Estimates for double Hilbert transform
Reconstructing a neural net from its output.
Neural nets were originally introduced as highly simplified systems of the neural system. Today they are widely used in technology and studied theoretically by scientists from several disciplines. (See e.g. [N]). However they remain little understood. (...)
Interpolation and extrapolation of smooth functions by linear operators.
Números, átomos y estrellas.
A generalized sharp Whitney theorem for jets.
Suppose that, for each point x in a given subset E ⊂ R, we are given an m-jet f(x) and a convex, symmetric set σ(x) of m-jets at x. We ask whether there exist a function F ∈ C(R) and a finite constant M, such that the m-jet of F at x belongs to f(x) + Mσ(x) for all x ∈ E. We give a necessary and sufficient condition for the existence of such F, M, provided each σ(x) satisfies a condition that we call "Whitnet w-convexity".
Formation of Singularities in Fluid Interfaces
Convergence on almost every line for functions with gradient in
We prove that if for certain values of , then
A local Bernstein inequality on real algebraic varieties.
Nodal sets of eigenfunctions on Riemannian manifolds.
Maximal seminorms on Weak
The canonical seminorm on Weak L¹
Bernstein's inequality on algebraic curves
Pseudo-differential operators with symbols admitting negative values
Lower bounds for Schrödinger equations
The atomic and molecular nature of matter.
The purpose of this article is to show that electrons and protons, interacting by Coulomb forces and governed by quantum statistical mechanics at suitable temperature and density, form a gas of Hydrogen atoms or molecules.
The mathematics of large atoms
The spin of the ground state of an atom.
In this paper we address a question posed by M. and T. Hoffmann-Ostenhof, which concerns the total spin of the ground state of an atom or molecule. Each electron is given a value for spin, ±1/2. The total spin is the sum of the individual spins.
Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential.
In [FS1] we announced a precise asymptotic formula for the ground-state energy of a non-relativistic atom. The purpose of this paper is to establish an elementary inequality that plays a crucial role in our proof of that formula. The inequality concerns the Thomas-Fermi potential VTF = -y(ar) / r, a > 0, where y(r) is defined as the solution of ⎧ y''(x) = x-1/2y3/2(x), ⎨ y(0) =...
Estimates of kernels on three-dimensional CR manifolds.
Waves in Honeycomb Structures
We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, . In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution , for data , which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion of the...
Page 1 Next