Future stability of FLRW solutions in expanding 3D space is shown for compact perturbations.
problem Future stability of expanding FLRW solutions with spatial topology R^3.
method Nonlinear stability analysis of spherically symmetric perturbations.
result Decay rates of energy momentum tensor components compared to Minkowski space.
MSFA clusters high-dimensional spatial data using spline-based covariance structures.
problem Clustering high-dimensional spatial data with flexible covariance structures.
method Mixture of spatial factor analyzers with spline-based covariance and matrix variate factor analyzers for dimensionality reduction.
result Proposed models accurately infer and differentiate distinct spatial patterns in tensor-variate data.
The 2008 financial crisis revealed banking consolidation paradoxically increased systemic fragility and global financial contagion with negligible spatial decay.
problem Fundamental vulnerabilities in interconnected banking systems during the 2008 financial crisis were inadequately addressed by existing frameworks.
method Developed a unified spatial-network framework using spectral analysis of network Laplacian operators combined with spatial difference-in-differences identification.
result Banking consolidation paradoxically increased systemic fragility and global financial contagion with negligible spatial decay.
New method finds precise late-time behavior of wave equations.
problem Analyzing late-time behavior of wave equations with inverse-square potentials.
method Physical-space-based method for deriving late-time asymptotics.
result Sharp, uniform decay estimates in time for asymptotic late-time tails.
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. We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.
Study on infinitely-wide CNNs and their adaptability to function spatial scales.
problem Understanding how CNNs efficiently learn high-dimensional functions and their adaptability to function spatial scales.
method Study infinitely-wide deep CNNs in the kernel regime, characterizing their spectrum and using generalisation bounds to prove adaptability.
result Deep CNNs adapt to the spatial scale of the target function, with error decay controlled by the effective dimensionality of function subsets.
In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at spatial infinity of an asymptotically flat initial data set. Similar to the small sphere limit of the Wang-Yau quasi-local mass, we prove that the leading order term of the quasi-local mass recovers the stress-energy tensor. For a va…
Diffusion Transformer captures spatial-temporal dependencies in sequential data.
problem Capturing rich spatial and temporal dependencies in sequential data.
method Established theoretical guarantees for diffusion transformers learning Gaussian process data.
result Spatial-temporal dependencies are captured within attention layers of diffusion transformers.
Stability proved for open Milne spacetime, showing gravity's long-term behavior.
problem Global stability of open Milne spacetime for Einstein-scalar field equations.
method Gaussian normal coordinates, exploiting expanding geometry of Milne spacetime.
result Spatial metric tends to hyperbolic metric as time goes to infinity.
Stability of catenoid in 4D Minkowski space proven without symmetry assumptions.
problem Stability of the catenoid as a nonflat stationary solution to the HVMC equation in 4D.
method Established asymptotic stability under codimension-1 assumption, using new commutator vector field estimates.
result Catenoid stability in 4D proven without symmetry assumptions, with improved pointwise decay.
A cubing strategy identifies stable hyperparameter regions for uncertainty quantification in spatial deep learning.
problem Uncertainty quantification in spatial deep learning models.
method Cubing-based diagnostic framework to recursively partition hyperparameter space and evaluate regions using scoring rules.
result Our approach produces competitive or superior predictive intervals compared to a statistical baseline model.
Kulldorff's (1997) seminal paper on spatial scan statistics (SSS) has led to many methods considering different regions of interest, different statistical models, and different approximations while also having numerous applications in epidemiology, environmental monitoring, and homeland security. SSS provides a way to …
New proof shows perturbed non-compact Einstein spaces attract to unique global solution.
problem Proving global solutions for perturbed non-compact negative Einstein spaces.
method Developed energy estimates for a hyperbolic system of Maxwell type.
result Global unique solution for perturbed non-compact negative Einstein spaces.
Proposes a new model for complex multivariate event data.
problem Modeling complex multivariate event data with spatio-temporal dynamics.
method Integrates spatial information into latent state evolution through learned temporal and spatial decay dynamics.
result Successfully recovers sensible temporal and spatial intensity structure in multivariate spatio-temporal point patterns.
The paper proves stability of certain cosmological models with negative spatial curvature.
problem Stability of Friedmann-Lemaître-Robertson-Walker cosmological models with negative spatial curvature.
method Linear stability analysis using Hodge decomposition and energy estimates.
result Uniform boundedness and decay of solutions to the linearized Einstein-Euler system.
Proposes a transformer model with geostatistical inductive bias for spatio-temporal forecasting.
problem Combining probabilistic rigor of geostatistics with flexible deep learning representations.
method Spatially-informed transformer with learnable covariance kernel.
result Successfully recovers spatial decay parameters end-to-end via backpropagation.
In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of n+1-dimensional, n≥3, spatially compact spacetimes which generalizes the k=−1 Friedmann--Robertson--Walker vacuum spacetime. Our results demonstrate causal geodes…
WBCP improves conformal prediction for distribution shifts using weighted Dirichlet posteriors.
problem Handling distribution shifts in conformal prediction.
method Generalizes Bayesian Quadrature Conformal Prediction (BQ-CP) to arbitrary importance-weighted settings.
result WBCP maintains coverage guarantees while providing richer uncertainty information.
The paper analyzes multivariate Hawkes processes and their induced population processes.
problem Analyzing the time-dependent joint probability distribution of multivariate Hawkes processes.
method Exact and asymptotic analysis of general multivariate Hawkes processes and their induced population processes.
result Full characterization of the time-dependent joint transform of the multivariate population process and its intensity process.
Novel heat flow estimates on ALE manifolds for Schrödinger operators.
problem Estimating heat flows on ALE manifolds with non-trivial L2-kernel. method Combining Fredholm theory for Dirac type operators and heat kernel advances.
result Established Lp−Lq decay estimates for heat flows. We study the persistence phenomenon in a socio-econo dynamics model using computer simulations at a finite temperature on hypercubic lattices in dimensions up to 5. The model includes a ` social\rq local field which contains the magnetization at time t. The nearest neighbour quenched interactions are drawn from a bin…
Global well-posedness and asymptotic convergence for vacuum Einstein's equations proved.
problem Proving global well-posedness and asymptotic convergence for vacuum Einstein's equations.
method Integrable damping mechanism induced by cosmological constant.
result Future-global solutions converge smoothly to a limiting metric of constant negative scalar curvature.
Study shows global oscillatory solutions for Yang-Mills heat flow in 4D space.
problem Investigating long-time dynamics of Yang-Mills heat flow with specific initial data.
method Analysis of SO(4)-equivariant Yang-Mills heat flow with SU(2) group in 4D space. result Global solutions can exhibit oscillatory behavior at time infinity.
Self-similar solutions to geometric flows are stable under small perturbations.
problem Stability of self-similar solutions in geometric flows.
method Global analytic solutions, compactness arguments, spatial equi-decay properties, and estimates of linearized operator.
result Perturbed solutions are asymptotically self-similar as time tends to infinity.
We study the persistence phenomenon in a socio-econo dynamics model using computer simulations at a finite temperature on hypercubic lattices in dimensions up to 5. The model includes a ` social\rq local field which contains the magnetization at time t. The nearest neighbour quenched interactions are drawn from a bin…
Study on Wasserstein gradient flow for MMD between Coulomb measures.
problem Analyzing the long-time behavior of MMD between probability and target measures using Coulomb kernels.
method Existence of global weak solutions, ultracontractive estimate, regularity analysis, exponential decay proof, defective Polyak-Lojasiewicz inequality.
result Exponential decay of squared MMD toward a uniformly positive target measure on flat torus.
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.
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 Matérn covariance function is a popular choice for prediction in spatial statistics and uncertainty quantification literature. A key benefit of the Matérn class is that it is possible to get precise control over the degree of mean-square differentiability of the random process. However, the Matérn class possesses e…
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.
A new EnKF method for elliptic PDEs reduces dimensionality for accurate state estimation.
problem Elliptic PDEs in fluid flows make traditional EnKF regularization ineffective.
method Low-rank factorization of the Kalman gain based on the Jacobian spectrum.
result Inference can be performed in a low-dimensional subspace of the state space.
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.
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Later, their decay assumptions were weakened by Metzger, Huang, Eichmair-M…
New non-separable covariance kernels for spatiotemporal data derived from harmonic oscillator physics.
problem Capturing complex spatiotemporal dependencies in Gaussian processes.
method Hybrid spectral method based on the harmonic oscillator, deriving explicit covariance kernels.
result Explicit non-separable covariance kernels with space-time interactions.
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.
Spatial blind source separation simplifies multivariate spatial prediction.
problem Predicting multivariate measurements at unobserved locations with spatial dependencies.
method Spatial blind source separation as a pre-processing tool compared to Cokriging and neural networks.
result Spatial blind source separation simplifies spatial prediction by avoiding cross-dependencies.
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 introduces groupoid racks for spatial surfaces.
problem Coloring diagrams of spatial surfaces for invariant calculation.
method Introduces groupoid racks with universal properties.
result Groupoid racks provide an invariant for spatial surfaces.
Machine learning algorithms find frequent application in spatial prediction of biotic and abiotic environmental variables. However, the characteristics of spatial data, especially spatial autocorrelation, are widely ignored. We hypothesize that this is problematic and results in models that can reproduce training data …
STICC clusters geographic objects considering both spatial contiguity and attributes.
problem Discovering repeated geographic patterns with spatial contiguity.
method Spatial Toeplitz Inverse Covariance-Based Clustering (STICC) method.
result STICC significantly outperforms baseline methods in adjusted rand index and macro-F1 score.