Physics

Closed-form expressions, parametrized by the Hurst exponent H and the length n of a time series, are derived for paths of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) in the A−T plane, composed of the fraction of turning points T and the Abbe value A . The exact formula for A fBm is expressed via Riemann ζ and Hurwitz ζ functions. A very accurate approximation, yielding a simple exponential form, is obtained. Finite-size effects, introduced by the deviation of fGn's variance from unity, and asymptotic cases are discussed. Expressions for T for fBm, fGn, and differentiated fGn are also presented. The same methodology, valid for any Gaussian process, is applied to autoregressive moving average processes, for which regions of availability of the A−T plane are derived and given in analytic form. Locations in the A−T plane of some real-world examples as well as generated data are discussed for illustration.

Permutation entropy techniques can be useful in identifying anomalies in paleoclimate data records, including noise, outliers, and post-processing issues. We demonstrate this using weighted and unweighted permutation entropy of water-isotope records in a deep polar ice core. In one region of these isotope records, our previous calculations revealed an abrupt change in the complexity of the traces: specifically, in the amount of new information that appeared at every time step. We conjectured that this effect was due to noise introduced by an older laboratory instrument. In this paper, we validate that conjecture by re-analyzing a section of the ice core using a more-advanced version of the laboratory instrument. The anomalous noise levels are absent from the permutation entropy traces of the new data. In other sections of the core, we show that permutation entropy techniques can be used to identify anomalies in the raw data that are not associated with climatic or glaciological processes, but rather effects occurring during field work, laboratory analysis, or data post-processing. These examples make it clear that permutation entropy is a useful forensic tool for identifying sections of data that require targeted re-analysis---and can even be useful in guiding that analysis.

Stochastic nonlinear dynamical systems can undergo rapid transitions relative to the change in their forcing, for example due to the occurrence of multiple equilibrium solutions for a specific interval of parameters. In this paper, we modify one of the methods developed to compute probabilities of such transitions, Trajectory-Adaptive Multilevel Sampling (TAMS), to be able to apply it to high-dimensional systems. The key innovation is a projected time-stepping approach, which leads to a strong reduction in computational costs, in particular memory usage. The performance of this new implementation of TAMS is studied through an example of the collapse of the Atlantic Ocean Circulation.

A new mathematical method for elucidating neutrino mass from beta decay is studied. It is based upon the solutions of transformed Fredholm and Volterra integral equations. In principle, theoretical beta-particle spectra can consist of several neutrino-mass eigenvalues. Integration of the theoretical beta spectrum with a normalized instrumental response function results in the Fredholm integral equation of the first kind. This equation is transformed in such a way that the solution of it is a superposition of the Heaviside step-functions, one for each neutrino mass eigenvalue. A series expansion leading to matrix linear equations is then derived to solve the transformed Fredholm equation. Another approach is derived when the theoretical beta spectrum is obtained by a separate deconvolution of the observed spectrum. It is then proven that the transformed Fredholm equation reduces to the Abel integral equation. The Abel equation has a general integral solution, which is proven in this work by using a specific function for the beta spectrum. As an example, a numerical solution of the Abel integral equation is also provided, which has a fractional sensitivity of about 0.001 for subtle neutrino eigenvalue searches, and can distinguish from experimental beta-spectrum discrepancies, such as shape and energy nonlinearities.

Least-squares fits are an important tool in many data analysis applications. In this paper, we review theoretical results, which are relevant for their application to data from counting experiments. Using a simple example, we illustrate the well known fact that commonly used variants of the least-squares fit applied to Poisson-distributed data produce biased estimates. The bias can be overcome with an iterated weighted least-squares method, which produces results identical to the maximum-likelihood method. For linear models, the iterated weighted least-squares method converges faster than the equivalent maximum-likelihood method, and does not require problem-specific starting values, which may be a practical advantage. The equivalence of both methods also holds for binomially distributed data. We further show that the unbinned maximum-likelihood method can be derived as a limiting case of the iterated least-squares fit when the bin width goes to zero, which demonstrates a deep connection between the two methods.

Frequency metrology outperforms any other branch of metrology in accuracy (parts in 10 −16 ) and small fluctuations ( < 10 −17 ). In turn, among celestial bodies, the rotation speed of millisecond pulsars (MSP) is by far the most stable ( < 10 −18 ). Therefore, the precise measurement of the time of arrival (TOA) of pulsar signals is expected to disclose information about cosmological phenomena, and to enlarge our astrophysical knowledge. Related to this topic, Pulsar Timing Array (PTA) projects have been developed and operated for the last decades. The TOAs from a pulsar can be affected by local emission and environmental effects, in the direction of the propagation through the interstellar medium or universally by gravitational waves from super massive black hole binaries. These effects (signals) can manifest as a low-frequency fluctuation over time, phenomenologically similar to a red noise. While the remaining pulsar intrinsic and instrumental background (noise) are white. This article focuses on the frequency metrology of pulsars. From our standpoint, the pulsar is an accurate clock, to be measured simultaneously with several telescopes in order to reject the uncorrelated white noise. We apply the modern statistical methods of time-and-frequency metrology to simulated pulsar data, and we show the detection limit of the correlated red noise signal between telescopes.

We derive asymptotic formulas for the mean exit time τ ¯ N of the fastest among N identical independently distributed Brownian particles to an absorbing boundary for various initial distributions (partially uniformly and exponentially distributed). Depending on the tail of the initial distribution, we report here a continuous algebraic decay law for τ ¯ N , which differs from the classical Weibull or Gumbell results. We derive asymptotic formulas in dimension 1 and 2, for half-line and an interval that we compare with stochastic simulations. We also obtain formulas for an additive constant drift on the Brownian motion. Finally, we discuss some applications in cell biology where a molecular transduction pathway involves multiple steps and a long-tail initial distribution.

Rapidly applying the effects of detector response to physics objects (e.g. electrons, muons, showers of particles) is essential in high energy physics. Currently available tools for the transformation from truth-level physics objects to reconstructed detector-level physics objects involve manually defining resolution functions. These resolution functions are typically derived in bins of variables that are correlated with the resolution (e.g. pseudorapidity and transverse momentum). This process is time consuming, requires manual updates when detector conditions change, and can miss important correlations. Machine learning offers a way to automate the process of building these truth-to-reconstructed object transformations and can capture complex correlation for any given set of input variables. Such machine learning algorithms, with sufficient optimization, could have a wide range of applications: improving phenomenological studies by using a better detector representation, allowing for more efficient production of Geant4 simulation by only simulating events within an interesting part of phase space, and studies on future experimental sensitivity to new physics.

This software performs the combination of m correlated estimates of n physics observables ( m≥n ) using the Best Linear Unbiased Estimate (BLUE) method. It is implemented as a C++ class, to be used within the ROOT analysis package. It features easy disabling of specific estimates or uncertainty sources, the investigation of different correlation assumptions, and allows performing combinations according to the importance of the estimates. This enables systematic investigations of the combination on details of the measurements from within the software, without touching the input.

The "backward simulation" of a stochastic process is defined as the stochastic dynamics that trace a time-reversed path from the target region to the initial configuration. If the probabilities calculated by the original simulation are easily restored from those obtained by backward dynamics, we can use it as a computational tool. It is shown that the naive approach to backward simulation does not work as expected. As a remedy, the Time Reverse Monte Carlo method (TRMC) based on the ideas of Sequential Importance Sampling (SIS) and Sequential Monte Carlo (SMC) is proposed and successfully tested with a stochastic typhoon model and the Lorenz 96 model. TRMC with SMC, which contains resampling steps, is shown to be more efficient for simulations with a larger number of time steps. A limitation of TRMC and its relation to the Bayes formula are also discussed.