arXiv research

Study stable capillary hypersurfaces with planar boundaries in half-spaces and domains.

New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.

Upper bounds on fixed points in PWL neural networks with hyperplane analysis.

Shows smoothness of varifolds with specific boundary angles.

Paper proves isoperimetric inequality for Minkowski spacetime.

Simple proof that stable minimal hypersurfaces in R^4 are hyperplanes.

The paper defines Dirichlet domains for Anosov subgroups in Lie groups.

Sharp lower bound on GHHs' representation power of CPWL functions.

A new reinforcement learning method improves Max-Cut solutions without needing training data.

Improved upper bound for mass partitioning problem using Gray codes.

The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We det…

For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain.

The projection of a compact oriented submanifold M^{n-1} in R^{n+1} on a hyperplane P^{n} can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S^1 in S^2 projects to zero in any plane containing the x_3-axis. Using currents to make this precise, we show: A lipschitz (homology)…

Grauert constructs complete Kähler metrics on complements of complex analytic sets.

The paper establishes a Miyaoka-Yau type inequality for hyperplane arrangements in complex projective space.

We consider an entire graph $S$ in $\mathbb R^{N+1}$ of a continuous real function $f$ over $\mathbb R^{N}$ with $N\ge 1$. Let $Ω$ be an unbounded domain in $\mathbb R^{N+1}$ with boundary $S$. Consider nonlinear diffusion equations of the form $\partial_t U= Δφ(U)$ containing the heat equation. Let $U$ be the solution…

The paper proves a margin inequality for separating hyperplanes, useful for analyzing algorithmic bias.

The paper revisits Markowitz's pseudodistance on pseudo-Riemannian manifolds.

We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $Σ^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We sho…

We study large-scale spatial systems that contain exogenous variables, e.g. environmental factors that are significant predictors in spatial processes. Building predictive models for such processes is challenging because the large numbers of observations present makes it inefficient to apply full Kriging. In order to r…

Let $\mathcal{A}$ be a central hyperplane arrangement in $\mathbb{C}^{n+1}$ and $H_i,i=1,2,...,d$ be the defining equations of the hyperplanes of $\mathcal{A}$. Let $f=\prod_i H_i$. There is a global Milnor fibration $F\hookrightarrow \mathbb{C}^{n+1} \setminus \mathcal{A} \xrightarrow{f} \mathbb{C}^*,$ where $F$ is ca…

Let $Σ$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $Σ$ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if $\partial Σ$ is embedded for $n=2$, or if $\partialΣ$ is convex…

Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: For $2 \leq n \…

Survey on hyperplane arrangements and their topology.

The paper introduces explainable $k$-means with axis-parallel hyperplanes for $d$-dimensional data.

Segre varieties' hyperplane sections are unstable under certain conditions.

Recently, considerable effort has been devoted to deep domain adaptation in computer vision and machine learning communities. However, most of existing work only concentrates on learning shared feature representation by minimizing the distribution discrepancy across different domains. Due to the fact that all the domai…

The paper generalizes a rigidity theorem for hypersurfaces with constant weighted mean curvature.

A translating soliton is a hypersurface $M$ in $\mathbb{R}^{n+1}$ such that the family $M_t= M- t \,\mathbf{e}_{n+1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf{e}_{n+1}^{\perp}.$ In this paper we obtain a cha…

We extend the theoretical analysis of a recently proposed single subspace learning algorithm, called Dual Principal Component Pursuit (DPCP), to the case where the data are drawn from of a union of hyperplanes. To gain insight into the properties of the $\ell_1$ non-convex problem associated with DPCP, we develop a geo…

Crowdsourcing is an effective tool for human-powered computation on many tasks challenging for computers. In this paper, we provide finite-sample exponential bounds on the error rate (in probability and in expectation) of hyperplane binary labeling rules under the Dawid-Skene crowdsourcing model. The bounds can be appl…

The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real …

We prove that the topological complexity of (a motion planning algorithm on) the complement of generic complex essential hyperplane arrangement of $n$ hyperplanes in an $r$-dimensional linear space is min$\{n+1,2r\}$.

We prove that the hyperplanes parallel to ${\bf e}_{n+1}$ are the unique examples of translating solitons $C^1-$asymptotic to two half-hyperplanes outside a vertical cylinder in $\R^{n+1}$.

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives …

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletio…

A theorem divides hyperplanes evenly with a line through the origin.

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the $(n+1)$-dimensional volume of regions between parallel hyperplanes, and the $n$-dimensional surf…

This paper explores how enforcing equivariance constraints limits neural network expressivity and proposes compensatory model size increases.

It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such solution its reciprocal is the radial function of only one o…

In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is th…

We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement.

We examine the existence of tangent hyperplanes to subriemannian balls. Strictly abnormal shortest paths are allowed

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $Δu + (1-u^2) u = 0 \hbox{in} \R^N$ with $\pp_{y_N}u >0$ must be such that its level sets $\{u=\la\}$ are all hyperplanes, {\em \bf at least} for dimension $N\le 8$. A counterexample for $N\ge 9$ has long been believed to exist. …

In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in $\mathbb{R}^{n+1}$($n=2,3)$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $Σ$ with nonnegative scalar curva…

Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.

Study hyperplanes in abelian groups and their signatures for manifold identification.

Study on recovering sparse linear classifiers from mixed binary responses.