The Statistics of the Points Where Nodal Lines Intersect a Reference Curve

We study the intersection points of a fixed planar curve $\Gamma$ with the nodal set of a translationally invariant and isotropic Gaussian random field \Psi(\bi{r}) and the zeros of its normal derivative across the curve. The intersection points form a discrete random process which is the object of this study. The field probability distribution function is completely specified by the correlation G(|\bi{r}-\bi{r}'|) = . Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point correlation function of the point process on the line, and derive other statistical measures (repulsion, rigidity) which characterize the short and long range correlations of the intersection points. We use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks. We apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of special interest is the observation that for monochromatic random waves, the number variance of the intersections with long straight segments grows like $L \ln L$, as opposed to the linear growth predicted by the percolation model, which was successfully used to predict other long range nodal properties of that field.Comment: 33 pages, 13 figures, 1 tabl

The influence of traffic, geometric and context variables on urban crash types: a grouped random parameter multinomial logit approach

Numerous road safety studies have been dedicated to the estimation of crash frequency and injury severity models. However, previous research has shown that different factors may influence the occurrence of crashes of different types. In this study, a dataset including information from crashes occurred at segments and intersections of urban roads in Bari, Italy was used to estimate the likelihood of occurrence of various crash types. The crash types considered are: single-vehicle, angle, rear-end and sideswipe. Models were estimated through a mixed logit structure considering various crash types as outcomes of the dependent variable and several traffic, geometric and context-related factors as explanatory variables (both site- and crash-specific). To account for systematic, unobserved variations among the crashes occurred on the same segment or intersection, the grouped random parameters approach was employed. The latter allows the estimation of segment- or intersection-specific parameters for the variables resulting in random parameters. This approach allows assessing the variability of results across the observations for individual segments/intersections.Segment type and the presence of bus lanes were included as explanatory variables in the model of crash types for segments. Traffic volume per entering lane, total entering lanes, total number of zebra crossings and the balance between major and minor traffic volumes at intersections were included as explanatory variables in the model of crash types for intersections. Area type was included in both segment and intersection models. The typical traffic at the moment of the crash (from on-line traffic prediction tools) and the period of the day were associated with different crash type likelihoods for both segments and intersections. Significant variations in the effect of several predictors across different segments or intersections were identified. The applicability of the study framework is demonstrated, in terms of identifying roadway sites with anomalous tendencies or high-risk sites with respect to specific crash types

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.Comment: 40 page

Fine-Grained Reliability for V2V Communications around Suburban and Urban Intersections

Safe transportation is a key use-case of the 5G/LTE Rel.15+ communications, where an end-to-end reliability of 0.99999 is expected for a vehicle-to-vehicle (V2V) transmission distance of 100-200 m. Since communications reliability is related to road-safety, it is crucial to verify the fulfillment of the performance, especially for accident-prone areas such as intersections. We derive closed-form expressions for the V2V transmission reliability near suburban corners and urban intersections over finite interference regions. The analysis is based on plausible street configurations, traffic scenarios, and empirically-supported channel propagation. We show the means by which the performance metric can serve as a preliminary design tool to meet a target reliability. We then apply meta distribution concepts to provide a careful dissection of V2V communications reliability. Contrary to existing work on infinite roads, when we consider finite road segments for practical deployment, fine-grained reliability per realization exhibits bimodal behavior. Either performance for a certain vehicular traffic scenario is very reliable or extremely unreliable, but nowhere in relatively proximity to the average performance. In other words, standard SINR-based average performance metrics are analytically accurate but can be insufficient from a practical viewpoint. Investigating other safety-critical point process networks at the meta distribution-level may reveal similar discrepancies.Comment: 27 pages, 6 figures, submitted to IEEE Transactions on Wireless Communication

Random planar graphs and the London street network

In this paper we analyse the street network of London both in its primary and dual representation. To understand its properties, we consider three idealised models based on a grid, a static random planar graph and a growing random planar graph. Comparing the models and the street network, we find that the streets of London form a self-organising system whose growth is characterised by a strict interaction between the metrical and informational space. In particular, a principle of least effort appears to create a balance between the physical and the mental effort required to navigate the city

Local geometry of random geodesics on negatively curved surfaces

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces with possibly variable negative curvatur