Stabilization operation for high-dimensional contact manifolds, proving many links are non-simple.
problem Understanding the structure and properties of high-dimensional contact manifolds.
method Definition and proof of stabilization operation for codimension 2 contact submanifolds in dim≥5 contact manifolds. result Many transverse links are non-simple.
The paper finds new eigenfunctions for minimal immersions and their stability index.
problem Finding new eigenfunctions for minimal immersions and their stability index.
method Explicitly showed new eigenfunctions for the stability operator.
result The stability index of minimal immersions is at least kℓ+3k+3ℓ+8. Study stability of operators on warped product manifolds.
problem Stability of operators on warped product manifolds.
method Examined the family of operators La=Δ−aS in a warped product of an infinite interval or real line by a compact manifold. result Stability of the operators La was studied in a specific type of manifold. Wave operators and spectral stability for Dirac operators under Ricci flow.
problem Stability of the absolutely continuous spectrum of Dirac operators under Ricci flow.
method Proving existence and completeness of wave operators for Dirac operators and their squares under Ricci flow.
result Criterion for spectral stability of Dirac operators and their squares under Ricci flow without injectivity radius assumptions.
We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.
Proves stability of gravitational instantons, proving operator positivity.
problem Stability of gravitational instantons.
method Riemannian analog of black hole mode stability for Hermitian, non-self-dual gravitational instantons.
result Teukolsky equation is a positive definite operator on Hermitian, non-self-dual gravitational instantons.
Study shows stability in X-ray transform on specific hyperbolic manifolds.
problem Stability of X-ray transform on asymptotically hyperbolic manifolds.
method Constructed a parametrix for the normal operator in 0-pseudodifferential calculus.
result Showed a stability estimate for the X-ray transform.
The study refines stability results for Yang-Mills fields and harmonic maps.
problem Stability of Yang-Mills fields and harmonic maps.
method Refinement of stability results using Jacobi operator over S^m.
result Refined stability results and Morse index estimates.
Study on catenoid stability using asymmetric potentials.
problem Stability of catenoids and related potentials.
method Exact solvable one-dimensional Schrödinger operator with asymmetric Darboux-Pöschl-Teller potential.
result Explicit construction of the unstable mode of the inner catenoid.
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.
The paper studies eigenvalues and stability of hypersurfaces in spheres.
problem Finding eigenvalues and stability of hypersurfaces in spheres.
method Derives equations for mean curvature and uses numerical methods to compute eigenvalues.
result Numerical computation of eigenvalues and stability indices for specific hypersurfaces.
SAM improves generalization by operating near the edge of stability.
problem Improving generalization in neural networks.
method Sharpness-Aware Minimization (SAM) approach to training neural networks.
result SAM operates near the 'edge of stability' identified by the analysis.
The paper defines and proves stabilization for 3-manifold decompositions with multibranched surface intersections.
problem Decomposing 3-manifolds with more than 3 handlebodies and multibranched surface intersections.
method Definition and proof of stabilization operations for these decompositions.
result Stable equivalence of handlebody decompositions with multibranched surface intersections.
Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.
problem Analyzing light ray transform in pseudo-Euclidean space.
method Investigate normal operator, derive inversion formula, analyze as Fourier Integral Operator.
result Derive an inversion formula and prove stability estimates.
Guillarmou extends X-ray transform to magnetic and thermostat flows.
problem Stability of magnetic X-ray transforms.
method Generalizes normal operator to thermostat and magnetic flows, proving ellipticity.
result Elliptic pseudodifferential operators of order -1 for generalized normal operators.
Single stabilization is not always enough to make exotic surfaces isotopic.
problem Determining if a single stabilization is sufficient to make exotic surfaces isotopic.
method Study of stabilization distance with satellite operations using Floer theoretic techniques.
result Found examples of exotic disks in the four-ball with arbitrarily large stabilization distance.
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.
Study on stability of minimal submanifolds in specific Einstein manifolds.
problem Investigating stability of minimal submanifolds in Einstein manifolds.
method Analyzing homogeneous minimal hypersurfaces in Page space and Sasaki-Einstein manifolds, computing stability operators and indices.
result Determined all homogeneous, minimal hypersurfaces and computed their stability operators and indices.
We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to r-stability as well as to almost-Einstein hypersurfaces.
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…
Unified treatment of stability problems in geometry and analysis.
problem Spherical closeness of hypersurfaces under geometric constraints.
method Estimate relating distance to geodesic spheres with norms of traceless Hessian operator.
result Unified treatment of stability problems in geometry and analysis.
Study on stability of 3D sessile drops, identifying degenerate kernel.
problem Linear stability of three-dimensional sessile drops with a free contact line.
method Derived constrained second variation, formulated Jacobi problem, combined geometric and Fourier analysis.
result Kernel of the constrained Jacobi operator is exactly the space of horizontal translations under pressure-volume nondegeneracy.
Flexible approach for normal approximations in geometric and topological statistics.
problem Normal approximation for complex statistics not expressible as sums of score functions.
method Flexible add-one cost operator combined with strong stabilization theory.
result Established normal approximation results for geometric and topological statistics.
Study on sine-cones' spectra and stability under Ricci-de Turck flow.
problem Analyzing stability and rigidity of sine-cones.
method Computed spectra of specific operators on sine-cones.
result Conditions for sine-cones' dynamic stability and rigidity.
Noise-robust Koopman operator framework for control with improved stability and performance.
problem Developing a stable and noise-robust Koopman operator for control tasks.
method Proposes a learning framework using Hankel matrix and neural network approximations for system dynamics, ensuring long-term stability and noise robustness.
result Demonstrates improved model performance and noise robustness in control tasks compared to existing methods.
Small deformations of marginally (outer) trapped surfaces are considered by using their stability operator. In the case of spherical symmetry, one can use these deformations on any marginally trapped round sphere to prove several interesting results. The concept of 'core' of a black hole is introduced: it is a minimal …
Stability of hypersurface immersions in Riemannian manifolds proved for Lp perturbations.
problem Stability of isometric immersions of hypersurfaces in Riemannian manifolds under Lp perturbations of their fundamental forms. method Young measure approach, relaxation of energy, regularity result for immersions.
result Sequence of immersions converges to an isometric immersion with the reference shape operator.
We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a re…
Gradient descent on neural nets often operates at the Edge of Stability, where loss behavior is complex but loss decreases over time.
problem Understanding the optimization dynamics of neural networks at the Edge of Stability.
method Empirical demonstration of gradient descent behavior in neural network training.
result Gradient descent on neural networks typically occurs at the Edge of Stability, where loss behavior is non-monotonic but loss decreases over time.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
Study on stability of Sasaki structures under deformations.
problem Stability of Sasaki structures under transverse holomorphic deformations.
method Analysis of transverse Kähler holonomy groups and stability properties.
result Stability of ${\oldmathcal S}$ under certain conditions on Sasaki 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.
The paper explores stability and generalization of deep GCNs.
problem Understanding the stability and generalization of deep GCNs from a theoretical perspective.
method Theoretical analysis of stability and generalization properties of deep GCNs.
result The stability and generalization of deep GCNs are influenced by the maximum absolute eigenvalue of the graph filter operators and the depth of the network.
The paper refines the stability index for a specific minimal hypersurface and verifies Yau's conjecture.
problem Stability of minimal hypersurfaces in spheres and eigenvalue multiplicity.
method Analytical and numerical methods to study eigenvalues and stability indices.
result The multiplicity of the eigenvalue for the Carlotto-Schulz minimal embedding is at least 2n+1+n^2.
New framework allows selective removal of stale data in option calibration.
problem Inability to remove old data from calibrated option pricing models without full retraining.
method Introduces operator-theoretic Gauss-Newton framework for selective forgetting.
result Provides stability guarantees and perturbation bounds for selective data removal.
We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the strong stability of such surfaces is studied. We also characterize constant mean curvature Hopf tori as the only ones att…
The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.
problem Classifying decompositions of 3-manifolds with handlebodies.
method Studied decompositions of 3-sphere and lens spaces with three handlebodies, using stabilizations.
result Determined whether decompositions are stabilized.
This is a survey of recent results on zeta- and eta-function poles and values for realizations of Laplace- and Dirac-type operators defined by pseudodifferential projection boundary conditions (including the Atiyah-Patodi-Singer operator and its square). Section 1 recalls some useful results for ps.d.o.s on closed mani…
Neural operators approximate Stackelberg game solutions.
problem Intractability of follower's best-response operator in dynamic Stackelberg games.
method Used attention-based neural operators to approximate the best-response operator.
result Approximate best-response operator yields close game value.
Paper introduces reinforcement learning for managing power grids.
problem Balancing power flows and maintaining grid stability in real-time.
method Reinforcement Learning applied to power network operations.
result Demonstrates feasibility of machine learning in power grid management.
Handles decompositions reveal new open book structures.
problem Understanding and manipulating open book structures on manifolds.
method Build handle decompositions and describe handle slides.
result Stabilization of open books to trivial monodromy.
We prove that the completed cohomology groups of SL_N(Z) in fixed degree stabilize as N goes to infinity. We also prove that the action of Hecke operators on stable cohomology is trivial, in a precisely defined sense.
ARBITER learns SPX-VIX term structures without arbitrage constraints.
problem Arbitrage-free modeling of SPX-VIX term structures.
method Risk-neutral neural operator mapping market states to operator outputs enforcing static arbitrage constraints.
result ARBITER outperforms other models in derivatives term structure evaluation metrics.
Paper calculates Morse index of Y-singular minimal surfaces.
problem Computing Morse index of Y-singular minimal surfaces.
method Utilized two simpler problems: fixed boundary problem and Dirichlet-to-Neumann map.
result Index of Y-catenoid is one.
We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G_2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on…
New method creates vacuum data at minimal and borderline decay thresholds.
problem Creating vacuum initial data at specific decay thresholds.
method Conical solution-operator method applied to vacuum asymptotically flat initial data.
result Demonstrates global and exterior stability of Minkowski spacetime.
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.
Local Neural Operators enable efficient system-level analysis of complex PDEs.
problem System-level analysis of large-scale dynamical systems using neural operators.
method Integrating local Neural Operators with Krylov subspace iterative methods for stability and bifurcation analysis.
result Demonstrated effectiveness of local Neural Operators in fixed-point, stability, and bifurcation analysis of nonlinear PDEs.