Ferreira R.S.,Federal University of Rio de Janeiro |
Borges C.L.T.,Federal University of Rio de Janeiro |
IEEE Transactions on Power Systems
In this paper, we propose a flexible mixed-integer linear programming formulation of the AC OPF problem for distribution systems, using convexification and linearization techniques. The proposed formulation allows the representation of discrete decisions via integer decision variables, captures the nonlinear behavior of the electrical network via approximations of controllable accuracy, and can be solved to global optimality with commercial optimization solvers. The formulation is based on conventional variables that describe network behavior, which ensures its flexibility and the possibility of application to various distribution system problems, as we indicate with case studies. © 2014 IEEE. Source
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Exciting news for space lovers and exploration seekers: researchers have discovered the first gamma-ray pulsar outside the Milky Way – and it sets the record of being the most luminous known gamma-ray pulsar to date. Imaged by NASA’s Fermi gamma-ray Space Telescope, the pulsar lies in the Tarantula Nebula’s outskirts in the Large Magellanic Cloud, a small satellite galaxy of the Milky Way found 163,000 light years away. The Tarantula Nebula is the biggest, most active, and most intricate star-formation area in the galactic community, identified as a bright gamma-ray source. Lead scientist and astrophysicist Pierrick Martin said PSR J0540-6919 is responsible for about half of the gamma-ray brightness originally believed to hail from the nebula. "That is a genuine surprise," he said. The new findings were announced Nov. 13 in the journal Science. The highest-energy light form, gamma-rays are deemed borne out of subatomic particles that collided in the wake of supernova explosions. Extremely condensed having collapsed in on itself, a supernova rotates quickly; during spinning, electromagnetic field shoots out energy pulses in the form of gamma rays, X-rays, radio waves, and visible light. The Tarantula Nebula is known for another pulsar, PSR J0537−6910 (J0537), discovered with the help of the Rossi X-ray Timing Explorer (RXTE) satellite and spins at nearly 62 times a second, the fastest-known rotation time for a young pulsar. J0540, on the other hand, whirls at just under 20 times per second. Co-author Lucas Guillemot said J0540’s gamma-ray pulses have 20 times the intensity of the pulsar in the Crab Nebula, the previous record-holder. “[Y]et they have roughly similar levels of radio, optical and X-ray emission,” he explained. J0540 also has an age of about 1,700 years, twice of the Crab Nebular pulsar’s and in contrast with most of over 2,500 known pulsars ages 10,000 to hundreds of millions of years. It took over six years for the telescope and for a reanalysis of telescope data to detect the pulsations. Before Fermi was launched in 2008, there were only seven gamma-ray pulsars detected, unlike today when the mission has found over 160 within the Milky Way. Few gamma rays reach the telescope to detect the pulsations without knowing the period ahead of time. Discoveries such as this one, according to Guillemot, offer “a better understanding of the extreme physics at work in young pulsars.”
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After decades of studies and research, scientists have estimated the age of the observable universe to be roughly 13.8 billion years old. The connection between distance and the speed of light -- explained by Albert Einstein's theory of relativity -- has allowed scientists to look at different regions of the vast outer space which lie 13.8 billion light-years away. The age and distance of the universe -- are these small hints to the possible existence of alien life? Scientists have yet to form a firm conclusion, but in late November last year, some experts were able to detect five mysterious radio bursts which may have all come from outside the Milky Way galaxy. These radio signals were discovered after an "alien megastructure" was reported to be orbiting around a distant star known as KIC 8462852. "It almost doesn't matter where you point your telescope, because there are planets everywhere. If there's somebody out there, there are going to be so many of them out there that I do think there's a chance," explained astronomer Seth Shostak of the Search for Extraterrestrial Intelligence (SETI) Institute in California. Now, a new study presented at the annual meeting of the American Astronomical Society in Florida suggests that an old, densely-packed and isolated group of stars located within the Milky Way may possibly sustain extraterrestrial life. These stars, collectively called globular clusters, may be a cradle of advanced civilizations, experts said. The Possibility Of Alien Life In Globular Star Clusters Scientists from the Harvard-Smithsonian Center for Astrophysics (CfA) and the Tata Institute of Fundamental Research in Mumbai believe that globular star clusters may be the first place in our galaxy to contain intelligent life beyond Earth. What exactly are globular star clusters? These are densely-packed and tight groups that contain thousands or millions of stars. These balls of star clusters may be about 100 light-years across each other on average, and are as old as the Milky Way galaxy itself. Our galaxy is home to about 150 globular star clusters, where most of them orbit the galactic outskirts. On average, these star clusters may be 10 to 12 billion years old, just a couple billion years younger than the observable universe. But Houston, We Have A Problem The stars within globular clusters have fewer of the essential elements considered as "building blocks" of planets, such as silicon (Si) and iron (Fe), because these elements must be formed in earlier generations of stars. This lack in heavy elements has led other scientists to argue that globular star clusters are less likely to contain planets. In fact, only one planet has been found within globular clusters: the oldest known exoplanet called PSR B1620-26 b or Methuselah. Still, astronomers Rosanne DiStefano and Alak Ray said these views are "too pessimistic." "It's premature to say there are no planets in globular clusters," said Ray. The duo explained that a lot of exoplanets have been discovered around host stars that are only one-tenth as rich with metals as our Sun. While planets that are Jupiter-sized are found more around stars that contained higher levels of Fe and Si, planets that are Earth-sized show no such bias. Another main problem: because globular clusters are too close-knit, this specific environment could threaten the possible formation and existence of planets within it. Scientists said a neighboring star could wander too close to a planetary system, consequently disrupting the gravitational forces and resulting to the unfortunate hurling of worlds into interstellar space. What Could Be the Right Clue? DiStefano and Ray explained that the habitable zone or the "Goldilocks" zone of a star varies greatly. The Goldilocks zone is the right distance at which planets would be not too warm or not too cold to have liquid water. Brighter stars have more distant Goldilocks zones, and have shorter life spans. Because globular clusters are old, these extremely bright stars have died out. In contrast, planets that orbit around dimmer stars huddle closer to each other. These dimmer stars are faint and closer, but they also live long enough to become red dwarfs. Potentially habitable planets that these faint stars host would orbit nearby and be relatively safe from stellar interactions. "Once planets form, they can survive for long periods of time, even longer than the current age of the universe," said DiStefano. What If Planets Within Globular Clusters Evolve? If livable planets could form within globular star clusters and survive for billions of years, extraterrestrial life in said planets would have enough time to become complex and even develop intelligence. The alien civilization would truly be different from our own. In our solar system, the nearest star is about four light-years (24 trillion miles) away. In a globular cluster, the nearest star may be 20 times closer or only one trillion miles apart. Interstellar exploration and communication, as well as space travel, would definitely be easier. DiStefano and Ray call this potential theory the "Globular Cluster Opportunity." "Sending a broadcast between the stars wouldn't take any longer than a letter from the U.S. to Europe in the 18th century," said DiStefano. Space missions would definitely take less time. NASA's Voyager probes are 100 billion miles away from our planet. In terms of globular cluster distance, this is one-tenth as far as it would take to reach the nearest star. A civilization at Earth's current technological level could easily send interstellar probes within the realm of a globular star cluster. DiStefano said the nearest globular cluster to our planet is thousand light-years away. This is why it is difficult for us to find planets, particularly in a space environment with a crowded core. However, it is possible to detect globular cluster planets on galactic outskirts. Through gravitational lensing, scientists might even spot free-floating planets or planets whose gravity magnifies light from a star. Lastly, scientists say that using SETI search methods to target globular clusters is an intriguing idea. SETI uses arrays of radio telescopes called Allen Telescope Array (ATA) to look for laser or radio broadcasts. Astronomer Frank Drake used the Arecibo radio telescope to broadcast the first deliberate message from our planet to outer space, a message directed to globular cluster Messier 13 (M13) or the Hercules Globular Cluster.
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This story has been updated throughout with further information from the sentencing. On Wednesday, the former Reuters journalist Matthew Keys was sentenced to two years in prison for computer hacking. Keys, who once worked for Tribune Company-owned Sacramento television station Fox 40, left that job in 2010 and went on to copy and paste login credentials for the Tribune Company’s content management system (CMS) into a chatroom where members of the hacking collective Anonymous planned out their operations. (Keys still denies all allegations.) An unknown person under the username “sharpie” then went on to log into the CMS and deface a Los Angeles Times article. The article’s headline and dek (the subtitle beneath the headline) remained defaced for about forty minutes before an editor noticed and changed it back. In October 2015, a jury found Keys guilty of three counts of the Computer Fraud and Abuse Act (CFAA), a conviction that carried with it a maximum sentence of 25 years. A pre-sentence report prepared by the probation office recommended 87 months. In its sentencing memorandum, the US Attorney’s Office ended up seeking a lower sentence of five years. Keys’s attorneys asked for probation instead, claiming that the defacement did not result in enough loss to the Tribune Company to warrant any prison time. Their contention goes to one of the most controversial aspects of the case. In order to be convicted of felony under the particular provisions of the Computer Fraud and Abuse Act which prosecutors used to charge Keys, the conduct must exceed a threshold of $5,000. But the evidence brought at trial with respect to the total loss—and which was likely cited in the sealed presentence report prepared by the probation office—is tied to the Tribune Company’s reaction in the wake of the hack, including an extensive assessment of the entire CMS, as well as emails, phone calls, and meetings made by both journalists and highly-paid executives. In the still-sealed pre-sentence report (PSR), loss to the Tribune Company was placed at about $249,000. At trial, prosecutors presented evidence of loss ranging between $10,206 and $13,147. All of these numbers are lowballs compared to the numbers that were tossed out at various stages before trial. One document calculated loss at $929,977. In an unexpected twist, while going over the defense’s objections to the PSR, Judge Kimberly Mueller limited the amount of loss (for purposes of sentencing) to whatever had been presented at trial, thus drastically reducing the amount of prison time recommended by the sentencing guidelines. In the end, by the judge’s own determination, the appropriate range for sentencing was between 37 and 46 months. Prosecutors said in their sentencing recommendation that a “sentence of five years imprisonment reflects Keys’s culpability and places his case appropriately among those of other white collar criminals who do not accept responsibility for their crimes.” But when pressed to provide a recommendation within the judge’s guidelines, Assistant US Attorney Matt Segal asked for 42 months. “We have not gone overboard on this case, and a midrange recommendation doesn’t go overboard either,” he said to the judge. Keys, third from the left, after his sentencing on April 13, 2016. Photo: Sarah Jeong Although Keys’s lawyers said that the defacement was a prank borne out of the “spirit of the time,” AUSA Segal said that Keys’s actions weren’t motivated by mischief (or as one would have it, the lulz), but rather a vindictive desire to harm his former employer. “This is not just a prank. We might be talking about a prank if this were sharpie’s sentencing,” Segal said, adding that “This was rage driven by profound narcissism.” Segal pointed to chat transcripts that showed that sharpie had planned to deface the entire front page of the LA Times website the next day, a plan that had come to nothing because the Tribune Company was on high alert after the first defacement, and had taken steps that cut off Keys’s access to the CMS. “The only thing that limited his crime was that sharpie didn’t want to hurt the LA Times as much as Keys did,” said Segal in court. The story that the prosecution told at trial was not of a one-off, regretful copy/paste into a chatroom, but rather of a weeks-long harassment campaign launched at a former employer. In a taped confession, the validity of which Keys contests, Keys admitted to sending a series of harassing emails under various pseudonyms taken from the TV show The X-Files. Pseudonymous emails were also were sent to viewers who had signed up for emails from the television station—some of whom were elderly, and reacted very poorly. Only later did he contact Anonymous. In court, Segal read out a victim statement written by Brandon Mercer, Keys’s former supervisor at Fox 40. Mercer spoke on behalf of Fox 40. Mercer said that as a journalist who had broken “the sacred trust of his employment,” Keys should be held to a higher standard. Dan Gaines, formerly of the LA Times, also gave a victim statement on behalf of that paper. Gaines said that Keys’s actions posed a dire threat to the LA Times, due to how difficult it was to differentiate between fake and real news on the web today. “We are one of the few points of stability on the web. The risk is real, and for our industry, it’s crucial that we be a beacon in a confusing world," he said. The sentencing recommendation memo suggests that if anything, both the prosecution and the ultimate recommended sentence were at least partly spurred on by how little Keys endeared himself to law enforcement. “Keys’s characteristics include narcissism and an arrogant indifference to the suffering of innocent and vulnerable people,” prosecutors wrote in the memorandum. Keys has said that he was targeted for his work as a journalist, and that the prosecution was politically motivated. In the sentencing recommendation, prosecutors denied this, stating outright, “Keys was not targeted because he was a journalist.” Prosecutors said that these statements amounting to lying and promoting “cynicism about the justice system,” pointing out numerous statements that Keys had made on Twitter and to news outlets after the verdict. Defense attorney Jay Leiderman defended Keys’s statements, saying that even if the record did not show that Keys had been targeted for being a journalist (something the judge acknowledged very clearly), their client was still entitled to believe that he had been targeted for that reason. Leiderman also said that Keys’s unapologetic stance could hardly be held against him, since their view was that Keys had been charged and convicted under an unjust and overbroad law, and they were planning on appealing. “Congress is regressive not progressive,” said Leiderman. “And in fact they are completely inactive nowadays. Although there have been proposals for remediation, no action has been taken.” He said that the CFAA was a “buggy law” that did not fit the present day, and “with that, the punishments do not fit the crime.” Judge Mueller showed a keen interest in contextualizing the Keys case within the larger universe that it inhabits. She asked both sides to compare the Keys case to other cases involving members of Anonymous (including the case against the Paypal 14 and the case against Jeremy Hammond), and to justify variances from the sentences that those people had received. “Is Keys himself a hacker?” she asked the prosecutor at one point. “Or is he an instigator with modest computer skills?” “I think the parties agree, it’s not the crime of the century,” said Judge Mueller. In the end, she went below the guidelines she had calculated out, considering mitigating factors such as Keys’s otherwise clean record, his history of complying with law enforcement, and the unlikeliness that he would reoffend. She sentenced him to 24 months in prison, with 24 months supervised release following. In sentencing Keys, Judge Mueller said that the effect of the defacement was “relatively modest and did not do much to actually damage the reputation of that publication,” but that she could not ignore that his “intent was to wreak further damage which could have had further consequences.” She said that his actions had harmed his former supervisor, coworkers, successors, as well as viewers of the TV station. “The mask that Mr. Keys put on appeared to allow a heartless character to utter lines that are unbecoming of the professional journalist that he holds himself to be, and is, in most respects, in fact.” Outside the courtroom, Leiderman expressed disappointment with the decision, saying, “We walked in today facing an 87 month sentence, we walked out with 24. I suppose we should be heartened by that. But I’m not.” Keys is to surrender to a facility—likely a prison in Lompoc, California—on June 15, 2016. His defense team plans to appeal, and to stay his sentence pending that appeal.
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Extended Data Table 1 summarizes all the observations made in both the initial 2013 follow-up and 2015 May/June observations (project code p2886). The ALFA beam 0 pointing positions in J2000 equatorial coordinates are summarized in Extended Data Table 2. In the p2030/p2886 observations, the major axis of the ALFA receiver was rotated 19°/90° with respect to North4. In 2015 May/June, we searched for additional bursts from FRB 121102 using a grid of six pointings using the seven-beam ALFA receiver to cover a generous ~9′ radius around the discovery beam position and side-lobe. The ALFA receiver was aligned East–West to optimize the sky coverage for this specific purpose. The centre beams of the six grid pointings are shown in red in Fig. 1, and the six outer ALFA beams are shown in blue. Each grid pointing position was observed at least four times for ~1,000 s. The beam positions of the discovery observation and 2013 follow-up gridding4 with ALFA (in that case rotated 19° with respect to North) are also indicated using the same colour scheme. The outer six ALFA beams in the multiple grid observations are only at roughly the same position because the projection of the ALFA beams on the sky depends on the position of the telescope feed with respect to the primary reflecting dish, and these do not overlap perfectly between independent observations. Two bursts on May 17 (bursts 2 and 3) and two on June 2 (bursts 4 and 5) were detected at a single grid position: FRBGRID2b in ALFA beam 6, which had positions of α = 05 h 32 min 01 s, δ = +33° 07′ 56′′ and α = 05 h 32 min 01 s, δ = +33° 07′ 53′′ (J2000) at the two epochs—that is, only a few arcseconds apart. Six more bursts (bursts 6–11) were detected on June 2 at a neighbouring grid position, FRBGRID6b in ALFA beam 0, ~1.3′ away at α = 05 h 31 min 55 s, δ = +33° 08′ 13′′. In all cases bursts were detected in only one beam of the seven-beam ALFA receiver at any given time. This shows that the bursts must originate beyond Arecibo’s Fresnel length of ~100 km (ref. 19). The intermittency of FRB 121102 makes accurate localization more challenging. Nonetheless, the detection in adjacent grid positions is informative, and to refine the position of FRB 121102, we simply take the average position between FRBGRID2b ALFA beam 6 and FRBGRID6b ALFA beam 0, which gives: α = 05 h 31 min 58 s, δ = +33° 08′ 04′′ (J2000) and, equivalently, Galactic longitude and latitude l = 174.89°, b = −0.23°. The approximate uncertainty radius of ~3′ is based on the amount of overlap between the two detection beam positions and the ALFA beam width at half power, which is ~3.5′. The distance from the initially reported burst 1 position is 3.7′, consistent with the interpretation that this burst was detected in a side-lobe. Although FRB 121102 bursts have been detected in beams with different central sky positions, all detections are consistent with a well defined sky position when one considers the imprint of the ALFA gain pattern on the sky during each observation4. Noteworthy is the fact that FRB 121102 lies directly in the Galactic plane, whereas the other claimed FRBs lie predominantly at high Galactic latitudes. The PALFA survey is only searching in the Galactic plane, however, and no comparable FRB survey at 1.4 GHz with Arecibo has been done at high Galactic latitudes. Therefore, this difference may simply be a consequence of where Arecibo has most deeply searched for FRBs and does not necessarily suggest that FRB 121102 is of Galactic origin. Furthermore, FRB 121102 was found in the Galactic anti-centre region of the PALFA survey, whereas searches in the inner-Galaxy region have thus far found no FRBs14. This may be because the Galactic foregrounds in the anti-centre region are comparatively low, so the deleterious effects of DM smearing and scattering, which may reduce our sensitivity to FRBs, are less important in the outer Galaxy than the inner Galaxy. The low Galactic latitude of FRB 121102 also contributes to its low DM excess factor β ≈ 3 compared to the β ≈ 1.2–40 range seen for the other 15 FRBs in the literature. Only FRB 010621 (ref. 27), with β ≈ 1.2, has a lower β than FRB 121102, and it has been proposed to be Galactic28. We note, however, that six of 16 FRBs have DMs comparable to or lower than FRB 121102. Furthermore, its total Galactic DM excess pc cm−3 is larger than that of the first-discovered FRB1. Lastly, within a generous 20-degree radius of FRB 121102, the highest-DM pulsar known is the millisecond pulsar PSR J0557+1550 (ref. 29; also a PALFA discovery), which has DM = 103 pc cm−3and β = 0.6, as well as the highest DM-inferred distance15 of any pulsar in this region, d = 5.7 kpc. FRB 121102’s DM is clearly anomalous, even when compared to this distant Galactic anti-centre pulsar. At an angular offset of 38°, we note the existence of PSR J0248+6021, with DM = 370 pc cm−3 and β = 1.8. Although the DM of this young, 217-ms pulsar is in excess of the maximum Galactic contribution in the NE2001 model15, this can be explained by its location within the dense, giant H ii region W5 in the Perseus arm30 at a distance of 2 kpc. A similar association for FRB 121102 has been sought to explain its β ≈ 3, but multi-wavelength investigations have as yet found no unmodelled Galactic structure4, 19. In summary, FRB 121102’s comparatively low β does not strongly distinguish it from other FRBs, or necessarily suggest it is more likely to be Galactic. Here we provide a brief description of the Arecibo Mock spectrometer data and search pipeline14 used for our follow-up observations of FRB 121102. The 1.4-GHz data were recorded with the Mock spectrometers, which cover the full ALFA receiver bandwidth in two subbands. Each 172-MHz subband was sampled with 16 bits, a time resolution of 65.5 μs, and frequency resolution of 0.34 MHz in 512 channels. The data were later converted to 4-bit samples to reduce the data storage requirements. Before processing, the two subbands were combined into a single band of 322 MHz (accounting for frequency overlap between the two subbands), which was centred at 1,375 MHz and spans 1,214.3–1,536.7 MHz. We used the PALFA PRESTO-based16 search pipeline14 to search for astrophysical signals in the frequency and time domains. These data were processed using the McGill University High Performance Computing Centre operated by Compute Canada and Calcul Québec. The presence of RFI can have a detrimental effect on our ability to detect bursts. We therefore applied PRESTO’s rfifind software tool to identify contaminated frequency channels and time blocks. Flagged channels and time blocks were masked in subsequent analyses. Time blocks contaminated by RFI are identified using data that are not corrected for dispersive delay (that is, DM = 0 pc cm−3), in order to avoid removing astrophysical signals. The data were corrected for dispersion using 7,292 trial DMs in the range 0–9,866.4 pc cm−3, generating a time series at each trial. We performed Fourier analyses of all the time series to look for periodic signals using PRESTO’s accelsearch software tool and detected no significant signal of a plausible astrophysical origin. We searched for single pulses in each dispersion-corrected time series by convolving a template bank of boxcar functions with widths ranging from 0.13 ms to 100 ms. This optimizes the detection of pulses with durations longer than the native sample time of the data. Single-pulse events at each DM were identified by applying a signal-to-noise ratio (S/N) threshold of 5. These single-pulse events were grouped and ranked using the RRATtrap sifting algorithm17. An astrophysical pulse is detected with maximum S/N at the signal’s true DM and is detected with decreasing S/N at nearby trial DMs. This is not generally the case for RFI, whose S/N does not typically peak at a non-zero trial DM. The RRATtrap algorithm ranks candidates based on this DM behaviour and candidate plots are produced for highly ranked single-pulse groups. These plots display the S/N of the pulse as a function of DM and time as well as an image of the signal as a function of time and observing frequency (for example, Fig. 2). The resulting plots were inspected for astrophysical signals, and pulses were found at a DM of ~559 pc cm−3 at a sky position consistent with the discovery position of FRB 121102 (ref. 4). It is possible that the analysed data contain weaker bursts, which cannot be reliably identified because their S/N is too low to distinguish them from RFI or statistical noise. If, in the future, the bursts are shown to have an underlying periodicity, then this would enable a deeper search for weak bursts. Using several approaches, we searched for an underlying periodicity matching the arrival times of the eight bursts detected in the 2015 June 2 observing session. There are no significant periodicities detected through a standard fast Fourier transform of the time series. We then carried out a similar analysis to that routinely used to detect periodicities in sporadically emitting radio pulsars31. In this analysis, we calculate differences between all of the burst arrival times and search for the greatest common denominator of these differences. We found several periods, not harmonically related, that fitted different subsets of bursts within a tolerance of 1% of the trial period, but none that fitted all of the bursts. We subsequently calculated residuals for the times-of-arrival for the eight bursts detected on 2015 June 2 for a range of trial periods using the pulsar timing packages TEMPO and PINT (see ‘Code availability’ section). We found that some of the periods returned by the differencing algorithm also resulted in residuals with root-mean-square value of less than 1% of the trial period. However, there were many non-harmonically related candidate periods resulting in residuals of a comparable root-mean-square value. Furthermore, given the number of trials necessary for this search, none of these trial periods was statistically significant. In addition, owing to the small number of detected bursts, and the widths of the pulses, we were not sensitive to periodicities much shorter than ~100 ms because our tolerance for a period match (or acceptable root-mean-square value) becomes a large fraction of the period and there are many possible fits. The 16-day gap between the 2015 May and June detections precluded us from including the May bursts in any search for periodicity in the single pulses. To produce the spectra shown in the right panels of Fig. 2, we corrected each spectrum for the bandpass of the receiver. We estimated the bandpass by taking the average of the raw data samples for each frequency channel. We then median-filtered that average bandpass with a width of 20 channels to remove the effects of narrow-band RFI and divided the observed spectrum of each burst by this median-filtered bandpass. The band-corrected burst spectra shown in the right sub-panels of Fig. 2 are still somewhat contaminated by RFI, however. The bottom and top ten channels (3.4 MHz) of the band were ignored owing to roll-off in the receiver response. To characterize the bandpass-corrected spectrum of each burst, we applied a power-law model using least-squares fitting. The power-law model is described by S ∝ να, where S is the flux density in a frequency channel, ν is the observing frequency, and α is the spectral index. These measured spectral indices and their uncertainties are shown in Table 1. We do not include a spectral index value for burst 6 because of the RFI in the lower half of the band. For bursts 7 and 9, we exclude data below 1,250 MHz, because of RFI contamination. For bursts 8 and 10, the power-law model was not a good descriptor, and therefore no value is reported in Table 1. We verified this technique by applying the bandpass correction to PALFA data of pulsar B1900+01. The measured spectral index was calculated for ten bright single pulses, and the values are consistent with the published value. We measured the DM for 10 of the 11 bursts and additionally the dispersion index ξ (from the dispersive delay Δt ∝ ν−ξ) for the brightest two. The DM and the dispersion index were calculated with a least-squares routine using the SIMPLEX and MIGRAD functions from the CERN MINUIT package (http://www.cern.ch/minuit). The user specifies the assumed form of the intrinsic pulse shape, which is then convolved with the appropriate DM smearing factor. For these fits a boxcar pulse template was used. Subbanded pulse profiles for each burst were generated by averaging blocks of frequency channels. The number of subbands generated depended on the S/N of the burst to ensure that there was sufficient S/N in each subband for the fit to converge. Subbands with no signal were excluded from the fit. Furthermore, the data were binned in time to further increase the S/N and reduce the effects of frequency-dependent flux evolution. As the true intrinsic pulse width is not known, each burst was fitted with a range of boxcar widths. The parameters corresponding to the input template yielding the cleanest residuals are reported. The DM value was fitted keeping the DM index fixed at 2.0. We note that burst 7 was too weak and corrupted by RFI to obtain reasonable fits. Additionally, for the brightest two bursts (8 and 11), we also did a joint fit of DM and dispersion index. The resulting dispersion index fits were 2.00 ± 0.02 and 1.999 ± 0.002 for bursts 8 and 11, respectively. These values are as expected for radio waves travelling through a cold, ionized medium. Frequency-dependent pulse profile evolution introduces systematic biases into the times of arrival in each subband. These biases in turn bias the DM determination. These systematics cannot be mitigated without an accurate model for the underlying burst shape versus frequency, which is not available in this case, and is further complicated by the fact that the burst morphology also changes randomly from burst to burst. We estimated the systematic uncertainty by considering what DM value would produce a delay across our observing band that is comparable to half the burst width in each case. Table 1 presents the results of the fits with the statistical and systematic uncertainties both quoted. The DM estimates do not include barycentric corrections (of the order of 0.01–0.1 pc cm−3). Although FRB 121102 is close to the ecliptic, the angular separation from the Sun was always much larger than 10°, and any annual contribution to the DM from the solar wind was small (<10−3 pc cm−3)32, 33. These effects are, therefore, much smaller than the aforementioned systematics in modelling the DMs of the bursts. The ±1σ range of DMs for the ten new bursts is 558.1 ± 3.3 pc cm−3, consistent with the discovery value4, 557.4 ± 2.0 pc cm−3. The DM values and dispersion indices reported here and previously4 were calculated using different methods. These two approaches fitted for different free parameters, so different co-variances between parameters may result in slightly different values. Also, different time and frequency resolutions were used. Nonetheless, the burst 1 parameters quoted here and previously4 are consistent within the uncertainties. The consistency of the DM values is conclusive evidence that a single source is responsible for the events. Some FRBs have shown clear evidence for multi-path propagation from scattering by the intervening interstellar or extragalactic material along the line of sight2, 6, 7, 8. However, the burst profiles from FRB 121102 show no obvious evidence for asymmetry from multi-path propagation. An upper bound4 on the pulse broadening time from burst 1 is 1.5 ms at 1.5 GHz. Using the NE2001 model for a source far outside the Galaxy, the expected pulse broadening is ~20 μs × ν−4.4 with ν in gigahertz, an order of magnitude smaller than the ~2-ms pulse widths and ~0.7-ms intra-channel dispersion smearing. The features of the spectra cannot be explained by diffractive interstellar scintillations; the predicted scintillation bandwidth for FRB 121102 is ~50 kHz at 1.5 GHz, which is unresolved by the 0.34-MHz frequency channels of our data. We would, therefore, also not expect to observe diffractive interstellar scintillation in our bursts. Additional scattering occurring in a host galaxy and the intergalactic medium is at a level below our ability to detect. However, observations at frequencies below 1.5 GHz may reveal pulse broadening that is not substantially smaller than the upper bound if we use as a guide the observed pulse broadening from other FRBs2, 6, 7, 8. Future observations that quantify diffractive interstellar scintillations can provide constraints on the location of extragalactic scattering plasma relative to the source, as demonstrated for FRB 110523 (ref. 8). The upper bound on pulse broadening for FRB 121102 implies that the apparent, scattered source size for radio waves incident on the Milky Way’s interstellar medium is small enough that refractive interstellar scintillation (RISS) from the interstellar medium is expected. For the line of sight to FRB 121102, we use the NE2001 model to estimate an effective scattering-screen distance of ~2 kpc from Earth and a scattering diameter of 6 milliarcseconds. The implied length scale for phase-front curvature is then l ≈ 2 kpc × 6 milliarcseconds = 12 au. For an effective, nominal velocity, V = 100 × V km s−1, the expected RISS timescale is days. At 1.5 GHz and with an effective velocity due to Galactic rotation of about 200 km s−1 in the direction of FRB 121102, RISS timescales of 20–40 days are expected. Modulation from RISS can be several tens of per cent34. This level of modulation could play a part in the detections of bursts in 2015 mid-May and 2015 June and their absence in 2015 early-May and at other epochs. However, the Solar System and the ionized medium have the same Galactic rotation, so the effective velocity could be smaller than 100 km s−1, leading to longer RISS timescales. The beam positions used in Fig. 1 and the data of the bursts used to generate Fig. 2 are provided as Source Data files (available online with the figures). The code used to analyse the data are available at the following sites: PRESTO (https://github.com/scottransom/presto), RRATtrap (https://github.com/ckarako/RRATtrap), TEMPO (http://tempo.sourceforge.net/), and PINT (http://github.com/nanograv/PINT).