The damped stochastic wave equation on p.c.f. fractals

A p.c.f. fractal with a regular harmonic structure admits an associated Dirichlet form, which is itself associated with a Laplacian. This Laplacian enables us to give an analogue of the damped stochastic wave equation on the fractal. We show that a unique function-valued solution exists, which has an explicit formulation in terms of the spectral decomposition of the Laplacian. We then use a Kolmogorov-type continuity theorem to derive the spatial and temporal H\"older exponents of the solution. Our results extend the analogous results on the stochastic wave equation in one-dimensional Euclidean space. It is known that no function-valued solution to the stochastic wave equation can exist in Euclidean dimension two or higher. The fractal spaces that we work with always have spectral dimension less than two, and show that this is the right analogue of dimension to express the "curse of dimensionality" of the stochastic wave equation. Finally we prove some results on the convergence to equilibrium of the solutions

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 $\mathbb{R}^n$ 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 $\mathbb{R}$ and $\mathbb{C}$.Comment: 11 page