arXiv research

Adaptive Bernstein copulas improve risk management by preventing overfitting and reducing simulation effort.

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

In this paper we study nonconvex penalization using Bernstein functions. Since the Bernstein function is concave and nonsmooth at the origin, it can induce a class of nonconvex functions for high-dimensional sparse estimation problems. We derive a threshold function based on the Bernstein penalty and give its mathemati…

In this paper we study nonconvex penalization using Bernstein functions whose first-order derivatives are completely monotone. The Bernstein function can induce a class of nonconvex penalty functions for high-dimensional sparse estimation problems. We derive a thresholding function based on the Bernstein penalty and di…

Ancient solutions to mean curvature flow have unique shapes.

The study models insurance dependence using Bernstein copulas.

Improved understanding of translating solitons using new techniques.

We summarize results concerning the Bernstein property of differential equations.

Explains Bernstein theorems for various geometric PDEs.

Improved analysis of UCRL2 with empirical Bernstein inequality reduces exploration-exploitation regret.

Ancient symplectic solutions to mean curvature flow are flat.

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 $C^n$

We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially new formulae for conformal symmetry breaking differential operators on functions, …

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

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

The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.

Sharp inequalities for matrix means with unknown variance.

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…

In this paper, we study the properties of potential function of the translating soliton $M$ in $R^{n+1}$ and the volume growth of the intersection of Euclidean balls with $M$. We give a condition to obtain the Bernstein theorem for the translating solitons. We also give an outline of a simple proof of the Bernstein the…

Paper proves stable minimal surfaces in 3D are flat.

We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.

Minimal hypertori found in 4D sphere, solving Bernstein conjecture.

New inequality for ternary variables improves on existing measures.

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.

New maximal surfaces solve Bernstein problems.

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 note, we propose Bernstein's problem and De Giorgi's conjecture for spatially inhomogeneous equations, as well as De Giorgi's conjecture for system of reaction-diffusion equations.

New minimal hypersphere found in 4-sphere solving Bernstein problem.

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.

In this note we present a short alternative proof for the Bernstein problem in the three-dimensional Heisenberg group ${\rm Nil}_3$ by using the loop group technique.

Study introduces new Bernstein inequalities for dependent data in Hilbert spaces.

Bernstein theorem proven for 2-valued minimal graphs in 4D.

We give an exposition, following joint works with J.-C. Zambrini, of the link between Euclidean Quantum Mechanics, Bernstein processes and isovectors for the heat equation. A new application to Mathematical Finance is then discussed.

Minimal surface equation results in constant solutions on RCD spaces.

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

Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.

This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.

A celebrated result of S. Bernstein states that every solution of the minimal surface equation over the entire plane has to be an affine linear function. Since the paper of Bernstein appeared in 1927, many different proofs and generalizations of this beautiful theorem were given, namely to higher dimensions and to more…

Classifies surfaces with no Gaussian curvature.

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.

The paper extends Bernstein Theorem for minimal spacelike surfaces in 4D Minkowski space.

We present a new PAC-Bayesian generalization bound. Standard bounds contain a $\sqrt{L_n \cdot \KL/n}$ complexity term which dominates unless $L_n$, the empirical error of the learning algorithm's randomized predictions, vanishes. We manage to replace $L_n$ by a term which vanishes in many more situations, essentially …

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

This paper improves flow models to better handle perturbations in real-world data.

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hy…