skip to main content


Title: Growth of Sobolev norms and loss of regularity in transport equations
We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data ρ ¯ ∈ H loc 1 ( R d ) , d ≥ 2 , we construct a divergence-free advecting velocity field v (depending on ρ ¯ ) for which the unique weak solution to the transport equation does not belong to H loc 1 ( R d ) for any positive time. The velocity field v is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space W s , p that does not embed into the Lipschitz class. The velocity field v is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE , 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.  more » « less
Award ID(s):
1909103 2108080 1814147 2043024 2124748
NSF-PAR ID:
10338847
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume:
380
Issue:
2225
ISSN:
1364-503X
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ABSTRACT

    Strongly lensed quasars can provide measurements of the Hubble constant (H0) independent of any other methods. One of the key ingredients is exquisite high-resolution imaging data, such as Hubble Space Telescope (HST) imaging and adaptive-optics (AO) imaging from ground-based telescopes, which provide strong constraints on the mass distribution of the lensing galaxy. In this work, we expand on the previous analysis of three time-delay lenses with AO imaging (RX J1131−1231, HE 0435−1223, and PG 1115+080), and perform a joint analysis of J0924+0219 by using AO imaging from the Keck telescope, obtained as part of the Strong lensing at High Angular Resolution Program (SHARP) AO effort, with HST imaging to constrain the mass distribution of the lensing galaxy. Under the assumption of a flat Λ cold dark matter (ΛCDM) model with fixed Ωm = 0.3, we show that by marginalizing over two different kinds of mass models (power-law and composite models) and their transformed mass profiles via a mass-sheet transformation, we obtain $\Delta t_{\rm BA}=6.89\substack{+0.8\\-0.7}\, h^{-1}\hat{\sigma }_{v}^{2}$ d, $\Delta t_{\rm CA}=10.7\substack{+1.6\\-1.2}\, h^{-1}\hat{\sigma }_{v}^{2}$ d, and $\Delta t_{\rm DA}=7.70\substack{+1.0\\-0.9}\, h^{-1}\hat{\sigma }_{v}^{2}$ d, where $h=H_{0}/100\,\rm km\, s^{-1}\, Mpc^{-1}$ is the dimensionless Hubble constant and $\hat{\sigma }_{v}=\sigma ^{\rm ob}_{v}/(280\,\rm km\, s^{-1})$ is the scaled dimensionless velocity dispersion. Future measurements of time delays with 10 per cent uncertainty and velocity dispersion with 5 per cent uncertainty would yield a H0 constraint of ∼15 per cent precision.

     
    more » « less
  2. Obeid, Iyad Selesnick (Ed.)
    The Temple University Hospital EEG Corpus (TUEG) [1] is the largest publicly available EEG corpus of its type and currently has over 5,000 subscribers (we currently average 35 new subscribers a week). Several valuable subsets of this corpus have been developed including the Temple University Hospital EEG Seizure Corpus (TUSZ) [2] and the Temple University Hospital EEG Artifact Corpus (TUAR) [3]. TUSZ contains manually annotated seizure events and has been widely used to develop seizure detection and prediction technology [4]. TUAR contains manually annotated artifacts and has been used to improve machine learning performance on seizure detection tasks [5]. In this poster, we will discuss recent improvements made to both corpora that are creating opportunities to improve machine learning performance. Two major concerns that were raised when v1.5.2 of TUSZ was released for the Neureka 2020 Epilepsy Challenge were: (1) the subjects contained in the training, development (validation) and blind evaluation sets were not mutually exclusive, and (2) high frequency seizures were not accurately annotated in all files. Regarding (1), there were 50 subjects in dev, 50 subjects in eval, and 592 subjects in train. There was one subject common to dev and eval, five subjects common to dev and train, and 13 subjects common between eval and train. Though this does not substantially influence performance for the current generation of technology, it could be a problem down the line as technology improves. Therefore, we have rebuilt the partitions of the data so that this overlap was removed. This required augmenting the evaluation and development data sets with new subjects that had not been previously annotated so that the size of these subsets remained approximately the same. Since these annotations were done by a new group of annotators, special care was taken to make sure the new annotators followed the same practices as the previous generations of annotators. Part of our quality control process was to have the new annotators review all previous annotations. This rigorous training coupled with a strict quality control process where annotators review a significant amount of each other’s work ensured that there is high interrater agreement between the two groups (kappa statistic greater than 0.8) [6]. In the process of reviewing this data, we also decided to split long files into a series of smaller segments to facilitate processing of the data. Some subscribers found it difficult to process long files using Python code, which tends to be very memory intensive. We also found it inefficient to manipulate these long files in our annotation tool. In this release, the maximum duration of any single file is limited to 60 mins. This increased the number of edf files in the dev set from 1012 to 1832. Regarding (2), as part of discussions of several issues raised by a few subscribers, we discovered some files only had low frequency epileptiform events annotated (defined as events that ranged in frequency from 2.5 Hz to 3 Hz), while others had events annotated that contained significant frequency content above 3 Hz. Though there were not many files that had this type of activity, it was enough of a concern to necessitate reviewing the entire corpus. An example of an epileptiform seizure event with frequency content higher than 3 Hz is shown in Figure 1. Annotating these additional events slightly increased the number of seizure events. In v1.5.2, there were 673 seizures, while in v1.5.3 there are 1239 events. One of the fertile areas for technology improvements is artifact reduction. Artifacts and slowing constitute the two major error modalities in seizure detection [3]. This was a major reason we developed TUAR. It can be used to evaluate artifact detection and suppression technology as well as multimodal background models that explicitly model artifacts. An issue with TUAR was the practicality of the annotation tags used when there are multiple simultaneous events. An example of such an event is shown in Figure 2. In this section of the file, there is an overlap of eye movement, electrode artifact, and muscle artifact events. We previously annotated such events using a convention that included annotating background along with any artifact that is present. The artifacts present would either be annotated with a single tag (e.g., MUSC) or a coupled artifact tag (e.g., MUSC+ELEC). When multiple channels have background, the tags become crowded and difficult to identify. This is one reason we now support a hierarchical annotation format using XML – annotations can be arbitrarily complex and support overlaps in time. Our annotators also reviewed specific eye movement artifacts (e.g., eye flutter, eyeblinks). Eye movements are often mistaken as seizures due to their similar morphology [7][8]. We have improved our understanding of ocular events and it has allowed us to annotate artifacts in the corpus more carefully. In this poster, we will present statistics on the newest releases of these corpora and discuss the impact these improvements have had on machine learning research. We will compare TUSZ v1.5.3 and TUAR v2.0.0 with previous versions of these corpora. We will release v1.5.3 of TUSZ and v2.0.0 of TUAR in Fall 2021 prior to the symposium. ACKNOWLEDGMENTS Research reported in this publication was most recently supported by the National Science Foundation’s Industrial Innovation and Partnerships (IIP) Research Experience for Undergraduates award number 1827565. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the official views of any of these organizations. REFERENCES [1] I. Obeid and J. Picone, “The Temple University Hospital EEG Data Corpus,” in Augmentation of Brain Function: Facts, Fiction and Controversy. Volume I: Brain-Machine Interfaces, 1st ed., vol. 10, M. A. Lebedev, Ed. Lausanne, Switzerland: Frontiers Media S.A., 2016, pp. 394 398. https://doi.org/10.3389/fnins.2016.00196. [2] V. Shah et al., “The Temple University Hospital Seizure Detection Corpus,” Frontiers in Neuroinformatics, vol. 12, pp. 1–6, 2018. https://doi.org/10.3389/fninf.2018.00083. [3] A. Hamid et, al., “The Temple University Artifact Corpus: An Annotated Corpus of EEG Artifacts.” in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2020, pp. 1-3. https://ieeexplore.ieee.org/document/9353647. [4] Y. Roy, R. Iskander, and J. Picone, “The NeurekaTM 2020 Epilepsy Challenge,” NeuroTechX, 2020. [Online]. Available: https://neureka-challenge.com/. [Accessed: 01-Dec-2021]. [5] S. Rahman, A. Hamid, D. Ochal, I. Obeid, and J. Picone, “Improving the Quality of the TUSZ Corpus,” in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2020, pp. 1–5. https://ieeexplore.ieee.org/document/9353635. [6] V. Shah, E. von Weltin, T. Ahsan, I. Obeid, and J. Picone, “On the Use of Non-Experts for Generation of High-Quality Annotations of Seizure Events,” Available: https://www.isip.picone press.com/publications/unpublished/journals/2019/elsevier_cn/ira. [Accessed: 01-Dec-2021]. [7] D. Ochal, S. Rahman, S. Ferrell, T. Elseify, I. Obeid, and J. Picone, “The Temple University Hospital EEG Corpus: Annotation Guidelines,” Philadelphia, Pennsylvania, USA, 2020. https://www.isip.piconepress.com/publications/reports/2020/tuh_eeg/annotations/. [8] D. Strayhorn, “The Atlas of Adult Electroencephalography,” EEG Atlas Online, 2014. [Online]. Availabl 
    more » « less
  3. Abstract

    We present extensive multifrequency Karl G. Jansky Very Large Array (VLA) and Very Long Baseline Array (VLBA) observations of the radio-bright supernova (SN) IIb SN 2004C that span ∼40–2793 days post-explosion. We interpret the temporal evolution of the radio spectral energy distribution in the context of synchrotron self-absorbed emission from the explosion’s forward shock as it expands in the circumstellar medium (CSM) previously sculpted by the mass-loss history of the stellar progenitor. VLBA observations and modeling of the VLA data point to a blastwave with average velocity ∼0.06cthat carries an energy of ≈1049erg. Our modeling further reveals a flat CSM density profileρCSMR−0.03±0.22up to a break radiusRbr≈ (1.96 ± 0.10) × 1016cm, with a steep density gradient followingρCSMR−2.3±0.5at larger radii. We infer that the flat part of the density profile corresponds to a CSM shell with mass ∼0.021M, and that the progenitor’s effective mass-loss rate varied with time over the range (50–500) × 10−5Myr−1for an adopted wind velocityvw= 1000 km s−1and shock microphysical parametersϵe= 0.1,ϵB= 0.01. These results add to the mounting observational evidence for departures from the traditional single-wind mass-loss scenarios in evolved, massive stars in the centuries leading up to core collapse. Potentially viable scenarios include mass loss powered by gravity waves and/or interaction with a binary companion.

     
    more » « less
  4. Abstract In this paper, we carry out a pilot parameter exploration for the collision-induced magnetic reconnection (CMR) mechanism that forms filamentary molecular clouds. Following Kong et al., we utilize Athena++ to model CMR in the context of resistive magnetohydrodynamics (MHD), considering the effect from seven physical conditions, including the ohmic resistivity ( η ), the magnetic field ( B ), the cloud density ( ρ ), the cloud radius R , the isothermal temperature T , the collision velocity v x , and the shear velocity v z . Compared to their fiducial model, we consider a higher and a lower value for each one of the seven parameters. We quantify the exploration results with five metrics, including the density probability distribution function ( ρ -PDF), the filament morphology (250 μ m dust emission), the B – ρ relation, the dominant fiber width, and the ringiness that describes the significance of the ringlike substructures. The exploration forms straight and curved CMR filaments with rich substructures that are highly variable in space and time. The variation translates to fluctuation in all five metrics, reflecting the chaotic nature of magnetic reconnection in CMR. A temporary B ∝ ρ relation is noticeable during the first 0.6 Myr. Overall, the exploration provides useful initial insights into the CMR mechanism. 
    more » « less
  5. Abstract

    We present a chemodynamical study of the Grus I ultra-faint dwarf galaxy (UFD) from medium-resolution (R∼ 11,000) Magellan/IMACS spectra of its individual member stars. We identify eight confirmed members of Grus I, based on their low metallicities and coherent radial velocities, and four candidate members for which only velocities are derived. In contrast to previous work, we find that Grus I has a very low mean metallicity of 〈[Fe/H]〉 = −2.62 ± 0.11 dex, making it one of the most metal-poor UFDs. Grus I has a systemic radial velocity of −143.5 ± 1.2 km s−1and a velocity dispersion ofσrv=2.50.8+1.3km s−1, which results in a dynamical mass ofM1/2(rh)=84+12×105Mand a mass-to-light ratio ofM/LV=440250+650M/L. Under the assumption of dynamical equilibrium, our analysis confirms that Grus I is a dark-matter-dominated UFD (M/L> 80M/L). However, we do not resolve a metallicity dispersion (σ[Fe/H]< 0.44 dex). Our results indicate that Grus I is a fairly typical UFD with parameters that agree with mass–metallicity and metallicity-luminosity trends for faint galaxies. This agreement suggests that Grus I has not lost an especially significant amount of mass from tidal encounters with the Milky Way, in line with its orbital parameters. Intriguingly, Grus I has among the lowest central densities (ρ1/23.52.1+5.7×107Mkpc−3) of the UFDs that are not known to be tidally disrupting. Models of the formation and evolution of UFDs will need to explain the diversity of these central densities, in addition to any diversity in the outer regions of these relic galaxies.

     
    more » « less