A radio-detected type Ia supernova with helium-rich circumstellar material

Type Ia supernovae (SNe Ia) are thermonuclear explosions of degenerate white dwarf stars destabilized by mass accretion from a companion star1, but the nature of their progenitors remains poorly understood. A way to discriminate between progenitor systems is through radio observations; a non-degenerate companion star is expected to lose material through winds2 or binary interaction3 before explosion, and the supernova ejecta crashing into this nearby circumstellar material should result in radio synchrotron emission. However, despite extensive efforts, no type Ia supernova (SN Ia) has ever been detected at radio wavelengths, which suggests a clean environment and a companion star that is itself a degenerate white dwarf star4,5. Here we report on the study of SN 2020eyj, a SN Ia showing helium-rich circumstellar material, as demonstrated by its spectral features, infrared emission and, for the first time in a SN Ia to our knowledge, a radio counterpart. On the basis of our modelling, we conclude that the circumstellar material probably originates from a single-degenerate binary system in which a white dwarf accretes material from a helium donor star, an often proposed formation channel for SNe Ia (refs. 6,7). We describe how comprehensive radio follow-up of SN 2020eyj-like SNe Ia can improve the constraints on their progenitor systems.

Based on our modeling, we conclude the CSM likely originates from a single-degenerate (SD) binary system where a WD accretes material from a helium donor star, an often hypothesized formation channel for SNe Ia 6,7 . We describe how comprehensive radio follow-up of SN 2020eyj-like SNe Ia can improve the constraints on their progenitor systems.
SN 2020eyj was first detected on 2020 March 07 UT (MJD = 58915.12, Sect. 1), at α = 11 h 11 m 47.19 s , δ = +29°23 06.5 (J2000). The SN was classified as a SN Ia 8 based on a low-resolution spectrum obtained on 2020 April 2, 25 days after the first detection. Comparisons with Type Ia and Ibc spectra from the literature support the SN Ia classification (Sect. 2 and Fig. 1). Unusual evolution of the later light curve prompted us to obtain a second spectrum on 2020 July 20, 131 days after first detection. The second spectrum was very similar to those of Type Ibn SNe (SNe Ibn), which are SNe that interact with helium-rich CSM and have spectra characterized by narrow (∼few×10 3 km s −1 ) He I emission lines while showing little to no H I 9,10 .
Based on the late time (tail phase) CSM-interaction dominated spectra (Fig. 2), SN 2020eyj falls in the category of the rare subclass of SNe Ia that show evidence of CSM interaction in their optical spectra (SNe Ia-CSM; 11 ). The narrow emission lines in the spectra of such interacting SNe arise from shock interaction between the fast-moving SN ejecta and the slow-moving CSM 12 . SNe Ia-CSM are strong contenders for the single degenerate (SD) SN Ia formation channel on account of the CSM, which is commonly assumed to originate from a non-degenerate donor star through stellar or accretion winds. Prior to SN 2020eyj, all the discovered SNe Ia-CSM exhibited prominent Balmer emission lines and only weak He emission features 11 .
Typically, CSM interaction contributes significantly to or even dominates the spectral and light curve evolution of SNe Ia-CSM from the start, hindering unambiguous classification as SNe Ia 13 . However, in some rare cases SNe Ia-CSM have shown a delay in CSM interaction 14-16 , suggesting the CSM was located far (> 10 15 cm) from the binary system at the time of explosion.
Notably, PTF11kx cemented SNe Ia-CSM as a bona fide SN Ia subclass by virtue of a delay of ∼60 days, allowing for an indisputable SN Ia classification prior to CSM interaction 15 . SN 2020eyj follows a similar evolution as PTF11kx, initially showing a typical SN Ia bell-shaped light curve (Fig. 3) and a spectrum consistent with a SN Ia of the 91T subgroup 17 without clear evidence for CSM interaction (Fig. 1). Then, at 50 days after first detection, the g-band light curve of SN 2020eyj diverges from a steady decline into a plateau that lasts ∼200 days. Such an evolution and color change is not expected for a normal SN Ia (Fig. 3), but is driven by the emergence of spectral features associated with CSM interaction (Sect. 3). We interpret the start of the plateau at 50 days as the epoch when CSM interaction starts to contribute significantly or to even dominate the light curve of SN 2020eyj. Assuming a SN ejecta velocity of 10 4 km s −1 18 , the delay corresponds to an inner boundary to the CSM of ∼ 4 × 10 15 cm. Save for the presence of He emission lines, the late-time spectra of SN 2020eyj are typical for the SN Ia-CSM class, with prominent broad Ca II emission from the near-infrared (near-IR) triplet and without any sign of O I λ7774 emission (Fig. 2). The compact and star-forming host galaxy of SN 2020eyj (Sect. 4) is also consistent with those of other SNe Ia-CSM 11 .
Despite the similarities between SN 2020eyj and other SNe Ia-CSM, the presence of He I lines and absence of prominent H I lines remains a striking difference with profound implications for the progenitor system. As H I is easier to ionize than He I, the absence of the lines indicates that the CSM around SN 2020eyj, and thus the companion star, is He-rich and H-poor. While the late-time spectra of SN 2020eyj are similar to those of SNe Ibn, these SNe are presumed to arise from the core collapse of massive (> 10 M ) stars 9,19,20 , which are unlikely to be in a binary system with a WD, as they would undergo core collapse long before the WD formed. A merger involving a degenerate He WD donor star is also disfavored, because in such merger models only a small amount of unburned He (∼ 0.03 M 21 ) is present close to ( 10 12 cm) the WD 22 , whereas the CSM around SN 2020eyj resides at > 10 15 cm. Instead, a strong candidate for the donor star in the SN 2020eyj progenitor system is a non-degenerate He star (initial mass 1−2 M , e.g. 23 ). WD + He star systems can be formed via binary evolution 24 , and this SD channel for SNe Ia has garnered recent interest because the very restrictive limits placed by radio non-detections and deep optical imaging 25 that exclude most H-rich donor star models, still allow for low CSM density WD + He star systems 25,26 . The possible detection in pre-explosion HST imaging of the progenitor system of the Type Iax (SNe Ia similar to SN 2002cx 27 ) SN 2012Z, a blue compact source interpreted as a He-star donor 28 , has further strengthened this hypothesis, although the thermonuclear nature of Type Iax SNe is debated 29 .
The CSM interaction in SN 2020eyj is also confirmed, for the first time in a Type Ia SN, through the detection of a radio counterpart, at a frequency of 5.1 GHz at 605 and 741 days after the first detection (Sect. 1). Follow-up in the X-rays did not yield a detection (Sect. 1). We model the radio synchrotron emission, which results from the shock interaction between the ejecta and the CSM, assuming two basic CSM configurations expected in a SD progenitor system; a constant density shell, and a wind-like density profile with density ρ ∝ r −2 (Fig. 4). A constant density shell could result from a mass ejection event such as a nova, whereas a wind-like CSM profile would be expected from an optically thick wind, where the mass-transfer rate from the donor star to the WD exceeds the maximum accretion rate of He-rich material that the WD can burn on its surface 26, 30 . In addition to CSM material arising from a SD scenario, we consider synchrotron emission resulting from the interaction of a SN Ia from a double degenerate (DD) white dwarf merger interacting with the local interstellar medium (ISM) 31 . For the SD shell model, the radio detections are best explained with a CSM mass of M csm = 0.36 M (Sect. 6), with the expectation that the radio light curve will start to drop off quite rapidly at ∼ 900 days. For the SD optically  Fig. 4) requires unusually high ISM densities and does not recover the observed decline in flux, ruling out the DD formation channel for SN 2020eyj (Sect. 6). The best fit radio light curves of the shell and wind models differ in particular at early phases (Fig. 4), but no radio data were obtained at these epochs. Instead, multi-frequency monitoring of the radio counterpart of SN 2020eyj until late phases (> 1000 days) will allow to discriminate between the rapid drop-off of the shell model, and a shallower decline expected in the case of a wind-like CSM.
A viable progenitor scenario for SN 2020eyj needs to explain not only the presence and properties of a He-rich CSM, but also its detached configuration. For the delayed Type Ia-CSM SN 2002ic, the CSM free cavity was attributed to a possible drop-off in mass-transfer rate or the emergence of a low-density fast wind evacuating the CSM 32 . In the case of PTF11kx, the delayed CSM interaction was explained by a scenario involving a symbiotic nova progenitor, where recurrent novae on the surface of the WD sweep up the wind-deposited CSM into shells 15 . SN 2020eyj shows strong similarities to PTF11kx, which may hint at a common progenitor scenario. Their light curves are virtually identical up until day 50 ( Fig. Extended Data Figure 1) with rise times of ∼14 days in g band, which is fast for a SN Ia 33 . And, except for the nature of the narrow emission lines, both SNe have similar spectra throughout their evolution (Figs. 1 and 2). For SN 2020eyj, a nova progenitor could look like V445 Puppis (V445 Pup, Sect. 6), the only known nova system that showed He-rich, but H-free, ejecta 34,35 . Notably, the V445 Pup system is considered a prime candidate progenitor system for the He star + WD SN Ia channel, as it is claimed to be host to a WD with a mass close to the Chandrasekhar limit 36 . Additionally, a prominent carbon rich equatorial dusty disc like the one in V445 Pup 34, 35 could explain (Sect. 6) the luminous IR counterpart of SN 2020eyj (Fig. Extended Data Figure 2), which we attribute to an IR echo from radiatively heated pre-existing dust with a dust mass of order 10 −2 M (Sect. 5).
The initial models invoking recurrent novae for the origin of PTF11kx 15 were challenged by the CSM masses involved 37 , which were too large by orders of magnitude for symbiotic nova mass build-up models 38 . Similarly, the mass resulting from a V445 Pup-like nova outburst ( 10 −3 M ; Sect. 3) is insufficient to explain the CSM mass observed in SN 2020eyj. However, a recent study of the radio evolution of V445 Pup suggests that the equatorial disk could have pre-dated (and survived) the nova outburst 39 , which would allow for mass build-up in the disc between nova eruptions. This scenario would require the SN to occur soon after the nova outburst, and before the resumption of mass-transfer between the donor and WD reforms the disc at small radii. We note that a nova similar to the year 2000 event of V445 Pup would not have been detectable at the distance of SN 2020eyj (Sect. 6).
SN 2020eyj represents the first observational example of the previously speculated class of SNe Ia-He CSM 40 . The presence of a dense CSM, supported by a radio detection, offers strong evidence for the SD scenario for SN 2020eyj, in particular for the He star + WD formation channel.
It is estimated ∼10% of all SD Type Ia SNe arise from this channel 7 , which is likely the dominant source of SNe Ia with short delay times 41 . Understanding the timescale of SN Ia activity is important for the chemical evolution of galaxies. The confirmed presence of a He-rich CSM in a SN Ia system also impacts SN Ia explosion modeling, as He plays a vital role in double detonation models where the WD explosion is triggered by the ignition of a massive ( 0.2 M ) He shell on its surface 30 . Constraining the rate of SNe Ia similar to SN 2020eyj would require systematic spectroscopic follow up of SNe Ia with long-lived light curves, as currently monitoring often stops after a seemingly normal SN Ia has been classified. Observational properties which SN 2020eyj shares with its H-analog PTF11kx, such as a fast rise and a 91T-like peak spectrum, can potentially guide such follow up efforts and allow for the discovery and study of more SN 2020eyj-like SNe Ia, including at radio wavelengths.

Extended Data
Extended Data Table 1 (Table Extended Data Table 3), which are not included in the model fits.
Extended Data were recovered in the ZTF data on 2020 March 07 UT (MJD = 58915.12) in both g and r filters.
For reference, we list some key characteristics of SN 2020eyj in Table Extended Data Table 1.
Optical photometry Follow-up photometry was obtained as part of public and partnership ZTF survey observations 48 with the ZTF camera 49 on the P48 telescope in the g and r bands, and later phases were also covered in the i band. The P48 data were reduced and host subtracted using the ZTF reduction and image subtraction pipeline 50 , which makes use of the ZOGY algorithm 51 for reference image subtraction. Following the rationale illustrated in 52 , we apply the difference image zero point magnitude to convert fluxes from units in detector data number (DN) to µJy, and translate fluxes to AB magnitudes. We apply a detection threshold of S/N 3, and for non-detections we compute 5 sigma upper limits. Table 5 lists the ZTF magnitudes and upper limits.
Additional photometric epochs were obtained with the Liverpool Telescope (LT) 53 , the SEDM on the P60, the LCO telescopes (program id. NOAO2020B-012), and ALFOSC on the NOT, with data reduced and host subtracted using the pipelines described in 54,55 or standard methods. In this work we also make use of the forced photometry service from the ATLAS survey 43 Table 5. The ATLAS and P48 light curves are shown in Fig. 3, binned into 1-night bins to enhance the signal to noise ratio (S/N).
Optical spectroscopy The first optical spectrum of SN 2020eyj was obtained with the Spectral Energy Distribution Machine (SEDM) 60  A log of the obtained spectra is provided in Table Extended Data Table 2, and the epochs of spectroscopy are indicated by the diamond markers on top of the light curves in Fig. 3. The spectra were absolute flux-calibrated against the r-band magnitudes using the Gaussian Process interpolated magnitudes and then corrected for MW extinction. All spectral data and corresponding information will be made available via WISeREP public database 69 . We present the peak SEDM spectrum in Fig. 1 and the later sequence of spectra in Fig. 2.
The initial spectrum obtained with SEDM is characterized by broad absorption features There is also no sign of material stripped from the donor star by the SN ejecta 77,78 , which is predicted to show up as narrow emission (< 1000 km −1 76 ).
The asymmetric line profile we associate with the SN also applies to the Hα emission line, epochs, with the earliest detection at 59 days after first detection ( Fig. 3 and Table Extended Data Table 3). The host is not detected in (stacked) WISE data prior to the SN explosion ( To convert the count-rate limit into a flux limit, we assumed a power-law spectrum with a photon index Γ of 2 and a Galactic neutral hydrogen column density of 1.

SN Ia classification
During the peak phase of SN 2020eyj, an optical spectrum was obtained with the low-resolution (R∼100) SEDM on the P60, 25 days after first detection. This high S/N spectrum was characterized by broad absorption features (Fig. 1), based on which SN 2020eyj was classified as a Type Ia SN at redshift z = 0.03 8 . Using SNIascore, a deep-learning-based classifier of SNe Ia based on low-resolution spectra, 92 noted that the SN could be a Type Ibc SN erroneously classified as SN Ia due to the degeneracy between peak spectra of SNe Ibc with those of SNe Ia at post-peak phases, but their classifier anyway favored a SN Ia classification. In general, based on the comparison study by 13  In terms of spectral features, the SEDM spectrum shows broad absorption lines that based on the spectral comparisons can be unambiguously identified as Si II, Fe II and Ca II (Fig. 1). An absence of oxygen lines is typical for Type Ia-CSM spectra, both as an absorption feature around peak and as emission in later epochs 11,107 , as seen in the early and late spectra of PTF11kx  96 , and the spectrum is dominated by CSM-interaction features.
In conclusion, based on its spectral features we classify SN 2020eyj as a Type Ia(-CSM) SN.
Furthermore, as we discuss in Sect. 3, the light curves of SN 2020eyj show strong similarities to those of PTF11kx, the SN that cemented SNe Ia-CSM as a subclass.

Light curve analysis
Light curve fits The light curve of SN 2020eyj (Fig. 3) can be divided into two phases, similar to its spectral evolution. In the first phase, lasting ∼50 days, the light curve follows a fairly typical bell-like shape, peaking at m∼17. are practically identical in g and r band for the first ∼45 days, even though the fits are independent ( Fig. Extended Data Figure 1). The r-band light curves peak at M r ∼ −19.3 for both SNe, consistent with both SNe Ia and SNe Ia-CSM, although both SNe are on the fainter end of the sample of SNe Ia-CSM described by 11 . From the light-curve fits we obtain for SN 2020eyj rise times in g and r band of 14 ± 2 and 16 ± 2 days since discovery, respectively. This is fast for a SN Ia 33 , but similar to PTF11kx (Fig. Extended Data Figure 1).
An important caveat about the light curve fit is that the intrinsic decline rate of SN 2020eyj could appear slower because of the contribution by CSM interaction. Based on the color evolution of the light curve, we know from day 50 onward that the CSM contribution is significant, but it is reasonable to assume that some CSM interaction already contributes to the light curve at earlier epochs. This means that the stretch parameter we measure should be regarded as an upper limit, and as a result so is the peak luminosity of the fit. SN 2020eyj, but also PTF11kx, are no typical SNe Ia, so the colors and peak magnitude could (to some extent) also be a property intrinsic to the class.

Host galaxy
The host of SN 2020eyj is a faint and compact galaxy with designation SDSS J111147.15+292305.9 ( Fig. Extended Data Figure 4). We retrieved science-ready co-added images from the Galaxy Evolution Explorer (GALEX) general release 6/7 116 Table Extended Data Table 4 lists all measurements. We fit the host galaxy SED with the software package Prospector version 0.3 120 to determine the host galaxy properties. We assumed a Chabrier initial mass function 121 and approximated the star formation history (SFH) by a linearly increasing SFH at early times followed by an exponential decline at late times (functional form t×exp (−t/τ ), where t is the age of the SFH episode and τ is the e-folding timescale). The model was attenuated with the 122 model. The priors were set identical to 119 123 . Adopting the parameterisation of the empirical oxygen calibration O3N2 by 124 , we obtain an oxygen abundance of 12 + log(O/H) = 8.14 ± 0.03. Such a low oxygen abundance is expected for a low mass galaxy 125 .
The host properties of 16 SNe Ia-CSM were reported in 11,126 . These authors concluded that all objects in their samples exploded in star-forming late-type galaxies (spiral and dwarf galaxies)

Dust properties
IR emission is commonly observed in interacting SNe, and can be attributed to the condensation of dust in the SN ejecta or in the shocked CSM, or to pre-existing dust in the CSM that is heated radiatively by the SN emission or by the ejecta/CSM shock interaction (e.g., [127][128][129][130]   Assuming optically thin dust, the flux F ν can be written as 131 : where M d is the mass of the dust, B ν the Planck blackbody function, T d the temperature of the dust, κ ν (a) the dust absorption coefficient as function of dust particle radius a, and d the distance to the observer. For simplicity, we assume a simple dust population of a single size composed entirely of amorphous carbon with grain size of 0.1 µm with the corresponding absorption coefficient κ as in 132,133 , and fit the WISE data to obtain an estimate of the dust temperature and mass. We note that the dust mass depends on assumed grain size, which we can not constrain on the available data. Varying the grain size from 0.01 to 1.0 µm changes the derived dust mass by an order of magnitude 132 . Over the first three epochs, up to 412 days, we derive a constant dust temperature of ∼800 K (Table Extended Data Table 3), consistent with a lack of color evolution in the WISE photometry (Fig. 3). Only at the fourth WISE epoch (614 days) do we see a significant drop in the dust temperature, to 608 ± 23 K. These dust temperatures are well below the expected evaporation temperature of dust (1500 K for silicates and 1900 K for graphite grains, e.g. 130 ). In addition to the dust temperatures, we obtain dust mass estimates of (1.8 ± 0.3) × 10 −3 M to (9.9 ± 2.1) × 10 −3 M for the first and the final WISE epochs, respectively (Table Extended Data Table 3).
The dust mass estimated for the final epoch corresponds to a CSM mass of 1 where r dg is the dust-to-gas ratio. The total integrated energy emitted in the IR is 9 × 10 49 erg (Table Extended Data Table 3), which is similar to the integrated energy emitted in the optical (Sect. 3).
In the case of optically thin dust that we consider here, the blackbody radius can be interpreted as a lower limit to the radius at which the dust resides. In the case of SN 2020eyj, the blackbody radius is (2.5 ± 0.2) × 10 16 cm in the first epoch, and increases thereafter to (6.4 ± 0.6) × 10 16 cm at 614 days (Table Extended Data Table 3 dense shell produced by the interaction of the ejecta with CSM also producing a substantial IR excess 128 . Interestingly, such line profile evolution has also been observed in the He nova V445 Pup, where it was attributed to dust obscuration within the shell 35 . In particular, for Type Ia-CSM SN 2005gj dust formation was inferred from line profiles 11 , while the bulk of the IR emission was also attributed to pre-existing dust 133   It is worth noting that the bolometric light curve only extends to 400 days, whereas the first detection of SN 2020eyj at 5 GHz took place at 605 days. Furthermore, it has been argued that the mass-transfer rates associated with the optically thick wind phase (> 10 −7 M yr −1 ) do not lead to SNe Ia, but rather to accretion induced collapse of the WD 139, 140 , although alternative wind models have been suggested to overcome this problem 141 .
CSM shells The CSM surrounding the H-rich analog of SN 2020eyj, PTF11kx, was argued to be concentrated in shells 15 . Other SNe Ia have shown evidence for CSM concentrated in thin shells, albeit at distances (∼ 10 16 cm) that no interaction with the ejecta is expected [142][143][144][145] . Shells have also been invoked for the configuration of the CSM in core-collapse H-rich Type IIn SNe, and typically attributed to ejection events by their massive progenitors. One noteworthy example is the the well studied SN 2014C, which transitioned from a stripped-envelope SN to a Type IIn SN due to interaction with a distant shell, and was detected in the radio 146,147 . Models for the radio emission of SNe Ia colliding with a constant-density shell of CSM have been previously presented in the literature, along with approximate functional forms to describe the evolution of the optically thick synchrotron light curve 148 . Since those models assume hydrogen-rich material, for our calculations we modify n e = ρ/m p to n e = ρ/(2m p ); otherwise we use the default parameters, notably B = 0.1. We explore shell models with a range of CSM masses M csm = (0.01 − 1) M and interaction end times from t end = 328 days (the spectrum that does not show prominent He I lines) to t end = 763 days (the second radio detection) -in this model, interaction must have ended before the second radio detection for the radio emission to have declined between the two observations. We assume a shell inner radius of R in = (30, 000 km s −1 )(50 days) = 1.3 × 10 16 cm to close the system of equations in the model; then, the ranges of M csm and t end correspond to a range of shell widths ∆R/R in = 3.4 − 7.5. For each model we calculate the representative model error as σ mod = max(|L ν,obs (t i ) − L ν,mod (t i )|/∆L ν,obs (t i )), where subscripts "obs" and "mod" refer to observed and modeled values, L ν is spectral luminosity, and ∆L ν is the error on the luminosity (flux error only; error in distance is not included). The best-fit model by this metric has M csm = 0.36 M and t end = 665 days, which is a very similar mass to what is found for PTF11kx based on analysis of its optical spectra 149 . We find models with σ mod ≤ 3 have Based on infrared spectra showing prominent carbon lines 34,151 , and a rapid decline in the light curve of V445 Pup, it was shown that a carbon-rich thick dust shell must have formed in the nova ejecta 34,150 . High resolution near-IR images resolved the nova event into an expanding narrow bipolar shell with bulk velocities of ∼6700 km s −1 , and a perpendicular central dust disc that strongly attenuates the optical He I emission lines arising from the receding shell 35 . Seven years after the outburst, the bipolar shell of V445 Pup, as imaged in the near-IR, extended to ∼ 10 17 cm, and the central dust torus had an outer radius (perpendicular to the lobes) of 10 16 cm 35 . An outer dust shell in a V445 Pup-like system could survive dust sublimation from a SN Ia explosion, depending on peak luminosity and grain composition 130 . A recent study of the long-lived radio evolution of V445 Pup showed the system was continuously synchrotron luminous for years after the outburst 39 . The synchrotron emission originated from the inner edge of the equatorial disc, and was interpreted as interaction between a wind coming off the WD from nuclear burning, and the surviving disc. The persistence of the disc through the nova outburst suggests the disc is at least comparable in mass with the mass of the nova ejecta, which was estimated to be ∼ 10 −4 M 36 . In turn, the mass of the WD in V445 Pup, close to the Chandrasekhar limit, limits the ejecta mass in the system to not more than ∼ 10 −3 M ( 36 , their Fig. 7).
ISM Radio emission can potentially arise from a Type Ia SN in the double-degenerate scenario as a result of interaction with the ISM. We have modeled the radio light curve from such a merger scenario in the same way as in 31,137 , i.e., we assume that the supernova is the result of two merging white dwarfs with masses 0.9 and 1.1 M as described by 152 . The outermost ejecta has a density slope ∝ ρ −n with n = 13 (see 31 (Fig. 4). In summary, our radio observations and their modeling argue strongly against an ISM scenario, which arises from a double degenerate progenitor system. Furthermore, the observed strong helium lines are also at odds with an ISM scenario 153 . We therefore conclude that SN 2020eyj did not result from the thermonuclear runaway of a WD in a DD progenitor system, leaving the SD scenario as the only viable alternative.  Code Availability Upon request, the corresponding author will provide code used to produce the figures.
The details of the models used in Sect. 3 and Sect. 6 can be found in ?, 26, 31, 148 and references therein.