We consider the zeros on the boundary @­ of a Neumann eigen- function '¸j of a real analytic plane domain ­. We prove that the number of its boundary zeros is O(¸j) where ¡¢'¸j = ¸2j '¸j . We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(¸j ). It follows that the number of nodal lines of '¸j (components of the nodal set) which touch the boundary is of order ¸j . This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.

We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure [..] on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of [..] or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.

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The structure of a stationary Gaussian process near a local maximum with a prescribed height$u$has been explored in several papers by the present author, see [5]–[7], which include results for moderate$u$as well as for$u$→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ$(t) t ∈ R$^{n}}, with mean zero and the covariance function$r$($t$). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.