Search

We study the spectral and temporal properties of the black hole X-ray transient binary MAXI J1820+070 during the 2018 outburst with Insight-HXMT observations. The outburst of MAXI J1820+070 can be divided into three intervals. For the two intervals of the outburst, we find that low-energy (below 140 keV) photons lag high-energy (140–170 keV) ones, while in the decay of the outburst, high-energy photons lag low-energy photons, both with a time-scale of the order of days. Based on these results, the canonical hysteresis effect of the ‘q’ shape in the hardness–intensity diagram can be reformed into a roughly linear shape by taking into account the lag corrections between different energy bands. Time analysis shows that the high-frequency break of hard X-rays, derived from the power-density spectrum of the first interval of the outburst, is in general larger and more variable than that of soft X-rays. The spectral fitting shows that the coverage fraction of the hard X-rays drops sharply at the beginning of the outburst to about 0.5, and then increases slightly. The coverage fraction drops to roughly zero once the source steps into soft state and increases gradually to unity when the source returns to low–hard state. We discuss the possible overall evolution scenario of corona hinted from these discoveries.

We studied the 2018 outburst of the black hole transient H 1743 − 322 with a series of Insight-HXMT, NICER, and NuSTAR observations, covering the 1–120 keV band. With our broad-band X-ray spectral modelling, we confirm that the source remained in the low/hard state throughout the month-long outburst, although it became marginally softer at peak flux. We detected Type-C quasi-periodic oscillations (QPOs) and followed the evolution of their properties. The QPO frequency increased from ∼0.1 to ∼0.4 Hz during the rising phase of the outburst and decreased again in the decline. Continuum X-ray flux, power-law photon index, QPO frequency, and QPO root-mean-square amplitude were positively correlated. The QPO amplitude was slightly higher in the soft X-ray band (typical values of 12–16 per cent, compared with 8–10 per cent in the hard band). Our spectral-timing results shed light on the initial rising phase in the low/hard state, which has rarely been monitored with such high cadence, time resolution, and broad-band coverage. Combining spectral and timing properties, we find that ‘failed’ (hard state only) and ‘successful’ outbursts follow the same initial evolutionary track, although the former class of outburst never reaches the threshold for a transition to softer (thermally dominated) accretion regimes.

Purpose To compare the outcomes of using an internal limiting membrane (ILM) flap and the conventional ILM peel technique for small- or medium-sized full-thickness macular hole (FTMH) repair. Design Retrospective, interventional case series. Methods Eyes with an FTMH ≤ 400 µm that underwent vitrectomy with either a single-layer inverted ILM flap (flap group, 55 eyes) or an ILM peel (peel group, 62 eyes) were enrolled. Best-corrected visual acuity (BCVA) and optical coherence tomography (OCT) measurements were obtained preoperatively and at 1, 3, 6, and 12 months postoperatively. Results Primary hole closure was achieved in 54 (98%) and 60 (97%) eyes in the flap and peel groups, respectively. The preoperative and postoperative 12-month BCVA were comparable between the groups but were significantly better in the flap than in the peel groups at 1 month (logMAR, mean ± SD, 0.83 ± 0.43 vs. 1.14 ± 0.50, P = 0.001), 3 months (0.58 ± 0.33 vs. 0.82 ± 0.43; P = 0.002), and 6 months (0.56 ± 0.32 vs. 0.72 ± 0.48; P = 0.028). In the flap group, foveal gliosis was less common than the peel group at 1 month (P = 0.030) and restored external limiting membrane and interdigitation zone was more common at 3 months (P = 0.046 and Conclusions Both the single-layer ILM flap and conventional ILM peel techniques closed FTMHs and improved vision. ILM flaps were associated with better visual outcomes up to 6 months postoperatively and should be considered in FTMHs ≤ 400 µm.

An incomplete dislocated intraocular lens (IOL) is often treated with IOL exchange because the IOL subluxates posteriorly during surgery and makes it difficult to fixate the IOL in situ. A trocar blade used for 23-gauge vitrectomy was used to lift and stabilize the IOL-capsular complex. The IOL can then be fixated using a suture loop fixation technique, which was originally limited to patients with decentered IOL. The advantage of this technique is that it allows the remaining zonular fibers and IOL to be preserved. The modified technique using a trocar blade to assist scleral fixation allows the incomplete dislocated IOL to be retrieved and fixated with a simplified surgical procedure.

We investigate systematically four outbursts of black hole system GX 339-4 observed by the Rossi X-ray Timing Explorer (RXTE) in both spectral and timing domains and find that these outbursts have some common properties although they experience different 'q' tracks in the hardness-intensity diagram (HID). While the spectral indices are around 1.5 in low hard state (LHS), 2.4 in soft intermediate state (SIMS) and high soft state (HSS), the spectral parameters of thermal, non-thermal and reflection components vary significantly in transitions from LHS to HIMS. Also the quasi periodic oscillation (QPO) shows a peculiar behavior during the state transition between LHS and HIMS: the RMS drop of type C fundamental QPO is accompanied with showing-up of the second harmonic. Interestingly, the QPO RMS is found to have a similar linear relationship with the non-thermal fraction of emission in different outbursts. These findings provide more clues to our understanding the outburst of the black hole X-ray binary system., 13 papes, 12 figures

To evaluate the safety of the domestic 13-valent pneumococcal polysaccharide conjugate vaccine-tetanus toxoid protein (PCV13-TT) after its licensure. The adverse event following immunization (AEFI) and the vaccination data of PCV13-TT in Zhejiang province from July 2020 to October 2021 were collected from national adverse event following immunization surveillance system and Zhejiang provincial immunization information system. Descriptive epidemiological method was used for this analysis. From July 2020 to October 2021, 302 317 doses of PCV13-TT were administered in children under 6 years old in Zhejiang Province and 636 AEFI case reports were received, with a reporting rate of 21.04 per 10 000 doses. Of these AEFI cases, 97.17% were mild vaccine product-related reaction (20.54 per 10 000 doses) and 95.44% occurred in the 0-1 d after vaccination (20.08 per 10 000 doses). The most common clinical diagnoses of AEFI included fever (224 cases), redness (204 cases), and induration (190 cases), while allergic rash (11 cases) was the most common diagnosis among the abnormal reactions. In conclusion,the present results bolstered that the domestic PCV13-TT was generally well tolerated in children under 6 years old in Zhejiang Province.评价国产13价肺炎球菌多糖结合疫苗（PCV13-TT）上市后使用的安全性。采用横断面描述性研究方法，由浙江省疾病预防控制中心收集国家疑似预防接种异常反应（AEFI）信息管理系统中2020年7月至2021年10月期间浙江省PCV13-TT的AEFI报告数据；从浙江省疫苗管理和预防接种信息综合管理系统中统计同期PCV13-TT的接种数据。采用描述流行病学方法对PCV13-TT的AEFI报告发生情况进行描述性分析。结果显示，2020年7月至2021年10月期间，浙江省5岁及以下儿童中PCV13-TT共接种302 317剂次，报告AEFI共636例（报告发生率为21.04/万剂次），97.17%为一般反应（20.54/万剂次），95.44%发生在接种后0~1 d内（20.08/万剂次）。一般反应中最常见的为发热（224例）、局部红肿（204例）及硬结（190例），异常反应中报告最多的是过敏性皮疹（11例）。综上，浙江地区PCV13-TT上市后在5岁及以下儿童中接种具有较好的安全性。.

GRO J1008-57, as a Be/X-ray transient pulsar, is considered to have the highest magnetic field in known neutron star X-ray binary systems. Observational data of the X-ray outbursts in GRO J1008-57 from 2017 to 2020 were collected by the Insight-HXMT satellite. In this work, the spin period of the neutron star in GRO J1008-57 was determined to be about 93.28 seconds in August 2017, 93.22 seconds in February 2018, 93.25 seconds in June 2019 and 93.14 seconds in June 2020. GRO J1008-57 evolved in the spin-up process with a mean rate of − ( 2.10 ± 0.05 ) × 10−4 s/d from 2009 – 2018, and turned into a spin down process with a rate of ( 6.7 ± 0.6 ) × 10−5 s/d from Feb 2018 to June 2019. During the type II outburst of 2020, GRO J1008-57 had the spin-up torque again. During the torque reversals, the pulse profiles and continuum X-ray spectra did not change significantly, and the cyclotron resonant scattering feature around 80 keV was only detected during the outbursts in 2017 and 2020. Based on the observed mean spin-up rate, we estimated the inner accretion disk radius in GRO J1008-57 (about 1 - 2 times of the Alfven radius) by comparing different accretion torque models of magnetic neutron stars. During the spin-down process, the magnetic torque should dominate over the matter accreting inflow torque, and we constrained the surface dipole magnetic field B ≥ 6 × 10 12 G for the neutron star in GRO J1008-57, which is consistent with the magnetic field strength obtained by cyclotron line centroid energy.

Phenotypic screening of an annotated small molecule library and initial SAR studies identified compound 2 as a robust enhancer of progranulin secretion. Detailed SAR development on conformationally restricted carbamate isosteres led to the identification of compound 60 with a 3-fold improvement in BV-2 potency and a 9-fold decrease in hERG inhibition over compound 2, substantially improving this important margin of safety relative to compound 2.

We report on the Insight-HXMT observations of the new black hole X-ray binary MAXI J1820+070 during its 2018 outburst. Detailed spectral analysis via the continuum fitting method shows an evolution of the inferred spin during its high soft sate. Moreover, the hardness ratio, the non-thermal luminosity and the reflection fraction also undergo an evolution, exactly coincident to the period when the inferred spin transition takes place. The unphysical evolution of the spin is attributed to the evolution of the inner disc, which is caused by the collapse of a hot corona due to condensation mechanism or may be related to the deceleration of a jet-like corona. The studies of the inner disc radius and the relation between the disc luminosity and the inner disc radius suggest that, only at a particular epoch, did the inner edge of the disc reach the innermost stable circular orbit and the spin measurement is reliable. We then constrain the spin of MAXI J1820+070 to be a*=0.2^{+0.2}_{-0.3}. Such a slowly spinning black hole possessing a strong jet suggests that its jet activity is driven mainly by the accretion disc rather than by the black hole spin., Comment: 14 pages, 13 figures, 5 tables, accepted for publication in MNRAS

Fast radio bursts (FRBs) are short pulses observed in radio band from cosmological distances. One class of models invoke soft gamma-ray repeaters (SGRs), or magnetars, as the sources of FRBs. Some radio pulses have been observed from some magnetars, however, no FRB-like events had been detected in association any magnetar burst, including one giant flare. Recently, a pair of FRB-like bursts (FRB 200428 hereafter) separated by milliseconds (ms) were detected from the general direction of the Galactic magnetar SGR J1935+2154. Here we report the detection of a non-thermal X-ray burst in the 1-250 keV energy band with the Insight-HXMT satellite, which we identify as emitted from SGR J1935+2154. The burst showed two hard peaks with a separation of 34 ms, broadly consistent with that of the two bursts in FRB 200428. The delay time between the double radio and X-ray peaks is about 8.57 s, fully consistent with the dispersion delay of FRB 200428. We thus identify the non-thermal X-ray burst is associated with FRB 200428 whose high energy counterpart is the two hard peaks in X-ray. Our results suggest that the non-thermal X-ray burst and FRB 200428 share the same physical origin in an explosive event from SGR J1935+2154., 24 pages, 9 figures, 6 tables; initial submission to a journal on May 9th, 2020. Significant changes include updated localization and detailed spectral evolution of the X-ray burst, and better determination of the two narrow X-ray peaks corresponding to the two radio pulses. Conclusions are strengthened. Nature Astronomy online on Feb. 18, 2021

A black hole X-ray binary produces hard X-ray radiation from its corona and disk when the accreting matter heats up. During an outburst, the disk and corona co-evolves with each other. However, such an evolution is still unclear in both its geometry and dynamics. Here we report the unusual decrease of the reflection fraction in MAXI J1820+070, which is the ratio of the coronal intensity illuminating the disk to the coronal intensity reaching the observer, as the corona is observed to contrast during the decay phase. We postulate a jet-like corona model, in which the corona can be understood as a standing shock where the material flowing through. In this dynamical scenario, the decrease of the reflection fraction is a signature of the corona's bulk velocity. Our findings suggest that as the corona is observed to get closer to the black hole, the coronal material might be outflowing faster., 11 pages, 5 figures; published in Nature Communications; An artist's impression of Figure 5 is updated here, and the animation is available at http://youbei.work/research/

Recently, the system parameters of Cygnus X-1, i.e., the black hole mass M, the orbital inclination i and the source distance D, have been updated. As the input constraints of the continuum-fitting method, it is necessary to refine the spin with these new parameters. The Hard X-ray Modulation Telescope (HXMT, also named Insight) was successfully launched on June 15th, 2017 in China. In this work, we analyzed spectra obtained by the Insight-HXMT during 2017-2018 and re-estimated the spin via the continuum-fitting method. We first re-estimated the spin using old parameters and found the dimensionless spin parameter to be a ⁎ > 0.934 (3σ), which is consistent with previous measurements both from the continuum-fitting and iron-line fitting methods ( Gou et al. 2011 , Gou et al. 2014 ; Fabian et al. 2012 , Walton et al. 2016 , Tomsick et al. 2018 ). Then, when the new parameters were applied, we obtained an increased spin: a ⁎ > 0.967 (3σ), which is also consistent with the recent measurement (Zhao et al. 2020, submitted to ApJ) and also confirms an extreme Kerr black hole lying in this system.

A new black hole X-ray binary (BHXRB) MAXI J1535-571 was discovered by MAXI during its outburst in 2017. Using observations taken by the first Chinese X-ray satellite, the Hard X-ray Modulation Telescope (dubbed as Insight-HXMT), we perform a joint spectral analysis (2-150 keV) in both energy and time domains. The energy spectra provide the essential input for probing the intrinsic Quasi-Periodic Oscillation (QPO) fractional rms spectra (FRS). Our results show that during the intermediate state, the energy spectra are in general consistent with those reported by Swift/XRT and NuSTAR. However, the QPO FRS become harder and the FRS residuals may suggest the presence of either an additional power-law component in the energy spectrum or a turn-over in the intrinsic QPO FRS at high energies.

We study the 2018 outburst of Aql X-1 via the monitor of all sky X-ray image (MAXI) data. We show that the outburst starting in February 2018 is a member of short-low class in the frame of outburst duration and the peak count rate although the outburst morphology is slightly different from the other fast-rise-exponential-decay (FRED) type outbursts with a milder rising stage. We study the partial accretion in the weak propeller stage of Aql X-1 via the MAXI data of the 2018 outburst. We report on the spectral analysis of 3 observations of Aquila X-1 obtained by Insight – hard X-ray modulation telescope (Insight-HXMT) during the late decay stage of the 2018 outburst. We discuss that the data taken by Insight-HXMT is just after the transition to the weak propeller stage. Our analysis shows the necessity of a comptonization component to take into account the existence of an electron cloud resulting photons partly up-scattered.

We present a detailed timing study of the brightest persistent X-ray source Sco X-1 using the data collected by the Hard X-ray Modulation Telescope ( I n s i g h t -HXMT) from July 2017 to August 2018. A complete Z-track hardness-intensity diagram (HID) is obtained. The normal branch oscillations (NBOs) at ∼ 6 Hz in the lower part of the normal branch (NB) and the flare branch oscillations (FBOs) at ∼ 16 Hz in the beginning part of the flaring branch (FB) are found in observations with the Low Energy X-ray Telescope (LE) and the Medium Energy X-ray Telescope (ME) of I n s i g h t -HXMT, while the horizontal branch oscillations (HBOs) at ∼ 40 Hz and the kilohertz quasi-periodic oscillations (kHz QPOs) at ∼ 800 Hz are found simultaneously above 20 keV for the first time on the horizontal branch (HB) by the High Energy X-ray Telescope (HE) and ME. We find that for all types of the observed QPOs, the centroid frequencies are independent of energy, while the root mean square (rms) increases with energy; the centroid frequencies of both the HBOs and kHz QPOs increase along the Z-track from the top to the bottom of the HB; and the NBOs show soft phase lags increasing with energy. A continuous QPO transition from the FB to NB in ∼ 200 s are also detected. Our results indicate that the non-thermal emission is the origin of all types of QPOs, the innermost region of the accretion disk is non-thermal in nature, and the corona is nonhomogeneous geometrically.

We have used model scaling so that the propagation of light through space could be studied using the well-known nonlinear Schrodinger equation. We have developed a set of numerical procedures to obtain a stable propagating wave so that it could be used to find out how wavelength could increase with distance travelled. We have found that broadening of wavelength, expressed as redshift, is proportional to distance, a fact that is in agreement with many physical observations by astronomers. There are other reasons for redshifts that could be additional to the transmission redshift, resulting in the deviation from the linear relationship as often observed. Our model shows that redshift needs not be the result of an expanding space that is a long standing view held by many astrophysicists. Any theory about the universe, if bases on an expanding space as physical fact, is open to question.

We present a spectral study of the black hole candidate MAXI J1348-630 during its 2019 outburst, based on monitoring observations with Insight-HXMT and Swift. Throughout the outburst, the spectra are well fitted with power-law plus disk-blackbody components. In the soft-intermediate and soft states, we observed the canonical relation L ~ T_in^4 between disk luminosity L and peak colour temperature T_in, with a constant inner radius R_in (traditionally identified with the innermost stable circular orbit). At other stages of the outburst cycle, the behaviour is more unusual, inconsistent with the canonical outburst evolution of black hole transients. In particular, during the hard rise, the apparent inner radius is smaller than in the soft state (and increasing), and the peak colour temperature is higher (and decreasing). This anomalous behaviour is found even when we model the spectra with self-consistent Comptonization models, which take into account the up-scattering of photons from the disk component into the power-law component. To explain both those anomalous trends at the same time, we suggest that the hardening factor for the inner disk emission was larger than the canonical value of ~1.7 at the beginning of the outburst. A more physical trend of radii and temperature evolution requires a hardening factor evolving from ~3.5 at the beginning of the hard state to ~1.7 in the hard intermediate state. This could be evidence that the inner disk was in the process of condensing from the hot, optically thin medium and had not yet reached a sufficiently high optical depth for its emission spectrum to be described by the standard optically-thick disk solution., 20 pages, 10 figures, 4 tables, accepted by ApJ

We present the third data release (DR3) of the X-shooter Spectral Library (XSL). This moderate-to-high resolution, near-ultraviolet-to-near-infrared ($350-2480$ nm, R $\sim$ 10 000) spectral library is composed of 830 stellar spectra of 683 stars. DR3 improves upon the previous data release by providing the combined de-reddened spectra of the three X-shooter segments over the full $350-2480$ nm wavelength range. It also includes additional 20 M-dwarf spectra from the ESO archive. We provide detailed comparisons between this library and Gaia EDR3, MILES, NGSL, CaT library, and (E-)IRTF. The normalised rms deviation is better than $D=0.05$ or 5$\%$ for the majority of spectra in common between MILES (144 spectra of 180), NGSL (112$/$116), and (E-)IRTF (55$/$77) libraries. Comparing synthetic colours of those spectra reveals only negligible offsets and small rms scatter, such as the median offset(rms) 0.001$\pm$0.040 mag in the (box1-box2) colour of the UVB arm,-0.004$\pm$0.028 mag in (box3-box4) of the VIS arm, and -0.001$\pm$0.045 mag in (box2-box3) colour between the UVB and VIS arms, when comparing stars in common with MILES. We also find an excellent agreement between the Gaia published (BP-RP) colours and those measured from the XSL DR3 spectra, with a zero median offset and an rms scatter of 0.037 mag for 449 non-variable stars. The unmatched characteristics of this library, which combine a relatively high resolution, a large number of stars, and an extended wavelength coverage, will help us to bridge the gap between the optical and the near-IR studies of intermediate and old stellar populations, and to probe low-mass stellar systems., Comment: 26 pages, 25 figures, accepted to Astronomy & Astrophysics. The data are available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ or on the XSL web-page http://xsl.astro.unistra.fr

We report results on the timing analysis of the 2020 giant outburst of 1A 0535+262, using broadband data from Insight-HXMT. The analysis of the pulse profile evolution from the subcritical-luminosity to the supercritical-luminosity regime is presented for the first time. We found that the observed pulse profile exhibits a complex dependence on both energy and luminosity. A dip structure at the energy of the cyclotron resonant scattering features is found for the first time in the pulse fraction–energy relation of 1A 0535+262, when the outburst evolves in a luminosity range from 4.8 × 1037 to 1.0 × 1038 erg s−1. The observed structure is luminosity dependent and appears around the source critical luminosity (∼6.7 × 1037 erg s−1).

The High Energy X-ray telescope (HE) on-board the Hard X-ray Modulation Telescope (\textit{Insight}-HXMT) can serve as a wide Field of View (FOV) gamma-ray monitor with high time resolution ($\mu$s) and large effective area (up to thousands cm$^2$). We developed a pipeline to search for Gamma-Ray Bursts (GRBs), using the traditional signal-to-noise ratio (SNR) method for blind search and the coherent search method for targeted search. By taking into account the location and spectrum of the burst and the detector response, the targeted coherent search is more powerful to unveil weak and sub-threshold bursts, especially those in temporal coincidence with Gravitational Wave (GW) events. Based on the original method in literature, we further improved the coherent search to filter out false triggers caused by spikes in light curves, which are commonly seen in gamma-ray instruments (e.g. \textit{Fermi}/GBM, \textit{POLAR}). We show that our improved targeted coherent search method could eliminate almost all false triggers caused by spikes. Based on the first two years of \textit{Insight}-HXMT/HE data, our targeted search recovered 40 GRBs, which were detected by either \textit{Swift}/BAT or \textit{Fermi}/GBM but too weak to be found in our blind search. With this coherent search pipeline, the GRB detection sensitivity of \textit{Insight}-HXMT/HE is increased to about 1.5E-08 erg/cm$^2$ (200 keV$-$3 MeV). We also used this targeted coherent method to search \textit{Insight}-HXMT/HE data for electromagnetic (EM) counterparts of LIGO-Virgo GW events (including O2 and O3a runs). However, we did not find any significant burst associated with GW events., Comment: 12 pages, 14 figures, 5 tables; accepted for publication in MNRAS

We report the energy-resolved broadband timing analysis of the black hole X-ray transient MAXI J1631-479 during its 2019 outburst from February 11 to April 9, using data from the Insight-Hard X-ray Modulation Telescope (Insight-HXMT), which caught the source from its hard intermediate state to the soft state. Thanks to the large effective area of Insight-HXMT at high energies, we are able to present the energy dependence of fast variability up to ~100 keV. Type-C quasi-periodic oscillations (QPOs) with frequency varying between 4.9 Hz and 6.5 Hz are observed in the 1-100 keV energy band. While the QPO fractional rms increases with photon energy from 1 keV to ~10 keV and remains more or less constant from ~10 keV to ~100 keV, the rms of the flat-top noise first increases from 1 keV to ~8 keV then drops to less than 0.1% above ~30 keV. We suggest that the disappearance of the broadband variability above 30 keV could be caused by the non-thermal acceleration in the Comptonizing plasma. At the same time, the QPOs could be produced by the precession of either a small-scale jet or a hot inner flow model., To be published in The Astrophysical Journal

The evidences for the influence of thermonuclear (type-I) X-ray bursts upon the surrounding environments in neutron star low-mass X-ray binaries (LMXB) were detected previously via spectral and timing analyses. Benefitting from a broad energy coverage of Insight-HXMT, we analyze one photospheric radius expansion (PRE) burst, and find an emission excess at soft X-rays. Our spectral analysis shows that, such an excess is not likely relevant to the disk reflection induced by the burst emission and can be attributed to an enhanced pre-burst/persistent emission. We find that the burst and enhanced persistent emissions sum up to exceed Eddington luminosity by ∼ 40 percentages. We speculate that the enhanced emission is from a region beyond the PRE radius, or through the Comptonization of the corona.

High energy X-ray telescope (HE) is one of the three instruments of Insight-HXMT (Hard X-ray Modulation Telescope) payload. The HE detector (HED) array is composed of 18 actively NaI(Tl)/CsI(Na) phoswich scintillators with a total geometric area of ∼ 5100 cm 2 and cover the energy range 20∼250 keV. In this paper we describe the on-ground detector-level calibration campaigns and present the principal instrument properties of HEDs.

We have developed an energy balance equation for the universe. The two system parameters involved in the equation could be “fine-tuned” so that the predicted temperature histories all lead to what is observed in the present cosmic microwave background. We have shown that various combinations of these two parameters are possible; in particular, the present background temperature needs not be the remnant of a very hot temperature in the far distance past. We also solved for the propagation of vortex solitons in optical fibres as contrasting examples to show how electromagnetic wave could be transmitted in a particular waveform under strictly controlled conditions. To avoid singularity, all vortexes have a black centre. We conclude that while numerical techniques can be used to account for an infinite quantity, it is unlikely that such a quantity could exist in reality.

We investigate under what conditions transient simulation could be used to integrate backward in time so that the initial field could be recovered from later histories. In this paper we use realistic examples and find that, in long histories, traces of the initial field would be present only in the exact analytical solutions. We conclude that the recovery of initial field is possible only if the equations could be solved analytically or only short time periods are involved. In practice, it is not possible to detect those traces by measurements or observations. If numerical procedures are used, truncation and discretization errors are always present. Fine-tuning of system parameters used or transforming time into another pseudo time frame may allow numerical integration to be carried out backward in time. But numerical instability is still a problem. Large spurious increases found by numerical procedures are most likely due to numerical inaccuracy and instability.