dataesr/scientific-paragraphs-categorization huggingface dataset:
5. Across all precipitation datasets, in 2005, an area ranging from 37 % to 51 % (mean 45 %) of the whole Amazon basin was moderately affected (Table  S1 , Fig. 2a ). ERA5 displayed the smallest area affected by moderate drought (2. 2 × 10 6 km 2 ; Table 1, Fig. 2 ), while CHIRPS and CRUNCEP showed a vast affected area (3. 0 × 10 6 km 2 ), an area about 36 % larger than displayed by ERA5. For severe and extreme drought conditions, ERA5 shows the smallest affected area with 3 % and 1 % of the basin affected. For severe drought conditions, CRUNCEP suggests that an area approximately 3 times larger was affected compared to ERA5 (0. 2 × 10 6 vs. 0. 6 × 10 6 km 2 ). CRUNCEP and GLDAS also encompass the largest area of extreme drought stress (0. 2 × 10 6 km 2 ; 3 % of the basin less than rMCWD < -2.
5; Fig. 2a ). During the 2010 drought, a larger area ranging between a minimum of 52 % (GPCC) and a maximum of 74 % (TRMM 6) was affected by moderate drought stress, which is about 36 % larger than during the 2005 drought (3. 6×10 6 vs. 2. 7 × 10 6 km 2 ; Table  S1 , Fig. 2 ). In addition, the area under severe drought stress was on average 25 % larger compared to 2005, and the area affected by extreme drought was double the size of the 2005 drought event. Particularly, GLDAS and TRMM 6 showed the largest area affected throughout the three drought classifications (Fig. 2b ). For 2016, two datasets (CHIRPS and CRUNCEP) showed with 38 % a considerably smaller area that was moderately affected by drought stress compared to GLDAS with 63 % of the area affected, respectively (datasets ranging between 2.
Spatial agreement of rainfall datasets using the rMCWD anomaly While the agreement of the total area affected by drought is relatively high (see Sect. 3. 1), the datasets only partly agree on the spatial extent and location of extreme drought conditions, particularly during the 2010 and 2016 events (Fig. 3 ). For 2005, all datasets agree on the drought epicentre being in central Amazonia. Datasets agree that an area of about 15 % of the Amazon basin was at least moderately affected (Fig. 3a ). Only a small overlap was found for the area affected by severe and extreme drought stress (Fig. 3b, c ). Here, only half of the datasets agreed on 4 % of central Amazonia being severely and 1. 5 % being extremely affected. For 2010, all datasets agreed on an affected area of 21 % in the Amazon basin, and half of the datasets agreed on an area of 60 % of the Amazon basin being moderately affected by drought stress (Fig.
3d ). The 2010 drought displayed no central hotspot, but rather there are three most affected areas in the eastern, southern and central parts of Amazonia on which most of the datasets agreed (Fig. 3d ). Severe drought stress in 2010 was in the southern part of Amazonia, where four datasets agreed (Fig. 3e ), while for extreme drought stress almost no overlap between datasets was found (Fig. 3f ). For 2016, all datasets agreed on an area of about 7 % of moderate drought stress, and half of the datasets agreed on 51 % of the basin being affected (Fig. 3g ). Agreement for severe and extreme drought stress was lower compared to the other drought years (Fig. 3h, i ). Most of the datasets located the epicentre of the drought in the northwestern Amazon basin. Some datasets also showed the south-central part of the basin being severely affected (Fig.
Constant vs. varying evapotranspiration rates: effects on drought severity and extent estimates We find that assuming a constant ET rate of 100 mm per month is only realistic in the northern part of the Amazon basin and only when compared to the DOLCE dataset (Fig. 4a, b ), which shows ET rates of about 100 mm per month during both the wettest (as averaged between June and August) and the driest months (as averaged between January and March). Using GLEAM, average ET rates are between 30 % and 50 % higher than 100 mm per month during the wettest months (Fig. 4c ) and remain higher than 100 mm per month also in the northern part during the dry season (Fig. 4f ). Evapotranspiration rates can be as low as 50 mm per month on average throughout the driest months for both ET datasets in the south of the basin (Fig.
4b  and f ). This spatial heterogeneity in evapotranspiration rates has implications for the extent and severity of drought stress expressed as the rMCWD anomaly when compared to using constant evapotranspiration. Using the two evapotranspiration datasets we find lower drought impacts across most parts of the Amazon basin for the 2 years of 2005 and 2010 (Fig. 4c, d, g, h ). In 2005 the mean area of moderate drought stress was lower when using variable ET: 44 % of the basin for GLEAM and 39 % for DOLCE, compared 46 % for a constant ET. Interestingly, these differences were not particularly located in the epicentre of drought during that year (see Fig. 3a, b, c ) but rather in the south and the high-latitude regions toward the Andes (Fig. 4c, g ). The total area of severe drought stress did only slightly decrease from 9 % (constant ET) to 8 % (GLEAM) and 7 % (DOLCE).
Comparison of drought indices: rMCWD, rscPDSI and rRAI anomalies Similar to rMCWD, there is variable agreement among datasets when evaluating the other two drought metrics, rRAI and rscPDSI (Fig. 5 ). The largest dry-season anomaly (rRAI) in 2005 was displayed by GPCC with 6. 5 % (0. 4 × 10 6 km 2 ; Table  2 ), followed by TRMM 7 with 5. 7 % of the Amazon basin being severely affected. ERA5 showed with 3 % the smallest area affected. In 2005, spatial patterns of rRAI matched with rMCWD anomalies despite rMCWD anomalies showing a larger area affected by severe drought stress (Fig. 5a, d ). rscPDSI displayed the smallest area affected by drought stress in 2005 also with GPCC and TRMM 7 showing the largest severely affected area with 5. 5 % and 3. 1 %, respectively. All other datasets showed less than 1 % of severely drought-affected areas in 2005.
The small spatial area of rscPDSI differed compared to the other two drought indicators (Fig. 5a, d, g ): some areas showed a strong disagreement between drought indices; e. g. central Amazonia was hit by severe drought stress according to rMCWD and rRAI (with three to four climate datasets in agreement), while, in contrast, rscPDSI did not indicate abnormally dry conditions there. In 2010, the differences in drought-affected areas were even more pronounced between the three indices (Fig. 5b, e, h ). Here, ERA5 and TRMM 7 showed the largest areas affected by severe drought stress based on the dry-season rRAI anomaly with 7 % and 5 %, respectively. Using rscPDSI all datasets showed an area between 1 % and 2. 5 % being severely affected. Interestingly, the area affected based on rMCWD roughly encompasses the area affected by rRAI but additionally shows a large area in the southeastern part of the basin being affected by severe drought stress (Fig.
5b, e ). In 2016, rscPDSI shows the largest area affected by drought stress with GLDAS showing 39 % (followed by TRMM 7, 16 %) of the basin being severely affected. Four datasets agreed on the affected area in the northeastern part of the basin (Fig. 5i ). Only one dataset (GLDAS) showed severe drought stress in 2016 when calculating dry-season rainfall anomalies (rRAI, Fig. 5c ), indicating no pronounced anomalies in dry-season rainfall according to all other datasets. rM-CWD and rscPDSI roughly agreed on the northern part of the basin being severely affected (Fig. 5f, i ). Average seasonal patterns are quite consistent across datasets but differ depending on the choice of drought index and drought event (Fig. 6 ). The strongest (most negative) rainfall anomaly was visible from May to July during the 2005 drought event (Fig.
6a ). Accumulating such low rain-   (d-f ) 2010 and (g-i) 2016. The colours indicate the number of datasets that agree on a specific drought level in a given pixel. Drought severity levels are defined as moderate (rMCWD < -0. 5), severe (rMCWD < -2. 0) and extreme (rMCWD < -2. 5). Orange pixels indicate areas where only one dataset shows the respective drought stress (no agreement is labelled "None"). White pixels represent areas where no dataset shows any drought signal. Note that in panels (a)-(f), TRMM 6 and GSWP3 were excluded, as they were either very similar to their successor (TRMM 7) or due to a similar reanalysis procedure (WATCH WFDEI). In panels (g)-(i), only six datasets were included which cover the full time period until 2016. fall estimates resulted in very low values of rCWD during that period (Fig.
Overall variability: precipitation datasets vs. drought indices vs. evapotranspiration datasets When assessing the variability in drought severity and extent across the nine different precipitation datasets, the two drought indices (rMCWD and rscPDSI) and the two evapotranspiration datasets (DOLCE and GLEAM), we find that across all drought events the choice of drought index accounts for roughly 60 % of the variability, while both the precipitation dataset and the evapotranspiration dataset account for 20 % each (Table  2 ).
Discussion We assessed the severity and spatial extent of the extreme drought years 2005, 2010 and 2015/16 in the Amazon region by computing different drought indices using a range of precipitation datasets. When analysing how drought conditions are captured in nine different precipitation datasets for the Amazon basin, we find that while the datasets mostly agree on the extent of the drought area, they differ in their location of drought 4. 1 Critical aspects regarding the detection of drought events in the Amazon basin.
Drought indices The idea of defining water deficit based on evapotranspiration rates goes back to  Stephenson (1998) , and the MCWD is now one of the most widely used indicators to assess drought stress in tropical forests  (Lewis et al. , 2011; Phillips et al. , 2009; Esquivel-Muelbert et al. , 2019). In its simplest form the calculation of MCWD only requires precipitation data and assumes a constant evapotranspiration (ET) rate of 100 mm per month  (Aragão et al. , 2007). Although the simplicity of rMCWD and aMCWD is a main advantage, a fixed ET (which we also used in our study) is inappropriate for regions other than the lowland tropics, where the lower sup- ply of energy may result in lower ET values. Most importantly, an approximated ET does not account for either seasonal variation (driven mainly by radiation, temperature and phenology) or spatial variation in ET related to soil and root properties  (Malhi et al.
Precipitation datasets For the three drought events in 2005, 2010 and 2016, CHIRPS, GLDAS and ERA5 diverted the most from the other datasets regarding the spatial drought extent. ERA5 shows the smallest area of moderate drought stress during 2005 but one of the largest areas in 2010 (Fig. 2 ). We found no obvious bias between the precipitation datasets regarding distribution and frequency of monthly rainfall (Fig. S2 ), with only ERA5 showing higher rainfall more frequently. Although TRMM 7 and CHIRPS are based on the same satellite data as their input, they differ regarding the size of the drought area, especially during 2016 (Fig. 2 ). Lewis et al. (2011)  estimated an area of 47 % (2. 5 × 10 6 km 2 ) of the Amazon basin moderately affected in 2005 using the TRMM 6 dataset, which compares well with the size of the affected area for most datasets analysed in our study (considering our 0.
6 × 10 6 km 2 larger study area; see Sect. 2). For 2010,  Lewis et al. (2011)  reported an area of 3. 2 × 10 6 km 2 being affected in comparison to 4. 5 × 10 6 km 2 in our analysis using TRMM 6 with very similar spatial patterns. The newer product, TRMM 7, however, shows less frequent rainfall but heavier rainfall than CHIRPS, maintaining a similar total amount of precipitation  (Giles et al. , 2020). Also, both versions (TRMM 6 and TRMM 7) differ regarding the total area affected by drought stress in 2005 and in particular in 2010, where TRMM 6 showed a 10 % larger area of the Amazon basin affected. This can be explained by the generally higher precipitation rates detected in the TRMM 7 dataset in comparison to TRMM 6  (Seto et al. , 2011) , leading to lower absolute values of rMCWD. Spatially, this difference was most pronounced in the western and northern parts of Amazonia, in the states of Acre and Roraima, and in Peru.
Because of such higher precipitation rates in TRMM 7 as compared to TRMM 6 and subsequently the much stronger drought response according to our analysis, studies based on TRMM 6 only might overstate the actual drought conditions and should be revisited. Precipitation datasets usually show remarkable differences in the representation of occurrence, frequency, intensity and location of events, mainly due to their nature of high spatial and temporal variability  (Covey et al. , 2016; Dirmeyer et al. , 2012). Generally, the sparse network of observations in the Amazon rainforest may explain the differences across precipitation datasets and drought indices for datasets that rely on station data. Within the last decade, the number of observations increased, due to a new denser network of stations.
This may improve the reanalysis models that are used for several precipitation datasets applied here; however, it does not improve datasets that only rely on gauge observations. Bias correction is also applied different across precipitation datasets. CRUNCEP and WATCH WFDEI use two different gridded bias-correction inputs, while the simulated precipitation fields of ERA5 do not use any bias corrections. Different datasets that are used for bias corrections can give very different results on regional scales  (Doblas-Reyes et al. , 2021). Jiménez-Muñoz et al. (  2016 ) quantified drought extent using scPDSI and found that 40 %, 25 % and 10 % of the Amazon basin were affected by moderate, severe and extreme drought stress, respectively, in March 2016. While we did not evaluate scPDSI directly but focused on rscPDSI to allow for a better cross-comparison to the other drought indicators, we found similar patterns for moderate drought stress (47 % of the basin affected) but different patterns under severe (11 %) and extreme (1 %) drought stress when evaluating rscPDSI using the ERA5 dataset.
Our estimation diverted from the results of Jiménez-Muñoz et al. (  2016 ) not only because of our different drought classification but also due to a different reference area (see Sect. 2). In addition, Jiménez-Muñoz et al. (  2016 ) used spatially resolved information on soil water capacity when calculating scPDSI and a longer baseline period (onset year is 1979 in their study vs. 2000 in our study). Furthermore, the choice of the precipitation dataset plays an important role. Compared to the datasets considered in our study, ERA5 showed the weakest drought impact during the 2016 drought event. GLDAS and TRMM 7 showed a much stronger drought impact with over 70 % of the area moderately and between 15 % and 39 % severely affected (Table  S2 ). This is particularly interesting because recent studies identify TRMM 7, CHIRPS and ERA5 as the best precipitation datasets when compared to gauge observations in South America  (Albergel et al.
Evapotranspiration datasets Using a dataset with variable ET instead of a constant ET of 100 mm per month leads to smaller areas affected by drought stress depending on the year and drought location (Fig. 4 ). According to our findings using a constant ET of 100 mm per month introduces not only a change in drought-affected areas but also a bias, as drought intensity and spatial extent are consistently higher for all drought years. The reason for this bias lies within the calculation of MCWD, which computes stronger deficits for higher values of ET (e. g. 100 mm per month) than for lower values (e. g. 50 mm per month) during months with low precipitation. This bias can be rather small during drought events that are located in the northern, wetter parts of the basin (as in 2005), but it can also be quite strong for droughts that are located in the southern parts which have a more pronounced dry season (as in 2010).
, 2018; Smith et al. , 2014). The simulated response of tropical forests in DGVMs is particularly sensitive to precipitation input under present and future climate change scenarios (e. g. Seiler et al. , 2015). Therefore, we recommend using multiple climate forcing datasets to test for climate data uncertainty also under present climate conditions. Particularly, studies based on TRMM 6 should possibly be revisited and complemented with more forcing datasets for their analysis.
Conclusions We find substantial variation in the spatial extent, location and timing of the extreme drought events in the years 2005, 2010 and 2016 in the Amazon basin. Depending on the precipitation dataset and drought index used, the area affected by severe (extreme) drought varied between 0 % and 39 % (0 % and 13. 7 %) for the 2016 event. Especially the area under severe drought conditions changed from almost no severe drought stress (five out of six datasets) when using rRAI to greater than 10 % when using rMCWD and rscPDSI instead. The variation partly results from the application of different drought metrics (rMCWD, rRAI and rscPDSI) and from differences in the underlying precipitation datasets. Such differences also propagate when quantifying the impacts of droughts on the carbon cycle of the Amazon rainforest and result in a large variability in biomass carbon losses for a particular drought year.
The estimated intensity of droughts depends predominantly on the selected drought indicator and to a lesser extent on the choices of precipitation and evapotranspiration dataset. We, therefore, recommend applying several drought metrics; climate (precipitation) datasets; and, if available, evapotranspiration datasets to account for model uncertainty when assessing the spatial extent, duration and location of droughts. We regard it as an important step when assessing drought impacts on tropical rainforests also under current climate conditions. Communicating the uncertainty in the estimation of drought events and their impacts on the Amazon rainforest is highly relevant, and thus, multiple datasets should be applied by any large-scale study on drought impacts on vegetation.
5 • ); by CRU gridded al. (2017) for Environmental Prediction (NCEP) daily, pentadal, observational monthly dataset European Centre for Medium-Range ERA5 Global Quarter-degree Reanalysis Albergel et Weather Forecasts (E) reanalysis resolution (0. 25 • ); al. (2018) (RA) product version 5 sub-daily, daily, monthly Global Land Data GLDAS 2. 1 Global Quarter-degree Geostationary satellite Rodell et Assimilation resolution (0. 25 • ); infrared cloud-top al. (2004) System daily, pentadal, temperature measurements monthly and microwave observation techniques Global Precipitation GPCC 2018 Global Quarter-degree Gridded in situ Schneider et Climatology Centre at resolution (0. 25 • ); observations al. (2018) Deutscher Wetterdienst monthly Global Soil Wetness GSWP3 Global Half-degree Reanalysis (20CR) Kim et al.
1 We also maintain a repository 2 containing the source code referred to in this paper. This paper is an extended version of a paper published in the ITP 2021 conference proceedings  [3]. The additions to this paper, apart from several small local changes throughout the text, mainly concern the following. • Code samples throughout have been updated to reflect parts of our formalization contributed to mathlib after the previous publication and to incorporate changes in mathlib after contribution. • Instead of only considering class groups of Dedekind domains, we briefly describe class groups for general domains; see the end of Section 2 and Section 7. 1. • The new Section 3. 1 gives a more detailed explanation of the use of typeclasses in mathlib. • The further evolution of fraction rings in mathlib is discussed at the end of Section 3.
In general, taking any root α of an irreducible polynomial of degree n over Q yields a number field of degree n: Q(α) = {c 0 + c 1 α +. + c n-1 α n-1 : c 0 , c 1 ,. , c n-1 ∈ Q}, and, up to isomorphism, all number fields of degree n arise in this way. The ring of integers O K of a number field K is defined as the integral closure of Z in K, namely O K := {x ∈ K : f (x) = 0 where f ∈ Z[x] is a monic polynomial} , 1 https://github. com/leanprover-community/mathlib/tree/dedekind-domain-dev 2 https://github. com/lean-forward/class-number-journal where we recall that a polynomial is called monic if its leading coefficient equals 1. While it might not be immediately obvious that O K is a ring, this follows from general algebraic properties of integral closures. Some examples of O K are the following.
The is scalar tower typeclass derives its name from its applicability to any three types between which exist scalar multiplication operations: class is_scalar_tower (M N α : Type*) [has_scalar M N] [has_scalar N α] [has_scalar M α] : Prop := (smul_assoc : ∀ (x : M) (y : N) (z : α), (x • y) • z = x • (y • z)) For example, if R is a ring, A is an R-algebra and M an A-module, we can state that M is also an R-module by adding a [is scalar tower R A M] parameter. Since x • y for an R-algebra A is defined as algebra map R A x * y, applying smul assoc for each x : K with y = (1 : L) and z = (1 : F ) shows that the algebra maps indeed commute in a tower of field extensions L/K/F. Common is scalar tower instances are declared in mathlib, such as for the maps R → S → B when S is a R-subalgebra of A and B is an A-algebra such that is scalar tower R A B; this also implies that the maps R → S → A form a tower.
If L is a field extension of K, we defined an intermediate field as a subfield that is also a K-subalgebra: a subfield that contains the image of algebra map K L. Other examples of subobjects available in mathlib are submonoids, subgroups and submodules (with ideals as a special case of submodules); all of these are provided with an instance of the set like typeclass that supplies notation such as a membership relation "x ∈ S". The new definitions found immediate use: soon after we contributed our definition of intermediate field to mathlib, the Berkeley Galois theory group used it in a formalization of the primitive element theorem. Soon after the primitive element theorem was merged into mathlib, we used it in our development of the trace form. This anecdote illustrates the decentralized development style of mathlib, with different groups and people building on each other's results in a collaborative process.
canonical equiv between two types of fractional ideals on R. We defined the addition, multiplication and intersection operations on fractional ideals, by showing that the corresponding operations on submodules map fractional ideals to fractional ideals. We also formalized that these operations give a commutative semiring structure on the type of fractional ideals. For example, multiplication of fractional ideals is defined as This justifies the notation I -1 = 1/I. In fact, we define this notation even for the ideal 0, by declaring that 0 -1 = 0. This reflects the existence of the typeclass group with zero in mathlib, consisting of groups endowed with an extra element 0 whose inverse is again 0. Moreover, mathlib used to define a/b := a * b -1 , but our definition of I -1 = 1/I would cause circularity.
Extending this proof to show the finiteness of Cl O K ′ is quite involved and does not result in a uniform proof for Cl OK and Cl O K ′. Our formalization instead adapted and generalized a classical approach to the finiteness of Cl OK , where the use of Minkowski's theorem is replaced by the pigeonhole principle. For an informal writeup of the proof, used in the formalization efforts, see https://github. com/lean-forward/class-number-journal/blob/jar-reviews/ FiniteClassGroup. pdf. The classical approach seems to go back to Kronecker and can be found, for instance, in  [13]. We note that some other "uniform" approaches can be found in  [14]  and  [15]. Let D be a Euclidean domain: in particular, it will be a PID and hence a Dedekind domain. Given a fraction field F of D, let K be a finite separable field extension of F.