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Reconstructing a neural net from its output.

Charles Fefferman — 1994

Revista Matemática Iberoamericana

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. (...)

A generalized sharp Whitney theorem for jets.

Charles Fefferman — 2005

Revista Matemática Iberoamericana

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".

The atomic and molecular nature of matter.

Charles L. Fefferman — 1985

Revista Matemática Iberoamericana

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 spin of the ground state of an atom.

Charles L. FeffermanLuis A. Seco — 1996

Revista Matemática Iberoamericana

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.

Charles L. FeffermanLuis A. Seco — 1993

Revista Matemática Iberoamericana

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) =...

Waves in Honeycomb Structures

Charles L. FeffermanMichael I. Weinstein — 2012

Journées Équations aux dérivées partielles

We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, V . In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of H V = - Δ + V and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution e - i H V t ψ 0 , for data ψ 0 , which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion of the...

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