This paper proves that regular hyperideal tetrahedra maximize volume under edge length constraints.
problem Understanding the volume of hyperideal tetrahedra with constrained edge lengths.
method Analyzes Schläfli formula and proves maximization of volume for regular tetrahedra.
result Regular hyperideal tetrahedra of edge length ℓ maximize volume among tetrahedra with all edges ≥ ℓ.
Critical nets in k-space have bounded edge lengths and vertices.
problem Understanding the structure and constraints of critical nets in k-dimensional space.
method Analyzing the properties of critical nets under fixed leaf positions and constraints.
result The total length of edges not incident with 1-valent vertices is bounded, and the degree and number of vertices are also bounded.
The paper provides conditions for realizing graphs and polytopes with specified edge lengths.
problem Proving the existence of planar embeddings or polyhedra with specified edge lengths.
method Practical sufficient conditions and software verification for non-self-intersecting perturbations of initial realizations.
result Existence of planar embeddings and polyhedra with specified edge lengths.
Fixed angles of convex polygons lead to combinatorially rich polytopes.
problem Understanding the structure of convex polygons with fixed vertex angles.
method Combining combinatorial and geometric approaches, including dual polytopes and Schwarz-Christoffel maps.
result Fixed-angles polytopes are dual to cyclic polytopes under certain conditions.
Computes volumes of metric maps on surfaces, linking to Weil-Petersson volumes.
problem Computing volumes of specific metric maps on surfaces.
method Using recent results on discrete maps with irreducibility constraints, computes volumes as homogeneous polynomials.
result Identifies volumes as homogeneous polynomials and satisfies string and dilaton equations.
Researchers found the Wigner derivative and its inverse are equal for spherical tetrahedra.
problem Computing the relationship between dihedral angles and edge lengths in tetrahedra.
method Computed the Wigner derivative and its inverse for spherical tetrahedra.
result The Wigner derivative and its inverse are equal for spherical tetrahedra.
The paper develops formulas for hyperbolic simplices based on edge lengths.
problem Understanding the geometry of hyperbolic simplices using only edge lengths.
method Develops geometric formulas for hyperbolic simplices based on edge lengths.
result Distance and projection formulas in hyperbolic simplices.
Symmetric TSP is structurally equivalent to a constrained Group Steiner Tree Problem.
problem Finding the shortest tour in a symmetric TSP.
method Structural equivalence between symmetric TSP and constrained Group Steiner Tree Problem.
result Maximizing net weight in the cGSTP is equivalent to minimizing the TSP tour length.
This note demonstrates how both the concept of distance and the concept of holonomy can be constructed from a suitable network with directed edges (and no lengths). The number of different edge types depends on the signature of the metric and the dimension of the holonomy group. If the holonomy group is of dimension on…
We show that if X is a minimal length carrier graph in a hyperbolic 3-manifold, M, then if X contains a sufficiently short edge, it must contain a short circuit, as well. The meaning of "short" depends only on the rank of the fundamental group of M. We also expand the class of manifolds which are known to have minimal …
Given a clover link, we construct a bottom tangle by using a disk/band surface of the clover link. Since the Milnor number is already defined for a bottom tangle, we define the Milnor number for the clover link to be the Milnor number for the bottom tangle and show that for a clover link, if Milnor numbers of length k …
Cube edges curves minimize systole length.
problem Finding the shortest closed curve on a cube.
method Combining exact calculations and estimates, including branched covers, elliptic integrals, geodesic trajectories, and conformal maps.
result The extremal length systole is realized by 12 curves surrounding the cube's edges.
Polyhedra's structure is uniquely defined by edge lengths and dihedral angles, even nonconvex.
problem Determining the structure of polyhedra based on edge lengths and dihedral angles.
method Proved rigidity under specific conditions in Euclidean, hyperbolic, and spherical geometries.
result Polyhedra's structure is uniquely defined by edge lengths and dihedral angles, even nonconvex.
Suppose C is a compact, n-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an effectively computable lower bound on the area of C, given an n-tuple of positive…
A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the configuration space of all planar realizations of a linkage (following work of K…
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.
Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an …
Kernel networks' stability edge linked to Fisher Information singularity.
problem Understanding the stability edge in high-capacity kernel Hopfield networks.
method Statistical manifold analysis and Riemannian geometry.
result The Ridge of Optimization corresponds to the Edge of Stability, revealing a dual equilibrium.
Study geodesics on graphs with random lengths, proving bi-infinite paths exist.
problem Existence of bi-infinite geodesic paths on graphs with random edge lengths.
method Sublinear Morse geodesics and first passage percolation analysis.
result Proves the existence of bi-infinite geodesic paths in graphs with specific properties.
Algorithm checks if geometrically triangulated manifolds are isometric.
problem Determining if two geometric triangulations of manifolds are isometric.
method Sequence of Pachner moves and barycentric subdivisions with bounds on lengths.
result Bounding the length of transformations between triangulations.
OL4EL optimizes edge learning on resource-constrained servers.
problem Resource constraints on edge servers hinder effective distributed machine learning.
method Online Learning for EL (OL4EL) framework using budget-limited multi-armed bandit model.
result OL4EL significantly improves learning performance while conserving resources.
The paper solves pentagon equations using triangulations and edge transformations.
problem Solving pentagon equations with triangulations and edge transformations.
method General data and transformation rule method applied to triangulations.
result Recovery of initial data after transformations.
Curves converge to circles under length constraints.
problem Understanding curve convergence under length constraints.
method Length-constrained curve diffusion to analyze curve behavior over time.
result Curves converge to circles in infinite time with exponential convergence.
Previously in 2014, we proposed the Nearest Descent (ND) method, capable of generating an efficient Graph, called the in-tree (IT). Due to some beautiful and effective features, this IT structure proves well suited for data clustering. Although there exist some redundant edges in IT, they usually have salient features …
In his PhD thesis, Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertic…
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.
The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-d…
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Ad…
We construct a function of the edge-lengths of a triangulated surface whose variation under a rescaling of all the edges that meet at a vertex is the defect angle at that vertex. We interpret this function as a gravitational effective action on the triangulation, and the variation as a trace anomaly.
We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant b on the space of those isometric deformations which, for conv…
The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle π in the middle points of a certain pair of its skew edges.
Edge language models show bias over time, especially on resource-constrained devices.
problem Bias in edge language models on resource-constrained devices.
method Comparative analysis of text-based bias across edge, cloud, and desktop environments; optimized Llama-2 model on Raspberry Pi 4; feedback loop mechanism to correct bias.
result Llama-2 on Raspberry Pi 4 shows 43.23% and 21.89% more bias over time compared to cloud and desktop models.
The paper studies stability of discrete planar curves using variational methods.
problem Stability of discrete planar curves under area constraints.
method Unified interpretation of discrete curvatures, determination of equilibrium curves, stability analysis.
result Equilibrium curves for the length functional under area-constraint conditions are determined and their stability is studied.
Study laws of cosines and sines for hyperbolic shapes with ideal vertices.
problem Formulating trigonometric laws for shapes with ideal vertices in hyperbolic geometry.
method Using hyperboloid model and Lorentzian geometry, establishing laws for quadrilaterals, pentagons, and partially truncated tetrahedra.
result Transversal lengths of partially truncated tetrahedra depend only on internal edge lengths at ideal vertices.
TOCO framework compresses neural networks based on tolerance analysis.
problem Deploying large neural networks on edge devices with limited resources.
method TOCO uses tolerance analysis to perform fine-grained compression, allowing flexibility to hardware changes.
result Fine-grained compression of neural networks on edge devices.
Graph-based method predicts edge flows from partial measurements.
problem Predicting edge flows from limited measurements.
method Graph-based semi-supervised learning with flow conservation constraints.
result Strong performance on synthetic and real-world flow networks.
The paper defines and studies discrete p-density and compression-radius profiles of lattice knots.
problem Understanding geometric properties of lattice knots.
method Develops a framework for discrete p-density and compression-radius profiles of lattice knots, studying them on length-filtered sets and finite move-graph exploration.
result Density and compression-radius values are not monotone, illustrating distinct optimization problems.
Study on polygons with fixed edge slopes and their perimeter function.
problem Characterizing and analyzing polygons with prescribed edge slopes.
method Configuration space description and perimeter as a Morse function.
result Characterization and computation of critical points and their Morse indices.
New method simplifies ideal curve flow with length constraint.
problem Analyzing ideal curve flow with length constraint.
method Introduced length constraint to simplify sixth order curvature flow.
result Flow exists for all time and converges to a round circle.
Derives conformal parameters of curves using inscribed circular polygons.
problem Characterizing conformal invariants of smooth curves in 3D.
method Limiting process with inscribed circular polygons, based on elementary geometry.
result Derives conformal length, curvature, and torsion via a novel method.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
Scl in groups acting on trees is rational and converges to limits.
problem Understanding stable commutator length in group actions on trees.
method Analyzing groups acting on trees with cyclic stabilizers, focusing on stable commutator length and its limits.
result Stable commutator length is rational and converges to limits in surgery families.
Optimizes pipelined computation and communication for edge learning within latency constraints.
problem Balancing data transmission and model training to meet latency requirements.
method Analyzes the optimal packet payload size tradeoff between bias and variance.
result Derives analytical bounds on the expected optimality gap for effective optimization.
The paper studies how curves evolve under area constraints and converges to a critical point.
problem Evolution of plane curves with fixed area under elastic energy gradient.
method Local and global existence of the flow, simplicity assumption, Łojasiewicz--Simon inequality.
result The evolving curve's length remains bounded and converges to a critical point.
Extends RSP model with net flow and capacity constraints for better network analysis.
problem Improving shortest path models with net flows and capacity constraints.
method Developed net flow RSP model and introduced capacity constraints. Proposed algorithms for computing expected routing costs and solving constrained problems using Lagrangian duality.
result Net flow RSP dissimilarity measure is competitive with state-of-the-art dissimilarities.
Unified description of tetrahedra in various spacetimes.
problem Characterizing tetrahedra with lightlike faces in different spacetimes.
method Using a generalized cross-ratio and edge lengths/dihedral angles to describe tetrahedra and their duals.
result Generalized ideal tetrahedra are the duals of tetrahedra with lightlike faces.
POET enables large neural network training on tiny devices with reduced energy.
problem Training large neural networks on memory-limited edge devices.
method Jointly optimizes rematerialization and paging for memory reduction, formulating an MILP for energy-efficient training.
result POET trains ResNet-18 and BERT within Cortex-M memory constraints, outperforming current methods in energy efficiency.
The abstract proves polygon inscriptions in curves with specific edge ratios.
problem Proving the existence of polygons inscribed in Jordan curves with prescribed edge ratios.
method Using the properties of differentiable curves and proportional side lengths.
result Existence of polygons inscribed in Jordan curves with prescribed edge ratios.