Triangulates permutahedra for Coxeter groups, revealing braid group connections.
problem Triangulating permutahedra for Coxeter groups.
method Constructs triangulations using total linear stability and height functions.
result Explicitly relates two braid group presentations.
One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds M+,M− introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symm…
New findings allow infinite mean intensity Hawkes processes to be stable.
problem Stability condition for Hawkes processes with infinite mean intensity.
method Analysis of Quadratic Hawkes processes with infinite mean intensity.
result Quadratic Hawkes processes are always stationary with infinite mean intensity when total endogeneity ratio exceeds unity.
New estimates show all stable Einstein manifolds are linear stable with respect to Perelman's ν-entropy.
problem Estimating the smallest eigenvalue of Laplace-Beltrami operator for stable Einstein manifolds.
method Estimating the smallest positive eigenvalue λ1 of the Laplace-Beltrami operator for standard Einstein manifolds (G/H,gst) and proving λ1>2E for all but 7 exceptions. result All stable Einstein manifolds found by Schwahn are linear stable with respect to Perelman's ν-entropy.
It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equa…
The study examines how twisting a knot affects its homology and stability properties.
problem Investigating the impact of twisting a knot on its homology and stability.
method Use bordered Floer homology and immersed curve invariants.
result Total dimension, τ(K_m), and thickness of K_m are linear functions of m for large m.
Study on stability of geodesic maps in non-isotropic manifolds.
problem Stability of totally geodesic wave maps in non-isotropic manifolds.
method Factorization property, PDE system in geodesic normal coordinates, global existence result via hyperboloidal foliation.
result Established global existence for small initial data, leading to geometric stability.
This work proves Kerr black holes are dynamically stable under certain perturbations.
problem Dynamical stability of Kerr black holes under axially symmetric perturbations.
method Dimensional reduction to 2+1 Einstein-wave map system, construction of positive-definite energy functional, proving boundary terms vanish.
result Strictly conserved positive energy for axially symmetric linear perturbations of Kerr black holes.
Geometrically classifies total stability spaces for Dynkin diagrams.
problem Classifying total stability spaces for triangulated categories.
method Constructing a geometric model of root categories as hQ-gons and proving isomorphisms. result Total stability spaces ToStDb(Q)/[2] are isomorphic to moduli spaces of stable hQ-gons. The study preserves lower bounds of total scalar curvature under specific metric convergence.
problem Preserving lower bounds of total scalar curvature on smooth manifolds.
method Used stability of Ricci flow and heat flow with Ricci flow background.
result Lower bound of weighted total scalar curvature is preserved under specified convergence conditions.
Stability of positive mass theorem for hyperbolic manifolds studied.
problem Stability of the positive mass theorem for asymptotically hyperbolic manifolds.
method Adapted intrinsic flat distance approach to show stability for a class of manifolds.
result Stability of the positive mass theorem for a class of asymptotically hyperbolic graphical manifolds.
The paper examines the stability of Killing cylinders in hyperbolic space.
problem Stability of Killing cylinders in hyperbolic space.
method Explicit computation of Morse index for Jacobi operator on various support surfaces.
result Delaunay surfaces can be bifurcated from Killing cylinders supported on geodesic planes.
Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.
problem K-stability of Calabi-Yau fibrations over curves.
method Adiabatic uniform K-stability and log-twisted K-stability of base curves.
result Uniform K-stability of Calabi-Yau fibrations if and only if base curves are K-stable.
New algorithms help machines forget old data efficiently.
problem Machine learning models can retain old data, hindering new learning.
method Developed TV-stable algorithms based on noisy SGD for convex and non-convex functions.
result Achieved efficient unlearning with upper and lower bounds on risk.
The paper examines stable capillary hypersurfaces in hyperbolic space.
problem Stability of capillary hypersurfaces with free boundary on a horosphere.
method Analysis of umbilical and totally geodesic hypersurfaces using stability criteria.
result Umbilical and totally geodesic hypersurfaces are the only stable capillary hypersurfaces with boundary on a horosphere.
The study classifies and characterizes totally symmetric sets in the general linear group.
problem Understanding the structure and properties of totally symmetric sets in the general linear group.
method Formulated a notion of irreducibility for totally symmetric sets in the general linear group and classified them.
result Classification of irreducible totally symmetric sets and those of maximal cardinality.
We can talk about two kinds of stability of the Ricci flow at Ricci flat metrics. One of them is a linear stability, defined with respect to Perelman's functional F. The other one is a dynamical stability and it refers to a convergence of a Ricci flow starting at any metric in a neighbourhood of a considere…
Study local properties of Chern-scalar curvature through linearization stability.
problem Local properties of Chern-scalar curvature function.
method Linearization analysis of the Chern-scalar curvature function.
result Stability of linearization and structure of metrics with prescribed curvature.
The paper improves SVM and localized SVM stability under triple perturbations.
problem Stability of SVMs and localized SVMs under triple perturbations.
method Generalizes and improves existing results, considering simultaneous variations in probability measure, regularization parameter, and kernel.
result Improved stability of SVMs and localized SVMs under triple perturbations.
The study bounds the stability of Gaussian mixtures under small perturbations.
problem Stability of Gaussian mixtures under small changes in distribution.
method Deriving an explicit bound on parameter stability of spherical Gaussian Mixture Models (sGMM) in a pre-defined model class.
result Upper bound on parameter distance of close sGMMs to the original sGMM, dependent only on the original model.
Study on totally symmetric sets with group applications.
problem Understanding totally symmetric sets and their group applications.
method Survey of existing theory and applications to various groups.
result Exploration of totally symmetric sets in multiple group contexts.
Study on sample complexity of policy gradient for stabilizing linear systems under multiplicative noise.
problem Learning optimal feedback gain for stabilizing linear systems with multiplicative noise.
method Analyzes the sample complexity of policy gradient methods, addressing the cusp obstruction and using symmetry to control divergent parts of the gradient.
result Proves that projected mini-batch policy gradient attains total sample complexity of O(1/η) when noise density is known and O(η^(-(2s+1)/(2s))) when estimated, for C^s noise densities with s ≥ 2.
Proofs and descriptions of totally geodesic submanifolds in symmetric spaces.
problem Classifying totally geodesic submanifolds in symmetric spaces.
method Independent proof and descriptions using algebraic and geometric properties.
result Natural descriptions and classifications of totally geodesic submanifolds.
The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.
problem Linear stability of Gradient Ricci Shrinker metrics.
method Found commutator formulas and generalized a stability theorem.
result Generalized a necessary condition for linear stability.
We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear stability of such gradient Ricci solitons.
The positive mass theorem states that the total mass of a complete asymptotically flat manifold with non-negative scalar curvature is non-negative; moreover, the total mass equals zero if and only if the manifold is isometric to the Euclidean space. Huang and Lee [2015] proved the stability of the Positive Mass Theorem…
The paper studies rigidity and stability of minimal submanifolds in hyperbolic space.
problem Conditions for a minimal submanifold to be totally geodesic.
method Analyzes the length of the second fundamental form and eigenvalues of the super stability operator.
result Sharp upper bounds for the first eigenvalue of the super stability operator for surfaces in hyperbolic 4-space.
The study examines spacelike foliations on Lorentz manifolds under specific conditions.
problem Investigating geometric properties of spacelike foliations on Lorentz manifolds.
method Analyzing conditions for stability, total geodesy, and total umbilicity of foliation leaves.
result Conditions for the stability, total geodesy, and total umbilicity of spacelike foliation leaves are established.
Random forest predicts catastrophe bond spreads with 93% accuracy.
problem Predicting spreads in the primary catastrophe bond market.
method Random forest approach using all information in offering circulars.
result Random forest explains 93% of spread variability, significantly better than linear regression (47%).
The paper studies stability of domains for the first eigenvalue on Riemannian manifolds.
problem Stability of extremal domains for the first eigenvalue of the Laplacian operator.
method Second variation of the first Dirichlet eigenvalue, stability criterion, classification of stable domains.
result Classification of stable extremal domains in spheres and topological bounds for general compact surfaces.
The conformal properties of complex Finsler metrics are studied. We give a characterization of a compact complex Finsler manifold to be globally conformal Kähler. The critical points of the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes are studied. The stability of crit…
Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with resp…
In this short note we extend an estimate due to J. Simons on the first stability eigenvalue of minimal hypersurfaces in spheres to the singular setting. Specifically, we show that any singular minimal hypersurface in Sn+1, which is not totally geodesic and satisfies the α-structural hypothesis, has first stability…
Proves stability of Einstein metrics on specific symmetric spaces.
problem Stability of Einstein metrics on symmetric spaces of compact type.
method Linear stability analysis of the Einstein-Hilbert action.
result Resolves stability problem for irreducible symmetric spaces of compact type.
Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative …
We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known ex…
We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compac…
New findings show score matching's accuracy doesn't ensure numerical stability in diffusion sampling.
problem Numerical stability issues in diffusion sampling despite small forward-marginal error.
method Constructing a smooth score field with arbitrarily small forward-marginal L2 error, showing nonexplosive behavior and moments of every order. result Euler--Maruyama discretizations can converge in probability even when moments diverge, demonstrating failure of weak convergence.
Study stability of Einstein manifolds with boundary.
problem Stability of Einstein manifolds with geometric boundary conditions.
method Using Ricci flow and calculus of variations, analyze stability with respect to the Einstein-Hilbert action.
result Introduce a new subspace of tensors (TVg tensors) for stability condition due to boundary constraints.
Stable GFlowNets prevent loss spikes and mode collapse in training.
problem Unstable training of GFlowNets leading to loss spikes and mode collapse.
method Assessed sensitivity of GFlowNet objectives, derived loss-to-TV bounds, and proposed Stable GFlowNets.
result Stable GFlowNets improve training behavior and distributional fidelity.
Study shows exponential sample complexity for stabilizing certain linear systems.
problem Statistical hardness of learning to stabilize linear time-invariant systems.
method Analysis of sample complexity and co-stabilizability using robust control ideas.
result Sample complexity increases exponentially with system dimension.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
Homological stability aids in computing group homology.
problem Computing homology of families of groups.
method Proving homological stability theorems and computing stable homology.
result Computation of Higman-Thompson groups' homology.
Paper proves zero stability for one-row colored sl₃-Jones polynomials.
problem Stability of coefficients in colored Jones polynomials.
method Linear skein theory based on Kuperberg's sl₃-webs.
result Zero stability for B-adequate links with anti-parallel twist regions.
New proof of Schwarzschild stability using geometric gauge.
problem Linear stability of Schwarzschild spacetime under gravitational perturbations.
method Employing a new geometric gauge and exploiting the structure of transport equations.
result Established both orbital and asymptotic stability for linearised quantities.
Non-Archimedean balanced metrics approximate cscK metrics for totally degenerate abelian varieties
problem Non-Archimedean balanced metrics for polarized abelian varieties
method Non-Archimedean analogue of the cscK metric
result Uniform estimate for Calabi-Yau metrics on fibers
Estimates parameters of interconnected linear systems using total variation penalization.
problem Joint estimation of parameters in interconnected linear dynamical systems.
method Total variation penalized least-squares estimator.
result The MSE goes to zero as the number of systems increases, even with constant trajectory length.
We prove the global non-linear stability, without symmetry assumptions, of slowly rotating charged black holes in de Sitter spacetimes in the context of the initial value problem for the Einstein-Maxwell equations: If one perturbs the initial data of a slowly rotating Kerr-Newman-de Sitter (KNdS) black hole, then in a …