Characterizes Lebesgue points using nearest neighbor methods.
problem Consistency of classification algorithms based on nearest neighbors.
method Characterization of Lebesgue points via 1-Nearest Neighbor regression.
result Proves convergence of 1-Nearest Neighbor classification algorithms in metric spaces.
Study shows twist tori equidistribute in moduli space, with other families having singular distributions.
problem Statistical behavior of twist tori in moduli space of hyperbolic surfaces.
method Analyzing expanding families of twist tori and their limiting distributions.
result Equidistribution of twist tori to a Lebesgue measure, with other families having singular distributions.
Uniqueness found for elliptic equations with drift on manifolds.
problem Finding unique solutions to elliptic equations with drift on manifolds.
method Investigation in weighted Lebesgue spaces, focusing on conditions for uniqueness.
result Sharp conditions on drift term for uniqueness in polynomial volume growth manifolds.
Alternative proof shows Kummer rigidity for K3 surfaces.
problem Proving automorphisms of K3 surfaces are Kummer examples.
method Exploits Ricci-flat metrics on K3 surfaces.
result Shows measure of maximal entropy automorphisms are Kummer.
The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that th…
Study shows singularity of stationary measure on Furstenberg boundary for certain random walks.
problem Singularity of stationary measure on Furstenberg boundary for random walks.
method Analysis of random walks on semisimple Lie groups with specific properties.
result Stationary measure is singular to Lebesgue measure in certain cases.
Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.
problem Limitation to static PDEs and different time-domain regularity.
method Extend spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, measure approximation error in Bochner-Sobolev norm.
result Established bound on approximation rate for functions in anisotropic weighted Fourier-Lebesgue spaces.
Introduces Lebesgue currents for real intersection theory.
problem Intersection of singular cycles in singular spaces.
method Introduces Lebesgue currents to define real intersection theory.
result Provides a foundation for real intersection theory.
Paper studies geometric properties of nonlinear Lebesgue spaces.
problem Geometric and analytic properties of nonlinear Lebesgue spaces.
method Formalizes pointwise description of geometric properties using a nonlinear Fubini-Lebesgue theorem.
result Definition of length structure, Alexandrov curvature bounds, and speed for absolutely continuous curves in nonlinear Lebesgue spaces.
We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports s…
Develops a new quadrature method for Lebesgue integrals.
problem Finding optimal values and weights for non-Gaussian processes.
method Solves a generalized eigenvalue problem to find value-nodes and weights.
result Advantages in analyzing irregular and stochastic processes.
Study generalizes Lebesgue curves to new space-filling and fractal sets.
problem Generating space-filling curves from planar substitutions.
method Generalized Lebesgue's construction to new curves and fractal sets.
result Some substitutions create relatively dense fractal-like sets.
RLF uses Riemann-Lebesgue cutting for better regression.
problem Improving regression accuracy through novel tree splitting.
method Develops Riemann-Lebesgue Tree (RLT) for partitioning response intervals.
result RLF achieves larger variance reduction compared to CART.
In this work we study the Lebesgue property for convex risk measures on the space of bounded càdlàg random processes (R∞). Lebesgue property has been defined for one period convex risk measures in \cite{Jo} and earlier had been studied in \cite{De} for coherent risk measures. We introduce and study th…
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …
For n>2, the action of the outer automorphism group of the rank n free group F_n on the SU(2)-character variety Hom(F_n,SU(2))/SU(2)$ is ergodic with respect to the Lebesgue measure class.
Fix a translation surface X, and consider the measures on X coming from averaging the uniform measures on all the saddle connections of length at most R. Then as R→∞, the weak limit of these measures exists and is equal to the Lebesgue measure on X. We also show that any weak limit of a subsequence of …
We prove a uniform Sobolev inequality along the Sasaki-Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.
Given a measure on the Thurston boundary of Teichmueller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones whic…
Let Q be a connected component of a stratum in the space of quadratic differentials for a non-exceptional Riemann surface of finite type. We show that the probability measure on Q in the Lebesgue measure class which is invariant under the Teichmueller flow is obtained by Bowen's construction.
Investigates stability properties of Haezendonck-Goovaerts premium principles in Orlicz spaces.
problem Stability properties of Haezendonck-Goovaerts premium principles in various Orlicz spaces.
method Analysis of stability properties including Fatou and Lebesgue properties, and continuity with respect to Φ-weak convergence. result Haezendonck-Goovaerts principles satisfy the Fatou property and Lebesgue property under certain conditions.
We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where p≥2, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure μ, …
Optimizes Lipschitz estimates for partitions of unity and characterizes spaces with Assouad-Nagata dimension.
problem Understanding the properties of partitions of unity and their Lipschitz bounds.
method Analyzes the standard partition of unity and its ℓp-generalizations, using the approximate midpoint property and Lebesgue number. result Optimal Lipschitz bounds for partitions of unity and characterizes metric spaces with Assouad-Nagata dimension.
In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.
Improved guarantees for misspecified kernelized bandit optimization.
problem Misspecification in kernelized bandit optimization.
method Localization and domain splitting techniques.
result Logarithmic or polylogarithmic growth of misspecification amplification.
Study small eigenvalues of Toeplitz operators and their relation to Mabuchi geodesics.
problem Analyzing small eigenvalues of Toeplitz operators on complex projective manifolds.
method Proving the existence of exponentially decaying eigenvalues for Toeplitz operators with specific symbols, and establishing a connection to Mabuchi geodesics.
result Logarithmic distribution of small eigenvalues correlates with Mabuchi geodesics between polarizations.
Develops numerical method for joint probability estimation from random processes.
problem Estimating joint probability distribution from random processes.
method Formulates and solves generalized eigenvalue problems for two random processes, then uses projections of eigenvectors to build a joint distribution estimator.
result Develops a new type of probability correlation, Pf[i];g[j], for random processes. The Cannon-Thurston map's measures become singular with respect to sphere measures.
problem Characterizing the behavior of measures under the Cannon-Thurston map.
method Analyzing the geometric properties of hyperbolic geodesics and quasi-geodesics.
result Natural measures on the circle become singular with respect to measures on the sphere.
Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution with finite first moment and whose support generates a non-elementary subgroup, converges almost surely to a point in the space PMF of projective measured foliations on the surface. This defines a harmonic measure on …
The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…
Automatic continuity of polynomial maps and cocycles proved.
problem Proving continuity of polynomial maps and cocycles.
method The approach involves proving continuity of polynomial maps and cocycles.
result Automatic continuity of polynomial maps and cocycles.
Study shows how geodesics behave in hyperbolic manifolds and proves measure singularity.
problem Understanding geodesic behavior and measure singularity in hyperbolic manifolds.
method Introduced kth excursion for geodesics, analyzed hitting and Lebesgue measures. result Proved hitting and Lebesgue measures on hyperbolic space are mutually singular.
Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.
problem Proving a generalized Gauss-Bonnet formula for conical metrics.
method Analyzing Gaussian curvature and Lebesgue integrability.
result Proved a generalized Gauss-Bonnet formula for conical metrics.
This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with its first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tilt-excess are established for those varifolds. The results are optimal in terms of…
We propose here a new discretization method for a class continuum gauge theories which action functionnals are polynomials of the curvature. Based on the notion of holonomy, this discretization procedure appears gauge-invariant for discretized analogs of Yang-Mills theories, and hence gauge-fixing is fully rigorous for…
Common perpendiculars equidistribute in negatively curved spaces.
problem Equidistribution of common perpendiculars in negatively curved spaces.
method Analyzing the Bowen-Margulis measure and geodesic flow properties.
result Lebesgue measures of common perpendiculars equidistribute to the Bowen-Margulis measure.
Analytic proof for minimal rank Sard conjecture.
problem Proving the minimal rank Sard conjecture in the analytic category.
method Using subanalytic abnormal distribution from [4], we establish a proof.
result The set of points accessible through singular horizontal curves of minimal rank has Lebesgue measure zero.
A novel approach to interpolation, classification, and clustering using Radon-Nikodym derivatives.
problem Interpolation, classification, and clustering problems in data analysis.
method Radon-Nikodym approach with Lebesgue quadrature for optimal clustering.
result The approach changes both probabilities and the probability space with new observations.
Generalizes Gauss-Bonnet to metrics with logarithmic singularities.
problem Calculating curvature for metrics with singularities on compact surfaces.
method Proves a generalized Gauss-Bonnet formula under Lebesgue integrability condition.
result Establishes formula for special Kähler metrics with meromorphic cubic differentials.
New measure defined on surface strata, invariant under scaling.
problem Defining an invariant measure on moduli spaces of dilation surfaces.
method Novel computation of cohomology with coefficients for mapping class group.
result SL(2,R)-invariant Lebesgue class measure on strata.
This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that the regular Gaussian distributions that factorize with respect to a chain graph G with d parameters have positive Lebesgue measure with respect to Rd, whereas those that factorize with respect…
Estimates for eigenfunctions and quasimodes on compact manifolds.
problem Characterizing eigenfunctions and quasimodes on compact manifolds.
method Sharp Lq-estimates for log-quasimodes, focusing on small Lebesgue exponents. result No characterization possible for q>qc. Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.
problem Establishing synthetic Ricci curvature conditions for Lipschitz manifolds.
method Uniform heat kernel bounds and synthetic Ricci curvature conditions.
result Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.
Neural networks can approximate rectifiable measures with small error.
problem Approximating complex rectifiable measures using neural networks.
method Using ReLU neural networks to approximate (countably) m-rectifiable measures as push-forwards of the Lebesgue measure.
result The approximation error in terms of Wasserstein distance can be made arbitrarily small.
Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney [{\it Geometric integration theory}, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure theorists because …
The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.
problem Understanding the behavior of geodesics and measures on fibered hyperbolic 3-manifolds.
method Properties of geodesics and measures on the circle and sphere are analyzed to prove singularity.
result Natural measures on the circle become singular with respect to measures on the sphere.
Derives integral formula for ReLU networks with limited weights.
problem Finding optimal neural network weights with limited L1-norm. method Derives integral representation formula for shallow ReLU networks under L1-norm constraint. result Explicitly solves the least L1-norm neural network representation for a given function. The paper provides consistency results for KDE on manifolds with irregular kernels.
problem Analyzing density estimation on manifolds with complex kernels.
method Strong uniform consistency with rates for KDE on Riemannian manifolds with Riemann integrable kernels.
result Strong uniform consistency with rates for KDE on manifolds.