The paper explores nonconvex penalties using Bernstein functions for sparse estimation.
problem Sparse estimation in high-dimensional problems.
method Nonconvex penalties based on Bernstein functions, with coordinate descent and proximal alternating linearized minimization methods.
result The Bernstein penalty leads to effective sparse estimation and classification.
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…
Improved understanding of translating solitons using new techniques.
problem Understanding translating solitons in geometry.
method Using a new test function and gradient estimate technique.
result Better Bernstein type result of translating solitons.
New formulae derived for conformal symmetry breaking operators.
problem Understanding conformal symmetry breaking operators.
method Bernstein-Sato identities for distribution kernels.
result New formulae for conformal symmetry breaking differential operators.
The paper proves Calabi-Bernstein type results for minimal and maximal surfaces in 3D and 3D-L spacetime.
problem Characterizing minimal and maximal surfaces in 3D and 3D-L spacetime.
method Analyzing surfaces with specific properties and using geometric and functional methods.
result Calabi-Bernstein type results for critical points of a weighted area functional in R 3 \mathbb{R}^{3} R 3 and L 3 \mathbb{L}^{3} L 3 . Simple proof shows Bernstein property fails for a minimal surface equation.
problem Whether a specific minimal surface equation has the Bernstein property.
method Simple argument to show the equation does not have the Bernstein property.
result The minimal surface equation does not have the Bernstein property.
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 M M in R n + 1 R^{n+1} R n + 1 and the volume growth of the intersection of Euclidean balls with M M 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…
Bernstein theorem proven for 2-valued minimal graphs in 4D.
problem Classifying 2-valued minimal graphs in 4D.
method Analyzing blowdown cones and combinatorial arguments.
result Two-valued minimal graphs in 4D are unions of two 3D planes.
Study on f f f -maximal graphs in a Lorentzian product, proving Bernstein theorem.
problem Characterize f f f -maximal graphs in a Lorentzian product. method Comparison of f f f -volumes and Bernstein theorem. result Essential condition on gradient for Bernstein theorem.
The paper proves a Bernstein property for a specific complex partial differential equation.
problem Investigating the Bernstein property for a complex partial differential equation.
method Analyzing a fourth order complex partial differential equation with specific conditions.
result The equation has a Bernstein property under certain conditions.
Minimal surface equation results in constant solutions on RCD spaces.
problem Analyzing minimal surfaces on RCD spaces.
method Using properties of RCD spaces and the minimal surface equation.
result Positive solutions to the minimal surface equation are constant on RCD spaces.
Abstract: Summarizes Bernstein property results for differential equations.
problem Summarizing Bernstein property results for differential equations.
method Not specified in the abstract, likely involves mathematical analysis and differential equations.
result Not specified in the abstract, likely involves proving or disproving the Bernstein property for specific equations.
Adaptive Bernstein copulas improve risk management by preventing overfitting and reducing simulation effort.
problem Overfitting and high simulation effort in estimating dependence models.
method Constructive approach to Bernstein copulas with an admissible discrete skeleton.
result Comparison of different copula approaches in risk management shows improved accuracy and efficiency.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
We prove new concentration inequalities for random variables.
problem Concentration of random variables in nonlinear functions.
method Efron-Stein inequalities and PAC-Bayesian approach.
result User-friendly concentration bounds for various applications.
Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed re…
Researchers solve a Riemannian geometry problem using warped products.
problem Solving a Moser-Bernstein problem in warped Riemannian manifolds.
method Study entire solutions to the minimal hypersurface equation in warped products.
result Solves the Moser-Bernstein problem in a broader class of Riemannian manifolds.
The study proves that certain minimal surfaces are flat under specific conditions.
problem Characterizing minimal surfaces in anisotropic spaces.
method Proving a Bernstein theorem for Φ Φ Φ -anisotropic minimal hypersurfaces. result The only entire smooth solutions to the Φ Φ Φ -anisotropic minimal hypersurfaces equation are linear functions. Minimal graph theorem proven for convex domains.
problem Characterizing minimal graphs over convex domains.
method Analyzing minimal surface equation solutions on convex domains.
result Minimal graphs over convex domains are linear.
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 R 4 \mathbb{R}^4 R 4 . Namely, under certain natural cond…
The paper proves that certain stationary hypersurfaces in high dimensions are essentially flat.
problem Characterizing stationary hypersurfaces in high-dimensional spaces.
method Analyzing the Euler-Dierkes-Huisken functional to prove the flatness of hypersurfaces.
result Smooth, complete, connected, embedded stationary hypersurfaces in high dimensions are linear.
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.
The study proves surfaces in a specific Heisenberg group must be simple planes.
problem Characterizing surfaces in a sub-Finsler Heisenberg group.
method Analyzes ( X , Y ) (X,Y) ( X , Y ) -Lipschitz surfaces in H 1 \mathbb{H}^1 H 1 with a sub-Finsler structure. result Complete, oriented, stable ( X , Y ) (X,Y) ( X , Y ) -Lipschitz surfaces are vertical planes. Ancient solutions to mean curvature flow have unique shapes.
problem Understanding unique shapes of ancient solutions.
method Proved a Bernstein theorem for ancient solutions.
result Ancient solutions to mean curvature flow have unique shapes.
A new method for high-dimensional classification using Bernstein polynomials.
problem Computational difficulties in high-dimensional SVM hinge loss.
method Proposes Bernstein support vector machine (BernSVM) and two efficient algorithms.
result Achieves a prediction accuracy rate of s log ( p ) / n \sqrt{s\log(p)/n} s log ( p ) / n with high probability. The study models insurance dependence using Bernstein copulas.
problem Modeling dependence structures in nonlife insurance data.
method Review and suggest fitting Bernstein copulas to empirical data.
result Monte Carlo simulation and PML estimation for aggregate losses.
Paper uses a new copula to model risk aggregation and capital allocation.
problem Modeling dependence between risks for risk aggregation and capital allocation.
method Uses a generalized Archimedean copula (mixed Bernstein copula) to define dependence structure and derives closed-form risk measures.
result Closed-form expressions for tail value-at-risk and allocations are derived.
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…
Explains Bernstein theorems for various geometric PDEs.
problem Bernstein problem for minimal surface, Monge-Ampère, and special Lagrangian equations.
method Expository review of existing theorems and systems.
result Discussion of Bernstein theorems for different geometric PDEs.
Improved analysis of UCRL2 with empirical Bernstein inequality reduces exploration-exploitation regret.
problem Exploration-exploitation in communicating Markov Decision Processes.
method Analysis of UCRL2 with Empirical Bernstein inequalities (UCRL2B).
result Regret bound of O ~ ( D Γ S A T ) \widetilde{O}(\sqrt{DΓS A T}) O ( D Γ S A T ) for UCRL2B. Ancient symplectic solutions to mean curvature flow are flat.
problem Understanding ancient solutions to mean curvature flow in symplectic geometry.
method Using a complex phase map to prove a Bernstein theorem.
result Ancient solutions to the symplectic mean curvature flow are flat.
In this paper, we prove some Bernstein type results for n n n -dimensional minimal Lagrangian graphs in quaternion Euclidean space H n ≅ R 4 n H^n\cong R^{4n} H n ≅ R 4 n . In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in C n C^n C n
The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.
problem Understanding local growth properties of Laplace eigenfunctions on compact Riemannian manifolds.
method Refined Donnelly-Fefferman method based on L 2 L^{2} L 2 --Carleman estimates, combined with elliptic regularity and patching of local Carleman estimates. result Almost sharp local L p L^{p} L p --Bernstein inequalities for p ∈ [ 1 , ∞ ] p\in[1,\infty] p ∈ [ 1 , ∞ ] . Article provides Bernstein gradient estimates for heat equations with potential terms.
problem Gradient estimates for heat equations with potential terms on weighted Riemannian manifolds.
method Derived Bernstein type gradient estimates for two systems of heat equations with linear, exponential, and combined potentials.
result Resolves part of the problem raised by Bhattacharyya et al. in \cite{SB-1}.
We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear e…
Bayesian optimisation for expensive experiments with shape prior.
problem Expensive experiments with time-varying control variables.
method Developed a novel Bayesian optimisation framework using Bernstein polynomial basis and dynamic polynomial degree adjustment.
result Demonstrated effectiveness on polymer fibre design and learning rate optimisation.
Study fast learning rates for heavy-tailed losses without boundedness.
problem Analyzing fast learning rates for heavy-tailed losses.
method Introducing two new conditions: envelope function and multi-scale Bernstein's condition.
result Proves learning rates faster than O ( n − 1 / 2 ) O(n^{-1/2}) O ( n − 1/2 ) and can be arbitrarily close to O ( n − 1 ) O(n^{-1}) O ( n − 1 ) . The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.
problem Bernstein problem for smooth maps to lower dimensions forming calibrated submanifolds.
method Established conditions for maps to be affine based on the slope's second elementary symmetric polynomial.
result Conditions ensuring maps are affine for coassociative and Cayley submanifolds in R 7 \mathbb{R}^7 R 7 and R 8 \mathbb{R}^8 R 8 . The paper extends a Bernstein theorem to codimension 2 minimal submanifolds.
problem Proving a Bernstein theorem for minimal submanifolds in higher codimension.
method Using convexity properties of Grassmannians and Allard's theorem.
result Minimal submanifolds in codimension 2 must be planes.
Complex analytic sets' Lipschitz geometry at infinity characterized.
problem Characterize entire complex analytic sets based on their Lipschitz geometry at infinity.
method Proved a complex non-parametric version of Moser's Bernstein Theorem and characterized algebraicity.
result Entire complex analytic sets at infinity are affine linear subspaces if and only if they are bi-Lipschitz homeomorphic to algebraic sets.
Improves Bernstein theorem for space-like graphs in Lorentz-Minkowski space.
problem Proves a new Bernstein-type theorem for space-like zero mean curvature graphs.
method Uses fluid mechanical duality between minimal surfaces and maximal surfaces.
result Shows that a zero mean curvature graph with only space-like and light-like points is a plane.
Sharp inequalities for matrix means with unknown variance.
problem Estimating matrix means with unknown variance.
method Empirical Bernstein inequalities for symmetric random matrices.
result Adapts to unknown variance with tight deviation bounds.
Improves Bernstein theorem for zero mean curvature hypersurfaces in Lorentz-Minkowski space.
problem Proving entire zero mean curvature graphs are hyperplanes in Lorentz-Minkowski space.
method Using line theorems at degenerate light-like points to generalize Bernstein theorem.
result Entire zero mean curvature graphs in Lorentz-Minkowski space are hyperplanes if they only contain space-like or light-like points.
Paper proves stable minimal surfaces in 3D are flat.
problem Understanding stable minimal surfaces in 3D.
method Analyzes quadratic area growth and stability conditions.
result Stable minimal Plateau surfaces in 3D are flat.
Study Bernstein results for self-shrinking solutions in Lagrangian flow.
problem Understanding entire self-shrinking solutions in Lagrangian flow.
method Prior estimates and barriers construction.
result Showed Bernstein type results for self-shrinking solutions.
Simple proof for graph curvature in Gauss space leads to new theorems.
problem Proving curvature bounds for graphs in Gauss space.
method Simple proof using weighted area estimates.
result Bernstein type theorems for self-shrinkers and graphic hypersurfaces.
Extends Onsager's conjecture to Besov spaces on manifolds with boundary.
problem Proving Onsager's conjecture on Riemannian manifolds with boundary.
method Constructing Hodge-Neumann heat kernel, obtaining off-diagonal decay and local Bernstein estimates.
result Extends Onsager's conjecture to Besov spaces B ^ 3 , V 1 3 \widehat{B}_{3,V}^{\frac{1}{3}} B 3 , V 3 1 .