Higher-order Parameters Research Articles

Computing Hypergeometric Functions Rigorously

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0 F 1 , 1 F 1 , 2 F 1 , and 2 F 0 (or the Kummer U -function) are supported for unrestricted complex parameters and argument, and, by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function p F q and computation of high-order parameter derivatives.

Design of Very High-Resolution Satellite Telescopes Part I: Optical System Design

The telescope design of a very high-resolution earth observation satellite is a challenging task, due to several constraints, including the size, mass, performance, tolerance, assembly, integrating, and environmental aspects. However, the demand for very high-resolution satellite imageries has increased day by day due to the numerous commercial and military applications. In this paper, two different optical configurations with ground sampling distance of 25 cm are investigated according to a definite mission statement. The first configuration represents an on-axis Korsch scheme, whereas the second design is based on the arrangement of the off-axis three-mirror anastigmatic configuration. Both designs have the advantage of small conic mirror constants without higher order parameters. Tolerance analysis of both designs utilizing a Monte Carlo algorithm is performed. The suggested baffling arrangements are simulated and tested according to a series of point sources with different off-axis angles. Finally, a comparison between the performance of the proposed designs and a recent study given by the French space agency is presented. The results show that the proposed designs can be implemented by the existing technologies and launched into space by the available launch vehicles.

Analysis of nuclear structure in a converging power expansion scheme

In the framework of the KIDS generalized energy density functional (EDF), the nuclear equation of state (EoS) is expressed as an expansion in powers of the Fermi momentum or the cubic root of the density ($\rho^{1/3}$). Although an optimal number of converging terms was obtained in specific cases of fits to empirical data and pseudodata, the degree of convergence remains to be examined not only for homogeneous matter but also for finite nuclei. One goal of the present work is to validate the minimal and optimal number of EoS parameters required for the description of homogeneous nuclear matter over a wide range of densities relevant for astrophysical applications. The major goal is to examine the validity of the adopted expansion scheme for an accurate description of finite nuclei. To this end we vary the values of the high-order derivatives of the EoS, namely the skewness of the energy of symmetric nuclear matter and the kurtosis of the symmetry energy, at saturation and examine the relative importance of each term in $\rho^{1/3}$ expansion for homogeneous matter. For given sets of EoS parameters determined in this way, we define equivalent Skyrme-type functionals and examine the convergence in the description of finite nuclei focusing on the masses and charge radii of closed-shell nuclei. The EoS of symmetric nuclear matter is found to be efficiently parameterized with only 3 parameters and the symmetry energy (or the energy of pure neutron matter) with 4 parameters when the EoS is expanded in the power series of the Fermi momentum. Higher-order EoS parameters do not produce any improvement, in practice, in the description of nuclear ground-state energies and charge radii, which means that they cannot be constrained by bulk properties of nuclei.

Higher-Order Assessment of OCT in Diabetic Macular Edema from the VISTA Study: Ellipsoid Zone Dynamics and the Retinal Fluid Index

To investigate retinal fluid features and ellipsoid zone (EZ) integrity dynamics on spectral-domain OCT (SD-OCT) in eyes with diabetic macular edema (DME) treated with intravitreal aflibercept injection (IAI) in the VISTA-DME study. A post hoc subanalysis of a phase III, prospective clinical trial. Eyes received either IAI 2 mg every 4 weeks (2q4) or every 8 weeks after 5 initial monthly doses (2q8). All eyes from the VISTA Phase III study in the IAI groups imaged with the Cirrus HD-OCT system (Zeiss, Oberkochen, Germany) were included. The OCT macular cube datasets were evaluated using a novel software platform to generate retinal layer and fluid boundary lines that were manually corrected for assessment of change in EZ parameters and volumetric fluid parameters from baseline. The retinal fluid index (i.e., proportion of the retinal volume consisting of cystic fluid) was also calculated at each time point. The feasibility of volumetric assessment of higher-order OCT-based retinal parameters and its correlation with best-corrected visual acuity (BCVA). Overall, 106 eyes of 106 patients were included. Specifically, 52 eyes of 52 patients were included in the IAI 2q4 arm, and 54 eyes of 54 patients were included in the IAI 2q8 arm. Ellipsoid zone integrity metrics significantly improved from baseline to week 100, including central macular mean EZ to retinal pigment epithelium (RPE) thickness (2q4: 26.6 μm to 31.6 μm, P < 0.001; 2q8: 25.2 μm to 31.4 μm, P < 0.001). At week 100, central macular intraretinal fluid volume was reduced by >65% (P < 0.001) and central macular subretinal fluid volume was reduced by >99% in both arms (P < 0.001). Central macular retinal fluid index (RFI) significantly improved in both arms (2q4: 17.9% to 7.2%, P < 0.001; 2q8: 19.8% to 4.2%, P < 0.001). Central macular mean EZ-RPE thickness (i.e., a surrogate for photoreceptor outer segment length) and central RFI were independently correlated with BCVA at multiple follow-up visits. Intravitreal aflibercept injection resulted in significant improvement in EZ integrity and quantitative fluid metrics in both 2q4 and 2q8 arms and correlated with visual function.

Gradient theory for crack problems in quasicrystals

Constitutive equations in gradient theory of quasicrystals are written for phonon and phason stresses, and the higher-order stress tensor. They are expressed by the phonon and phason strains and the gradient of phonon strains. The higher-order elastic parameters are proportional to the conventional elastic stiffness coefficients by the internal length material parameter. Material parameters in constitutive equations correspond to the decagonal quasicrystals. The principle of virtual work is applied to derive governing equations and corresponding boundary conditions. The finite element method (FEM) is developed to solve general 2D boundary value problems in problems described by governing equations for strain-gradient theory of quasicrystals. The path-independent J-integral is also derived for fracture mechanics analysis of such solids. Numerical examples are presented to demonstrate the veracity of the formulations.

Visitor safety in recreational protected areas: Exploring responsibility-sharing from a management perspective

The management of visitor safety in protected areas is characterised by complex responsibility-sharing relationships. Utilising the Delphi technique as a qualitative research tool, this paper explores responsibility from the perspective of protected area managers within an Australian context. Through an iterative series of email-based exchanges, managers described their responsibilities illustrating a relationship between visitors (representing the party at risk) and themselves (the managing agency representing the party in authority). A three-dimensional framework was developed through a consensus-driven process, defining the context within which risk management decisions are made. The three dimensions comprise geographical, service and marketing. The results of this study offer conceptual guidance for managing agencies to systematically establish higher-order responsibility parameters in order to plan visitor risk management interventions. The usefulness of the results to guide future research is discussed.

An enactive approach to appropriation in the instrumented activity of trail running.

The incorporation of external tools during a sports activity can be analyzed through the dynamics of appropriation. In this study, we assumed that appropriation could be documented at both the phenomenological and behavioral scales and aimed to characterize trail runners' interactions with five carrying systems (i.e., backpacks proposing different ways of carrying water) in an ecological setting. The runners ran a 3-km trail running loop, equipped with inertial sensors to quantify both their vertical oscillations and those of the carrying systems. After the trials, phenomenological data were collected in enactive interviews. Results showed that (1) the runners encountered issues related to the carrying system, whose emergence in their experiences while running revealed the interplay between the tool's transparency (i.e., when runners provided no account of the carrying system) and opacity (i.e., when runners mentioned perceptions of disturbing system elements), and (2) when the runners carried the water bottles on the pectoral straps, they felt the system bouncing in an uncomfortable way, especially in the less technical parts of the route. We therefore investigated the low- and high-order parameters of coordination by computing the vertical accelerations and the acceleration couplings between the carrying system and the runners in order to identify coordination modes. The congruence between the runners' experiences and the behavioral data was noted in terms of (1) the system's vertical oscillations (i.e., low-order parameters) and (2) the couplings between the accelerations of the runners and the backpacks (i.e., high-order parameters). Our results demonstrated that the appropriation process was shaped by the interactions between the runners' activity, the environment and the physical properties of the tool. These interactions occurred in fluctuating phases where the runners perceived the carrying systems as more or less incorporated. Our results highlighted how tool incorporation is revealed through changes in its transparency/opacity in the actor's activity.

Quantifying the uncertainties on spinodal instability for stellar matter through meta-modeling

The influence of the uncertainties of the equation of state empirical parameters on the neutron stars crust-core phase transition is explored within a meta-modeling approach, in which the energy per particle is expanded as a Taylor series in density and asymmetry around the saturation point. The phase transition point is estimated from the intersection of the spinodal instability region for dynamical fluctuations with the chemical equilibrium curve. Special attention is paid to the inclusion of high-order parameters of the Taylor series and their influence on the transition point. An uncorrelated prior distribution is considered for the empirical parameters, with bulk properties constrained through effective field theory predictions, while the surface parameters are controlled from a fit of nuclear masses using the extended Thomas Fermi approximation. The results show that the isospin dependent curvature Ksym and skewness Qsym have the most significant correlations with the transition point, along with the well known influence of the Lsym parameter. The estimated density and pressure of the crust-core transition are nt = (0.071 ± 0.011) fm−3 and Pt = (0.294 ±0.102) MeV fm−3.

Scalable Synthesis of Switchable Assemblies of Gold Nanorod Lyotropic Liquid Crystal Nanocomposites.

A new class of solvent free, lyotropic liquid crystal nanocomposites based on gold nanorods (AuNRs) with high nanorod content is reported. Application of shear results in switchable, highly ordered alignment of the nanorods over several centimeters with excellent storage stability for months. For the synthesis, AuNRs are surface functionalized with a charged, covalently tethered corona, which induces fluid-like properties. This honey-like material can be deposited on a substrate and a high orientational order parameter of 0.72 is achieved using a simple shearing protocol. Switching shearing direction results in realignment of the AuNRs. For a film containing 75 wt% of AuNRs the alignment persists for several months. In addition to the lyotropic liquid crystal characteristics, the AuNRs films also exhibit anisotropic electrical conductivity with an order of magnitude difference between the conductivities in direction parallel and perpendicular to the alignment of the AuNRs.

Characterization of Mismatch Constraint Effect for the Central Crack Plate under Biaxial Loading

The effects of biaxial loading and creep mismatch on creep fracture parameter C(t) and constraint effect are investigated for the plane strain central crack plate. The studied biaxialities in this paper are limited to the range of [-1,1]. Based on the three-term asymptotic solution for creep crack tip field, the higher-order terms coefficient parameter A2(t) is introduced to characterize the creep mismatch constraint effect. The impacts of loading level on the parameter A2(t) for different biaxialities and creep mismatch factors are also examined in this paper. The result shows that the steady creep fracture parameter C* decreases with the increments of mismatch factor and biaxiality B. The creep constraint parameter A2(t) is insensitive to loading level for the creep mismatch crack plate under biaxial loading. The steady creep constraint parameter increases with the increments of biaxialities and decreases as m increases, which indicates the higher under-match plate with higher biaxiality has higher creep constraint effect.

Effect of high-order empirical parameters on the nuclear equation of state

A quantitative knowledge of the nuclear Equation of State (EoS) requires an accurate estimation of the uncertainties on the EoS parameters and their mutual correlations. Such correlations are empirically observed in a large set of EoS models by different authors, but they are not always fully understood. We show that some of these correlations can be interpreted from basic physical constraints imposed on a simple Taylor expansion of the binding energy around saturation density. In particular, we investigate the correlations among the following empirical parameters: the symmetry energy $E_{sym}$, the slope and curvature of the symmetry energy $L_{sym}$ and $K_{sym}$, and the curvature and skewness of the binding energy in symmetric matter $K_{sat}$ and $Q_{sat}$. The uncertainties on these correlations is estimated through analytical modelling as well as a meta-modelling analysis of the EoS subject to physical constraints. We show that a huge dispersion of the correlations among low order empirical parameters is induced by the unknown higher order empirical parameters, such as $K_{sym}$ (second order) and $Q_{sat}$ and $Q_{sym}$ (third order). We also propose an explanation of the reason why $Q_{sat}$ is weakly constrained by present experimental data. We conclude that selective observables on high order parameters, such as $K_{sym}$ and $Q_{sat}$, should be better determined before the present uncertainties on the EoS can be further reduced.

Prediction of high-order line-shape parameters for air-broadened O2 lines using requantized classical molecular dynamics simulations and comparison with measurements

Line-shape models such as the Hartmann-Tran (HT) profile have adjustable high-order parameters that are usually determined by fits to experimental spectra. As an alternative approach, we demonstrate that fitting the HT profile to theoretical spectra provides high-order line-shape parameters for O2 transitions that are consistent with experimentally determined values. To this end, normalized absorption spectra of air-broadened O2 lines were computed without adjustable parameters using requantized classical molecular dynamics simulations (rCMDS). These theoretical calculations were made at a pressure of 203 kPa and for values of the Doppler width that cover near-Doppler-limited to collisional-broadened pressure conditions. Hartmann-Tran (HT) line profiles with adjustable line-shape parameters were then simultaneously fit to the set of rCMDS-calculated spectra in a global multispectrum analysis. The retrieved high-order line-shape parameters (i.e. the speed dependence of the line broadening and the Dicke narrowing coefficient) were subsequently used as fixed HT parameters in the analysis of seven air-broadened O2 lines of the a1Δg←X3Σg−(0,0) band. The spectra were measured over a fifteen-fold range of total gas pressure at high spectral resolution and signal-to-noise ratio with a frequency-stabilized cavity ring-down spectroscopy system. We show that these predicted parameters enable all the measured lines to be fit to within 1%, which is much better than best fits of the Voigt line profile to the measured spectra. This approach opens the route for predicting high-order line-shape parameters from first-principles calculations and for their inclusion in spectroscopic databases. Furthermore, the temperature dependences of the broadening coefficient and its speed dependent component for air-broadened O2 lines were also calculated using rCMDS.

Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model

A higher-order nonlocal strain-gradient model is presented for the damped vibration analysis of single-layer graphene sheets (SLGSs) in hygrothermal environment. Based on Kirchhoff plate theory in conjunction with a higher-order (bi-Helmholtz) nonlocal strain gradient theory, the equations of motion are obtained using Hamilton's principle. The higher-order nonlocal strain gradient theory has lower- and higher-order nonlocal parameters and a material characteristic parameter. The presented model can reasonably interpret the softening effects of the SLGS, and indicates a reasonably good match with the experimental flexural frequencies. Finally, the roles of viscous and structural damping coefficients, small-scale parameters, hygrothermal environment and elastic foundation on the vibrational responses of SLGSs are studied in detail.

Investigation of Cross-Phase Modulation-Induced Crosstalk with Sub-Planck Higher-Order Dispersion Parameters in Optical Transmission Systems

Abstract Cross-phase modulation (XPM) is one of the major limiting effects in wavelength division multiplexed (WDM) optical transmission systems. This paper mathematically investigates the impact of XPM-induced crosstalk with second (2OD), third (3OD) and combination of sub-planck higher-order dispersion (HOD) parameters for WDM systems at different combinations of effective core areas, optical powers, modulating frequencies and dispersion values. It has been observed that XPM-induced crosstalk increases with increase in optical powers, modulating frequencies and dispersion values, but it reduces for higher effective core areas. The results for 2OD, 3OD and combination of dispersion parameters up to eighth order (8OD) have been reported. For instance, with a transmission length of 50 km for different values of optical power ranging from 0.1 mW to 2 mW, crosstalk due to XPM was observed to vary from (−75 to −44 dB), (−172 to −144 dB) and (−275 to −250 dB) in the presence of 2OD, 3OD and ∑ n = 4 8 n O D \mathop \sum \limits_{n = 4}^8 n{\rm{OD}} combination, respectively. On the basis of analysis, one can select the appropriate frequency as well as the value of dispersion or type of fibre to achieve minimum crosstalk.

Reynolds and dispersive shear stress contributions above highly skewed roughness

The roughness functions induced by irregular peak- and/or pit-dominated surfaces in a fully developed turbulent channel flow are studied by direct numerical simulation. A surface generation algorithm is used to synthesise an irregular Gaussian height map with periodic boundaries. The Gaussian height map is decomposed into ‘pits-only’ and ‘peaks-only’ components, which produces two additional surfaces with similar statistical properties, with the exception of skewness, which are equal and opposite $({\mathcal{S}}=\pm 1.6)$. While the peaks-only surface yields a roughness function comparable to that of the Gaussian surface, the pits-only surface exhibits a far weaker roughness effect. Analysis of results is aided by deriving an equation for the roughness function that quantitatively identifies the mechanisms of momentum loss and/or gain. The statistical contributions of ‘form-induced’ and stochastic fluid motions to the roughness function are examined in further detail using quadrant analyses. Above the Gaussian and peaks-only surfaces, the contributions of dispersive and Reynolds shear stresses show a compensating effect, whereas above the pits-only surface, an additive effect is observed. Overall, the results emphasise the sensitivity of the near-wall flow with respect to higher-order topographical parameters, which can, in turn, induce significant differences in the roughness function above a peak- and/or pit-dominated surface.

Mid-infrared supercontinuum generation using As2Se3 photonic crystal fiber and the impact of higher-order dispersion parameters on its supercontinuum bandwidth

A dispersion engineered As2Se3chalcogenide hexagonal photonic crystal fiber which can produce a mid-infrared supercontinuum (SC) spectral evolution spanning from 2 μm to beyond 15 μm with a low peak power of 3 kW is numerically designed and demonstrated. Numerical analysis is carried out to investigate the impact of higher-order dispersion (HOD) parameters on the output SC bandwidth and shows that the SC spectral broadening at the output of the proposed design depends on the convergence of the Taylor approximation with increasing fitting parameters, which implies a sufficient number of HOD parameters must be included during numerical simulations. Four designs with different structural parameters are optimized for pumping, each operating at a different pump wavelength to test the convergence of output SC by the successive addition of HOD parameters. To realize spurious free SC spectral evolution by the proposed designs, HOD terms up to the sixteenth-order are included during all SC simulations. The proposed design can be used in molecular finger print spectroscopy, bio-medical imaging as well as various mid-infrared region applications.

A simulation study of water property changes using geometrical alteration in SPC/E**Project supported by the National Natural Science Foundation of China (Grant Nos. 11635003, 11025524, and 11161130520), the National Basic Research Program of China (Grant No. 2010CB832903), and the European Commission’s 7th Framework Programme (Fp7-PEOPLE-2010-IRSES) (Grant Agreement Project No. 269131).

We present a systematic investigation of the impact of changing the geometry structure of the SPC/E water model by performing a series of molecular dynamic simulations at 1 bar (1 bar = 105 Pa) and 298.15 K. The geometric modification includes altering the H–O–H angle range from 90° to 115° and modifying the O–H length range from 0.90 Å to 1.10 Å in the SPC/E model. The former is achieved by keeping the dipole moment constant by modifying the O–H length, while in the latter only the O–H length is changed. With the larger bond length and angle, we find that the liquid shows a strong quadrupole interaction and high tetrahedral structure order parameter, resulting in the enhancement of the network structure of the liquid. When the bond length or angle is reduced, the hydrogen bond lifetime and self-diffusion constant decrease due to the weakening of the intermolecular interaction. We find that modifying the water molecular bond length leading to the variation of the intermolecular interaction strength is more intensive than changing the bond angle. Through calculating the average reduced density gradient and thermal fluctuation index, it is found that the scope of vdW interaction with neighbouring water molecules is inversely proportional to the change of the bond length and angle. The effect is mainly due to a significant change of the hydrogen bond network. To study the effect of water models as a solvent whose geometry has been modified, the solutions of ions in different solvent environments are examined by introducing NaCl. During the dissolving process, NaCl ions are ideally dissolved in SPC/E water and bond with natural water more easily than with other solvent models.

Forty-four New and Known M-dwarf Multiples in the SDSS-III/APOGEE M-dwarf Ancillary Science Sample

Abstract Binary stars make up a significant portion of all stellar systems. Consequently, an understanding of the bulk properties of binary stars is necessary for a full picture of star formation. Binary surveys indicate that both multiplicity fraction and typical orbital separation increase as functions of primary mass. Correlations with higher-order architectural parameters such as mass ratio are less well constrained. We seek to identify and characterize double-lined spectroscopic binaries (SB2s) among the 1350 M-dwarf ancillary science targets with APOGEE spectra in the SDSS-III Data Release 13. We measure the degree of asymmetry in the APOGEE pipeline cross-correlation functions (CCFs) and use those metrics to identify a sample of 44 high-likelihood candidate SB2s. At least 11 of these SB2s are known, having been previously identified by Deshpande et al. and/or El-Badry et al. We are able to extract radial velocities (RVs) for the components of 36 of these systems from their CCFs. With these RVs, we measure mass ratios for 29 SB2s and five SB3s. We use Bayesian techniques to fit maximum-likelihood (but still preliminary) orbits for four SB2s with eight or more distinct APOGEE observations. The observed (but incomplete) mass-ratio distribution of this sample rises quickly toward unity. Two-sided Kolmogorov–Smirnov tests find probabilities of 18.3% and 18.7%, demonstrating that the mass-ratio distribution of our sample is consistent with those measured by Pourbaix et al. and Fernandez et al., respectively.

Focusing High-Resolution Highly-Squinted Airborne SAR Data with Maneuvers

Maneuvers provide flexibility for high-resolution highly-squinted (HRHS) airborne synthetic aperture radar (SAR) imaging and also mean complex signal properties in the echoes. In this paper, considering the curved path described by the fifth-order motion parameter model, effects of the third- and higher-order motion parameters on imaging are analyzed. The results indicate that the spatial variations distributed in range, azimuth, and height directions, have great impacts on imaging qualities, and they should be eliminated when designing the focusing approach. In order to deal with this problem, the spatial variations are decomposed into three main parts: range, azimuth, and cross-coupling terms. The cross-coupling variations are corrected by polynomial phase filter, whereas the range and azimuth terms are removed via Stolt mapping. Different from the traditional focusing methods, the cross-coupling variations can be removed greatly by the proposed approach. Implementation considerations are also included. Simulation results prove the effectiveness of the proposed approach.

Polarized neutron scattering study of the multiple order parameter system NdB4

Neutron polarization analysis has been carried out in order to clarify the magnetic structures of multiple order parameter $f$-electron system ${\mathrm{NdB}}_{4}$. We confirmed the noncollinear ``all-in all-out'' structure (${\mathrm{\ensuremath{\Gamma}}}_{4}$) of the in-plane moment, which is in good agreement with our previous neutron powder diffraction study. We found that the magnetic moment along the $c$-axis ${m}_{c}$ showed diagonally antiferromagnetic structure (${\mathrm{\ensuremath{\Gamma}}}_{10}$), inconsistent with previously reported ``vortex'' structure (${\mathrm{\ensuremath{\Gamma}}}_{2}$). The microscopic mixture of these two structures with ${\stackrel{P\vec}{q}}_{0}=(0,0,0)$ appears in phase II and remains stable in phases III and IV, where an incommensurate modulation coexists. The unusual magnetic ordering is phenomenologically understood via Landau theory with the primary order parameter ${\mathrm{\ensuremath{\Gamma}}}_{4}$ coupled with higher-order secondary order parameter ${\mathrm{\ensuremath{\Gamma}}}_{10}$. The magnetic moments were estimated to be $1.8\ifmmode\pm\else\textpm\fi{}0.2$ and $0.2\ifmmode\pm\else\textpm\fi{}0.05{\ensuremath{\mu}}_{\mathrm{B}}$ at $T=7.5\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ for ${\mathrm{\ensuremath{\Gamma}}}_{4}$ and ${\mathrm{\ensuremath{\Gamma}}}_{10}$, respectively. We also found a long-period incommensurate modulation of the ${\stackrel{P\vec}{q}}_{1}=(0,0,1/2$) antiferromagnetic structure of ${m}_{c}$ with the propagation ${\stackrel{P\vec}{q}}_{s1}\phantom{\rule{0.16em}{0ex}}=(0.14,0.14,0.1)$ and ${\stackrel{P\vec}{q}}_{s2}=(0.2,0,0.1)$ in phase III and IV, respectively. The amplitude of sinusoidal modulation was about ${m}_{c}=1.0\ifmmode\pm\else\textpm\fi{}0.2{\ensuremath{\mu}}_{\mathrm{B}}$ at $T=1.5$ K. The local ($0,0,1/2$) structure consists of in-plane ferromagnetic and out-of-plane antiferromagnetic coupling of ${m}_{c}$, opposite to the coexisting ${\mathrm{\ensuremath{\Gamma}}}_{10}$. The ${m}_{c}$ of ${\mathrm{\ensuremath{\Gamma}}}_{10}$ is significantly enhanced up to $0.6{\ensuremath{\mu}}_{\mathrm{B}}$ at $T=1.5$ K, which is accompanied by the incommensurate modulations. The Landau phenomenological approach indicates that the higher-order magnetic and/or multipole interactions based on the pseudoquartet $f$-electron state play important roles.