Quartic Coefficients Research Articles

DEM simulation and optimization of crushing chamber shape of gyratory crusher based on Ab-t10 model

To study and optimize the crushing chamber performance of the gyratory crusher, the impact of the crushing chamber shape on the gyratory crusher performance is explored in this paper. The discrete element method (DEM) analysis model of ore particle and gyratory crusher is established, where ore particle crushing is realized based on the Ab-t10 breakage model, and its validity is verified based on actual working conditions. On this basis, the performance of the gyratory crusher under different crushing chamber shapes is studied by combining the DEM simulation with the response surface method (RSM), and the performance prediction model is obtained based on multiple nonlinear regression. Further, the productivity is considered as the main optimization objective, the mass percentage of effective particles and the crushing force are used as auxiliary optimization objectives. The multi-objective optimization is implemented and the optimized effects are analysed by DEM simulation, and the optimized quartic equation coefficients of the concave curve are obtained. Finally, the results show that the productivity, mass percentage of effective particles, and crushing force are increased to varying degrees. Meanwhile, the clogged layer area increases to some extent, which promotes the crushing of ore particles.

Modulation instability gain and localized waves in the modified Frenkel–Kontorova model with high-order nonlinearities

In this paper, modulation instability and nonlinear supratransmission are investigated in a one-dimensional chain of atoms using cubic–quartic nonlinearity coefficients. As a result, we establish the discrete nonlinear evolution equation by using the multi-scale scheme. To compute the modulation instability gain, the linearizing technique is employed. The effect of the higher nonlinear component on modulation instability is particularly examined. Following that, full numerical integration was performed to identify modulated wave patterns as well as the appearance of a rogue wave. Through the nonlinear supratransmission phenomenon, one end of the discrete model is driven into the forbidden bandgap. As a consequence, for driving amplitudes above the supratransmission threshold, the bright soliton and modulated wave patterns are satisfied. An important behavior is observed in the transient range of time of propagation when the bright soliton wave turns into a chaotic soliton wave. These results corroborate our analytical investigations on the modulation instability and show that the one-dimensional chain of atoms is a fruitful medium to generate long-lived modulated waves.

Cascade of vestigial orders in two-component superconductors: Nematic, ferromagnetic, s -wave charge- 4e , and d -wave charge- 4e states

Electronically ordered states that break multiple symmetries can melt in multiple stages, similarly to liquid crystals. In a partially melted phase, known as vestigial phase, a bilinear made out of combinations of the multiple components of the primary order parameter condenses. Multicomponent superconductors are thus natural candidates for vestigial order since they break both the $\text{U}(1)$-gauge and also time-reversal or lattice symmetries. Here, we use group theory to classify all possible real-valued and complex-valued bilinears of a generic two-component superconductor on a tetragonal or hexagonal lattice. While the more widely investigated real-valued bilinears correspond to vestigial nematic or ferromagnetic order, the little explored complex-valued bilinears correspond to a vestigial charge-$4e$ condensate, which itself can have an underlying $s$-wave, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave, or ${d}_{xy}$-wave symmetry. To properly describe the fluctuating regime of the superconducting Ginzburg-Landau action and thus access these competing vestigial phases, we employ both a large-$N$ and a variational method. We show that while vestigial order can be understood as a weak-coupling effect in the large-$N$ approach, it is akin to a moderate-coupling effect in the variational method. Despite these distinctions, both methods yield similar results in wide regions of the parameter space spanned by the quartic Landau coefficients. Specifically, we find that the nematic and ferromagnetic phases are the leading vestigial instabilities, whereas the various types of charge-$4e$ order are attractive albeit subleading vestigial channels. The only exception is for the hexagonal case, in which the nematic and $s$-wave charge-$4e$ vestigial states are degenerate. We discuss the limitations of our approach, as well as the implications of our results for the realization of exotic charge-$4e$ states in material candidates.

On the time evolution of cosmological correlators

Developing our understanding of how correlations evolve during inflation is crucial if we are to extract information about the early Universe from our late-time observables. To that end, we revisit the time evolution of scalar field correlators on de Sitter spacetime in the Schrödinger picture. By direct manipulation of the Schrödinger equation, we write down simple “equations of motion” for the coefficients which determine the wavefunction. Rather than specify a particular interaction Hamiltonian, we assume only very basic properties (unitarity, de Sitter invariance and locality) to derive general consequences for the wavefunction’s evolution. In particular, we identify a number of “constants of motion” — properties of the initial state which are conserved by any unitary dynamics — and show how this can be used to partially fix the cubic and quartic wavefunction coefficients at weak coupling. We further constrain the time evolution by deriving constraints from the de Sitter isometries and show that these reduce to the familiar conformal Ward identities at late times. Finally, we show how the evolution of a state from the conformal boundary into the bulk can be described via a number of “transfer functions” which are analytic outside the horizon for any local interaction. These objects exhibit divergences for particular values of the scalar mass, and we show how such divergences can be removed by a renormalisation of the boundary wavefunction — this is equivalent to performing a “Boundary Operator Expansion” which expresses the bulk operators in terms of regulated boundary operators. Altogether, this improved understanding of the wavefunction in the bulk of de Sitter complements recent advances from a purely boundary perspective, and reveals new structure in cosmological correlators.

Algorithm 1010

Aiming to provide a very accurate, efficient, and robust quartic equation solver for physical applications, we have proposed an algorithm that builds on the previous works of P. Strobach and S. L. Shmakov. It is based on the decomposition of the quartic polynomial into two quadratics, whose coefficients are first accurately estimated by handling carefully numerical errors and afterward refined through the use of the Newton-Raphson method. Our algorithm is very accurate in comparison with other state-of-the-art solvers that can be found in the literature, but (most importantly) it turns out to be very efficient according to our timing tests. A crucial issue for us is the robustness of the algorithm, i.e., its ability to cope with the detrimental effect of round-off errors, no matter what set of quartic coefficients is provided in a practical application. In this respect, we extensively tested our algorithm in comparison to other quartic equation solvers both by considering specific extreme cases and by carrying out a statistical analysis over a very large set of quartics. Our algorithm has also been heavily tested in a physical application, i.e., simulations of hard cylinders, where it proved its absolute reliability as well as its efficiency.

Rotating tensor-multiscalar solitons

In the context of a special class of tensor-multi-scalar theories of gravity for which the target-space metric admits an isometry under which the theory is invariant, we present rotating vacuum solutions, namely with no matter fields. These objects behave like nontopological solitons, whose primary stability is due to the conserved charge arising from the global symmetry. We consider theories with two and three scalar fields, different Gauss curvatures and quartic interaction coefficients. As it occurs for boson stars, their angular momentum is quantized.

The unusual symmetry of hexafluoro-o-xylene-A microwave spectroscopy and computational study.

The rotational constants and quartic centrifugal distortion coefficients of hexafluoro-o-xylene were precisely derived from the 8 GHz to 18 GHz gas phase microwave spectrum. In addition, the rotational constants of all singly substituted 13C isotopologues were determined. Instead of the intuitively expected symmetry of C2v, as in o-xylene, calculations with a variety of methods (B3LYP, CAM-B3LYP, ωB97XD, MP2, and coupled-cluster singles, doubles, and perturbative triples) predict a C2 symmetry structure in which the two CF3 groups rotate in opposite directions by about 16°. While the experimental results in this study are not capable of proving this unusual symmetry, they can support future microwave, gas phase electron diffraction or nuclear magnetic resonance studies. From the presented data, a preliminary r0 structure was determined, reproducing the experimental rotational constants with deviations of no more than 15 kHz. Analysis of the interactions between the two CF3 groups using an effective fragment potential approach identifies two major contributions to their interaction, due to exchange-repulsion and electrostatic repulsion, with electrostatic repulsion responsible for the barrier at the C2v geometry.

Probabilistic moveout analysis by time warping

Parameter estimation from reflection moveout analysis represents one of the most fundamental problems in subsurface model building. We have developed an efficient moveout inversion method based on the process of automatic flattening of common-midpoint (CMP) gathers using local slopes. We find that as a by-product of this flattening process, we can also estimate reflection traveltimes corresponding to the flattened CMP gathers. This traveltime information allows us to construct a highly overdetermined system and subsequently invert for moveout parameters including normal-moveout velocities and quartic coefficients related to anisotropy. We use the 3D generalized moveout approximation (GMA), which can accurately capture the effects of complex anisotropy on reflection traveltimes as the basis for our moveout inversion. Due to the cheap forward traveltime computations by GMA, we use a Monte Carlo inversion scheme for improved handling of the nonlinearity between the reflection traveltimes and moveout parameters. This choice also allows us to set up a probabilistic inversion workflow within a Bayesian framework, in which we can obtain the posterior probability distributions that contain valuable statistical information on estimated parameters such as uncertainty and correlations. We use synthetic and real data examples including the data from the SEAM Phase II unconventional reservoir model to demonstrate the performance of our method and discuss insights into the problem of moveout inversion gained from analyzing the posterior probability distributions. Our results suggest that the solutions to the problem of traveltime-only moveout inversion from 2D CMP gathers are relatively well constrained by the data. However, parameter estimation from 3D CMP gathers associated with more moveout parameters and complex anisotropic models are generally nonunique, and there are trade-offs among inverted parameters, especially the quartic coefficients.

Effects of lateral heterogeneity on time-domain processing parameters

SUMMARY Time-domain processing of seismic reflection data has always been an important engine that is routinely utilized to produce seismic images and to expeditiously construct subsurface models. The conventional procedure involves analysing parameters related to the derivatives of reflection traveltime with respect to offset including normal moveout (NMO) velocities (second-order derivatives) and quartic coefficients (fourth-order derivatives). In this study, we propose to go beyond the typical assumption of 1-D laterally homogeneous medium when relating those ‘processing’ parameters to the subsurface medium parameters and take into account the additional influences from lateral heterogeneity including curved interfaces and smoothly variable velocities. We fill in the theoretical gap from previous studies and develop a general framework for such connection in layered anisotropic media. We show that in general, the influences of lateral heterogeneity get accumulated from all layers via a recursive relationship according to the Fermat’s principle and can be approximately quantified in terms of the lateral derivatives of the layer interface surfaces and velocities. Based on the same general principle, we show that our approach can also be used to study the lateral heterogeneity effects on diffraction traveltime and its second-order derivative related to time-migration velocity. In this paper, we explicitly specify expressions for NMO and time-migration velocities with the influences from both types of heterogeneity suitable for 2-D data sets and also discuss possible extensions of the proposed theory to 3-D data sets and to parameters related to higher-order traveltime derivatives. Using numerical examples, we demonstrate that the proposed theory can lead to more accurate reflection and diffraction traveltime predictions in comparison with those obtained based on the 1-D assumption. Both the proposed theoretical framework and its numerical testing for forward traveltime computation presented in this study aid in understanding the effects from lateral heterogeneity on time-processing parameters and also serve as an important basis for designing an efficient technique to separate those influences in important processes such as Dix inversion for a more accurate subsurface model in the future.

An analytical investigation on the dynamic stability of a rotor filled with liquid

This paper deals with the dynamic stability of a rigid rotor arbitrarily filled with liquid. On the basis of the established coupled-field equations of the rotor system, the general whirling eigenequation, which is a quartic complex coefficients equation, is derived. In order to obtain the solutions of the general whirling eigenequation, a mathematical method is proposed. To illustrate the precision of calculating results, a comparison is carried out between the present analysis and the numerical results. The results show that two calculation results are in good agreement. Then the stability of the rotor system is analyzed. It is shown that the dynamic instability occurs at a particular bound of the spinning speed. Moreover, the effects of system parameters, such as fluid-fill ratio and mass ratio, on the unstable regions are discussed.

Negative thermal expansion behavior in MZrF6(M=Ca,Mg,Sr) : Ab initio lattice dynamical studies

The thermal expansion behavior of metal fluorides can be tuned by choosing an appropriate metal cation. We present ab initio lattice dynamical studies on the metal fluorides $(\mathrm{CaZr}{\mathrm{F}}_{6},\mathrm{MgZr}{\mathrm{F}}_{6}$, and $\mathrm{SrZr}{\mathrm{F}}_{6})$ and identify the anharmonic phonon modes responsible for the negative thermal expansion (NTE) in these materials. These phonon modes involve $\mathrm{Zr}{\mathrm{F}}_{6}$ polyhedral rotational motion, which leads to large transverse amplitude of the vibrations of the fluorine atom in the Zr-F-M bond in these compounds. The compounds with larger anisotropy in the thermal amplitude of the fluorine atom show larger NTE behavior. This has enabled us to understand the substantial variation in thermal expansion behavior of these compounds at high temperature. We have also calculated the potential energy profile of selected phonon modes, which enables us to derive the cubic and quartic anharmonic coefficients.

Transient chirality of anilides - The rotational spectrum of trans-benzanilide

The rotational spectrum of trans-benzanilide C13H11NO is reported for the most part of the cm-wave range (3–19 GHz). Heavy atom frame conversion between the two possible enantiomers surprisingly leads to significant tunnelling splittings in the observed spectrum. The large amplitude motion parameters for the occurring tunnelling as well as the nuclear quadrupole coupling constants for the 14N were determined in addition to the rotational constants and quartic centrifugal distortion coefficients. Contrary to other trans-anilides, calculations indicate the existence of a non planar amide group in trans-benzanilide.

Constraints on both the quadratic and quartic symmetry energy coefficients by 2β− -decay energies

In this Rapid Communication, the $2{\ensuremath{\beta}}^{\ensuremath{-}}$-decay energies $Q(2{\ensuremath{\beta}}^{\ensuremath{-}})$ given in the atomic mass evaluation are used to extract not only the quadratic volume symmetry energy coefficient ${c}_{\mathrm{sym}}^{v}$, but also the quartic one ${c}_{\mathrm{sym},4}^{v}$. Based on the modified Bethe-Weizs\"acker nuclear mass formula of the liquid-drop model, the decay energy $Q(2{\ensuremath{\beta}}^{\ensuremath{-}})$ is found to be closely related to both the quadratic and quartic symmetry energy coefficients ${c}_{\mathrm{sym}}^{v}$ and ${c}_{\mathrm{sym},4}^{v}$. There are totally 449 data of decay energies $Q(2{\ensuremath{\beta}}^{\ensuremath{-}})$ used in the present analysis where the candidate nuclei are carefully chosen by fulfilling the following criteria: (1) large neutron-proton number difference $N\ensuremath{-}Z$, (2) large isospin asymmetry $I$, and (3) limited shell effect. The values of ${c}_{\mathrm{sym}}^{v}$ and ${c}_{\mathrm{sym},4}^{v}$ are extracted to be 29.345 and 3.634 MeV, respectively. Moreover, the quadratic surface-volume symmetry energy coefficient ratio is determined to be $\ensuremath{\kappa}={c}_{\mathrm{sym}}^{s}/{c}_{\mathrm{sym}}^{v}=1.356$.

Soliton collisions for a higher-order nonlinear Schrödinger–Maxwell–Bloch system in an erbium-doped fiber

Under investigation in this paper is a higher-order nonlinear Schrödinger–Maxwell–Bloch system with quintic terms, which describes the ultrashort optical pulses, up to the attosecond duration, in an erbium-doped fiber. Via the Hirota method and symbolic computation, we derive the one- and two-soliton solutions, while soliton propagation and collision are graphically analyzed: A soliton is shown to maintain its amplitude and shape during the propagation. Soliton collisions are elastic, while bright and dark two solitons are also observed. Head-on collision between the two solitons is illustrated, and shapes of the two solitons do not change during the collision. We discuss the influence of the quadratic, cubic, quartic and quintic coefficients on the collision features: With the value of the quadratic coefficient decreases, widths of the two solitons become larger. Amplitudes of the two solitons decrease with the value of the cubic coefficient reducing. Amplitudes of the two solitons become lower when we reduce the value of the quartic coefficient, and amplitudes of the two solitons decrease with the value of the quintic coefficient decreasing. Overtaking collision between the two solitons can occur, and the two solitons keep their shapes invariant except for the phase shifts before and after the collision. Bound states are also seen between the equal-velocity bright two solitons: They attract and repulse each other periodically. A bound state is formed between the dark two equal-velocity solitons.

Theory of interval traveltime parameter estimation in layered anisotropic media

Moveout approximations for reflection traveltimes are typically based on a truncated Taylor expansion of traveltime squared around the zero offset. The fourth-order Taylor expansion involves normal moveout velocities and quartic coefficients. We have derived general expressions for layer-stripping second- and fourth-order parameters in horizontally layered anisotropic strata and specified them for two important cases: horizontally stacked aligned orthorhombic layers and azimuthally rotated orthorhombic layers. In the first of these cases, the formula involving the out-of-symmetry-plane quartic coefficients has a simple functional form and possesses some similarity to the previously known formulas corresponding to the 2D in-symmetry-plane counterparts in vertically transversely isotropic (VTI) media. The error of approximating effective parameters by using approximate VTI formulas can be significant in comparison with the exact formulas that we have derived. We have proposed a framework for deriving Dix-type inversion formulas for interval parameter estimation from traveltime expansion coefficients in the general case and in the specific case of aligned orthorhombic layers. The averaging formulas for calculation of effective parameters and the layer-stripping formulas for interval parameter estimation are readily applicable to 3D seismic reflection processing in layered anisotropic media.

Divergence of the isospin-asymmetry expansion of the nuclear equation of state in many-body perturbation theory

The isospin-asymmetry dependence of the nuclear matter equation of state obtained from microscopic chiral two- and three-body interactions in second-order many-body perturbation theory is examined in detail. The quadratic, quartic and sextic coefficients in the Maclaurin expansion of the free energy per particle of infinite homogeneous nuclear matter with respect to the isospin asymmetry are extracted numerically using finite differences, and the resulting polynomial isospin-asymmetry parametrizations are compared to the full isospin-asymmetry dependence of the free energy. It is found that in the low-temperature and high-density regime where the radius of convergence of the expansion is generically zero, the inclusion of higher-order terms beyond the leading quadratic approximation leads overall to a significantly poorer description of the isospin-asymmetry dependence. In contrast, at high temperatures and densities well below nuclear saturation density, the interaction contributions to the higher-order coefficients are negligible and the deviations from the quadratic approximation are predominantly from the noninteracting term in the many-body perturbation series. Furthermore, we extract the leading logarithmic term in the isospin-asymmetry expansion of the equation of state at zero temperature from the analysis of linear combinations of finite differences. It is shown that the logarithmic term leads to a considerably improved description of the isospin-asymmetry dependence at zero temperature.

Global sea level trends in the presence of variable sea level velocities, and variable accelerations

Abstract This study investigates, using a new variable-acceleration model, the validity of the implicit assertion in previous studies regarding global constant sea level rise accelerations. Thirteen out of twenty seven globally distributed tide gauge stations, with records longer than 80 years, exhibit statistically significant quartic coefficients (p < 0.05) revealing the presence of variable sea level accelerations though not as a global phenomenon. Most of these stations initially exhibit decreasing negative velocities until early 20th century and increasing positive velocities after 1970’s following a period of constant velocities. It is shown that, for those locations experiencing statistically significant variable sea level accelerations, the estimates based on the conventional linear representation of linear sea level trends are not appropriate, and are notably biased for a number of stations. All solutions account for serial correlations, which otherwise induce biases in solution statistics. It is also demonstrated that the omission of non-linearities in sea level changes will bias the sea level trends for short records, such as those from satellite altimetry, as large as 3 mm/yr.

High-resolution Raman spectroscopy of ν2and associated hot bands of13CH12CH: global analysis of the anharmonic resonances

The Raman spectrum of the 13CH12CH molecule in the region of its ν2 vibrational band was recorded at a resolution of 2.6 × 10−3 cm−1 and analyzed. Several Q branches were observed and identified, corresponding to the ν2 fundamental and the associated hot bands ν2 + ν4 – ν4, ν2 + ν5 – ν5, ν2 + 2ν4 – 2ν4 and 2ν2 – ν2. The usual normal modes of vibration in acetylene, with the symmetric (ν1) and antisymmetric (ν3) CH stretchings, the CC stretching (ν2), the degenerate trans (ν4) and cis (ν5) bendings were adopted. All the spectra were obtained using the experimental technique of quasi-continuous stimulated Raman spectroscopy. The fundamental ν2 band and the ν2 + ν4 – ν4 and ν2 + ν5 – ν5 hot bands were recorded at room temperature, while ν2 + 2ν4 – 2ν4 was recorded at a temperature of 355 K. Finally, the spectrum of the 2ν2 – ν2 band was recorded at 170 K using a Raman-Raman double resonance technique in which a stimulated Raman pumping process is used to populate a single rotational level of the v2 = 1 state, prior to the spectroscopy stage. A 15 ns delay between pump and spectroscopy is introduced to allow partial collisional relaxation of the rotational population of v2 = 1 among rotational levels. The assigned Raman transitions and several previously observed infrared bands have been simultaneously fitted in order to derive a set of deperturbed molecular parameters, including the quartic anharmonic coupling coefficients K1,255 and K3,245. The ro-vibrational pattern of the stretching and stretching-bending combination states and an accurate description of the anharmonic resonances in this molecule have been obtained. Copyright © 2013 John Wiley & Sons, Ltd.

Nonequilibrium steady state of a stochastic system driven by a nonlinear drift force

We investigate the properties of the nonequilibrium steady state for the stochastic system driven by a nonlinear drift force and influenced by noises that are not identically and independently distributed. The nonequilibrium steady state (NESS) current results from a residual part of the drift force that is not canceled by the diffusive action of noises. From our previous study for the linear drift force, the NESS current was found to circulate on the equiprobability surface with the maximum at a stable fixed point of the drift force. For the nonlinear drift force, we use the perturbation theory with respect to the cubic and quartic coefficients of the drift force. We find an interesting potential landscape picture where the probability maximum shifts from the fixed point of the drift force and, furthermore, the NESS current has a nontrivial circulation that flows off the equiprobability surface and has various centers not located at the probability maximum. The theoretical result is well confirmed by the computer simulation.

Effects on the Dental Arch Form Using a Preadjusted Appliance with Premolar Extraction in Class I Crowding

To determine the dental arch form effects of treatment with a preadjusted appliance (0.022'' slot) performed concomitantly with extraction of premolars in Class I crowding. Twenty-six patients (20.17 +/- 12.15 years) with Class I crowding who attained a favorable occlusion after treatment were divided into nonextraction and extraction groups. The three-dimensional coordinates of the FA point of each tooth were determined. The dental arch form was expressed as a quartic polynomial expression with log F value used to represent the dental arch form and calculated using the quadratic and quartic coefficients. Changes in the dental arch form before and after treatment were compared between the groups. In the extraction group, log F value was significantly higher after treatment, and the upper dental arch became more tapered. The U1-APo was significantly lower after treatment. No change was observed in U1-FH. The anterior teeth demonstrated posterior movement due to sliding mechanics, and torque was controlled. The anterior length of the dental arch became significantly longer after treatment for the maxilla and mandible in both groups. Results demonstrated that the upper dental arch might become tapered after treatment used concomitantly with premolar extraction, as a result of the increase in anterior length while maintaining intercanine width for dealing with crowding of the anterior teeth. It is necessary to anticipate that the upper dental arch form will become tapered during extraction treatment for Class I crowding and to select an appropriate arch form.