Improved neural network approximates analytic and L^p functions efficiently.
problem Efficiently approximating analytic and L^p functions using neural networks.
method Three-dimensional ReLU network architecture for sawtooth functions, improving approximation rates.
result Substantially improved exponential approximation rates for analytic functions and general L^p functions.
Study rigidity in Riemannian manifolds using Pohozoaev and P-function approaches.
problem Rigidity in Serrin's overdetermined problems in Riemannian manifolds.
method Prove a Pohozoaev-type identity, use conformal vector field, and apply P-function approach.
result Show Serrin's type rigidity result in Riemannian manifolds.
Study proves symmetry of bounded domains in Riemannian manifolds.
problem Symmetry of bounded domains in Riemannian manifolds.
method Integral identities and P-function method. result Equality implies the domain is isometric to a Euclidean ball.
Minimum width for ReLU networks to approximate L^p functions is max(d_x+1, d_y).
problem Characterizing the minimum width for ReLU networks to approximate L^p functions.
method Analyzing networks with ReLU activation functions and proving the minimum width required.
result The minimum width required for the universal approximation of L^p functions is exactly max(d_x+1, d_y).
Study on p-Laplace equation in convex cones, proving rigidity under specific conditions.
problem Overdetermined problem for p-Laplace equation in convex cones. method Established properties of capacitary potential, used P-function, isoperimetric inequality, and Heintze-Karcher inequality. result Rigidity result under orthogonal intersection assumption.
We consider an overdetermined Serrin's type problem in space forms and we generalize Weinberger's proof in [Arch. Rational Mech. Anal., 43 (1971)] by introducing a suitable P-function.
The paper proves positivity preservation and self-adjointness for Schrödinger operators on incomplete Riemannian manifolds.
problem Positivity preservation and self-adjointness for Schrödinger-type operators on incomplete Riemannian manifolds.
method Control of potential behavior near the Cauchy boundary, essential self-adjointness proof, core of smooth compactly supported functions.
result Positivity preservation and essential self-adjointness of Schrödinger operators on Lp functions on incomplete Riemannian manifolds. Regularity results for geodesic X-ray transform on nonsmooth manifolds
problem Geodesic X-ray transform on nonsmooth simple manifolds
method Symbol smoothing arguments and pseudodifferential operators with low regularity symbols
result Improved injectivity results for Lp functions Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.
problem Overdetermined problem for p-Laplacian equation on compact Riemannian manifolds.
method Introduced a new P-function related to the first nonzero eigenvalue for p-Laplacian, derived integral identities, and applied them to achieve inequalities and the Soap Bubble Theorem.
result Achieved the Heintze-Karcher type inequality and the Soap Bubble Theorem.
On 7D quaternionic contact manifolds, eigenvalue bounds imply special structure.
problem Eigenvalue bounds on quaternionic contact manifolds.
method Lower Ricci curvature bound and non-negative P-function. result Eigenvalue is smallest if and only if manifold is qc-Einstein.
New integral estimates on substatic manifolds improve Alexandrov Theorem.
problem Improving integral estimates on substatic manifolds.
method Introducing a new vector field with nonnegative divergence.
result Generalization and improvement of integral estimates leading to Alexandrov Theorem.
Paper studies unique interior points and estimates for generalized translating soliton problems.
problem Generalized translating soliton type problems.
method Proves uniqueness of interior critical points, derives C0 and C1 estimates using minimum principles. result Derives a priori C0 and C1 estimates for solutions. Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Mo…
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…
Two signature-based methods solve optimal stopping in non-Markovian frameworks.
problem Optimal stopping in non-Markovian frameworks, particularly pricing American options.
method Primal and dual formulations using linear functionals of rough path signatures.
result Both primal and dual methods converge and provide numerical examples.
Neural networks can approximate any L^p functions on R^n.
problem Approximating functions on unbounded domains with neural networks.
method Monotone sigmoid, ReLU, ELU, Softplus, LeakyReLU activation functions.
result Shallow neural networks can arbitrarily well approximate L^p functions on R^n.
The paper proves a conjecture about positivity preserving in Riemannian manifolds.
problem Proving positivity preserving for Lp functions on Riemannian manifolds. method New a-priori regularity result, Liouville type theorem, Brezis-Kato inequality.
result Proves a conjecture by M. Braverman, O. Milatovic, and M. Shubin (2002).
The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.
problem Proving Liouville-type theorems for positive harmonic functions on specific types of manifolds.
method Employing the P-function method and a closed conformal vector field inherent to such manifolds.
result Confirms some cases of Wang's conjecture and provides a partial verification of Wang's conjecture on warped product manifolds.
These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R^3 with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, and Ros states that such an M must…
Minimum width for ReLU networks on compact domain is exactly max{d_x, d_y, 2}
problem Characterizing the minimum width for ReLU networks to approximate functions on compact domains
method Analyzing the minimum width for Lp approximation of Lp functions from [0,1]d to Rdy using ReLU-like activation functions result The minimum width for Lp approximation on a compact domain is exactly max{d_x, d_y, 2} for ReLU-like activation functions Study proves deep narrow RNNs can approximate any function, with minimum width independent of data length.
problem Proving universality of deep narrow RNNs with bounded widths.
method Analyzing RNNs as dynamical systems, proving universality for deep narrow structures with specific widths.
result Minimum width for universality of deep narrow RNNs is independent of data length.
Study nonexistence and gradient estimates for solutions on manifolds with bounded Ricci curvature.
problem Nonexistence and gradient estimates for solutions of a specific quasi-linear equation on manifolds.
method Utilizes Sobolev inequalities and geometric properties to establish results.
result Extends and improves previous results on nonexistence and gradient estimates.
Let M be a compact Riemannian submanifold of Rm of dimension d and let X1,...,Xn be a sample of i.i.d. points in M with uniform distribution. We study the random operators Δhn,nf(p):=nhnd+21∑i=1nK(hnp−Xi)(f(Xi)−f(p)),p∈M where ${K(u…
The paper proves rigidity theorems for space-like hypersurfaces in Minkowski space.
problem Stability and rigidity of space-like hypersurfaces in Minkowski space.
method Weinberger-type approach with P-functions and integral identities.
result Hypersurfaces with constant higher-order mean curvature ratios and specific boundary conditions are parts of hyperboloids.
The paper proves conditions for positivity preservation on Riemannian manifolds.
problem Conditions for positivity preservation on Riemannian manifolds.
method Smooth monotonic approximation, subharmonic distributions, and manifold versions of inequalities.
result Conditions for Lp-Positivity Preservation on Riemannian manifolds.