Over twenty years ago, Marc Kac posed what is arguably one of the simplest inverse problems in pure mathematics: "Can one hear the shape of a drum?" [19]. Mathematically, the question is formulated as follows. Let /2 be a simply connected, plane domain (the drumhead) bounded by a smooth curve 7, and consider the wave equation on /2 with Dirichlet boundary condition on 7 (the drumhead is clamped...

A method of patterning surfaces for liquid-crystal alignment using a polarization holography exposure on a linear photopolymerizable polymer alignment layer is demonstrated. Three configurations are demonstrated which include registered planar-periodic surface boundary conditions on both surfaces true polarization gratings , planar-periodic and uniform planary surface boundary conditions, and p...

In this paper we consider the one-parameter family of Bona-Smith systems, which belongs to the class of Boussinesq systems modelling two-way propagation of long waves of small amplitude on the surface of water in a channel. We study three initial-boundary-value problems for these systems, corresponding, respectively, to nonhomogeneous Dirichlet, reflection, and periodic boundary conditions pose...

Diophantine criteria occur naturally in the theory of partial differential equations through the notorious problem of small denominators. An extensive treatment of such problems in the theory of PDEs can be found, e.g., in [6]. In this paper, we are interested in a Diophantine problem related to an inhomogeneous wave equation in n spatial and one temporal dimension with periodic boundary condit...

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy (L-norm) is studied. It is proved that the interaction of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately sat...

In [2], two of the authors and W. Meeks found examples of translation-invariant, embedded minimal surfaces with an infinite number of topological ends. For each k > 0, a surfaceMk was constructed, which was invariant with respect to a translation parallel to the x3-axis, and under a rotation group of order k+ 1 around the x3-axis. The method of construction was generalized in [3] to obtain the ...

The response of a soap film to the continuous deformation of its wire boundary is considered, with particular attention to the topological transitions that can occur at critical stages of the deformation process. Two well-known examples that have been studied by both theory and experiment are the catenoid suspended between circular wires in parallel planes, and the Möbius-strip soap film spanni...

No abstract. Erratum After publication of the original article [1] it came to the Publisher's attention that the first author's name had been listed incorrectly. The list of authors has now been corrected. We apologise for any inconvenience this has caused. which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.