New rigidity result for metrics with positive scalar curvature and specific decay.
problem Understanding metrics with positive scalar curvature and C0 decay. method Analyzing metrics with non-negative scalar curvature and C0 decay properties. result Metrics with specific decay properties must be flat.
Paper proves Hamilton-Tian conjecture using partial C0-estimate.
problem Existence of Kähler-Einstein metrics on Fano manifolds.
method Using Liu-Székelyhidi's partial C0-estimate for polarized Kähler metrics with Ricci bounded below.
result Proves Hamilton-Tian conjecture for Kähler-Ricci flow.
Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.
problem Approximating smooth 2-tori in high-dimensional spaces.
method Polyhedral approximation using Lagrangian and isotropic tori.
result Smooth 2-tori can be approximated by polyhedral Lagrangian or isotropic tori in C0 or C1 sense.
Proves some flat spacetimes can't be extended smoothly.
problem Proving smooth extension of certain FLRW spacetimes.
method Utilizing Sbierski's C0-inextendibility techniques. result Proves C0-inextendibility of spatially flat FLRW spacetimes. We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph on N vertices and with connection probability p0; under the …
A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
In this paper we study the stochastic evolution equation (1.1) in martingale-type 2 Banach spaces (with the linear part of the drift being only a generator of a C0-semigroup). We prove the existence and the uniqueness of solutions to this equation. We apply the abstract results to the Heath-Jarrow-Morton-Musiela (HJMM)…
Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.
problem Solving the quaternionic Monge-Ampère equation for (n−1)-quaternionic plurisubharmonic functions on a hyperKähler manifold. method Proves existence and uniqueness of solutions using a Cherrier-type inequality and C1 and C2 estimates. result Obtains smooth solutions to the quaternionic Monge-Ampère equation.
Study shows a mass quantity for C0 metrics that agrees with ADM mass.
problem Understanding ADM mass for C0 metrics and its behavior under Ricci-DeTurck flow. method Developed a C0 mass quantity and analyzed its behavior under Ricci-DeTurck flow. result The C0 mass at infinity is independent of coordinate charts and has controlled distortion under Ricci-DeTurck flow. Simplified proof and new C0 estimate for Kähler-Einstein metrics.
problem Existence of Kähler-Einstein metrics on Calabi-Yau manifolds.
method Alternative C0 a priori estimate for the Monge-Ampère equation. result Established a new uniform bound for the solution of the Monge-Ampère equation.
The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.
problem Proving the existence of nodal solutions to a specific type of partial differential equation.
method First, a C0−estimate for positive f-invariant solutions is proven. Then, the existence of mountain pass solutions with arbitrarily large energy is established. result The existence of infinitely many nodal solutions to the equation Δ2u−αΔu+βu=uq is proven. We propose a simple change to existing neural network structures for better defending against gradient-based adversarial attacks. Instead of using popular activation functions (such as ReLU), we advocate the use of k-Winners-Take-All (k-WTA) activation, a C0 discontinuous function that purposely invalidates the neural …
Box Thirding identifies the best arm efficiently under limited samples.
problem Efficiently identifying the best arm with limited sampling.
method Iterative ternary comparison of arms, discarding the weakest and exploring the best.
result Achieves comparable performance to Successive Halving with less predefined parameters.
Extends Minkowski stability proof to minimal decay assumptions.
problem Global stability of Minkowski spacetime with minimal decay.
method Extends Christodoulou-Klainerman's proof to minimal decay assumptions.
result Exterior stability of Minkowski holds with borderline decay.
Study shows Hamiltonian diffeomorphisms form a connected component in C0-topology for most symplectic rational surfaces.
problem Understanding the C0-topology of symplectic diffeomorphisms on rational surfaces. method Combining techniques from symplectic mapping class groups and C0-symplectic topology, establishing C0-distance estimates. result Hamiltonian diffeomorphisms form a connected component in C0-topology for all but a few exceptions on rational surfaces. 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.
Unique solutions found for wave-like decaying null infinity equations.
problem Wave-like decaying null infinity equations with spherically symmetric Einstein-scalar-field.
method Local and global unique solutions for small initial data.
result Sharp decaying condition for unique solutions.
New findings on flatness of certain metrics with fast decay.
problem Rigidity of positive mass theorem under fast metric decay.
method Considered metrics with nonnegative scalar curvature and rapid decay at infinity.
result Any such metric is necessarily flat in dimensions 4 and higher if decay rate exceeds Schwarzschild metric.
Study on curvature decay in steady Ricci solitons, proving dichotomy.
problem Curvature decay in steady Ricci solitons.
method Established a dichotomy for curvature decay in specific types of solitons.
result Proved a dichotomy on curvature decay for certain steady Ricci solitons.
Cautious Weight Decay modifies weight decay for better optimization.
problem Improving optimization in deep learning models.
method Applies weight decay selectively based on parameter sign alignment.
result Consistently improves model performance across various tasks and scales.
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
Study on massless Vlasov equation on Reissner-Nordström spacetimes, showing decay rates and non-decay phenomena.
problem Analyzing decay and non-decay rates of solutions to the massless Vlasov equation on Reissner-Nordström spacetimes.
method Quantitative analysis of geodesic flow and comparison to wave equation instability results.
result Exponential decay rates in subextremal cases and polynomial rates in extremal cases, with non-decay of transversal derivatives in extremal cases.
Introduces gradient decay in Softmax for better generalization.
problem Improving generalization performance in neural networks.
method Gradient decay hyperparameter in Softmax for varying gradient rates based on probability.
result Gradient decay rate affects generalization performance and can be tuned for better optimization.
Study examines wave equation decay and Strichartz estimates on conic manifolds.
problem Analyzing wave equation behavior on conic spaces with critical electromagnetic potentials.
method Established decay and Strichartz estimates through localized spectral measure construction.
result Extended and improved previous results on wave equation behavior with critical potentials.
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
Weight decay stabilizes training dynamics by slowing progressive sharpening.
problem Understanding how weight decay affects training stability in deep learning models.
method Analyzing weight decay effects at the Edge of Stability, developing a mathematical framework.
result Weight decay dampens oscillations and stabilizes sharpness in CNNs, causing a phase transition in MLPs.
Polynomial decay of correlations shown for curved surfaces.
problem Analyzing geodesic flows on curved surfaces.
method Proving polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
result Polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
The paper examines robustness of topological entropy in geodesic flows.
problem Entropy robustness in geodesic flows under C0 perturbations. method Study of topological entropy on Riemannian metrics with C0 topology. result Metrics with contractible closed geodesics have robust entropy.
Last SGD iterate bounds for overparameterized linear regression.
problem Analyzing the last iterate risk bounds of SGD with decaying stepsize for overparameterized linear regression.
method Problem-dependent analysis of last iterate risk bounds of SGD with geometrically decaying stepsize.
result Proved nearly matching upper and lower bounds on the excess risk for last iterate SGD with geometrically decaying stepsize.
Optimal learning rates decay to zero in easy tasks and maintain a warmup phase in hard tasks.
problem Optimizing learning rates under functional scaling laws for model training.
method Deriving optimal learning-rate schedules based on exponents s and β. result Sharp phase transition between easy and hard tasks, with different decay behaviors.
We found that factors decay over time, with momentum fitting best.
problem Understanding how factors decay over time and their impact on performance.
method Derived a hyperbolic decay model for factors, tested against linear and exponential alternatives.
result Momentum exhibits hyperbolic decay, outperforming linear and exponential models.
The paper predicts and explains the decay of stock anomaly performance over time.
problem Predicting and explaining the drop in risk-adjusted performance of stock anomalies.
method The authors propose ex-ante characteristics based on hypotheses of out-of-sample decay and in-sample overfitting.
result The year of publication explains 30% of the variance in Sharpe decay across factors.
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadractic curvature decay and some properties of the asymptotic cones of such a manifold.
The paper extends decay estimates to graphs with positive spectrum.
problem Proving decay estimates for nonnegative functions on graphs.
method Sharp ℓ2 decay estimates for nonnegative generalized subharmonic functions. result Extends Li and Wang's result to graphs with positive Laplacian spectrum.
The paper studies harmonic map heat flow stability and decay rates.
problem Analyzing stability and decay rates of harmonic map heat flow solutions.
method Use of homogeneous Besov space B˙p,∞pd(Rd) for small initial data and self-similar decay assumption. result Decay rates for solutions of the harmonic map flow of the form ∥ablau(t)∥L∞(Rd)≤Ct−21 and self-similar decay under stronger initial conditions. In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional κ-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.
Extends global stability of Minkowski spacetime to minimal decay assumptions.
problem Global stability of Minkowski spacetime under minimal decay assumptions.
method Uses rp-weighted estimates instead of vectorfield method. result Proves global stability of Minkowski spacetime under minimal decay assumptions.
SHIFT framework identifies subgroups with large ML model performance decay.
problem Large model performance decay in subgroups when deployed.
method Subgroup-scanning Hierarchical Inference Framework (SHIFT) for performance drift.
result SHIFT identifies interpretable subgroups with large performance decay and suggests targeted actions to mitigate it.
New Sobolev inequalities found for curved spaces.
problem Sobolev inequalities in curved spaces with specific decay conditions.
method Used ABP method developed by Cabré and Brendle.
result Established Sobolev inequalities for compact domains and submanifolds.
New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.
Study on optimal ReLU networks with weight decay for interpolation.
problem Interpolating data with radially symmetric distributions using shallow ReLU networks.
method Weight decay regularization in infinite neuron, infinite data limit; analysis of growth rates.
result Existence and growth rates of unique radially symmetric minimizers with weight decay.
We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such s…
Sharp decay constant for positive scalar curvature metrics on manifolds.
problem Finding the optimal decay constant for positive scalar curvature metrics.
method New exhaustion result using μ-bubbles.
result Decay constant of 2/3 is sharp and necessary for complete metrics.
Novel Adam-family method with decoupled weight decay for training neural networks.
problem Training nonsmooth neural networks with weight decay.
method Proposes a novel Adam-family method with decoupled weight decay, establishing convergence properties and demonstrating superior performance.
result Asymptotically approximates SGD and enhances generalization performance.
Constructs index for elliptic operators using rapidly decaying kernels.
problem Index of elliptic operators in Fréchet algebra.
method Uses heat operators and heat kernel asymptotics.
result Index can be represented by an idempotent involving heat operators.
Regularization in the optimization of deep neural networks is often critical to avoid undesirable over-fitting leading to better generalization of model. One of the most popular regularization algorithms is to impose L-2 penalty on the model parameters resulting in the decay of parameters, called weight-decay, and the …
Study shows decay of correlations on specific types of flows.
problem Analyzing decay of correlations in specific flow types.
method Asymptotic expansion of correlation function on Abelian covers.
result Established an expansion in inverse powers of time.