17,801 research outputs found
Strongly Charged Polymer Brushes
Charged polymer brushes are layers of surface-tethered chains. Experimental
systems are frequently strongly charged. Here we calculate phase diagrams for
such brushes in terms of salt concentration n_s, grafting density and polymer
backbone charge density. Electrostatic stiffening and counterion condensation
effects arise which are absent from weakly charged brushes. In various phases
chains are locally or globally fully stretched and brush height H has unique
scaling forms; at higher salt concentrations we find H ~ n_s^(-1/3), in good
agreement with experiment.Comment: 5 pages, 3 Postscript figure
Existence and space-time regularity for stochastic heat equations on p.c.f. fractals
We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped
with regular harmonic structures. We show that if the spectral dimension of the
set is less than two, then function-valued "random-field" solutions to these
SPDEs exist and are jointly H\"older continuous in space and time. We calculate
the respective H\"older exponents, which extend the well-known results on the
H\"older exponents of the solution to SHE on the unit interval. This shows that
the "curse of dimensionality" of the SHE on depends not on the
geometric dimension of the ambient space but on the analytic properties of the
operator through the spectral dimension. To prove these results we establish
generic continuity theorems for stochastic processes indexed by these
p.c.f.s.s. sets that are analogous to Kolmogorov's continuity theorem. We also
investigate the long-time behaviour of the solutions to the fractal SHEs
Bounds of incidences between points and algebraic curves
We prove new bounds on the number of incidences between points and higher
degree algebraic curves. The key ingredient is an improved initial bound, which
is valid for all fields. Then we apply the polynomial method to obtain global
bounds on and .Comment: 11 page
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