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.
Formula for harmonic current dimension on foliated surfaces, extending Brunella's inequality.
problem Calculating the dimension of harmonic currents on foliated complex surfaces.
method Proving a formula involving Furstenberg entropy and Lyapunov exponent.
result Hausdorff dimension of harmonic current is bounded and can be calculated precisely.
The paper proves a generalized Stokes' Theorem for certain singular submanifolds.
problem Validity of Stokes' Theorem for singular submanifolds and differential forms.
method Combines Lebesgue integration with gauge integration techniques.
result Proves a generalized Stokes' Theorem for integral currents with finite Minkowski content.
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.
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.
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 …
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.
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.
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…
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.
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 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.
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.
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…
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.
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.
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…
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.
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. 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. Researchers prove hitting measure singularity for most Fuchsian and Kleinian groups.
problem Singularity of hitting measure for random walks on discrete subgroups.
method Algebraic and geometric convergence, hyperbolic Dehn filling.
result Proved singularity conjecture for certain measures on cocompact Fuchsian and Kleinian groups.
Proves inequality linking function deviation to gradient norm on compact manifolds.
problem Analyzing coupled elliptic systems on compact manifolds.
method Develops a new Poincaré-Sobolev inequality with a density-free reference average.
result Poincaré constant depends on the density's gradient norm.
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)…
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, …
Study shows a subset of foliations on Pn has all singular points linearizable.
problem Characterizing singularities of foliations on projective spaces.
method Analyzes the space of singular foliations by curves on Pn with degree d. result Subset of foliations has all singular points linearizable and no invariant algebraic curves if degree is at least 2.
New framework measures systemic risk with variable market volatility.
problem Classical risk measures fail to capture market volatility complexity.
method Proposes a new framework on Lp(⋅) space with random exponents. result Derives dual representations of systemic risk quantification.
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 …
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.
Proves positive mass theorem for non-spin manifolds with distributional curvature.
problem Proving the positive mass theorem for non-spin manifolds with distributional curvature.
method Using manifolds with asymptotically flat metrics and distributional curvature, the authors show non-negative ADM mass under specific conditions.
result The generalized ADM mass is non-negative for the specified conditions.
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.
Paper proves diffusion models work on manifolds.
problem Current diffusion models assume densities are w.r.t. Lebesgue measure, limiting their applicability.
method Introduced convergence results for diffusion models on more general target distributions.
result Quantitative bounds on Wasserstein distance for target and generated distributions.
New statistical Minkowski distances for Gaussian mixtures with closed-form formulas.
problem Computing distances for Gaussian mixture models efficiently.
method Proposed novel statistical distances based on Minkowski's inequality for Gaussian mixtures.
result Closed-form formula for Gaussian mixture models with integer exponents.
Can you fill R^n with a froth of "soap bubbles" that meet at most n at a time? Not if they have bounded diameter, as follows from Lebesgue's Covering Theorem. We provide some related results and conjectures.
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.
The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.
problem Improving energy decay estimates for Dir-stationary Q-valued functions.
method Establishing improved decay estimates and applying them to derive Liouville-type theorems and continuity.
result Dir-stationary Q-valued functions exhibit the Lebesgue property and reside in a generalized Campanato-Morrey space.