Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
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48 results for “expanding shapes”
New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.
problem Understanding the limiting shape of hypersurfaces expanding in hyperbolic space.
method Introduced shifted inverse curvature flow with positive power for a smooth curvature function.
result For , limiting shape is always round as maximal existence time is approached.
Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves, surfaces, activities, character motions and many other objects. In …
The study examines evolving star-shaped hypersurfaces in hyperbolic spaces, influenced by ambient geometry.
problem Evolution of star-shaped hypersurfaces in hyperbolic spaces.
method Nonhomogeneous expanding curvature flows in hyperbolic spaces.
result The asymptotic behavior of the flow depends on the ambient space's geometry, leading to different limiting metrics.
The paper improves GP regression for sparse sensor data in structural mode shape reconstruction.
problem Reconstructing full-field structural mode shapes from sparse sensor data.
method Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework.
result The proposed method provides more accurate and reliable mode shapes.
We present new computations of tight shapes obtained using the constrained gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of…
The article explores derivatives of shapes other than circles and spheres.
problem Understanding derivatives of various shapes.
method First-year calculus approach.
result Derivatives of shapes other than circles and spheres are explored.
The study proves topological rigidity for certain geometric shapes using Poincaré inequalities.
problem Proving topological rigidity for translators and self-expanders in mean curvature flow.
method Abstract structure theorem for weighted manifolds, Poincaré inequality, and topological control.
result Full topological control on translators and self-expanders under stability or curvature assumptions.
Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spacesmath.DG
The study proves uniqueness and symmetry of self-similar solutions in warped product spaces.
problem Uniqueness and symmetry of self-similar solutions in warped product spaces.
method Analysis of curvature flows with homogeneous speed functions in warped product spaces.
result Compact star-shaped self-similar solutions in warped product spaces are slices.
Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.
problem Understanding the asymptotic behavior of anisotropic mean curvature flow.
method Established local gradient estimates for anisotropic -harmonic functions and weak solutions of IAMCF.
result Weak IAMCF is asymptotic to the expanding Wulff shape solution at infinity.
Smooths out complex shapes into simpler forms.
problem Transforming complex shapes into simpler, smooth forms.
method Perturbing minimizing hypercones and viscosity mean convex cones into smooth, properly embedded hypersurfaces.
result Properly embedded smooth minimizing hypersurfaces and self-expanders are achieved.
In this paper, we first investigate the flow of convex surfaces in the space form expanding by , where is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power for and for …
Polygonal meshes provide an efficient representation for 3D shapes. They explicitly capture both shape surface and topology, and leverage non-uniformity to represent large flat regions as well as sharp, intricate features. This non-uniformity and irregularity, however, inhibits mesh analysis efforts using neural networ…
New algorithm broadens BART models applicability.
problem Limited applicability of Bayesian additive regression trees (BART) models due to conditional conjugacy.
method Introduces a reversible jump Markov chain Monte Carlo algorithm for generalized BART models.
result Extends BART models to arbitrary generalized BART models without conditional conjugacy.
In this paper, we present experimental results obtained from retraining the last layer of the Inception v3 model in classifying images of human faces into one of five basic face shapes. The accuracy of the retrained Inception v3 model was compared with that of the following classification methods that uses facial landm…
TIR expands LLM capabilities by enabling problem-solving strategies.
problem Lack of a principled theory explaining why LLMs with tools are more capable.
method Formal proof and Advantage Shaping Policy Optimization (ASPO) algorithm.
result TIR model decisively outperforms pure-text models on challenging benchmarks.
The paper studies curvature flows in Euclidean and hyperbolic spaces, proving smooth convergence to spheres.
problem Analyzing curvature flows in Euclidean and hyperbolic spaces.
method Introduced a class of expanding flows with specific speed functions and proved their longtime existence and smooth convergence.
result The flows converge smoothly to spheres in Euclidean and hyperbolic spaces under certain conditions.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
New expanders found using origami surfaces with spectral gap.
problem Constructing expanders with spectral gap on surfaces of arbitrary genus.
method Affine actions on origami surfaces to achieve spectral gap.
result New expanders distinct from classical ones.
The paper extends rigidity results for -self-expanders to hyperplanes, spheres, and cylinders.
problem Characterizing -self-expanders as hyperplanes, spheres, and cylinders.
method Extending results on self-expanders to -self-expanders, proving rigidity results.
result Characterizes hyperplanes, spheres, and cylinders as -self-expanders.
The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some traces of amoebas were appearing from time to time, even before the formal introduct…
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
Operationalizes engagement as human behavior, linking theory and data.
problem Fuzziness of engagement concept.
method Formal framework, Melchoir Model, model comparison, theory-driven hypothesis.
result Engagement can be shaped and interpreted using data-driven methods.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecturemath.DG
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
New expanding Ricci solitons found starting in dimension four.
problem Finding expanding Ricci solitons in specific dimensions.
method Constructing gradient expanding Ricci solitons asymptotic to cones and on trivial vector bundles.
result Continuous families of expanding Ricci solitons on products of Einstein manifolds.
Study on the spectrum of drift Laplacian on Ricci expanders.
problem Analyzing the spectrum of the drift Laplacian on Ricci expanders.
method Investigation of discrete spectrum under proper potential function, asymptotic behavior of potential function, and computation of eigenvalues.
result Discrete spectrum of the drift Laplacian on Ricci expanders with bounded Ricci curvature.
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…
Triviality, Rotational Symmetry, and Classification of Complete Expanding Gradient Yamabe Solitonsmath.DG
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
Constructs self-expanders of positive genus for cones in R^3.
problem Creating self-expanders of positive genus for cones in R^3.
method Constructs self-expanders asymptotic to cones, uses mean curvature flow.
result Constructs self-expanders with unbounded genus asymptotic to a rotationally symmetric cone.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
Catastrophic forgetting continues to severely restrict the learnability of controllers suitable for multiple task environments. Efforts to combat catastrophic forgetting reported in the literature to date have focused on how control systems can be updated more rapidly, hastening their adjustment from good initial setti…
Study finds unique self-expanders for mean curvature flow.
problem Finding solutions to mean curvature flow.
method Derived equation based on generalized Lawson-Osserman cone and modified equilibria theory.
result Existence and uniqueness of self-expanders proved.
We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We also give an upper bound for the scalar curvature of noncompact expander when the Ricci curvature is nonpositive…
Study cohomogeneity one expanding Ricci solitons on specific topologies.
problem Characterize and analyze cohomogeneity one expanding Ricci solitons.
method Analyze ODEs, define expander degree, calculate cohomogeneity one expander degree.
result Reconstruct and calculate cohomogeneity one expander degree for specific topologies.
Expanders and Property Amath.GT
We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.
We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …
The paper examines properties and rigidity of self-expanders in Euclidean space.
problem Characterizing and estimating properties of self-expanders in Euclidean space.
method Analyzing mean curvature flow, volume growths, and stability of self-expanders.
result Proves the uniqueness of certain self-expanders in 3D space.
The paper classifies 2D complete Lagrangian self-expanders in complex 2-space.
problem Classifying complete Lagrangian self-expanders in complex 2-space.
method Obtained a classification theorem.
result A classification of 2D complete Lagrangian self-expanders with constant squared norm of the second fundamental form.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for -dimensional self-expanders.
Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flowmath.DG
We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expa…
A note on the classification of four-dimensional gradient steady and expanding Ricci solitonsmath.DG
Study classifies 4D Ricci solitons with specific curvature conditions.
problem Classifying 4D gradient steady and expanding Ricci solitons with given curvature properties.
method Asymptotically cylindrical and conical assumptions; half-harmonic and half-nonnegative isotropic curvature conditions.
result Partial classification of 4D gradient expanding Ricci solitons with half-nonnegative isotropic curvature.
We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain …
We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat unified approach to construct expanders. We also show that the graphs we obtain …
In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to -invariant co…
Proves uniqueness of small entropy self-expanders.
problem Uniqueness of self-expanders with small entropy.
method Mountain-pass theorem and integer degree argument.
result Proves uniqueness result for self-expanders.
Study self-expanding solutions of mean curvature flow in various dimensions.
problem Characterize complete mean convex self-expanding hypersurfaces and their properties.
method Analyzing the function and to understand the structure of self-expanders.
result Complete mean convex self-expanders are products of self-expanding curves and flat subspaces under certain conditions.