Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.
problem Finding optimal radii for packing circles in various plane regions.
method Deterministic analytic formulae and recurrence relations.
result Formulated analytic formulae for 2D circle packing on various plane shapes.
Recently, Freedman [arXiv:2301.00295] introduced the idea of packing a maximal number of links into a bounded region subject to geometric constraints, and produced upper bounds on the packing number in some cases, while commenting that these bounds seemed far too large. We show that the smallest of these "extravagantly…
The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.
We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finite…
The paper connects Apollonian packings to knot theory and improves link representations.
problem Realizing algebraic links in Apollonian packings.
method Introducing new representations of links in tangency graphs of sphere packings, proving link realizability, and improving upper bounds.
result Any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing.
The paper studies circle packings using renormalization and subdivision rules.
problem Characterizing and proving properties of circle packings with specific subdivision rules.
method Iterations of skinning maps on Teichmüller spaces, renormalization theory, subdivision rules.
result Uniformly contracting renormalization operator and geometric inflexibility of circle packings.
Study on packing links with geometric constraints.
problem Maximizing link density in space with geometric restrictions.
method Investigates packing essential links within Euclidean space.
result Upper bounds on maximal density are found, but are large.
For any given natural number k, this paper gives upper bounds on the radius of a packing of a complete hyperbolic surface of finite area by k equal-radius disks in terms of the surface's topology. We show that the bounds given here are sharp in some cases and not sharp in others.
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.
We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples.…
The main result of this paper is that given a group G acting geometrically by isometries on a CAT(0) space X and a cyclic subgroup H of G generated by a rank-1 isometry of X, H has bounded packing in G. We give two proofs of this result. The first one is by a characterization of rank-1 isometries by Ham…
Circle packings on translation surfaces are consistent across different surfaces.
problem Existence and consistency of circle packings on translation surfaces.
method Analysis of circle packings on translation surfaces, including strata with varying genus.
result Same circle packing can be realized on varying translation surfaces in a certain stratum.
New algorithm for learning functions with bounds on error and sample complexity.
problem Learning [0,1]-valued functions in a prediction model. method General-purpose algorithm with upper and lower bounds on expected error and sample complexity.
result Improved bounds on sample complexity and agnostic learning conditions.
After having investigated the regular prisms and prism tilings in the $\SLR$ space in the previous work \cite{Sz13-1} of the second author, we consider the problem of geodesic ball packings related to those tilings and their symmetry groups pq21. $\SLR$ is one of the eight Thurston geometries that can be de…
New graph feedback model for bandits with improved regret bounds.
problem Understanding how graph structure affects regret in bandit problems.
method Introduced fractional weak domination number and k-packing independence number to capture upper and lower bounds on regret. Used strong duality theorem to derive upper and lower bounds. result Proved general upper and lower bounds on regret for various graph structures, showing tightness up to a logarithmic factor.
Study counts and equidistributes tori in Kleinian group self-joinings.
problem Counting and equidistribution of tori in Kleinian group self-joinings.
method Analyzes d-dimensional torus packings invariant under a self-joining of a Kleinian group. result Equidistribution results for tori with small volume in a class of d-dimensional torus packings. Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.
problem Understanding sphere packings and their arithmetic origins in various dimensions.
method Introduces Kleinian Sphere Packings and Bugs, extending Arithmeticity Theorem.
result Kleinian packings and Bugs come from Q-arithmetic lattices of simplest type.
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.
The paper studies rigidity of sphere packings on 3D manifolds with boundary.
problem Rigidity of sphere packings on 3D manifolds with boundary.
method Introduced generalized Thurston's sphere packings and proved their rigidity properties.
result Generalized Thurston's sphere packings are locally determined by combinatorial scalar curvatures and cannot be deformed while keeping combinatorial Ricci curvatures fixed.
Study generates infinite circle packings with a specific property.
problem Generating infinite circle packings with a unique property.
method Investigates an infinite family of circle packings and uses them to create Apollonian packings.
result Created an infinite set of circle packings with the Apollonian property.
Paper proves circle packings converge to Riemann mapping for Jordan domains.
problem Proving discrete conformal maps converge to Riemann mapping.
method Establishing solvability theorem for inversive distance circle packings.
result Bowers-Stephenson's conjecture for Jordan domains is proven.
Proves rigidity of sphere packings on 3D manifolds.
problem Rigidity of sphere packings on 3D manifolds.
method Combining combinatorial scalar curvature and Ricci curvature to prove rigidity.
result Proves infinitesimal rigidity of Thurston's Euclidean sphere packing.
The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fun…
Paper introduces new flows to find circle packings with specific curvature.
problem Finding circle packings with prescribed total geodesic curvatures.
method Introduces combinatorial Calabi flow, fractional combinatorial Calabi flow, and combinatorial p-th Calabi flow.
result Establishes conditions for the longtime behaviors of these flows.
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.
problem Proving the Discrete Schwarz-Pick Lemma for circle packings with various inversive distances.
method Using a variational principle for circle packings with inversive distances, the paper extends the lemma to a broader range of packings.
result The Discrete Schwarz-Pick Lemma holds for circle packings with inversive distances in (−1,1], provided an additional condition on triangle weights. 3D Ricci flows have bounded diameter before Type I singularities.
problem Bounding the diameter of 3D Ricci flows before Type I singularities.
method Introduced a neck-region concept and proved packing measure Ahlfors regularity.
result Uniformly bounded diameter up to Type I singular time.
The paper solves the existence problem of sphere packings in higher dimensions.
problem Existence of crystallographic sphere packings in certain higher dimensions.
method Geometric doubling procedure and computations with Lorentzian quadratic forms.
result Solves the existence problem of crystallographic sphere packings in higher dimensions.
Projective rigidity of circle packings on complex surfaces proved.
problem Proving rigidity of circle packings on complex projective surfaces.
method Proved projective rigidity through triangulations and complex projective structures.
result Space of circle packings is projectively rigid on complex projective surfaces.
The paper studies rigid sphere packings on 3D manifolds with boundary.
problem Investigating rigid sphere packings on 3D manifolds with boundary.
method Introducing generalized sphere packings, proving rigidity, introducing combinatorial curvature flows.
result Generalized sphere packing metrics are determined by combinatorial scalar curvature.
Study on non-orientable surfaces for maximal disc packings.
problem Maximizing disc packings in non-orientable surfaces.
method Analyzing compact non-orientable surfaces of genus g≥3 for maximal k-packings. result Characterization of maximal disc packings in non-orientable surfaces.
Grassmannian packings improve CNN kernels' diversity and reduce sparsity.
problem Kernel sparsity and lack of diversity in CNNs decrease model capacity.
method Initialize CNN kernels with Grassmannian packings to maximize diversity and minimize sparsity.
result Grassmannian packings lead to diverse features and improved classification accuracy.
This paper optimizes neural network training by packing multiple models on a single GPU.
problem Efficiently sharing limited training resources among multiple neural network models.
method Proposes a primitive called 'pack' to jointly train multiple models on a single GPU.
result Significant performance improvements for hyperparameter tuning, up to 40% for two models.
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
Study of rod packings in 3-torus using 3-manifold geometry.
problem Understanding crystal structures in crystallography through rod packings in 3-torus.
method Use of 3-manifold geometry and topology to analyze complements of rod packings.
result Find families of complements that are hyperbolic and Seifert fibred.
Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.
problem Whether a prescribed total geodesic curvature can be realized by a degenerated circle packing.
method Introduced combinatorial Ricci flow to find the desired degenerated circle packed surface, analogous to Chow-Luo and Takatsu methods.
result Fully characterized sufficient and necessary conditions for the existence of degenerated circle packings and showed their uniqueness.
Minimal surfaces created from tiny circle packings changes.
problem Creating minimal surfaces from circle packings.
method Parametrizing deformations of circle packings and relating them to discrete minimal surfaces.
result Every minimal surface of Koebe type can be extended to a general type minimal surface.
Paper constructs hyperbolic metrics using circle packings and curvature parameters.
problem Creating polyhedral metrics for surfaces of various topologies.
method Using circle packings and curvature parameters, the paper constructs hyperbolic polyhedral metrics.
result Unified approach to producing polyhedral metrics for surfaces of broader topological types.
Paper calculates ball number of links using Lorentz geometry and circle packing.
problem Calculating the minimum number of balls needed to represent a link.
method Lorentz geometry and circle packing theorem applied to ball packings.
result Shows ball(L)≤5cr(L) for any link L. Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.
problem Sarnak's spectral gap question for hyperbolic packings.
method Analysis of Patterson-Sullivan base eigenfunctions and spectral gaps.
result Unique square-integrable eigenfunction has maximal spectral gap.
New theorem proves rigidity of circle packings in hyperbolic geometry.
problem Rigidity of circle packings in hyperbolic geometry.
method Established maximum principles and applied them to prove rigidity.
result Proved infinite rigidity of weighted Delaunay triangulations in the Poincaré disk.
The Piyavskii-Shubert algorithm is analyzed for global optimization of Lipschitz functions.
problem Maximizing a non-concave Lipschitz function over a compact domain.
method Sequential function evaluations using a bandit-optimization approach.
result New bounds on the number of evaluations needed for optimization accuracy.
The paper solves circle packings on surfaces with boundaries.
problem Circle packing on surfaces with boundaries and finite genus.
method Using Thurston's algorithm and discrete Schwarz-Pick lemma.
result A unique solution to the boundary value problem exists.
The paper deforms circle packings on surfaces with constant curvature.
problem Deforming circle packings on surfaces to constant curvature.
method Using Ge-Xu's α-flow to deform initial inversive distance circle packings.
result The inversive distance circle packing with constant α-curvature is unique under certain conditions.
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.
We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector. When the set of unit vectors …
Circle packings on compact surfaces simplified.
problem Simplifying circle packings on complex surfaces.
method Uniformisation of weighted maps.
result Unified approach to circle packings.
Properness proven for circle packings and Delaunay patterns on complex projective structures.
problem Proving properness for circle packings and Delaunay patterns on complex projective structures.
method Considering circle packings and Delaunay circle patterns on surfaces with complex projective structures, proving properness of the forgetful map.
result Proved properness of the forgetful map sending circle packings and Delaunay patterns to underlying complex structures.