793 research outputs found
Mean Field Limit for Coulomb-Type Flows
We establish the mean-field convergence for systems of points evolving along
the gradient flow of their interaction energy when the interaction is the
Coulomb potential or a super-coulombic Riesz potential, for the first time in
arbitrary dimension. The proof is based on a modulated energy method using a
Coulomb or Riesz distance, assumes that the solutions of the limiting equation
are regular enough and exploits a weak-strong stability property for them. The
method can handle the addition of a regular interaction kernel, and applies
also to conservative and mixed flows. In the appendix, it is also adapted to
prove the mean-field convergence of the solutions to Newton's law with Coulomb
or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson
type system.Comment: Final version with expanded introduction, to appear in Duke Math
Journal. 35 page
Torsion order of smooth projective surfaces
To a smooth projective variety whose Chow group of -cycles is -universally trivial one can associate its torsion index ,
the smallest multiple of the diagonal appearing in a cycle-theoretic
decomposition \`a la Bloch-Srinivas. We show that is the
exponent of the torsion in the N\'eron-Severi-group of when is a
surface over an algebraically closed field , up to a power of the
exponential characteristic of .Comment: A few more minor changes in Colliot-Th\'el\`ene's appendi
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