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At least 199 records · Page 11
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Defect-induced displacement of topological surface state in quantum magnet MnBi2Te4

The topological magnet MnBi2⁢Te4 (MBT), with gapped topological surface state, is an attractive platform for realizing quantum anomalous Hall and axion insulator states. However, the experimentally observed surface state gaps fail to meet theoretical predictions, although the exact mechanism behind the gap suppression has been debated. Recent theoretical studies suggest that intrinsic antisite defects push the topological surface state away from the MBT surface, closing its gap and making it less accessible to scanning probe experiments. Here, we report on the local effect of defects on the MBT surface states and demonstrate that high defect concentrations lead to a displacement of the surface states well into the MBT crystal, validating the theorized mechanism. The local and global influence of antisite defects on the topological surface states are studied with samples of varying defect densities by combining scanning tunneling microscopy, angle-resolved photoemission spectroscopy, and density functional theory. Our findings identify a combination of increased defect density and reduced defect spacing as the primary factors underlying the displacement of the surface states and suppression of surface gap, guiding further development of topological quantum materials.

Lupke, Felix [Carnegie Mellon University (CMU)]↗
NASA NTRS1989

Barotropic instability of basic states with a realistic jet and a wave

The stability of basic states consisting of a jet similar to the stratospheric polar night jet and a traveling wave with a single zonal wavenumber is examined in a linearized nondivergent barotropic model on a sphere. Basic state waves are chosen to resemble observed traveling and stationary features in the winter stratosphere. Results are presented for disturbance growth rates, propagation characteristics, and energy conversion as a function of the basic state wave amplitude. The effects of small amplitude basic state waves on unstable disturbances arising from a zonally symmetric jet are discussed; results are shown where a small amplitude basic state wave dramatically affects the stability characteristics. Evidence is shown that the presence of a traveling wave may favor the appearance of disturbances that include other zonal wavenumbers which move with the basic state wave; this result is discussed in relation to the origin of observed quasi-nondispersive features in the polar winter stratosphere. Results for a stationary wavenumber 1 basic state wave suggest that a distorted polar vortex may be unstable to disturbances that would lead to further distortion. An unstable disturbance for a basic state with an eastward moving wavenumber 2 has components which resemble, in period and location, traveling waves that are observed in the winter stratosphere.

Manney, Gloria L.↗
NASA NTRS1991

A theoretical study of the low-lying states of Ti2 and Zr2

The low-lying states of Ti2 and the valence isoelectronic Zr2 are examined theoretically by means of a multireference configuration-interaction (MRCI) method. MRCI calculations demonstrate that two of the Zr2 states are very low-lying and that the resulting vertical excitation is consistent with the optical spectrum of Zr2. The ground state is predicted for Ti2 on the basis of valence correlation with the MRCI method and the average coupled-pair functional technique. Calculations of the inner-shell correlation effects are estimated and found to lower the 3Delta g state to a ground state, and another to a very low-lying state. The ground state of Ti2 is assigned to 3Delta g since it is lower than the other state at all levels of correlation and is derived from the same atomic asymptote. This conclusion is supported by the lack of an electron-spin resonance signal but contradicts the absence of subcomponents on the Raman spectral lines.

Bauschlicher, Charles W., Jr.↗
NASA NTRS1992

Performance and state-space analyses of systems using Petri nets

The goal of any modeling methodology is to develop a mathematical description of a system that is accurate in its representation and also permits analysis of structural and/or performance properties. Inherently, trade-offs exist between the level detail in the model and the ease with which analysis can be performed. Petri nets (PN's), a highly graphical modeling methodology for Discrete Event Dynamic Systems, permit representation of shared resources, finite capacities, conflict, synchronization, concurrency, and timing between state changes. By restricting the state transition time delays to the family of exponential density functions, Markov chain analysis of performance problems is possible. One major drawback of PN's is the tendency for the state-space to grow rapidly (exponential complexity) compared to increases in the PN constructs. It is the state space, or the Markov chain obtained from it, that is needed in the solution of many problems. The theory of state-space size estimation for PN's is introduced. The problem of state-space size estimation is defined, its complexities are examined, and estimation algorithms are developed. Both top-down and bottom-up approaches are pursued, and the advantages and disadvantages of each are described. Additionally, the author's research in non-exponential transition modeling for PN's is discussed. An algorithm for approximating non-exponential transitions is developed. Since only basic PN constructs are used in the approximation, theory already developed for PN's remains applicable. Comparison to results from entropy theory show the transition performance is close to the theoretic optimum. Inclusion of non-exponential transition approximations improves performance results at the expense of increased state-space size. The state-space size estimation theory provides insight and algorithms for evaluating this trade-off.

Watson, James Francis, III↗
NASA NTRS1995

Low-state disks and low-beta disks

Stellar black hole candidates (BHCs) exhibit bimodal spectral states. We calculate nonthermal disk spectra, demonstrating that a large photon index (alpha (sub x) approximately 2-3) observed in the soft (high) state is due to a copious soft photon supply, whereas soft photon starvation leads to a smaller index (alpha (sub x) approximately 1.5-2) in the hard (low) state. Thus, the absence of the soft component flux in the low state cannot be due to obscuration. A possible disk configuration during the low state is discussed. We proposed that a low-state disk may be a low-beta disk in which magnetic pressure may exceed gas pressure becuase of the suppression of field escape by a strong shear. As a result, disk material will take the form of blobs constricted by mainly toroidal magnetic fields. Fields are dissipated mainly by occasional reconnection events with a huge energy release. This will account for large-amplitude, aperiodic X-ray variations (flickering) and high-energy radiation with small alpha(sub x) from hard state BHCs and possibly from active galactic nuclei. Further, we propose a hysteretic relation between the mass-flow rate and plasma-beta, a ratio of gas pressure to magnetic pressure, for the spectral evolution of transient BHCs. The disk is in the low-beta state in quiescence and early rise. The low-beta disk is optically thin and affected by advection. A hard-to-soft transition occurs before the peak luminosity, since there is no advection-dominated branch at higher luminosities. An optically thick, high-beta disk appears at small radii. In the decay phase of the light curve, the standard-type disk becomes effectively optically thin, when a soft-hard transition is triggered. High-beta plasmas in the main body shrink to form minute blobs, and low-beta coronal plasma fills interblob space.

Mineshige, Shin↗
NASA NTRS1999

Quasi-Periodic Oscillations in Black Hole Candidates as an Indicator of Transition Between Low and High States

By comparing positions on a spectral color-color diagram from 10 black hole candidates (BHCS) observed with Ginga (1354-64, 1826-24, 1630-47, LMC X-1, LMC X-3, GS 2000+25, GS 2023+33, GS 1124-68, Cyg X-1, and GX 339-4) with the observed broadband noise (BBN) (0.001-64 Hz) and quasi-periodic oscillation (QPO) variability, we find that the "very high state" is spectrally intermediate to the soft/high state and hard/low state. We find a transition point in spectral hardness where the dependence of the BHC QPO centroid frequency (of GS 1124-68 and GX 339-4) on spectral hardness switches from a correlation to an anticorrelation; where the BBN variability switches from high state to low state; and where the spectral hardness of the QPO relative to that of the BBN variability is a maximum. This coincidence of changing behavior in both the QPO and the broadband variability leads us to hypothesize that the QPO is due to interaction between the physical components which dominate the behaviors of BHCs when they occupy the hard/low and soft/high states. We conclude that these QPOs should be observed from BHCs during transition between these two states. Comparison with QPO and BBN behavior observed during the 1996 transition of Cyg X-1 supports this hypothesis. We also report 1-3 Hz QPOs observed in GS 2000+25 and Cyg X-1 in the hard/low state, and we compare these to the QPOs observed in GS 1124-68 and GX 339-4.

Rutledge, Robert E.↗
NASA NTRS1995

The Low-Lying Electronic States of LiB

The spectroscopic constants for the triplet and singlet states of LiB below about 30 000/ cm are determined using an internally contracted multireference configuration interaction approach in conjunction with [6s 5p 3d 2f] atomic natural orbital basis sets. The ground state is (sup 3)Pi as found in previous work. No excited triplet states are found to be ideal for characterizing the ground state; the (1)(sup 3)Sigma(sup -) state has a transition energy that is too small for many experimental approaches and the (2)(sup 3)Pi and (3)(sup 3)Pi states have bond lengths that are significantly longer than the ground state, resulting in transition intensities that are spread out over many vibrational levels of the ground state.

Ricca, Alessandra↗
NASA NTRS2003

SNC Oxygen Fugacity Recorded in Pyroxenes and its Implications for the Oxidation State of the Martian Interior: An Experimental and Analytical Study

Knowledge of the oxidation state of a magma is critical as it is one of the parameters which controls the nature and composition of the resulting crystals. In terrestrial magmatic systems, oxygen fugacity (fo2) is known to vary by over nine orders of magnitude. With variations of this magnitude, understanding the compositional differences, phase changes, and crystallization sequence variations, caused by the magma fo2, is essential in deciphering the origin of all igneous rocks. Magmatic oxidation state is of great importance in that it reflects the degree of oxidation of the source region and can provide insight into magmatic processes, such as metasomatism, degassing, and assimilation, which may have changed them. Carmichael [1991] argues that most magmas are unlikely to have their redox states altered from those of their source region. This assumption allows for estimation of the oxidation state of planetary interiors. Conversely, it is known that the fo2 of the magma can be affected by other processes, which occur outside of the source region and therefore, the oxidation state may record those too. Processes which could overprint source region fugacities include melt dehydrogenation or other volatile loss, water or melt infiltration, or assimilation of oxidized or reduced wallrock. Understanding which of these processes is responsible for the redox state of a magma can provide crucial information regarding igneous processes and other forces active in the region. The composition of the SNC basalts and their widely varying proposed oxidation states raise some interesting questions. Do the SNC meteorites have an oxidized or reduced signature? What was the oxygen fugacity of the SNC source region at the time of melt generation? Is the fugacity calculated for the various SNC samples the fugacity of the magma source region or was it overprinted by later events? Are there different oxidation states in the Martian interior or a single one? This proposal seeks to address all of these questions.

McCanta, M. C.↗
NASA NTRS2001

State-space receptive fields of semicircular canal afferent neurons in the bullfrog

Receptive fields are commonly used to describe spatial characteristics of sensory neuron responses. They can be extended to characterize temporal or dynamical aspects by mapping neural responses in dynamical state spaces. The state-space receptive field of a neuron is the probability distribution of the dynamical state of the stimulus-generating system conditioned upon the occurrence of a spike. We have computed state-space receptive fields for semicircular canal afferent neurons in the bullfrog (Rana catesbeiana). We recorded spike times during broad-band Gaussian noise rotational velocity stimuli, computed the frequency distribution of head states at spike times, and normalized these to obtain conditional pdfs for the state. These state-space receptive fields quantify what the brain can deduce about the dynamical state of the head when a single spike arrives from the periphery. c2001 Elsevier Science B.V. All rights reserved.

Non-NASA Center↗
NASA NTRS2011

The Precipitation Characteristics of ISCCP Tropical Weather States

We examine the daytime precipitation characteristics of the International Satellite Cloud Climatology Project (ISCCP) weather states in the extended tropics (35 deg S to 35 deg N) for a 10-year period. Our main precipitation data set is the TRMM Multisatellite Precipitation Analysis 3B42 data set, but Global Precipitation Climatology Project daily data are also used for comparison. We find that the most convective weather state (WS1), despite an occurrence frequency below 10%, is the most dominant state with regard to surface precipitation, producing both the largest mean precipitation rates when present and the largest percent contribution to the total precipitation of the tropical zone of our study; yet, even this weather state appears to not precipitate about half the time. WS1 exhibits a modest annual cycle of domain-average precipitation rate, but notable seasonal shifts in its geographic distribution. The precipitation rates of the other weather states tend to be stronger when occurring before or after WS1. The relative contribution of the various weather states to total precipitation is different between ocean and land, with WS1 producing more intense precipitation on average over ocean than land. The results of this study, in addition to advancing our understanding of the current state of tropical precipitation, can serve as a higher order diagnostic test on whether it is distributed realistically among different weather states in atmospheric models.

Lee, Dongmin↗
NASA NTRS2012

Using ISCCP Weather States to Decompose Cloud Radiative Effects

The presentation will examine the shortwave (SW) and longwave (LW) cloud radiative effect CRE (aka "cloud radiative forcing") at the top-of-the-atmosphere and surface of ISCCP weather states (aka "cloud regimes") in three distinct geographical zones, one tropical and two mid-latitude. Our goal is to understand and quantify the contribution of the different cloud regimes to the planetary radiation budget. In the tropics we find that the three most convectively active states are the ones with largest SW, LW and net TOA CRE contributions to the overall daytime tropical CRE budget. They account for 59%, 71% and 55% of the total CRE, respectively. The boundary layer-dominated weather states account for only 34% of the total SW CRE and 41% of the total net CRE, so to focus only on them in cloud feedback studies may be imprudent. We also find that in both the northern and southern midlatitude zones only two weather states, the first and third most convectively active with large amounts of nimbostratus-type clouds, contribute ",40% to both the SW and net TOA CRE budgets, highlighting the fact that cloud regimes associated with frontal systems are not only important for weather (precipitation) but also for climate (radiation budget). While all cloud regimes in all geographical zones have a slightly larger SFC than TOA SW CRE, implying cooling of the surface and slight warming of the atmosphere, their LW radiative effects are more subtle: in the tropics the weather states with plentiful high clouds warm the atmosphere while those with copious amounts of low clouds cool the atmosphere. In both midlatitude zones only the weather states with peak cloud fractions at levels above 440 mbar warm the atmosphere while all the rest cool it. These results make the connection of the contrasting CRE effects to the atmospheric dynamics more explicit - "storms" tend to warm the atmosphere whereas fair weather clouds cool it, suggesting a positive feedback of clouds on weather systems. The breakdown of CRE by cloud regime are however not entirely similar between the two midlatitude zones. Despite the existence of an additional state in the nort!lern midlatitudes, only four weather states have net daytime CREs with absolute values above 100 Watts per square meter compared to six in the south. This reminds us that the environment where clouds occur also has a crucial role in determining their radiative effects. All the above make evident that reproducing grand averages of current CRE by climate models in only part of the challenge. If existing cloud regimes and shifts in their distributions and frequency of occurrence in a changed climate are not properly simulated, the radiative role of clouds will not be adequately predicted.

Oreopoulos, L.↗
NASA NTRS2016

Magnetospheric State of Sawtooth Events

Magnetospheric sawtooth events, first identified in the early 1990s, are named for their characteristic appearance of multiple quasiperiodic intervals of slow decrease followed by sharp increase of proton differential energy fluxes in the geosynchronous region. The successive proton flux oscillations have been interpreted as recurrences of stretching and dipolarization of the nightside geomagnetic field. Due to their often extended intervals with 210 cycles, sawteeth occurrences are sometimes referred to as a magnetospheric mode. While studies of sawtooth events over the past two decades have yielded a wealth of information about such events, the magnetospheric state conditions for the occurrence of sawtooth events and how sawtooth oscillations may depend on the magnetospheric state conditions remain unclear. In this study, we investigate the characteristic magnetospheric state conditions (specified by Psw interplanetary magnetic field (IMF) Btot, IMF Bz Vsw, AE, Kp and Dst, all time shifted with respect to one another) associated with the intervals before, during, and after sawteeth occurrences. Applying a previously developed statistical technique, we have determined the most probable magnetospheric states propitious for the development and occurrence of sawtooth events, respectively. The statistically determined sawtooth magnetospheric state has also been validated by using out-of-sample events, confirming the notion that sawtooth intervals might represent a particular global state of the magnetosphere. We propose that the sawtooth state of the magnetosphere may be a state of marginal stability in which a slight enhancement in the loading rate of an otherwise continuous loading process can send the magnetosphere into the marginally unstable regime, causing it to shed limited amount of energy quickly and return to the marginally stable regime with the loading process continuing. Sawtooth oscillations result as the magnetosphere switches between the marginally stable (loading) and unstable (unloading) phases.

Fung, Shing F.↗
NASA NTRS2019

System and Method for Training of State-Classifiers

Method and systems are disclosed for training state-classifiers for classification of cognitive state. A set of multimodal signals indicating physiological responses of an operator are sampled over a time period. A depiction of operation by the operator during the time period is displayed. In response to user input selecting a cognitive state for a portion of the time period, the one or more state-classifiers are trained. In training the state-classifiers, the set of multimodal signals sampled in the portion of the time period are used as input to the one or more state-classifiers and the selected one of the set of cognitive states is used as a target result to be indicated by the one or more state-classifiers.

Stephens, Chad L.↗
NASA NTRS2019

Psychophysiological Monitoring of Aerospace Crew State

As next-generation space exploration missions necessitate increasingly autonomous systems, there is a critical need to better detect and anticipate crewmember interactions with these systems. The success of present and future autonomous technology in exploration spaceflight is ultimately dependent upon safe and efficient interaction with the human operator. Optimal interaction is particularly important for surface missions during highly coordinated extravehicular activity (EVA), which consists of high physical and cognitive demands with limited ground support. Crew functional state may be affected by a number of variables including workload, stress, and motivation. Real-time assessments of crew state that do not require a crewmember’s time and attention to complete will be especially important to assess operational performance and behavioral health during flight. In response to the need for objective, passive assessment of crew state, the aim of this work is to develop an accurate and precise prediction model of human functional state for surface EVA using multi-modal psychophysiological sensing. The psychophysiological monitoring approach relies on extracting a set of features from physiological signals and using these features to classify an operator’s cognitive state. This work aims to compile a non-invasive sensor suite to collect physiological data in real-time. Training data during cognitive and more complex functional tasks will be used to develop a classifier to discriminate high and low cognitive workload crew states. The classifier will then be tested in an operationally relevant EVA simulation to predict cognitive workload over time. Once a crew state is determined, further research into specific countermeasures, such as decision support systems, would be necessary to optimize the automation and improve crew state and operational performance.

Wusk, Grace C.↗
NASA NTRSDate not supplied

NASA/GEWEX SRB Surface Radiation Budget Fluxes Through the Prism of Weather States

The NASA/GEWEX Surface Radiation Budget Version 4 (SRB Rel4) product is now publicly available, covering the period from July 1983 through June 2017. This supersedes the SRB Rel3 (Stackhouse et al., 2011) product which has been used in the community for a wide variety of applications, including climate model validation, agriculture, solar energy, and architecture. SRB Rel4 uses the newly recalibrated and processed ISCCP HXS product as its primary input for cloud and radiance data, replacing ISCCP DX with a ninefold increase in pixel count (10km instead of 30km). This version retains a 1°x1° resolution but benefits from a much larger number of samples per grid box than the SRB Rel3. ISCCP also provides an atmospheric temperature and moisture dataset known as nnHIRS which we use here, along with Seaflux and Landflux surface and near-surface meteorological parameters. Rel4 incorporates several important algorithm improvements. These include recalculated shortwave (SW) atmospheric transmissivities and reflectivities yielding a somewhat less transmissive atmosphere. Both shortwave and longwave (LW) now also include variable aerosol composition and radiative properties, allowing for the use of a detailed aerosol history from the Max Planck Institute Aerosol Climatology (MACv1). LW and SW algorithms now produce pristine sky fluxes, allowing the aerosol flux effects to be quantified. For SW, ocean albedo and snow/ice albedo are improved from Release 3. Total solar irradiance is now variable, and reduced to an average of 1361 Wm-2. The radiative treatment of ice cloud is improved. For the LW, a climatological monthly varying spectral surface emissivity is added. Here we evaluate the SRB Rel4 top of atmosphere (TOA) and surface fluxes in the context of weather states and oceanic regions. Weather states (Tselioudis et al., 2013, Tselioudis et al., 2021) are defined by K-means clustering 2-D histograms of satellite-retrieved cloud optical depths and cloud top pressures. The most recent (Tselioudis et al., 2021) weather states derived from the International Satellite Cloud Climatology Project (ISCCP) H-series data results in eight cloud weather states and one clear sky weather state, for a total of nine weather states. We examine the statistics of SRB Rel4 fluxes for each weather state and a variety of relevant land and ocean regions. We compare to similar analyses performed on CERES EBAF and SYN1Deg flux products, and validate against Baseline Surface Radiometer Network (BSRN) land stations and Pacific Marine Environmental Laboratory (PMEL) ocean buoy data. It is demonstrated that the weather state concept is valuable when examining the comparative strengths and weaknesses of satellite radiative flux algorithms.

Atmospheric Radiation↗
NASA NTRS2022

GBaTSv2: a revised synthesis of the likely basal thermal state of the Greenland Ice Sheet

The basal thermal state (frozen or thawed) of the Greenland Ice Sheet is under-constrained due to few direct measurements, yet knowledge of this state is becoming increasingly important to interpret modern changes in ice flow. The first synthesis of this state relied on inferences from widespread airborne and satellite observations and numerical models, for which most of the underlying datasets have since been updated. Further, new and independent constraints on the basal thermal state have been developed from analysis of basal and englacial reflections observed by airborne radar sounding. Here we synthesize constraints on the Greenland Ice Sheet's basal thermal state from boreholes, thermomechanical ice-flow models that participated in the Ice Sheet Model Intercomparison Project for CMIP6 (ISMIP6; Coupled Model Intercomparison Project Phase 6), IceBridge BedMachine Greenland v4 bed topography, Making Earth Science Data Records for Use in Research Environments (MEaSUREs) Multi-Year Greenland Ice Sheet Velocity Mosaic v1 and multiple inferences of a thawed bed from airborne radar sounding. Most constraints can only identify where the bed is likely thawed rather than where it is frozen. This revised synthesis of the Greenland likely Basal Thermal State version 2 (GBaTSv2) indicates that 33 % of the ice sheet's bed is likely thawed, 40 % is likely frozen and the remainder (28 %) is too uncertain to specify. The spatial pattern of GBaTSv2 is broadly similar to the previous synthesis, including a scalloped frozen core and thawed outlet-glacier systems. Although the likely basal thermal state of nearly half (46 %) of the ice sheet changed designation, the assigned state changed from likely frozen to likely thawed (or vice versa) for less than 6 % of the ice sheet. This revised synthesis suggests that more of northern Greenland is likely thawed at its bed and conversely that more of southern Greenland is likely frozen, both of which influence interpretation of the ice sheet's present subglacial hydrology and models of its future evolution. The GBaTSv2 dataset, including both code that performed the analysis and the resulting datasets, is freely available at https://doi.org/10.5281/zenodo.6759384 (MacGregor, 2022).

Joseph A. MacGregor↗
NASA NTRSDate not supplied

Preparing Quantum Many-Body Scar States on Quantum Computers

Quantum many-body scar states are highly excited eigenstates of many-body systems that exhibit atypical entanglement and correlation properties relative to other eigenstates at the same energy density. Scar states also give rise to infinitely long-lived coherent dynamics when the system is prepared in a special initial state having finite overlap with them. Many models with exact scar states have been constructed, but the fate of scarred eigenstates and dynamics when these models are perturbed is difficult to study with classical computational techniques. In this work, we propose state preparation protocols for individual scar states in a particular model, as well as superpositions of them that give rise to coherent dynamics. For superpositions of scar states, we propose both a linear depth unitary and a finite-depth nonunitary state preparation protocol, the latter of which uses measurement and postselection to reduce the circuit depth. For individual scarred eigenstates, we propose a circuit architecture with polynomial depth. We also provide proof of principle implementations of these protocols on superconducting quantum hardware.

Quantum Computing↗
DOE OSTI2025

Properties of states near 𝐸 𝑥 = 6 MeV in 18 Ne through 17 F +𝑝 scattering

The rate of energy production in the hot-CNO cycle and breakout to the rapid-proton capture process in Type I X-ray bursts is strongly related to the 14 O⁡(𝛼,𝑝)⁢ 17 F reaction rate. The properties of states in 18 Ne near 𝐸 𝑥 = 6.1–6.3 MeV are important for understanding the 14 O⁡(𝛼,𝑝)⁢ 17 F reaction rate. In order to study 18 Ne resonances around this energy region, the RESOLUT radioactive-ion beam facility at Florida State University was used to perform 17 F⁡(𝑝,𝑝)⁢ 17 F elastic scattering on a polypropylene target under inverse kinematics. Scattered protons were detected in a silicon-strip detector array while recoiling 17 F ions were detected in coincidence in a gas ionization detector. An 𝑅-matrix analysis of measured cross sections was conducted along with a reanalysis of data from previous 17 F +𝑝 measurements. All the data analyzed are well described by a consistent set of parameters with a 1 − assignment for a state at 6.14(1) MeV. A second comparable solution is also found with a 3 − assignment for the 6.14(1) MeV state. There is no conclusive evidence supporting one of the two solutions over the other. The rates of the 14 O⁡(𝛼,p)⁢ 17 F reaction that are determined from the two solutions differ by up to an order of magnitude.

6 ≤ A ≤ 19↗