10,807 research outputs found
Poincar\'e and log-Sobolev inequalities for mixtures
This work studies mixtures of probability measures on and
gives bounds on the Poincar\'e and the log-Sobolev constant of two-component
mixtures provided that each component satisfies the functional inequality, and
both components are close in the -distance. The estimation of those
constants for a mixture can be far more subtle than it is for its parts. Even
mixing Gaussian measures may produce a measure with a Hamiltonian potential
possessing multiple wells leading to metastability and large constants in
Sobolev type inequalities. In particular, the Poincar\'e constant stays bounded
in the mixture parameter whereas the log-Sobolev may blow up as the mixture
ratio goes to or . This observation generalizes the one by Chafa\"i and
Malrieu to the multidimensional case. The behavior is shown for a class of
examples to be not only a mere artifact of the method.Comment: 13 page
Euler class groups, and the homology of elementary and special linear groups
We prove homology stability for elementary and special linear groups over
rings with many units improving known stability ranges. Our result implies
stability for unstable Quillen K-groups and proves a conjecture of Bass. For
commutative local rings with infinite residue fields, we show that the
obstruction to further stability is given by Milnor-Witt K-theory. As an
application we construct Euler classes of projective modules with values in the
cohomology of the Milnor Witt K-theory sheaf. For d-dimensional commutative
noetherian rings with infinite residue fields we show that the vanishing of the
Euler class is necessary and sufficient for a projective module P of rank d to
split off a rank 1 free direct summand. Along the way we obtain a new
presentation of Milnor-Witt K-theory.Comment: 64 pages. Revised Section 5. Comments welcome
Gluing Riemannian manifolds with curvature operators at least k
We glue two manifolds which have curvature operators at least k (in the sense
of eigenvalues) along their common boundary. We show that if the sum of the
second fundamental forms of the boundary is positive semidefinite, then the
curvature operator of the resulting manifold is at least k up to an arbitrarily
small error term. Similar results hold for Ricci, scalar, bi, isotropic and
flag curvature, respectively
Macroscopic limit of the Becker-D\"oring equation via gradient flows
This work considers gradient structures for the Becker-D\"oring equation and
its macroscopic limits. The result of Niethammer [17] is extended to prove the
convergence not only for solutions of the Becker-D\"oring equation towards the
Lifshitz-Slyozov-Wagner equation of coarsening, but also the convergence of the
associated gradient structures. We establish the gradient structure of the
nonlocal coarsening equation rigorously and show continuous dependence on the
initial data within this framework. Further, on the considered time scale the
small cluster distribution of the Becker--D\"oring equation follows a
quasistationary distribution dictated by the monomer concentration
Charge conservation in RHIC and contributiuons to local parity violation observables
Relativistic heavy ion collisions provide laboratory environments from which
one can study the creation of a novel state of matter, the quark gluon plasma.
The existence of such a state is postulated to alter the mechanism and
evolution of charge production, which then becomes manifest in charge
correlations. We study the separation of balancing charges at kinetic
freeze-out by analyzing recent result on balancing charge correlations for
Au+Au collisions at . We find that balancing charges
are emitted from significantly smaller regions in central collisions compared
to peripheral collisions. The results indicate that charge diffusion is small
and that the centrality dependence points to a change of the production
mechanism. In addition we calculate the contributions from charge-balance
correlations to STAR's local parity violation observable. We find that local
charge conservation, when combined with elliptic flow, explains much of STAR's
measurement.Comment: 11 pages, 10 figure
The shape of the proton at high energies
We present first calculations of the fluctuating gluon distribution in a
proton as a function of impact parameter and rapidity employing the functional
Langevin form of the JIMWLK renormalization group equation. We demonstrate that
when including effects of confinement by screening the long range Coulomb field
of the color charges, the evolution is unitary. The large-x structure of the
proton, characterized by the position of three valence quarks, retains an
effect on the proton shape down to very small values of x. We determine the
dipole scattering amplitude as a function of impact parameter and dipole size
and extract the rapidity evolution of the saturation scale and the proton
radius.Comment: 8 pages, 6 figure
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