New stability thresholds detect K-stability in Fano manifolds.
problem Detecting K-stability in Fano manifolds.
method Introducing new stability thresholds and studying geodesic rays in Kähler potentials.
result New entropy functional relates to radial entropy functional.
Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.
problem Understanding the homotopy type of stabilizers of smooth functions on surfaces.
method Analyzing the homotopy properties of stabilizers for a specific class of smooth functions.
result The homotopy type of the connected component of the identity map of the stabilizer is completely described for Morse-Bott functions.
Study max- and min-stability under first-order stochastic dominance, finding new functional characterizations.
problem Understanding max- and min-stability in stochastic dominance.
method Representation theorem for functionals satisfying max-stability, combining max- and min-stability to define Lambda-quantiles.
result New characterizations of functionals, including Lambda-quantiles, in finance and political science.
Paper relaxes stability and generalization assumptions for SGD.
problem Stability and generalization for SGD under restrictive assumptions.
method Introduces on-average model stability and develops novel bounds.
result First-ever-known fast bounds in low-noise setting using stability approach.
Homological stability fails for Cremona groups, rational varieties, and function fields.
problem Homological stability in Cremona groups fails in both possible ways.
method Explained the failure of homological stability for Cremona groups.
result Homological stability fails for Cremona groups in both possible ways.
Mathematical approach defines stability conditions for ML models.
problem Ensuring stability of machine learning models.
method Adopted topological and metric spaces theory to define stability.
result Stability of ML models depends on topological properties of classification sets.
The study identifies conditions under which algorithmic stability explains generalization in interpolating learning systems.
problem Understanding when algorithmic stability explains generalization in interpolating learning systems.
method Modeling training as a function-space trajectory and measuring sensitivity to single-sample perturbations.
result There exist interpolating regimes with small risk where contractive sensitivity cannot hold, showing that stability is not a universal explanation.
Stability of mapping spaces is shown to be related to the D-topology.
problem Understanding the relationship between stability and the D-topology of mapping spaces.
method Reformulation and proof of stability theorems in diffeological étale manifolds.
result Stable classes of mapping spaces are D-open.
The paper proves stability for scalar-flat metrics on manifolds with boundary.
problem Stability of scalar-flat metrics on manifolds with boundary.
method Reduced problem to boundary functional and used deficit control.
result Deficit controls distance to minimizing set on manifolds with boundary.
Stabilizes complex systems using diffusion models trained on Lyapunov functions.
problem Generating stabilizing controllers for complex dynamical systems.
method Trains a diffusion model on pairs of asymptotically stable vector fields and their Lyapunov functions to identify the closest stable field and adjust control functions.
result Efficient and rapid stabilization of unseen systems, showcasing generalizability.
The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.
problem Minimizing nonsmooth nonconvex functions with convergence guarantees.
method Developed a framework for stochastic subgradient methods with global stability guarantees.
result Iterates are uniformly bounded and asymptotically stabilize around the stable set of the differential inclusion.
Establishes relationships between prudence and stability properties of risk functionals.
problem Stability properties of risk functionals
method General relationships and preservation of prudence under cash-additive hulls and inf-convolutions
result General methods for constructing prudent risk measures
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.
We study partition functions of random Bergman metrics, with the actions defined by a class of geometric functionals known as `stability functions'. We introduce a new stability invariant - the critical value of the coupling constant - defined as the minimal coupling constant for which the partition function converges.…
Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.
problem Existence of Kähler-Einstein metrics on manifolds with big anticanonical class.
method Developed a theory of Deligne functionals and slope formulas for singular metrics, proving a slope formula for the Ding functional in the big setting.
result Existence of a unique Kähler-Einstein metric implies uniform Ding stability.
We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the …
The paper generalizes K-stability results to singular and weighted settings.
problem Generalizing K-stability to singular and weighted settings.
method Generalization of results in \cite{Li22a} to singular and weighted settings.
result The \(\mathbb{G}\)-uniform weighted K-stability for models implies \(\mathbb{G}\)-coercivity of the weighted Mabuchi functional.
The study examines rigidity and stability of gradient estimates on surfaces and manifolds.
problem Rigidity and stability of gradient estimates for positive harmonic functions and solutions to heat equations.
method Sharp gradient estimates for positive harmonic functions and solutions to heat equations on surfaces and manifolds with nonnegative curvature.
result Obtained rigidity and stability results for gradient estimates.
Paper studies stability of generalized Ricci solitons with mathematical rigor.
problem Stability of generalized Ricci solitons under curvature assumptions.
method Computed second variation formula, proved stability conditions.
result Equivalence of dynamical and linear stability on steady gradient solitons.
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
problem Stability and instability of ALE Ricci-flat metrics.
method Use of a modified Perelman's λ-functional and Lojasiewicz inequality.
result Demonstrates dynamical instability of ALE Ricci-flat metrics.
Wiatowski and Bölcskei, 2015, proved that deformation stability and vertical translation invariance of deep convolutional neural network-based feature extractors are guaranteed by the network structure per se rather than the specific convolution kernels and non-linearities. While the translation invariance result appli…
The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
The study examines stability of specific geometric flows.
problem Stability of Pluriclosed and Generalized Ricci solitons.
method Analyzes the second variation of generalized Einstein--Hilbert functional and infinitesimal deformations.
result Stability of the flows and solitons under specific conditions.
KCRL learns stable policies for nonlinear systems with formal guarantees.
problem Lack of stabilization guarantees in RL methods for safety-critical systems.
method KCRL uses Krasovskii's Lyapunov functions as a stability constraint and a primal-dual approach to learn stabilizing policies.
result KCRL guarantees learning a stabilizing policy in a finite number of interactions.
Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
Stability of Dehn functions proven for ultralimits of Sobolev maps.
problem Stability of Dehn functions under ultralimit of Sobolev maps.
method Using ultralimits of Sobolev maps and properties of ultralimits of Lipschitz maps.
result Stability of Dehn functions under ultraconvergence of pointed length spaces.
Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is…
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
Proves results on K-stability using arcs and Mabuchi functional.
problem K-stability in Fano manifolds and uniform K-polystability.
method Arcs and numerical criterion for stability of pairs.
result Characterizes coercivity of Mabuchi functional in terms of K-polystability.
Paper stabilizes persistent homology rank functions for statistical inference.
problem Stability issues in persistent homology rank functions.
method Derive stability results for rank functions under FDA metrics.
result Rank functions stabilize, improving statistical inference.
Paper improves stability analysis of SGD for various loss functions and data distributions.
problem Improving stability analysis of SGD for non-convex loss functions and data distributions.
method Analyzes stability of SGD for convex and non-convex loss functions, and improves data-dependent bounds.
result Improved stability bounds for non-convex loss functions and convex regularized loss functions.
We present sufficient conditions for topological stability of continuous functions f:R→R having finitely many local extrema with respect to averagings by discrete measures with finite supports.
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…
We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existe…
Hermitian-Einstein metrics linked to stability of bundles on orbifolds.
problem Existence of Hermitian-Einstein metrics on stable vector bundles over compact Kähler orbifolds.
method Equivalence of slope stability to the existence of Hermitian-Einstein metrics and properness of a functional.
result Equivalence of Hermitian-Einstein metrics and slope stability for stable vector bundles.
The identity map of certain Einstein manifolds is stable in both energy and bienergy.
problem Stability of the identity map in Einstein manifolds.
method Investigation of conformal-biharmonic stability compared to harmonic stability.
result The conformal-biharmonic index coincides with the harmonic index, except for the 4D Euclidean sphere.
New cyclicity measures defined in weighted Besov spaces, with stability and geometric analysis.
problem Characterizing cyclicity in weighted Besov spaces.
method Defining cyclicity indices based on potential theory and capacity, studying stability under perturbations, and linking zero set structure to cyclicity.
result Novel invariants and conditions for cyclicity in various function spaces.
The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney C∞-topology). We show that a Morse function is stable if it is end-trivial at any point in its discriminant, where end-triviality (which is also called local triviality …
We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of non-compact real reductive Lie groups on topological spaces that admit functions similar to the Kempf-Ness function. The point of this…
Study shows how close functions are to optimal in Riemannian manifolds.
problem Understanding how close functions are to optimal in Riemannian manifolds.
method Analyzes quantitative stability of Sobolev inequalities on compact Riemannian manifolds.
result Functions that nearly saturate a critical Sobolev inequality are quantitatively close to extremal functions.
New algorithms learn stability certificates from data, avoiding complex dynamics.
problem Synthesizing stability certificates from complex dynamical systems.
method Developed algorithms to learn certificate functions from trajectory data, establishing generalization error bounds.
result Efficiently learned certificates can be used for adaptive control.
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.
In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of F-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…
Adaptive optimal control using value iteration (VI) initiated from a stabilizing policy is theoretically analyzed in various aspects including the continuity of the result, the stability of the system operated using any single/constant resulting control policy, the stability of the system operated using the evolving/ti…
Extends fractional Lp uncertainty principles with extremizers and stability results.
problem Investigating uncertainty principles in fractional Lp settings. method Analyzing the fractional Schrödinger equation to find extremal functions and sharp constants.
result Proves stability of extremizers for fractional uncertainty inequalities.
The study characterizes and studies stability of biharmonic hypersurfaces in complex space forms.
problem Characterizing and studying biharmonic hypersurfaces in complex space forms.
method Characterizing hypersurfaces as critical points of a higher order energy functional.
result Existence and non-existence results for CPn and CHn. Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.
problem Optimal functional inequalities and their stability.
method Compactness theorems, characterization of optimizers, and quantitative stability analysis.
result Characterization and stability of optimizers for Sobolev inequalities and their fractional generalizations.