We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.
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48 results for “minimal configurations”
Solves Plateau-Douglas problem for singular configurations in general metric spaces.
problem Existence of minimal surfaces for singular configurations.
method Generalized approach via minimal sequences in metric spaces.
result Existence of minimal surfaces for singular configurations in general metric spaces.
Minimal elastic networksmath.AP
Minimal elastic networks minimize energy and length at fixed angles.
problem Finding optimal network configurations under elastic constraints.
method Minimizing a combination of elastic energy and length.
result Existence and regularity of minimizers with prescribed angles.
The study finds conditions for area-minimizing cones over submanifolds.
problem Conditions for area-minimizing cones over submanifolds.
method General configuration results for area-minimizing cones.
result Cone over the minimal product of submanifolds and spheres are area-minimizing.
Study on tilings of the plane with two types of tiles of varying areas.
problem Classifying tilings with minimal interface length.
method Analysis of isoperimetric configurations for different lattice types and tile areas.
result Three distinct tilings configurations found based on tile area ratio.
The study finds optimal minimum distances for Green's energy points on compact manifolds.
problem Finding optimal minimum distances for Green's energy points on compact Riemannian manifolds.
method Analyzing point configurations minimizing discrete energy with the Green's function for the Laplacian.
result Every point in a minimizing configuration lies inside a harmonic ball, and the minimum distance has optimal asymptotic order.
Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.
problem Finding the hyperbolic 3-orbifold with minimal volume among non-arithmetic ones.
method Utilized the tetrahedral Coxeter group and horoball configuration to prove minimal volume.
result The 1-cusped quotient of hyperbolic space by the tetrahedral Coxeter group has minimal volume.
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].
Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.
The study of polygon areas with fixed perimeter.
problem Finding the minimum number of critical points for polygon areas.
method Analysis of the configuration space and critical points of the area function.
result Computed indices of critical points (regular stars) on the configuration space.
Researchers find optimal configurations of complex knots and links.
problem Finding the most efficient configurations of complex knots and links.
method Minimizing Möbius and Minimum Distance energies by describing them with a small number of free parameters.
result Optimal geometries for Hopf links, Borromean rings, and chain links are found.
Smectic liquid crystals are materials formed by stacking deformable, fluid layers. Though smectics prefer to have flat, uniformly-spaced layers, boundary conditions can impose curvature on the layers. Since the layer spacing and curvature are intertwined, the problem of finding minimal configurations for the layers bec…
Minimal surfaces from simple polynomials solve a geometric problem.
problem Constructing minimal surfaces using Traizet's method.
method Using polynomials that satisfy a hypergeometric differential equation.
result Simple minimal surfaces are described by these polynomials.
Two surfaces minimize variance of Gaussian curvature.
problem Minimizing the variance of Gaussian curvature on triply periodic minimal surfaces.
method Interpreting branch values of Gauss map, expressing variance as integrals of exponentials of Green's functions, analyzing Hessian.
result The P and D surfaces are local minimizers of the variance of Gaussian curvature.
Researchers analyze geodesic complexity in robot paths on tree graphs.
problem Understanding optimal paths for robots on tree graphs.
method Examined geodesic complexity in ordered and unordered configuration spaces of graphs in and metrics, finding explicit geodesics and families.
result Geodesic complexity matches topological complexity in all cases studied.
Role mining tackles the problem of finding a role-based access control (RBAC) configuration, given an access-control matrix assigning users to access permissions as input. Most role mining approaches work by constructing a large set of candidate roles and use a greedy selection strategy to iteratively pick a small subs…
Bayesian optimization (BO) aims to minimize a given blackbox function using a model that is updated whenever new evidence about the function becomes available. Here, we address the problem of BO under partially right-censored response data, where in some evaluations we only obtain a lower bound on the function value. T…
Kernel-based algorithm optimizes cellular network configuration through multi-task learning.
problem Optimizing network configuration based on field experience and minimizing exploration cost.
method Kernel-based multi-BS contextual bandit algorithm leveraging conditional kernel embedding for multi-task learning.
result The proposed algorithm reduces exploration cost and improves network performance.
Study lens spaces' definite fillings, classifying those with specific inequalities.
problem Classifying lens spaces with certain inequalities for definite fillings.
method Combinatorial framework and forbidden configurations.
result Classification of lens spaces based on forbidden configurations.
Classifies low energy maps from curved surfaces into spheres.
problem Classifying maps from surfaces of constant curvature into spheres.
method Analyzes maps with low energy and degree ±1, focusing on bubble configurations.
result Maps are quantitively close to a bubble configuration with specific radii.
New minimal surfaces found using Toda lattice and integrable systems.
problem Constructing new minimal surfaces with specific genus.
method Using Toda lattice and integrable systems techniques.
result New singly periodic minimal surfaces with genus .
The paper proves the openness of K-semistability for Fano varieties.
problem Stability of K-semistability in families of log Fano pairs.
method By showing the stability threshold is a constructible function and proving special test configurations arise from log canonical places.
result The stability threshold is a constructible function on fibers, and any minimizer of the stability threshold exists.
In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the result of the rational blow-down of a symplectic -4-sphere in X, then it is non…
For any flat projective family $(\mX,\mL)\rightarrow C$ such that the generic fibre $\mX_η$ is a klt Q-Fano variety and $\mL|_{\mX_η}\sim_{Q}-K_{X_η}$, we use the techniques from the minimal model program (MMP) to modify the total family. The end product is a family such that every fiber is a klt Q-Fano variety. Moreov…
We study a recent general criterion for the injectivity of the conformal immersion of a Riemannian manifold into higher dimensional Euclidean space, and show how it gives rise to important conditions for Weierstrass-Ennerper lifts defined in the unit disk endowed with a conformal metric. Among the corollar…
In this paper we prove the existence of families of n-dimensional complete embedded minimal submanifolds of C^n with a prescribed configuration of k>1 asymptotic planes. These submanifolds are obtained by desingularizing the intersection of the asymptotes, using a gluing theorem applied to a generalization of a special…
Develops a new theory of width for embedded circles in Riemannian manifolds.
problem Defining and understanding the width of embedded circles in Riemannian manifolds.
method Morse-Lusternik-Schnirelmann theory applied to geodesics and minimising configurations.
result Classifies configurations of minimising geodesics intersecting embedded circles.
The paper disproves compactness for high-energy Willmore immersions and finds minimal bubbles on Willmore surfaces.
problem Compactness for high-energy Willmore immersions of Willmore energy above .
method Explicit construction of minimal bubbles and analysis of limit sequences of Willmore immersions.
result Compactness for immersed Willmore tori of energy below is proven.
In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules,…
Study minimal networks on spheres and balls near standard metrics.
problem Existence of minimal networks in spheres and balls with metrics close to standard.
method Finite-dimensional reduction method, inspired by configuration of networks and triods.
result Existence of minimal networks in spheres and balls for metrics close to standard.
Rigidity of critical eigensections on spheres proven.
problem Rigidity of critical eigensections on spheres.
method Proved rigidity of critical eigensections through SO(3)-rotations.
result Minimal non-degenerate critical eigensections are deformation rigid.
Defines and studies positive configurations in affine buildings.
problem Understanding the geometry of positive configurations in affine buildings.
method Elementary definition and study of properties, conjectures involving minimal networks and max-flow/min-cut theorem.
result Conjectures about the tropicalized canonical functions in terms of the geometry of affine buildings.
BONSAI optimizes parameters while respecting a default configuration, reducing unnecessary changes.
problem Standard BO pushes weakly relevant parameters to the boundary, making it hard to distinguish between important and spurious changes.
method BONSAI is a default-aware BO policy that prunes low-impact deviations from a default configuration while controlling acquisition value loss.
result BONSAI matches the GP-UCB regret rate while recovering the minimal- solution, reducing the number of non-default parameters in recommended configurations.
Study of convergence of point-object configurations to a charged dust continuum.
problem Understanding the convergence of discretized point-object configurations to a charged dust continuum.
method Establishing existence and uniqueness of horizons/minimal surfaces, studying geometries of regions exterior to minimal surfaces, and discussing limits.
result Examples of scalar curvature jumps upon taking Gromov-Hausdorff and intrinsic flat limits.
Minimal generating sets found for Kim-Manturov groups.
problem Understanding the structure of groups related to surface triangulations.
method Provided minimal generating sets and determined abelianizations.
result Minimal generating sets and abelianization results for the groups.
In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring's problem to deal with links in a fixed link-homotopy …
We consider a generic configuration of regions, consisting of a collection of distinct compact regions in which may be either smooth regions disjoint from the others or regions which meet on their piecewise smooth boundaries in a generic way. We introduce a skeletal linking …
Study finds minimal length networks connecting three points in Heisenberg group.
problem Finding minimal length networks connecting three points in the Heisenberg group.
method Proved existence of minimal horizontal triods, formulated curve shortening flow, used numerical experiments.
result Characterized and deformed minimal horizontal triods into critical points for length functional.
This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. The most interesting phenomena found in these experiments is the dependence of the trajectories of energy descent upon the initial geometric conditions of the knotted embedd…
Solves the quintuple bubble problem on spheres and Euclidean spaces.
problem Minimizing total perimeter of multiple bubbles enclosing fixed volumes.
method Developed spectral theory of Jacobi operator and new bubble deformation method.
result Confirmed quintuple bubble conjecture on spheres and Euclidean spaces.
Constructs classifiers for neural networks with specific data configurations.
problem Finding global minima of deep ReLU neural networks on sequentially separable data.
method Explicitly constructs zero loss neural network classifiers using cumulative parameters and truncation maps.
result Global minimizers can be described with a limited number of parameters based on the data structure.
New differential equation generates more doubly periodic minimal surfaces.
problem Finding new doubly periodic minimal surfaces.
method Derived a differential equation equivalent to balance equations.
result Generates many more solutions to balance equations.
Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheetscond-mat.soft
Complex wrinkling patterns emerge in non-Euclidean elastic sheets due to energy minimization.
problem Understanding hierarchical buckling patterns in non-Euclidean elastic sheets.
method Minimizing elastic energy to explain complex wrinkling patterns.
result Branch-point singularities are key to generating complex wrinkling patterns.
We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-s…
We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in . Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity…
Minimal networks minimize length and mass in certain configurations.
problem Finding minimal networks that minimize length and mass.
method Global and local calibrations to prove minimization properties.
result Minimal networks minimize mass and interfaces in partitions.
We derive a dimensionally-reduced limit theory for an -dimensional nonlinear elastic body that is slender along dimensions. The starting point is to view an elastic body as an -dimensional Riemannian manifold together with a not necessarily isometric -immersion in -dimensional Euclidean space. The…
Framework optimizes expensive manufacturing processes efficiently.
problem Optimizing input parameters for advanced manufacturing methods.
method Bayesian optimization with tailored acquisition function and parallel acquisition.
result Framework efficiently finds optimal parameters with minimal process cost.