We study the reduction of certain PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe ``good'' p-adic integral models of these Shimura varieties and study their étale local structure. In particular, we exhibit a stratification of their (singular) special fibers and give a partial calculation of the sheaf of nearby cycles.

For partial differential equations of mixed elliptic-hyperbolic and degenerate types which are the Euler-Lagrange equations for an associated Lagrangian, invariance with respect to changes in independent and dependent variables is investigated, as are results in the classification of continuous one-parameter symmetry groups. For the variational and divergence symmetries, conservation laws are derived via the method of multipliers. The conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems, and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems.

In our paper [GSV], we discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper, we consider the case of a general matrix of transition exponents. Our leading idea is that a relevant geometric object in this case is a certain closed 2-form compatible with the cluster algebra structure. The main example is provided by Penner coordinates on the decorated Teichmüller space, in which case the above form coincides with the classical Weil-Petersson symplectic form.

We consider the Cauchy problem for the system ∂_tu_i + div_z(g(|u|)u_i) = 0, i ∈ {1,…, k}, in m space dimensions and with g ∈ C³. When k ≥ 2 and m = 2, we show a wide choice of g's for which the bounded variation (BV) norm of admissible solutions can blow up, even when the initial data have arbitrarily small oscillation and arbitrarily small total variation, and are bounded away from the origin. When m ≥ 3, we show that this occurs whenever g is not constant, that is, unless the system reduces to k decoupled transport equations with constant coefficients.

The classical theory of conformal mappings involves best possible pointwise estimates of the derivative, thus supplying a measure of the extremal expansion/contraction possible for a conformal mapping. It is natural to consider also the integral means of |ϕ'|^t along circles |z| = r, where ϕ is the conformal mapping in question and t is a real parameter (0 < r < 1 if ϕ is defined in the unit disk, while 1 < r < +∞ if ϕ is defined in the exterior disk). The extremal growth rate as r → 1 of the integral means which follows from the classical pointwise estimates is by far too fast. Better estimates were found by Clunie, Makarov, Pommerenke, Bertilsson, Shimorin, and others. Here we introduce a new method—based on area-type estimates—which discards as little as possible of the information supplied by the area methods. The result is a considerable improvement in the estimates of the integral means spectrum known up to this point.