dataesr/scientific-paragraphs-categorization huggingface dataset:
Additionally, we ran three PCAs within each family (Alligatoridae, Crocodylidae and Gavialidae) to test if the principal axes of variance changed from one subclade to another, and if they could enable a better differentiation between species than in the total Crocodylia dataset (each clade-specific analysis being independent from the total morphospace deformation induced by the differences of variance observed in each family). Statistical tests and linear regression on log-transformed centroid size were performed both in the total dataset and in each subclade. Both sides of the intertympanic sinus system were landmarked for visualisation purposes. To check how much variation was influenced by the asymmetry in our data, we used the function bilat. symmetry to quantify the symmetric and asymmetric component of shape variation and compare their statistical significance.
| Ancestral state reconstruction and phylogenetic mapping A calibrated molecular phylogenetic tree was produced with Timetree 5  (Kumar et al. , 2022)  containing only the species for which we had at least three specimens to reconstruct an ontogenetic trend: Alligator mississippiensis, Caiman crocodilus, Caiman latirostris, Tomistoma schlegelii, Gavialis gangeticus, Osteolaemus tetraspis, Mecistops cataphractus, Crocodylus siamensis/porosus (considered as one) and Crocodylus niloticus. This topology was imported into R in Newick format using ape package 5. 4-1. Reconstructions of the ancestral ontogenetic trajectories were undertaken with phytools package 1. 2-0. Coefficients of speciesspecific linear models of PC1 and PC2 versus log-centroid size were used to reconstruct ancestral states with anc.
| Comparative description of the intertympanic sinus system A diverticulum is a soft-tissue pocket of epithelial origin filled with air, forming a pneumatic sinus. Its development hollows out a 'recess', a cavity in the bone, generated by the growth of the pneumatic diverticulum  (Kuzmin et al. , 2021). In order to help appreciate the morphological variability occurring in our morphometric dataset, a comparative description of the intertympanic sinus system is provided for all modern crocodylian genera except Paleosuchus. Tw specimen of Osteolaemus tetraspis (MZS Cro 040) is used as a reference model for the description (Figures  1  and 2 ) because it displays well developed and differentiated diverticula compared to other species. Here, the term 'intertympanic sinus system' refers to the association of three pneumatic diverticula: the intertympanic diverticulum, parietal diverticulum and otoccipital diverticulum, which are more closely connected to each other than with the pharyngotympanic sinus system  (Dufeau & Witmer, 2015) , making it possible to study them as a separate unit.
niloticus and Mecistops, while C. siamensis and C. porosus are moderately impacted (Figure  8b , Figure  S22 ). Crocodylus palustris, Osteolaemus and Voay are weakly impacted, showing thicker diverticula in adults (Figure  10a ). In Indomalayan Crocodylus species C. siamensis, C. porosus and C. palustris, the parietal recess also displays an increased rate of development, being greatly developed vertically in adults (Figure  8e , Figure  S5 ) and revealing a common developmental event in this lineage (character 5, Figure  13 ). Finally, the anteroposterior compression seems to start earlier during the ontogeny of Gavialidae (pre-displacement at node VII, Figure  8c ). This would result in the simultaneous development of the parietal and intertympanic recesses; however, the shift in the calculated trajectory at node VII could be mainly driven by the shift in the intercept of the Tomistoma trajectory, which must be taken cautiously in the absence of a hatchling in our dataset.
In contrast, Gavialis shows the most aquatic lifestyle of all extant crocodylians, spending most of its time in water and being the sole extant species to not be able to walk in a semi-upright stance on land  (Stevenson & Whitaker, 2010). The impact of lifestyle on intertympanic sinus shape was thus tested by partitioning the crocodylian species in three classes (semiaquatic for most genera, sub-aquatic for Gavialis and semi-terrestrial for Osteolaemus), revealing a significant correlation (p = 0. 003), but accounting for only 7. 5% of the overall variation (Table  S2 ). Thus, it is likely that intertympanic sinus characteristics were modulated by ecological parameters during the evolution of the crocodylian lineages. However, we were not able to precisely assess which shape changes could be linked to the living environment with our dataset.
which corroborate molecular data at high-and low-level phylogenetic relationships in extant taxa. Alligatoridae are clearly differentiated from Longirostres by the absence of a prootic-parietal 14697580, 2023, 6, Downloaded from https://onlinelibrary. wiley. com/doi/10. 1111/joa. 13830 by Cochrane France, Wiley Online Library on [07/06/2023]. 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 ostium in the latter. Sinus post-hatching development begins with a common hatchling shape shared by all studied species, from which ontogenetic trajectories diverge, following different heterochronic modifications resulting in distinct adult characteristics.
ACK N O WLE D G E M ENTS The authors thank  Didier Berthet  (MHNL),  Blandine Bartschi  and  Emmanuel Robert  ( UCBL ),  Ronan Allain ,  Florent Goussard  and  Nour-Eddine Jalil (MNHN) ,  Patrick Campbell and Susannah Maidment (NHMUK) ,  Marie Meister  and  Elisabeth Ludes-Fraulob  (MZS),  Margarethe Maillart  ( ENS Lyon ),  Christophe Borrely  (MHNM),  Bruno Vila  and  Baptiste Suchéras-Marx  ( AMU ),  Medhi Mouana ,  Anne-Lise Charruault (UM) ,  Thomas van de Kamp and Marcus Zuber (SMNK) ,  Jessie Maisano ,  Laurence Witmer ,  David Blackburn ,  Cody W. Thompson  (Morphosource),  Christopher Brochu ,  Thimothy Rowe  ( Digimorph ) for access to the specimens. This work was supported by the  Agence Nationale de la Recherche  ( SEBEK  project  N° ANR-19-CE31-0006-01  to JEM). Irena Raselli  was supported by the  Swiss National Science Foundation  ( SNF 205321 175978 ).
Acknowledgements We thank  Pietro Corvaja  and  Umberto Zannier  for several discussions. ER was supported by  Institut Universitaire de France  and the  ANR  project " FOLIAGE ",  ANR-16-CE40-0008. AT was partially supported by  PRIN "Advances   in Moduli Theory and Birational Classification " and is a member of  GNSAGA-INdAM. JW was supported in part by  Taiwan's MoST  grant  108-2115-M-001-001-MY2.
Horses Morphological data were recorded on 2053 young jumping horses, at the end of a show jumping competition, and on 36 of their sires. A first sample of 784 horses was recorded in 2002 and subsequent samples in 2015  (734)  and in 2016 (571). Recording was performed at 24 different events, with between 15 and 201 horses per event (average: 87. 0). Gender was evenly distributed: 49% females and 51% males (including 26% geldings). Age was also evenly distributed among the young horses: 52% were 4 years old and 48% were 5 or 6 years old (with only four horses aged 6). The 36 sires were between 9 and 25 years old. The breed of the horses was mostly Selle Français (SF, 91%) with a few foreign sport horses (5%), Anglo Arabs (2%), and crossed sport horses (2%). Among those that were not SF, 39% had at least one SF parent.
Morphometric data The 3D morphometric method was developed by  [33, 37, 38]  and has already been described in detail in  [39]. Briefly, horses were filmed when led by hand while walking back and forth on a 1-m-wide and 25-m-long level track. Horses were filmed from four angles, after calibration of the cameras with an aluminum structure of known dimensions: from the front, from the back, and from two angles from the right side, slightly from the front and slightly from behind (Fig. 1 ). Data processing was then performed off-line with synchronized and digitized films. For each horse, two reference moments were identified: one for analysis of the forehand and one for the hindquarters. References were chosen according to the vertical position of the fore and hind cannon bone, respectively. For each reference moment, the four angle views of the horse were then downloaded onto computers.
Competition data Analysis of the jumping competition data was performed with the complete dataset of official jumping competitions from 2001 to 2019 since the birth year 1997, including 150,817 horses and 654,340 annual performances. Horses born abroad with a partial career in France were excluded from this dataset. Success in competition was measured by the repeated observations of an annual performance, which was the logarithm of an annual sum of points. Points were distributed in each competition according to rank and technical difficulty. Among the young horses that were measured for morphometry, nearly all (95%) performed in jumping competitions.
Judges' score data Morphology was judged during young horse breeding shows organized by the French stud book of Selle Français. From 2005 to 2019, 28,412 horses aged 2 or 3 were recorded, from one to nine times per horse, with a mean of 1. 6, for a total of 46,349 records. Recorded horses were 46% female. Horses were scored on a scale from 1 to 20 for nine criteria: neck, body forehand (shoulder, arm, forearm), withers-back-loins, body hindquarters (croup, pelvis, thigh), joints, stance of forelegs, stance of hindlegs, impression, and chic. Only 19% of the horses that were measured for morphometry were also judged in breeding shows but 92% of them had paternal half-sibs that were judged in breeding shows (mean number of siblings: 188. 5).
0 reference sequence was used to specify SNP positions on the reference map; EquCab 2. 0 used as alternative reference when this information was not available (N = 2127 SNPs). After quality control, 375,687 SNPs were retained, which represented 73. 2, 73. 2, and 55. 7% of the SNPs on, respectively, the Illumina Equine SNP50 chip, the Equine SNP70 Bead-Chip, and the Affymetrix Axiom Equine chip. Imputation was then performed using the Fimpute 3. 0 software  [43] , and by adding pedigree information for four generations  (18,682 horses). The genomic data were used only for genome-wide association analysis (GWAS).
Because of differences in the walking phase when the image was selected, the actual position of the neck was not relevant for the morphological description. Each horse was therefore finally characterized by the three coordinates for 43 points (Fig. 3 ). A Procrustes analysis was performed in order to center and scale the 3D landmark coordinates  [34]  using the Geomorph package  [44]  in the R environment, v. 4. 0. 2  [45]. The Procrustes analysis gave a centroid size and Procrustes coordinates. For the genetic analysis, only unconstrained variables were retained: Procrustes coordinates of points before mirroring and those that were not used as a reference or fixed by construction (for example, the X coordinate for point 4 was the same as that of point 3 because the verticality of the canon bone was used to define the reference image).
The Wombat software  [46]  was used to estimate variance components by restricted maximum likelihood (REML). Bivariate analyses were performed for all possible pairs of the 63 coordinates. Then, estimates of 2 × 2 (co-)variance matrices from these 1953 analyses were combined to provide an overall 63 × 63 positive semi-definite (co-)variance matrix with the following requirements: elements 'similar' to those from individual analyses, no substantially change in estimates of variance ratios, and result in estimates of phenotypic covariances components that are little distorted. The method used is a likelihood-based (1) y = Xb + Zu + e, approach which implicitly treats estimates from individual analyses as if they were matrices of corrected sums of squares and cross-products due to some pseudo observations  [47].
GWAS for morphometric shapes A mixed linear animal model similar to Eq. 1 was used to analyze the summarized shapes y • , using the BLUPF90 software  [48]. The relationship matrix between genetic values was constructed using both pedigree and genomic data in a single-step genomic best linear unbiased prediction (GBLUP) method  [49] [50] [51] [52] , as well as the rules outlined by  [53]  to construct the relationship matrix. GWAS was performed using back solutions of SNP effects and associated p-values, calculated as described in  [54]. Genomic control  [55]  was applied to the p-values to prevent incorrect distribution of the test statistics. The effective number of independent tests in our data was calculated as the inverse of the mean of the linkage disequilibrium (r 2 ) between all available pairs of SNPs by chromosome  [56].
Joint analysis of morphometry and performance in competition and judges scores Bivariate models for each of the ten initial summarized shapes y • were used to estimate genetic correlations of morphometry with competition results and judges scores, using the Wombat software  [46]. The model used for each y • was that of Eq. (  1 ) with a supplementary conceptual permanent environmental effect to separate, in the residual, the part correlated with every performance in competition or breeding shows from the uncorrelated part. The model used for competition data and judges scores was: where y c is the vector of annual competition perfor- mances or judges scores; b c is the vector of fixed effects, the combined effect (134 levels) of year (from 2001 to 2019), age class (from 4 to 10 years old in steps of 1 year, 11-12 years old, and 13 years old and older), and gender (male: stallions and geldings together; female) for competition performance, and class of age (five levels according to age in days: ≤ 2.
Special features of genetic analysis of geometric morphometrics Morphology of an organism must be studied in the form of synthetic criteria, which are the combinations of the  coordinates of the elementary points placed anatomically on the organism. However, the possible number of anatomical landmarks used to describe morphology is necessarily large. In our case, we recorded the position in space (three coordinates) of 25 points, i. e. , 75 variables. Since the position of some was fixed beforehand, 63 variables remained. Estimating the variance-covariance matrix of 63 variables is a challenge since it involves 2016 genetic (co-)variances and as many residuals (co-)variances. In order to estimate these variance components while reducing the size of the system, Meyer  [46]  developed an algorithm and a software program, WOMBAT, that performs an eigenvalue and eigenvector decomposition of the genetic and residual variance-covariance matrix simultaneously with likelihood maximization of the variables subject to the fitted linear mixed model.
Acknowledgements The authors wish to thank all the owners, riders, grooms and helpers, show organizers and  Selle Français Stud  Book representatives who made the data collection possible. We thank the team who measured the horses and the position of the anatomical marks:  E. Bezombes ,  D. Billerey Lemen ,  P. Bleteau ,  M. Chassot ,  P. Fontaine ,  N. Garcia-Marinez ,  L. Gosselin ,  F. Guillot ,  N. Marechal ,  C. Ruhlmann ,  J. M. Yvon.
We illustrate our results by explicit examples where the computations were done using the NonA package, developed in Maple, in which we have implemented the results of this paper. Details on the package as well as a complete worked example are given in Section 4. 3. The paper is organized as follows. In Section 2, we recall the main results of  [Chartouny et al. (2021) ] concerning the case of a perfect measurement. Section 3 contains the main contribution of the present paper for a class of standard structured perturbations. Section 4 contains the details concerning the implementation in Maple of the results developed in the paper and a worked example that illustrates the main commands. Finally, all the mathematical prerequisites (Laplace transform, convolution product, Weyl algebra, Gröbner basis, annihilator) are recalled in Section A.
The NonA package The NonA package 2 is dedicated to the effective study of the algebraic parameter estimation problem, introduced by M. Fliess and H. Sira-Ramirez in  [Fliess et al. (2003) ] and investigated in the former Non-A project-team (Inria Lille-Nord Europe). This Maple package, based on the OreModules package  [Chyzak et al. (2007) ], has been developed by A. Quadrat (Inria Paris, Ouragan projectteam). The binary is freely downloadable at https://who. rocq. inria. fr/Alban. Quadrat/Non-A/. Let us shortly describe the main commands of this package.
Development of the NonA package For some examples, Maple 2020 is not able to calculate the inverse Laplace transform. For instance, if we want to compute the inverse Laplace transform of 𝑍 ′ (𝑠) 2 using Maple, i. e. , the square of the first derivative of 𝑍 (𝑠) with respect to 𝑠, the output is "Error, (in collect) invalid 1st argument normal". We encounter the same problem with expressions of the form 𝑍 (𝑠) 2 , 𝑍 (𝑠) 3 , 𝑍 ′′ (𝑠) 4 , 𝑍 (𝑠) 𝑍 ′ (𝑠) 3. To solve this problem, we have implemented a convolution procedure. This function takes as input an expression 𝑒 that can contain powers of functions in 𝑠 and the parameter 𝑡. The output of the function is the convolution of 𝑒. For instance, the command convolution(Z(s)^2,t); yields the following output 𝑡 0 𝑍 (𝑡 -𝜏 1 ) 𝑍 (𝜏 1 ) 𝑑𝜏 1. Moreover, if we want to compute the degree in 𝑠 of an expression that has the following form 𝑒 := 𝑧(𝑠) 5 𝑠 9 + 𝑠 4 + 𝑧 ′′ (𝑠) 𝑠 2 , the function degree will display FAIL.
As input, the function takes the matrix 𝑈 and gives 𝑈 -1 as output. This function enables us to save time in the computation process since the explicit expression of 𝑈 -1 can be given (see  [Chartouny et al. (2021) ]). For instance, when using the regular function of Maple to compute the inverse of 𝑈 for 𝑛 = 9, the function takes 0. 183 CPU seconds. On the other hand, the InvMatrix only takes 0. 005 CPU seconds. Of course, the difference highly increases with the size 𝑛 of the matrix 𝑈 to inverse. More importantly, we are collecting different procedures that make automatic the computation of the initial conditions and the parameters of a signal defined by a linear ODE with polynomial coefficients affected by a structured perturbation of the form given in Section 3. Finally, our second contribution to the NonA package is the development of more explicit examples that have been added to the NonA library of examples (e.
Worked example We illustrate the NonA package with Examples 3. 1, 3. 4, 3. 7, 3. 9, and 3. 11. For more complex and therefore longer examples (e. g. , sums of exponentials, sums of sinusoids, sums of orthogonal polynomials), see the library of examples of the NonA package. We first start by loading the following two packages: > with(OreModules): with(NonA): We then define the OD operator annihilating the signal 𝑥(𝑡) = 𝑎 sin(𝜔 𝑡 + 𝜙) and its order: > L := dt^2+omega^2; n := 2: 𝐿 := dt 2 + 𝜔 2 We can then define the following two Weyl algebras 𝐴 = Q(𝜔, 𝜗 0 , 𝜗 1 , 𝑐)⟨𝜕 𝑡 , 𝑡 | 𝜕 𝑡 𝑡 = 𝑡 𝜕 𝑡 + 1⟩, 𝐵 = Q(𝜔, 𝜗 0 , 𝜗 1 , 𝑐)⟨𝜕 𝑠 , 𝑠 | 𝜕 𝑠 𝑠 = 𝑠 𝜕 𝑠 + 1⟩ in which we shall make the computations (for the time-domain and for the frequency-domain): > A := DefineOreAlgebra(diff=[ds,s],polynom=[s],comm=[omega,vartheta[0], > vartheta[1],c]): > B := DefineOreAlgebra(diff=[dt,t],polynom=[t],comm=[omega,vartheta[0], > vartheta[1],c]): We can now apply the command ParameterEstimation of the NonA package that generates the OD operators and polynomials defining the different terms of Equation (  13 ): > ParEst := ParameterEstimation(L,B); 𝑃𝑎𝑟 𝐸𝑠𝑡 := [𝜔 2 + 𝑠 2 , [-𝑠, -1], -𝜔 2 -𝑠 2 ] We state again that we consider the case of a ramp perturbation, i.
Moreover, we can also check again that 𝑃1 𝑄 = 0: Next, we apply 𝐺 1 to 𝑠 2 eq to get the equation from which we can compute the initial conditions 𝜗 0 and 𝜗 1. Note that, as expected, this equation does not depend on the unknown constant 𝑐. > EQ1 := ApplyMatrix(P1,s^2*eq,A); We then apply the command ExplicitExpression of the NonA package to obtain the following expression of 𝜗 0 in the time-domain: > vartheta0_t := ExplicitExpression(vartheta0_s,s); 𝐸𝑄1 := 𝑠 2 𝜔 2 + 𝑣𝑎𝑟𝑡ℎ𝑒𝑡𝑎0-𝑡 := 30 1 𝑡 6 ( 𝑡 0 -1/24 𝜔 4 (𝑡 -𝜏 1 ) 4 𝜏 1 5 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -𝜔 2 (𝑡 -𝜏 1 ) 2 𝜏 1 5 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -𝜏 1 5 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 1/12 𝜔 4 (𝑡 -𝜏 1 ) 5 𝜏 1 4 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 6 𝜔 2 (𝑡 -𝜏 1 ) 3 𝜏 1 4 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 (26 𝑡 -26 𝜏 1 ) 𝜏 1 4 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -1/36 𝜔 4 (𝑡 -𝜏 1 ) 6 𝜏 1 3 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -8 𝜔 2 (𝑡 -𝜏 1 ) 4 𝜏 1 3 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -110 (𝑡 -𝜏 1 ) 2 𝜏 1 3 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 14 𝜔 2 (𝑡-𝜏 1 ) 5 𝜏 1 2 𝑧 (𝜏 1 ) 5 d𝜏 1 + 𝑡 0 120 (𝑡 -𝜏 1 ) 3 𝜏 1 2 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -1/5 𝜔 2 (𝑡 -𝜏 1 ) 6 𝜏 1 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -35 (𝑡 -𝜏 1 ) 4 𝜏 1 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 2 (𝑡 -𝜏 1 ) 5 𝑧 (𝜏 1 ) d𝜏 1 ) Using this formula for 𝜗 0 , we can compute an expression for 𝜗 1.
We do not print the result here. > vartheta1_s := normal(subs(vartheta[0]=vartheta0_s,solve(EQ1,vartheta[1]))): > vartheta1_t := ExplicitExpression(vartheta1_s,s): The above expressions involved the unknown coefficient 𝜔 2 of the ODE satisfied by 𝑥(𝑡). We shall now estimate it. To achieve this task, we first apply 𝜕 4 𝑠 to the equation 𝑠 2 eq (see Example 3. 11): > P3 := ds^4: > EQ2 := ApplyMatrix(P3,s^2*eq,A); 𝐸𝑄2 := d 4 d𝑠 4 𝑧 (𝑠) 𝜔 2 𝑠 2 + d 4 d𝑠 4 𝑧 (𝑠) 𝑠 4 + 8 d 3 d𝑠 3 𝑧 (𝑠) 𝜔 2 𝑠 + 16 d 3 d𝑠 3 𝑧 (𝑠) 𝑠 3 +12 d 2 d𝑠 2 𝑧 (𝑠) 𝜔 2 + 72 d 2 d𝑠 2 𝑧 (𝑠) 𝑠 2 + 96 𝑠 d d𝑠 𝑧 (𝑠) + 24 𝑧 (𝑠) and we then solve the result for 𝜔 2 to obtain > omega2_s := solve(EQ2,omega^2); 𝑜𝑚𝑒𝑔𝑎2 -𝑠 := - d 4 d𝑠 4 𝑧 (𝑠) 𝑠 4 +16 d 3 d𝑠 3 𝑧 (𝑠) 𝑠 3 +72 d 2 d𝑠 2 𝑧 (𝑠) 𝑠 2 +96 𝑠 d d𝑠 𝑧 (𝑠)+24 𝑧 (𝑠) d 4 d𝑠 4 𝑧 (𝑠) 𝑠 2 +8 𝑠 d 3 d𝑠 3 𝑧 (𝑠)+12 d 2 d𝑠 2 𝑧 (𝑠) Then, we use the ExplicitExpression command of the NonA package again to obtain the following expression of 𝜔 2 in the time domain: > omega2_t := ExplicitExpression(omega2_s,s); Finally, if we replace 𝑧(𝑡) by 𝑥(𝑡) = 𝑎 sin(𝜔 𝑡 + 𝜙) in the formulas obtained for 𝜔 2 , 𝜗 0 , and 𝜗 1 , then we can check that we find again 𝜔 2 , 𝑎 sin(𝜙), and 𝜔 𝑎 cos(𝜙) as expected: 𝑜𝑚𝑒𝑔𝑎2 -𝑡 := 𝑡 0 -𝜏 1 4 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -(-16 𝑡 +16 𝜏 1 )𝜏 1 3 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -36 (𝑡 -𝜏 1 ) 2 𝜏 1 2 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 16 (𝑡 -𝜏 1 ) 3 𝜏 1 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -(𝑡 -𝜏 1 ) 4 𝑧 (𝜏 1 ) d𝜏 1 𝑡 0 1/2 (𝑡 -𝜏 1 ) 2 𝜏 1 4 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 -4/3 (𝑡 -𝜏 1 ) 3 𝜏 1 3 𝑧 (𝜏 1 ) d𝜏 1 + 𝑡 0 1/2 (𝑡 -𝜏 1 ) 4 𝜏 1 2 𝑧 (𝜏 1 ) > z := t -> a*sin(omega*t+phi): > simplify(convert(omega2_t,int)); > simplify(convert(vartheta0_t,int)); > simplify(convert(vartheta1_t,int)); 𝜔 2 𝑎 sin (𝜙) 𝜔 𝑎 cos (𝜙).
Study design and setting This descriptive study relied on data collected with a questionnaire during a national congress on general medicine in Bordeaux, France, from 16-18 June 2021. This was the first face-to-face congress after the lifting of the COVID-19 related restrictions. Participants (~1300 GPs and GP trainees) came from all over France. An invitation to complete the questionnaire was sent by email to all congress participants. The questionnaire was completed anonymously. The study was carried out following the Commission Nationale de l'Informatique et des Libertés (CNIL) methodology MR-004 for the use of personal data for research. According to French regulations, this study did not require approval by an ethics committee. Respondents gave their informed and written consent at the beginning of the online questionnaire.