dataesr/scientific-paragraphs-categorization huggingface dataset:
mm database for MetaMath  [21]  and the Mizar Mathematical Library  [22]. The field of algebraic numbers, or more generally algebraic closures of arbitrary fields, are also available in many provers. For example, Blot  [23]  formalized algebraic numbers in Coq, Cohen  [24]  constructed the subfield of real algebraic numbers in Coq, Thiemann, Yamada and Joosten  [25]  formalized algebraic numbers in Isabelle/HOL, Carneiro  [26]  in MetaMath, and Watase  [27]  in Mizar. To our knowledge, the Coq Mathematical Components library is the only formal development beside ours specifically dealing with number fields  [19, field/algnum. v]. Apart from the general theory of algebraic numbers, there are formalizations of specific rings of integers. For instance, the Gaussian integers Z[i] have been formalized in Isabelle/HOL by Eberl  [28] , in MetaMath by Carneiro  [29]  and in Mizar by Futa, Mizushima, and Okazaki  [30].
) on many occasions. An important aspect we had to take into account was to consistently combine different descriptions of mathematical objects from different sources. The formalization project described in this paper resulted in the contribution of thousands of lines of Lean code involving hundreds of declarations. A rough estimate concerning the former would be that about five thousand lines of project specific code were added, and about half of that number of lines of more generic background code. We validated existing design choices used in mathlib, refactored those that did not scale well and contributed our own set of designs. The real achievement was not to complete each proof, but to build a better foundation for formal mathematics.
Acknowledgements. We would like to thank  Jasmin Blanchette  and the anonymous reviewers for useful comments on previous versions of the manuscript, which found their way into this paper. A. N. would like to thank  Prof. Kevin Buzzard  for his constant support and encouragement, and for introducing her to the other co-authors. A. N. and F. N. wish to express their deepest gratitude to  Anne Baanen  for the generosity shown along all stages of the project. Without Anne 's neverending patience, it would have been impossible for them to contribute to this project, and to overcome several difficulties. Finally, we would like to thank the whole mathlib community for invaluable advice all along the project. Declarations.
Background: Sequencing viruses in many specimens is hindered by excessive background material from hosts, microbiota, and environmental organisms. Consequently, enrichment of target genomic material is necessary for practical high-throughput viral genome sequencing. Hybridization probes are widely used for enrichment in many fields, but their application to viral sequencing faces a major obstacle: it is difficult to design panels of probe oligo sequences that broadly target many viral taxa due to their rapid evolution, extensive diversity, and genetic hypervariability. To address this challenge, we created ProbeTools, a package of bioinformatic tools for generating effective viral capture panels, and for assessing coverage of target sequences by probe panel designs in silico. In this study, we validated ProbeTools by designing a panel of 3,600 probes for subtyping the hypervariable haemagglutinin (HA) and neuraminidase (NA) genome segments of avian-origin influenza A viruses (AIVs).
Unfortunately, these have largely been implemented on a study-by-study basis and have not resulted in general-purpose software tools that can be easily used by others. Meanwhile, among the handful of published probe design packages, there is only one option that specifically addresses viral hypervariability  [13]. The rest are intended for comparatively conserved eukaryotic genomes and are inadequate for many viral applications  [14] [15] [16] [17]. This leaves virologists with limited options for designing their own hybridization probes, especially if they have minimal capacity for custom programming, sophisticated mathematics, and experimental bioinformatics. Here, we present ProbeTools, a user-friendly command line software package for designing compact probe panels against diverse viral taxa and other hypervariable genomic targets.
It provides easy-to-use modules for generating probes and assessing panel coverage of provided target sequences. We demonstrate ProbeTools' effectiveness by designing capture panels for avian-origin influenza A viruses (AIVs). These viruses are subtyped by two hypervariable viral surface proteins called haemagglutinin (HA) and neuraminidase (NA), making them an appropriately challenging case study for ProbeTools. The genome segments encoding these proteins have diversified into 16 avian-origin HA subtypes and 9 avian-origin NA subtypes, giving rise to 144 possible combinations and the HxNx nomenclature used in both animal and human contexts (e. g. H1N1, H3N2, H5N1, H7N9). Furthermore, each of these subtypes has diverged into numerous clades, many of which have been only partially characterized  [12, [18] [19].
Assessing basic k-mer clustering and marginal improvements to target coverage with additional probes: We began by assessing probe design against hypervariable targets with a basic k-mer clustering algorithm, wherein all 120-mers were enumerated from a target space of AIV reference sequences then clustered based on 90% nucleotide sequence identity. We used this strategy, implemented in the ProbeTools clusterkmers module, to generate probe panels of increasing size against 14,772 HA segment reference sequences and 11,967 NA segment reference sequences. We then used the ProbeTools capture module, which aligns probe sequences against target sequences, to assess target space coverage, i. e. the percentage of nucleotide positions in each target sequence covered by at least one probe in the panel (Figure  1A , solid lines).
As expected, panels with more probe sequences provided better target space coverage, however we observed diminishing marginal improvements for both HA and NA genome segments. We also noted that reference sequences with no probe coverage remained in the target space past the point of diminishing marginal returns. These results highlighted two limitations of the basic k-mer clustering approach: HA and NA segments remained undetected despite designing additional probes, and additional probes provided only modest and diminishing improvements to the distribution of target coverage. were designed against target spaces of 14,772 haemagglutinin (HA) and 11,967 neuraminidase (NA) genome segment reference sequences. The ProbeTools clusterkmers module was used to make panels using basic k-mer clustering and the makeprobes module was used to make panels with an incremental strategy.
For each panel, probe coverage of reference sequences was assessed in silico using the ProbeTools capture module. A) For both strategies, A B C increasing panel size improved the 10 th percentile of reference sequence coverage with diminishing marginal increases, but incrementally designed panels achieved more extensive coverage at larger panel sizes. Incrementally designed panels also provided better coverage of the worst-covered reference sequence using fewer probes. B) Incrementally designed panels shifted coverage distributions upwards for the worst-covered reference sequences. Each reference sequence in the target space is represented as a dot, plotted according to its probe coverage. Coverage of the worst-covered reference sequence and 10 th percentile of all reference sequences are indicated above the axis.
Improving target coverage with incremental panel design focused on poorly covered targets: To address the limitations we observed with basic k-mer clustering, we devised an incremental design strategy to improve marginal coverage increases, especially for poorly covered targets. In this strategy, basic k-mer clustering was used to design panels in smaller batches of 100 probes. After adding each batch to the growing panel, target space regions without probe coverage were identified using the capture module. These low coverage regions were then extracted with another ProbeTools module called getlowcov and used as a new target space for designing the next batch. In this way, each subsequent batch of probes was focused on regions not already covered by the panel. We compared target space coverage for panels designed with this incremental strategy against panels designed above using basic k-mer clustering (Figure  1 ).
The incremental strategy provided higher 10 th percentiles of coverage, especially for HA panels larger than 2000 probes and NA panels larger than 1200 probes (Figure  1A ). Furthermore, the incremental strategy provided better coverage for the worst-covered reference sequences (Figure  1AB ). We also compared depth of probe coverage, i. e. the number of probes covering each nucleotide position in target sequences (Figure  1C ). This comparison suggested that the incremental strategy improved target coverage by redistributing probes from positions with deep coverage to shallow coverage. Based on the observed performance improvements of the incremental strategy, it was implemented as an additional self-contained ProbeTools module called makeprobes (Figure  2 ). Predicted coverage of HA and NA subtypes by AIV_v1 panel: Using the incremental strategy implemented in the ProbeTools makeprobes module, we generated an AIV-targeting probe panel called AIV_v1.
It was composed of 1,935 HA-specific probes and 1,435 NA-specific probes. We also included 184 probes targeting the highly conserved matrix segment (M) which is the standard AIV diagnostic target  [24, 38]. We then used the ProbeTools capture module to predict probe coverage using the AIV_v1 panel for all 36,313 AIV reference sequences in the target space. The minimum, maximum, and 10 th percentile of reference sequence coverage was calculated for each HA and NA subtype and the M segment (Figure  3A ). We observed that M segments had the best coverage followed by NA subtypes then HA subtypes, reflecting the comparative levels of genomic diversity within these genome segments. No reference sequence had less than 59. 6% coverage, which is sufficient for segment and subtype identification. HA subtypes H5, H7, and H9 are considered high priority for AIV surveillance because they frequently cause agricultural outbreaks and novel influenza infections in humans  [23] [24] [25] [26] 38] ; 90% of H5, H7, and H9 reference sequences had at least 94.
A B against 36,313 reference sequences were generated with the ProbeTools capture module and stratified by subtype for HA and NA segments. The minimum, 10 th percentile, and maximum of probe coverage against reference sequences from each subtype/segment are indicated. B) A significant positive monotonic association was observed between the number of sequences from a subtype in the target space and that subtype's 10 th percentile of coverage. Each dot represents an HA or NA subtype, and the results of Spearman's rank correlation test are indicated on the plots. In vitro capture of diverse egg-cultured influenza isolates: After assessing the AIV_v1 panel in silico, we had it synthesized and used it to perform in vitro captures on a collection of diverse egg-cultured AIV isolates (Table  1 ).
0fold (Figure  4A ). We also calculated the percentage of the capture pools composed of background material from the mock-infected egg cultures, then compared these percentages preand post-capture (Figure  4B ). Before capture, the mean background percentage was 99. 17%, but this was reduced to 0. 03% following capture. Together, these data demonstrate effective enrichment of AIV material and removal of background by probe capture with the AIV_v1 panel. (described in Table  1 ). AIV libraries were pooled together, diluted 1:100 (ng/ng) in libraries of background material made from mock-infected egg cultures, then captured three times independently. A) Pre-and post-capture pools were sequenced to calculate fold-enrichment at each nucleotide position in the haemagglutinin (HA), neuraminidase (NA), and matrix (M) genome segments of these isolates (mean of three independent replicates).
3% breadth of enrichment to understand why they were apparently not captured in full. First, we used the ProbeTools capture module to assess if the AIV_v1 panel lacked probes targeting their particular genome segment sequences. We observed that most positions without significant enriched were nonetheless extensively covered by the probe panel (Figure  5A ). This indicated that insufficient design by ProbeTools was not a major explanation for the lack of significant capture of these segments. enrichment between three independent replicates was significantly higher for positions that did not achieve significant enrichment (p<0. 0001, Levene's test). D) Most positions with insignificant enrichment narrowly failed the enrichment test's pre-determined alpha level of 5%. Next, we assessed whether experimental factors were responsible for nucleotide positions in these segments failing to achieve statistically significant enrichment.
Comparison of in silico probe coverage prediction and in vitro probe capture enrichment: ProbeTools relies on in silico coverage assessment by the capture module, both for final panel evaluation and for identifying poorly covered sequences during incremental design. To validate ProbeTools' coverage assessment algorithm, we examined how closely its in silico predictions agreed with in vitro capture results on egg-cultured AIV isolates. Using the ProbeTools capture module, we determined which nucleotide positions in the egg-cultured AIVs were predicted to be covered by the AIV_v1 probe panel. We then compared these predictions to our in vitro capture results to see if significant enrichment had actually occurred at these nucleotide positions (Figure  6 ). Predicted probe coverage and significant enrichment results were concordant for 89.
DISCUSSION This study highlighted some important considerations when designing panels using ProbeTools. Foremost among these was the effect of target space composition on panel inclusivity. In this AIV case study, we noted a significant positive monotonic association between panel coverage and the number of reference sequences representing a particular subtype in the target space. Based on how the ProbeTools algorithm ranks probe candidates by the number of k-mers in the cluster they represent, it stands to reason that over-representing similar taxa (which would contain many similar k-mers) would bias the resulting panel towards these taxa. Consequently, ProbeTools users should have a thorough knowledge of the contents of their target space and the possible sources of sampling bias in the databases from which they obtain their reference sequences.
In the case of AIVs, the agricultural impacts and public health threats of certain HA subtypes have led to more frequent sequencing of these subtypes and accessioning of their genome sequences in popular databases. For our panel, this contributed to bias towards subtypes like H5, H7 and H9. Whether this is a benefit or limitation will depend on the intended application. In the context of outbreak prevention and pandemic preparedness, a panel biased towards taxa that are known for their agricultural impact and zoonotic potential is beneficial. If the objective is to characterize viral diversity and ecology in wildlife, however, this could be a limitation. To obtain the best results, ProbeTools users should purposefully curate their target space to serve their probe capture objectives. Users may want to identify taxa whose detection is a priority and over-represent them in the target space.
Conversely, users may want to 'flatten' their target space to ensure no particular taxa, clades, subtypes, etc dominate. This could be done manually, by selecting specific sequences to represent relevant groups, or it could be attempted bioinformatically by pre-clustering target sequences, providing the number and length of target sequences do not make this computationally prohibitive. Another strategy could be to use the various ProbeTools modules to extract low coverage sequences from specific groups whose target sequences have poor probe coverage after a core panel is designed. For instance, had H15 subtype AIVs been a surveillance priority in this study, supplemental H15-specific probes could have been designed by running the capture, getlowcov, and makeprobes modules on the H15 subset of target sequences after noting their comparatively low coverage by the main panel.
Genomic sequencing is becoming central to these One Health ventures, and viral capture panels will need designing and updating as our knowledge of viral threats continues to expand  [43] [44]. The on-going COVID-19 pandemic has also demonstrated the value of viral genomics to public health  [45] [46] [47] [48] , resulting in unprecedented investments in sequencing capacity at public health laboratories. This will expand routine genomics for numerous common pathogens, requiring the development of new target enrichment protocols. ProbeTools can facilitate probe design tasks for all of these endeavours. The clusterkmers module enumerates and clusters probe-length k-mers from userprovided target sequences. 1) K-mers are enumerated using a sliding window that advances by a specified number of bases.
METHODS The capture module predicts how well user-provided probe sequences cover userprovided target sequences. 1) Each probe sequence is locally aligned against each target sequence using BLASTn  [35]. 2) Alignments are filtered, retaining those with a minimum sequence identity over a minimum alignment length. 3) Subject alignment start and end coordinates are extracted from the BLASTn results to determine which nucleotide positions in the target sequences are covered by probes. By default, capture requires 90% sequence identity over at least 60 bases to assign probe coverage to the aligned positions. The getlowcov module uses the output of capture to extract genomic regions with low coverage from the provided targets. This allows for additional probe design focused on poorly covered regions of the target space.
This module returns all sub-sequences where a minimum number of consecutive bases were covered by fewer than a specified number of probes. By default, getlowcov returns all sub-sequences over 40 bases in length where all bases in the subsequence were covered by zero probes. The stats module uses the output of capture to calculate coverage statistics. For each provided target, it calculates the percentage of nucleotide positions covered by varying numbers of probes ("target coverage" and "probe depth"). The makeprobes module chains the previous modules together to implement a generalized incremental design strategy (Figure  2 ). In this strategy, probes are designed in batches, and regions of the target space with probe coverage are removed between batches so that additional probes are focused on poorly covered areas.
This module can be used as a convenient departure point for custom designs. The user is only required to provide target sequences and select a batch size. They can optionally specify a maximum panel size and target space coverage goal. The makeprobes module iterates through its design loop, adding batches of probes to the panel until the maximum panel size is met, the target space coverage goal is achieved, or no further probes can be generated. Preparation of AIV target space: All available full-length influenza A virus genome segment sequences from avian hosts were downloaded from the Influenza Research Database (www. fludb. org) on Dec 5, 2017  [36]. Sequences containing degenerate bases were removed to avoid low quality entries. Sequences were then clustered using VSEARCH cluster_fast (v1.
ACKNOWLEDGMENTS We thank  Dorde Masovic  for realizing Fig. 2 with vector graphics. We thank  Alexandre Wadoux  for insightful exchanges on spatial statistics and  Simon Gascoin  for discussions on along-track undulations in Pléiades DEMs. We also thank  Eyjólfur Magnússon  and  Joaquín Belart  for providing insights into Gaussian simulations for DEMs. We thank  Jonathan Bamber ,  César Deschamps-Berger  and  Matthias Huss  for comments on early versions of the manuscript. SPOT6/7 data were obtained from  GEOSUD  ( ANR-10-EQPX-20 ,  programme 'Investissements d'Avenir '). INRAE is a member of Labex OSUG. R. H. acknowledges a fellowship from the  University of Toulouse. R. H. , A. D. , E. M and D. F. acknowledge funding from the  Swiss National Science Foundation , grant number  184634. E. B.
DATA AND CODE AVAILABILITY STATEMENT The code developed for the processing and analysis of all data, and the generation of figures and tables in this article is available at https: //github. com/rhugonnet/dem error study, with routines implemented in the DEM analysis Python package xDEM available at https: //github. com/GlacioHack/xdem with supporting documentation at https://xdem. readthedocs. io, and the spatial statistics Python package SciKit-GStat available at https://github. com/mmaelicke/scikit-gstat with supporting documentation at https://scikit-gstat. readthedocs. io.
Secondly, they provide the building block to accurately predict the expected measurable signal, also in the case where GW events are used along with other external datasets, (e. g. , those provided by LSS surveys). Since the typical astrophysical processes responsible for GW emission are located in galaxies, the GWB of astrophysical origin traces the LSS of our Universe, almost as precisely as galaxies do. By cross-correlating GW maps of the sky with galaxy number count maps we can establish how GW sources are spatially clustered, i. e. , how they trace the underlying dark matter field, and how they are temporally distributed when most of the emission is generated. Moreover, by using different galaxy catalogs, we can also measure how GW sources are correlated with different types of galaxies observed in frequency bands ranging from radio to optical and UV, highlighting the importance of a multi-wavelength astronomy.
Note that the idea of cross-correlating different datasets has already been extensively applied in Cosmology, for instance by using different LSS surveys  [133, 134]  or LSS surveys in combination with the cosmic microwave background  [135] [136] [137] , neutrinos  [138] , ultra-high energy cosmic rays  [139, 140]  and so on; however accurate predictions of the expected signal are still required to be able to use this technique. On the other hand the low resolution of GW sky maps should be viewed as an opportunity instead of a limitation: even if new GW observatories will improve the overall angular resolution, these maps can be used only to study the clustering properties of GW sources at large scales. However this regime has not been fully and thoroughly probed yet and can potentially contain exciting new physics features.
Acknowledgments NB is partially supported by the  Spanish MINECO  under grant  BES-2015-073372. DB and SM acknowledge partial financial support by  ASI  Grant No. 2016-24-H. 0. ACJ is supported by  King's College London  through a  Graduate Teaching Scholarship. AR acknowledges funding from  MIUR  through the  "Dipartimenti di eccellenza" project Science of the Universe. MS is supported in part by the  Science and Technology Facility Council (STFC) ,  United Kingdom , under the research grant  ST/P000258/1.