Non-monotonic Feature Research Articles

Quantitative susceptibility mapping for susceptibility source separation with adaptive relaxometric constant estimation (QSM-ARCS) from solely gradient-echo data

To separate the contributions of paramagnetic and diamagnetic sources within a voxel, a magnetic susceptibility source separation method based solely on gradient-echo data has been developed. To measure the opposing susceptibility sources more accurately, we propose a novel single-orientation quantitative susceptibility mapping method with adaptive relaxometric constant estimation (QSM-ARCS) for susceptibility source separation. Moreover, opposing susceptibilities and their anisotropic effects were determined in healthy volunteers in the white matter. Multiple spoiled gradient echo and diffusion tensor imaging of ten healthy volunteers was obtained using a 3 T magnetic resonance scanner. After the opposing susceptibility and fractional anisotropy (FA) maps had been reconstructed, the parametric maps were spatially normalized. To evaluate the agreements of QSM-ARCS against the susceptibility source separation method using R2 and R2* maps (χ-separation) by Bland–Altman plots, the opposing susceptibility values were measured using white and deep gray matter atlases. We then evaluated the relationships between the opposing susceptibilities and FAs in the white matter and used a field-to-fiber angle to assess the fiber orientation dependencies of the opposing susceptibilities. The susceptibility maps in QSM-ARCS were successfully reconstructed without large artifacts. In the Bland–Altman analyses, the opposing QSM-ARCS susceptibility values excellently agreed with the χ-separation maps. Significant inverse and proportional correlations were observed between FA and the negative and positive susceptibilities estimated by QSM-ARCS. The fiber orientation dependencies of the negative susceptibility represented a nonmonotonic feature. Conversely, the positive susceptibility increased linearly with the fiber angle with respect to the B0 field. The QSM-ARCS could accurately estimate the opposing susceptibilities, which were identical values of χ-separation, even using gradient echo alone. The opposing susceptibilities might offer direct biomarkers for assessment of the myelin and iron content in glial cells and, through the underlying magnetic sources, provide biologic insights toward clinical transition.

Z-shaped dejamming phase diagram of colloidal gels

For physically gelled colloidal suspensions, there are two routes to transform the gel from solid to liquid. One is to raise the temperature, and the other is to increase the shear deformation. In this investigation, we found that the phase boundary of this solid-to-liquid transformation exhibits a surprising Z-shaped curve in the strain-temperature plane. This nonmonotonic feature in phase transition appears to be present in various nanoparticle-filled colloidal gels with significant differences in chemical composition, filler type, structure, particle shape, average diameter, and particle size distribution. By applying the Kraus model to the breakage and restoration of filler networks and comparing our findings to nonequilibrium glassy behavior, we found that this nonmonotonic phenomenon can be theoretically predicted by combining the glassy melting kinetics of filler networks at high temperatures with the viscosity-retarded dissociation between particles at low temperatures.

A comparative study of chlorine and bromine species addition on the explosion limits of hydrogen-oxygen mixtures

In this work, we investigate the effects of chlorine (Cl) and bromine (Br) containing species additions on the pressure-temperature explosion limits of hydrogen-oxygen (H2-O2) mixtures computationally. It is shown that, with the addition of HCl, Cl2, HBr and Br2, the explosion limit evolves from being a non-monotonic Z-shaped curve to monotonic or maintains its non-monotonic response depending on the different types of additives. Specifically, with the addition of HCl to the mixtures, the limit always maintains the non-monotonic structure, while with the addition of around 18% of Cl2, it loses its Z-shaped structure. Compared with HCl and Cl2, more sensitive responses were found for HBr and Br2. With the addition of around 1% of HBr and 0.7% of Br2 to the H2-O2 mixtures, the limit's non-monotonic feature disappears rapidly, and becomes a monotonic curve. Further detailed kinetics analysis explains that after the addition of the halogen species, the radical competitions are different at the low- (P = 1.0 × 102 Pa), intermediate- (P = 1.0 × 104 Pa) and high- (P = 1.0 × 106 Pa) pressures respectively, resulting in the change of the dominant reactions. These findings enhance the understanding of Cl and Br containing species sensitization for the explosion limits of the H2-O2 mixtures and help to understand the kinetic interaction between the H2-O2 system and halogen.

Pressure induced insulator to metal transition in quantum spin liquid candidate NaYbS2

Pressure induced insulator to metal transition followed by the appearance of superconductivity has been observed recently in inorganic quantum spin liquid candidate NaYbSe2. In this paper, we study the properties of isostructural compound NaYbS2 under pressure. It is found that the resistance of NaYbS2 single crystal exhibits an insulating state below 82.9 GPa, but with a drop of more than six orders of magnitude at room temperature. Then a minimum of resistance is observed at about 100.1 GPa and it moves to lower temperature with further compression. Finally, a metallic state in the whole temperature range is observed at about 130.3 GPa accompanied by a non-Fermi liquid behavior below 100 K. The insulator to metal transition, non-monotonic resistance feature and non-Fermi liquid behavior of NaYbS2 under pressure are similar to those of NaYbSe2, suggesting that these phenomena might be the universal properties in NaLnCh 2 (Ln = rare earth, Ch = O, S, Se) system.

Modulation of Steady-State Heat Transport in a Dissipative Multi-Mode Qubit-Photon System

Quantum heat transport is considered as an indispensable branch of quantum thermodynamics to potentially improve performance of thermodynamic devices. We theoretically propose a dissipative qubit-photon system composed of multiple coupled resonators interacting with a single two-level qubit, to explore the steady-state heat transport by tuning both the inter-resonator photon hopping and the qubit-photon coupling. Specifically in the three-mode case, the dramatic enhancement and suppression of the heat current into the central resonator can be modulated by the corresponding frequency, compared to the currents into two edge resonators. Moreover, fruitful cycle current components are unraveled at weak qubit-photon coupling, which are crucial to exhibit the nonmonotonic feature with increase of the reservoir temperature bias. In the one-dimensional case under the mean-field framework, the influence of the photon hopping on heat transport is analyzed. The steady-state heat current is comparatively enhanced to the single-mode limit at weak qubit-photon coupling, stemming from the nonvanishing mean-field photon excitation parameter and the additional cycle current component. We hope these obtained results may have possible applications in quantum thermodynamic manipulation and energy harvesting.

Subsurface Ocean Temperature Responses to the Anthropogenic Aerosol Forcing in the North Pacific

AbstractSeparating the climate response to external forcing from internal climate variability is a key challenge. While most previous studies have focused on surface responses, here we examine zonal‐mean patterns of North Pacific subsurface temperature responses. In particular, the changes since 1950 driven by anthropogenic aerosol emissions are found by using a pattern recognition method. Based on the single‐forcing large‐ensemble simulations from two models, we show that aerosol forcing caused a nonmonotonic temporal response and a characteristic zonal‐mean pattern within North Pacific, which is distinct from the pattern associated with internal variability. The aerosol‐forced pattern with the nonmonotonic temporal feature shows a substantial temperature change in subpolar regions and a reversed change on the southern flank of the subtropical gyre. A similar characteristic pattern and nonmonotonic time evolution are extracted from the subsurface observations, which likely reflect the subsurface responses to the aerosol forcing, although differences exist with the simulated responses.

Effect of a novel generalized incidence rate function in [formula omitted] model: Stability switches and bifurcations

When a disease outbreaks in a population, the incidence rate function may increase initially with an increase of infective and then decrease due to inhibitory effects before getting saturated. For this purpose, we propose a new generalized incidence rate function and demonstrate the impact of this incidence rate function’s non-monotonic feature on the stability of endemic equilibria. In this paper, an SIR compartmental model with this new nonlinear incidence rate function and saturated treatment rate function is proposed. We obtain a disease free equilibrium which is locally stable for R0<1 and unstable otherwise. The system may have multiple endemic equilibria which can be either anti-saddle (node, focus or center) or saddle. We discover that when the incidence rate function is increasing at an anti-saddle endemic equilibrium, the endemic equilibrium may become either stable, unstable or switch stability via a parameter change. However, when the incidence rate function is non-monotonic and if the incidence rate decreases at an anti-saddle endemic equilibrium, the endemic equilibrium is always stable. Due to the limited treatment, we find forward (transcritical) and backward bifurcations. We also establish Hopf bifurcation and saddle–node bifurcation in the model system. Further, we incorporate the incubation delay in the model system. We obtain the same number of equilibria in the delay model. The disease free equilibrium is locally asymptotically stable for all delay lengths for R0<1. The endemic equilibrium switches its stability due to change in delay. Due to delay variation, the increasing incidence rate at an anti-saddle endemic equilibrium causes the multiple stability switches of the equilibrium from both stable and unstable initial states. Further, when the incidence rate decreases at the stable anti-saddle endemic equilibrium, it switches its stability once or multiple times and finally becomes unstable due to delay variation. We explicitly, show multiple stability switches of an endemic equilibrium due to existence of multiple Hopf bifurcations. When we further analyzed model system numerically, we could establish the existence of Hopf–Hopf bifurcation of codimension 2. In this case, the increase in the delay parameter causes the existence of two oscillatory solutions around the unstable endemic equilibrium. Hence, our results show that the delay system either destabilize developing the same frequency oscillations via Hopf bifurcation or two different frequency oscillations via Hopf–Hopf bifurcation of codimension 2. The existence and exhibition of these two frequency oscillations are novel and have not been explored much in epidemiological models.

A ductile fracture model incorporating stress state effect

To understand comprehensively the relationship between material ductility and stress state, the new shear, shear-compression, and shear-tension specimens were specially designed, and seven types of specimens were used in total to achieve precise control of stress states over wide ranges of stress triaxialities and Lode angle parameters. A hybrid experimental-numerical method was employed to determine the equivalent plastic strain, stress triaxiality, and Lode angle parameter, and hence to construct the 3D fracture locus of the Ti-6Al-4V alloy. Different micromechanisms were found to dominate the failure process under various stress states. Based on the experimental results, a new ductile fracture model coupling both stress triaxiality and Lode angle parameter was proposed and implemented into the commercial finite element program ABAQUS/Explicit via the user material subroutine VUMAT. Comparative studies with some other fracture criteria indicate that the proposed fracture model can capture the non-monotonic feature of fracture strain, and predict the fracture strain with better accuracy in a wide range of stress states. A verification test was performed to check the predictive capability of the present model. The results show the excellent agreement between experiment and simulation with respect to the fracture displacement and fracture morphology. This study paves the way for characterizing and predicting the ductile behavior of metallic materials under complex stress states, and provides a fundamental databank for the analysis and design of Ti-6Al-4V alloy structures.

Feature selection based on multiview entropy measures in multiperspective rough set

The performance of the neighborhood rough set model in feature selection is limited by nonobjective parameter selection method, the uncertainty measures considered only from a single view, and high time cost caused by processing high-dimensional data. To solve the above problems, this study first defines the interclass boundary to granulate the samples in different classes, and three types of neighborhood concepts—negative perspective, neutral perspective, and positive perspective—are put forward based on different cognitive perspectives. Then, the concept of the multiperspective rough set model is developed. The most prominent feature of this model is the discovery of differences between classes from the given data, without any parameters. Second, by integrating the information theory and algebraic views under the multiperspective rough set model, multiview entropy measures are proposed to effectively measure the uncertainty in data. Moreover, a nonmonotonic feature selection algorithm based on the mutual information in the multiview entropy measures under the neutral perspective as the evaluation function of feature importance is designed to resolve the disadvantages of the algorithms based on the monotone evaluation function. Finally, Information Gain is introduced to preliminarily decrease the dimension of high-dimensional data sets to promote classification accuracy and reduce time consumption. The experimental results confirm that the proposed algorithm is efficient in eliminating noise and increasing classification accuracy.

Tuning nonequilibrium heat current and two-photon statistics via composite qubit-resonator interaction

Quantum thermal transport and two-photon statistics serve as two representative nonequilibrium features in circuit quantum electrodynamics (cQED) systems. Here we investigate quantum heat flow and two-photon correlation function at steady state in a composite qubit-resonator model, where one qubit shows both transverse and longitudinal couplings to a single-mode optical resonator. With weak qubit-resonator interaction, we unravel two microscopic transport pictures, i.e., cotunneling and cyclic heat exchange processes corresponding to transverse and longitudinal couplings, respectively. The nonmonotonic behavior of the heat current is exhibited by tuning the temperature bias with the weak longitudinal coupling. At strong qubit-resonator coupling, the heat current also exhibits a nonmonotonic feature by increasing qubit-resonator coupling strength, which tightly relies on the scattering processes between the qubit and the corresponding thermal bath. Furthermore, the longitudinal coupling is preferred to enhance heat current in the strong qubit-resonator coupling regime. For two-photon correlation function, it exhibits an antibunching-to-bunching transition by tuning the composite angle, which is mainly dominated by the modulation of the energy gap between the first and the second excited eigenstates. Our results are expected to deepen the understanding of nonequilibrium thermal transport and nonclassical photon radiation based on the cQED platform.

Bulging-to-Budding Transition of Lipid Droplets Confined within Vesicle Membranes.

Lipid droplets (LDs) are intracellular organelles that act as reservoirs for energy homeostasis and phospholipid balance between supply and consumption. In comparison with extensive studies on LD biogenesis from a biological viewpoint, little is known about the mechanical interaction between LDs and vesicles. Here we perform a systematic theoretical study on the budding and morphological evolution of an artificial LD embedded within the lipid membrane of a pressurized vesicle. It is found that LD bulging and budding depend on the bending rigidity and spontaneous curvature of the vesicle membrane, LD-vesicle interfacial interaction energy strength and size ratio, and osmotic pressure of the vesicle. Beyond critical interfacial interaction strength, the embedded LD undergoes a discontinuous shape transition from a lens-shaped bulge to a spherical protrusion connecting to the nearly spherical vesicle lumen via an infinitesimally small monolayer neck. Moreover, a positive monolayer spontaneous curvature promotes budding transition. As the vesicle becomes smaller, higher cost of the monolayer stretching energy is required for an LD to achieve budding transition. Budding phase diagrams distinguishing the embedded and budding states of the LD-vesicle complex accounting for osmotic pressure and interfacial interaction strength are established with the budding transition boundary displaying a nonmonotonic feature. Our results reveal how embedded LDs overcome soft membrane confinement and protrude, and provide fundamental insights into the clustering of nanoparticles between vesicle monolayers.

Formation of L10-FeNi hard magnetic material from FeNi-based amorphous alloys

L10-FeNi hard magnetic alloy with coercivity reaching 861 Oe was synthesized through annealing Fe42Ni41.3Si8B4P4Cu0.7 amorphous alloy, and the L10-FeNi formation mechanism has been studied. It is found the L10-FeNi in annealed samples at 400 °C mainly originated from the residual amorphous phase during the second stage of crystallization which could take place over 60 °C lower than the measured onset temperature of the second stage with a 5 °C/min heating rate. Annealing at 400 °C after fully crystallization still caused a slight increase of coercivity, which was probably contributed by the limited transformation from other high temperature crystalline phases towards L10 phase, or the removal of B from L10 lattice and improvement of the ordering quality of L10 phase due to the reduced temperature from 520 °C to 400 °C. The first stage of crystallization has hardly direct contribution to L10-FeNi formation. Ab initio simulations show that the addition of Si or Co in L10-FeNi has the effect of enhancing the thermal stability of L10 phase without seriously deteriorating its magnetic hardness. The non-monotonic feature of direction dependent coercivity in ribbon segments resulted from the combination of domain wall pinning and demagnetization effects. The approaches of synthesizing L10-FeNi magnets by adding Si or Co and decreasing the onset crystallization temperature have been discussed in detail.

Effects of pressure on propagation characteristics of methane-air edge flames within two-dimensional mixing layers: A numerical study

The effects of pressure on propagation of laminar edge flames were numerically investigated using an OpenFOAM-6 platform developed with multicomponent transport properties. With various equivalence ratio gradients (∇ϕ) as inlet conditions, propagation of non-stationary edge flames in two-dimensional mixing layers at three pressures (0.5 bar, 1.0 bar and 2.0 bar) were studied. The heat release rate increases with increasing pressure and decreases with increasing ∇ϕ due to flame curvature and stretch. The linearity between flame curvature K and ∇ϕ is revealed. The K and slope of this linearity are both larger under high pressure. Flame stretch is dominated by flow strain and flame curvature rather than unsteady flame motion. With increasing ∇ϕ and pressure, stretch rate κ shows increasing feature due to larger curvature and stronger flow strain. The obvious negative linear dependence of local flame speed Sd on κ is revealed under three pressures. The range of Karlovitz number under three pressures are 0.2~1.4, indicating that edge flames are weakly stretched and the linear correlation between Sd and κ could be explained by weakly stretched flame theory. Compared with positive dependence of mass diffusion term Sdd on K, the dominating negative dependence of reaction term Sdr on K leads to negative correlation of Sd with K. Global flame speed UF shows non-monotonic feature with increasing ∇ϕ under three pressures, which is mainly due to non-monotonic upstream velocity reduction Udrop. Flame stretch is important for the shift from critical gradient ∇ϕcd (for maximum Udrop) to smaller ∇ϕcF (for maximum UF) and stronger decreasing feature of UF thanUdrop under large ∇ϕ.

Enhanced Heterogeneous Diffusion of Nanoparticles in Semiflexible Networks.

The transport of nanoparticles in semiflexible networks, which form diverse principal structural components throughout living systems, is important in biology and biomedical applications. By combining large-scale molecular simulations as well as theoretical analysis, we demonstrate here that nanoparticles in polymer networks with semiflexible strands possess enhanced heterogeneous diffusion characterized by more evident hopping dynamics. Particularly, the hopping energy barrier approximates to linear dependence on confinement parameters in the regime of moderate rigidity, in contrast to the quadratic dependence of both its soft and hard counterparts. This nonmonotonic feature can be attributed to the competition between the conformation entropy and the bending energy regulated by the chain rigidity, captured by developing an analytical model of a hopping energy barrier. Moreover, these theoretical results agree reasonably well with previous experiments. The findings bear significance in unraveling the fundamental physics of substance transport confined in network-topological environments and would provide an explanation for the dynamics diversity of nanoparticles within various networks, biological or synthetic.

Incremental approaches for heterogeneous feature selection in dynamic ordered data

Feature selection can identify essential features and reduce the dimensionality of features, improving the classification ability of a learning model. In this study, we consider data with a preference-order relation, i.e., ordered data. In the big data era, ordered data contain noise and exhibit heterogeneous features (including numerical and categorical features) and dynamic characteristics (i.e., new objects are added and obsolete objects are removed with evolving time). The dominance-based neighborhood rough set (DNRS) considers the preference order relation of heterogeneous features and demonstrates fault tolerance; thus, it can be applied well to heterogeneous feature selection in ordered data. At present, DNRS-based heterogeneous feature selection methods are only used for static ordered data. For dynamic ordered data, existing heterogeneous feature selection approaches are highly time-consuming because they are required to recalculate knowledge from scratch when multiple objects vary. Motivated by this issue, we utilize a matrix-based method in this work to study incremental heterogeneous feature selection based on DNRS in dynamic ordered data. First, we define neighborhood dominance conditional entropy (NDCE) as the uncertainty measure and introduce a non-monotonic feature selection strategy based on this measure. Second, the neighborhood dominance relation matrix and its diagonal matrix are defined to calculate NDCE in matrix form. Third, the updating mechanisms of the diagonal matrix are studied when objects vary and used to update NDCE. Lastly, two incremental feature selection algorithms are proposed when multiple objects are added to or deleted from heterogeneous ordered data. Experiments are performed on public data sets. Experimental results verify that the proposed incremental algorithms are effective and efficient for updating feature subsets in dynamic heterogeneous ordered data.

Nonmonotonic traveling waves in an unsaturated medium induced by a solid force and fluid mass sources

The traveling wave responses in an unsaturated medium induced by a solid force and fluid mass sources are derived in 3D Green's functions using the Dirac Delta function decompositions. The wave behaviors are described in pore fluid pressures and solid frame displacement along with fluctuations of capillary pressure in response to time variant fluid injections and solid force loadings. These solutions are numerically calculated and analyzed for unconsolidated sands as a function of water saturation as representative examples. Our results reveal and validate four wave modes, a fast compressional wave (P1), two slow compressional waves (P2 and P3), and a shear wave. The P1 wave and shear wave can be effectively observed. The slow waves with diffusive characters bear low amplitudes and low velocities. The traveling waves in the pore fluid pressures and solid displacement are sensitive to the different sources and the water saturation. The solid force induces the total stress perturbation while the fluid mass sources directly make the volume strains in the fluid phases. This fluid-solid coupling difference causes the distinct traveling wave signature contrasts in the pore-fluid and solid-frame response amplitudes and tendencies with water retention. For the variation of water saturation, the traveling wave shows a nonmonotonic feature. The fluctuation of capillary pressure exhibits a concave upward relationship, which can further interpret the changes of wave amplitude with water saturation. A solid force may enhance the fluctuation. However, for high water saturation, the interactive fluid-solid effect by the solid force may break the fluid contact due to the extremely compressible air.

Cellular Uptake of Active Particles.

Active particles are widely recognized to potentially revolutionize technologies in numerous biomedical applications. However, the physical origin behind cellular uptake of these particles in the nonequilibrium state remains scarcely understood. Here we combine Brownian dynamics simulation as well as theoretical analysis to provide the criterion for cellular uptake of active particles, related to various physical attributes. Upon enhancing the activity, the uptake efficiency for the active particles with tilted orientation is examined to be nonmonotonic, in stark contrast to the monotonic dependence for active particles orientated normally to the membrane. This can be attributed to the interplay between membrane adhesion energy and kinetic energy of active particles, resulting in unique kinetic pathways. Furthermore, a theoretical model that captures the essential physics of the cellular endocytosis process is developed to reproduce this nonmonotonic feature. The results are of immediate interest to understand and tune activity-mediated cellular interaction and internalization of such emerging colloids.

Runaway electron beam dynamics at low plasma density in DIII-D: energy distribution, current profile, and internal instability

Parameters of the post-disruption runaway electron (RE) beam in the low density background plasma achieved after secondary deuterium injection are investigated in DIII-D. The spatially resolved RE energy distribution function is measured for the first time during the RE plateau stage by inverting hard x-ray bremsstrahlung spectra. It has maximum energy up to 20 MeV and a non-monotonic feature at 5–6 MeV observed only in the core of the beam supporting the possibility of kinetic instabilities. Results of Fokker-Plank modelling qualitatively support the formation of the non-monotonic distribution function. The RE current profile is reconstructed for the first time using the spatially resolved RE energy distribution. It is found to be more peaked than the pre-disruption plasma current, with higher internal inductance, suggesting preferential formation of REs in the core plasma or potentially a radially inward motion of REs. The accessed relatively low current (180 kA) RE beam is found to be MHD stable, likely due to its elevated safety factor profile. From this base stable equilibrium, an internal beam instability is accessed by ramping up the current. The instability leads to a sawtooth-like relaxation of the RE current profile, but drives no RE loss. An internal kink mode proposed as a candidate instability is supported by results of MARS-F modelling. Electron cyclotron emission (ECE) spectrum measured during the low density RE plateau is found to be bifurcated, with a break point at ≈ 100 GHz, suggesting resonant absorption of the ECE at low frequencies.

Effective feature reduction for link prediction in location-based social networks

In this study, we investigated feature-based approaches for improving the link prediction performance for location-based social networks (LBSNs) and analysed their performances. We developed new features based on time, common friend detail and place category information of check-in data in order to make use of information in the data which cannot be utilised by the existing features from the literature. We proposed a feature selection method to determine a feature subset that enhances the prediction performance with the removal of redundant features by clustering them. After clustering features, a genetic algorithm is used to determine the ones to select from each cluster. A non-monotonic and feasible feature selection is ensured by the proposed genetic algorithm. Results depict that both new features and the proposed feature selection method improved link prediction performance for LBSNs.

The detailed shapes of equal-loudness-level contours at low frequencies.

High-resolution equal-loudness-level contours (ELCs) were measured over the frequency range 10-250 Hz using 19 normal-hearing subjects. Three levels of the 50-Hz reference sound were used, corresponding to the levels at 50 Hz of the 30-, 50-, and 70-phon standardized ELCs given in ISO-226:2003. The dynamic range of the contours generally decreased with increasing reference level, and the slope was shallow between 10 and 20 Hz, consistent with previous studies. For the lowest level, the ELCs were sometimes but not always smooth and on average followed the standardized 30-phon contour for frequencies above 40 Hz. For the two higher levels, the individual ELCs showed a distinct non-monotonic feature in a "transition region" between about 40 and 100 Hz, where the slope could reach near-zero or even positive values. The pattern of the non-monotonic feature was similar across levels for the subjects for whom it was observed, but the pattern varied across subjects. Below 40 Hz, the slopes of the ELCs increased markedly for all loudness levels, and the levels exceeded those of the standardized ELCs. Systematic deviations from the standardized ELCs were largest for frequencies below 40 Hz for all levels and within the transition region for the two higher levels.