dataesr/scientific-paragraphs-categorization huggingface dataset:
Computing explanations To use the certified code (see section 4. 5), the axioms N , T , T k, Tk_eq_dec, λ, and µ need to be implemented for each dataset. The N axiom is provided in figure  3. For the four datasets T = f loat, T k = int, and Tk_eq_dec is the equality on int. λ and µ are extracted from the datasets. The experiment are run in a Dell Inc. Latitude 7400 with 32Gio RAM and 8 Intel® Core™ i7-8665U CPU @ 1. 90GHz. A summary of the explanation properties is given in Figure  5. As can be observed, the explanation sizes are generally small, which confirms the interest in computing AXp and CXp explanations and helps to explain the model. The algorithm makes 2 * N calls to κ, which is linear in the number of features. This enables the code to scale effectively with larger feature sets.
The number of features explains the difference in computation times for the different models and in particular why the execution time, reduced to 100 instances, is more important for the pokemon dataset which has three times more features than the others. Even if the car model has only six features, the execution time is higher than expected. We assume that this is due to the size of the generated model. Despite this, the computation time for calculating explanations of several hundred instances is reasonable considering that the code is written for certification rather than optimization. As an example, Figure  6  shows the distribution of features in the explanations for the heart dataset and model. Our analysis reveals that, certain features are found to only contribute to the abductive explanations (platelets), while others are only relevant to the contrastive explanations (diabetes).
Comparison with direct mono-language implementation To evaluate the effectiveness of the certified code, the algorithms were coded in Python (file XGBoost/find_xp. py) and run on the same models and dataset as the certified code. The comparison of the execution time is presented in the table in Figure  7. Except for the model car, we can notice that the code generated by Coq takes around 25% more user time and 5% more system time. The difference in execution time is due to the fact that the code was written to facilitate proof, not efficiency, and calls to classifiers are cross-language. However, despite this, the execution times are still reasonable and demonstrate the dual benefits of the work: proving that the algorithm is correct and providing a proof of concept for the use of certified correct code.
The operating system is Ubuntu 22. 04. 2 LTS (64 bits). The execution time is compute using the Linux time command. Name Average Average Time for the entire Time for 100 instances size AXp size CXp dataset (proportion) Heart 2. 45 1. 61 user 0m24. 991s 0m8. 358s sys 0m0. 909s 0m0. 304s Car 2. 39 1. 27 user 6m1. 489s 0m20. 919s sys 0m1. 704s 0m0. 098s Placement 2. 14 1. 51 user 0m8. 493s 0m8. 739s sys 0m0. 864s 0m0. 584s Pokemon 1. 71 2. 76 user 2m47. 066s 0m20. 883 sys 0m1. 600s 0m0. 200s.
In a very deep Regge kinematics, the two-dimensional hyperbola-shaped domain of integration of the Mandelstam integral becomes very skewed and requires care in the numerical evaluation, see Figure  7. For these integrals, we have found that we obtained reliable results using the standard GlobalAdaptive method of Mathematica, supplied with various PrecisionGoal, which we would in general keep to 5 to keep good runtimes and control on the numerics. We did not give much importance to the estimated error of the numerical integrations, because we have a precise and independent way to check our results. which is to measure violation of the imposed unitarity condition, and we have found that it is satisfied to a very good precision, e. g. 1 -|S 0 | ∼ 10 -10 for the 2QE amplitudes. We plan to release the code in the near future  16  , once we have optimized it fully.
Computation times. To close this section, let us just comment briefly on the computation times. We chose our typical grids for ρ(s, t) 11,455 points and ρ(s) with 100 points, respectively, such that computation times were manageable. As a the data plotted in this paper could be reran on a modern laptop in less a week. Furthermore, the codes are easily parallelized, since the iteration proceeds by evaluating integrals of the same functions ρ (n) (s), ρ (n) (s, t) at all the grid points (s i , t j ) and are completely independent of each other. To produce a typical dataset, with 10 different couplings, and 10 iterations (with hotstart to achieve good precision), can by done in O(10h) with 50 CPUs, using mathematica's native lightweight grid-computing environment. Given how easy it is to parallelize this code, which essentially performs thousands of independent integrals at each step, we expect a huge speed-up if it were adapted to run on a GPU.
B Grid generation In this section, we explain the Mathematica routines that we used to generate our meshes. They rely on the use of the Finite Element Methods package, that can be called with <<NDSolve'FEM'. Mathematica documentation is provided at https://reference. wolfram. com/language/FEMDocumentation/tutorial/ElementMeshCreation. html. This package is very handy and in particular it provides interpolating routines via the command ElementMeshInterpolation. 1d meshes The one-dimensional meshes are straightforward to generate. A grid with Nelem elements and Gradient Ratio r is generated as follows: In  [1] := <<NDSolve'FEM' In  [2] := meshx=ToGradedMesh[Line[{{0},{1}}],<|"Alignment"->"BothEnds","ElementCount" ->Nelem,"GradingRatio"->r|>,"MeshOrder"->1]; The grading ratio r, is quantity that enforces some exponential density in the grid, near the extremities (through the "Alignment"->"BothEnds" option, which can also be set to Right or Left, see documentation).
However due to elastic unitarity the spectral densities decay close to the leading Landau curve in a simple known fashion, see  [38] , and correspondingly we have observed in our analysis that this region does not require any special care. The FEM package has a useful command ProductMesh that takes two meshes and produce a tensorial product of them. The elements of this tensor-product are quadrilateral elements which are rectangles. However, we are not interested in a mesh that covers the whole range [0; 1] × [0; 1], we only need to cover the wedge below the Landau curve y = (1 -x)/4 for the double spectral density ρ el (x, y), see Section 2. 3. The construction starts from a product of two meshes covering [0; 1/4] and [0; 1]. Then one removes points above the Landau curve, add the boundary Landau curve.
Acknowledgments We thank  Nima Arkani-Hamed ,  David Atkinson ,  Pierre Aubert ,  Lucia Cordova ,  Miguel Correia ,  Joan Elias Miro ,  Giulia Isabella ,  Hofie Hannesdottir ,  Kelian Häring ,  Enrico Herrmann ,  Andreas Juettner ,  Denis Karateev ,  Madalena Lemos ,  Andrew McLeod ,  Sebastian Mizera ,  Julio Parra-Martinez ,  Joao Penedones ,  Jiaxin Qiao ,  Balt van Rees ,  Matthew Schwartz ,  Amit Sever ,  Jaroslav Trnka ,  Pierre Vanhove ,  Pedro Vieira ,  Matt Walters ,  Xiang Zhang  for useful discussions. We thank the authors of [45,46] for sharing and explaining their detailed results to us. We thank  Dmitrii Vorobev  and  Dmitri Vasiliev  for exploring the d-dimensional formulae as a part of the Physics Practicum 2022. This project has received funding from the  European Research Council (ERC)  under the European Union's  Horizon 2020 research and innovation programme  (grant agreement number  949077 ) and from  Agence Nationale de la Recherche (ANR) , project  ANR-22-CE31-0017.
| 16s rRNA gene sequencing analysis Raw read quality was assessed with FastQC V. 0. 11. 9 (Andrews, 2010; Table  S3 ), and reports were generated using MultiQC V. 1. 6  (Ewels et al. , 2016). Reads were cleaned and adaptors removed with Trimmomatic V. 0. 39  (Bolger et al. , 2014;  Illuminaclip 2:30:10; leading 30, trailing 30, sliding window 26:30 and minlen 150). Cleaned read quality was checked with the same software (Table  S2 ). Microbiome analysis was performed using the QIIME 2 framework V. 2021. 4  (Bokulich et al. , 2018). Dereplicated and trimmed sequences were imported into the framework as PairedEndFastqManifestPhred33V2 format and denoised using the DADA2 plugin, based on the DADA2 V. 1. 8 R library  (Callahan et al. , 2016) , which removed singletons, chimeras and sequencing errors and processed the sequences into a table of exact amplicon sequence variants (ASVs;  Callahan et al.
The Bray-Curtis dissimilarity  (Sørensen, 1948)  and Jaccard similarity index  (Jaccard, 1908)  matrices were also calculated with the q2-diversity tool. Naive Bayes feature classifiers were trained using the q2-feature-classifier tool to assign taxonomy to the sequences  (Bokulich et al. , 2018). For bacterial classifier training, the SILVA-138-SSURef-Full-Seqs QIIME pre-formatted database (SILVA-138-SSURef-Full-Seqs. qza as DataSeq. qza and Silva-v138-full-length-seq-taxonomy. qza as RefTaxo. qza; see https://github. com/miker obeso n/make_SILVA_db) was used, and for hooked V1V3 parts of the 16S region, we used the forward 27f sequence ('GTGCC AGC MGC CGC GGTAA') and the reverse 534r sequence ('GTNTT ACN GCG GCK GCTG') on the feature-classifier extract-reads tool to limit the rate of bad assignments following  Bokulich et al.
(2018). All our raw sequences have been deposited in the Sequence Read Archive (SRA) under submission number SUB11051098 (BioProject IDRJNA804658), from SAMN25761711 to SAMN25761723 (9 samples). All scripts can be found at https://github. com/PLSte nger/Rhabd astre lla_globo stell ata_and_Hyrti os_erect us_micro biome.
| Statistical analysis For the community diversity analysis, indices (see Section 2. 3) were calculated with the R package multcomp V. 1. 4-16  (Hothorn et al. , 2008). For bacterial community composition, relative abundance was visualized through a barplot using the ggplot2 R package V. 3. 3. 3  (Wickham et al. , 2016). A Principal Component Analyses (PCA) biplot was used to explain variation within bacterial communities in both sponges using the R packages multcomp V. 1. 4-16  (Hothorn et al. , 2008) , factoextra V. 1. 0. 7  (Kassambara & Mundt, 2020 ) and vegan V. 2. 5-7  (Oksanen et al. , 2019).
that are equivalent to species richness; DeSantis et al. , 2006) , Shannon entropy (richness and diversity that accounts for both F I G U R E 1 The Bouraké semi-enclosed lagoon study site. The top panel shows the location of the study sites in the Grande Terre of New Caledonia. At the bottom a zoom on the Bouraké semienclosed lagoon with the innermost and the outermost zone of the lagoon. Géorep New Caledonian database and qgis software were used to build the figure (modified from Maggioni et al. , 2021). 14390485, 2023, 5, Downloaded from https://onlinelibrary. wiley. com/doi/10. 1111/maec. 12757 by IFREMER Centre Bretagne BLP, Wiley Online Library on [23/10/2023]. See the Terms and Conditions (https://onlinelibrary. wiley. com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License abundance and evenness of taxa; Shannon & Weaver, 1949) and.
2. 1. 10 (Price   et al. , 2009), According to Bokulich et al. (2018) , generating a tree allows phylogenetic diversity analyses, including Faith's Phylogenetic Diversity(Faith PD)  and weighted and unweighted UniFrac. In addition to counts of features per sample, these metrics require a rooted phylogenetic tree relating the features to one another. To characterize the diversity of this new and now rarefied dataset, we used the q2-diversity tool to produce a set of diversity indices, including the observed number of ASVs (same as above; DeSantis.
More than 61,000 proteins have up-to-date correspondence between their amino acid sequence (UniProtKB) and their 3D structures (PDB), enabled by the Structure Integration with Function, Taxonomy and Sequences (SIFTS) resource. SIFTS incorporates residue-level annotations from many other biological resources. SIFTS data is available in various formats like XML, CSV and TSV format or also accessible via the PDBe REST API but always maintained separately from the structure data (PDBx/mmCIF file) in the PDB archive. Here, we extended the wwPDB PDBx/mmCIF data dictionary with additional categories to accommodate SIFTS data and added the UniProtKB, Pfam, SCOP2, and CATH residue-level annotations directly into the PDBx/mmCIF files from the PDB archive. With the integrated UniProtKB annotations, these files now provide consistent numbering of residues in different PDB entries allowing easy comparison of structure models.
Introduction As of March 2023, the Protein Data Bank (PDB)  1  contains over 200,000 entries representing over 61,000 unique entries in the Universal Protein Resource Knowledgebase (UniProtKB)  2. Often, the PDB archive has the same protein in multiple entries under different experimental conditions or interacting with different macromolecules (proteins, DNA, RNA) or ligand molecules  [3] [4] [5]. Multiple 3-dimensional coordinates of the same protein are invaluable for comparative structure-function studies  3, 6, 7. Linking structure data with annotations available in other data resources such as UniProtKB 2 and to the structural and functional annotations is critical in order to understand biological function and processes at a molecular level. However, one of the barriers to comparative analysis or data integration is the independent, depositor-provided residue numbering in the coordinate files, which may not be the same as the protein sequence numbering  8.
While solving a protein 3D structure, many times the experiments are carried out only on a part of complete protein molecules (e. g. a domain) to make the sample amenable to experimental methods, especially in cases where there are highly flexible linker regions or intrinsically disordered regions  9, 10. Around 58% of the structures in the PDB contain smaller fragments (e. g. a domain) corresponding to different regions of a protein sequence. To determine where these fragments are located on the full-length protein sequence, these fragments need to be mapped to a common reference e. g. protein sequence numbering from a relevant entry in the UniProtKB database. The situation becomes complicated as often the flexible regions in the protein molecules are not modelled leading to unobserved residues i.
e. residues without atomic coordinates in protein structures. The occurrence of missing residues makes structure-to-sequence mapping even more challenging. To address this fundamental problem of standardising residue numbering to make protein structure data more accessible to the broader scientific community, the PDBe  11  and UniProtKB 2 teams collaborated to establish the Structure Integration with Function, Taxonomy and Sequences (SIFTS) resource in 2002  12, 13. SIFTS provides up-to-date residue level mapping, with each weekly PDB release, between UniProtKB protein sequences and PDB protein structures allowing better integration of annotations based on protein sequence and structure. In addition to mapping PDB structures to UniProtKB sequences, SIFTS also maps to other biological resources such as Pfam  14  , InterPro  15  , SCOP  16  , CATH  17  , IntEnz  18  , GO  19, 20  , Ensembl  21  , NCBI taxonomy database  22  and Homologene  23.
In the past 20 years, SIFTS has become an essential resource, and its data provides the foundation of many data services and web pages. SIFTS is fundamental to the PDBe and PDBe-KB data resources  24  and other databases, such as UniProtKB 2 , Pfam  14  , RCSB PDB  25  , PDBj  26  , SCOP2  27  , InterPro  15  and MobiDB  28  , rely on SIFTS to fetch cross-references between PDB structures and other biological databases. SIFTS data is distributed as summary flat files in CSV/TSV formats and also as a detailed per-entry XML files with residue-level information available from the EMBL-EBI FTP area (ftp://ftp. ebi. ac. uk/pub/databases/msd/sifts/). SIFTS data is also accessible via the PDBe API  29. While SIFTS data has significantly improved the interoperability of PDB structure data with other key data resources, it still requires to be accessed separately from the 3D coordinates data in the PDB.
The SIFTS output format is incompatible with 3D visualisation software that use the PDBx/mmCIF standard  30  and requires an additional step of parsing the data to display SIFTS annotations on protein 3D structure. To boost the FAIRness  31  (Findable, Accessible, Interoperable and Reusable) by further improving the findability and interoperability of PDB structures, the next logical step is to integrate SIFTS annotations alongside the 3D coordinates in the PDBx/mmCIF files. Moreover, with the availability of numerous high-quality, predicted protein structure models from resources like SWISS-MODEL  32  and AlphaFold DB  33, 34  , which generally follow the protein sequence numbering scheme, it was timely and essential to augment the protein sequence numbering for the experimentally determined 3D coordinates in the PDB.
Taking only the PDB-UniProtKB pairs with the same source organism or atleast having a common ancestor within one or two levels up to species level in the taxonomy tree and having at least 90% sequence identity, the pair with the highest sequence identity is annotated as the best mapping. Once we have established the mapping between UniProtKB and PDB protein residues, the cross-references from other resources such as Pfam 14 , InterPro  15  , SCOP  16  , CATH  17  , IntEnz  18  , GO  19, 20  , Ensembl  21  and Homologene 23 are added. The SIFTS annotations are stored in the SIFTS database, which is used to make the data accessible via the PDBe REST API. Individual XML files for each PDB entry with residue-level information are exported and the summary files are generated in CSV/TSV formats.
An additional process was designed that reads the data from the SIFTS database and augments the PDB structure files with UniProtKB numbering and structure (SCOP2, and CATH resource) and sequence (Pfam resource) domain annotations. This update yields more consistent, standardised metadata. It is important to note that none of the core PDB information, such as atomic coordinates and experimental data, are altered in any way. Figure  1  shows the schematic overview of the data flow of the SIFTS process and highlights the additional process that was developed to export these data into PDBx/mmCIF files. The process helps researchers and data services access SIFTS data directly from the PDBx/mmCIF 37 files. To facilitate this update, additional "SIFTS-specific" mmCIF data categories were designed and integrated into the core PDBx/mmCIF data dictionary.
_pdbx_sifts_unp_segments This new category describes residue range-based cross-references specific to the UniProtKB database. It shows segments/regions of PDB residues mapped to the canonical UniProtKB accession and all its isoforms. The residue mapping is established by aligning the PDB sequence to each UniProtKB accession (canonical and all the isoforms) and the sequence identity between the aligned PDB-UniProtKB pair is provided. This category also indicates the best mapped UniProtKB accession.
_pdbx_sifts_xref_db PDB structures often have missing residues, expression tags or linker regions, making the expansion of mappings from segments (residue range) to individual residues cumbersome. An essential category, "_ pdbx_sifts_xref_db", therefore describes residue level cross-references to external databases. This category provides annotations specific to the best mapped UniProtKB accession and can be used to identify all the mappings for each residue to external databases (Fig. 3 ).
This also facilitates comparing proteins with similar domain structures and identifies potential functional relationships. All the new SIFTS-specific or modified PDBx/mmCIF categories are shown in grey boxes. The new SIFTSspecific PDBx/mmCIF categories introduced to show per-segment annotations from UniProtKB and all the other external data resources (Pfam, SCOP2, CATH) are "_pdbx_sifts_unp_segments" and "_pdbx_sifts_ xref_db_segments" respectively. "_pdbx_sifts_xref_db" is another new SIFTS-specific PDBx/mmCIF category introduced to show per-residue annotations. We also modified the "_atom_site" category to indicate the best mapped UniProtKB sequence. Fig. 3  Single placeholder in PDBx/mmCIF files to find all the annotations associated with any residue from external databases. This figure shows the "_pdbx_sifts_xref_db" category for PDB 4daj.
This critical new data category can describe residue-level cross-references to external databases. The items specific to the UniProtKB database and other cross-reference databases are marked in beige and green coloured boxes respectively. SIFTS is not only widely used in scientific research but also by several data resources  12. For example, UniProtKB exploits SIFTS information to provide structure mapping in the UniProtKB database. SCOP  55  and Pfam 14,56 also use SIFTS to map protein domains and connect sequence domains with their corresponding structures. The web resource Kincore relies on SIFTS to map protein kinases to their respective structures, extract relevant information such as domain boundaries and ligand binding sites, and provide a structural classification of protein kinases and their inhibitors  50.
These files are already being used by Mol* 67 (https:// molstar. org/viewer/) to perform extremely fast superpositions using the SIFTS UniProtKB mapping. This superposition functionality in Mol*  67  is very powerful as it gives users the means to directly superimpose protein structures in their web browser without downloading any data or software. Mol* uses the SIFTS specific new Fig. 5  Distinguishing between multiple instances of the same protein in the PDBx/mmCIF file. The data item ". instance_id" enables users to identify the two copies of the same protein, Streptavidin (UniProtKB accession P22629), in the dimeric Streptavidin structure (PDB 6s50). Fig. 6  Identification of split domains from PDBx/mmCIF file. The "_pdbx_sifts_xref_db_segments" category in the PDBx/mmCIF file of PDB 4daj helps to clearly identify discontinuous domains.
It is important to note that adding additional data to a PDBx/mmCIF file, such as augmenting best mapped UniProtKB residue mapping in the "atom_site" category can come with a trade-off of an increase in the file size. While this may not be an issue for smaller PDB entries, it can become problematic for larger entries with significant file size. To address this issue, wwPDB provides binaryCIF  74  (bcif) files as an alternative to traditional PDBx/mmCIF files. The bcif format is a compressed binary version of the PDBx/mmCIF format that significantly reduces the file size, making it easier to handle and share large amounts of structural data. The Mol*, an open-source software for 3D molecular visualisation and analysis, also supports the bcif file format, allowing users to easily access and analyse structural data in this format.
Methods PDBx/mmCIF file format and PDBx/mmCIF dictionary. The PDBx/mmCIF(Protein Data Bank exchange/macromolecular Crystallographic Information File) is a well-established data format utilised for storing and sharing information related to the three-dimensional structure of macromolecules, including proteins and nucleic acids. Widely considered as the master format for the PDB archive, it is extensively used for representing structural data. The format uses a text-based file format that encodes data and metadata utilising data items grouped into categories. The PDBx/mmCIF dictionary 30 defines a standardised set of categories and data items, along with controlled vocabularies and explicit relationships between different categories and data items. This format is extensible, allowing the incorporation of new data items and categories, as demonstrated by the IHM  75  and ModelCIF 76 extensions.