Abstract:The formation of new threats and the increasing complexity of urban built infrastructures underline the need for more robust and sustainable systems, which are able to cope with adverse events. Achieving sustainability requires the strengthening of resilience. Currently, a comprehensive approach for the quantification of resilience of urban infrastructure is missing. Within this paper, a new generalized mathematical framework is presented. A clear definition of terms and their interaction builds the basis of this resilience assessment scheme. Classical risk-based as well as additional components are aligned along the timeline before, during and after disruptive events, to quantify the susceptibility, the vulnerability and the response and recovery behavior of complex systems for multiple threat scenarios. The approach allows the evaluation of complete urban surroundings and enables a quantitative comparison with other development plans or cities. A comprehensive resilience framework should cover at least preparation, prevention, protection, response and recovery. The presented approach determines respective indicators and provides decision support, which enhancement measures are more effective. Hence, the framework quantifies for instance, if it is better to avoid a hazardous event or to tolerate an event with an increased robustness. An application example is given to assess different urban forms, i.e., morphologies, with consideration of multiple adverse events, like terrorist attacks or earthquakes, and multiple buildings. Each urban object includes a certain number of attributes, like the object use, the construction type, the time-dependent number of persons and the value, to derive different performance targets. The assessment results in the identification of weak spots with respect to single resilience indicators. Based on the generalized mathematical formulation and suitable combination of indicators, this approach can quantify the resilience of urban morphologies, independent of possible single threat types and threat locations.
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A relation between the number of bound collective excitations of an atomic Bose-Einstein condensate and the phase shift of elastically scattered atoms is derived. Within the Bogoliubov model of a weakly interacting Bose gas this relation is exact and generalises Levinson's theorem. Specific features of the Bogoliubov model such as complex-energy and continuum bound states are discussed and a numerical example is given. In this Letter, we consider the scattering of identical atoms from a spherically symmetric, weakly-interacting atomic Bose-Einstein condensate (BEC), held in a finite, localised trapping potential. In this situation, the phase shift δ l (k) of l-wave scattering at momentum k = 0 can be related to the number n c of bound collective excitations of the condensate bywhere σ = 1 for a bound state exactly at the threshold for s-wave scattering and σ = 0 otherwise. Equation (1) is the generalisation of Levinson's theorem to the Bogoliubov equations describing the excitations of a weakly interacting BEC and is the main result of this Letter. Scattering of cold atoms is a fundamental physical process relevant to high-precision atomic spectroscopy and quantum information processing. As temperatures are lowered, condensation of bosonic atoms is inevitable and the scattering of single atoms from condensates needs to be understood. Scattering experiments involving condensates have already been demonstrated in the context of four-wave mixing experiments [3,4]. With the development of atomic lasers [5] as coherent matter wave sources and with the flexibility introduced by trapping and guiding of cold atoms with micro-fabricated electrical circuits [6,7], precision measurement of scattering properties becomes feasible. Moreover, interferometric measurements should allow a direct access of the phase shifts and thereby confirm Levinson's theorem experimentally in contrast to the case of conventional atomic scattering experiments where usually intensities are measured only.Theoretical attention has been given to identical particle scattering from BECs at low energy where transparency effects have been predicted [8] and at high energy where density distributions [9] and quantum correlations can be probed [10]. Very recently, negative time delays in one-dimensional scattering from atomic BECs have been predicted [11].In the following, we will apply a multi-channel scattering formalism to the Bogoliubov equations and prove the relation (1) by contour integration. We will discuss special situations that can occur like unstable complexenergy collective modes and continuum bound states. An instructive numerical example of a realistic scattering situation is given and the role of the condensate and channel coupling for the Levinson theorem are discussed.In the standard Bogoliubov approach [12], the weakly interacting Bose gas is described by a condensate with a small amount of coherent quantum depletion and a gas of non-interacting quasi-particles describing small thermal or externally induced collective excitations. The q...
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