Financial ``Anti-Bubbles'': Log-Periodicity in Gold and Nikkei collapses

We propose that imitation between traders and their herding behaviour not only lead to speculative bubbles with accelerating over-valuations of financial markets possibly followed by crashes, but also to ``anti-bubbles'' with decelerating market devaluations following all-time highs. For this, we propose a simple market dynamics model in which the demand decreases slowly with barriers that progressively quench in, leading to a power law decay of the market price decorated by decelerating log-periodic oscillations. We document this behaviour on the Japanese Nikkei stock index from 1990 to present and on the Gold future prices after 1980, both after their all-time highs. We perform simultaneously a parametric and non-parametric analysis that are fully consistent with each other. We extend the parametric approach to the next order of perturbation, comparing the log-periodic fits with one, two and three log-frequencies, the latter one providing a prediction for the general trend in the coming years. The non-parametric power spectrum analysis shows the existence of log-periodicity with high statistical significance, with a prefered scale ratio of $\lambda \approx 3.5$ for the Nikkei index $\lambda \approx 1.9$ for the Gold future prices, comparable to the values obtained for speculative bubbles leading to crashes.Comment: 14 pages with 4 figure

Three-body properties of low-lying 12^{12}Be resonances

We compute the three-body structure of the lowest resonances of $^{12}$Be considered as two neutrons around an inert $^{10}$Be core. This is an extension of the bound state calculations of $^{12}$Be into the continuum spectrum. We investigate the lowest resonances of angular momenta and parities, $0^{\pm}$, $1^{-}$ and $2^{+}$. Surprisingly enough, they all are naturally occurring in the three-body model. We calculate bulk structure dominated by small distance properties as well as decays determined by the asymptotic large-distance structure. Both $0^{+}$ and $2^{+}$ have two-body $^{10}$Be-neutron d-wave structure, while $1^{-}$ has an even mixture of $p$ and d-waves. The corresponding relative neutron-neutron partial waves are distributed among $s$, $p$, and d-waves. The branching ratios show different mixtures of one-neutron emission, three-body direct, and sequential decays. We argue for spin and parities, $0^{+}$, $1^{-}$ and $2^{+}$, to the resonances at 0.89, 2.03, 5.13, respectively. The computed structures are in agreement with existing reaction measurements.Comment: To be published in Physical Review

Stochastics theory of log-periodic patterns

We introduce an analytical model based on birth-death clustering processes to help understanding the empirical log-periodic corrections to power-law scaling and the finite-time singularity as reported in several domains including rupture, earthquakes, world population and financial systems. In our stochastics theory log-periodicities are a consequence of transient clusters induced by an entropy-like term that may reflect the amount of cooperative information carried by the state of a large system of different species. The clustering completion rates for the system are assumed to be given by a simple linear death process. The singularity at t_{o} is derived in terms of birth-death clustering coefficients.Comment: LaTeX, 1 ps figure - To appear J. Phys. A: Math & Ge

A Simple Approach to Maximum Intractable Likelihood Estimation

Approximate Bayesian Computation (ABC) can be viewed as an analytic approximation of an intractable likelihood coupled with an elementary simulation step. Such a view, combined with a suitable instrumental prior distribution permits maximum-likelihood (or maximum-a-posteriori) inference to be conducted, approximately, using essentially the same techniques. An elementary approach to this problem which simply obtains a nonparametric approximation of the likelihood surface which is then used as a smooth proxy for the likelihood in a subsequent maximisation step is developed here and the convergence of this class of algorithms is characterised theoretically. The use of non-sufficient summary statistics in this context is considered. Applying the proposed method to four problems demonstrates good performance. The proposed approach provides an alternative for approximating the maximum likelihood estimator (MLE) in complex scenarios

Nonparametric identification of linearizations and uncertainty using Gaussian process models – application to robust wheel slip control

Gaussian process prior models offer a nonparametric approach to modelling unknown nonlinear systems from experimental data. These are flexible models which automatically adapt their model complexity to the available data, and which give not only mean predictions but also the variance of these predictions. A further advantage is the analytical derivation of derivatives of the model with respect to inputs, with their variance, providing a direct estimate of the locally linearized model with its corresponding parameter variance. We show how this can be used to tune a controller based on the linearized models, taking into account their uncertainty. The approach is applied to a simulated wheel slip control task illustrating controller development based on a nonparametric model of the unknown friction nonlinearity. Local stability and robustness of the controllers are tuned based on the uncertainty of the nonlinear models’ derivatives

Protoplanetary Disk Turbulence Driven by the Streaming Instability: Non-Linear Saturation and Particle Concentration

We present simulations of the non-linear evolution of streaming instabilities in protoplanetary disks. The two components of the disk, gas treated with grid hydrodynamics and solids treated as superparticles, are mutually coupled by drag forces. We find that the initially laminar equilibrium flow spontaneously develops into turbulence in our unstratified local model. Marginally coupled solids (that couple to the gas on a Keplerian time-scale) trigger an upward cascade to large particle clumps with peak overdensities above 100. The clumps evolve dynamically by losing material downstream to the radial drift flow while receiving recycled material from upstream. Smaller, more tightly coupled solids produce weaker turbulence with more transient overdensities on smaller length scales. The net inward radial drift is decreased for marginally coupled particles, whereas the tightly coupled particles migrate faster in the saturated turbulent state. The turbulent diffusion of solid particles, measured by their random walk, depends strongly on their stopping time and on the solids-to-gas ratio of the background state, but diffusion is generally modest, particularly for tightly coupled solids. Angular momentum transport is too weak and of the wrong sign to influence stellar accretion. Self-gravity and collisions will be needed to determine the relevance of particle overdensities for planetesimal formation.Comment: Accepted for publication in ApJ (17 pages). Movies of the simulations can be downloaded at http://www.mpia.de/~johansen/research_en.ph

Circularizing Planet Nine through dynamical friction with an extended, cold planetesimal belt

Unexpected clustering in the orbital elements of minor bodies beyond the Kuiper belt has led to speculations that our solar system actually hosts nine planets, the eight established plus a hypothetical "Planet Nine". Several recent studies have shown that a planet with a mass of about 10 Earth masses on a distant eccentric orbit with perihelion far beyond the Kuiper belt could create and maintain this clustering. The evolutionary path resulting in an orbit such as the one suggested for Planet Nine is nevertheless not easily explained. Here we investigate whether a planet scattered away from the giant-planet region could be lifted to an orbit similar to the one suggested for Planet Nine through dynamical friction with a cold, distant planetesimal belt. Recent simulations of planetesimal formation via the streaming instability suggest that planetesimals can readily form beyond 100au. We explore this circularisation by dynamical friction with a set of numerical simulations. We find that a planet that is scattered from the region close to Neptune onto an eccentric orbit has a 20-30% chance of obtaining an orbit similar to that of Planet Nine after 4.6Gyr. Our simulations also result in strong or partial clustering of the planetesimals; however, whether or not this clustering is observable depends on the location of the inner edge of the planetesimal belt. If the inner edge is located at 200au the degree of clustering amongst observable objects is significant.Comment: Accepted to MNRA

Magnetostrictive behaviour of thin superconducting disks

Flux-pinning-induced stress and strain distributions in a thin disk superconductor in a perpendicular magnetic field is analyzed. We calculate the body forces, solve the magneto-elastic problem and derive formulas for all stress and strain components, including the magnetostriction $\Delta R/R$. The flux and current density profiles in the disk are assumed to follow the Bean model. During a cycle of the applied field the maximum tensile stress is found to occur approximately midway between the maximum field and the remanent state. An effective relationship between this overall maximum stress and the peak field is found.Comment: 8 pages, 6 figures, submitted to Supercond. Sci. Technol., Proceed. of MEM03 in Kyot

The destruction of inner planetary systems during high-eccentricity migration of gas giants

Hot Jupiters are giant planets on orbits a few hundredths of an AU. They do not share their system with low-mass close-in planets, despite these latter being exceedingly common. Two migration channels for hot Jupiters have been proposed: through a protoplanetary gas disc or by tidal circularisation of highly-eccentric planets. We show that highly-eccentric giant planets that will become hot Jupiters clear out any low-mass inner planets in the system, explaining the observed lack of such companions to hot Jupiters. A less common outcome of the interaction is that the giant planet is ejected by the inner planets. Furthermore, the interaction can implant giant planets on moderately-high eccentricities at semimajor axes $<1$ AU, a region otherwise hard to populate. Our work supports the hypothesis that most hot Jupiters reached their current orbits following a phase of high eccentricity, possibly excited by other planetary or stellar companions.Comment: Replaced with accepted versio

The effects of external planets on inner systems: multiplicities, inclinations, and pathways to eccentric warm Jupiters

We study how close-in systems such as those detected by Kepler are affected by the dynamics of bodies in the outer system. We consider two scenarios: outer systems of giant planets potentially unstable to planet--planet scattering, and wide binaries that may be capable of driving Kozai or other secular variations of outer planets' eccentricities. Dynamical excitation of planets in the outer system reduces the multiplicity of Kepler-detectable planets in the inner system in $\sim20-25\%$ of our systems. Accounting for the occurrence rates of wide-orbit planets and binary stars, $\approx18\%$ of close-in systems could be destabilised by their outer companions in this way. This provides some contribution to the apparent excess of systems with a single transiting planet compared to multiple, however, it only contributes at most $25\%$ of the excess. The effects of the outer dynamics can generate systems similar to Kepler-56 (two coplanar planets significantly misaligned with the host star) and Kepler-108 (two significantly non-coplanar planets in a binary). We also identify three pathways to the formation of eccentric warm Jupiters resulting from the interaction between outer and inner systems: direct inelastic collision between an eccentric outer and an inner planet, secular eccentricity oscillations that may "freeze out" when scattering resolves in the outer system; and scattering in the inner system followed by "uplift", where inner planets are removed by interaction with the outer planets. In these scenarios, the formation of eccentric warm Jupiters is a signature of a past history of violent dynamics among massive planets beyond $\sim1$ au.Comment: 24 pages, 19 figures. Accepted to MNRA