Study of Ricci flow convergence on surfaces with boundary.
problem Convergence of singular solutions to Ricci flow on compact surfaces with boundary.
method Subsequential convergence analysis of Ricci flow with prescribed geodesic curvature.
result Convergence does not depend on the sign of geodesic curvature of the boundary in the case of rotational symmetry.
We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature f…
We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and deri…
We study the behaviour of the Kähler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kähler class, then the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base.
The study examines the behavior of Gaussian processes' minimums and overshoots.
problem Understanding the behavior of Gaussian processes' minimums and overshoots.
method Analyzing conditional distributions and subsequential limits of minimizers.
result The scaled overshoot converges to an exponential random variable with mean σ_*^2.
Given a sequence of properly embedded minimal surfaces in a 3-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity, and we analyze what happens when one blows up the surfaces near a point where…
Given a Hermitian line bundle L→M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as ε→0, of couples (uε,∇ε) critical for the rescalings \begin{align*} &E_ε(u,\nabla)=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{1}{4ε^2}(1-|u|^2)^2\Big) \end{align*} of the self-dua…
We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and 2π, where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angl…
Compactness theory for super Ricci flows provides convergence results.
problem Understanding convergence of super Ricci flows.
method Developed a compactness theory for super Ricci flows.
result Subsequential convergence to a metric flow under certain conditions.
We consider the continuous W2,d immersions of d-dimensional hypersurfaces in Rd+1 with second fundamental forms uniformly bounded in Ld. Two results are obtained: first, a family of such immersions is constructed, whose limit fails to be an immersion of a manifold. This addresses the endpoint ca…
We propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. Our methods possess three main advantages compared to current state-of-the-art accelerated first-order methods: (1) they allow using two different extrapolation points to evaluate the grad…
Compactness theorem for Fueter sections yields non-zero harmonic 1-forms.
problem Counting monopoles and special Lagrangians in Calabi-Yau 3-folds.
method Proving compactness of Fueter sections and analyzing their renormalized sequences.
result Renormalized sequence of Fueter sections converges to a non-zero Z2-harmonic 1-form. Proposes BMME for optimizing nonsmooth nonconvex problems with block structure.
problem Optimizing nonsmooth nonconvex problems with block structure.
method Block Alternating Bregman Majorization Minimization with Extrapolation (BMME).
result Subsequential convergence to a first-order stationary point under mild assumptions, global convergence under stronger conditions.
Study shows translators can have non-removable singularities at infinity but eventually converge to unique planes.
problem Understanding singularities and convergence of translators at infinity.
method Global analysis of quasilinear soliton equations, sharp non-standard elliptic decay estimates, and potential theory.
result Finite entropy, finite genus translators converge to uniquely determined planes at infinity.
Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.
problem Uniform diameter and volume estimates for Chern-Ricci flow on Hermitian minimal models.
method Uniform diameter and volume estimates, local Kähler assumption, Perelman's reduced length, almost monotonicity formula for reduced volume.
result Gromov-Hausdorff convergence of the Chern-Ricci flow on Hermitian minimal models.
This paper studies the convergence of penalized energy to harmonic maps in Riemannian manifolds.
problem Analyzing the convergence of penalized energy to harmonic maps in Riemannian manifolds.
method Using the penalized energy functional and weak convergence techniques, the paper proves the energy identity for Ginzburg-Landau approximation of harmonic maps.
result The defect measure ν can be expressed as the sum of energies of harmonic spheres for arbitrary manifolds.
Study examines Yang-Mills-Higgs energy convergence to codimension-three area functional.
problem Asymptotic behavior of Yang-Mills-Higgs energy in large mass limit.
method Investigates the asymptotic behavior of Yang-Mills-Higgs energy in the large mass limit, proving convergence to the codimension-three area functional.
result The (n−3)-currents dual to the Yang-Mills-Higgs energy converge to a relative integral (n−3)-cycle. This paper improves MI-based BCIs by applying transfer learning across all components.
problem Reducing calibration effort for new subjects in MI-based BCIs.
method Proposes TL in spatial filtering, feature engineering, and classification blocks, and adds data alignment.
result Integrating data alignment and sophisticated TL significantly improves classification performance and reduces calibration effort.
The study finds critical points of Yang-Mills-Higgs energy on 3-manifolds.
problem Finding critical points of Yang-Mills-Higgs energy on 3-manifolds.
method 2-parameter min-max construction and energy gap analysis.
result Existence of non-trivial critical points on 3-manifolds with bounded geometry.
For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.
problem Energy quantization in Ginzburg-Landau vortices for higher dimensions.
method Analyzing normalized energy measures and vorticity sets.
result Energy quantization only holds when density is less than 2.
Proves weak convergence equals mean convergence in GGC.
problem Proving convergence in GGC distributions.
method Using generalized gamma convolution (GGC) and expected utility maximization.
result Weak convergence implies mean convergence in GGC.
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean…
The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.
problem Convergence of Lipschitz functions and sets defined by equations.
method Painlevé-Kuratowski convergence applied to Lipschitz functions and sets defined by equations.
result Generalizations and reverses of classical theorems on convergence of functions and sets.
Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.
problem Distance between Lorentzian manifolds.
method Intrinsic timed Hausdorff convergence.
result Intrinsic timed Hausdorff convergence implies Gromov-Hausdorff and big bang convergence.
Studied SGD convergence under weak conditions.
problem Convergence of SGD in nonconvex optimization.
method Analyzed biased nonconvex SGD under mild conditions.
result Provided convergence rates and complexities.
Study on convergence rate of Q-curvature flow in 6 dimensions.
problem Analyzing the convergence rate of Q-curvature flow in 6 dimensions. method Provided an example of a slowly converging Q6-curvature flow in dimension 6. result The Q-curvature flow in 6 dimensions does not always converge exponentially, unlike in 2 dimensions. The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce …
Establishes geometric convergence of iterative optimization algorithms.
problem Analyzes convergence of iterative optimization algorithms under general assumptions.
method General framework for iterative optimization algorithms, proving asymptotic geometric convergence and providing convergence rates.
result Asymptotic geometric convergence of iterative optimization algorithms with exact rate.
Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
The paper explores null distance convergence for warped product spacetimes.
problem Defining convergence for sequences of spacetimes as metric spaces.
method Using the null distance to define convergence of spacetimes.
result Optimal convergence theorem for warped product spacetimes.
New quasi-Newton method guarantees global superlinear convergence.
problem Global convergence and superlinear convergence of quasi-Newton methods.
method Hybrid proximal extragradient method with online learning for Hessian approximation.
result First globally convergent quasi-Newton method with explicit superlinear convergence rate.
We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.
problem Achieving decoupled convergence in nonlinear two-time-scale stochastic approximation.
method Nested local linearity assumption, suitable step size selection, convergence analysis of matrix cross term, fourth-order moment convergence rates.
result Finite-time decoupled convergence rates can be achieved in nonlinear two-time-scale stochastic approximation with proper step size selection.
The article introduces a new convergence concept for Lorentzian spaces and applies it to generalized cones.
problem Stability of curvature bounds in generalized Lorentzian cones.
method Introduces ℓ-convergence for Lorentzian pre-length spaces, applies it to generalized cones, and proves stability of curvature bounds. result Sharp timelike curvature and curvature-dimension bounds for generalized cones are established.
AdaBoost's classifier and margins converge to a known value.
problem Convergence properties of AdaBoost algorithm.
method Formal proofs of convergence properties of AdaBoost's classifier and margins.
result AdaBoost's classifier and margins converge to a known value.
Study shows gap between uniform convergence and test error in random feature models.
problem Understanding the gap between uniform convergence and test error in random feature models.
method Analytical expressions for uniform convergence over norm balls, interpolators, and minimum norm interpolator risk derived and proved.
result Uniform convergence over interpolators still gives a non-trivial bound of test error even when classical uniform convergence is vacuous.
The Sinkhorn-Knopp derivatives converge with linear rate.
problem Optimal transport problem with entropic regularization.
method Iterative proportional fitting procedure.
result Derivatives converge with linear rate.
The paper examines convergence of distances in Lipschitz structures on manifolds.
problem Convergence of distances in Lipschitz vector fields and norms on manifolds.
method Analysis of convergence of distances associated to converging structures of Lipschitz vector fields and norms.
result Under mild controllability assumption, distances converge locally uniformly to the limit Carnot-Carathéodory distance.
New approach to geometric quantization for symplectic manifolds.
problem Quantization of symplectic manifolds with non-singular Lagrangian fibrations.
method Using spectral convergence of metric measure spaces, the authors develop a new geometric quantization approach.
result Spectral and quantum Hilbert space convergence results for Kähler and almost Kähler quantizations.
We introduce a natural definition of Lp-convergence of maps, p≥1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of …
We prove a criterion of convergence in the augmented Teichmueller space that can be phrased in terms of convergence of the hyperbolic metrics or of quasiconformal convergence away from the nodes.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
DCDC calculates convergence rates for Markov chains using neural networks.
problem Computing precise convergence rates for Markov chains is hard.
method Developed a neural network-based algorithm (DCDC) to bound convergence rates in Wasserstein distance.
result Demonstrated effective convergence bounds for real-world Markov chains.
Proves curvature tensor convergence for smoothable spaces.
problem Curvature tensor behavior in smoothable Alexandrov spaces.
method Weak convergence of curvature tensors in noncollapsing sequences.
result Proves convergence of curvature tensors in smoothable Alexandrov spaces.
Develops clustering methods based on likelihood and convergence proved.
problem Hard clustering based on likelihood.
method k-MLE, k-Bregman, k-VARs approaches.
result Convergence proved for clustering methods.
Study utility maximization with costs, proving convergence and strategies.
problem Utility maximization with proportional transaction costs.
method Extended weak convergence theory and Meyer--Zheng topology.
result Prove convergence of utility maximization problems and optimal trading strategies.
New technique explains convergence in ML models with data modifications.
problem Understanding convergence of ML models under data changes.
method Analogue of Fatou's lemma and gamma-convergence.
result Relevance and applications in general ML tasks and domain adaptation.
Characterizes problems solvable via linear convergence algorithms.
problem Optimization problems solvable with linear convergence.
method Riemannian gradient descent.
result Characterized problems solvable via linear convergence.
Equivalence shown between two mathematical concepts for hyperbolic surfaces.
problem None explicitly stated, but related to mathematical equivalence of concepts.
method Benjamini-Schramm convergence and zeta functions equivalence demonstration.
result Equivalence of Benjamini-Schramm convergence and zeta functions for compact hyperbolic surfaces.