Expected Number Of Intersections Research Articles

Asymptotic distribution of nodal intersections for arithmetic random waves

We study the nodal intersections number of random Gaussian toral Laplace eigenfunctions (‘arithmetic random waves’) against a fixed smooth reference curve. The expected intersection number is proportional to the the square root of the eigenvalue times the length of curve, independent of its geometry. The asymptotic behaviour of the variance was addressed by Rudnick–Wigman; they found a precise asymptotic law for ‘generic’ curves with nowhere vanishing curvature, depending on both its geometry and the angular distribution of lattice points lying on circles corresponding to the Laplace eigenvalue. They also discovered that there exist peculiar ‘static’ curves, with variance of smaller order of magnitude, though did not prescribe what the true asymptotic behaviour is in this case.In this paper we study the finer aspects of the limit distribution of the nodal intersections number. For ‘generic’ curves we prove the central limit theorem (at least, for ‘most’ of the energies). For the aforementioned static curves we establish a non-Gaussian limit theorem for the distribution of nodal intersections, and on the way find the true asymptotic behaviour of their fluctuations, under the well-separatedness assumption on the corresponding lattice points, satisfied by most of the eigenvalues.

Nodal intersections for random waves against a segment on the 3-dimensional torus

We consider random Gaussian eigenfunctions of the Laplacian on the three-dimensional flat torus, and investigate the number of nodal intersections against a straight line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on the arithmetic properties of the straight line. The considerations made establish a close relation between this problem and the theory of lattice points on spheres.

Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus

We consider random Gaussian eigenfunctions of the Laplacian on the standard torus, and investigate the number of nodal intersections against a line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on whether the slope of the straight line is rational or irrational. Our findings exhibit a close relation between this problem and the theory of lattice points on circles.

Nodal intersections for random eigenfunctions on the torus

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard two-dimensional flat torus (``arithmetic random waves'') with a fixed smooth reference curve with nonvanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. In particular, this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability.

Nodal intersections for random waves on the 3-dimensional torus

We investigate the number of nodal intersections of ran- dom Gaussian Laplace eigenfunctions on the standard three-dimensional at torus with a xed smooth reference curve, which has nowhere vanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenum- ber, independent of the geometry. Our main result gives a bound for the variance, if either the torsion of the curve is nowhere zero or if the curve is planar.

Stochastic modelling of stress processes in power plant boiler walls

A stochastic model of the integrated firing and supply system was set up, using the theory of semi-Markov processes and assuming a stochastically identified firing system model. On the basis of the results of the paper a proposed basic statistics, mainly the expected intersection number, of the repeating mechanical stress level of the combustion chamber's structure can be determined. According to the introduced method, the characteristic mechanical stress level can be limited, which prevents early fatigue and rupture of the important structural elements of the combustion chamber.