We describe some problems, observations, and conjectures concerning thickness and packing density of knots and links in $\sp^3$ and . We prove the thickness of a nontrivial knot or link in $\sp^3$ is no more than , the thickness of a Hopf link. We also give arguments and evidence supporting the conject…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
Trend · papers per month
48 results for “packing density”
Packing Meets Topologymath.GT
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.
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.
We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study th…
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 . $\SLR$ is one of the eight Thurston geometries that can be de…
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 diameter function is a topological Morse function.
problem The relationship between systole and diameter functions on Teichmüller space.
method Mapping class group-equivariant topological Morse function approach.
result The diameter function on Teichmüller space is a topological Morse function.
The paper analyzes geometric densities and compression radii for knot types.
problem Optimizing geometric quantities associated with knot types.
method Develops a factorization framework for scale-covariant size functionals.
result Different minimizing sequences for density, compression, packing, and ropelength problems.
The paper explores the longest slopes of exceptional Dehn fillings in hyperbolic 3-manifolds.
problem Finding the longest slopes of exceptional Dehn fillings in hyperbolic 3-manifolds.
method Constructing hyperbolic 3-manifolds and proving equivalence to maximal density horoball packings of planar surfaces.
result The bound of six for slopes of exceptional fillings is not realized by small Seifert fibered space fillings.
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 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.
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.
A game of packingsmath.MG
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.
This paper improves neural network efficiency by combining filter columns and retraining, boosting array utilization and accuracy.
problem Efficient implementation of sparse convolutional neural networks on systolic arrays.
method Column combining of filter matrices, retraining of remaining weights, joint optimization for high utilization and accuracy.
result Significantly increased systolic array utilization efficiency (e.g., ~4x) and maintained high classification accuracy.
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.
Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packingsmath.DG
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 explores properties of curves and surfaces of constant width using Fourier series.
problem Investigating properties of curves and surfaces of constant width.
method Utilizing Fourier series to derive properties and proving theorems.
result The density of a certain packing of Reuleaux triangles exceeds the maximum density for circles.
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 , provided an additional condition on triangle weights.
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.
Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundarymath.DG
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 for maximal -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.
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.
Paper proves rigidity of inversive distance circle packings.
problem Proving rigidity of inversive distance circle packings.
method Variational principles and combinatorial curvature study.
result Global rigidity of inversive distance circle packings proved.
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 generalizes Ricci flow for inversive distance circle packings.
problem Deforming circle packings with prescribed cone angles.
method Generalized combinatorial Ricci flow for inversive distance.
result Generalized flow deforms any inversive distance circle packing to a unique packing with prescribed cone angles.
New tube manifolds model hyperbolic crystallography with dense ball packings.
problem Finding dense ball packings in hyperbolic space by specific tube manifolds.
method Using tube or cobweb manifolds $Cw = \HYP/\BCw$ with -rotational symmetry, derived from Coxeter orthoscheme reflection groups.
result Derived minimal tube manifolds that are not covered by smaller manifolds, with dense ball packings.
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.
Ball packings for linksmath.CO
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 for any link .
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 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.
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.
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.
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.
Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometrymath.DG
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.
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.
From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are giv…
New proof simplifies existing proofs of Bowers-Stephenson conjecture.
problem Proving the global rigidity of circle packing with inversive distance.
method Variational proof simplifying previous methods.
result Global rigidity of circle packing with inversive distance in (-1, +∞).
The study finds upper bounds on disk packings on hyperbolic surfaces.
problem Finding optimal disk packings on hyperbolic surfaces.
method Analyzes the radius of packings of a complete hyperbolic surface by equal-radius disks.
result Bounds on disk packings are sharp in some cases and not in others.