The paper examines uniform perfectness of diffeomorphism groups on open manifolds.
problem Uniform perfectness of diffeomorphism groups on open manifolds.
method Study of uniform perfectness, boundedness, and simplicity of diffeomorphism groups of compact and open manifolds.
result Obtained upper bounds of diameters for commutator length, balls, and conjugation-generated norm.
Uniform boundedness of Riesz transforms on Riemannian manifolds is established with a dichotomy.
problem Establishing uniform boundedness of Riesz transforms on Riemannian manifolds.
method Constructing a complete Riemannian manifold M to demonstrate the dichotomy. result A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds.
Study of waves on Reissner-Nordström-AdS black holes, proving uniform boundedness and continuity.
problem Analyzing waves on black holes in AdS spacetimes, focusing on uniform boundedness and continuity.
method Initial data on a spacelike hypersurface, Dirichlet boundary conditions at infinity, proving uniform boundedness and continuity.
result Uniform boundedness and continuity of waves at the Cauchy horizon on Reissner-Nordström-AdS black holes.
Study on curvature invariants near singularities of wavefronts.
problem Conditions for extendibility and boundedness of curvature invariants.
method Investigation of Gaussian curvature, Mean curvature, and principal curvatures near singularities.
result Relationship between convergence to infinity and uniform approximation of fronts.
We consider Kerr spacetimes with parameters a and M such that |a|<< M, Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally, stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a Schwarzschild metric with parameter M>0, with appropriate geometric assumptions on …
Paper refines Chen-Cheng's estimates for Kähler metrics.
problem Uniform boundedness of scalar curvature assumption.
method Replacing uniform boundedness with Lp-boundedness. result Improved estimates for Kähler metrics under Lp-boundedness. The paper proves boundedness and decay of Teukolsky equations on Kerr backgrounds.
problem Analyzing boundedness and decay of Teukolsky equations on Kerr backgrounds.
method Adapting techniques from scalar waves, uniform-in-frequency estimates for Teukolsky PDEs were obtained.
result Solutions of Teukolsky equation on subextremal Kerr backgrounds remain bounded and decay in time.
Uniform estimates for complex Monge-Ampère equations on hermitian varieties.
problem Uniform boundedness of Chern-Ricci flat potentials in conifold transitions.
method Proving uniform a priori estimates for degenerate complex Monge-Ampère equations.
result Generalization of a theorem to hermitian contexts.
We prove a local graphical theorem for two-dimensional self-shrinkers away from the origin. As applications, we study the asymptotic behavior of noncompact self-shrinkers with finite genus. Also, we show uniform boundedness on the second fundamental form of two-dimensional noncompact self-shrinkers with bounded mean cu…
In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow…
New stability framework relaxes boundedness assumptions for generalization bounds.
problem Overly restrictive assumptions for modern learning settings with heavy-tailed or unbounded losses.
method Develops a stability-based framework requiring only finite Lp moment conditions. result Sharp generalization bounds derived for various learning paradigms.
We study pointwise and Lp gradient estimates of the heat kernel, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. We also prove uniform boundedness results on Lp spaces for the heat operator of the Hodge Laplacian on differenti…
The paper examines the boundedness of bundle diffeomorphism groups over a circle.
problem Investigating the boundedness of bundle diffeomorphism groups over a circle.
method Distinguishing an integer k and constructing a function to analyze the bundle diffeomorphism group.
result The bundle diffeomorphism group is uniformly perfect when k ≥ 1 and unbounded when k = 0.
Uniform bounds found for extremal subsets in specific Alexandrov spaces.
problem Bounding properties of extremal subsets in Alexandrov spaces with constraints on dimension, curvature, and diameter.
method Application of essential coverings introduced by Yamaguchi.
result Uniform upper bounds on the number, Betti numbers, and volume of extremal subsets.
Proof of boundedness of quasimorphisms for certain Lie groups.
problem Bounding quasimorphisms on lattices in Lie groups.
method Thermodynamic formalism applied to bounded cohomology.
result Every π-quasimorphism on irreducible uniform lattices is bounded.
The paper analyzes Teukolsky equations on Kerr backgrounds, proving boundedness and decay of solutions.
problem Analyzing boundedness and decay of solutions to Teukolsky equations on Kerr backgrounds.
method Frequency space analysis of transformed Teukolsky equations on Kerr backgrounds.
result Fixed frequency solutions remain bounded and decay in time for subextremal Kerr backgrounds.
By using the De Giorgi iteration method we will give a new simple proof of the recent result of B.Kotschwar, O.Munteanu, J.Wang [KMW] and N.Sesum [S] on the local boundedness of the Riemmanian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci…
The present paper provides a new generic strategy leading to non-asymptotic theoretical guarantees on the Leave-one-Out procedure applied to a broad class of learning algorithms. This strategy relies on two main ingredients: the new notion of Lq stability, and the strong use of moment inequalities. Lq stability e…
Unified error analysis for discrete flow models.
problem Error analysis of discrete flow models.
method Stochastic calculus theory, Girsanov theorem, generator matching, uniformization.
result First error analysis for discrete flow models.
Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical ℓ∞-semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in …
The paper proves the law of one price in a continuous-time setting without friction.
problem Identifying conditions under which the law of one price holds in a continuous-time setting without frictions.
method Formulating a new mechanism for LOP failure and proving a novel variant of the uniform boundedness principle.
result Establishes the equivalence of the economic concept of LOP with the probabilistic property of the existence of a local $\scr{E}$-martingale state price density.
We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain …
A new sequential test for unnormalized densities.
problem Testing unnormalized densities with adaptive stopping.
method Sequential kernelized Stein discrepancy test, using non-uniform Stein kernels.
result Valid test with asymptotic lower bound for growth.
Let C(Sg,p) denote the curve complex of the closed orientable surface of genus g with p punctures. Masur-Minksy and subsequently Bowditch showed that C(Sg,p) is δ-hyperbolic for some δ=δ(g,p). In this paper, we show that there exists some δ>0 independent of g,p such that the cu…
Uniform TD(0) bound derived for function approximation with Markov noise.
problem Uniform concentration bound for TD(0) with function approximation.
method Contractive stochastic approximation, martingale and Markov noises, Poisson equation, relaxed concentration inequalities.
result Uniform all-time concentration bound for TD(0) with linear function approximation.
New algorithms improve distributed optimization under mild variance conditions.
problem Improving distributed optimization for large-scale machine learning problems.
method Revisited Federated Averaging and SCAFFOLD algorithms under a general variance condition.
result Established convergence results for smooth nonconvex objective functions under mild variance conditions.
Paper removes bounded gradient assumption for SGD in nonconvex learning.
problem Existing theoretical results for SGD in nonconvex learning require uniform boundedness of gradients, which is hard to verify.
method Establishes sufficient conditions for SGD convergence without bounded gradient assumption.
result SGD achieves optimal convergence rates for nonconvex and gradient-dominated objectives.
This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, …
The paper studies boundedness of pseudo-differential operators on smooth manifolds.
problem Boundedness of pseudo-differential operators in Lp-Lq spaces on smooth manifolds. method Using global symbols and extending Hörmander's condition, the paper investigates Lp-boundedness, L∞-BMO estimates, and Lp-Lq boundedness for Fourier multipliers and pseudo-differential operators. result The paper proves Lp-Lq boundedness for the range 1<p≤2≤q<∞. Compact metrics found near Kähler manifold's canonical class.
problem Finding unique cscK metrics near Kähler manifold's canonical class.
method Proved existence and uniqueness of cscK metrics for Kähler classes near canonical class.
result Metric spaces are pre-compact in Gromov-Hausdorff sense.
Study proves boundedness of operators in variable exponent Morrey spaces.
problem Boundedness of operators in global Morrey-type spaces with variable exponents.
method Analysis of Hardy-Littlewood maximal operator and potential type operator in variable exponent Morrey spaces.
result Boundedness of the Hardy-Littlewood maximal operator and potential type operator in global Morrey-type spaces with variable exponents.
Study on droplet flow on uneven surfaces, proving existence and properties.
problem Understanding droplet movement on irregular surfaces.
method Existence of smooth flow and 1/2-Hölder continuous minimizing movement solutions.
result Properties of minimizing movements including comparison principles and uniform boundedness.
We give an algorithmically efficient version of the learner-to-compression scheme conversion in Moran and Yehudayoff (2016). In extending this technique to real-valued hypotheses, we also obtain an efficient regression-to-bounded sample compression converter. To our knowledge, this is the first general compressed regre…
Study on Hardy-Littlewood maximal operators on manifolds with bounded geometry.
problem Boundedness and mapping properties of Hardy-Littlewood maximal operators on Riemannian manifolds.
method Analysis of Lp boundedness, conformal invariance, and weak type estimates. result Sharp Lp estimates for the centred operator on Riemannian models with pinched negative scalar curvature. We show that \emph{No unbounded profit with bounded risk} (NUPBR) implies \emph{predictable uniform tightness} (P-UT), a boundedness property in the Emery topology which has been introduced by C. Stricker \cite{S:85}. Combining this insight with well known results from J. Mémin and L. Słominski \cite{MS:91} leads to a …
Study examines Lp-boundedness of Hodge projection on manifolds with ends.
problem Understanding Lp-boundedness of Hodge projection on manifolds with ends. method Investigates the relationship between Hodge projection, Riesz transform, and bounded harmonic functions.
result Connects Lp-boundedness of Hodge projection to the structure of L2 harmonic one-forms and bounded harmonic functions. The study shows boundedness and constructs a moduli space for Calabi-Yau fibrations.
problem Boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves.
method Fixed volumes and Iitaka volumes of general fibers, adiabatic sense construction.
result Constructs a separated coarse moduli space of K-stable Calabi-Yau fibrations.
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions V for bounded Schrödinger operator Δ−V. method Investigates weighted L2-boundedness of Hodge projector. result Characterizes function V for Schrödinger operator boundedness. Establishes a principle for metric convergence in Kähler geometry.
problem Convergence of evolving Riemannian metrics in Kähler geometry.
method General 'boundedness implies convergence' principle applied to collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows.
result Obtains convergence results for collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows.
Study examines boundedness of oscillating singular integrals on specific Lie groups.
problem Investigating boundedness of oscillating singular integrals on Lie groups of polynomial growth.
method Presented kernel criteria in terms of sub-Riemannian structure and Fourier analysis.
result Extended classical oscillating conditions for boundedness of oscillating convolution operators.
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.
This paper proves geodesic curvature measures are bounded for curves near cross cap singularities.
problem Boundedness of geodesic curvature measures near cross cap singularities.
method Analyzes intrinsic cross cap singularities and extends Gauss-Bonnet formula.
result Proves boundedness of geodesic curvature measures for curves near cross cap singularities.
Paper presents a novel method to assess boundedness and stability of nonlinear systems with variable delays.
problem Challenges in assessing boundedness and stability of vector nonlinear systems with variable delays and coefficients.
method Develops a novel framework to evaluate the evolution of solution norms in such systems by constructing scalar counterparts.
result Introduces new criteria for boundedness and stability and estimates the radii of containing balls for history functions.
Proves boundedness and decay for spin 1 Teukolsky equation on Reissner-Nordström spacetime.
problem Analyzing stability of charged black holes in perturbations.
method Gauge-invariant quantities and Fackerell-Ipser-type equations.
result Proves boundedness and polynomial decay for ℓ=1 spherical mode. Study on flat connections with controlled irregularity.
problem Boundedness of algebraic flat connections with limited irregularity.
method Analysis of families of algebraic flat connections and holonomic D-modules.
result Established boundedness of families of algebraic flat connections with controlled irregularity.
Study confirms boundedness of certain singularities in log Fano geometry.
problem Boundedness of log Fano cone singularities and minimal log discrepancies.
method Analyzing local volumes and minimal log discrepancies of Kollár components.
result Boundedness of K-semistable log Fano cone singularities confirmed in dimension three.
Paper characterizes foliated bundle classes via quasi-morphisms and studies their boundedness.
problem Characterizing bounded characteristic classes of foliated bundles.
method Using non-descendible quasi-morphisms on the universal covering of the structure group.
result Non-existence of foliated structures on some Hamiltonian fibrations and non-triviality of the second bounded cohomology group.
The paper proves boundedness of a Riesz transform on weighted manifolds.
problem Establishing \(L^p\)-boundedness of the covariant Riesz transform on differential forms.
method Heat-kernel criterion, volume doubling, heat kernel estimates, curvature control, gradient bounds.
result The covariant Riesz transform is \(L^p\)-bounded for \(p>2\) on weighted Riemannian manifolds.