions: 1) the Chern connection remains holomorphic at the singular point, 2) the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal wei…
Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.
problem Determining the stick number and edge length of knots in a hexagonal lattice.
method Introducing a linear transformation between lattices to prove strict inequalities and classifying knots.
result Only trefoil and figure-eight knots are 11-stick knots in the hexagonal lattice.
HexaConv improves CNN performance by using hexagonal filters and group convolutions.
problem Improving CNN performance by exploiting more symmetries.
method Implementing planar and group convolutions over hexagonal lattices.
result HexaConv outperforms conventional CNNs on aerial scene classification.
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.
New geometric perspective for optimal learning on hexagonal structures.
problem Optimal learning process on hexagonal structures.
method Local trivial fibrations and Ceva's theorem.
result Learning can be defined on hexagonal structures.
The Phi- relationship also known as Phi-factor appears in a number of lattice structures, mostly considering the lines within several separate circles or polygons. The paper considers a regular hexagonal tessellation as a lattice with the highest specific mechanical stiffness.
Hexagonal tilings minimize perimeter with unequal volumes.
problem Finding optimal tessellations with unequal cell volumes.
method Minimizing perimeter functionals for different classes of problems.
result Hexagonal tilings are optimal among partitions with almost equal areas.
This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.
We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration spa…
For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, w…
Hexagonal diagrams link complex curves in CP2 to minimal genus surfaces.
problem Understanding the relationship between complex curves and surfaces in CP2. method Hexagonal lattice diagrams and trisection of CP2. result Positive genus surfaces in CP2 are isotopic to complex curves if they admit hexagonal lattice diagrams. New knots found that can only fit in non-reduced projections.
problem Finding knots that can only fit in non-reduced projections.
method Infinite family of knots, systematic flype finding tool.
result Knots with hexagonal mosaic number realized only in non-reduced projections.
There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves analogous to the domino flip move. The triple diagrams that result have connections to …
The paper studies how grid cell patterns emerge in neural networks.
problem Understanding how grid cells in the brain form hexagonal firing patterns.
method Training recurrent neural networks with conformal normalization of velocity inputs.
result Conformal normalization is crucial for the emergence of hexagonal grid patterns in neural networks.
Study motion of discrete interfaces on triangular lattice using Almgren, Taylor, and Wang's approach.
problem Motion of discrete interfaces on triangular lattice driven by ferromagnetic interactions.
method Coupling Almgren, Taylor, and Wang's minimizing movements approach with Braides, Gelli, and Novaga's discrete-to-continuum analysis.
result Limit motion of origin-symmetric convex hexagons compared to crystalline curvature evolution.
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these for…
It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant appr…
The Ollivier Ricci flow with prescribed curvature on infinite graphs.
problem Ricci flow with prescribed curvature on infinite graphs.
method Existence and uniqueness of the solution to the Ricci flow.
result Convergence of the Ricci flow for graphs with girth at least 6.
The paper analyzes how grid cells perform path integration and learns hexagon grid patterns.
problem Understanding how grid cells perform path integration calculations.
method Theoretical analysis of a general representation model of path integration by grid cells, identifying group representation and isotropic scaling conditions.
result The learned model of hexagon grid patterns is capable of accurate long distance path integration.
New curvature measure on graphs improves diameter and eigenvalue bounds.
problem Improving curvature bounds on graph structures.
method Hybrid curvature definition on variable neighborhoods.
result Gradient estimates and curvature bounds proven.
We prove that every spherical football (also known as a spherical soccer ball) is a branched cover, branched only in the vertices, of the standard football made up of 12 pentagons and 20 hexagons. We also give examples showing that the corresponding result is not true for footballs of higher genera. Moreover, we classi…
We generalize arc coordinates for maximal representations on a pair of pants.
problem Maximal representations of reflection groups on hyperbolic surfaces.
method Introducing geometric parameters and reflections in Siegel space.
result Natural parametrization of maximal representations into PSp(4, R).
Geodesic 3-webs on surfaces are characterized by integrable flow equations.
problem Characterizing surfaces with hexagonal geodesic 3-webs.
method Integrable flow equations and generalized hodograph transform method.
result Geodesic flow on surfaces admits cubic first integrals for hexagonal 3-webs.
Drinfeld associator is a key tool in computing the Kontsevich integral of knots. A Drinfeld associator is a series in two non-commuting variables, satisfying highly complicated algebraic equations - hexagon and pentagon. The logarithm of a Drinfeld associator lives in the Lie algbera L generated by the symbols a,b,c mo…
The paper connects dilogarithm identities to Pell's equation solutions via continued fractions.
problem Connecting dilogarithm identities to solutions of Pell's equation.
method Using continued fraction expansions of units in the ring of integers of quadratic fields.
result Ramanujan's dilogarithm identities correspond to a hexagonal identity.
New examples challenge extended Courant property for linear combinations of eigenfunctions.
problem Extended Courant property for linear combinations of eigenfunctions.
method Numerical computations and examples of equilateral rhombus and regular hexagon.
result Counterexamples to the Extended Courant property for linear combinations of eigenfunctions.
A note on the uniqueness of differential characters and K-theory via homological algebra.
problem Existence and uniqueness of differential characters and differential K-theory.
method Observation and application of Rakesh Pawar's results in homological algebra.
result The hexagon diagram uniquely determines differential K-theory groups up to isomorphism.
Solutions of an implicit ODE form a web. Already for cubic ODEs the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. Here the most symmetric case of hexagona…
Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.
problem Understanding the algebraic, geometric, and topological properties of grid cells.
method Investigating recurrent neural network models of grid cells, focusing on Lie group and Lie algebra representations, conformal isometry, and hexagon periodic patterns.
result Conformal isometry leads to hexagon periodic patterns in grid cell responses and accurate path integration.
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.
Sharp density threshold for Property (T) found in random groups models.
problem Density threshold for Kazhdan's Property (T) in random groups.
method Quotient of free groups by random reduced words, new geometrical tools.
result Sharp density threshold for Property (T) equals 1/3.
Proof of injection from double shuffle to Kashiwara-Vergne Lie algebra.
problem Injecting double shuffle Lie algebra into Kashiwara-Vergne Lie algebra.
method Inclusion of brunnian braids group on different genus 0 surfaces, using lower central series of brunnian Lie algebras, and explicit links between maps.
result Injection of double shuffle Lie algebra into symmetric Kashiwara-Vergne Lie algebra.
Formula counts all fullerenes with given vertices.
problem Counting all fullerenes with a given number of vertices.
method Used modular forms to derive an exact formula.
result Exact formula for the number of oriented fullerenes.
A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. We consider the zigz…