arXiv research

A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussi…

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.

In this work we study the Lebesgue property for convex risk measures on the space of bounded càdlàg random processes ($\mathcal{R}^\infty$). 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…

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.

Characterizes Lebesgue points using nearest neighbor methods.

RLF uses Riemann-Lebesgue cutting for better regression.

Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.

Study shows twist tori equidistribute in moduli space, with other families having singular distributions.

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 introduce the notion of Lebesgue currents. They are a special type of currents involving Lebesgue measure. We apply it to define the intersection of singular cycles, which provides the foundation to the real intersection theory.

Study shows singularity of stationary measure on Furstenberg boundary for certain random walks.

Common perpendiculars equidistribute in negatively curved spaces.

Proves inequality linking function deviation to gradient norm on compact manifolds.

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 …

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some known examples coming from the theory of metric measured spaces and also from oscill…

Researchers prove hitting measure singularity for most Fuchsian and Kleinian groups.

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 $\mathbb{R}^d$, whereas those that factorize with respect…

The Cannon-Thurston map's measures become singular with respect to sphere measures.

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 $R\to\infty$, 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 generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk.…

Analytic proof for minimal rank Sard conjecture.

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 $p\ge 2$, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $μ$, …

One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe h…

In this effort, we derive a formula for the integral representation of a shallow neural network with the ReLU activation function. We assume that the outer weighs admit a finite $L_1$-norm with respect to Lebesgue measure on the sphere. For univariate target functions we further provide a closed-form formula for all po…

Systemic risk measures are crucial for the stability of financial markets, yet classical formulations fail to capture the complexity of market volatility. We propose a new framework for systemic risk measurement on the variable-exponent Bochner-Lebesgue space $L^{p(\cdot)}$, where the exponent $p(\cdot)$ is a random va…

New integration theory on topological spaces, including fractals.

The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue 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…

Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.

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…

The paper analyzes Laplace learning for Gaussian measure data in infinite dimensions, proving convergence.

Study shows a subset of foliations on $\mathbb{P}^n$ has all singular points linearizable.

Study permeable sets and their dimensions, with applications to fractals.

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.

Proves stability of cone-volume measure with nearly constant density.

This paper focuses on using the first curvature $κ(t)$ of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems [Y. Wang, H. Sun, Y. Song et al., arXiv:1808.00290] to $n$-dimensional systems. We prove that for a system $\dot{r}(t)=Ar(t)$, (i) i…

Study shows how optimal transport behaves in higher dimensions.

Problems of interpolation, classification, and clustering are considered. In the tenets of Radon--Nikodym approach $\langle f(\mathbf{x})ψ^2 \rangle / \langleψ^2\rangle$, where the $ψ(\mathbf{x})$ is a linear function on input attributes, all the answers are obtained from a generalized eigenproblem $|f|ψ^{[i]}\rangle =…

Neural networks can approximate rectifiable measures with small error.

This paper proposes a new approach to describe the stability of linear time-invariant systems via the torsion $τ(t)$ of the state trajectory. For a system $\dot{r}(t)=Ar(t)$ where $A$ is invertible, we show that (1) if there exists a measurable set $E_1$ with positive Lebesgue measure, such that $r(0)\in E_1$ implies t…

Paper studies geometric properties of nonlinear Lebesgue spaces.

This paper deals with both complex dynamical systems and conformal iterated function systems. We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a $d$-parameter family of such semigroups satisfies the transversality condition, then for almost every par…

Study generalizes Lebesgue curves to new space-filling and fractal sets.

The paper proves conditions for non-uniform expansion in partially hyperbolic systems.

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to s…

We consider Hilbert and Funk geometries on a strongly convex domain in the Euclidean space. We show that, with respect to the Lebesgue measure on the domain, Hilbert (resp. Funk) metric has the bounded (resp. constant negative) weighted Ricci curvature. As one of corollaries, these metric measure spaces satisfy the cur…