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 . This paper improves flow models to better handle perturbations in real-world data.
problem Flow models amplify initial errors in perturbed data, leading to poor generalization.
method Utilizes Bernstein-type polynomials to construct Normalizing Flows (NF) for higher robustness.
result Proposed NF framework provides theoretical upper bounds and practical advantages.
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.
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 , ∞ ] . 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.
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. BF-VI improves posterior approximation in complex models.
problem Inefficient posterior approximations in complex models.
method Combines normalizing flows and Bernstein polynomial transformations.
result BF-VI outperforms other VI methods in approximating complex multivariate posteriors.
New method estimates density functionals using polynomial basis without full distribution knowledge.
problem Estimating quantities like information divergence functions requires complete distribution knowledge and integration.
method Introduces data-driven basis functions and develops methods for basis expansions of functionals of two distributions.
result Approximates functions of distributions as closely as desired using the new basis set.
In this paper we use Bernstein and Chebyshev polynomials to approximate the price of some basket options under a bivariate Black-Scholes model. The method consists in expanding the price of a univariate related contract after conditioning on the remaining underlying assets and calculating the mixed exponential-power mo…
In this note, we prove that smooth self-shrinkers in $\Real^{n+1}$ , that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this note is that no growth assumption at infinity is need…
The paper equidistributes zeros of random polynomials and sections on manifolds.
problem Equidistribution of zeros of random polynomials and sections on manifolds.
method Weighted pluripotential theory, asymptotic Bernstein-Markov measures, variance estimation.
result Equidistribution holds for non-i.i.d. random coefficients and non-homogeneous manifolds.
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.
Develops a nonparametric method to estimate isotropic covariance functions efficiently.
problem Estimating isotropic covariance functions without assuming a specific parametric form.
method Uses Bernstein polynomials and sieve maximum likelihood estimation.
result Consistent estimator with improved performance compared to parametric and nonparametric alternatives.
We utilize copulas to constitute a unified framework for constructing and optimizing variational proposals in hierarchical Bayesian models. For models with continuous and non-Gaussian hidden variables, we propose a semiparametric and automated variational Gaussian copula approach, in which the parametric Gaussian copul…
We give improved constants for data dependent and variance sensitive confidence bounds, called empirical Bernstein bounds, and extend these inequalities to hold uniformly over classes of functionswhose growth function is polynomial in the sample size n. The bounds lead us to consider sample variance penalization, a nov…
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.
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of ℓ \ell ℓ -form-valued k k k -forms on triangulations. result Generalizes existing finite element methods for various tensor fields.
Derives a new maximum principle for Riemannian manifolds with volume growth constraints.
problem Maximum principles for functions on Riemannian manifolds with specific volume growth conditions.
method Derives a new maximum principle using vector fields and divergence conditions.
result Applies the principle to Bernstein-type results and existence of minimal submanifolds.
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.
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.
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…
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.
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 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.
Analytic networks with bounded coefficients can't outperform polynomial approximations.
problem Approximation limits of neural networks with analytic activation functions under coefficient constraints.
method Deterministic analysis using comparison argument and Bernstein-type estimates.
result Networks with analytic activation functions and controlled coefficients cannot outperform classical polynomial approximation rates on non-analytic targets.
We will prove that \emph{there are no stable complete hypersurfaces of R 4 \mathbb{R}^4 R 4 with zero scalar curvature, polynomial volume growth and such that ( − K ) H 3 ≥ c > 0 \dfrac{(-K)}{H^3}\geq c>0 H 3 ( − K ) ≥ c > 0 everywhere, for some constant c > 0 c>0 c > 0 }, where K K K denotes the Gauss-Kronecker curvature and H H H denotes the mean curvature of the immersion. …
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.
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 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 . 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}.
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.
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.
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…
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.
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…
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.
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.
New inequality for ternary variables improves on existing measures.
problem Analyzing excess losses and weighted majority votes with ternary random variables.
method Developed a split-kl inequality and its PAC-Bayes extension.
result Outperforms existing inequalities in certain regimes.
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.
Proves properties of hypersurfaces in Euclidean space.
problem Bernstein type problem for λ-hypersurfaces.
method Analyzes the Gauss image and applies it to derive results.
result Affirmative answer to Bernstein type problem for λ-hypersurfaces.