Counting unicellular maps on non-orientable surfaces

A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable unicellular maps of fixed topology. In particular, we give exact formulas for the precubic case (all vertices of degree 1 or 3), and asymptotic formulas for the general case, when the number of edges goes to infinity. Our strategy is inspired by recent results obtained by the second author for the orientable case, but significant novelties are introduced: in particular we construct an involution which, in some sense, "averages" the effects of non-orientability

Small volume expansions for elliptic equations

This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form \nabla \cdot D^\eps \nabla u^\eps=0 or (-\Delta + q^\eps) u^\eps=0 with prescribed Neumann conditions. The theory is well-known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order \eps^{2d}, where $d$ is dimension and \eps is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most \eps^{d+1} rather than the expected \eps^d. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables.Comment: 42 page