arXiv research

This paper provides a theoretical justification of the superior classification performance of deep rectifier networks over shallow rectifier networks from the geometrical perspective of piecewise linear (PWL) classifier boundaries. We show that, for a given threshold on the approximation error, the required number of b…

Study proves existence of a specific type of flow in geometry.

Integral currents with boundary of finite mass are integral.

Proves rigidity of boundaries with constant mean curvature in warped product manifolds.

Study the singular limit of a boundary reaction equation, showing energy concentration and varifold support.

The paper solves a partial Plateau problem using $H$-mass.

We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.

Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.

The paper shows how to make certain sets on a sphere smooth and flat.

Study boundary behavior of limit interfaces in Riemannian manifolds without convexity assumptions.

This note is devoted to the study of sets of finite perimeter over RCD$(K,N)$ metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of …

We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,α}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such $T$ is supported in a s…

We prove a version of Smirnov type theorem and Charatheodory type theorem for a harmonic homeomorphism of the unit disk onto a Jordan surface with rectifiable boundary. Further we establish the classical isoperimetric inequality and Riesz--Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptio…

The paper examines the structure and stability of boundaries in noncollapsed RCD spaces.

In this paper we study the singular set of Dirichlet-minimizing $Q$-valued maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski b…

The Riemannian hemisphere has a lower bound for its mass.

New varifolds with capillary boundary properties studied.

Study on high-codimensional minimal surfaces in hyperbolic space.

New Grunsky operator for disk maps to complex plane.

In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $Γ$ that, similarly…

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up…

New study analyzes security of neural network data reconstruction attacks.

Solves a problem posed by Brezis and Mironescu about least mass of area-minimizing currents.

Study rectifying curves in 3D multiplicative Euclidean space.

Investigates Darboux rectifying curves on smooth surfaces.

A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n…

Study characterizes $k$-rectifiable sets in homogeneous groups.

Recalls and refines the concept of algebraically rectifiable curves.

Shows smoothness of varifolds with specific boundary angles.

Harmonic maps to Euclidean buildings have rectifiable singular strata.

We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and su…

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserma…

New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.

Rectifies singular set of harmonic maps into complex.

For two disjoint rectifiable star-shaped Jordan curves (including round circles) in the asymptotic boundary of hyperbolic 3-space, if the distance (see Definition 1.8) between these two Jordan curves are bounded from above by some constant, then there exists an annulus-type area minimizing (or equivalently least area) …

The paper generalizes rectifying and normal curves in Lorentzian n-space.

In this paper we investigate the performance of different types of rectified activation functions in convolutional neural network: standard rectified linear unit (ReLU), leaky rectified linear unit (Leaky ReLU), parametric rectified linear unit (PReLU) and a new randomized leaky rectified linear units (RReLU). We evalu…

Neural networks can approximate rectifiable measures with small error.

We show that C^1 hypersurfaces in the Heisenberg group are countably N-rectifiable. As a corollary, this shows that all C^1_H graphs over the xy-plane are countable N-rectifiable, showing the equivalence of this notion of rectifiability with that of Franchi, Serra Cassano and Serapioni for such surfaces.

Suppose that $Γ_0\subset\mathbb R^{n+1}$ is a closed countably $n$-rectifiable set whose complement $\mathbb R^{n+1}\setminus Γ_0$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of $Γ_0$ is finite or grows at most exponentially near infinity. Under these assumptions, we…

The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and n_2 of the curve. In this study, we have obtained some characterizations of semi-real spatial quaternionic rectifyin…

In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a …

In Heisenberg groups, rectifiability is studied for subsets using $C^{1,α}$-regular surfaces.

The paper characterizes rectifying curves on smooth surfaces using isometries and Darboux frames.

The paper proves smoothness of stationary varifolds.

In this paper, we introduce a new class of curves αcalled a f-rectifying curves, which its f-position vector defined by α_{f}(s)=\int f(s)T(s)ds always lie in the rectifying plane of α, where f is an integrable function and T is the speed curve of α. In particular case, when the function f=0 or constant, the class of f…

Study anisotropic obstacle problem for minimal surfaces using Cahn-Hoffman transform.

Rectified flows achieve optimal sample complexity for generating data.