Signs of the cusps in binary lenses

The cusps of the caustics of any gravitational lens model can be classified into positive and negative ones. This distinction lies on the parity of the images involved in the creation/destruction of pairs occurring when a source crosses a caustic in a cusp. In this paper, we generalize the former definition of the sign of the cusps. Then we apply it to the binary lens. We demonstrate that the cusps on the axis joining the two lenses are positive while the others are negative. To achieve our objective, we combine catastrophe theory, usually employed in the derivation of the properties of caustics, with perturbative methods, in order to simplify calculations and get readable results. Extensions to multiple lenses are also considered.Comment: 18 pages, 2 figures. Accepted by Journal of Mathematical Physics. After it is published, it will be found at http://ojps.aip.org/jmp

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A

From Smart Cities To Playable Cities. Towards Playful Intelligence In The Urban Environment

In the last decade, we have seen the rise of urban play as a tool for community building, and city-making and Western society is actively focusing on play/playfulness and intelligent systems as a way to approach complex challenges and emergent situations. In this paper, we aim to initiate a dialogue between game scholars and architects. Like many creative professions, we believe that the architectural practice may benefit significantly from having more design methodologies at hand, thus improving lateral thinking. We aim at providing new conceptual and operative tools to discuss and reflect on how games and smart systems facilitate long-term the shift from the Smart Cities to the Playable one, where citizens/players have the opportunity to hack the city and use the smart city’s data and digital technology for their purposes to reactivate the urban environment

Secondary caustics in close multiple lenses

We investigate the caustic structure of a lens composed by a discrete number of point-masses, having mutual distances smaller than the Einstein radius of the total mass of the system. Along with the main critical curve, it is known that the lens map is characterized by secondary critical curves producing small caustics far from the lens system. By exploiting perturbative methods, we derive the number, the position, the shape, the cusps and the area of these caustics for an arbitrary number of close multiple lenses. Very interesting geometries are created in some particular cases. Finally we review the binary lens case where our formulae assume a simple form.Comment: 9 pages with 5 figures. Accepted by A&

A Survey on the Classical Limit of Quantum Dynamical Entropies

We analyze the behavior of quantum dynamical entropies production from sequences of quantum approximants approaching their (chaotic) classical limit. The model of the quantized hyperbolic automorphisms of the 2-torus is examined in detail and a semi-classical analysis is performed on it using coherent states, fulfilling an appropriate dynamical localization property. Correspondence between quantum dynamical entropies and the Kolmogorov-Sinai invariant is found only over time scales that are logarithmic in the quantization parameter.Comment: LaTeX, 21 pages, Presented at the 3rd Workshop on Quantum Chaos and Localization Phenomena, Warsaw, Poland, May 25-27, 200