Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactly k crossings as a rational function of the generating function of crossing-free configurations. Using these expressions, we...

We consider mathematical aspects of the logarithmic spiral and its utility in turbulence modeling. We consider mathematically the set of point crossings resulting from a linear intersection through the center of a logarithmic spiral. We derive analytically the fractal dimension as a function of scale for this set of crossings. We also derive analytically the power spectrum of the thresholded fu...

to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function r. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is sti...

I n New Zealand, as well as many other countries, unsignalized, marked pedestrian crossings are known as zebra crossings. Last year, the New Zealand National Roads Board (NRB) published a technical recommendation (TR1l ) that sets out standards for the installation and upgrading of zebra crossings.’ The report also includes statutory requirements and recommendations on warrants, location criter...

The sociotechnical systems approach argues that systems with adaptive capacity will be safer. However, this approach may conflict with current practice in safety management. We applied cognitive work analysis to understand the problem of pedestrian safety at rail level crossings and used the findings to evaluate the current design against the values of sociotechnical systems theory. The evaluat...

fic safety problem for road users and a major safety issue for railways [1–2]. The developed in-vehicle warning system aims to reduce the number of accidents at level crossings by increasing the alertness and awareness of the car driver approaching a level crossing [3]. The system is expected to improve safety especially at passive level crossings with occasional vehicle drivers where existing ...

The syntactic structure of sentences exhibits a striking regularity: dependencies tend to not cross when drawn above the sentence. We investigate two competing explanations. The traditional hypothesis is that this trend arises from an independent principle of syntax that reduces crossings practically to zero. An alternative to this view is the hypothesis that crossings are a side effect of depe...

We observe strong modal coupling between the TE00 and TM00 modes in Si3N4 ring resonators revealed by avoided crossings of the corresponding resonances. Such couplings result in significant shifts of the resonance frequencies over a wide range around the crossing points. This leads to an effective dispersion that is one order of magnitude larger than the intrinsic dispersion and creates broad w...