Linear response theory in stock markets

Linear response theory relates the response of a system to a weak external force with its dynamics in equilibrium, subjected to fluctuations. Here, this framework is applied to financial markets; in particular we study the dynamics of a set of stocks from the NASDAQ during the last 20 years. Because unambiguous identification of external forces is not possible, critical events are identified in the series of stock prices as sudden changes, and the stock dynamics following an event is taken as the response to the external force. Linear response theory is applied with the log-return as the conjugate variable of the force, providing predictions for the average response of the price and return, which agree with observations, but fails to describe the volatility because this is expected to be beyond linear response. The identification of the conjugate variable allows us to define the perturbation energy for a system of stocks, and observe its relaxation after an event.

Black holes and Boyle's law — The thermodynamics of the cosmological constant

When the cosmological constant, Λ, is interpreted as a thermodynamic variable in the study of black hole thermodynamics a very rich structure emerges. It is natural to interpret Λ as a pressure and define the thermodynamically conjugate variable to be the thermodynamic volume of the black hole (which need not bear any relation to the geometric volume). Recent progress in this new direction for black hole thermodynamics is reviewed.

Exploitation of chemical profiles by conjugate variable analysis: application to the dating of a tropical ice core (Nevado Illimani, Bolivia)

Ice core dating is a key parameter for the interpretation of the ice archives. However, the relationship between ice depth and age can generally not be easily established and requires to combine a large number of investigations and/or modeling effort. This paper presents a new approach of ice core dating based on conjugate variable (depth and spatial frequency) analysis of chemical profiles. The relationship between the depth of a given ice layer and the date it was deposited is determined using ion concentration depth profiles obtained along a one hundred-meters deep ice core recovered in the summit area of the Nevado Illimani (6350 m a.s.l.), located in the Eastern Bolivian Andes (16°37' S, 67°46' W). The results of Fourier conjugate analysis and wavelet tranforms are first compared. Both methods are applied to nitrate concentration depth profile. The resulting chronologies are checked by comparison with the multi-proxy year-by-year dating published by de Angelis et al. (2003) and with volcanic tie points, demonstrating the efficiency of Fourier conjugate analysis when tracking the natural variability of chemical proxies. The Fourier conjugate analysis is then applied to concentration depth profiles of seven other ions thus providing information on the suitability of each of them for dating studies of tropical Andean ice cores.

ANALYTICAL CONDITIONS FOR AMPLITUDE DEATH INDUCED BY CONJUGATE VARIABLE COUPLINGS

We construct local and global conjugate variable coupled chaotic systems with an arbitrary number of identical chaotic oscillators. Amplitude death phenomena are found in these two kinds of systems. General methods are proposed to theoretically analyze conditions for amplitude death under local and global conditions, respectively. Taking the Rössler chaotic system as the oscillator unit, we apply these methods to determine the analytical conditions for amplitude death. And the transition relations between local and global coupling induced amplitude deaths are also given. All theoretical results are well confirmed by numerical simulations.

The uncertainty relation for joint measurement of position and momentum

We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (\Delta P)(\Delta Q)\geq C\hbar, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision \Delta Q of the position measurement, and the perturbation \Delta P of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.