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# Thunderstorm charge structures producing gigantic jets

## Abstract

Gigantic jets are atmospheric electrical discharges that propagate from the top of thunderclouds to the lower ionosphere. They begin as lightning leaders inside the thundercloud, and the thundercloud charge structure primarily determines if the leader is able to escape upward and form a gigantic jet. No observationally verified studies have been reported on the thundercloud charge structures of the parent storms of gigantic jets. Here we present meteorological observations and lightning simulation results to identify a probable thundercloud charge structure of those storms. The charge structure features a narrow upper charge region that forms near the end of an intense convective pulse. The convective pulse produces strong storm top divergence and turbulence, as indicated by large values of storm top radial velocity differentials and spectrum width. The simulations show the charge structure produces leader trees closely matching observations. This charge structure may occur at brief intervals during a thunderstorm’s evolution due to the brief nature of convective pulses, which may explain the rarity of gigantic jets compared to other forms of atmospheric electrical discharges.

## Introduction

Nearly all gigantic jets have been found to originate in tall (14–18 km altitude), intense thunderstorms featuring overshooting tops that form in maritime tropical environments1,2,3,4,5,6,7,8,9. There has been one documented case of a gigantic jet emerging from a low topped (6.5 km altitude) winter thunderstorm over the Mediterranean Sea, but it also had an overshooting top as the environmental tropopause was near 6 km altitude10. When the emerging location of gigantic jets at the storm top can be accurately determined, they appear to escape from or near the convective core of the thunderstorm2,3,5,6,7,8. Often gigantic jets occur during or near the end of a convective ‘pulse’, which corresponds to a period of rapid thunderstorm intensification4,6,8. It has also been found that gigantic jets escape the thundercloud along an axis that marks the center of their parent storm’s divergent outflow8 and that some of these storms form in environments with large horizontal wind speeds near the altitudes of their respective thundercloud tops8,10,11.

Because gigantic jet observations are rare12,13, it is nearly impossible to obtain in-situ measurements (e.g., balloons, etc.) to understand the charge structure producing those unusual electrical phenomena. Our current knowledge of the charge structure is mainly obtained from modeling studies and analysis of meteorological and lightning data. Previous modeling studies have shown that for a normal polarity thunderstorm that has a classic tripolar charge structure (vertically stacked regions of upper positive, middle negative, and lower positive charge), gigantic jets are initiated between the middle negative and upper positive charge regions as a normal intracloud flash, with the negative leader subsequently escaping the upper positive charge14,15. This result is consistent with the observed gigantic jets being predominately of negative polarity and transferring negative charge to the ionosphere1,2,3,4,5,7,9,16. In order for the negative leader to escape the cloud, the upper positive charge must be weakened - likely by mixing with the upper negative screening charge layer14,15. This creates a charge imbalance between the main thundercloud charge regions, enabling the leader to escape.

The charge structures used in those modeling studies were not formulated from direct observations of gigantic jet producing convection, but were assumed to resemble a classic tripolar charge structure17,18. They featured a wide, weakened upper positive charge region over a similarly sized middle negative charge region14,15, augmented by a small lower positive charge region. The simulations were conducted with the top boundary set right above the cloud top, and their purpose was to determine if a lightning leader from a normal intracloud flash could escape the upper charge region. They did not answer where this leader would propagate once it left the upper charge and whether it would form a bolt-from-the-blue, jet, or gigantic jet. These studies, however, successfully demonstrated that a charge imbalance inside the thundercloud is required for a leader to escape.

Past observational studies have analyzed meteorological data6,8 of gigantic jet producing storms and analyzed the parent lightning16 associated with gigantic jets. Gigantic jet producing convection was found to have moderate to high convective available potential energy values (1200–3500 J kg−1) and 10 dBZ radar reflectivity values reaching 14–17 km altitude6,8. The storms analyzed by Meyer et al.6 had peak altitudes of very high frequency lightning data and radar reflectivity near the times of the gigantic jets, which is consistent with the storms undergoing a convective pulse. Lazarus et al.8 found that storm top turbulence, as inferred from eddy dissipation rate19, was at a maximum when the gigantic jets occurred. The gigantic jets also escaped along the center axis of divergent outflow near storm top. Lu et al.16 found the parent lightning as observed by very high frequency networks resembled an ordinary intracloud flash between the middle negative and upper positive charge regions, but with an ‘attempted’ bolt-from-the-blue discharge16 preceding the gigantic jets.

This paper attempts to identify the thundercloud charge structure that produces gigantic jets, and aims to determine how a lightning discharge develops after it leaves the thundercloud charge. The charge structure is formulated by combining lightning and radar data analysis of gigantic jet producing convection with an emphasis on thunderstorm charge structure, and from lightning simulations performed using a three dimensional probabilistic model. The probabilistic lightning simulations are performed over a larger simulation domain than those used in previous studies to study where the escaped lightning leader propagates. Simulations for two other thunderstorm charge structures, which are either proposed by previous modeling studies or possibly formed during the parent storm of gigantic jets, are also performed to identify the most likely charge structure that produces gigantic jets. Finally, this study seeks to understand the meteorological processes that form this charge structure.

## Results

To better understand the common thunderstorm features near cloud top during gigantic jets, base reflectivity, radial velocity, and spectrum width derived from Weather Surveillance Radar 88-Doppler (WSR-88D) radar scans were analyzed for four gigantic jet producing thunderstorms (Fig. 1 and Table 1). Reflectivity is a measure of the power scattered back to the radar from the target, radial velocity is the inbound/outbound velocity along a radial path extending from the radar, and spectrum width is a measure of the radial velocity spectrum in a radar bin. Spectrum width can be used as a proxy for turbulence, where large spectrum width is associated with increased turbulence20,21. The parent thunderstorm dates, approximate times, and locations are listed in Table 1. Some of the storms analyzed here were studied in detail before concerning their meteorological features and lightning activity, as obtained by radar measurements and lightning detection data6,8,16. Here we focus on analyzing the meteorological and lightning data that gives information of the temporal and spatial properties of the charge structure of those storms.

All storms had reflectivity values greater than 30 dBZ at high altitudes (>12 km) and tall thundercloud tops (Fig. 1a–d, column 1). Strong horizontally diverging winds near the thundercloud top were present in all cases (Fig. 1a–d, column 2), with radial velocity differentials of 26–55 m s−1 (Table 1). The vertical columns of high reflectivity were collocated with the centers of divergent outflow. Large values of spectrum width (9–11 m s−1) were horizontally displaced from and just outside the convective core (Fig. 1a–d, column 3). These large values of spectrum width indicate turbulent mixing20 near the thundercloud top, and are indicators of where upper negative screening charge may be mixed with upper positive charge11. These values are particularly large, as past studies have shown that spectrum width values greater than 4 m s−1 are considered turbulent22. Lastly, the National Lightning Detection Network intracloud (NLDN IC) events and very high frequency (VHF) lightning sources associated with each gigantic jet were located near the axis of divergent outflow and near the convective core - except for the attempted bolt-from-the-blue (discussed in Lu et al.16 - open white circles in Fig. 1a).

The charge structure producing gigantic jets is found from a combination of different radar variables, lightning data, and lightning simulations, but information about the charge structure is first obtained from re-analyzing available VHF lightning mapping data at different periods of the storms. The evolution of the VHF inferred charge structure for the 28 September 2010 Florida gigantic jet is investigated at four time periods, each about one minute long (Fig. 2 and Table 2). This case is chosen as it is closest to the VHF mapping system (about 60 km away). The details of determining the thundercloud charge structure are described in the Methods section. This storm underwent a convective surge or pulse near the time of the gigantic jet, as identified by Meyer et al.6 and supported by Fig. 2 and Table 2. The VHF charge analysis times are: before the convective pulse (pre-pulse), near the beginning of the convective pulse (initial pulse), near the end of the convective pulse and during the time of the gigantic jet (final pulse), and after the convective pulse (post-pulse). The specific time periods analyzed are shown in Table 2, and each period corresponds to a panel in Fig. 2. A similar analysis was completed for the 09 September 2010 Oklahoma thunderstorm that produced two gigantic jets6,16 and the 03 August 2013 Florida thunderstorm that produced four gigantic jets7,8, both of which had a charge structure evolution similar to the 28 September 2010 Florida thunderstorm (see Supplementary Information).

The thunderstorm that produced the 28 September 2010 Florida gigantic jet had a wide, diffuse upper positive charge region over a similarly sized middle negative charge region before the convective pulse (Fig. 2a). The large horizontal extent of the upper positive charge is indicated by the large azimuthal variation of VHF sources (ΔX), which was 25.5 km (Table 2). The mean altitude of VHF sources ($$\overline{Z}$$) in the upper positive charge region was 11.5 km altitude. The charge configuration during this time closely resembled a classic tripolar thunderstorm charge structure, but with a very small or nonexistent lower positive charge region as indicated by the lack of VHF sources at lower altitudes. Consistently, there were only three NLDN reported negative cloud-to-ground discharges in the time spanning fifteen minutes before the gigantic jet. There was a relatively weak divergence couplet present at storm top during this time, with a radial velocity differential of 29 m s−1 (not shown in Table 2). During the onset of the convective pulse, the number of VHF sources in the upper positive charge increased, and the upper positive charge climbed higher in altitude (Fig. 2b). The majority of VHF sources were contained within the relatively narrow reflectivity column (>30 dBZ). The intensity of the pulse was reflected by the altitude of the 45 dBZ echo, which reached a local maximum during this time (about 10 km, compared with 5 km before and after the pulse). The radial velocity differential at storm top increased to 38 m s−1. Near the end of the convective pulse and during the gigantic jet (Fig. 2c), the upper positive charge reached its highest altitude ($$\overline{Z}$$ of 13.2 km) and became very narrow, with a ΔX of 4.9 km. The majority of the VHF sources were confined within the boundaries of maximum spectrum width (marked as vertical lines in Fig. 2c) surrounding the reflectivity column (>30 dBZ), and there was a significant decrease in the number of VHF sources. The radial velocity differential at storm top reached its maximum during this time, with a value of 40 m s−1. The open white circles shown in Fig. 2c were from the ‘attempted bolt-from-the-blue’16, which was part of the parent gigantic jet flash. After the convective pulse, the upper positive charge widened again (ΔX of 12.7 km) and began to subside ($$\overline{Z}$$ of 12.9 km). However, the radial velocity differential at storm top remained large (39 m s−1) until the next radar volume scan (four minutes later), when the radial divergence couplet disappeared altogether.

### Probabilistic lightning simulations

Simulations using a three dimensional probabilistic lightning model14,15 were performed in conjunction with the data analysis discussed above in order to find the charge structure of gigantic jet producing storms (Fig. 3). Three charge structures were tested and simulations for each charge structure were run a total of ten times to find the discharge patterns that were most prevalent for each charge structure, so we would not report on outliers. If the discharge reaches the top boundary of the simulation domain, it is categorized as a possible gigantic jet. It should be noted that the model does not simulate the temporal evolution of the discharges, but the spatial characteristics of the discharges are simulated, similar to previous studies14,15,23,24.

Two charge structures feature narrow upper positive charge (Fig. 3a,d), but one has a distribution of upper negative screening charge around the top of the upper positive charge (Fig. 3a). The charge structure with upper negative screening charge is considered because it is probable that upper negative screening charge exists around the highest cloud tops of convective cells that produce negative gigantic jets. Considering the relaxation time at storm top is very short (about 15 s at 15 km altitude15,25), negative screening charge can form quickly at high storm tops. But, the strong storm top diverging winds should push away the screening charge from the center axis of the convective core, which would make a ‘hole’ in the upper negative screening charge. In nature this screening charge is likely not perfectly symmetric, but for the sake of simplicity, perfect axial symmetry is assumed. The altitudes and dimensions of the upper positive and middle negative charge regions in Fig. 3a,d are set according to the lightning and radar data in the previous section during the time of the gigantic jets. A charge structure with a wide upper positive charge region, similar to previous modeling studies of escaped leaders14,15 is also considered (Fig. 3g), which models the charge structure shown by Fig. 2a,d. Lower positive charge is likely very small immediately preceding and during gigantic jets6,11, as shown in Fig. 2 and discussed above, and therefore are not included in the charge structures. However, simulations with lower positive charge were also completed (see Supplementary Information), and the results indicate that the inclusion of small lower positive charge does not change the conclusions of this study. Considering the convective systems producing gigantic jets are very wide (about 40–60 km for the storms presented here), the lateral cloud edges are far from the parent GJ flash, so the lateral screening charges are not included in the simulations.

After specifying the geometry of the charge regions, the only key parameter left to specify the thunderstorm charge structure for simulation is the amount of charge in each region. Three constraints are used: (1) the leader discharge is initiated at the location where the electric field exceeds the threshold value for leader initiation (about 200 kV m−1 at ground pressure)26,27,28,29 by 1–10% (2) the charge densities must be less than the maximum values found from observations30,31,32,33 - on the order of a few tenths to a few nCm−3, and (3) the amount of net negative charge in the system that allows a negative leader to escape is minimized. The minimum amount of net negative charge is chosen because any additional amount of negative charge would always produce an escaped leader. This results in net charges of −65 C for Fig. 3a, −55 C for Fig. 3d, and −40 C for Fig. 3g. The minimum amount of net negative charge is different for each charge structure due to the different charge region geometries.

The discharge patterns shown in Fig. 3a,d closely resemble the parent lightning both inside and outside the thunderstorms producing gigantic jets. Within the confines of the cloud, the parent lightning has little lateral extension inside the upper positive charge region, which is consistent with the observations of the initial in-cloud discharge activity of gigantic jets16, as also shown by Fig. 2c. Upon exiting the thundercloud, the discharges escape as negative leaders extending upward above the convective core, similar to observations2,3,5,6,7,8. Finally, the discharges reach the upper boundary of the simulation domains. All ten (100%) simulations for each charge structure produced results closely resembling the discharge patterns presented in Fig. 3a,d, demonstrating the effectiveness of these charge structures to produce upward negative leaders with the capability to form gigantic jets.

The discharges simulated with a wide upper positive charge region (Fig. 3g) have large lateral extension and significant branching inside the upper positive charge region. The negative leader network in the upper positive charge extends significantly farther horizontally than the positive leader network in the middle negative charge, which contrasts with the initiating lightning observed by VHF sensors for gigantic jets16. When the discharge exits the upper positive charge, it bends significantly and terminates on the lateral boundary of the simulation domain. This is the dominant discharge pattern for this charge structure, with seven out of ten (70%) simulations giving such a discharge pattern. This indicates that this charge structure is conducive to propagating discharges laterally above the cloud instead of directly upward, possibly turning into a bolt-from-the-blue. The other three simulations (30%) did reach the top boundary of the simulation domain, but the discharge still showed significant bending, terminating near the upper corners of the domain. The charge structure shown by Fig. 3g also often produced multiple leaders exiting the upper positive charge, contrary to observations of gigantic jets.

Figure 3b,e,h shows the direction of −E overlaid on the thundercloud charge structure and discharge trees as the negative leaders exit the upper charge regions (note: in order to clearly show the direction of −E at every point the length of the arrow does not scale with its magnitude). We choose to show −E because its direction shows where a negative leader is most likely to propagate. The colored lines in these panels correspond to Fig. 3c,f,i and denote paths to calculate scaled electric potential (discussed below). In Fig. 3b,e, −E above the upper charge points toward the vertical symmetry axis (Distance = 10 km), and then upward, constraining the escaped negative leaders to propagate directly upward above the thundercloud. For the charge structure with wide upper positive charge (Fig. 3h), −E points outward from the vertical symmetry axis, encouraging the escaped negative leader to propagate laterally above the cloud.

Profiles of scaled electric potential (ϕscaled) above the upper charge regions are shown in Fig. 3c,f,i, with the paths indicated by the colored lines in Fig. 3b,e,h. Since the lightning propagation threshold field is linearly dependent on air density26,27,28,29, the potential along each path is normalized to ground pressure to allow for meaningful comparisons of leader propagation along different paths. This scaled potential is calculated by

$${\varphi }_{scaled}={\int }_{path}\,-\,\frac{{N}_{0}}{N}{\bf{E}}\cdot {\bf{dl}}$$
(1)

with all paths beginning at the box marked A and ending at the corresponding boxed letter, with N (N0) the density at a given altitude (ground). Each path is 10 km in length. The path from A → B is plotted twice in panels 3c,f,i for ease of viewing. For the charge structure with narrow upper positive charge and upper negative screening charge (Fig. 3a), two minima in ϕscaled exist along the lateral paths of A → F and A → D (Fig. 3c). Considering the escaping leader is of negative polarity, it propagates towards increasing potential. Thus, the escaping leader avoids the lateral paths (A → F and A → D). The diagonal paths (A → E, A → C) and the vertical path (A → B) have large values of ϕscaled, so the negative leader travels in the upward direction. But, the largest potential difference occurs along the vertical path (A → B), which prompts the escaped negative leader to propagate along the vertical symmetry axis. Figure 3f shows the scaled potential profile for the charge structure with narrow upper positive charge. The variation of ϕscaled along each path is similar to the previous case, but the vertical and diagonal paths have a smaller potential difference when compared to the charge structure of Fig. 3a. The escaped negative leader still propagates along the vertical symmetry axis where the largest potential difference is.

The charge structure with wide upper positive charge has a scaled potential profile above the cloud (Fig. 3i) that is significantly different from the charge structures with narrow upper positive charge. The largest directional derivative and largest potential difference occur for the lateral paths (A → F, A → D). Thus, the escaping negative leader propagates in the lateral direction above the cloud, instead of upward. Also, notice that the vertical and horizontal paths have a narrower spread of values in ϕscaled for this charge structure, compared with the other two cases, which increases the probability for the escaped leader to propagate in a random direction. However, the largest potential differences are along the lateral directions producing a dominant discharge pattern that extends laterally above the cloud.

Considering the amount of net charge is different among the three cases, additional simulations were completed for the charge structures of Fig. 3a,g, only with the net charge amounts switched (Fig. 4) to investigate the effect of net charge on the discharge leader discharge tree. The charge structure in Fig. 4a does not produce any escaped negative leaders out of the additional ten simulations, forming normal IC discharges for every simulation. For the charge structure in Fig. 4b, the escaped negative leader terminates on the lateral boundary of the simulation domain. This is the dominant discharge pattern for this charge structure, which occurs for seven (70%) of the simulations and is similar to the charge structure presented in Fig. 3g. From the results shown in Figs 3 and 4, it is clear that wide, weakened upper positive charge can produce escaped leaders at a smaller amount of net charge than narrow, weakened upper positive charge, but those escaped leaders propagate in the lateral direction once they escape. Narrow, weakened upper positive charge requires more net negative charge for a leader to escape, but once a negative leader escapes, it will propagate upward. This is also true when there is more net negative charge than shown in Fig. 3a,d. Thus, whether a successfully escaped leader will propagate upward, not laterally, to potentially form a gigantic jet is primarily determined by the geometry of the charge structure and not by the amount of net charge.

## Discussion

Due to the short time scale of screening charge formation at high altitudes15,25, the upper negative screening charge can approach the magnitude of the upper positive charge15, indicating the negative screening charge in Fig. 3a could potentially be much stronger. Simulations indicate (not shown) that when the upper negative screening charge in Fig. 3a is doubled to −50 C, the absolute value of the leader tip is about 60 MV when the ascending negative leader reaches the top boundary of the simulation (30 km altitude), which corresponds to a jump altitude of about 36 km (for negative streamers). This is similar to the jump altitudes of the last two gigantic jets produced by tropical depression Dorian7, which were about 35 km. Other things being equal under the constraints on the amount of charge in each region discussed in the previous section, the charge structure in Fig. 3a can allow accumulation of more net negative charge in the system, making it potentially easier to produce gigantic jets for this charge structure.

The wide, weakened upper positive charge structure (Fig. 3g) has a leader network above the cloud that extends significantly in the lateral direction. This charge structure assumes relatively uniform mixing of the upper negative screening charge throughout the entire volume of the upper positive charge, with a fully symmetrical configuration of the charge regions. Such a perfect symmetry likely does not occur in nature. A similar charge structure to that in Fig. 3g but with laterally displaced, weakened upper positive charge has been shown to produce bolt-from-the-blue discharges11, and the leader network shown in Fig. 3g resembles a bolt-from-the-blue discharge, consistent with that study.

The charge structure producing negative gigantic jets seems to be a result of the convective pulse. During the convective pulse, the radial velocity differentials and values of spectrum width at storm top reached a maximum (see Supplementary Figure S3). The maximum spectrum width values were located on the outer edge of the convective core (convective core as defined by reflectivity >30 dBZ). We theorize that the diverging winds push the negative screening charge away from the center axis of the highest cloud tops, and large turbulent eddies form around the convective core mixing negative screening charge with upper positive charge. This may explain how the narrow upper positive charge region found in Fig. 2c is formed. This hypothesis is supported by the electrical measurements made at high altitudes by U-2 airplanes above thunderstorms with high cloud tops37,38. These measurements show the vertical component of the electric field becomes increasingly positive as the airplane passes above the highest cloud tops of the thunderstorm. This suggests (for a normal polarity storm) the upper negative screening charge is being pushed to the sides of the overshooting top, creating a ‘hole’ in the screening charge layer, which reveals the upper positive charge (see Fig. 9 of Vonnegut et al.37). Mixing of upper negative screening charge with upper positive charge near high cloud tops has been hypothesized before, from observations of anomalous VHF activity in the upper regions of thunderstorms39,40,41,42,43. The authors of those studies speculated that this was caused by the upper negative screening layer being folded into the top of the thunderstorm41,42,43. This motion near cloud top is similar to the entrainment studies described by Blyth et al.44 and Stith45.

The findings reported here suggest that convective pulses creating overshooting tops are a primary driver in creating the charge structures that produce negative gigantic jets. However, convective pulses and overshooting tops are commonly found in supercell and multicell convection throughout the mid-latitudes, where gigantic jet observations are infrequent. The question is then: why are there not more gigantic jet observations from mid-latitude convection where intense updrafts and overshooting tops are commonplace? The answer to this question is likely related to the differences between the charge structures of mid-latitude and maritime tropical convection that have intense, pulsating updrafts that produce overshooting tops. Our limited results may be suggestive that (also see Supplementary Information) maritime tropical convection with pulsating overshooting tops exhibit relatively simple charge structures similar to the normal tripolar configuration with upper positive charge, middle negative charge, and possibly a small lower positive charge. This is supported by the dominant positive IC and negative CG discharges from gigantic jet producing storms6,7,8. In contrast, the charge structures in supercell convection have been found to be very complex, with anywhere from three to twelve charge regions existing simultaneously. These charge regions are often adjacent to each other, which results in small intracloud flashes that occur at very high rates, with total flash rates in supercell convection often reaching several hundred per minute40,42,46,47. Total flash rates from gigantic jet producing convection have been found to be an order of magnitude lower6,7,8 or even less. Mid-latitude supercell and multicell convection often have anomalous charge structures that have huge areas of middle/lower positive charge48,49,50. These storms exhibit large percentages of positive CG discharges (50–100% compared with <10% for normal convection51) and IC discharges between the middle negative and middle/lower positive charge regions (negative IC discharges). For these anomalously charged storms, the middle/lower positive charge participates in the majority of discharges, so few discharges take place in the upper parts of the thundercloud. Thus, for mid-latitude supercell and multicell convection that have intense updrafts and overshooting tops, other forms of discharges often win the competition to neutralize charge, such as discharges between adjacent pockets of charge and negative IC discharges, rather than the normal positive IC discharge between vertically stacked regions of upper positive and middle negative charge that is associated with the initiation of gigantic jets14,15,16.

## Methods

Due to the smaller azimuthal uncertainty of the VHF network52, the vertical cross sections in Fig. 2 were taken along the azimuthal direction. The statistics in Table 1 were completed by projecting the VHF points onto a plane perpendicular to a radial path extending from the VHF network. The value of ΔX was found by taking the one standard deviation σx about the mean in the azimuthal direction. The mean altitude was found by taking the mean of the VHF points in the vertical direction. The NLDN events associated with each gigantic jet were chosen as the closest positive IC event (negative charge moving upward) to each gigantic jet in space and time. Figures S1 and S2 and Table S1 were similarly created this way. However, considering the VHF mapping systems detected fewer points due to the large radial distance from the mapping systems and their respective thunderstorms, the analysis times only included points associated with the upper positive charge. Signatures of the positive leaders in the middle negative charge region were sparse and not included in Figures S1 and S2. Also, the mean altitude of the VHF points was not calculated, as the points had a large variation in the vertical direction.

### Lightning Simulation Model

The three dimensional probabilistic fractal model used in this study has been described in detail in other studies14,15,24. The simulations presented here used equidistant grid points of 400 m in the x, y, z directions and used open boundaries24 over a perfectly electrically conducting flat ground plane with zero potential. Simulations with a smaller or larger grid size gave similar results (see Fig. S5). When a smaller (larger) grid size was used, the net amount of negative charge needed to form an escaped leader was less (more). This is due to the leader trees with fine resolution being able to occupy the regions of thundercloud charge better than the leader trees with coarse resolution, which resulted in more charge on the leader tree for the fine resolution cases (this effect of the grid resolution was noted previously by Mansell et al.23). The simulation also uses an internal channel field of 1.0 kVm−1 to account for leader resistivity57, which is similar in magnitude to other lightning modeling studies23. The internal electric field is assumed to scale with neutral density as $$\frac{N}{{N}_{0}}$$, as it has been shown that the reduced electric field in the channel formed after streamer-to-leader transitions is about the same at 20 km and 40 km altitudes (Fig. 16 of da Silva and Pasko58), but it should be noted that the scaling law of the leader channel field has not been well studied. A larger vertical domain was also used (30 km altitude) compared to previous simulations of gigantic jets14,15. The top boundary of the simulation (terminal altitude) was chosen because the model can only simulate a conducting leader channel, and the gigantic jet is predominantly composed of leaders below this altitude58. The simulation with lower positive charge (Fig. S4) required more middle negative charge for the leader to escape.

The constraints to determine the amount of charge in each region were (1) the leader discharge is initiated at the location where the electric field exceeds by 1–10% of the threshold value of the ambient field for leader initiation (about 200 kVm−1 at ground pressure)26,27,28,29 (2) the charge densities must be less than the maximum values found from observations30,31,32,33 (a few tenths to a few nCm−3), and (3) the charge structures must have the minimum amount of net negative charge in the thundercloud for a negative leader to escape.

Each charge region has a Gaussian distribution. The charge structures in Fig. 3a,d,g all have middle negative charge regions centered at 7.5 km altitude, with a full width half maximum (FWHM) of 10 km in horizontal and 2 km in the vertical direction. The upper positive charge in Fig. 3a,d,g is centered at 13 km altitude. The FWHM of the upper positive charge in Fig. 3a,d is 5 km in the horizontal and 2.15 km in the vertical direction. The negative screening charge is centered at 14.0 km altitude, has a FWHM of 10 km in the horizontal and 0.80 km in the vertical direction. The ‘hole’ in the screening charge is created by slightly penetrating the narrow upper positive charge into the large wide screening layer. Lastly, the upper positive charge in Fig. 3g has a FWHM of 12 km in the horizontal and 2 km in the vertical direction.

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## References

1. 1.

Pasko, V. P., Stanley, M. A., Matthews, J. D., Inan, U. S. & Wood, T. G. Electrical discharge from a thundercloud top to the lower ionosphere. Nature 416, 152–154 (2002).

2. 2.

Su, H. T. et al. Gigantic jets between a thundercloud and the ionosphere. Nature 423, 974–976 (2003).

3. 3.

Cummer, S. A. et al. Quantification of the troposphere-to-ionosphere charge transfer in a gigantic jet. Nat. Geosci. 2, 617–620 (2009).

4. 4.

Soula, S. et al. Gigantic jets produced by an isolated tropical thunderstorm near Réunion Island. J. Geophys. Res. 116 (2011).

5. 5.

Huang, S.-M. et al. Optical and radio signatures of negative gigantic jets: Cases from Typhoon Lionrock (2010). J. Geophys. Res. 117 (2012).

6. 6.

Meyer, T. C. et al. Radar and lightning analyses of gigantic jet-producing storms. J. Geophys. Res. Atmos. 118, 2872–2888, https://doi.org/10.1002/jgrd.50302 (2013).

7. 7.

Liu, N. Y. et al. Upward electrical discharges observed above Tropical Depression Dorian. Nat. Commun. 6 (2015).

8. 8.

Lazarus, S., Splitt, M., Brownlee, J., Spiva, N. & Liu, N. Y. A thermodynamic, kinematic and microphysical analysis of a jet and gigantic jet-producing Florida thunderstorm. J. Geophys. Res. 120, 8469–8490, https://doi.org/10.1002/2015JD023383 (2015).

9. 9.

Liu, N. Y., McHarg, M. G. & Stenbaek-Nielsen, H. C. High-altitude electrical discharges associated with thunderstorms and lightning. J. Atmos. Solar Terr. Phys (2015).

10. 10.

van der Velde, O. A. et al. Multi-instrumental observations of a positive gigantic jet produced by a winter thunderstorm in Europe. J. Geophys. Res. 115 (2010).

11. 11.

Boggs, L. D. et al. An analysis of five negative sprite-parent discharges and their associated thunderstorm charge structures. J. Geophys. Res. 121, 759–784, https://doi.org/10.1002/2015JD024188 (2016).

12. 12.

Liu, N. Y., Dwyer, J. R., Stenbaek-Nielsen, H. C. & McHarg, M. G. Sprite streamer initiation from natural mesospheric structures. Nat. Commun. 6 (2015).

13. 13.

Chen, A. B. et al. Global distributions and occurrence rates of transient luminous events. J. Geophys. Res. 113 (2008).

14. 14.

Krehbiel, P. R. et al. Upward electrical discharges from thunderstorms. Nat. Geosci. 1, 233–237 (2008).

15. 15.

Riousset, J. A., Pasko, V. P., Krehbiel, P. R., Rison, W. & Stanley, M. A. Modeling of thundercloud screening charges: Implications for blue and gigantic jets. J. Geophys. Res. 115 (2010).

16. 16.

Lu, G. et al. Lightning development associated with two negative gigantic jets. Geophys. Res. Lett. 38 (2011).

17. 17.

Krehbiel, P. R. The electrical structure of thunderstorms. In The Earth’s Electrical Environment, 90–113 (National Academy Press, Washington, D.C., 1986).

18. 18.

Williams, E. R. The tripolar structure of thunderstorms. J. Geophys. Res. 94, 13151–13167 (1989).

19. 19.

Cohn, S. A. Radar measurements of turbulent eddy dissipation rate in the troposphere: A comparison of techniques. Journal of Atmospheric and Oceanic Technology 12, 85–95 (1995).

20. 20.

Istok, M. J. & Doviak, R. J. Analysis of the relation between doppler spectral width and thunderstorm turbulence. Journal of the atmospheric sciences 43, 2199–2214 (1986).

21. 21.

Kollias, P., Albrecht, B., Lhermitte, R. & Savtchenko, A. Radar observations of updrafts, downdrafts, and turbulence in fair-weather cumuli. Journal of the atmospheric sciences 58, 1750–1766 (2001).

22. 22.

Melnikov, V. M. & Doviak, R. J. Turbulence and wind shear in layers of large doppler spectrum width in stratiform precipitation. Journal of Atmospheric and Oceanic Technology 26, 430–443 (2009).

23. 23.

Mansell, E. R., MacGorman, D. R., Ziegler, C. L. & Straka, J. M. Simulated three-dimensional branched lightning in a numerical thunderstorm model. J. Geophys. Res. 107 (2002).

24. 24.

Riousset, J. A., Pasko, V. P., Krehbiel, P. R., Thomas, R. J. & Rison, W. Three-dimensional fractal modeling of intracloud lightning discharge in a New Mexico thunderstorm and comparison with lightning mapping observations. J. Geophys. Res. Accepted (2007).

25. 25.

MacGorman, D. R. & Rust, W. D. The Electrical Nature of Storms (Oxford Univ. Press, New York, NY, 1998).

26. 26.

Marshall, T. C., McCarthy, M. P. & Rust, W. D. Electric field magnitudes and lightning initiation in thunderstorms. J. Geophys. Res. 100, 7097–7104 (1995).

27. 27.

Dwyer, J. R. A fundamental limit on electric fields in air. Geophys. Res. Lett. 30 (2003).

28. 28.

Behnke, S. A., Thomas, R. J., Krehbiel, P. R. & Rison, W. Initial leader velocities during intracloud lightning: Possible evidence for a runaway breakdown effect. J. Geophys. Res. 110 (2005).

29. 29.

Williams, E. R. Problems in lightning physics–the role of polarity asymmetry. Plasma Sources Sci. Technol. 15, S91–S108 (2006).

30. 30.

Winn, W. P., Moore, C., Holmes, C. & Byerley, L. Thunderstorm on july 16, 1975, over langmuir laboratory: A case study. Journal of Geophysical Research: Oceans 83, 3079–3092 (1978).

31. 31.

Winn, W. P., Moore, C. B. & Holmes, C. R. Electric Field Structure in an active part of a small, isolated thundercloud. J. Geophys. Res. 86, 1187–1193 (1981).

32. 32.

Byrne, G. J., Few, A. A. & Weber, M. E. Altitude, thickness and charge concentration of charged regions of four thunderstorms during trip 1981 based upon in situ balloon electric field measurements. Geophys. Res. Lett. 10, 39–42 (1983).

33. 33.

Marshall, T. C. & Rust, W. D. Electric field soundings through thunderstorms. J. Geophys. Res. 96, 22297–22306 (1991).

34. 34.

Uman, M. A. The earth and its atmosphere as a leaky spherical capacitor. American Journal of Physics 42, 1033–1035 (1974).

35. 35.

Markson, R. Ionospheric potential variations obtained from aircraft measurements of potential gradient. Journal of Geophysical Research 81, 1980–1990 (1976).

36. 36.

da Silva, C. L. & Pasko, V. P. Vertical structuring of gigantic jets. Geophys. Res. Lett. 40, 3315–3319 (2013).

37. 37.

Vonnegut, B., Moore, C., Espinola, R. & Blau, H. Jr. Electric potential gradients above thunderstorms. Journal of the Atmospheric Sciences 23, 764–770 (1966).

38. 38.

Blakeslee, R. J., Christian, H. J. & Vonnegut, B. Electrical measurements over thunderstorms. J. Geophys. Res. 94, 13135–13140 (1989).

39. 39.

Krehbiel, P. R. et al. Gps-based mapping system reveals lightning inside storms. Eos, Transactions American Geophysical Union 81, 21–25 (2000).

40. 40.

Bruning, E. C., Rust, W. D., MacGorman, D. R., Biggerstaff, M. I. & Schuur, T. J. Formation of charge structures in a supercell. Monthly Weather Review 138, 3740–3761 (2010).

41. 41.

Emersic, C., Heinselman, P., MacGorman, D. & Bruning, E. Lightning activity in a hail-producing storm observed with phased-array radar. Monthly Weather Review 139, 1809–1825 (2011).

42. 42.

Calhoun, K. M., MacGorman, D. R., Ziegler, C. L. & Biggerstaff, M. I. Evolution of lightning activity and storm charge relative to dual-doppler analysis of a high-precipitation supercell storm. Monthly Weather Review 141, 2199–2223 (2013).

43. 43.

MacGorman, D. R., Elliott, M. S. & DiGangi, E. Electrical discharges in the overshooting tops of thunderstorms. J. Geophys. Res. (2017).

44. 44.

Blyth, A. M., Cooper, W. A. & Jensen, J. B. A study of the source of entrained air in montana cumuli. Journal of the atmospheric sciences 45, 3944–3964 (1988).

45. 45.

Stith, J. L. Observations of cloud-top entrainment in cumuli. Journal of the atmospheric sciences 49, 1334–1347 (1992).

46. 46.

MacGorman, D. R. et al. Telex the thunderstorm electrification and lightning experiment. Bulletin of the American Meteorological Society 89, 997–1013 (2008).

47. 47.

Wiens, K. C., Rutledge, S. A. & Tessendorf, S. A. The 29 june 2000 supercell observed during steps. part ii: Lightning and charge structure. Journal of the atmospheric sciences 62, 4151–4177 (2005).

48. 48.

Lang, T. J. et al. The severe thunderstorm electrification and precipitation study. Bulletin of the American Meteorological Society 85, 1107–1125 (2004).

49. 49.

Tessendorf, S. A., Rutledge, S. A. & Wiens, K. C. Radar and lightning observations of normal and inverted polarity multicellular storms from steps. Monthly Weather Review 135, 3682–3706 (2007).

50. 50.

Weiss, S. A., Rust, W. D., MacGorman, D. R., Bruning, E. C. & Krehbiel, P. R. Evolving complex electrical structures of the steps 25 june 2000 multicell storm. Monthly Weather Review 136, 741–756 (2008).

51. 51.

Orville, R. E. Cloud-to-ground lightning flash characteristics in the contiguous united states: 1989–1991. Journal of Geophysical Research: Atmospheres 99, 10833–10841 (1994).

52. 52.

Thomas, R. J. et al. Accuracy of the Lightning Mapping Array. J. Geophys. Res. 109 (2004).

53. 53.

Boccippio, D., Heckman, S. & Goodman, S. A diagnostic analysis of the kennedy space center ldar network: 1. data characteristics. Journal of Geophysical Research: Atmospheres (1984–2012) 106, 4769–4786 (2001).

54. 54.

Thomas, R. J. et al. Observations of vhf source powers radiated by lightning. Geophysical research letters 28, 143–146 (2001).

55. 55.

Velde, O. A. & Montanyà, J. Asymmetries in bidirectional leader development of lightning flashes. Journal of Geophysical Research: Atmospheres 118 (2013).

56. 56.

Shao, X. M. & Krehbiel, P. R. The spatial and temporal development of intracloud lightning. J. Geophys. Res. 101, 26641–26668 (1996).

57. 57.

Williams, E. & Heckman, S. Polarity asymmetry in lightning leaders: The evolution of ideas on lightning behavior from strikes to aircraft. AerospaceLab p –1 (2012).

58. 58.

da Silva, C. L. & Pasko, V. P. Dynamics of streamer-to-leader transition at reduced air densities and its implications for propagation of lightning leaders and gigantic jets. J. Geophys. Res. 118(13), 561–13,590 (2013).

## Acknowledgements

The authors would like to thank Gaopeng Lu and Ted Mansell for their discussion. We express thanks for Robert Brown and Jennifer Wilson from the NASA Kennedy Space Center for providing the VHF LMA data. We thank Ron Holle (ron.holle@vaisala.com) at Vaisala for providing the NLDN lightning data and his support on this project. We express great gratitude to Timothy Lang and Eric Bruning for their input on high altitude airplane measurements. This research was supported in part by NSF grants AGS-0955379, AGS-1552177 and AGS-1348046.

## Author information

### Affiliations

1. #### Aerospace, Physics and Space Sciences Department, Florida Institute of Technology, Melbourne, FL, USA

• Levi D. Boggs
• , Jeremy A. Riousset
•  & Hamid K. Rassoul
2. #### Department of Physics and Space Science Center, University of New Hampshire, Durham, NH, USA

• Ningyu Liu
•  & Feng Shi
3. #### Ocean Engineering and Marine Sciences, Florida Institute of Technology, Melbourne, FL, USA

• Steven Lazarus
4. #### College of Aeronautics, Florida Institute of Technology, Melbourne, FL, USA

• Michael Splitt

### Contributions

L.D.B. and N.Y.L. drafted the manuscript and N.Y.L. supervised the project. J.A.R. developed the lightning simulation model. L.D.B., J.A.R., and F.S. performed the lightning simulations. L.D.B., M.E.S., S.M.L. and H.K.R. performed the meteorological analysis. All authors contributed to the discussion of the results and preparation of the manuscript.

### Competing Interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Levi D. Boggs or Ningyu Liu.


text:
# Is there a rules-based way to make a dragon player character? [duplicate]

I got to thinking Wouldn't it be cool to play some kind of proper dragon with Character levels?

So I wonder, is there a rules-based way to make a dragon player character?

My guess is that this is one of those DM fiat sort of things, and that it could be unbalanced. But is there any way?

# A metallic dragon's Change Shape doesn't give them class features

The dragon magically polymorphs into a humanoid or beast that has a challenge rating no higher than its own, or back into its true form... Its statistics and capabilities are... replaced by those of the new form, except any class features or legendary actions of that form.

[emphasis mine]

## However...

A GM can add class features to any monster using the rules for Creating a Monster in the Dungeon Master's Guide

See this question for more details on this option.

• I read the question as about making a PC dragon (and it's conflating “having class levels” with “being a player character”). If you see that too, it may be worth addressing directly. – SevenSidedDie Jun 2 '18 at 17:00
• @SevenSidedDie I didn't read it that way but I can see that as an interpretation. I'll ask a clarifying comment – David Coffron Jun 2 '18 at 17:01

format: plain


text:
# Integral $\int \sqrt{4+4x^2+1/x^2} dx$

I'm totally lost on how I would integrate the following function:

$f(x) = \sqrt{4+4x^2+1/x^2}$

If anyone could even just point me to the method of integration that would be grand.

-
If I have understood it correctly (see my edit), you could notice that it is a perfect square. –  Ross Millikan Nov 20 '12 at 1:14
You are probably solving an arclength problem. Typically, when you have a function $y$, differentiate, square the derivative, add $1$, take the square root, you get an awful mess that can't be integrated in elementary terms. But with very careful choice of constants, one can produce artificial situations where things magically simplify. –  André Nicolas Nov 20 '12 at 1:25
Note that it is non-trivial to compute general integrals of this form when the function under the radical is not a quadratic perfect square of linear factors. –  tacos_tacos_tacos Nov 20 '12 at 1:26

$$4 + 4x^2 + \dfrac1{x^2} = \left(2x + \dfrac1x \right)^2$$ Hence, \begin{align} \sqrt{4 + 4x^2 + \dfrac1{x^2}} & = \begin{cases} 2x + \dfrac1x & x > 0\\ -\left( 2x + \dfrac1x\right) & x < 0\end{cases}\\ & = 2 \vert x \vert + \dfrac1{\vert x \vert} \end{align}
Compute the square root!${}{}{}{}{}$


text:
[OS X TeX] List of feature requests on this list

Alain Schremmer schremmer.alain at gmail.com
Sun Feb 23 22:07:24 EST 2014

On Feb 23, 2014, at 9:58 PM, Alain Schremmer wrote Re: [OS X TeX] Wish
for TeXShop : attach custom icon to a given file:

> I even used to keep track of these requests. Eventually, it seemed
> that most of them would be incorporated, if rather inconspicuously.

May 11, 2009 4:08:00 PM EDT, Alain Schremmer wrote:

Here it is:

In order to facilitate later archive searches, features being
suggested/requested on this list should probably be suggested/
requested and discussed IN SEPARATE THREADS, UNDER THEIR OWN NAME,
rather than in this thread. Note that this List of feature requests on
this list does refer to such posts for details.

Suggesting a TeXShop feature on this list is absolutely not an
alternative to making a request at  http://sourceforge.net/tracker/?group_id=155796&atid=797241
.

This "wish list" is a compilation of TeXShop features being suggested/
requested on this list. Omissions result from my negligence rather
than from my malevolence. (This thread started as a way to gauge the
importance accorded to features suggested/requested on this list but,
while "voting" didn't seem to generate much interest, "keeping track"
seems to have.)

============================================
Newly Granted Wish(es)"".
––––––––––––––––––––––––––––––––––––––––––––
New Wish(es):
============================================

Features requests that are indented were granted and remain here only
for the record.

Hopeful regards
--schremmer

—A: Line numbers in the source file window (that the Console refers
to) as an option (to turn it off on simple documents) TeXShop now has
optional line numbers (At least as of v2.25)

—B: Change font size in the source window (Available: Source > Font
> Bigger/Smaller)

—C: Change Command I  from meaning Source > Font > Italic to a
keyboard command for \emph (Available under Leopard: Goedde, March 24,
2008 11:35:58 AM EDT & Also works for Tiger: Munn March 30, 2008
7:26:50 PM EDT)

—D: Change Command B from meaning Source > Font > Bold to a keyboard
command for \textbf (Available under Leopard: Goedde, March 24, 2008
2:43:20 PM EDT & Also works for Tiger: Munn March 30, 2008 7:26:50 PM
EDT)

E—A way to collapse one or more sections' text / better yet an
outliner mode

F1—A SECOND Typeset button, one that does NOT save at the same time
F2—A preference option to control whether Typeset saves or not - and
link that to the keyboard typeset command.
F3—Have a Command-click/Option-click/Control-click choice that runs
the file but doesn't save it. (See Litchfield, March 3, 2009 4:14:29
PM EST)

—G: Let the bottom of the source window be standard, with a
horizontal scroll bar (Available: Source > Wrap Lines > None)

—H: The ability to split the output window as well as the source
window (if possible with sync) TeXShop can split the output window
with sync support. TeXShop does not sync the top portion of source to
the top portion of output and the bottom portion of source to the
bottom portion of output. Instead, sync from either portion of source
syncs to the active portion of output. Thus, wishes R1 and R2 have
been considered as granted.

—I: A button in the source window to open the output window (To
bypass the menu's File > Open for Preview)( Available: a) Window >
Source<>Preview, b) with a key press: Apple-1)

—J: Cursor staying put (Available: Pref > Document > Select on
Activate)

K1—Colored matched brackets.
K2—Indenting of the nested environments (Loukanova March 26, 2008
2:49:30 PM EDT)

L—Emacs <-> TeXShop. (Loukanova, March 24, 2008 5:51:47 PM EDT + 25
Mar 2008 07:47:57 +0100 (CET)

M1—An option to color $...$ strings a solid color.
M2—An option to color  $$...$$ strings a solid color. (Andretta
March 31, 2008 9:57:31 AM EDT)

N1—An option to always open documents (.tex and .pdf) in the same
position where they were last closed. (Schremmer March 30, 2008
6:50:45 PM EDT)
N2—When opening tex file, position at the same line number as when
last closed (Seguin, March 31, 2008 11:30:06 PM EDT)
N3—When opening pdf file, display the same page as when last closed
(Seguin, March 31, 2008 11:30:06 PM EDT)

O—An option that allows the Tags menu drop down to always display at
the same scroll position from which the last tag was selected.

P—Autocompletion based on recently used words . (Rosinski March 26,
2008 8:04:20 AM EDT & 8:44:30 AM EDT)

Q1—Easy access to "Finder Info" for the PDF output. (Moore March 30,
2008 5:39:42 PM EDT & Kew-Moore March 30, 2008 7:14:03 PM EDT)
Q2—Easy access to "PDF Info". (Moore March 30, 2008 5:39:42 PM EDT &
Kew-Moore March 30, 2008 7:14:03 PM EDT)

—R1: Ability to split the PDF preview window like you can split the
source window. ("There seems to be a work-around". Dyballa July 7,
2008 11:14:13 AM EDT) "It may require support from Apple developers"
Moore July 7, 2008 5:52:36 PM EDT). See H.
—R2: Ability to split the PDF preview window like you can split the
source window with, in addition, the two top panes being synchronized
and the two bottom panes being synchronized. See H.

S1—Ability to take "snapshots" à la Skim. (Goldstein July 7, 2008
1:49:49 PM EDT)
S2—A "spyglass" floating window that is the target of cross-reference
hyperlinks in the main PDF viewing window. (Moore July 7, 2008 5:52:36
PM EDT)

T—"An extensible tag list would be a nice addition". (Schulz March 2,
2009 9:24:07 AM EST)

U—"recognize the structure of math constructs so that consecutive
clicks of the mouse select increasingly larger contexts" (See Themis
Matsoukas, March 3, 2009 3:59:46 PM EST)

V — "highlight current line with cursor" (Arnold May 11, 2009 12:33:34
AM EDT)

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# Matrices and Evaluation

## Homework Statement

Hi all. I am doing this work and cant seem to find any information on this in any of my notes or text books. The question is, "Evaluate (if possible) AB, BA, CD and DC", this is what i need some help with.
I also have further on the question, "Evaluate | u |, | v |, u . v and u * v", but this one i can do already so i dont need help with this one.

I know that a single matrix can't be evaluated exactly and since i cant find anything on evaluating either a single matrix or multiple matrices, i assume at the moment that, it is not possible to evaluate these from the first question either?

The relevant equations are below:

$$A = \left(\begin{array}{c} 21 \ 60 & 2 \ 7 \end{array}\right)$$
$$B = \left(\begin{array}{c} 4 \ -5 & 3 \ -2 \end{array}\right)$$
$$C = \left(\begin{array}{c} 15 \ 3 \ 7 & 5 \ 12 \ 4 \end{array}\right)$$
$$D = \left(\begin{array}{c} 3 \ 5 \ 15 & 1 \ -1 \ 7 & -5 \ 1 \ 8 \end{array}\right)$$

Any help/ information would be greatly appreciated. Thanks

Last edited:

HallsofIvy
Homework Helper
You seem to be having a problem with the word "evaluate". You certainly can evaluate a single matrix- its value is itself. Similarly, "evaluating" AB simply means finding the product of matrices A and B. AB, CD, and DC also mean just "multiply the matrices".

Last edited by a moderator:
Ahh ok, i get it now. Thanks a lot.
Just done them all and they seem correct, ive still come up with DC not being possible due to there being a different number of columns in the first matrix compared to the number of rows in the second matrix.

Thanks again

HallsofIvy


text:
# setparts: Set partitions In RobinHankin/partitions: Additive Partitions of Integers

## Description

Enumeration of set partitions

## Usage

 1 2 3 4 setparts(x) listParts(x,do.set=FALSE) vec_to_set(vec) vec_to_eq(vec)

## Arguments

 x If a vector of length 1, the size of the set to be partitioned. If a vector of length greater than 1, return all equivalence relations with equivalence classes with sizes of the elements of x. If a matrix, return all equivalence classes with sizes of the columns of x do.set Boolean, with TRUE meaning to return the set partitions in terms of sets (as per sets package) and default FALSE meaning to present the result in terms of equivalence classes vec An integer vector representing a set partition

## Details

A partition of a set \mjeqnS=\left\lbrace 1,...,n\right\rbraceS=1,...,n is a family of sets \mjeqnT_1,...,T_kT1,...,Tk satisfying

• \mjeqn

i\neq j\longrightarrow T_i\cap T_j=\emptysetunion(Ti,Tj) empty if i != j

• \mjeqn\cup

_i=1^kT_k=Sunion(T1,T2,...,Tk)=S

• \mjeqn

T_i\neq\emptysetTi not empty for \mjeqni=1,..., k1,...,k

The induced equivalence relation has \mjeqni\sim ji~j if and only if i and j belong to the same partition. Equivalence classes may be listed using listParts()

There are exactly fifteen ways to partition a set of four elements:

 (1234) (123)(4), (124)(3), (134)(2), (234)(1) (12)(34), (13)(24), (14)(23) (12)(3)(4), (13)(2)(4), (23)(1)(4), (24)(1)(3), (34)(1)(2) (1)(2)(3)(4)

Note that (12)(3)(4) is the same partition as, for example, (3)(4)(21) as the equivalence relation is the same.

Consider partitions of a set S of five elements (named 1,2,3,4,5) with sizes 2,2,1. These may be enumerated as follows:

 1 2 3 4 5 6 7 8 > u <- c(2,2,1) > setparts(u) [1,] 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 [2,] 2 2 3 1 1 1 2 2 3 2 2 3 1 1 1 [3,] 3 2 2 3 2 2 1 1 1 3 2 2 2 1 2 [4,] 2 3 2 2 3 2 3 2 2 1 1 1 2 2 1 [5,] 1 1 1 2 2 3 2 3 2 2 3 2 1 2 2

See how each column has two 1s, two 2s and one 3. This is because the first and second classes have size two, and the third has size one.

The first partition, x=c(1,2,3,2,1), is read “class 1 contains elements 1 and 5 (because the first and fifth element of x is 1); class 2 contains elements 2 and 4 (because the second and fourth element of x is 2); and class 3 contains element 3 (because the third element of x is 3)”. Formally, class i has elements which(x==u[i]).

You can change the print method by setting, eg, option(separator="").

Functions vec_to_set() and vec_to_eq() are low-level helper functions. These take an integer vector, typically a column of a matrix produced by setparts() and return their set representation.

## Value

Returns a matrix each of whose columns show a set partition; an object of class "partition". Type ?print.partition to see how to change the options for printing.

## Note

The clue package by Kurt Hornik contains functionality for partitions (specifically cl_meet() and cl_join()) which might be useful. Option do.set invokes functionality from the sets package by Meyer et al.

## Author(s)

Luke G. West (C++) and Robin K. S. Hankin (R); listParts() provided by Diana Tichy

## References

• R. K. S. Hankin 2006. Additive integer partitions in R. Journal of Statistical Software, Code Snippets 16(1)

• R. K. S. Hankin 2007. “Set partitions in R”. Journal of Statistical Software, Volume 23, code snippet 2

• Kurt Hornik (2017). clue: Cluster ensembles. R package version 0.3-53. https://CRAN.R-project.org/package=clue

• Kurt Hornik (2005). A CLUE for CLUster Ensembles. Journal of Statistical Software 14/12. https://www.jstatsoft.org/v14/i12/

parts, print.partition
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 setparts(4) # all partitions of a set of 4 elements setparts(c(3,3,2)) # all partitions of a set of 8 elements # into sets of sizes 3,3,2. jj <- restrictedparts(5,3) setparts(jj) # partitions of a set of 5 elements into # at most 3 sets listParts(jj) # induced equivalence classes setparts(conjugate(jj)) # partitions of a set of 5 elements into # sets not exceeding 3 elements setparts(diffparts(5)) # partitions of a set of 5 elements into # sets of different sizes


text:
# Neutrino 2020

June 22, 2020 to July 2, 2020
US/Central timezone

## Search for muon-philic new light gauge boson at Belle II

Not scheduled
10m
Poster

### Speaker

Mr Yongsoo Jho (Yonsei University)

### Description

Motivated by the long-lasting 3.5σ discrepancy in the anomalous magnetic moment of muon, we consider a new muon-specific force mediated by a light gauge boson, $X$, with mass $m_X < 2 m_\mu$ and the coupling constant $g_X∼(10^{−4},10^{−3})$. We show that the Belle II experiment has a robust chance to probe such a light boson in $e^+e^- \to \mu^+ \mu^- X$ channel and cover the most interesting parameter space explaining the discrepancy with the planned target luminosity, $\int dt \ \mathcal{L} = 50 \ \text{ab}^{−1}$. From the signal of $\mu^- \mu^+ + E_{miss}$ at Belle II, we expect that the (invisibly decaying) muon-philic light ($m_X < 2m_\mu$) gauge boson can be probed down to $g_X \geq 1.5 \times 10^{−4} (4.6 \times 10^{−4}, 2.3 \times 10^{−4})$ for 50 (1, 10) ab$^{−1}$ search.

### Mini-abstract

Light muonic force search at Belle II by using "Dimuon+Missing Energy($\mu^+ \mu^+ + E_{miss}$)".

### Primary authors

Dr Po-Yan Tseng (Yonsei University) Prof. Seong Chan Park (Yonsei University) Mr Yongsoo Jho (Yonsei University) Prof. Youngjoon Kwon (Yonsei University)


text:
# How to Sample Approval Elections?

We study the multifaceted question of how to sample approval elections in a meaningful way. Our analysis aims to discern the properties of various statistical cultures (both established and new ones). Based on the map-of-elections framework by Szufa et al. [2020], we graphically represent statistical cultures; and, by that, provide an intuitive understanding of their differences and properties.

• 9 publications
• 29 publications
• 2 publications
• 17 publications
• 8 publications
• 7 publications
• 42 publications
05/17/2021

### Putting a Compass on the Map of Elections

Recently, Szufa et al. [AAMAS 2020] presented a "map of elections" that ...
09/24/2019

### Improved Summation from Shuffling

A protocol by Ishai et al. (FOCS 2006) showing how to implement distribu...
01/12/2022

### On the Statistical Complexity of Sample Amplification

Given n i.i.d. samples drawn from an unknown distribution P, when is it ...
08/13/2021

### Limitations of Local Quantum Algorithms on Random Max-k-XOR and Beyond

Motivated by a question of Farhi et al. [arXiv:1910.08187, 2019], we stu...
07/07/2020

### Exploring Consequences of Simulation Design for Apparent Performance of Statistical Methods. 2: Results from simulations with normally and uniformly distributed sample sizes

This report continues our investigation of effects a simulation design m...
04/15/2020

### A Philosophy of Data

We argue that while this discourse on data ethics is of critical importa...
08/09/2018

### Sample size estimation for power and accuracy in the experimental comparison of algorithms

Experimental comparisons of performance represent an important aspect of...

## 1 Introduction

In an approval election bra-fis:b:approval-voting, each voter indicates which candidates he or she finds acceptable for a certain task (e.g., to be a president, to join the parliament, or to enter the final round of a competition) and a voting rule aggregates this data into the final outcome. In the single-winner setting (e.g., when choosing the president), the most popular rule is to pick the candidate with the highest number of approvals. In the multiwinner setting (e.g., in parliamentary elections or when choosing finalists in a competition) there is a rich spectrum of rules to choose from, each with different properties and advantages. Approval voting is particularly attractive due to its simplicity and low cognitive load imposed on the voters. Indeed, its practical applicability has already been tested in a number of field experiments, including those in France las-str:j:approval-experiment; bau-ige:b:french-approval-voting; bou-bla-bau-dur-ige-lan-lar-las-leb-mer:t:french-approval-voting-2017 and Germany alo-gra:j:german-approval-voting. Over the recent years there was also tremendous progress regarding its theoretical properties (see, e.g., the overviews of las-san:chapter:approval-multiwinner and lac:sko:t:approval-survey).

In spite of all these achievements, numerical experiments regarding approval voting are still challenging to design. One of the main difficulties is caused by the lack of consensus on which statistical election models to use. Below we list a few models (i.e., statistical cultures) that were recently used:

1. In the impartial culture setting, we assume that each vote is equally likely. Taken literally, this means that each voter approves each candidate with probability

bar-lan-yok:c:hamming-approval-manipulation. As this is quite unrealistic, several authors treat the approval probability as a parameter bre-fal-kac-nie2019:experimental_ejr; fal-sli-tal:c:vnw or require that all voters approve the same (small) number of candidates lac-sko:j:av-vs-cc. A further refinement is to choose an individual approval probability for each candidate lac-mal:c:vnw-shortlisting.

2. In Euclidean models, each candidate and voter is a point in , where is a parameter, and a voter approves a candidate if they are sufficiently near. Such models are used, e.g., by bre-fal-kac-nie2019:experimental_ejr and god-bat-sko-fal:c:2d. Naturally, the distribution of the candidate and voter points strongly affects the outcomes.

3. Some authors consider statistical cultures designed for the ordinal setting (where the voters rank the candidates from the most to the least desirable one) and let the voters approve some top-ranked candidates (e.g., a fixed number of them). This approach is taken, e.g., by lac-sko:j:av-vs-cc on top of the ordinal Mallows model (later on, caragiannis2022evaluating provided approval-based analogues of the Mallows model).

Furthermore, even if two papers use the same model, they often choose its parameters differently. Since it is not clear how the parameters affect the models, comparing the results from different papers is not easy. Our goal is to initiate a systematic study of approval-based statistical cultures and attempt to rectify at least some of these issues. We do so by extending the map-of-elections framework of szu-fal-sko-sli-tal:c:map and boe-bre-fal-nie-szu:c:compass to the approval setting.

Briefly put, a map-of-elections is a set of elections with a distance between each pair. Such a set of elections can then be embedded in a plane, by representing each election as a point and embedding the distance between the elections into Euclidean distances in the plane. Such a map-of-elections is useful as, by analyzing the distances between elections generated from various models, we can often obtain some insights about their nature.

To create a map-of-elections for approval elections, we start by identifying two metrics between approval elections, the isomorphic Hamming distance and the approvalwise distance. The first one is accurate, but difficult to compute, whereas the second one is less precise, but easily computable. Fortunately, in our election datasets the two metrics are strongly correlated; thus, we use the latter one.

Next, we analyze the space of approval elections with a given number of candidates and voters. For each , by -identity (-ID) elections we mean those where all the votes are identical and approve the same -fraction of candidates; and by -impartial culture (-IC) elections those where each voter chooses to approve each candidate with probability . We view -ID and -IC elections as two extremes on the spectrum of agreement between the voters and, intuitively, we expect that every election (where each voter approves on average a fraction of candidates) is located somewhere between these two. In particular, for , we introduce the -resampling model, which generates elections whose expected approvalwise distance from -ID is exactly the fraction of the distance between -ID and -IC (and the expected distance from -IC is the fraction).

Armed with these tools, we proceed to draw maps of elections. First, we consider -ID, -IC, and -resampling elections, where the and values are chosen to form a grid, and compute the approvalwise distances between them. We find that, for a fixed value of , the -resampling elections indeed form lines between the -ID and -IC ones, whereas for fixed values they form lines between -ID and -ID ones (which we refer to as the empty and full elections). We obtain more maps by adding elections generated according to other statistical cultures; the presence of the -resampling grid helps in understanding the locations of these new elections. For each of our elections we compute several parameters, such as, e.g, the highest number of approvals that a candidate receives, the time required to compute the results of a certain multiwinner voting rule, or the cohesiveness level (see Section 2 for a definition). For each of the statistical cultures, we present maps where we color the elections according to these values. This gives further insight into the nature of the elections they generate. Finally, we compare the results for randomly generated elections with those appearing in real-life, in the context of participatory budgeting.

## 2 Preliminaries

For a given positive integer , we write to denote the set , and as an abbreviation for .

#### Elections.

A (simple) approval election consists of a set of candidates and a collection of voters . Each voter casts an approval ballot, i.e., he or she selects a subset of candidates that he or she approves. Given a voter , we denote this subset by . Occasionally, we refer to the voters or their approval ballots as votes; the exact meaning will always be clear from the context. An approval-based committee election (an ABC election) is a triple , where is a simple approval election and is the size of the desired committee. We use simple elections when the goal is to choose a single individual, and ABC elections when we seek a committee.

Given an approval election (be it a simple election or an ABC election) and a candidate , we write to denote the number of voters that approve . We refer to this value as the approval score of . The single-winner approval rule (called AV) returns the candidate with the highest approval score (or the set of such candidates, in case of a tie).

For two voters and , their Hamming distance is , i.e., the number of candidates approved by exactly one of them. Other distances include, e.g., the Jaccard one, defined as . For other examples of such distances, we point to the work of caragiannis2022evaluating.

#### Approval-Based Committee Voting Rules.

An approval-based committee voting rule (an ABC rule) is a function that maps an ABC election to a nonempty set of committees of size . If an ABC rule returns more than one committee, then we consider them tied.

We introduce two prominent ABC rules. Multiwinner Approval Voting (AV) selects the candidates with the highest approval scores. Given a committee , its approval score is the sum of the scores of its members; . If there is more than one committee that achieves a maximum score, AV returns all tied committees. The second rule is Proportional Approval Voting (PAV). PAV outputs all committees with maximum PAV-score:

 scorepav(W)=∑v∈Vh(|A(v)∩W|),

where is the harmonic function. Intuitively, AV selects committees that contain the “best” candidates (in the sense of most approvals) and PAV selects committees that are in a strong sense proportional justifiedRepresentation. In contrast to AV, which is polynomial-time computable, PAV is NP-hard to compute azi-gas-gud-mac-mat-wal:c:approval-multiwinner; sko-fal-lan:j:collective

. In practice, PAV can be computed by solving an integer linear program

jair/spoc or by an approximation algorithm DudyczMMS20-tight-pav-apx.

#### Cohesive Groups.

Intuitively, a proportional committee should represent all groups of voters in a way that (roughly) corresponds to their size. To speak of proportional committees in ABC elections, justifiedRepresentation introduced the concept of cohesive groups.

###### Definition 1.

Consider an ABC election with voters and some non-negative integer . A group of voters is -cohesive if (i) and (ii) .

An -cohesive group is large enough to deserve representatives in the committee and is cohesive in the sense that there are candidates that can represent it. A number of proportionality notions have been proposed based on cohesive groups, such as (extended) justified representation justifiedRepresentation, proportional justified representation Sanchez-Fernandez2017Proportional, proportionality degree sko:c:prop-degree, and others. For our purposes, it is sufficient to note that all these concepts guarantee cohesive groups different types of representations represented (see also the survey of lac:sko:t:approval-survey for a comprehensive overview).

## 3 Statistical Cultures for Approval Elections

In the following, we present several statistical cultures (probabilistic models) for generating approval elections. Our input consists of the desired number of voters and a set of candidates . For models that already exist in the literature, we provide examples of papers that use them.

#### Resampling, IC, and ID Models.

Let and be two numbers in . In the -resampling model, we first draw a central ballot , by choosing approved candidates uniformly at random; then, we generate each new vote by initially setting and executing the following procedure for every candidate : With probability , we leave ’s approval intact and with probability we resample its value (i.e., we let be approved with probability ). The resampling model is due to this paper and is one of our basic tools for analyzing approval elections. By fixing , we get the -impartial culture model (-IC) where each candidate in each vote is approved with probability ; it was used, e.g., by bre-fal-kac-nie2019:experimental_ejr and fal-sli-tal:c:vnw. By fixing , we ensure that all votes in an election are identical (i.e., approve the same fraction of the candidates). We refer this model as -identity (-ID).

#### Disjoint Model.

The -disjoint model, where and  are numbers in and is a non-negative integer, works as follows: We draw a random partition of into sets, , and, to generate a vote, we choose uniformly at random and sample the vote from a -resampling model with the central vote that approves exactly the candidates from  (while the central votes are independent of , we still need this parameter for resampling).

#### Noise Models.

Let and be two numbers from and let be a distance between approval votes. We require that  is polynomial-time computable and, for each two approval votes and , depends only on , , and ; both distances from Section 2 have this property. In the -Noise model we first generate a central vote as in the resampling model and, then, each new vote  is generated with probability proportional to . Such noise models are analogous to the Mallows model for ordinal elections and were studied, e.g., by caragiannis2022evaluating. In particular, they gave a sampling procedure for the case of the Hamming distance. We extend it to arbitrary distances.

###### Proposition 1.

There is a polynomial-time sampling procedure for the -noise models (as defined above).

###### Proof.

Let be the central vote and let . Consider non-negative integers and such that and . The probability of generating a vote that contains candidates from and candidates from is proportional to the following value (abusing notation, we write to mean the value ; indeed, depends only on , , and ):

 f(x,y)=(zx)(m−zy)ϕd(x,y,z).

Next, let . To sample a vote, we draw values and with probability and form the vote as approving random members of and random members of . ∎

In the reminder, we only use the noise model with the Hamming distance and we refer to it as the -noise model. Note that the roles of and in this model are similar but not the same as in the -resampling model (for example, for we get the -ID model, but for get the -IC one).

#### Euclidean Models.

In the -dimensional Euclidean model, each candidate and each voter is a point from and a voter  approves candidate  if the distance between their points is at most (this value is called the radius); such models were discussed, e.g., in the classical works of Enelow and Hinich [enelow1984spatial,enelow1990advances], and more recently by elk-lac:c:ci-vi--approval, elk-fal-las-sko-sli-tal:c:2d-multiwinner, bre-fal-kac-nie2019:experimental_ejr, and god-bat-sko-fal:c:2d. We consider -dimensional models for , where the agents’ points are distributed uniformly at random on . We refer to them as 1D-Uniform and 2D-Square models (note that to fully specify each of them, we also need to indicate the radius value).

#### Truncated Urn Models.

Let be a number in and let  be a non-negative real number (the parameter of contagion). In the truncated Pólya-Eggenberger Urn Model berg1985paradox we start with an urn that contains all possible linear orders over the candidate set. To generate a vote, we (1) draw a random order from the urn, (2) produce an approval vote that consists of top candidates according to (this is the generated vote), and (3) return copies of to the urn. For , all votes with approved candidates are equally likely, whereas for large values of all votes are likely to be identical (so the model becomes similar to -ID).

## 4 Metrics

Next we describe two (pseudo)metrics used to measure distances between approval elections. Since we are interested in distances between randomly generated elections, our metrics are independent of renaming the candidates and voters.

Consider two equally-sized candidate sets and , and a voter with a ballot over . For a bijection , by we mean a voter with an approval ballot . In other words, is the same as , but with the candidates renamed by . We write to denote the set of all bijections from to . For a positive integer , by we mean the set of all permutations over . Next, we define the isomorphic Hamming distance (inspired by the metrics of fal-sko-sli-szu-tal:c:isomorphism).

###### Definition 2.

Let and be two elections, where , and . The isomorphic Hamming distance between and , denoted , is defined as:

 minσ∈Π(C,D)minρ∈Sn(∑ni=1ham(σ(vi),uρ(i))).

Intuitively, under the isomorphic Hamming distance we unify the names of the candidates in both elections and match their voters to minimize the sum of the resulting Hamming distances. We call this distance isomorphic because its value is zero exactly if the two elections are identical, up to renaming the candidates and voters. Computing this distance is -hard (see also the related results for approximate graph isomorphism arv-koe-kuh-vas:c:approximate-graph-isomorphism; gro-rat-woe:c:approximate-graph-isomorphism).

###### Proposition 2.

Computing the isomorphic Hamming distance between two approval elections is NP-hard.

Thus we compute this distance using a brute-force algorithm (which is faster than using, e.g., ILP formulations). Since this limits the size of elections we can deal with, we also introduce a simple, polynomial-time computable metric.

###### Definition 3.

Let be an election with candidate set and

voters. Its approvalwise vector, denoted

, is obtained by sorting the vector , in non-increasing order.

###### Definition 4.

The approvalwise distance between elections and with approvalwise vectors and is defined as:

 dapp(E,F)=|x1−y1|+⋯+|xm−ym|.

In other words, the approvalwise vector of an election is a sorted vector of the normalized approval scores of its candidates, and an approvalwise distance between two elections is the distance between their approvalwise vectors. We sort the vectors to avoid the explicit use of a candidate matching, as is needed in the Hamming distance. Occasionally we will speak of approvalwise distances between approvalwise vectors, without referring to the elections that provide them.

It is easy to see that the approvalwise distance is computable in polynomial time. Indeed, its definition is so simplistic that it is natural to even question its usefulness. However, in Section 6.3 we will see that in our election datasets it is strongly correlated with the Hamming distance. Thus, in the following discussion, we focus on approvalwise distances.

## 5 A Grid of Approval Elections

To better understand the approvalwise metric space of elections, next we analyze expected distances between elections generated according to the -resampling model.

Fix some number of candidates and parameters , such that is an integer, and consider the process of generating votes from the -resampling model. In the limit, the approvalwise vector of the resulting election is:

 ((1−ϕ)+(ϕ⋅p),…,(1−ϕ)+(ϕ⋅p)p⋅m,ϕ⋅p,…,ϕ⋅p(1−p)⋅m).

Indeed, each of the candidates approved in the central ballot either stays approved (with probability ) or is resampled (with probability , and then gets an approval with probability ). Analogous reasoning applies to the remaining candidates. With a slight abuse of notation, we call the above vector . Furthermore, we refer to as the -ID vector, to as the -IC vector, and to -ID and -ID vectors as the empty and full ones, respectively (note that 0-ID 0-IC and 1-ID 1-IC).

Now, consider two additional numbers, , such that is an integer. Simple calculations show that:

 dapp(empty,full) =m, dapp(p-IC,p-ID) =2mp(1−p), dapp(av(p,ϕ),av(p′,ϕ)) =m⋅|p−p′|, dapp(av(p,ϕ),av(p,ϕ′)) =2mp(1−p)⋅|ϕ−ϕ′|.

Thus is a  fraction of the distance between empty and full, and is a fraction of the distance between -IC and -ID (see also Figure 2). Furthermore, is the largest possible approvalwise distance.

Intuitively, -resampling elections form a grid that spans the space between extreme points of our election space; the larger the  parameter, the more “chaotic” an election becomes (formally, the closer it is to the -IC elections), and the larger the  parameter, the more approvals it contains (the closer it is to the full election). We use -resampling elections as a background dataset, which consists of elections with 100 candidates and 1000 voters each, with the following and parameters:

1. is chosen from and is chosen from the interval ,111By generating elections with a parameter from interval , we mean generating one election for each value , for .

2. is chosen from and is chosen from the interval .

For each of these elections we compute a point in , so that the Euclidean distances between these points are as similar to the approvalwise distances between the respective elections as possible. For this purpose, similarly to szu-fal-sko-sli-tal:c:map, we use the Fruchterman-Reingold force-directed algorithm fruchterman1991graph. For the resulting map, see the clear grid-like shape at the left side of Figure 3.222While our visualizations fit nicely into the two-dimensional embedding, our election space has a much higher dimension. Whenever we present maps of elections later in the paper, we compute them in the same way as described above (but for datasets that include other elections in addition to the background ones).

## 6 Experiments

In this section, we use the map-of-elections approach to analyze quantitative properties of approval elections generated according to our models. In particular, we will see how an election’s position in the grid influences each of the properties, and what parameters to use to generate elections with the quantitative property in a desired range.

### 6.1 Experimental Design

Concretely, we consider the following four statistics:

Max. Approval Score.

The highest approval score among all candidates in a given election, normalized by the maximum possible score, i.e., the number of voters.

Cohesiveness Level.

The largest integer such that there exists an -cohesive group (for committee size ).

Voters in Cohesive Groups.

Fraction of voters that belong to at least one 1-cohesive group (committee size ).

PAV Runtime.

Runtime (in seconds) required to compute a winning committee under the PAV rule, by solving an integer linear program provided by the abcvoting library abcvoting, using the Gurobi ILP solver.

We use six datasets. Five of them are generated using our statistical cultures and consist of 100 candidates and 1000 voters (except for the experiments related to the cohesiveness level, where we have 50 candidates and 100 voters, due to computation time). We have: 250 elections from the Disjoint Model (50 for each with ); 225 elections from the Noise Model with Hamming distance (25 for each with ); 225 elections from the Truncated Urn Model (25 for each with ); 200 elections from Euclidean Model (100 for 1D-Uniform, with radius in , and 100 for 2D-Square, with radius in ); these parameters are as used by bre-fal-kac-nie2019:experimental_ejr. The sixth dataset uses real-life participatory budgeting data and contains 44 elections from Pabulib pabulib, where for each (large enough) election we randomly selected a subset of 50 candidates and 1000 voters.

### 6.2 Experimental Results

Our visualizations are shown in Figures 3 and 4. We use the grid structure of the background dataset for comparison with other datasets. Notably, some of them do not fill this grid: the disjoint model (Figure 4b) is restricted to the lower half (i.e., the disjoint model does not yield elections with very many approvals), the Euclidean model (Figure 4

a) is restricted to the left half (due to the uniform distribution of points, its elections are rather “chaotic”), and the real-world dataset Pabulib (Figure

4d) is placed very distinctly in the bottom left part.

To get an intuitive understanding of the four statistics, let us consider the background dataset in Figure 3a. We see that the highest approval score is lowest in the lower left side and increases towards up and right. This is sensible: If the average number of approved candidates increases, so does this statistic; also, if voters become more homogeneous, high-scoring candidates are likely to exist. Moreover, regarding voters in cohesive groups, it turns out that in most elections almost all voters belong to a 1-cohesive group, with the left lower part as an exception (where there are not enough approvals to form -cohesive groups). The time needed to find a winning committee under PAV is correlated with the distance from 0.5-IC. We see that it takes the longest to find winning committees if the election is unstructured. Similar to the highest approval score, the cohesiveness level increases when moving up or right in the diagram. Cohesive groups with levels close to the committee size only exist in very homogeneous elections (rightmost path) and elections with many approvals (top part).

We move on to the results for the five other datasets. Note that each figure also contains the background dataset (gray dots) for reference. These results help to understand the differences between our statistical cultures.

The maximum approval score statistic provides an insight into whether there is a candidate that is universally supported. Instances with a value close to 1 possess such a candidate. In a single-winner election, this candidate is likely to be a clear winner. This is undesirable when simulating contested elections or shortlisting. Also note that in the real-world data set (Pabulib) we do not observe the existence of such a candidate.

When looking at the PAV runtime, we find some statistical cultures that generate computationally difficult elections, e.g., the -resampling model with parameter values close to and (0.5-IC), the noise model with parameters and , and the disjoint model with . Yet, instances from the real-world dataset, as well as from the Euclidean and urn ones, can be computed very quickly.333Less than 1 second on a single core (Intel Xeon Platinum 8280 CPU @ 2.70GH) of a 224 core machine with 6TB RAM. In contrast, the worst-case instance (0.3-IC) required 25 minutes on 13 cores.

Concerning voters in cohesive groups, whenever this statistic is close to 1, it is easy to satisfy most voters with at least one approved candidate in the committee; such committees are easy to find justifiedRepresentation. Since many proportional rules take special care of voters that belong to cohesive groups, in such elections there are no voters that are at a systematic disadvantage. In many of our generated elections (almost) all voters belong to -cohesive groups, but this is not the case for the real-world, Pabulib data. Indeed, to simulate Pabulib data well, we would likely need to provide some new statistical culture(s). For the cohesiveness level, we see that all models generate a full spectrum (i.e., ) of cohesiveness levels. That said, we expect realistic elections to appear in the “lower left” part of our grid (with few approvals), and such elections tend to have low cohesiveness levels. Indeed, this is also the case for Pabulib elections; hence it is important how proportional rules treat -cohesive groups with small .

### 6.3 Correlation

Figures 3 and 4 are based on the approvalwise distance. We argue that they would not change much if we used the (computationally intractable) isomorphic Hamming distance. To this end, we generated 363 elections with 10 candidates and 50 voters from the statistical cultures used in the previous experiment (detailed dataset description is in the appendix C). We compare Hamming and approvalwise distances; the results are presented in Figure 2. Each dot there represent a pair of elections and its coordinates are the distances between them, according to the Hamming and approvalwise metrics. The Pearson Correlation Coefficient is 0.989, and for 67% of elections the distances are identical.

## 7 Future Work

An important task for future work is to broadly study real-world datasets with the methods proposed in this paper. Most of these come from political elections with few candidates. As our analysis may be influenced by the number of candidates, a direct comparison with the figures in this paper is not possible; instead one has to rerun the experiments for a similar number of candidates.

## Acknowledgements

Martin Lackner was supported by the Austrian Science Fund (FWF), project P31890. Nimrod Talmon was supported by the Israel Science Foundation (ISF; GrantNo.630/19). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 101002854).

## Appendix A Proof of Proposition 2

###### Proof of Proposition 2.

We describe a reduction from the NP-hard Clique problem. An instance of Clique is , where is a graph, with set of vertices and set of edges .

Given such an instance of Clique, create two elections, and , with voters and candidates each. The intuitive idea of the reduction is for to correspond to and for to correspond to a -clique; then, a minimal Hamming distance would correspond to finding a -clique in . Details follows.

Denote the voters of by and the candidates of by . For each , let approve . Furthermore, for each , let approve each for which there is an edge in .

Denote the voters of by and the candidates of by . For each , let approve , and disapprove all other candidates.

Set the bound on the Hamming distance to be . This completes the description of the reduction. Next we show the two directions for correctness.

First, assume that there is a -clique in . Without loss of generality let be the vertices of the -clique. Then, match the voters of so that , , is matched to ; and match the candidates of so that , , is matched to . That way, all the approvals of correspond to approvals in the resulting ; in particular, for each and such that approves , the matched voter is approving the matched candidate. The number of approvals of the matched that correspond to disapprovals in are exactly .

For the other direction, note that the Hamming distance corresponding to any matching of the voters and candidates of equals to , where is the number of disapprovals between the first matched voters and candidates, thus only when corresponding to a -clique. ∎

## Appendix B Cohesive Groups

In this section we describe algorithms we used to obtain results regarding cohesive groups. The algorithm is based on the one provided by Anonymous [2022].

### b.1 Calculating Maximum Cohesiveness Level

We show an algorithm using Integer Linear Programming (ILP) that we used to calculate the maximum cohesiveness level of a group of voters.

Let be an approval election instance and be the committee size. We ask what is the maximum value of such that there exists an -cohesive group of voters, that is, a subset of at least voters that have at least common candidates, that is, approved by all of them.

First of all, let us point out that we can find by using binary search, because each -cohesive group is also an -cohesive group for any . We can also search for by looping through each possible value, that is, from to , and stop as soon as there is no cohesive group of a given the cohesiveness level. The second option may be better when we expect the cohesiveness level to be small, however, the first is better in the sense of theoretical computational complexity. In our algorithm, we focus on checking if there exists an -cohesive group for a given cohesiveness level .

Let us assume we are given an election , a committee size and a cohesiveness level . We show how to construct an ILP instance that indicates whether there exists an -cohesive group in . For the sake of brevity, we set and . Furthermore, let be the binary matrix of approvals for , that is, we have if the -th voter approves the -th candidate, and we have otherwise.

We note that, if there exists an -cohesive group of any size, then there also exists an -cohesive group of size exactly , that is, with the lowest possible size (we can just remove redundant voters, because the set of commonly approved candidates would not decrease). Thus it is enough to ask whether there exists an -cohesive group of size .

To construct our ILP instance, we create the specified variables:

1. For each voter

, we create a binary variable

, with the intention that if the -th voter belongs to the cohesive group, and otherwise.

2. For each candidate , we create a binary variable , with the intention that if all the selected voters (that is, specified by variables ) approve the -th candidate, and otherwise.

For convenience, we refer to the voters (to the candidates) whose () variables are set to as selected.

Now let us specify the constraints for these variables. First of all, we need to ensure that exactly voters and at least candidates have been selected:

 ∑ni=1xi=s, and ∑mj=1yj≥ℓ.

Furthermore, we need to ensure that each selected voter approves all the selected candidates. Thus, for each , we form a constraint:

 ∑ni=1aij⋅xi≥s⋅yj.

Now let us show how the above inequality works. On the one hand, if the -th candidate is not selected, then this inequality is satisfied trivially, because the right-hand side is equal to . On the other hand, if the -th candidate is selected, then the sum on the left-hand side must be at least , i.e., there must be at least selected voters who approve this candidate. Since there are exactly selected voters, all of them must approve the -th candidate.

Now we see that, if there exists an assignment which satisfies the above constraints, then the selected voters form an -cohesive group. Otherwise, there is no -cohesive group.

## Appendix C Correlation

The dataset used for comparing metrics in Figure 2 consists of: 40 elections from the Disjoint Models, 45 elections from the Noise Models with Hamming distance, 50 elections from the Truncated Urn Models, 50 elections from Euclidean Models, 134 elections from Resampling Models, 20 elections from IC, 20 elections from ID, and four extreme elections (i.e., 0.5-IC, 0.5-ID, Empty, Full).


text:
# Quandle properties

Let $G$ be a group, take Conjugacy class of $g$, let us denote it by $g^G$ and let $g^G=X$

Let $x, y \in X$, define $xy=xyx^{-1}$, then $X$ is closed under this operation

How I can proof this axiom if quandle group? for each $x, y \in X$, there exist a unique $z \in X$ such that $xz=y$, by using the above definition.

If we suppose there are two elements of $z_1,z_2 \in G$ such that $xz_1=y$ and $xz_2=y$ if we prove $z_1=z_2$ then it will be proved. for this

if we equate the above two equation $xz_1=xz_2$ then cancellation property does not hold how we can prove $z_1=z_2$? or is it possible to apply $x^{-1}$ on both sides in Quandle groups.

Let's normalize the notation a little since your use of juxtaposition to stand for two different operations makes your question hard to read. We'll only use juxtaposition for the group multiplication. $$x \rhd y = xyx^{-1}$$ Let $x,y \in X$. We want to find a $z \in X$ such that $x \rhd z = y$. Using that for all $\alpha \in X$, there is a $h(\alpha) \in G$ such that $\alpha = g^{h(\alpha)}$, we compute: \begin{align} && x \rhd z &= y \\ \iff&& xzx^{-1} &= y\\ \iff&& g^{h(x)}z(g^{h(x)})^{-1} &= g^{h(y)} \\ \iff&& z &= (g^{h(x)})^{-1}g^{h(y)}g^{h(x)} \\ \iff&& z &= (h(x)gh(x)^{-1})^{-1}h(y)gh(y)^{-1}h(x)gh(x)^{-1} \\ \iff&& z &= \left(h(y)^{-1}h(x)gh(x)^{-1}\right)^{-1}g\left(h(y)^{-1}h(x)gh(x)^{-1}\right) \in X \end{align} In short, the only group element that $z$ can be happens to be in $X$, as was to be shown.