Inverse Power Law Research Articles

Stochastic properties of musical time series

Musical sequences are correlated dynamical processes that may differ depending on musical styles. We aim to quantify the correlations through power spectral analysis of pitch sequences in a large corpus of musical compositions as well as improvised performances. Using a multitaper method we extend the power spectral estimates down to the smallest possible frequencies optimizing the tradeoff between bias and variance. The power spectral densities reveal a characteristic behavior; they typically follow inverse power laws (1/f β-noise), yet only down to a cutoff frequency, where they end in a plateau. Correspondingly the pitch autocorrelation function exhibits slow power law decays only up to a cutoff time, beyond which the correlations vanish. We determine cutoff times between 4 and 100 quarter note units for the compositions and improvisations of the corpus, serving as a measure for the degree of persistence and predictability in music. The histogram of exponents β for the power law regimes has a pronounced peak near β = 1 for classical compositions, but is much broader for jazz improvisations.

Tapping strength variability in sensorimotor experiments on rhythmic tapping.

We report psychophysical experiments and time series analyses to investigate sensorimotor tapping strength fluctuations in human periodic tapping with and without a metronome. The power spectral density of tapping strength fluctuations typically decays in an inverse power law (1/fβ-noise) associated with long-range correlations, i.e., with a slow power-law decay of tapping strength autocorrelations and scale-free behavior. The power-law exponents β are scattered around β=1 ranging from 0.67 to 1.8. A log-linear representation of the power spectral densities reveals rhythmic peaks at frequencies f=0.25 (and f=0.5) and a tendency to slightly accentuate every fourth (and second) stroke when subjects try to synchronize their tapping with a metronome.

Revisiting empirical solar energetic particle scaling relations

Aims. The space radiation environment conditions and the maximum expected coronal mass ejection (CME) speed are assessed by investigating scaling laws between the peak proton flux and fluence of solar energetic particle (SEP) events with the speed of the CMEs. Methods. We used a complete catalog of SEP events, covering the last ∼25 years of CME observations (i.e., 1997–2017). We calculated the peak proton fluxes and integrated event fluences for events that reached an integral energy of up to E > 100 MeV. For a sample of 38 strong SEP events, we first investigated the statistical relations between the recorded peak proton fluxes (IP) and fluences (FP) at a set of integral energies of E > 10 MeV, E > 30 MeV, E > 60 MeV, and E > 100 MeV versus the projected CME speed near the Sun (VCME) obtained by the Solar and Heliospheric Observatory/Large Angle and Spectrometric Coronagraph (SOHO/LASCO). Based on the inferred relations, we further calculated the integrated energy dependence of both IP and FP, assuming that they follow an inverse power law with respect to energy. By making use of simple physical assumptions, we combined our derived scaling laws to estimate the upper limits for VCME, IP, and FP by focusing on two cases of known extreme SEP events that occurred on 23 February 1956, (GLE05) and in AD774/775, respectively. Based on the physical constraints and assumptions, several options for the upper limit VCME associated with these events were investigated. Results. A scaling law relating IP and FP to the CME speed as VCME5 for CMEs ranging between ∼3400–5400 km/s is consistent with values of FP inferred for the cosmogenic nuclide event of AD774/775. At the same time, the upper CME speed that the current Sun can provide possibly falls within an upper limit of VCME ≤ 5500 km/s.

Theoretical analysis of the structure, thermodynamics, and shear elasticity of deeply metastable hard sphere fluids.

The structure, thermodynamics, and slow activated dynamics of the equilibrated metastable regime of glass-forming fluids remain a poorly understood problem of high theoretical and experimental interest. We apply a highly accurate microscopic equilibrium liquid state integral equation theory, in conjunction with naïve mode coupling theory of particle localization, to study in a unified manner the structural correlations, thermodynamic properties, and dynamic elastic shear modulus in deeply metastable hard sphere fluids. Distinctive behaviors are predicted including divergent inverse critical power laws for the contact value of the pair correlation function, pressure, and inverse dimensionless compressibility, and a splitting of the second peak and large suppression of interstitial configurations of the pair correlation function. The dynamic elastic modulus is predicted to exhibit two distinct exponential growth regimes with packing fraction that have strongly different slopes. These thermodynamic, structural, and elastic modulus results are consistent with simulations and experiments. Perhaps most unexpectedly, connections between the amplitude of long wavelength density fluctuations, dimensionless compressibility, local structure, and the dynamic elastic shear modulus have been theoretically elucidated. These connections are more broadly relevant to understanding the slow activated relaxation and mechanical response of colloidal suspensions in the ultradense metastable region and deeply supercooled thermal liquids in equilibrium.

A mathematical model for bedrock incision in near‐threshold gravel‐bed rivers

AbstractGravel‐bed rivers that incise into bedrock are common worldwide. These systems have many similarities with other alluvial channels: they transport large amounts of sediment and adjust their forms in response to discharge and sediment supply. At the same time, the occurrence of bedrock incision implies behaviour that falls on a spectrum between fully detachment‐limited ‘bedrock channels’ and fully transport‐limited ‘alluvial channels’. Here, we present a mathematical model of river profile evolution that integrates bedrock erosion, gravel transport and the formation of channels whose hydraulic geometry is consistent with that of near‐threshold alluvial channels. We combine theory for five interrelated processes: bedload sediment transport in equilibrium gravel‐bed channels, channel width adjustment to flow and sediment characteristics, abrasion of bedrock by mobile sediment, plucking of bedrock and progressive loss of gravel‐sized sediment due to grain attrition. This model contributes to a growing class of models that seek to capture the dynamics of both bedrock incision and alluvial sediment transport. We demonstrate the model's ability to reproduce expected fluvial features such as inverse power law scaling between slope and area, and width and depth consistent with near‐threshold channel theory, and we discuss the role of sediment characteristics in influencing the mode of channel behaviour, erosional mechanism, channel steepness and profile concavity.

Silence is golden, but my measures still see—why cheaper-but-noisier outcome measures in large simple trials can be more cost-effective than gold standards

ObjectiveTo assess the cost-effectiveness of using cheaper-but-noisier outcome measures, such as a short questionnaire, for large simple clinical trials.BackgroundTo detect associations reliably, trials must avoid bias and random error. To reduce random error, we can increase the size of the trial and increase the accuracy of the outcome measurement process. However, with fixed resources, there is a trade-off between the number of participants a trial can enrol and the amount of information that can be collected on each participant during data collection.MethodsTo consider the effect on measurement error of using outcome scales with varying numbers of categories, we define and calculate the variance from categorisation that would be expected from using a category midpoint; define the analytic conditions under which such a measure is cost-effective; use meta-regression to estimate the impact of participant burden, defined as questionnaire length, on response rates; and develop an interactive web-app to allow researchers to explore the cost-effectiveness of using such a measure under plausible assumptions.ResultsAn outcome scale with only a few categories greatly reduced the variance of non-measurement. For example, a scale with five categories reduced the variance of non-measurement by 96% for a uniform distribution. We show that a simple measure will be more cost-effective than a gold-standard measure if the relative increase in variance due to using it is less than the relative increase in cost from the gold standard, assuming it does not introduce bias in the measurement. We found an inverse power law relationship between participant burden and response rates such that a doubling the burden on participants reduces the response rate by around one third. Finally, we created an interactive web-app (https://benjiwoolf.shinyapps.io/cheapbutnoisymeasures/) to allow exploration of when using a cheap-but-noisy measure will be more cost-effective using realistic parameters.ConclusionCheaper-but-noisier questionnaires containing just a few questions can be a cost-effective way of maximising power. However, their use requires a judgement on the trade-off between the potential increase in risk of information bias and the reduction in the potential of selection bias due to the expected higher response rates.

From Zipf to Price and beyond

Consider a society comprising n members that are ranked in a decreasing order of their personal wealths. For this society: Zipf’s Law manifests the case in which the members’ ranks and wealths display an inverse power-law relation; and Price’s Law manifests the case in which the n richest members possess, collectively, half of all the wealth. This paper goes from Zipf’s Law to Price’s Law and beyond, and it does so by introducing and exploring a novel Generalized Price’s Law (GPL): a general allometric scaling of the quantiles of rank distributions. Akin to multifractals, the GPL is governed by spectrums of powers. The GPL spectra are investigated, and are shown to be fractal objects: from a socioeconomic perspective, the spectra are power-law Lorenz curves that characterize poor fractality and rich fractality; from a probabilistic perspective, the spectra are characterized by Pareto statistics and by Lindy Laws. The intersection of Zipf’s Law and of the GPL is also investigated, and it is shown to be: (i) the phase transition between the two markedly different fractal regimes of the GPL spectra; (ii) the phase transition between two markedly different macroscopic regimes of Zipf’s Law; (iii) Price’s Law. Metaphorically, this paper establishes navigation directions in the space of rank distributions: how to get from the ‘Zipf street’ to the ‘Price junction’, from there to the new ‘GPL avenue’, and back.

The breakdown of the direct relation between the density scaling exponent and the intermolecular interaction potential for molecular systems with purely repulsive intermolecular forces

In this work, we question the generally accepted statement that the character of intermolecular interactions can be directly determined from the scaling exponent. We show that the scaling exponent evidently differs on the molecular architecture for pure inverse power law systems. Since these systems are characterized by a well-defined dependency between the intermolecular virial and potential energy, the results contradict theoretical predictions and previous reports. Moreover, we suggest that the recently proposed “molecular force” method also returns the value that varies from the one scaling the dynamics. Finally, the obtained result allows us to conclude that the value of the density scaling exponent for real liquids depends on intramolecular interactions.

Quantum gravity effects on the tachyon inflation from thermodynamic perspective

By considering the Friedmann equations emerging from the entropy-area law of black hole thermodynamics in the context of the generalized uncertainty principle, we study tachyon inflation in the early universe. The presence of a minimal length modifies the Friedmann equations and hence the slow-roll and perturbation parameters in the tachyon model. These modifications, though small, affect the viability of the tachyon inflation in confrontation with observational data. We compare the numerical results of the model with Planck2018 TT, TE, EE +lowE+lensing+BAO+BK14(18) data and Planck2018 TT, TE,EE +lowE+lensing+BK14(18) +BAO+LIGO & Virgo2016 data at 68% and 95% CL. We show that while the tachyon inflation with power-law, inverse power-law and inverse exponential potentials is not observationally viable in comparison with the 1σ and 2σ confidence levels of the new joint data, in the presence of the minimal length the model becomes observationally viable.

Bayesian estimation and prediction under progressive-stress accelerated life test for a log-logistic model

Accelerated life tests play a very critical role in reliability analysis because highly reliable products are being produced with recent advanced technologies to sustain the market demand and competition. A progressive-stress accelerated life test is one of the kinds of accelerated life tests that allows applied stress to change continuously. Considering the importance of progressive-stress accelerated life tests, this paper deals with progressive-stress accelerated life tests when the lifetimes of units follow the log-logistic distribution and are progressively type-II censored where the associated scale parameter conforms to the inverse power law. Different estimators of model parameters are derived using maximum likelihood and Bayesian methods. Interval estimation is also considered. In the sequel, approximate confidence, bootstrap, and Bayes credible intervals are constructed. Bayes estimators are obtained under squared error, LINEX, and entropy loss functions using proper and improper prior distributions. A simulation study is conducted based on various censoring schemes. The coverage percentages and interval widths are computed via Monte Carlo simulations. Bayes predictive estimates and intervals are also obtained. Finally, two different accelerated life test data are analyzed for illustration purposes.

Stringlet excitation model of the boson peak.

The boson peak (BP), a low-energy excess in the vibrational density of states over the Debye contribution, is often identified as a characteristic of amorphous solid materials. Despite decades of efforts, its microscopic origin still remains a mystery. Recently, it has been proposed, and corroborated with simulations, that the BP might stem from intrinsic localized modes involving one-dimensional (1D) string-like excitations ("stringlets"). We build on a theory originally proposed by Lund that describes the localized modes as 1D vibrating strings, but we specify the stringlet size distribution to be exponential, as observed in simulations. We provide an analytical prediction for the BP frequency ωBP in the temperature regime well below the observed glass transition temperature Tg. The prediction involves no free parameters and accords quantitatively with prior simulation observations in 2D and 3D model glasses based on inverse power law potentials. The comparison of the string model to observations is more uncertain when compared to simulations of an Al-Sm metallic glass material at temperatures well above Tg. Nonetheless, our stringlet model of the BP naturally reproduces the softening of the BP frequency upon heating and offers an analytical explanation for the experimentally observed scaling with the shear modulus in the glass state and changes in this scaling in simulations of glass-forming liquids. Finally, the theoretical analysis highlights the existence of a strong damping for the stringlet modes above Tg, which leads to a large low-frequency contribution to the 3D vibrational density of states, observed in both experiments and simulations.

81‐2: Automotive OLED Life Extension Utilizing Automatic Luminance Control

Understanding OLED life characteristics is crucial towards implementation of OLED technology into automotive applications. The use of the Arrhenius equation together with the Inverse Power Law may be utilized to estimate the improvement in OLED life when automatic luminance control methods are utilized.

The Inflationary with Inverse Power-Law Potential in Tsallis Entropy

The Inflationary with Inverse Power-Law Potential in Tsallis Entropy

Phase-space analysis in non-minimal symmetric-teleparallel dark energy

We modify the symmetric-teleparallel dark energy through the addition of a further Yukawa-like term, in which the non-metricity scalar, Q, is non-minimally coupled to a scalar field Lagrangian where the phion acts as quintessence, describing dark energy. We investigate regions of stability and find late-time attractors. To do so, we conduct a stability analysis for different types of physical potentials describing dark energy, namely the power-law, inverse power-law, and exponential potentials. Within these choices, we furthermore single out particular limiting cases, such as the constant, linear and inverse potentials. For all the considered scenarios, regions of stability are calculated in terms of the signs of the coupling constant and the exponent, revealing a clear degeneracy among coefficients necessary to ensure stability. We find that a generic power-law potential with α>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha > 0$$\\end{document} is not suitable as a non-minimal quintessence potential and we put severe limits on the use of inverse potential, as well. In addition, the equations of state of each potential have been also computed. We find the constant potential seems to be favored than other treatments, since the critical point appears independent of the non-minimal coupling.

Cell motility in cancer, crucial events, criticality, and Lévy walks

The analysis of glioblastoma (GB) cell locomotion and its modeling inspired by Lévy random walks is presented herein. We study such walks occurring on a two-dimensional plane where the walk is similar to the motion of a bird flying with a constant velocity, but with random changes of direction in time. The intelligence of the bird is signaled by the instantaneous changes of flying direction, which become invisible in the time series obtained by projecting the 2D walk either on the x-axis or y-axis. We establish that the projected 1D time series share the statistical complexity of time series frequently used to monitor physiological processes, shedding light on the role of crucial events (CE-s) in pathophysiology. Such CE-s are signified by abrupt changes of flying direction which are invisible in the 1D physiological time series. We establish a connection between the complex scaling index δ generated by the CE-s through μR=2−δ, where μR is the inverse power law index of the probability density function of the time interval between consecutive failures of the process of interest. We argue that the identification of empirical indices along with their theoretical relations afford important measures to control cancer.

Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds.

We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter λ, and on the other hand our numerics show that it follows a power law for a broad range of values of λ. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.

Wear life prediction of piston ring and cylinder liner based on integration of experiment and bench test data

The wear data of piston ring-cylinder liner (PRCL) of high-power diesel engines faces the problems of small samples, randomness, and underutilization, which leads to the wear life prediction of PRCL has been a difficult task in the engineering field. A wear life prediction method of PRCL based on the integration of experimental data and bench test data is proposed. First, a large amount of wear data is rapidly obtained through accelerated wear tests on specimens, and the distribution type of wear depth of PRCL is determined. Secondly, based on Rhee's wear equation, the nonlinear relationship between time, load, temperature, and wear depth is considered, and the acceleration models of temperature and load were introduced as the Arrhenius model and the inverse power law model, respectively. The wear life prediction model (WLPM) for specimens and parts with stochasticity is developed. The eigenvalues of WLPM’s parameters of specimens and parts were utilized to construct the mapping relationship from the specimen to the part. Finally, the Bayesian statistical method is used to merge the wear data of specimens and the parts, and the WLPM of specimens and the mapping relationship are taken as the a priori information, the bench data are taken as the new sample information, and the MCMC-Gibbs sampling method is used to estimate the mapping-based WLPM of parts. The results show that through the validation test, the method predicts the wear depth of PRCL with 100 % falling into the 95 % confidence interval better than the WLPM prediction results of a single specimen or part, and the errors of predicting the mean wear life of PRCL are all within 10 %. The method can provide a reference for the quality evaluation and engineering application of PRCL.

On probing turbulence in core-collapse supernovae in upcoming neutrino detectors

Neutrino propagation through a turbulent medium can be highly non-adiabatic leading to distinct signatures in the survival probabilities. A core-collapse supernova can be host to a number of hydrodynamic instabilities which occur behind the shockfront. Such instabilities between the forward shock and a possible reverse shock can lead to cascades introducing turbulence in the associated matter profile, which can imprint itself in the neutrino signal. In this work, we consider realistic matter profiles and seed in the turbulence using a randomization scheme to study its effects on neutrino propagation in an effective two-flavor framework. We focus on the potential of upcoming neutrino detectors — DUNE and Hyper-Kamiokande to constrain the parameters characterizing turbulence in a supernova. We find that these experiments can effectively constrain the parameter space for the amplitude of the spectra, they will only have mild sensitivity to the spectral index, and cannot inform on deviations from the usual Kolmogorov 5/3 inverse power law. Furthermore, we also confirm that the double-dip feature, originally predicted in the neutrino spectra associated with forward and reverse shocks, can be completely washed away in the presence of turbulence, leading to total flavor depolarization.

Maintenance and Reliability Improvement of Roller Bearings Operating at High Temperature: Thermal Stress Analysis Approach

In this paper the authors present the outcome of an experimental maintenance and reliability investigation conducted to prevent operation failure of pinion bearings caused by excessive temperature with a case study of a 1360kW ball mill driveline system in mineral processing application. Aiming to improve the overall plant performance, the lubrication regime of pinion bearings as key component within the ball mill driveline system was studied to determine possible causes of temperature increase at 80 to over 100°C with a 25 - 30°C gap over the maximum operating temperature. An analytical review of maintenance data is conducted to mitigate future operational risks by implementing a new lubrication regime and shutdown frequency for overall statutory inspection. Accelerated life test based inverse power law has been applied to identify the root causes of excessive bearings temperature to establish immediate short-term solution paths towards a guaranteed less required maintenance asset improvement practice, less operational cost, and plant availability enhancement. Furthermore, long-term solutions have been proposed based on modelling by thermal network approach of pinion bearings for thermomechanical stress prevention. A single parameter based accelerated test on pinion bearings was applied to establish operational fault tolerant factors to be considered for asset capability improvement, which remains an open question for future studies. These actions have shown significant improvement with temperature decrease between 15 and 25°C below the maximum required temperature.

Surface roughness characterization using representative elementary area (REA) analysis

We proposed the Representative Elementary Area (REA) analysis method and illustrated how it is needed to evaluate representative roughness parameters of surfaces. We used mean height (Sa) roughness to study how its variations converge to a steady state as we expanded the area of investigation (AOI) using combined scan tiles obtained through Confocal Laser Scanning Microscopy. We tested quartz and glass surfaces, subjecting them to various levels of polishing with grit sizes ranging between # 60 and #1200. The scan tiles revealed a multiscale roughness texture characterized by the dominance of valleys over peaks, lacking a fractal nature. REA analysis revealed Sa variations converged to a steady state as AOI increased, highlighting the necessity of the proposed method. The steady-state Sa, denoted as SaREA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ ext{Sa}}}_{{\ ext{REA}}}$$\\end{document}, followed an inverse power law with polishing grit size, with its exponent dependent on the material hardness. The REA length representing SaREA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ ext{Sa}}}_{{\ ext{REA}}}$$\\end{document} of glass surfaces, followed another inverse power law with polishing grit size and an indeterminate relationship for quartz surfaces. The multiscale characteristics and convergence to steady state were also evident in skewness, kurtosis, and autocorrelation length (Sal) parameters. Sal increased to a maximum value before decreasing linearly as AOI was linearly increased. The maximum Sal, termed as Salmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ ext{Sal}}}_{{\ ext{max}}}$$\\end{document}, exhibited a linear relationship with REA. In the absence of REA analysis, the magnitude of uncertainty depended on the polishing grit size. Finely polished surfaces exhibited a 10–20% variability, which increased to up to 70% relative to the steady-state Sa with coarser polishing.