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We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the surface in question to its four closest neighbors, forming quads. We then focus on the simplices of $d=2$, or triangles embedded in these quads, which make up a pure simplicial complex with $d=2$. This allows us to numerically compute the local metric along with zweibeins, which subsequently leads to a derivation of discrete curvature defined at every triangle or face. We provide an efficient algorithm with $\mathcal{O}(N \log{N})$ complexity that first orients two-dimensional surfaces, solves the nonlinear system of equations of the spin-connections resulting from the torsion condition, and returns the value of curvature at each face.

We study the metric corresponding to a three-dimensional coset space $SO(4)/SO(3)$ in the lattice setting. With the use of three integers $n_1, n_2$, and $n_3$, and a length scale, $l_{\mu}$, the continuous metric is transformed into a discrete space. The numerical outcomes are compared with the continuous ones. The singularity of the black hole is explored and different domains are studied.

We study numerically the curvature tensor in a three-dimensional discrete space. Starting from the continuous metric of a three-sphere, we transformed it into a discrete space using three integers $n_1, n_2$, and $n_3$. The numerical results are compared with the expected values in the continuous limit. We show that as the number of cells in the lattice increases, the continuous limit is recovered.

We focus on studying, numerically, the scalar curvature tensor in a two-dimensional discrete space. The continuous metric of a two-sphere is transformed into that of a lattice using two possible slicings. In the first, we use two integers, while in the second we consider the case where one of the coordinates is ignorable. The numerical results of both cases are then compared with the expected values in the continuous limit as the number of cells of the lattice becomes very large.

In this paper, we propose a general framework for optimal control measures, which follows the evolution of COVID-19 infection counts collected by Surveillance Units on a country level. We employ an autoregressive model that allows to decompose the mean number of infections into three components that describe: intra-locality infections, inter-locality infections, and infections from other sources such as travelers arriving to a country from abroad. We identify the inter-locality term as a time-evolving network and when it drives the dynamics of the disease we focus on its properties. Tools from network analysis are then employed to get insight into its topology. Building on this, and particularly on the centrality of the nodes of the identified network, a strategy for intervention and disease control is devised.

Diesel generators are emerging as community-initiated solutions to compensate for electricity shortage in cities marred by economical crisis and/or conflict. The resulting pollution distribution in dense urban environments is a major source of concern to the population. In the absence of periodic observations from properly distributed sensors, as is the case in Beirut, physically based computational modeling stand out as an effective tool for predicting the pollutant distribution in complex environments, and a cost-effective framework for investigating what-if scenarios and assessing mitigation strategies. Here, we present a Lagrangian transport model-based study of PM2.5 dispersion originating from a large number of diesel generators in Beirut. We explore large and small scale dispersion patterns in selected smalls domains and over the entire city. The scenarios considered investigate the impact of topography, atmospheric stability, presence of buildings, diesel generators distribution, and stacks elevations for representative meteorological conditions. Assessment of these scenarios is carried out in terms of small and large scale dispersion patterns and the mean concentration at street level and population exposure proxy indicators. We also report on the efficacy of elevating the stack height as a mitigation measure at different representative wind and atmospheric stability conditions.

In this paper, we explore the analogy between the refugees' drownings in the sea and the earthquakes' occurrences and focus on the aspect that characterizes the statistics of their spatial and temporal successions. The former is shown to parallel the spatial distribution of consecutive drowning events with the difference that the latter exhibits short-range behavior below $\kappa = 4km$ and it is characterized by scale-free statistics, with a critical exponent $\delta \approx 0.5$, falling within the range of the earthquakes' $\delta = 0.65 \pm 0.20$, as well as finite size scaling beyond $\kappa = 4km$, while the distribution of events' rates exhibits no similarity with that of the earthquakes. Finally, the events' velocity distribution is also recovered. $\kappa$ is suspected to be related to the radar and mobile network's coverage ranges and thus effectively represents a cut-off in the ability of picking up signals on drownings in the sea.

We consider the flow of Burgers' equation on an open set of (small) functions in $L^2([0,1])$. We derive explicitly the Koopman decomposition of the Burgers' flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for $t>0$ for small Cauchy data, and up to $t=0$ for regular Cauchy data. The convergence up to $t=0$} leads to a `completeness' property for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by (Page & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compared to the eigenvalues of a Dynamic Mode Decomposition (DMD).

In this paper, we focus on a critical component of the city: its building stock, which holds much of its socio-economic activities. In our case, the lack of a comprehensive database about their features and its limitation to a surveyed subset lead us to adopt data-driven techniques to extend our knowledge to the near-city-scale. Neural networks and random forest are applied to identify the buildings' number of floors and construction periods' dependencies on a set of shape features: area, perimeter, and height along with the annual electricity consumption, relying on a surveyed data in the city of Beirut. The predicted results are then compared with established scaling laws of urban forms, which constitutes a further consistency check and validation of our work ow.

\begin{abstract} In this Letter we follow the asymptotic spatial correlation of buildings' heights $G_{\infty}(r)$ in the whole of the Netherlands \cite{bag3d}, which comprises $\approx 10,000,000$ buildings, for the purpose of recovering its scaling with respect to space and time given respectively by the exponents $r^{2 \alpha}$ and $t^{2\beta}$. This allows us to identify the universality class of the evolution of the urban skyline seen as a dynamically evolving interface. Two major classes of cities were identified based on the recovered value of $\alpha =0.4$ and $\alpha = 0$, which correspond respectively to the KPZ and the EW universality classes. Picking a discrete model from each of these classes and mapping it to physical rules for constructions in cities we conclude that imposed restrictions on buildings' heights are reflected in the exponent $\alpha$ and thus have implications on how the skyline evolves.

We look at buildings' competition over space in cities through the lens of ecology. Adopting the convex hull of the building's footprint perimeter as a definition of species yields parallels to forest trees' competition, which we expound on. Their perimeter distribution $p(r)$ follows a power-law behavior beyond a critical threshold of the density of the built environment. In this regime, the species coexistence likelihood $p(d)$, where $d$ is the distance to the nearest competitor, which we define to be a building with a larger $r$, bifurcates with the buildings' number $n$. This reveals two different predation laws: a vicious predatory one which is linked spatial homogeneity and segregation, as opposed to another favoring spatial diversity and intermixing between species.

In this Letter we follow the short-ranged Syrian refugees' migration to Lebanon as documented by the UNHCR. We propose a model inspired by the Debye-H\"{u}ckel theory and show that it properly predicts the refugees' mobility while the gravity model fails. We claim that the interaction between origin cities attenuates and/or extenuates the flux to destinations, and consequently, in analogy with the effective charges of interacting particles in a plasma, these source cities are characterized by effective populations determined by their pairwise remoteness/closeness and defined by areas of control between the fighting parties.

We explore the relation between urban road network characteristics particularly circuitry, street orientation entropy and the city's topography on the one hand and the building's orientation entropy on the other in order to quantify their effect on the city's solar potential. These statistical measures of the road network reveal the interplay between the built environment's design and its sustainability.