2nd Order Terms Research Articles

Retinal image quality in myopic children undergoing orthokeratology alone or combined with 0.01% atropine

BackgroundThe retinal image quality derived from lower-order (LOA) and higher-order aberrations (HOA) for fixed 3-mm and photopic pupil diameters, in children undergoing combined 0.01% atropine and orthokeratology (AOK) versus those receiving orthokeratology alone (OK) over two years was evaluated.MethodsThe visual Strehl ratio based on the optical transfer function (VSOTF), derived from 2nd- to 4th-order terms (LOA and HOA combined), 2nd-order terms (LOA only), and 3rd- to 4th-order terms (HOA only) for fixed 3-mm and natural photopic pupil diameters, was compared between the two treatment groups. The individual Zernike coefficients for a fixed 3-mm pupil size of 2nd- to 4th-orders, root mean square (RMS) of LOA ({Z}_{2}^{0}, {Z}_{2}^{-2}, and {Z}_{2}^{2} combined), HOA (3rd to 4th orders inclusive), and Coma ({Z}_{3}^{-1}{mathrm{and }Z}_{3}^{1} combined) were also compared between the two groups.ResultsRight eye data of 33 AOK and 35 OK participants were analysed. Under photopic conditions, significantly lower VSOTF based on HOA only was observed in the AOK group compared with that in the OK group at all post-treatment visits (all P < 0.05); however, interactions between HOA and LOA resulted in comparable overall retinal image quality (i.e., VSOTF based on LOA and HOA combined) between the two groups at all visits (all P > 0.05). For a fixed 3-mm pupil size, the VSOTF based on HOA only, LOA only, or HOA and LOA combined, were not different between the two groups (all P > 0.05). AOK participants had slower axial elongation (mean ± SD, 0.17 ± 0.19 mm vs. 0.35 ± 0.20 mm, P < 0.001), a larger photopic pupil size (4.05 ± 0.61 mm vs. 3.43 ± 0.41 mm, P < 0.001) than OK participants, over two years.ConclusionsHOA profile related to an enlarged pupil size may provide visual signal influencing eye growth in the AOK group.

A Novel Progressive Response Surface Method for High-order polynomial Metamodel

The response surface method (RSM) model as a general metamodel has already widely been applied to many practical problems. However, the majority of studies only focus on the 2nd-order terms RSM model instead of how to determine the high-order terms. This paper introduced a novel progressive response surface method (NPRSM) combining optimal Latin hypercube design (OLHD), moving least square method (MLSM) and statistical test. This NPRSM model can decide the high-order terms by progressive procedure. Then three numerical function can be proposed to validate the accuracy and the approximated performance of the NPRSM model.

C-band 120-Gb/s PAM-4 transmission over 50-km SSMF with improved weighted decision-feedback equalizer

An improved weighted decision-feedback equalizer (IWDFE) is adopted and experimentally demonstrated to mitigate chromatic dispersion (CD) induced power fading for C-band four-level pulse amplitude modulation (PAM-4) system. Based on a compressed sigmoid nonlinear function, the IWDFE is superior to the conventional DFE, rule-1 and rule-2 based WDFEs in reducing both the error propagation probability and percentage of errors. The IWDFE is combined with a feed-forward equalizer (FFE) to deal with both pre-cursor and post-cursor interference simultaneously. We also introduce 2nd-order polynomial nonlinear terms in both FFE and IWDFE at receiver side, constructing a nonlinear FFE-IWDFE (N-FFE-IWDFE) to simultaneously equalize both the CD-induced power fading and nonlinear distortions. Experimental results show that compared with nonlinear FFE-DFE (N-FFE-DFE), rule-1 and rule-2 based nonlinear FFE-WDFEs (N-FFE-WDFEs), the N-FFE-IWDFE can significantly reduce the bit error ratio (BER), which is close to that of the error-propagation-free N-FFE-DFE. By utilizing the N-FFE-IWDFE, 120-Gb/s PAM-4 transmission system over 50-km standard single-mode fiber (SSMF) is realized with the BER below 7% hard-decision forward error correction (HD-FEC) limit of 3.8 × 10−3, achieving around 1.1-dB improvement of the receiver sensitivity over the N-FFE-DFE. To the best of our knowledge, this is the first time that an IWDFE is employed for C-band IM/DD optical transmission system.

Paleoclimate records reveal elusive ~200-kyr eccentricity cycle for the first time

The main cycles of Earth's eccentricity as recorded in paleoclimate archives signify 2nd order terms, combining two g-frequencies associated with the precession of perihelia of the inner planets and Jupiter. However, many weaker cycles are present and may be documented as well. Here we report on a thus far unnoticed ~200-kyr cycle observed in rhythmically bedded marine marls and limestones of early Paleocene age in NE Spain and adjacent France. This cycle is expressed as alternating distinct and less distinct ~100-kyr CaCO3 maxima and associated magnetic susceptibility (MS) minima, which correspond to successive minima of the short ~100-kyr eccentricity cycle. The most plausible explanation is that this ~200-kyr cycle represents a weak but real cycle in eccentricity. Due to its weakness, this cycle is only expressed as alternatingly strong and weak ~100-kyr minima in eccentricity time series, a pattern that is identical to what we observe in the proxy records and outcrop. Its origin as eccentricity cycle is complex as it is composed of 4 to 6 individual components. A simplified solution solely based on the five leading g-frequencies reveals that most of these components originate from combinations of these 5 frequencies. However, they do not correspond to 2nd order terms, which underlie the principal short and long eccentricity cycles, but to much weaker higher order terms. Three of them have been identified as 4th order terms and two as 6th order terms, representing combinations of 2nd order terms, such as the first harmonic, 2(g2 – g5), of the 405-kyr cycle and related components. It is for the first time that the combined effect of such weak components has been identified in paleoclimatic records, which is only possible because they are not overwhelmed by much stronger 2nd order terms in this frequency band. Finally, the typical pattern that results from interference with the much stronger ~100-kyr cycle might be used as a template for testing the accuracy of astronomical solutions in the future.

Robust design optimization of a nonlinear monostable energy harvester with uncertainties

Based on the improved interval extension, a robust optimization method for nonlinear monostable energy harvesters with uncertainties is developed. In this method, the 2nd order terms in the interval extension formula of the objective function (output voltage) are kept so this approach is suitable for a nonlinear energy harvesting system. To illustrate this method, uncertain mass, uncertain capacitance and uncertain electromechanical coupling coefficient are optimized to maximize the central point of output voltage whose deviation of which is simultaneously minimized. Then, an optimal design with different robustness is obtained. The results also show that the robustness of the optimal design of nonlinear monostable energy harvesters is increased, but the cost of performance has to be paid for. Overall, the framework provides the optimal design for nonlinear energy harvesters (monostable, bistable, tristable, multistable harvesters).

On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes equations in polar coordinates

Abstract The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier–Stokes system involving additional 2nd order terms with a regularizing parameter τ &gt; 0. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier–Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced. The main results are valid for τ = 0 as well, i.e., for the barotropic compressible Navier–Stokes system.

A family of metric gravities

.The goal of this paper is to take a completely fresh approach to metric gravity, in which the metric principle is strictly adhered to but its properties in local space-time are derived from conservation principles, not inferred from a global field equation. The global field strength variation then gains some flexibility, but only in the regime of very strong fields (2nd-order terms) whose measurement is now being contemplated. So doing provides a family of similar gravities, differing only in strong fields, which could be developed into meaningful verification targets for strong fields after the manner in which far-field variations were used in the 20th century. General Relativity (GR) is shown to be a member of the family and this is demonstrated by deriving the Schwarzschild metric exactly from a suitable field strength assumption. The method of doing so is interesting in itself because it involves only one differential equation rather than the usual four. Exact static symmetric field solutions are also given for one pedagogical alternative based on potential, and one theoretical alternative based on inertia, and the prospects of experimentally differentiating these are analyzed. Whether the method overturns the conventional wisdom that GR is the only metric theory of gravity and that alternatives must introduce additional interactions and fields is somewhat semantical, depending on whether one views the field strength assumption as a field and whether the assumption that produces GR is considered unique in some way. It is of course possible to have other fields, and the local space-time principle can be applied to field gravities which usually are weak-field approximations having only time dilation, giving them the spatial factor and promoting them to full metric theories. Though usually pedagogical, some of them are interesting from a quantum gravity perspective. Cases are noted where mass measurement errors, or distributions of dark matter, can cause one theory to mimic another implying that such estimates or distributions should be first obtained from weak-field measurements before being used to discriminate verification candidates. By this method theorists gain insight into the local constraints on space-time, and GR verification gains strong-field comparative objectives.

The virial theorem and ground state energy estimates of nonlinear Schrödinger equations in $$\mathbb {R}^2$$ R 2 with square root and saturable nonlinearities in nonlinear optics

The virial theorem is a nice property for the linear Schrodinger equation in atomic and molecular physics as it gives an elegant ratio between the kinetic and potential energies and is useful in assessing the quality of numerically computed eigenvalues. If the governing equation is a nonlinear Schrodinger equation with power-law nonlinearity, then a similar ratio can be obtained but there seems to be no way of getting any eigenvalue estimates. It is surprising as far as we are concerned that when the nonlinearity is either square-root or saturable nonlinearity (not a power-law), one can develop a virial theorem and eigenvalue estimates of nonlinear Schrodinger (NLS) equations in $${{\mathbb {R}}^{2}}$$ with square-root and saturable nonlinearity, respectively. Furthermore, we show here that the eigenvalue estimates can be used to obtain the 2nd order term (which is of order $$\ln \Gamma $$ ) of the lower bound of the ground state energy as the coefficient $$\Gamma $$ of the nonlinear term tends to infinity.

Interatomic inelastic current

In order to identify the location of an inelastic event and to distinguish between situations that are before or after this event, we derive equations for the interatomic inelastic transmission as a perturbation series in the electron-phonon interaction. This series contains both even and odd ordered corrections, and while the even ordered corrections can be thought as a Dyson’s expansion of the interatomic elastic transmission in the electron-phonon self-energy, the odd ordered corrections represent something new. We explicitly derive expressions for the interatomic inelastic transmission up to second order and the 1st order correction represents the lowest order term of this new family of terms. We apply this to three model systems and are able to distinguish between situations before and after the inelastic event as steps in the 2nd order transmission. We also see that when the transmission is evaluated between atoms that are coupled by the electron-phonon interaction, the 1st and 2nd order terms must be added together to form a meaningful transmission. Within the limited scope of the models considered here, the 1st order term appears to be the signature of the inelastic event.

A Fourier spectrum-based strain energy damage detection method for beam-like structures in noisy conditions

In this paper, the Fourier spectrum-based strain energy damage detection method for beam-like structures is proposed based on the discrete Fourier transform. The classical strain energy damage detection method localizes damage by the comparison of the strain energy between the intact and inspected structures. The evaluation of the 2nd-order derivative term in the strain energy plays a crucial part in the comparison. The classical methods are mostly based on a numerical derivative estimation for this term. The numerical derivative, however, introduces additional disturbances into damage indications. To address this problem, a discrete Fourier transform-based strain energy is proposed with the emphasis of enhancing the performance in noisy condition. The validations conducted on the simulated and experimental data show that the developed method is effective enough for composite beam damage detection in noisy environments.

Advanced ponderomotive description of electron acceleration in ICRF discharge initiation

This contribution proposes a new approach for the ponderomotive description of electron acceleration in ICRF discharge initiation. The motion of electrons in the parallel electric field E z is separated into a fast oscillation and a slower drift around the oscillation centre. Three terms are maintained in the Taylor expansion of the electric field (0th , 1st and 2nd order). The efficiency for electron acceleration by E z (z , t ) is then assessed by comparing the values of these terms at the slow varying coordinate z 0 . When (i) the 0th order term is not significantly larger than 1st order term at the reflection point, or when (ii) the 2nd order term is negative and not sufficiently small compared to the 1st order term at the reflection point, then the electron will gain energy in the reflection.An example for plasma production by the TOMAS ICRF system is given. Following the described conditions it can be derived that plasma production is (i) most efficient close to the antenna straps (few cm's) where the field gradient and amplitude are large, and (ii) that the lower frequency field accelerates electrons more easily for a given antenna voltage.

Coupling Realistic Computational Neural Model and Human Experimental Data: the Cerebellar Role in Eye Blink Conditioning and its Alteration due to Transcranial Magnetic Stimulation

Coupling Realistic Computational Neural Model and Human Experimental Data: the Cerebellar Role in Eye Blink Conditioning and its Alteration due to Transcranial Magnetic Stimulation

Gedanken Experiment for Delineating the Regime for the Start of Quantum Effects, and Their End, Using Turok’s Perfect Bounce Criteria and Radii of a Bounce Maintaining Quantum Effects, as Delineated by Haggard and Rovelli

Haggard and Rovelli delineated an outer radius as to the range of quantum effects, which extends past the Schwartzshield radius. This is defined as 7/3 times the mass of the initial cosmological system. We also have a range of perturbative effects as delineated by Turok’s article which gives a range of values of for which second order perturbative terms in cosmological evolution may play a role, where we have second order perturbation terms for which . Right afterwards, there are no perturbative behavior and no perturbation if . This is the 2nd order term for perturbing term for GW (Gravitational wave) as denoted by , and near the “zero point” of cosmological expansion, and from there we determine the size of the quantum effects, i.e. when they initiate, the relevant initial entropy, so as to determine the radii of initial cosmology, so quantum gravity may initiate its activity, in our toy universe. The criteria of Turok is used to obtain the relevant mass, m, used in the initial radii so that it is 7/3 times the mass of the initial cosmological system. We use the “Criteria of Turok” to delineate the start of quantum gravity effects. Mass m is done via appealing to graviton mass, and that times initial entropy, which is commented upon in Equation (9).

A classical mechanics model for the interpretation of piezoelectric property data

In order to provide a means of understanding, the relationship between the primary electromechanical coefficients and simple crystal chemistry parameters for piezoelectric materials, a static analysis of a 3 atom, dipolar molecule has been undertaken to derive relationships for elastic compliance sE, dielectric permittivity εX, and piezoelectric charge coefficient d in terms of an effective ionic charge and two inter-atomic force constants. The relationships demonstrate the mutual interdependence of the three coefficients, in keeping with experimental evidence from a large dataset of commercial piezoelectric materials. It is shown that the electromechanical coupling coefficient k is purely an expression of the asymmetry in the two force constants or bond compliances. The treatment is extended to show that the quadratic electrostriction relation between strain and polarization, in both centrosymmetric and non-centrosymmetric systems, is due to the presence of a non-zero 2nd order term in the bond compliance. Comparison with experimental data explains the counter-intuitive, positive correlation of k with sE and εX and supports the proposition that high piezoelectric activity in single crystals is dominated by large compliance coupled with asymmetry in the sub-cell force constants. However, the analysis also shows that in polycrystalline materials, the dielectric anisotropy of the constituent crystals can be more important for attaining large charge coefficients. The model provides a completely new methodology for the interpretation of piezoelectric and electrostrictive property data and suggests methods for rapid screening for high activity in candidate piezoelectric materials, both experimentally and by novel interrogation of ab initio calculations.

Numerical study of nonlinear Peregrine breather under finite water depth

Though the focusing method can effectively generate waves which satisfy the definition of rogue waves at a specific position and moment, however, being inherently not nonlinear, the focusing model is still a controversial rogue wave generation method. Recently, nonlinear models are becoming more popular for rogue wave generation in physical and numerical tanks. In this paper, a weakly nonlinear model known as the Peregrine breather solution of the cubic Schrödinger equation is studied under finite water depth. In contrast with the focusing model, nonlinearity is considered throughout the simulation process, i.e., from the wave model to the generated waves. The numerical results are validated against theoretical solutions as well as experimental measurements. To further investigate their temporal-frequency characteristics, a wavelet analysis is performed on the generated Peregrine breather, and the concepts of life time and traveling distance are studied. The influence of higher order nonlinearity, i.e., the 2nd-order Stokes component of the perturbed expansion under finite water depth, is taken into account. We also discuss the influence of 2nd-order term on the life time, travel distance, and energy distribution.

Comparison among Magnus/Floquet/Fer expansion schemes in solid-state NMR.

We here revisit expansion schemes used in nuclear magnetic resonance (NMR) for the calculation of effective Hamiltonians and propagators, namely, Magnus, Floquet, and Fer expansions. While all the expansion schemes are powerful methods there are subtle differences among them. To understand the differences, we performed explicit calculation for heteronuclear dipolar decoupling, cross-polarization, and rotary-resonance experiments in solid-state NMR. As the propagator from the Fer expansion takes the form of a product of sub-propagators, it enables us to appreciate effects of time-evolution under Hamiltonians with different orders separately. While 0th-order average Hamiltonian is the same for the three expansion schemes with the three cases examined, there is a case that the 2nd-order term for the Magnus/Floquet expansion is different from that obtained with the Fer expansion. The difference arises due to the separation of the 0th-order term in the Fer expansion. The separation enables us to appreciate time-evolution under the 0th-order average Hamiltonian, however, for that purpose, we use a so-called left-running Fer expansion. Comparison between the left-running Fer expansion and the Magnus expansion indicates that the sign of the odd orders in Magnus may better be reversed if one would like to consider its effect in order.

Note on linearized “new massive gravity” in arbitrary dimensions

By means of a triple master action we deduce here a linearized version of the "New Massive Gravity" (NMG) in arbitrary dimensions. The theory contains a 4th-order and a 2nd-order term in derivatives. The 4th-order term is invariant under a generalized Weyl symmetry. The action is formulated in terms of a traceless $\eta^{\mu\nu}\Omega_{\mu\nu\rho}=0$ mixed symmetry tensor $\Omega_{\mu\nu\rho}=-\Omega_{\mu\rho\nu}$ and corresponds to the massive Fierz-Pauli action with the replacement $e_{\mu\nu}=\p^{\rho}\Omega_{\mu\nu\rho}$. The linearized 3D and 4D NMG theories are recovered via the invertible maps $\Omega_{\mu\nu\rho} = \epsilon_{\nu\rho}^{\quad\beta}h_{\beta\mu} $ and $\Omega_{\mu\nu\rho} = \epsilon_{\nu\rho}^{\quad \gamma\delta}T_{[\gamma\delta]\mu} $ respectively. The properties $h_{\mu\nu}=h_{\nu\mu}$ and $T_{[[\gamma\delta]\mu]}=0$ follow from the traceless restriction. The equations of motion of the linearized NMG theory can be written as zero "curvature" conditions $\p_{\nu}T_{\rho\mu} - \p_{\rho}T_{\nu\mu}=0$ in arbitrary dimensions.

Emission lineshapes of the B850 band of light-harvesting 2 (LH2) complex in purple bacteria: A second order time-nonlocal quantum master equation approach

The emission lineshape of the B850 band in the light harvesting complex 2 of purple bacteria is calculated by extending the approach of 2nd order time-nonlocal quantum master equation [S. Jang and R. J. Silbey, J. Chem. Phys. 118, 9312 (2003)]. The initial condition for the emission process corresponds to the stationary excited state density where exciton states are entangled with the bath modes in equilibrium. This exciton-bath coupling, which is not diagonal in either site excitation or exciton basis, results in a new inhomogeneous term that is absent in the expression for the absorption lineshape. Careful treatment of all the 2nd order terms are made, and explicit expressions are derived for both full 2nd order lineshape expression and the one based on secular approximation that neglects off-diagonal components in the exciton basis. Numerical results are presented for a few representative cases of disorder and temperature. Comparison of emission line shape with the absorption line shape is also made. It is shown that the inhomogeneous term coming from the entanglement of the system and bath degrees of freedom makes significant contributions to the lineshape. It is also found that the perturbative nature of the theory can result in negative portion of lineshape in some situations, which can be removed significantly by inclusion of the inhomogeneous term and completely by using the secular approximation. Comparison of the emission and absorption lineshapes at different temperatures demonstrates the role of thermal population of different exciton states and exciton-phonon couplings.

Frequency response areas in the inferior colliculus: nonlinearity and binaural interaction.

The tuning, binaural properties, and encoding characteristics of neurons in the central nucleus of the inferior colliculus (CNIC) were investigated to shed light on nonlinearities in the responses of these neurons. Results were analyzed for three types of neurons (I, O, and V) in the CNIC of decerebrate cats. Rate responses to binaural stimuli were characterized using a 1st- plus 2nd-order spectral integration model. Parameters of the model were derived using broadband stimuli with random spectral shapes (RSS). This method revealed four characteristics of CNIC neurons: (1) Tuning curves derived from broadband stimuli have fixed (i. e., level tolerant) bandwidths across a 50–60 dB range of sound levels; (2) 1st-order contralateral weights (particularly for type I and O neurons) were usually larger in magnitude than corresponding ipsilateral weights; (3) contralateral weights were more important than ipsilateral weights when using the model to predict responses to untrained noise stimuli; and (4) 2nd-order weight functions demonstrate frequency selectivity different from that of 1st-order weight functions. Furthermore, while the inclusion of 2nd-order terms in the model usually improved response predictions related to untrained RSS stimuli, they had limited impact on predictions related to other forms of filtered broadband noise [e. g., virtual-space stimuli (VS)]. The accuracy of the predictions varied considerably by response type. Predictions were most accurate for I neurons, and less accurate for O and V neurons, except at the lowest stimulus levels. These differences in prediction performance support the idea that type I, O, and V neurons encode different aspects of the stimulus: while type I neurons are most capable of producing linear representations of spectral shape, type O and V neurons may encode spectral features or temporal stimulus properties in a manner not easily explained with the low-order model. Supported by NIH grant DC00115.

Towards the ground state of the supermembrane

The explicit, near the origin, form of the ground state of the SU(2) supermembrane matrix model is studied. We evaluate the 2nd order terms of the Taylor expansion of the wave-function, which together with the 0th and the 1st order terms can be used to determine other terms by recurrence equations coming from the Schrödinger equation.