Finite Probability Research Articles

Earthquake precursors: activation or quiescence?

SUMMARY We discuss the long-standing question of whether the probability for large earthquake occurrence (magnitudes m > 6.0) is highest during time periods of smaller event activation, or highest during time periods of smaller event quiescence. The physics of the activation model are based on an idea from the theory of nucleation, that a small magnitude earthquake has a finite probability of growing into a large earthquake. The physics of the quiescence model is based on the idea that the occurrence of smaller earthquakes (here considered as magnitudes m > 3.5) may be due to a mechanism such as critical slowing down, in which fluctuations in systems with long-range interactions tend to be suppressed prior to large nucleation events. To illuminate this question, we construct two end-member forecast models illustrating, respectively, activation and quiescence. The activation model assumes only that activation can occur, either via aftershock nucleation or triggering, but expresses no choice as to which mechanism is preferred. Both of these models are in fact a means of filtering the seismicity time-series to compute probabilities. Using 25 yr of data from the California–Nevada catalogue of earthquakes, we show that of the two models, activation and quiescence, the latter appears to be the better model, as judged by backtesting (by a slight but not significant margin). We then examine simulation data from a topologically realistic earthquake model for California seismicity, Virtual California. This model includes not only earthquakes produced from increases in stress on the fault system, but also background and off-fault seismicity produced by a BASS–ETAS driving mechanism. Applying the activation and quiescence forecast models to the simulated data, we come to the opposite conclusion. Here, the activation forecast model is preferred to the quiescence model, presumably due to the fact that the BASS component of the model is essentially a model for activated seismicity. These results lead to the (weak) conclusion that California seismicity may be characterized more by quiescence than by activation, and that BASS–ETAS models may not be robustly applicable to the real data.

Geminate-ion escape probability for arbitrary mean free path: Use of the Metropolis method

Of the various methods of improving the Onsager theory of geminate electron–ion recombination for arbitrary mean free path, an especially promising one involves the use of the Metropolis rejection criterion [A. Mozumder, Radiat. Phys. Chem. 72 (2005) 73; 74 (2005) 227]. The present Letter further includes the effect of finite reaction probability at the reaction radius by a tridiagonal matrix method. Model calculations suggest usefulness when the overall mobility exceeds ∼1cm2V−1s−1. Variations of the escape probability with mean free path, initial separation and final reaction rate are discussed.

Efficient heralding of photonic qubits with applications to device-independent quantum key distribution

We present an efficient way of heralding photonic qubit signals using linear optics devices. First we show that one can obtain asymptotically perfect heralding and unit success probability with growing resources. Second, we show that even using finite resources, we can improve qualitatively and quantitatively over earlier heralding results. In the latte r scenario, we can obtain perfect heralded photonic qubits while maintaining a finite success probability. We demonstrate the advantage of our heralding scheme by predicting key rates for device independent quantum key distribution, taking imperfections of sources and detectors into account.

Bootstrapping asset price bubbles

In this paper we propose a method that allows to test for asset price bubbles. The method is mainly based on a bootstrap methodology which helps to compute the finite sample probability distribution of the asymptotic tests which were recently proposed in Phillips et al. (2011) and Phillips and Yu (2009). We apply the method to the Nasdaq stock price index and Case-Shiller house price index. The results indicate that speculation was behind the upsurge in both asset prices.

Estimation of the Maximum Possible Magnitude in the Framework of a Doubly Truncated Gutenberg-Richter Model

We discuss to what extent a given earthquake catalog and the assumption of a doubly truncated Gutenberg–Richter distribution for the earthquake magnitudes allow for the calculation of confidence intervals for the maximum possible magnitude M . We show that, without further assumptions such as the existence of an upper bound of M , only very limited information may be obtained. In a frequentist formulation, for each confidence level α the confidence interval diverges with finite probability. In a Bayesian formulation, the posterior distribution of the upper magnitude is not normalizable. We conclude that the common approach to derive confidence intervals from the variance of a point estimator fails. Technically, this problem can be overcome by introducing an upper bound ![Graphic][1] for the maximum magnitude. Then the Bayesian posterior distribution can be normalized, and its variance decreases with the number of observed events. However, because the posterior depends significantly on the choice of the unknown value of ![Graphic][2] , the resulting confidence intervals are essentially meaningless. The use of an informative prior distribution accounting for preknowledge of M is also of little use, because the prior is only modified in the case of the occurrence of an extreme event. Our results suggest that the maximum possible magnitude M should be better replaced by M T , the maximum expected magnitude in a given time interval T , for which the calculation of exact confidence intervals becomes straightforward. From a physical point of view, numerical models of the earthquake process adjusted to specific fault regions may be a powerful alternative to overcome the shortcomings of purely statistical inference. [1]: /embed/inline-graphic-1.gif [2]: /embed/inline-graphic-2.gif

KAM TORI AND ABSENCE OF DIFFUSION OF A WAVE-PACKET IN THE 1D RANDOM DNLS MODEL

When nonlinearity is added to an infinite system with purely discrete linear spectrum, Anderson modes become coupled with one another by terms of higher order than linear, allowing energy exchange between them. It is generally believed, on the basis of numerical simulations in such systems, that any initial wave-packet with finite energy spreads down chaotically to zero amplitude with second moment diverging as a power law of time, slower than standard diffusion (subdiffusion). We present results which suggest that the interpretation of spreading cannot be described as initially believed and that new questions arise and still remain opened. We show that an initially localized wave-packet with finite norm may generate two kinds of trajectories both obtained with nonvanishing probability.The first kind consists of KAM trajectories which are recurrent and do not spread. Empirical investigations suggest that KAM theory may still hold in infinite systems under two conditions: (1) the linearized spectrum is purely discrete, (2) the considered solutions are square summable and not too large in amplitude. We check numerically that in appropriate regions of the parameter space, indeed many initial conditions can be found with finite probability that generate (nonspreading) infinite dimension tori (almost periodic solutions) in a fat Cantor set in (projected) phase space.The second kind consists of trajectories which look initially chaotic and often spread over long times. We first rigorously prove that initial chaos does not necessarily imply complete spreading e.g. for large norm initial wave-packet. Otherwise, in some modified models, no spreading at all is proven to be possible, despite the presence of initial chaos in contradiction with early beliefs. The nature of the limit state is still unknown.However, we attempt to present empirical arguments suggesting that if a trajectory starts chaotically spreading, there will necessarily exist (generally large) critical spreading distances that depend on the disorder realization where the trajectory will be sticking to a dense set of KAM tori. This effect should induce drastic slowing down of the spreading which could be viewed as "inverse Arnold diffusion" since the trajectory approaches KAM tori regions instead of leaving them. We suggest that this effect should self-organize the chaotic behavior and that at long time, the wave-packet might not be spread down to zero, but could have a limit profile with marginal chaos (with singular continuous spectrum), despite a long spatial tail. Further analytical and numerical investigations are required.

THE ORIGIN OF VARIABILITY OF THE INTERMEDIATE-MASS BLACK-HOLE ULX SYSTEM HLX-1 IN ESO 243-49

The ultra-luminous intermediate-mass black-hole system HLX-1 in the ESO 243-49 galaxy exhibits variability with a possible recurrence time of a few hundred days. Finding the origin of this variability would constrain the still largely unknown properties of this extraordinary object. Since it exhibits an intensity-hardness behavior characteristic of black-hole X-ray transients, we have analyzed the variability of HLX-1 in the framework of the disk instability model that explains outbursts of such systems. We find that the long-term variability of HLX-1 is unlikely to be explained by a model in which outbursts are triggered by thermal-viscous instabilities in an accretion disc. Possible alternatives include the instability in a radiation-pressure dominated disk but we argue that a more likely explanation is a modulated mass-transfer due to tidal stripping of a star on an eccentric orbit around the intermediate-mass black hole. We consider an evolutionary scenario leading to the creation of such a system and estimate the probability of its observation. We conclude, using a simplified dynamical model of the post-collapse cluster, that no more than 1/100 to 1/10 of Mbh < 10^4 Msun IMBHs - formed by run-away stellar mergers in the dense collapsed cores of young clusters - could have a few times 1 Msun Main-Sequence star evolve to an AGB on an orbit eccentric enough for mass transfer at periapse, while avoiding collisional destruction or being scattered into the IMBH by 2-body encounters. The finite but low probability of this configuration is consistent with the uniqueness of HLX-1. We note, however, that the actual response of a standard accretion disk to bursts of mass transfer may be too slow to explain the observations unless the orbit is close to parabolic (and hence even rarer) and/or additional heating, presumably linked to the highly time-dependent gravitational potential, are invoked.

Agnostic about in Vivo Inverse Agonism of Agouti-Related Peptide

Agnostic about in Vivo Inverse Agonism of Agouti-Related Peptide

Precise large deviations for a customer-based individual risk model

In this paper, we propose a customer-based individual risk model, in which potential claims by customers are described as i.i.d. heavy-tailed random variables, but different insurance policy holders are allowed to have different probabilities to make actual claims. Some precise large deviation results for the prospective-loss process are derived under certain mild assumptions, with emphasis on the case of heavy-tailed distribution function class ERV (extended regular variation). Lundberg type limiting results on the finite time ruin probabilities are also investigated.

Erlangian approximation to finite time ruin probabilities in perturbed risk models

In this paper, we consider a class of perturbed risk processes that have an underlying Markov structure, including Markov-modulated risk processes, and Sparre–Andersen risk processes when both inter-claim times and claim sizes are phase-type. We apply the Erlangization method to the risk process in the class in order to obtain an accurate approximation of the finite time ruin probability. In addition, we develop an efficient recursive procedure by recognizing a repeating structure in the probability matrices we work with. We believe the present work is among the first to either compute or approximate finite time ruin probabilities in the perturbed risk model.

Existence of shadow prices in finite probability spaces

A shadow price is a process \({\widetilde{S}}\) lying within the bid/ask prices \({\underline{S},\overline{S}}\) of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with price process \({\widetilde{S}}\) leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow price.

Recombination of hydrogen atoms on fine-grain graphite

The probability of recombination of hydrogen atoms on surfaces of fine-grain graphite EK98 was investigated as a function of surface roughness. The source of hydrogen atoms used in this experiment was weakly ionised plasma created with an inductively coupled radiofrequency generator at pressures from 30Pa to 175Pa in hydrogen. Hydrogen atom density was measured by means of fibre optic catalytic probes. The recombination coefficient of the graphite samples was determined by observing their impact on the spatial distribution of the atom density in a closed side-arm of the reactor. Smith's diffusion model was used to calculate the values of the recombination coefficient. The measured recombination coefficient was found to increase much faster than the measured effective surface. This discrepancy is explained by the fact that on a surface which is not perfectly flat, there is a finite probability for multiple collisions. Impinging atoms collide more than once with the surface before they are reflected into the surface, which results in a larger probability of recombination.

Modeling the Coalescence Kinetics of Cell Surface Receptor Clusters

Cells in a multicellular organism must be ready to respond to a variety of extracellular signals such as ligands on the surface of cells or parasites. To understand these signaling mechanisms, many current experiments investigate transmembrane signaling of immunoreceptors on B cells, T cells, or mast cells, initiated by binding to specific monovalent ligands in fluid membranes. In mast cells, receptor clusters initially form at cellular protrusions through diffusion mediated trapping, undergo primarily diffusive motion with a partial directed element as well, and eventually coalesce at a finite rate to form a large central receptor patch termed synapse (Spendier et al. 2010. Biophys. J. 99:388–397). We are currently developing a coalescence theory to investigate the kinetics of receptor cluster coalescence in detail, which is important in understanding the mechanism and dynamics of cellular signaling. Our coalescence theory is split into trapping considerations, which generalize Smoluchowski's well known theory for arbitrary melding probability, and a feedback idea. Trapping considerations were developed with a unified approach constructed earlier for excitons (Kenkre and Reineker. 1982. Exciton Dynamics in Molecular Crystals and Aggregates. Springer, Berlin) to compute the particle survival probability for any dimension and motion due to a closed or open trapping surface. To build a coalescence theory, we use our trapping prescription to compute the rate at which particles disappear and solve for the time dependent trap radius through a self-consistent approach. We find that our coalescence theory for a finite melding probability is in good agreement with simulations.

N-Terminal Fragment of Cardiac Myosin Binding Protein-C (CMYBP-C) Reduces Actomyosin Power Output in the Laser Trap Assay

Mutations in cMyBP-C cause hypertrophic cardiomyopathy, but cMyBP-C's modulation of cardiac contractility is poorly understood. The N-terminus of cMyBP-C binds actin and myosin S2, while the C-terminus binds the myosin rod. Thus, cMyBP-C may act as a tether between the thick and thin filament, imposing a load against which myosin must operate. We used a laser trap force clamp assay (Debold et al. B.J., 2005) to characterize actomyosin's force velocity relationship (F:V) in the absence and presence of a bacterially expressed N-terminal cMyBP-C fragment, C0C3. In this assay a small ensemble of skeletal myosin (∼8 heads) translocates a skeletal actin filament under a series of constant loads (1-8 pN), resulting in a classic hyperbolic F:V with a maximum isometric force of 10 ± 2 pN. Adding C0C3 at a 1:1 ratio to myosin depressed actomyosin's F:V: i.e. 48% reduction in maximum unloaded velocity with no effect on maximum force. This reduced actomyosin's maximum power output by 40%. The depressed velocities suggest C0C3 may impose as much as 2 pN of opposing load to the myosin ensemble. As limited numbers of myosin translocate the actin filament under load, a finite probability exists that no myosin will be attached, causing rapid (∼2 ms duration) backward movements (34 ± 1 nm) of the actin filament, i.e. slips. The frequency of slips rises with increasing load, which is diminished significantly in the presence of C0C3, suggesting C0C3 prevents slips by its attachment to the actin filament. Based on the depression of the F:V and fewer actin filament slips in the presence of C0C3, we propose that the N-terminus of cMyBP-C can tether actin and impose an effective viscous load in the sarcomere to modulate cardiac power output.

Iterative Reclassification in Agglomerative Clustering

In model-based clustering of complex data, a probability model, typically a finite mixture probability model, forms the basis of the distance measure between any pair of clusters. The idea of model-based clustering was popularized by the framework and accompanying software of Fraley and Raftery (2002). In particular, model-based agglomerative hierarchical clustering is now a frequently used approach for probabilistic grouping of data, due to the speed and simplicity of implementation. This article investigates deficiencies in the clusterings proposed from this popular approach, and presents a review of small refinements and extensions to the procedure with differing performance gains and computational costs. The improvements are illustrated through application to simulated and real data examples, including the clustering of gene expression time profiles. Some of the proposed improvements to agglomerative clustering are, like the procedure itself in its usual form, deterministic; perhaps surprisingly though, the best overall results here are obtained via a stochasticized version of the entire procedure. While the focus of this article is probability model-based clustering, many of the schemes presented are equally applicable to agglomerative clustering under any distance measure.The simulated data from this article along with the C++ code used for implementing the algorithms for all of the examples can be obtained online from the Supplemental Material.

Risk associated with managed investment primary production projects

The paper presents a model of investment risk associated with projects that promote primary production, such as plantations, forests, aquaculture and agribusiness generally, commonly through managed investment schemes (MISs) set up as businesses. An analysis of business failures leads to a finite probability that the investor will lose money. The attractiveness of the investment is further diminished because of the uncertainty in the downstream return on investment and time of the return. Based on probabilistic arguments, the paper presents a robust model and method on which to make investment decisions. The approach is practical and will be useful to people interested in managed agribusiness investments. The approach is significant in that it quantifies what heretofore was largely undertaken qualitatively.

Stochastic Aeroelastic Analysis of Composite Helicopter Rotor

The effect of uncertainty in composite material properties on the aeroelastic response, vibratory loads, and stability of a hingeless helicopter rotor is investigated. The uncertainty impact on rotating natural frequencies of the blade is studied with Monte Carlo simulations and first-order reliability methods. The stochastic aeroelastic analyses in hover and forward flight are carried out with Monte Carlo simulations. The flap, lag, and torsion responses show considerable scatter from their baseline values, and the uncertainty impact varies with the azimuth angle. Furthermore, the blade response shows finite probability of resonance-type conditions caused by modal frequencies approaching multiples of the rotor speed. The 4/rev vibratory forces show large deviations from their baseline values. The lag mode damping shows considerable scatter due to uncertain material properties with an almost 40% probability of instability in hover.

The Dual BKR Inequality and Rudich's Conjecture

Let be a set of terms over an arbitrary (but finite) number of Boolean variables. Let U() be the set of truth assignments that satisfy exactly one term in . Motivated by questions in computational complexity, Rudich conjectured that there exist ∊, δ &gt; 0 such that, if is any set of terms for which U() contains at least a (1−∊)-fraction of all truth assignments, then there exists a term t ∈ such that at least a δ-fraction of assignments satisfy some term of sharing a variable with t [8].We prove a stronger version: for any independent assignment of the variables (not necessarily the uniform one), if the measure of U() is at least 1 − ∊, there exists a t ∈ such that the measure of the set of assignments satisfying either t or some term incompatible with t (i.e., having no satisfying assignments in common with t) is at least $\Gd = 1- \Ge - \frac{4\Ge}{1-\Ge}$. (A key part of the proof is a correlation-like inequality on events in a finite product probability space that is in some sense dual to Reimer's inequality [11], a.k.a. the BKR inequality [5], or the van den Berg–Kesten conjecture [3].)

Fast Magnetic Reconnection in the Plasmoid-Dominated Regime

A conceptual model of resistive magnetic reconnection via a stochastic plasmoid chain is proposed. The global reconnection rate is shown to be independent of the Lundquist number. The distribution of fluxes in the plasmoids is shown to be an inverse-square law. It is argued that there is a finite probability of emergence of abnormally large plasmoids, which can disrupt the chain (and may be responsible for observable large abrupt events in solar flares and sawtooth crashes). A criterion for the transition from the resistive magnetohydrodynamic to the collisionless regime is provided.

Ramified optimal transportation in geodesic metric spaces

An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.