We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in 2 or more dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with multiplicity N tending to infinity. The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant. Our main result is that the variance of Leray measure is asymptotically 1 (4 pi N), as N tends to infinity, at least in dimensions 2 and at least 5.
A number of statistical properties of a random, moving surface are obtained in the special case when the surface is Gaussian and isotropic. The results may be stated with special simplicity for a ‘ring9 spectrum when the energy in the spectrum is confined to one particular wavelength y. In particular, the average density of maxima per unit area equals pie (2 3y 2 ), and the average length, per unit area, of the contour drawn at the mean level equals pie ( 2y) The following statistical properties are derived for a random, moving, Gaussian surface: (1) the probability distribution of the surface elevation and of the magnitude and orientation of the gradient; (2) the average number of zero-crossings per unit distance along a line in an arbitrary direction; (3) the average length of the contours per unit area, and the distribution of their direction; (4) the average density of maxima and minima per unit area of the surface, and the average density of specular points (i.e, points where the two components of gradient take given values); (5) the probability distribution of the velocities of zero-crossings along a given line; (6) the probability distribution of the velocities of contours and of specular points; (7) the probability distribution of the envelope and phase angle, and hence (8) when the spectrum is narrow, the probability distribution of the heights of maxima and minima and the distribution of the intervals between successive zero-crossings along an arbitrary line. All the results are expressed in terms of the two-dimensional energy spectrum of the surface, and are found to involve the moments of the spectrum up to a finite order only. (1), (3), (4), (5) and (6) are discussed in detail for the special case of a narrow spectrum. The converse problem is also studied and solved: given certain statistical properties of the surface, to find a convergent sequence of approximations to the energy spectrum. The problems arise in connexion with the statistical analysis of the sea surface. (More detailed summaries are given at the beginning of each part of the paper.)   For real (time-reversal symmetric) quantum billiards, the mean length L of nodal line is calculated for the nth mode (n>>1), with wavenumber k, using a Gaussian random wave model adapted locally to satisfy Dirichlet or Neumann boundary conditions. The leading term is of order k (i.e. √n), and the first (perimeter) correction, dominated by an unanticipated long-range boundary effect, is of order log k (i.e. log n), with the same sign (negative) for both boundary conditions. The leading-order state-to-state fluctuations δL are of order √log k. For the curvature κ of nodal lines, |κ| and √κ2 are of order k, but |κ|3 and higher moments diverge. For complex (e.g. Aharonov-Bohm) billiards, the mean number N of nodal points (phase singularities) in the mode has a leading term of order k2 (i.e. n), the perimeter correction (again a long-range effect) is of order klog k (i.e. √nlog n) (and positive, notwithstanding nodal depletion near the boundary) and the fluctuations δN are of order k√log k. Generalizations of the results for mixed (Robin) boundary conditions are stated. The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function. An umbrella is convertible between an expanded and contracted position to adjust the area of coverage thereof when the umbrella is in the opened condition. The umbrella is formed with a pair of sticks which extend from the handle thereof with the sticks being pivoted at the handle for movement between a spread position whereby the sticks may be placed in a generally V-shaped configuration, and a position wherein the sticks are brought together to be generally parallel with each other. Each stick includes a slider which may move relative to the stick in order to effect opening and closing of the canopy of the umbrella, which canopy is generally supported upon the ends of the sticks by ribs. With the sticks in a generally parallel position, a smaller area of coverage is provided by the umbrella. By spreading the sticks apart into the V-shaped configuration by pivotal motion thereof about the handle, the canopy of the umbrella is spread to cover a larger area and the umbrella may be used, for example, by two persons.