In this paper, we study the properties of potential function of the translating soliton in and the volume growth of the intersection of Euclidean balls with . 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…
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48 results for “Bernstein condition”
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 and .
The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
problem Conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
method Proving Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in Euclidean and Minkowski contexts.
result Zero Gaussian curvature convex hypersurfaces must be hyperplanes if the mean curvature goes to zero at infinity.
New bounds for non-convex estimators without Bernstein condition.
problem Sharp excess risk bounds for non-convex and improper estimators.
method Exponential-tail local Rademacher complexity risk bounds with offset condition.
result Sharp bounds for non-convex and improper estimators without Bernstein condition.
We present a new PAC-Bayesian generalization bound. Standard bounds contain a $\sqrt{L_n \cdot \KL/n}$ complexity term which dominates unless , the empirical error of the learning algorithm's randomized predictions, vanishes. We manage to replace by a term which vanishes in many more situations, essentially …
We establish the following theorem of Bernstein type for the first Heisenberg group: Let S be a C^2 connected H-minimal surface which is a graph over some plane P, then S is either a non-characteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition.
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 . Namely, under certain natural cond…
Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.
problem Uniqueness of hypersurfaces with constant higher order mean curvature in hyperbolic space.
method Generalization of Bernstein theorem and proof of Bernstein type results for immersed hypersurfaces.
result Rigidity of horospheres and equidistant spheres in terms of their higher order mean curvatures.
Bayes meets Bernstein at the Meta Level: an Analysis of Fast Rates in Meta-Learning with PAC-Bayesstat.ML
Paper shows how meta-learning can reduce prior learning cost.
problem Learning the prior in meta-learning with fast rates.
method Examined Gibbs algorithm in meta-learning context.
result Bernstein's condition holds at meta level, reducing prior learning cost.
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…
We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is parabolic) and satisfy the null convergence condition. In particular, we provide severa…
The paper extends Bernstein Theorem for minimal spacelike surfaces in 4D Minkowski space.
problem Analyzing Bernstein property for minimal spacelike surfaces in 4D Minkowski space.
method Study of an extension of the Bernstein Theorem for minimal spacelike surfaces in R^4_1.
result The Bernstein property does not hold in general for graphic spacelike surfaces in R^4_1.
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…
In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in , namely, if the $L^n…
Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flowscs.LG
The paper proposes a new method for probabilistic load forecasting using Bernstein-Polynomial Normalizing Flows.
problem High variability in short-term load forecasting at the low-voltage level due to fluctuating demand and increasing electrification.
method Flexible conditional density forecasting based on Bernstein polynomial normalizing flows with neural network control.
result Density predictions outperform traditional methods for 24h-ahead load forecasting.
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.
Paper estimates curvature of minimal surfaces in a specific geometric space.
problem Estimating curvature of minimal hypersurfaces in Heisenberg groups.
method Extending Simons formula and Kato inequality to sub-Riemannian setting, applying to stable hypersurfaces.
result Integral curvature estimates for stable hypersurfaces in Heisenberg groups.
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.
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…
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…
Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.
problem Cohomology of BGG complexes on bounded Lipschitz domains.
method Computes cohomology of conformal deformation and Hessian complexes in Sobolev spaces, allowing multiple input complexes.
result Establishes conformal Korn inequality and proposes generalizations of continuum models with microstructures.
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.
Modelling and simulation of dependence structures in nonlife insurance with Bernstein copulasstat.ME
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.
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.
On Bernstein-type theoremsmath.DG
We summarize results concerning the Bernstein property of differential equations.
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 for UCRL2B.
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 -anisotropic minimal hypersurfaces.
result The only entire smooth solutions to the -anisotropic minimal hypersurfaces equation are linear functions.
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 -dimensional minimal Lagrangian graphs in quaternion Euclidean space . In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in
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, …
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 with high probability.
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 and .
Bernstein type gradient estimate for system of weighted local heat equations with potential termmath.AP
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}.
Solves index problem for curved BGG sequences in parabolic geometry.
problem Index theory of curved Bernstein-Gelfand-Gelfand sequences.
method Utilizes K-homology and noncommutative geometry.
result Solves the index problem for BGG-sequences on flat parabolic geometry.
A new method improves uncertainty quantification in Bayesian inference.
problem Poor uncertainty quantification in traditional Gibbs posteriors.
method Sequential Gibbs posteriors with a Bernstein-von Mises theorem.
result Sequential Gibbs posteriors provide better frequentist coverage.
We obtain necessary conditions for the existence of complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. In the two-dimensional case we prove Bernstein-type results in each of these ambient spaces.
Study of spacelike hypersurfaces in twisted product spacetimes with specific conditions.
problem Characterizing and understanding spacelike hypersurfaces in twisted product spacetimes.
method Analysis of conditions ensuring global hyperbolicity, use of conformal geometry results.
result Non-existence results for constant mean curvature hypersurfaces under certain conditions.
We improve bounds for stochastic processes, especially those with heavy tails.
problem Bounding the concentration of sub- processes with heavy tails.
method Variational approach to concentration, focusing on sub-Gaussian and other tail conditions.
result First dimension-free self-normalized empirical Bernstein inequality.
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.
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…
Synthesizes robust estimators for domain adaptation.
problem Improving prediction accuracy in target domain with limited data.
method Synthesizes a family of robust least squares estimators using convex optimization.
result Robust strategies can outperform non-robust interpolations.
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.
This paper gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a counter-example on the problem of Bernstein type.
Proof that certain complex equations have only simple solutions.
problem Characterizing solutions to complex equations.
method Analyzing fully nonlinear elliptic operators.
result Entire smooth solutions are quadratic polynomials.
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.
problem Finding minimal embedded hypertori in 4D sphere.
method Analyzing minimally embedded and immersed hypertori and hyperspheres.
result Infinitely many non-isometric minimally embedded hypertori and hyperspheres found.