Higher-order Coefficients Research Articles

Higher-order QCD corrections to hadronic \u03c4 decays from Pad\xe9 approximants

Perturbative QCD corrections to hadronic τ decays and e+e− annihilation into hadrons below charm are obtained from the Adler function, which at present is known in the chiral limit to five-loop accuracy. Extractions of the strong coupling, αs, from these processes suffer from an ambiguity related to the treatment of unknown higher orders in the perturbative series. In this work, we exploit the method of Padé approximants and its convergence theorems to extract information about higher-order corrections to the Adler function in a systematic way. First, the method is tested in the large-β0 limit of QCD, where the perturbative series is known to all orders. We devise strategies to accelerate the convergence of the method employing renormalization scheme variations and the so-called D-log Padé approximants. The success of these strategies can be understood in terms of the analytic structure of the series in the Borel plane. We then apply the method to full QCD and obtain reliable model-independent predictions for the higher-order coefficients of the Adler function. For the six-, seven-, and eight-loop coefficients we find c5,1 = 277 ± 51, c6,1 = 3460±690, and c7,1 = (2.02±0.72)×104, respectively, with errors to be understood as lower and upper bounds. Our model-independent reconstruction of the perturbative QCD corrections to the τ hadronic width strongly favours the use of fixed-order perturbation theory (FOPT) for the renormalization-scale setting.

Expected precision of Europa Clipper gravity measurements

The primary gravity science objective of NASA’s Clipper mission to Europa is to confirm the presence or absence of a global subsurface ocean beneath Europa’s Icy crust. Gravity field measurements obtained with a radio science investigation can reveal much about Europa’s interior structure. Here, we conduct extensive simulations of the radio science measurements with the anticipated spacecraft trajectory and attitude (17F12v2) and assets on the spacecraft and the ground, including antenna orientations and beam patterns, transmitter characteristics, and receiver noise figures. In addition to two-way Doppler measurements, we also include radar altimeter crossover range measurements. We concentrate on ± 2 h intervals centered on the closest approach of each of the 46 flybys. Our covariance analyses reveal the precision with which the tidal Love number k2, second-degree gravity coefficients C¯20 and C¯22, and higher-order gravity coefficients can be determined. The results depend on the Deep Space Network (DSN) assets that are deployed to track the spacecraft. We find that some DSN allocations are sufficient to conclusively confirm the presence or absence of a global ocean. Given adequate crossover range performance, it is also possible to evaluate whether the ice shell is hydrostatic.

Higher-order clustering in networks.

A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucial to understanding complex networks, and the clustering behavior with respect to such higher-order network structures is not well understood. Here we introduce higher-order clustering coefficients that measure the closure probability of higher-order network cliques and provide a more comprehensive view of how the edges of complex networks cluster. Our higher-order clustering coefficients are a natural generalization of the traditional clustering coefficient. We derive several properties about higher-order clustering coefficients and analyze them under common random graph models. Finally, we use higher-order clustering coefficients to gain new insights into the structure of real-world networks from several domains.

A model of Saturn inferred from its measured gravitational field

We present an interior model of Saturn with an ice-rock core, a metallic region, an outer molecular envelope and a thin transition layer between the metallic and molecular regions. The shape of Saturn’s 1 bar surface is irregular and determined fully self-consistently by the required equilibrium condition. While the ice-rock core is assumed to have a uniform density, three different equations of state are adopted for the metallic, molecular and transition regions. The Saturnian model is constrained by its known mass, its known equatorial and polar radii, and its known zonal gravitational coefficients, J2n, n = 1, 2, 3. The model produces an ice-rock core with equatorial radius 0.203 RS, where RS is the equatorial radius of Saturn at the 1-bar pressure surface; the core density ρc = 10388.1 kgm−3 corresponding to 13.06 Earth masses; and an analytical expression describing the Saturnian irregular shape of the 1-bar pressure level. The model also predicts the values of the higher-order gravitational coefficients, J8, J10 and J12, for the hydrostatic Saturn and suggests that Saturn’s convective dynamo operates in the metallic region approximately defined by 0.2 RS < re < 0.7 RS, where re denotes the equatorial radial distance from the Saturnian center of figure.

Research on the calibration technology of an underwater camera based on equivalent focal length

Calibration is one of the most crucial and inevitable processes in vision technology, whose performance can significantly affect the measurement accuracy of the vision system. To compensate for coefficients of water factors, the conventional calibration technology for the underwater camera generally introduces several higher-order distortion coefficients, which, unfortunately, suffer poor stability, low efficiency, as well as computational complexity. To overcome these drawbacks, a newish calibration method based on equivalent focal length is proposed in this paper. The method takes the refraction distortion of water factors as equivalent focal length changing to cause. Applying the invariance principle of linear perspective projection, distortion correction should first be carried out. Subsequently, the internal and external parameters of the camera are calculated by using Zhang's calibration method based on the camera linear model, which simplifies the process of underwater camera calibration. Finally, the experimental results show that the calibration method can effectively improve the accuracy of the underwater camera.

Peripheral Monochromatic Aberrations in Young Adult Caucasian and East Asian Eyes.

Myopia prevalence rates differ between racial groups. If the growth of the eye is sensitive to differences in optical input, the difference in spherical aberration between East Asian and Caucasian eyes found in this study may be important in understanding myopia development. The aim of this study was to determine differences in peripheral wavefront aberrations between two racial groups. Wavefront aberrations were measured using a COAS-HD aberrometer across the 42 × 32° central visual field on 37 right eyes of young adults (18 Caucasians, 19 East Asians; mean age 21.5 ± 2.4 years). The mean spherical equivalent refraction was -1.94 ± 1.63 diopters (D) with a range of -5.87 to +0.16 D. Effect of race and visual field position on refractions, individual Zernike aberration coefficients up to the fourth order, higher-order root-mean-square aberration, and total root-mean-square aberration were assessed by repeated-measures analysis of covariance. Caucasians and East Asians had similar relative peripheral myopia across the visual field. All higher-order aberration coefficients were affected by visual field position. Race had no significant effect on any higher-order Zernike coefficient, but the difference in mean vertical coma coefficient (Equation is included in full-text article.)across the visual field (i.e., average of 38 field locations) approached significance, being less positive in Caucasians than in East Asians (P = .08). When correction was made for the Caucasian group being slightly less myopic than the East Asian group, spherical aberration coefficient (Equation is included in full-text article.)was less positive in Caucasians than in East Asians by 0.04 μm (P = .001). The rates of change of coma coefficients across the field were not affected by race. Caucasians and East Asians had similar relative peripheral myopia, but with less positive spherical aberration coefficient in Caucasians than in East Asians. It remains to be determined whether aberrations have a role in the difference of myopia prevalence rates in different countries.

Experimental characterization of nonlinear interference noise as a process of intersymbol interference.

We demonstrate a method for experimentally characterizing the second order statistics of nonlinear interference noise (NLIN) as an intersymbol interference (ISI) process. The method enables measurement of the properties of high-order ISI coefficients, which have been largely overlooked in the past. The ability of measuring these statistics is imperative for designing effective NLIN mitigation schemes. The variance, temporal correlation times, and cross correlations of the various ISI coefficients are evaluated in several system implementations.

Series expansion method for determination of order of 3-dimensional bar-joint mechanism with arbitrarily inclined hinges

A general method for evaluating the order of infinitesimal mechanism is presented for 3-dimensional bar-joint mechanism, which is defined as assembly of rigid bars connected to nodes by hinges with arbitrarily inclined directions. Compatibility conditions of translations and rotations are derived at bar-ends with respect to the generalized displacements consisting of displacements and rotations of nodes and bars. Degree of kinematic indeterminacy is computed using singular value decomposition of the linearized compatibility matrix. The order of infinitesimal mechanism is determined by the existence condition of higher-order coefficients of the generalized displacements with respect to the path parameter of deformation. The coefficients are obtained in a similar manner as stability analysis of geometrically nonlinear structures, where symbolic computation software package is used for analytical derivation of equations. The detailed procedure of analysis is shown through the numerical examples of a two-bar and a four-bar linkages, and the results are verified using a finite element analysis software.

On the bispectra of very massive tracers in the Effective Field Theory of Large-Scale Structure

The Effective Field Theory of Large-Scale Structure (EFTofLSS) provides a consistent perturbative framework for describing the statistical distribution of cosmological large-scale structure. In a previous EFTofLSS calculation that involved the one-loop power spectra and tree-level bispectra, it was shown that the k-reach of the prediction for biased tracers is comparable for all investigated masses if suitable higher-derivative biases, which are less suppressed for more massive tracers, are added. However, it is possible that the non-linear biases grow faster with tracer mass than the linear bias, implying that loop contributions could be the leading correction to the bispectra. To check this, we include the one-loop contributions in a fit to numerical data in the limit of strongly enhanced higher-order biases. We show that the resulting one-loop power spectra and higher-derivative plus leading one-loop bispectra fit the two- and three-point functions respectively up to k≃ 0.19 h Mpc−1 and k≃ 0.14 h Mpc−1 at the percent level. We find that the higher-order bias coefficients are not strongly enhanced, and we argue that the gain in perturbative reach due to the leading one-loop contributions to the bispectra is relatively small. Thus, we conclude that higher-derivative biases provide the leading correction to the bispectra for tracers of a very wide range of masses.

Uniqueness for &lt;i&gt;L&lt;sub&gt;p&lt;/sub&gt;&lt;/i&gt;-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms

In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be Hölder continuous in $x$ with exponent slightly less than $1/2$. This case is treated by using stability of maximal and minimal $L_{p}$-viscosity solutions.

Hadron thermodynamics from imaginary chemical potentials

We use 4stout improved staggered lattice data at imaginary chemical potentials to calculate fugacity expansion coefficients in finite temperature QCD. We discuss the phenomenological interpretation of our results within the hadron resonance gas (HRG) model, and the hints they give us about the hadron spectrum. We also discuss features of the higher order coefficients that are not captured by the HRG. This conference contribution is based on our recent papers [1, 2].,

Volterra approach – a new method to accurately calculate the non-linear and time-varying roll restoring arm of ships in irregular longitudinal seas

ABSTRACTWith several incidents of large amplitude roll motion of ships observed at sea over the past two decades, the dynamic stability of modern ship types has assumed significant importance. The move by the International Maritime Organization to develop a second generation level-2 criteria has resulted in considerable attention to the development of simplified analytical models for roll motion. Especially for the problem of parametric roll, there is still no analytical model which accurately captures the non-linear roll restoring stiffness taking into account the combined effect of incident waves and dynamic heave and pitch motions of the vessel. This paper extends our previous work to accurately capture both the linear and higher-order time-varying roll stiffness coefficients through an analytical model while taking into account incident waves and dynamic motion of the vessel. This approach called the Volterra method is compared against the existing simplified models and a non-linear time-domain simulation program to demonstrate better agreement of the proposed approach.

Homotopy-Analysis-Method-Based Solutions of Quintic Nonlinear Angular Motion for Projectiles

The current research literature has shown that the nonlinear aerodynamic coefficients have a significant impact on the angular motion of projectiles. However, high-order aerodynamic coefficients, such as quintic terms, are seldom considered due to their complexity and the lack of an appropriate analytical workhorse. The homotopy analysis method is becoming a popular analytical tool for nonlinear problems because it can address strong nonlinearities without relying on any small or large physical parameters. An efficient approach to determine the convergence-control parameter is proposed and incorporated into the basic algorithm of the homotopy analysis method. This new algorithm is applied to two problems from the ballistics community literature to which quintic static moments are deliberately added and analytical solutions in general forms are obtained. By presenting some numerical examples, it is demonstrated that this algorithm is highly efficient and accurate over a wide region of convergence. As a result, the effects of the nonlinear aerodynamic parameters and initial angle of attack on the property of angular motion are well described mathematically using these formulas. The results of this research provide a reference for studying more complicated problems related to nonlinear projectile motion.

Determination of the diffusivity, dispersion, skewness and kurtosis in heterogeneous porous flow. Part I: Analytical solutions with the extended method of moments.

Abstract The extended method of moments (EMM) is elaborated in recursive algorithmic form for the prediction of the effective diffusivity, the Taylor dispersion dyadic and the associated longitudinal high-order coefficients in mean-concentration profiles and residence-time distributions. The method applies in any streamwise-periodic stationary d-dimensional velocity field resolved in the piecewise continuous heterogeneous porosity field. It is demonstrated that EMM reduces to the method of moments and the volume-averaging formulation in microscopic velocity field and homogeneous soil, respectively. The EMM simultaneously constructs two systems of moments, the spatial and the temporal, without resorting to solving of the high-order upscaled PDE. At the same time, the EMM is supported with the reconstruction of distribution from its moments, allowing to visualize the deviation from the classical ADE solution. The EMM can be handled by any linear advection-diffusion solver with explicit mass-source and diffusive-flux jump condition on the solid boundary and permeable interface. The prediction of the first four moments is decisive in the optimization of the dispersion, asymmetry, peakedness and heavy-tails of the solute distributions, through an adequate design of the composite materials, wetlands, chemical devices or oil recovery. The symbolic solutions for dispersion, skewness and kurtosis are constructed in basic configurations: diffusion process and Darcy flow through two porous blocks in “series”, straight and radial Poiseuille flow, porous flow governed by the Stokes–Brinkman–Darcy channel equation and a fracture surrounded by penetrable diffusive matrix or embedded in porous flow. We examine the moments dependency upon porosity contrast, aspect ratio, Peclet and Darcy numbers, but also for their response on the effective Brinkman viscosity applied in flow modeling. Two numerical Lattice Boltzmann algorithms, a direct solver of the microscopic ADE in heterogeneous structure and a novel scheme for EMM numerical formulation, are called for validation of the constructed analytical predictions.

Closed-form tidal approximants for binary neutron star gravitational waveforms constructed from high-resolution numerical relativity simulations

We construct closed-form gravitational waveforms (GWs) with tidal effects for the coalescence and merger of binary neutron stars. The method relies on a new set of eccentricity-reduced and high-resolution numerical relativity (NR) simulations and is composed of three steps. First, tidal contributions to the GW phase are extracted from the time-domain NR data. Second, those contributions are employed to fix high-order coefficients in an effective and resummed post-Newtonian expression. Third, frequency-domain tidal approximants are built using the stationary phase approximation. Our tidal approximants are valid from the low frequencies to the strong-field regime and up to merger. They can be analytically added to any binary black hole GW model to obtain a binary neutron star waveform, either in the time or in the frequency domain. This work provides simple, flexible, and accurate models ready to be used in both searches and parameter estimation of binary neutron star events.

Combined effects of frequency and higher-order effects on soliton conversion in an erbium fiber with inhomogeneous broadening

In this paper, soliton propagation in an erbium-doped fiber with inhomogeneous broadening is considered which can be described by the inhomogeneous Hirota– Maxwell–Bloch equations. The Darboux transformation method is employed to generate the soliton solution through a linear eigenvalue problem. In particular, our results demonstrate explicitly that a soliton can be converted into various nonlinear waves such as periodic wave, anti-kink soliton and flat-top soliton in the presence of higher-order effects with inhomogeneous broadening. Additionally, we found that higher-order coefficients have strong influence on the soliton transition while frequency is only responsible for phase shift. The results might be of certain value for the study of the soliton management and soliton conversion.

Computing aggregate properties of preimages for 2D cellular automata.

Computing properties of the set of precursors of a given configuration is a common problem underlying many important questions about cellular automata. Unfortunately, such computations quickly become intractable in dimension greater than one. This paper presents an algorithm-incremental aggregation-that can compute aggregate properties of the set of precursors exponentially faster than naïve approaches. The incremental aggregation algorithm is demonstrated on two problems from the two-dimensional binary Game of Life cellular automaton: precursor count distributions and higher-order mean field theory coefficients. In both cases, incremental aggregation allows us to obtain new results that were previously beyond reach.

Physical uniqueness of higher-order Korteweg-de Vries theory for continuously stratified fluids without background shear

The 2nd-order Korteweg-de Vries (KdV) equation and the Gardner (or extended KdV) equation are often used to investigate internal solitary waves, commonly observed in oceans and lakes. However, application of these KdV-type equations for continuously stratified fluids to geophysical problems is hindered by nonuniqueness of the higher-order coefficients and the associated correction functions to the wave fields. This study proposes to reduce arbitrariness of the higher-order KdV theory by considering its uniqueness in the following three physical senses: (i) consistency of the nonlinear higher-order coefficients and correction functions with the corresponding phase speeds, (ii) wavenumber-independence of the vertically integrated available potential energy, and (iii) its positive definiteness. The spectral (or generalized Fourier) approach based on vertical modes in the isopycnal coordinate is shown to enable an alternative derivation of the 2nd-order KdV equation, without encountering nonuniqueness. Comparison with previous theories shows that Parseval’s theorem naturally yields a unique set of special conditions for (ii) and (iii). Hydrostatic fully nonlinear solutions, derived by combining the spectral approach and simple-wave analysis, reveal that both proposed and previous 2nd-order theories satisfy (i), provided that consistent definitions are used for the wave amplitude and the nonlinear correction. This condition reduces the arbitrariness when higher-order KdV-type theories are compared with observations or numerical simulations. The coefficients and correction functions that satisfy (i)-(iii) are given by explicit formulae to 2nd order and by algebraic recurrence relationships to arbitrary order for hydrostatic fully nonlinear and linear fully nonhydrostatic effects.

Parameter identification of pedestrian's spring-mass-damper model by ground reaction force records through a particle filter approach

The spring-mass-damper (SMD) model with a pair of internal biomechanical forces is the simplest model for a walking pedestrian to represent his/her mechanical properties, and thus can be used in human-structure-interaction analysis in the vertical direction. However, the values of SMD stiffness and damping, though very important, are typically taken as those measured from stationary people due to lack of a parameter identification methods for a walking pedestrian. This study adopts a step-by-step system identification approach known as particle filter to simultaneously identify the stiffness, damping coefficient, and coefficients of the SMD model's biomechanical forces by ground reaction force (GRF) records.After a brief introduction of the SMD model, the proposed identification approach is explained in detail, with a focus on the theory of particle filter and its integration with the SMD model. A numerical example is first provided to verify the feasibility of the proposed approach which is then applied to several experimental GRF records. Identification results demonstrate that natural frequency and the damping ratio of a walking pedestrian are not constant but have a dependence of mean value and distribution on pacing frequency. The mean value first-order coefficient of the biomechanical force, which is expressed by the Fourier series function, also has a linear relationship with pacing frequency. Higher order coefficients do not show a clear relationship with pacing frequency but follow a logarithmic normal distribution.

Mixed higher-order flow harmonics and nonlinear response coefficients in PbPb collisions at 2.76 and 5.02 TeV with CMS

The mixed higher-order flow harmonics and nonlinear response coefficients of charged particles are presented as a function of pT and centrality in PbPb collisions at sNN=2.76 TeV and 5.02 TeV with the CMS detector. The results are obtained using the scalar product method, and cover a pT range from 0.3 GeV/c to 8.0 GeV/c, pseudorapidity |η|<2.4, and a centrality range of 0-60%. The mixed harmonic results at 5.02 TeV are compared to the matching higher-order flow harmonics from two-particle correlations, which measure vn values with respect to the n-th order event plane. It is observed that the nonlinear response coefficients of the odd harmonics are larger than the even harmonics ones. The results are compared with hydrodynamic predictions using different shear viscosity to entropy density ratios and different initial conditions.