Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes

We present the first exact analysis of some of the temporal properties of multivariate self-excited Hawkes conditional Poisson processes, which constitute powerful representations of a large variety of systems with bursty events, for which past activity triggers future activity. The term "multivariate" refers to the property that events come in different types, with possibly different intra- and inter-triggering abilities. We develop the general formalism of the multivariate generating moment function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the "shock") as a function of the current time $t$. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays $\sim 1/t^{1-(m+1)\theta}$ of the rate of triggered events as a function of the distance $m$ of the events to the initial shock in the type space, where $0 < \theta <1$ for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via a kind of inter-breeding genealogy.Comment: 40 pages, 8 figure

Vere-Jones' Self-Similar Branching Model

Motivated by its potential application to earthquake statistics, we study the exactly self-similar branching process introduced recently by Vere-Jones, which extends the ETAS class of conditional branching point-processes of triggered seismicity. One of the main ingredient of Vere-Jones' model is that the power law distribution of magnitudes m' of daughters of first-generation of a mother of magnitude m has two branches m'm with exponent beta+d, where beta and d are two positive parameters. We predict that the distribution of magnitudes of events triggered by a mother of magnitude $m$ over all generations has also two branches m'm with exponent beta+h, with h= d \sqrt{1-s}, where s is the fraction of triggered events. This corresponds to a renormalization of the exponent d into h by the hierarchy of successive generations of triggered events. The empirical absence of such two-branched distributions implies, if this model is seriously considered, that the earth is close to criticality (s close to 1) so that beta - h \approx \beta + h \approx \beta. We also find that, for a significant part of the parameter space, the distribution of magnitudes over a full catalog summed over an average steady flow of spontaneous sources (immigrants) reproduces the distribution of the spontaneous sources and is blind to the exponents beta, d of the distribution of triggered events.Comment: 13 page + 3 eps figure

Power Law Distributions of Seismic Rates

We report an empirical determination of the probability density functions $P_{\text{data}}(r)$ of the number $r$ of earthquakes in finite space-time windows for the California catalog. We find a stable power law tail $P_{\text{data}}(r) \sim 1/r^{1+\mu}$ with exponent $\mu \approx 1.6$ for all space ($5 \times 5$ to $20 \times 20$ km$^2$) and time intervals (0.1 to 1000 days). These observations, as well as the non-universal dependence on space-time windows for all different space-time windows simultaneously, are explained by solving one of the most used reference model in seismology (ETAS), which assumes that each earthquake can trigger other earthquakes. The data imposes that active seismic regions are Cauchy-like fractals, whose exponent $\delta =0.1 \pm 0.1$ is well-constrained by the seismic rate data.Comment: 5 pages with 1 figur

Evolution of iron core white dwarfs

Recent measurements made by Hipparcos (Provencal et al. 1998) present observational evidence supporting the existence of some white dwarf (WD) stars with iron - rich, core composition. In this connection, the present paper is aimed at exploring the structure and evolution of iron - core WDs by means of a detailed and updated evolutionary code. In particular, we examine the evolution of the central conditions, neutrino luminosity, surface gravity, crystallization, internal luminosity profiles and ages. We find that the evolution of iron - rich WDs is markedly different from that of their carbon - oxygen counterparts. In particular, cooling is strongly accelerated as compared with the standard case. Thus, if iron WDs were very numerous, some of them would have had time enough to evolve at lower luminosities than that corresponding to the fall - off in the observed WD luminosity function.Comment: 8 pages, 21 figures. Accepted for publication in MNRA

Intercluster Correlation in Seismicity

Mega et al.(cond-mat/0212529) proposed to use the ``diffusion entropy'' (DE) method to demonstrate that the distribution of time intervals between a large earthquake (the mainshock of a given seismic sequence) and the next one does not obey Poisson statistics. We have performed synthetic tests which show that the DE is unable to detect correlations between clusters, thus negating the claimed possibility of detecting an intercluster correlation. We also show that the LR model, proposed by Mega et al. to reproduce inter-cluster correlation, is insufficient to account for the correlation observed in the data.Comment: Comment on Mega et al., Phys. Rev. Lett. 90. 188501 (2003) (cond-mat/0212529

Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes

We present a stability study of the class of multivariate self-excited Hawkes point processes, that can model natural and social systems, including earthquakes, epileptic seizures and the dynamics of neuron assemblies, bursts of exchanges in social communities, interactions between Internet bloggers, bank network fragility and cascading of failures, national sovereign default contagion, and so on. We present the general theory of multivariate generating functions to derive the number of events over all generations of various types that are triggered by a mother event of a given type. We obtain the stability domains of various systems, as a function of the topological structure of the mutual excitations across different event types. We find that mutual triggering tends to provide a significant extension of the stability (or subcritical) domain compared with the case where event types are decoupled, that is, when an event of a given type can only trigger events of the same type.Comment: 27 pages, 8 figure

Variable-delay feedback control of unstable steady states in retarded time-delayed systems

We study the stability of unstable steady states in scalar retarded time-delayed systems subjected to a variable-delay feedback control. The important aspect of such a control problem is that time-delayed systems are already infinite-dimensional before the delayed feedback control is turned on. When the frequency of the modulation is large compared to the system's dynamics, the analytic approach consists of relating the stability properties of the resulting variable-delay system with those of an analogous distributed delay system. Otherwise, the stability domains are obtained by a numerical integration of the linearized variable-delay system. The analysis shows that the control domains are significantly larger than those in the usual time-delayed feedback control, and that the complexity of the domain structure depends on the form and the frequency of the delay modulation.Comment: 13 pages, 8 figures, RevTeX, accepted for publication in Physical Review

Dissociation of Relativistic Projectiles with the Continuum-Discretized Coupled-Channels Method

Relativistic effects in the breakup of weakly-bound nuclei at intermediate energies are studied by means of the continuum-discretized coupled-channels method with eikonal approximation. Nuclear coupling potentials with Lorentz contraction are newly included and those effects on breakup cross sections are investigated. We show that relativistic corrections lead to larger breakup cross sections. Coupled-channel effects on the breakup cross sections are also discussed.Comment: 9 pages, 7 figures, to be published in Prog. Theo. Phy