Related papers: A Rough Divergence Theorem

We prove an abstract Fubini-type theorem in the context of monoidal and enriched category theory, and as a corollary we establish a Fubini theorem for integrals on arbitrary convergence spaces that generalizes (and entails) the classical…

For normalized sums $Z_n$ of i.i.d. random variables, we explore necessary and sufficient conditions which guarantee the normal approximation with respect to the R\'enyi divergence of infinite order. In terms of densities $p_n$ of $Z_n$,…

A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $\alpha$-mixing…

We consider integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in \mathbb N \setminus \{0\}$ and $\Gamma$ is sufficiently smooth. We prove that, if $q\in \Gamma$…

Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…

Classically, theorems of Fatou and Julia describe the boundary regularity of functions in one complex variable. The former says that a complex analytic function on the disk has non-tangential boundary values almost everywhere, and the…

We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central…

After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well…

In a simple drawing of a graph, any two edges intersect in at most one point (either a common endpoint or a proper crossing). A simple drawing is generalized twisted if it fulfills certain rather specific constraints on how the edges are…

Standard sequential inference architectures are compromised by a normalizability crisis when confronted with extreme, structured outliers. By operating on unbounded parameter spaces, state-of-the-art estimators lack the intrinsic geometry…

This work introduces a novel and general class of continuous transforms based on hierarchical Voronoi based refinement schemes. The resulting transform space generalizes classical approaches such as wavelets and Radon transforms by…

We prove uniform convergence of metrics $g_k$ on a closed surface with bounded integral curvature (measure) in the sense of A.D. Alexandrov, under the assumption that the curvature measures $\mathbb{K}_{g_k}=\mu^1_k-\mu^2_k$, where…

We establish a general uniqueness theorem for subharmonic functions of several variables on a domain. A corollary from this uniqueness theorem for holomorphic functions is formulated in terms of the zero subset of holomorphic functions and…

In the first part of the paper boundary-value problems are considered under weak assumptions on the smoothness of the domains. We assume nothing about smoothness of the boundary $\partial D$ of a bounded domain $D$ when the homogeneous…

In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under…

We introduce a new notion of "regularity structure" that provides an algebraic framework allowing to describe functions and / or distributions via a kind of "jet" or local Taylor expansion around each point. The main novel idea is to…

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show…

In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals.…

We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $\Psi^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of…

The theory underlying robust distributed learning algorithms, designed to resist adversarial machines, matches empirical observations when data is homogeneous. Under data heterogeneity however, which is the norm in practical scenarios,…