arXiv research

CascadeBAI identifies best arms in cascading bandits with fixed confidence.

The paper introduces methods to quantify uncertainty in sampling without replacement.

Improved exploration in factored average-reward MDPs reduces regret.

A new method for streaming PCA provides confidence intervals for eigenvector entries.

Improved understanding of translating solitons using new techniques.

Paper estimates curvature of minimal surfaces in a specific geometric space.

New algorithm learns optimal path in reinforcement learning with linear approximations.

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

The paper proves Calabi-Bernstein type results for minimal and maximal surfaces in 3D and 3D-L spacetime.

Based on a calibration argument, we prove a Bernstein type theorem for entire minimal graphs over Gauss space $\mathbb{G}^n$ by a simple proof.

In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^n\cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^n$

A weighted area estimate for entire graphs with bounded weighted mean curvature in Gauss space is given by a simple proof. Bernstein type theorems for self shrinkers (\cite {wa}) as well as for graphic $λ$-hypersurfaces (\cite{ chwe2}) follow immediately as consequences.

In this paper, our purpose is to study rigidity theorems for $λ$-hypersurfaces in Euclidean space under Gauss map. As a Bernstein type problem for $λ$-hypersurfaces, we prove that an entirely graphic $λ$-hypersurface in Euclidean space is a hyperplane.

Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space $\boldsymbol L^3$ which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space $\boldsymbol E^3$ and maximal surfaces in Lorentz-Minko…

Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\boldsymbol R^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved a line theorem for hypersurfaces at their degenerate light-like poi…

We proved that any complete hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ whose Gauss image is contained in an open hemisphere has to be proper. As applications, we derive a counterpart of Hoffman-Osserman-Schoen's result for $λ$-hypersurfaces, which gives an affirmative answer to the Bernstein type problem pr…

For entire spacelike stationary 2-dimensional graphs in Minkowski spaces, we establish Bernstein type theorems under specific boundedness assumptions either on the W-function or on the total (Gaussian) curvature. These conclusions imply the classical Bernstein theorem for minimal surfaces in 3-dimensional Euclidean spa…

We introduce a Bernstein-type inequality which serves to uniformly control quadratic forms of gaussian variables. The latter can for example be used to derive sharp model selection criteria for linear estimation in linear regression and linear inverse problems via penalization, and we do not exclude that its scope of a…

We summarize results concerning the Bernstein property of differential equations.

The paper provides bounds for high-dimensional U-statistics with novel order-explicit inequalities.

The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.

New theorems prove uniqueness of solutions to geometric PDEs.

Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore…

Develops a deep learning framework for various data types.

Article provides Bernstein gradient estimates for heat equations with potential terms.

Paper proves a theorem about constant mean curvature surfaces in isotropic 3-space.

The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.

In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…

In this paper, we discuss the self-shrinking systems in higher codimensional spaces. We mainly obtain several Bernstein type results and a sharp growth estimate.

We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.

In this note we prove that every two-dimensional entire Willmore graph in $R^3$ with square integrable mean curvature is a plane.

Self-shrinkers are important geometric objects in the study of mean curvature flows, while the Bernstein Theorem is one of the most profound results in minimal surface theory. We prove a Bernstein type result for graphical self-shrinker surfaces with codimension two in $\mathbb{R}^4$. Namely, under certain natural cond…

This paper studies the geometry of minimum-volume confidence sets for multinomial parameters.

Algorithm finds Nash equilibria in complex games with function approximation.

We introduce the beta function of a knot in euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues.

Study of spacelike hypersurfaces in twisted product spacetimes with specific conditions.

Algorithm constructs confidence sets for deep neural networks with PAC guarantees.

We obtain a Bernstein type result for entire two dimensional minimal graphs in $\mathbb{R}^{4}$, which extends a previous one due to L. Ni. Moreover, we provide a characterization for complex analytic curves.

The study proves properties of capillary graphs in half-spaces.

We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter θ, i.e. we want to construct a confidence set for theta that contains theta with high probability, and that is as small as possible. The l_2 diameter of a …

We identify a region $\Bbb{W}_{\f{1}{3}}$ in a Grassmann manifold $\grs{n}{m}$, not covered by a usual matrix coordinate chart, with the following important property. For a complete $n-$submanifold in $\ir{n+m} \, (n\ge 3, m\ge2)$ with parallel mean curvature whose image under the Gauss map is contained in a compact su…

In the lorentzian product $\Bbb G^n\times\Bbb R_1,$ we give a comparison between the $f$-volume of an entire $f$-maximal graph and the $f$-volume of the hyperbolic $H_r^+$ under the assumption that the gradient of the function defining the graph is bounded away from 1. As a consequence, we obtain a Bernstein type theor…

We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in $(\mathbb{R}^n\times\mathbb{R}^n, g_τ)$. The proofs rely on a priori estimates and barriers construction.

The aim of this expository article is to reveal the equivalence of Jörgens' Theorem on the two dimensional unimodular Hessian equation and Fu's Theorem on the two dimensional special Lagrangian equation.

Let Σbe a complete minimal Lagrangian submanifold of \C^n. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σlies in one of these regions, then Σis an affine space.

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

New method constructs confidence sets for GLMs via game theory.

New methods improve confidence set calibration in complex models.