Xiran Cai
email: xirancai@stanford.edu

Laura Peralta

Renald Brenner

Gianluca Iori

Didier Cassereau

Kay Raum

Pascal Laugier

Quentin Grimal

Anisotropic elastic properties of human cortical bone tissue inferred from inverse homogenization and resonant ultrasound spectroscopy

Keywords: Bone extravascular matrix, Anisotropic stiffness, Inverse homogenization, Resonant ultrasound spectroscopy

Introduction

Bone size, shape and chemical composition have evolved to reach a compromise between mass, resistance to fracture and amplitude of elastic deformations [START_REF] Currey | Bones: structure and mechanics[END_REF]. The elastic properties of bone tissue at the millimeter scale (or mesoscale) [START_REF] Grimal | A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties[END_REF], which play a key role in bone mechanics, depend on the properties and organization of the constituents at the nanoscale (mineral crystals and collagen molecules), at the micrometer scale (aligned mineralized collagen fibers forming lamellae) [START_REF] Rho | Mechanical properties and the hierarchical structure of bone[END_REF], and at the scale of a few tenth of microns where osteons and vascular pores are the main microarchitectural features.

Bone tissue at the micrometer scale, i.e., the solid material in which vessels and nerves are embedded is often referred to as bone extravascular matrix (EVM), shortly bone matrix [START_REF] Fritsch | universal' microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: Micromechanics-based prediction of anisotropic elasticity[END_REF][START_REF] Grimal | A two-parameter model of the effective elastic tensor for cortical bone[END_REF][START_REF] Parnell | Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone[END_REF]. Bone material at the mesoscale is accordingly modelled as a network of vascular canals (Haversian and Volkman canals) embedded in a homogeneous bone matrix [START_REF] Hellmich | Can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independent phase properties and their interactions?[END_REF][START_REF] Baumann | The relative influence of apatite crystal orientations and intracortical porosity on the elastic anisotropy of human cortical bone[END_REF][START_REF] Cai | Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity[END_REF]. The elastic properties of EVM deserve a careful study because, combined with the micro-architecture of the vascular porosity, they determine bone elasticity at the mesoscale, which has a direct impact on the macroscopic mechanical behavior of whole bone. Also, for a given macroscopic loading, bone matrix elastic properties play a role in mechanotransduction as they govern the amplitude of matrix deformations which determine the interstitial fluid flow and the local strains sensed by osteocytes [START_REF] Hemmatian | Aging, osteocytes, and mechanotransduction[END_REF].

While, the inter-individual variations of EVM elastic properties in the general population are expected to be small [START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF][START_REF] Hoffler | Heterogeneity of bone lamellar-level elastic moduli[END_REF], considerable alteration of matrix elastic properties have been observed as a consequence of bone pathologies [START_REF] Currey | Role of collagen and other organics in the mechanical properties of bone[END_REF][START_REF] Grabner | Age-and genotype-dependence of bone material properties in the osteogenesis imperfecta murine model (oim)[END_REF][START_REF] Bala | Bone mineralization: from tissue to crystal in normal and pathological contexts[END_REF] and drugs [START_REF] Bala | Bone micromechanical properties are compromised during long-term alendronate therapy independently of mineralization[END_REF][START_REF] Ma | Long-term effects of bisphosphonate therapy: perforations, microcracks and mechanical properties[END_REF]. Several experimental approaches have been proposed to measure EVM elastic properties, which can be categorized as direct and indirect approaches. Direct approaches are nanoindentation [START_REF] Zysset | Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur[END_REF][START_REF] Franzoso | Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation[END_REF], scanning acoustic microscopy (SAM) [START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF][START_REF] Raum | Site-matched assessment of structural and tissue properties of cortical bone using scanning acoustic microscopy and synchrotron radiation µct[END_REF] and mechanical testing on µm-sized bone specimens [START_REF] Yamada | Micro-cantilever bending for elastic modulus measurements of a single trabecula in cancellous bone[END_REF][START_REF] Schwiedrzik | In situ micropillar compression reveals superior strength and ductility but an absence of damage in lamellar bone[END_REF][START_REF] Luczynski | Extracellular bone matrix exhibits hardening elastoplasticity and more than double cortical strength: evidence from homogeneous compression of non-tapered single micron-sized pillars welded to a rigid substrate[END_REF]. For instance, using nano-indentation, elastic properties of trabecular and cortical bone tissues were found to be similar [START_REF] Turner | The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques[END_REF] and the interstitial tissue was found to be stiffer than osteonal tissue [START_REF] Rho | Elastic properties of microstructural components of human bone tissue as measured by nanoindentation[END_REF]. Using SAM, it was found that collagen fiber orientation has an impact on tissue elastic anisotropy probed at the microscale [START_REF] Granke | Microfibril orientation dominates the microelastic properties of human bone tissue at the lamellar length scale[END_REF][START_REF] Schrof | Multimodal correlative investigation of the interplaying micro-architecture, chemical composition and mechanical properties of human cortical bone tissue reveals predominant role of fibrillar organization in determining microelastic tissue properties[END_REF].

These direct approaches have several limitations. Several assumptions on tissue elasticity (isotropy, assumed Poisson's ratio) are necessary to obtain the Young's modulus from a nanoindentation load-displacement curve [START_REF] Oyen | A practical guide for analysis of nanoindentation data[END_REF]. Also, to get a representative value of bone matrix elasticity, several tens of indentation locations must be considered in the osteonal and interstitial tissues. SAM yields a map of the acoustic reflectivity of a bone surface using a high frequency ultrasound probe. Precisely, the method yields acoustic impedance values which can only be converted to Young's modulus using assumptions on tissue composition and Poisson's ratio [START_REF] Preininger | Spatial-temporal mapping of bone structural and elastic properties in a sheep model following osteotomy[END_REF]. Mechanical testing on µm-sized specimens requires dedicated experimental apparatus, long specimen's preparation time and it is not always possible to achieve a testing environment mimicking physiological conditions, e.g., as for experiments conducted in an electronic microscope [START_REF] Schwiedrzik | In situ micropillar compression reveals superior strength and ductility but an absence of damage in lamellar bone[END_REF].

An indirect approach to measure trabecular tissue elasticity was introduced by van Rietbergen et al. [START_REF] Van Rietbergen | Determination of trabecular bone tissue elastic properties by comparison of experimental and finite element results[END_REF]; Tissue elasticity is retrieved by back-calculation combining experimental mechanical testing on a specimen of several millimeters and a corresponding micro-finite element model accounting for the specimen-specific trabecular microarchitecture. A linear relationship between tissue Young's modulus and bulk (measured) modulus can be assumed, which makes it possible to infer tissue modulus by searching for a good match between model and experimental data. A similar approach was proposed in which mechanical testing was replaced by ultrasound resonance testing [START_REF] Daoui | Assessment of trabecular bone tissue elasticity with resonant ultrasound spectroscopy[END_REF].

Direct methods are designed to probe the EVM elastic modulus in one single direction (typically along the osteon axis), and inverse methods, as used to retrieve matrix elasticity in trabecular bone, assume isotropic properties. This is a major limitation of both direct and indirect approaches: they hardly give access to elastic anisotropy unless very specific and elaborated techniques are implemented [START_REF] Franzoso | Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation[END_REF][START_REF] Lakshmanan | Assessment of anisotropic tissue elasticity of cortical bone from high-resolution, angular acoustic measurements[END_REF]. However, EVM is not isotropic [START_REF] Rho | Elastic properties of microstructural components of human bone tissue as measured by nanoindentation[END_REF][START_REF] Saïed | Spatial distribution of anisotropic acoustic impedance assessed by time-resolved 50-mhz scanning acoustic microscopy and its relation to porosity in human cortical bone[END_REF][START_REF] Wolfram | Transverse isotropic elastic properties of vertebral trabecular bone matrix measured using microindentation under dry conditions (effects of age, gender, and vertebral level)[END_REF].

The present article introduces a novel indirect approach to retrieve the anisotropic EVM elastic properties.

The new method is applied to a collection of human specimens to document the EVM elastic properties in an elderly population. Our approach, coined inverse homogenization, is similar to the back calculation procedure introduced by van Rietbergen et al. [START_REF] Van Rietbergen | Determination of trabecular bone tissue elastic properties by comparison of experimental and finite element results[END_REF] for trabecular bone but it uses anisotropic elastic information at the mesoscale (experimental data) and the computation of a specimen's elastic tensor with homogenization. The method combines resonant ultrasound spectroscopy (RUS) which accurately measures the transverse isotropic elastic tensor of millimeter-sized bone specimens [START_REF] Bernard | Accurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopy[END_REF][START_REF] Bernard | Elasticity-density and viscoelasticity-density relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements[END_REF] and Fast Fourier Transform (FFT) homogenization method [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF][START_REF] Brenner | Numerical computation of the response of piezoelectric composites using fourier transform[END_REF], which has been widely used in engineering material mechanics to evaluate the effective properties of composites and porous materials.

Materials and methods

Sample preparation

Bone specimens were harvested from the left femur of 29 human cadavers, provided by the Départment Universitaire d'Anatomie Rockefeller (French body donation to science program, declaration number: DC-2015-2357; Laboratory of Anatomy, Faculty of Medicine Lyon Est, University of Lyon, France) through the French program on voluntary corpse donation to science. Among the donors, 16 were females and 13 were males (50 -95 years old, 77.8 ± 11.4, mean ± SD). The nominal specimen size was 3 × 4 × 5 mm 3 in radial (axis 1), circumferential (axis 2) and axial directions (axis 3), respectively, defined by the anatomical shape of the femoral diaphysis. Details of sample preparation can be found in Cai et al. [START_REF] Cai | Anisotropic elastic properties of human femoral cortical bone and relationships with composition and microstructure in elderly[END_REF].

Bulk elasticity of the bone specimens

Cortical bone was assumed to be a transversely isotropic material [START_REF] Yoon | Ultrasonic wave propagation in human cortical bone -ii. measurements of elastic properties and microhardness[END_REF][START_REF] Orías | Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur[END_REF]; using Voigt notation, the matrix form of the stiffness tensor writes

C =               C 11 C 12 C 13 0 0 0 C 12 C 11 C 13 0 0 0 C 13 C 13 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 66               , (1)
where, C 12 = C 11 -2C 66 and (1-2) is the isotropy plane; C 11 and C 33 are the longitudinal stiffness constants, C 12 and C 13 are the off-diagonal stiffness constants; and C 44 and C 66 are the shear stiffness constants.

In the present study, we use already published anisotropic elasticity data of cortical bone specimens measured with RUS [START_REF] Cai | Anisotropic elastic properties of human femoral cortical bone and relationships with composition and microstructure in elderly[END_REF]. The main sources of experimental errors are related to the irregularities of specimen's geometry and the uncertainties on the values of extracted resonant frequencies. These errors have been documented in a previous study and are of the order of 1.7% for shear stiffness constants, 3.1% for longitudinal and off-diagonal stiffness constants [START_REF] Cai | Quantification of stiffness measurement errors in resonant ultrasound spectroscopy of human cortical bone[END_REF]. Hereafter, the stiffness constants measured by RUS are referred to as C EXP ij .

Bone microstructure and degree of mineralization

Degree of mineralization of bone (DMB) and vascular porosity (φ) data used in this work has been published and was obtained following the imaging protocol described in detail in Cai et al. [START_REF] Cai | Anisotropic elastic properties of human femoral cortical bone and relationships with composition and microstructure in elderly[END_REF]. Briefly, specimens were imaged with a pixel size of 6.5 µm with synchrotron radiation micro-computed tomography (SR-µCT) at the European Synchrotron Radiation Facility (ESRF) using a SR-µCT setup based on a 3-D parallel beam geometry acquisition [START_REF] Salomé | A synchrotron radiation microtomography system for the analysis of trabecular bone samples[END_REF][START_REF] Weitkamp | Status and evolution of the esrf beamline id19[END_REF]. The calibrated gray levels of each specimen were converted into volumetric DMB as detailed in Nuzzo et al. [START_REF] Nuzzo | Synchrotron radiation microtomography allows the analysis of three-dimensional microarchitecture and degree of mineralization of human iliac crest biopsy specimens: Effects of etidronate treatment[END_REF].

For the purpose of homogenization (see Section 2.5), the 3D volume of each specimen was slightly rotated using Fiji [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] so that the image reference frame coincides with the orientation of the specimen's faces. In each specimen, the largest possible rectangular parallelepiped volume of interest (VOI) was selected manually to achieve the largest available volume (average size about 2.8 × 3.9 × 4.8 mm 3 ). The VOIs were binarized by simple thresholding treating the void volumes as a solid and the bone phase as a background, from which φ of each specimen was measured. Finally, the voxel size was downsampled to 35 µm after having conducted convergence tests reported in Appendix B. The average total number of voxels in each VOI was about 1.2 million.

Bone matrix acoustic impedance

A subset of 23 specimens (all from the lateral quadrant) were measured with a custom SAM to provide an independent measurement of tissue elastic properties for comparison with the results of inverse homogenization. The SAM operating with a spherically focused 100-MHz transducer (KSI 100/60 • , Krämer Scientic Instruments, Herborn, Germany) was used to probe the acoustic impedance normal to the samples surfaces according to the measurement procedure extensively detailed in [START_REF] Raum | Bone microstructure and elastic tissue properties are reflected in qus axial transmission measurements[END_REF]. Calibrated impedance (Z) maps were obtained with a lateral resolution of 19.8 µm for the faces perpendicular to radial and axial directions. The Z-maps were segmented allowing the separation of vascular porosity and bone EVM. The acoustic impedance of the matrix was determined from the segmented maps of the specimens. The small pores (Volkmann's canal, osteocyte lacunae) could not be resolved so that they contributed to the probed bulk matrix properties.

Matrix impedance in the probing direction, denoted Z i (i = 1, 3), was defined as the mean value of the Z-map scanned in the face of direction i. Matrix impedance was then converted to EVM stiffness using the following equation:

C Z ii = Z i ρ t , (2)
where ρ t = 1.12 + 0.73 • DMB -0.033 • DMB 2 is the tissue density converted from DMB values [START_REF] Raum | Derivation of elastic stiffness from site-matched mineral density and acoustic impedance maps[END_REF].

Forward problem: homogenization to simulate bulk stiffness

Each specimen's VOI was considered as a representative volume for homogenization containing two homogeneous phases: vascular porosity and EVM. The elastic properties of the pore voxels were defined with a bulk modulus of 2.2 GPa and a null shear modulus, corresponding to elastic properties of water.

The EVM voxels were allocated transverse isotropic elastic properties C m ij . The isotropy plane of the EVM material was taken to be the plane 1 -2. The terms C m ij are the unknowns of the inverse homogenization problem.

The bulk stiffness tensor C F F T ij of each VOI was evaluated by the FFT homogenization method [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF][START_REF] Brenner | Numerical computation of the response of piezoelectric composites using fourier transform[END_REF].

Briefly, the bulk stiffness tensor is obtained after solving the local mechanical problem in the VOI (V ) consisting of the equilibrium equation, generalized Hooke's law as constitutive equation, compatibility of the displacement field and boundary conditions. The phases are assumed to be perfectly bonded. The local problem closed by periodic boundary conditions [START_REF] Suquet | Homogenization Techniques for Composite Media[END_REF][START_REF] Michel | Effective properties of composite materials with periodic microstructure: a computational approach[END_REF] can be solved by the FFT-based numerical approach proposed by Moulinec and Suquet [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF].

The implementation of the method has been described in [START_REF] Cai | Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity[END_REF]. Note that no elastic symmetry assumption was made for the calculation of the effective stiffness tensor. For the requirements of the present study, we implemented a parallelized version of the FFT method. One loading in the FFT homogenization method took around 50 iterations in our specimens and the effective stiffness tensor could be computed in about 1 minute on a workstation (CPU, Intel Xeon E5-2695 v3, 8 threads per loading).

For all VOIs, the values of the off-diagonal terms of the homogenized stiffness tensor C F F T , which are null in the case of orthotropy when the tensor is expressed in the material basis, were at least two orders of magnitude smaller than the other terms. These off-diagonal terms were, therefore, disregarded, i.e., the 12 coefficients with index kl = 11, 22, 33, 44, 55, 66, 12, 21, 13, 31, 23, 32 were kept.

Inverse homogenization to recover EVM stiffness

The inverse homogenization consists in determining the matrix stiffness tensor C m which provides an optimum match between C EXP and C FFT (C m ) Two approaches have been implemented: (1) The 'calibration approach' assumes that one mesoscale coefficient C F F T ij is fully determined by the porous microstructure and

its EVM counterpart C m ij . (2)
The 'optimization approach' determines the EVM stiffness constants without such assumption, i.e., it accounts for the possible dependence of one term of the mesoscale tensor C F F T ij on all EVM stiffness constants. The 'calibration approach' is straightforward and less demanding in terms of computational resources, but intrinsically less precise to determine C m ij . We found that the two approaches lead to only slightly different results. Only the details and the results of the 'optimization approach' are detailed in the paper. The methodology and results of the 'calibration approach' are reported in Appendix A.

EVM stiffness constants C m of each specimen were obtained by minimizing the objective function defined as the root-mean-square-error (RMSE) between experimental and homogenized bulk stiffness constants. That is, for each specimen, we solve

RM SE(C m ) = 1 12 kl C EXP kl -C F F T kl (C m ) C EXP kl 2 × 100%, (3)
Note that the 12 non-zero coefficients (labeled by kl ) are used in eq. GPa, C m0 66 = 5.4 GPa. These values, reported in [START_REF] Cai | Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity[END_REF], were found by minimizing the differences between experimental and homogenized stiffness constants for the entire collection of specimens.

Estimation of the precision of inverse homogenization to recover EVM stiffness

C m is recovered from experimental elasticity data and the images of each specimen's microstructure.

The images have a high contrast and a high resolution so that there is little concern about the precision of Three specimens with porosities covering the entire porosity range (5.0%, 9.9% and 15.4%) were selected for the MC simulations to account for an expected dependence of the error on porosity. The experimental uncertainties on stiffness constants were modeled with independent normal distributions. The distributions were centred at mesoscale stiffness values obtained from the FFT homogenization of each of the three specimen's VOI with EVM stiffness C m0 . The width of the distributions were consistent with the typical experimental uncertainties for longitudinal and off-diagonal stiffness constants (SD of 3.0%) and shear stiffness constants (SD of 1.5%). A set of 1000 independent realizations of mesoscale stiffness tensors were drawn from the normal distributions. The EVM stiffness C m ij was calculated for each realization of mesoscale stiffness tensor using inverse homogenization and compared with C m0 to calculate the relative error on each coefficient, defined as different. The level of significance for all the tests was set to p = 0.05.

δC m ij = (C m ij -C m0 ij )/C m0 ij × 100%, (4) 2

Results

Descriptive statistics

The specimens of one subject with a porosity higher than 30% was not included in the analysis. Another specimen was broken during sample preparation. Finally 55 specimens were used for this study.

The bulk elasticity of five specimens with a porosity larger than 14.5% were found to be slightly orthothropic due to the shape of the pore network [START_REF] Cai | Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity[END_REF]. Precisely, the relative difference between C F F T exceeded 3% for these five specimens. This violates the assumption of a transverse isotropic material used to process RUS measurements. Despite the fact that, for these specimens, the inverse homogenization optimization lead to a relatively good fit with a RMSE (eq. 3) between 1 and 2%, these specimens were discarded. The results presented below are consequently given for 50 specimens.

Experimental stiffness constants, porosity, DMB and matrix impedance are summarized in Table 1. The coefficient of variation (CV) of DMB was about 2% which is smaller than the CV of Z i (4.2 -4.3%) and The RMSEs after inversion (eq. 3) were small, between 0.06 -1.01% (Mean ± SD = 0.26 ± 0.19%).

C EXP ij (7 -14%).
The EVM stiffness constants C m ij and the engineering moduli (obtained by inverting the stiffness tensor) are summarized in Table 2. The CVs of C m ij (3.0 -7.1%, Table 2), EVM Young's moduli (4.5 -5.5%) and EVM Poisson's ratios (5.3 -11.5%) were greater than that of ρ m (2%, Table 1).

Error propagation analysis

The distribution of each δC m ij (eq. 4) was found to be normally distributed (Shapiro-Wilk's test). The results are summarized in Table 3. The offset (bias) of δC m ij from zero was small, less than 0.8%. The uncertainty characterized by SDs of δC m ij were between 3.1-4.7% for longitudinal and off-diagonal coefficients and 1.5 -1.9% for shear coefficients (Table 3). These uncertainties are comparable with the SDs of the errors modeled in MC simulations and corresponding to experimental uncertainty on C EXP ij . We noted that the SD of δC m ij increases as porosity increases. is comparable with that of Z i . Significant positive correlations were observed between Z 1 and the C m 44 , C m 66 (0.52 < r < 0.60, Figure 1), and between Z 3 and C m 44 (r = 0.42, Figure 2). Trends of positive (not significant) correlations were also observed between Z 1 and C m 11 , C m 33 and C m 13 (0.34 < r < 0.38, Figure 1), and between Z 3 and C m 66 (r = 0.36, Figure 2). negatively with ν m 12 (r = -0.61) (Figure 4). The linear regression equations between ρ m and EVM stiffness constants are given in Table 5.

Discussion

In this work, we introduced an inverse homogenization approach to evaluate the anisotropic elastic properties C m ij of cortical bone EVM. The approach uses a measurement of the transverse isotropic stiffness tensor at the mesoscale (with RUS) and assumes that the EVM is itself transverse isotropic. The EVM stiffness values obtained are to be understood as 'mean' values representative of the entire volume of the specimen as the forward problem (homogenization) assumes a homogeneous EVM throughout the specimen's volume. One key aspect of the work is that the homogenization is performed with the real vascular pore structure obtained with the state-of-the-art 3D µ-CT imaging. Also, the VOI used as a representative volume element for homogenization almost matches the specimen volume assessed by RUS, enabling a direct comparison between predicted and measured mesocale stiffness. It is noteworthy that the method provides the complete elastic tensor, from which the engineering moduli (i.e., Young's moduli, shear moduli and Poisson's ratios) can be obtained.

The fine knowledge of experimental errors on the stiffness constants measured by RUS [START_REF] Cai | Quantification of stiffness measurement errors in resonant ultrasound spectroscopy of human cortical bone[END_REF] allows performing a realistic estimation of the errors on C m ij determined by inverse homogenization. The magnitude of the errors (SD= 1.5 -1.9% for shear coefficients and SD = 3.1 -4.7% for longitudinal and off-diagonal coefficients, Table 3) were close to the errors on the bulk stiffness constants measured with RUS. Note that the errors on C m 11 were a bit larger than on C m 33 , which may be explained by the fact that small anisotropy in the transverse plane is disregarded in the inverse homogenization process, forcing an 'artificial' adjustment of C m 11 . The errors on the EVM stiffness may be minimized by improving the precision and accuracy of RUS [START_REF] Cai | Quantification of stiffness measurement errors in resonant ultrasound spectroscopy of human cortical bone[END_REF].

The average values of EVM stiffness constants and Young's moduli agreed well with the values reported in previous studies measured on human femoral bone as well [START_REF] Franzoso | Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation[END_REF][START_REF] Turner | The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques[END_REF][START_REF] Lakshmanan | Assessment of anisotropic tissue elasticity of cortical bone from high-resolution, angular acoustic measurements[END_REF] (Table 6). These values from the literature were evaluated by various approaches, including acoustic, indentation and micromechanical model-based methods. Specifically, the average values of C m 33 , C m 44 , C m 66 and all the engineering moduli fell inside the range of values measured by SAM and nano-indentation. For C m 11 and the two off-diagonal coefficients C m 13 and C m 12 , our values were a bit higher. As far as we are aware, only two articles reported the measurement of the entire anisotropic elastic tensor of the EVM [START_REF] Franzoso | Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation[END_REF][START_REF] Lakshmanan | Assessment of anisotropic tissue elasticity of cortical bone from high-resolution, angular acoustic measurements[END_REF]. The method used by Lakshmanan et al. [START_REF] Lakshmanan | Assessment of anisotropic tissue elasticity of cortical bone from high-resolution, angular acoustic measurements[END_REF], based on SAM measurements relies on empirical laws (relating acoustic impedance to several stiffness constants) and idealized micromechanical models (for the off-diagonal coefficients) to retrieve the elastic tensor. The method used by Franzoso and Zysset [START_REF] Franzoso | Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation[END_REF], based on nano-indentation measurements, uses an indenter shape-dependent model [START_REF] Swadener | Indentation of elastically anisotropic half-spaces by cones and parabolae of revolution[END_REF] and a fabric tensor incorporating important assumptions (fibers organization and simplified intrinsic material model).

For instance, our data and the data from [START_REF] Lakshmanan | Assessment of anisotropic tissue elasticity of cortical bone from high-resolution, angular acoustic measurements[END_REF] respect the inequality relation ν 23 < ν 31 < ν 12 noted in Cowin [START_REF] Cowin | Bone mechanics handbook[END_REF], while not the data from Franzoso and Zysset [START_REF] Franzoso | Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation[END_REF]. In our approach, we only assume transverse isotropic elastic symmetry for EVM. Furthermore, we provide data for a relatively large number of subjects. We found significant correlations between impedance data and EVM shear stiffness (Figure 1) and 2).

For the other stiffness constants, the data suggest a relationship might exist with acoustic impedance but no significant correlations were observed. The differences between the mean values of the longitudinal EVM stiffness evaluated by inverse homogenization and derived from SAM were relatively small (< 10.2%, Table 4) considering that the two methodologies are different. However, stiffness derived with SAM and inverse homogenization were not correlated. The low and non-significant correlation may in part be due to the fact that only 23 among 55 specimens were measured with SAM.

The reason for the lack of correlation is likely due to experimental errors of both techniques. RUS is intrinsically more accurate for shear stiffness compared to longitudinal stiffness [START_REF] Migliori | Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids[END_REF][START_REF] Landa | Modal resonant ultrasound spectroscopy for ferroelastics[END_REF], which would explain that correlations are found between impedance and shear stiffness. Note that, SAM requires a flat and polished surface to measure the acoustic impedance. This is obtained by a manual polishing of the specimens which is especially difficult to realize when the sample size is small, as is the case for our specimens (Z 1 and Z 3 were measured on the 4 × 5 mm 2 and 3 × 4 mm 2 surfaces, respectively). In addition, SAM is a surface measurement, whereas the C m ij evaluated by inverse homogenization are representative of the entire volume of a specimen. The relatively narrow range of variation of the EVM stiffness, combined with measurement errors are likely to weaken correlations.

Significant correlations between DMB (averaged in the specimen's volume) and EVM shear stiffness (Fig-

ure 3), and most of the EVM engineering moduli (Figure 4) were observed, which has not been reported before. Similar to the observation for impedance data, the correlation between DMB and the EVM longitudinal stiffness were not significant. This is consistent with the magnitude of the errors in stiffness measured by RUS as previously mentioned. Furthermore, opposite correlations between DMB and the Poisson's ratios were observed suggesting different deformation behavior in different directions may exist at the tissue level as bone is mineralizing, which may be interesting to investigate in the future. Note that the repeatability of the DMB measured by SR-µCT varying beam energy was reported to be as small as 0.26% using the same setup on beamline ID 19 at ESRF [START_REF] Nuzzo | Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography[END_REF], which may be a reason why correlations with stiffness are found although the range of variation of DMB is small.

The bone specimens used in this study are from donors with undocumented medical history. Overall, the presented EVM stiffness data set (Table 2) may only represent a group of elderly population without a specific bone pathology. As expected for such a population, the EVM stiffness values vary in a relatively small range (from 4.8% to 15.4%). This data may be used as a reference data set to compare with EVM stiffness in bone pathology.

Some limitations in this study should be mentioned. Although the entire EVM stiffness tensor of the specimens is provided in this work, the accuracy of the values need further investigation. The comparison with SAM-derived stiffness shows a relatively good agreement, and the comparison with the values obtained from nanoindentation indicates that the values of EVM stiffness obtained with inverse homogenization are in the expected range. Most importantly, the correlations between some stiffness constants and DMB demonstrate that the proposed method is sufficiently precise to probe small differences of elasticity due to small changes of DMB.

We assumed that the EVM is homogeneous. In reality, bone matrix mechanical properties are heterogeneous depending on the local variations of the degree of mineralization (differences of elastic modulus between osteonal and interstitial region in cortical bone [START_REF] Rho | Elastic properties of microstructural components of human bone tissue as measured by nanoindentation[END_REF][START_REF] Rho | Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nano-indentation[END_REF]) and microfibril orientation [START_REF] Granke | Microfibril orientation dominates the microelastic properties of human bone tissue at the lamellar length scale[END_REF]. Though it has been shown in trabecular bone that the bulk elastic properties only changed about 2% considering an heterogeneous model [START_REF] Gross | Mineral heterogeneity has a minor influence on the apparent elastic properties of human cancellous bone: a srµct-based finite element study[END_REF], the effect of neglecting tissue heterogeneity has not been quantified in cortical bone.

We also assumed that the EVM stiffness is transverse isotropic. This was a necessary assumption because the RUS experimental data itself was obtained with this assumption at the mesoscale level. This assumption proved to be valid for most of the specimens, however it may not be valid for specimens taken at other anatomical sites.

Conclusion

To conclude, we introduced an inverse homogenization method to evaluate cortical bone EVM anisotropic elastic properties. Our approach provides the entire EVM stiffness tensor with a precision of a few percent.

The EVM stiffness tensor is provided for 50 specimens from 26 elderly donors exhibiting the range of variation for this population. The present work is unprecedented documentation of EVM anisotropic elasticity.

One important perspective is the quantification of in situ strains of the EVM under loading which has consequences for the mechanotransduction process. The RMSE (eq. 3) was also calculated for each specimen after C m was obtained to compare with the ones from optimization approach. The RMSEs from the calibration approach were between 0.12 -2.52% (Mean ± SD = 0.76 ± 0.46%) greater than that from the optimization approach in general which is expected.

The optimization approach takes all the coupling between the coefficients into account, i.e., it allows more degrees of freedom. Therefore, it is expected that the optimization approach would produce smaller RMSEs.

  the reconstruction of the pore vascular network. The method of solution used for the forward problem yields the exact solution of the continuum mechanics problem with a controlled numerical precision. It follows that the uncertainties on C m ij mostly stem from the experimental uncertainties on C EXP ij . We conducted an error propagation analysis using Monte-Carlo (MC) simulations to estimate the errors on C m ij due to the uncertainties on C EXP ij .

22 andC F F T 11 ,

 2211 or between C F F T 44 and C F F T 55

Figure 1

 1 Figure 1 and 2 summarize the relationships observed between C m ij and Z i . The CV of C m ij (3.0 -7.1%)

Figure 1 :

 1 Figure 1: Relationships between Z 1 and C m ij , and the corresponding linear regression model (solid line) with the 95% CI (dot line). r is the Pearson's correlation coefficient.

Figure 2 :

 2 Figure 2: Relationships between Z 3 and C m 44 , C m 66 , and the corresponding linear regression model (solid line) with the 95% CI (dot line). r is the Pearson's correlation coefficient.

3. 4 .

 4 Variations of EVM stiffness with DMB Significant correlations were observed between ρ m and C m 44 , C m 66 (0.65 < r < 0.70, Figure 3). The data suggest a positive relationship might exist between C m 11 , C m 33 , C m 13 and ρ m , but the correlations (0.24 < r < 0.27) were not significant. Also, ρ m positively correlated with E m 1 , ν m 23 and ν m 31 (0.32 < r < 0.59), and

Figure 3 :

 3 Figure 3: Relationships between ρm and C m ij , and the corresponding linear regression model (solid line) with the 95% CI (dot line). r is the Pearson's correlation coefficient.

Figure 4 :

 4 Figure 4: Relationships between ρm and EVM Young's moduli and Poisson's ratio, and the corresponding linear regression model (solid line) with the 95% CI (dot line). r is the Pearson's correlation coefficient.

  (as an example) and C m 11 was established by varying the values of C m 11 starting from C m0 11 while fixing C m0 33 , C m0 13 , C m0 44 , C m0 66 (see Figure A.1). The range of variations were C m0 11 ± 10 GPa, C m0 33 ± 12 GPa, C m0 13 ± 6 GPa, C m0 44 ± 4 GPa and C m0 66 ± 3 GPa (5 evenly distributed points in the interval for each coefficient), respectively. Then, the C EXP 11 was projected to C m 11 given the established relationship (Figure A.1). It costed about 30 minutes (5 × 6 = 30 forward computations) to calculate EVM stiffness constants of a specimen in the calibration approach.

  3. The Nelder-Mead Simplex Method implemented in Matlab Optimization Toolbox Release 2017b (MathWorks, Natick, MA), which does not require derivative information, was used to solve the minimization problem. The iterative solver stopped when, for two consecutive iterations, both the difference of the objective function values (eq. 3), and the difference of the norms of C m ij , were smaller than 0.001. For most of the specimens, the optimization stopped after about 130 iterations which costed about 130 minutes per specimen. The optimization procedure was initialized at the values (C m0 ) of C m0 11 = 24.4 GPa, C m0 33 = 33.5 GPa, C m0 13 = 14.7 GPa, C m0 44 = 6.9

  .8. Data analysis Linear least squares regressions were conducted to analyze the relationships between elastic properties, impedance and DMB. Pearson's correlation coefficient are reported. Normality of the distributions of the C m ii and C Z ii was verified using the ShapiroWilk test. Paired t-test was used to test if C m ii and C Z ii were

Table 1 :

 1 The average, SD and range of the experimental data of the stiffness constants (C EXP

		C EXP 11	(GPa)	C EXP 33	(GPa)	C EXP 13	(GPa)	C EXP 44	(GPa)	C EXP 66	(GPa)
	Mean±SD	20.1 ± 1.5	29.5 ± 1.3	12.0 ± 0.9	6.0 ± 0.4	4.5 ± 0.5
	Range	16.9 -23.8	25.7 -32.1	9.3 -14.1	4.7 -6.9	3.5 -5.9
		φ (%)	ρm (g/cm 3 )	Z 1 (Mrayl)	Z 3 (Mrayl)
	Mean±SD	6.5 ± 2.6	1.02 ± 0.02	6.8 ± 0.3	7.4 ± 0.3
	Range	1.8 -12.1	0.95 -1.06	6.2 -7.5	6.9 -8.1

ij

), porosity (φ), DMB (ρm) and matrix acoustic impedance (Z i ). Note that Z i was measured on 23 specimens from the lateral quadrant.

Table 2 :

 2 The mean, SD and range of the EVM stiffness constants C m ij and engineering moduli.

		C m 11 (GPa)	C m 33 (GPa)	C m 13 (GPa)	C m 44 (GPa)	C m 66 (GPa)
	Mean±SD	24.5 ± 1.2	33.6 ± 1.1	14.7 ± 1.0	7.0 ± 0.3	5.4 ± 0.4
	Range	21.7 -27.5 31.4 -36.2 12.1 -16.7	6.1 -7.5	4.5 -6.3
		E m 1 (GPa)	E m 3 (GPa)	ν m 23	ν m 31	ν m 12
	Mean±SD	15.1 ± 0.8	22.4 ± 1.0	0.26 ± 0.02 0.38 ± 0.02 0.41 ± 0.03
	Range	13.2 -17.0 20.3 -24.9 0.19 -0.31 0.34 -0.41 0.32 -0.48

Table 3 :

 3 The errors (Mean±SD in %) on EVM stiffness constants affected by the experimental errors on the bulk stiffness constants. Calculation were performed for 3 specimens of 5.0%, 9.9% and 15.4% porosity.

		φ = 5.0%	φ = 9.9%	φ = 15.4%
	δC m 11	0.21 ± 3.49	-0.01 ± 3.96 -0.76 ± 4.66
	δC m 33	0.02 ± 3.05	-0.09 ± 3.06 -0.17 ± 2.99
	δC m 13	0.17 ± 3.12	0.03 ± 3.55	-0.35 ± 3.90
	δC m 44	-0.12 ± 1.49 -0.07 ± 1.58	0.08 ± 1.59
	δC m 66	-0.16 ± 1.62	0.07 ± 1.70	0.69 ± 1.87
	C m ii and C Z ii were significantly different (p < 0.002). The mean value of C m 11 was about 5.1% smaller than
	that of C Z 11 , while the mean value of C m 33 was about 10.2% greater than that of C Z 33 . C Z ii exhibited a greater
	variability (CV was about 8%) compared to C m ii (CV between 3 -5%). No significant correlation was found
	between C m 11 and C Z 11 , nor between C m 33 and C Z 33 .

3.3. Comparison between EVM stiffness derived by RUS and SAM

SAM provides a direct assessment of acoustic impedance Z which is a good proxy for stiffness. Impedance can be converted to stiffness C Z ii according to eq. 2 which can be directly compared to C m ii . EVM longitudinal stiffness constants evaluated by inverse homogenization and SAM are compared in Table

4

. The values of

Table 4 :

 4 Comparison between EVM stiffness measured by RUS and SAM.

		Mean±SD	Range
	C m 11 (GPa)	24.0 ± 1.2	21.7 -26.7
	C Z 11 (GPa)	25.4 ± 2.0	21.5 -30.6
	C m 33 (GPa)	33.5 ± 1.1	31.4 -36.2
	C Z 33 (GPa)	30.4 ± 2.4	25.8 -35.3

Table 5 :

 5 Empirical laws between ρm and C m ij .

		Laws
	C m 11	14.96 × ρm + 9.28
	C m 33	13.99 × ρm + 19.39
	C m 13	11.79 × ρm + 2.69
	C m 44	8.82 × ρm -2.02
	C m 66	11.13 × ρm -5.97

Table 6 :

 6 Comparison between the bone EVM elastic properties found in this work and in previous studies. The values are summarized as mean ± SD except those converted by the mean values of the coefficients (elastic tensor to Engineering moduli or conversely). Franzoso: Franzoso and Zysset[START_REF] Franzoso | Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation[END_REF]; Lakshmanan: Lakshmanan et al.[START_REF] Lakshmanan | Assessment of anisotropic tissue elasticity of cortical bone from high-resolution, angular acoustic measurements[END_REF]; Turner:[START_REF] Turner | The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques[END_REF].SAM provided impedance data and combined with a model to derive mass density from DMB, provided an independent measure of EVM stiffness. SAM is seemingly the most appropriate technique to perform a comparison with our EVM stiffness values because it scans entire surfaces of the cuboid specimens; SAM data can then be averages, yielding values directly comparable with C m ij derived from inverse homogenization.

			Acoustic		Nano-indentation
		This study	Lakshmanan	Turner		Franzoso
	C m 11 (GPa)	24.5 ± 1.2	21.9 ± 2.1	22.4 ± 0.8	-	14.3
	C m 33 (GPa)	33.6 ± 1.1	29.9 ± 5.0	28.3 ± 0.3	-	38.6
	C m 13 (GPa)	14.7 ± 1.0	9.7 ± 1.6	-	-	12.3
	C m 12 (GPa)	13.8 ± 0.9	9.2 ± 1.5	-	-	10.3
	C m 44 (GPa)	7.0 ± 0.3	6.7 ± 1.2	-	-	7.7 ± 0.5
	C m 66 (GPa)	5.4 ± 0.4	6.4	-	-	4.7 ± 0.4
	E m 1 (GPa)	15.1 ± 0.8	16.8	14.9 ± 0.5 16.6 ± 0.3	9.2 ± 0.6
	E m 3 (GPa)	22.4 ± 1.0	23.8	20.6 ± 0.2 23.5 ± 0.2	24.7 ± 2.7
	ν m 23	0.26 ± 0.02	0.22	-	-	0.29 ± 0.02
	ν m 31	0.38 ± 0.02	0.31	-	-	0.56 ± 0.02
	ν m 12	0.41 ± 0.03	0.32	-	-	0.25 ± 0.01

Acknowledgment

This work has received financial support from the Agency National Research under the ANR-13-BS09-0006 MULTIPS project. The authors would like to thank ESRF for the access of beamline at ID 19 and 17 and the help from Cécile Olivier and Françoise Peyrin for performing SR-µCT experiments.

Appendix A. Calibration approach

From several preliminary tests, we found that the coupling between C F F T ij and its counterpart C m ij (e.g., between C F F T 11 and C m 11 ) is much stronger than the other EVM stiffness constants which means that specimen-specific relationships can be built between them. Specifically, the relationship between C F F T Bland-Altman's plots between the C m ij from the two approaches showed that only a small bias existed for C m 11 (0.15 GPa), but not for the other coefficients (< 0.07 GPa) and the relative differences are quite small (CV was between 0.35 -0.96%) compared to experimental errors. Therefore, one can conclude that there is no significant difference between the two inverse homogenization approaches.

Appendix B. Convergence study

The convergence study was carried out on the three specimens mentioned in Section 2.7 whose original pixel size (ps) of the 3D images was 6.5 µm.

Appendix B.1. Forward problem

The convergence study for the forward problem was mainly used to determine the pixel size for the VOIs.

Details can be found in Cai et al. [START_REF] Cai | Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity[END_REF] (Appendix B.2.).

Appendix B.2. Inverse problem

The ps of the images of each specimen was increased to 10, 15, 20, 25, 30 and 35 µm, respectively, in order to study how the image resolution affects the result of the inverse homogenization. The bulk stiffness constants of each specimen, calculated by the FFT homogenization when ps = 10 µm and using C m0 ij as the EVM stiffness constants, were used as the input. Applying the inverse homogenization on the images at the other ps (15 -35 µm) for each specimen, the bias between EVM stiffness constants C m,ps ij evaluated at different ps and C m0 ij was quantified by the RMSE denoted below:

Results showed that when ps 35 µm, all the RMSEs were less than 0.5%. Compared with the experimental errors in RUS, these values are smaller. To this end, ps = 35 µm was considered as an appropriate value for the 3D images of the specimens and was chosen in this work.

Appendix C. Validation of the inversion

The effective stiffness constants C F F T 0 ij of all the 55 specimens were calculated by the FFT homogenization with the values C m0 ij for bone EVM and the pixel size of the images was 35 µm. Using C F F T 0 ij as the input, both the calibration and optimization approaches were implemented to calculate the EVM stiffness constants C m ij and the RMSE of each specimen was evaluated as:

The results are depicted in Figure C.2 in which we observed that all the RMSE are less than 0.15% and 0.018% for the calibration and optimization approaches, respectively, and most are less than 0.05% and 0.005%, which means that the inverse methods themselves are reliable.


Tulika Bose
email: tulika.bose@loria.fr

Irina Illina
email: illina@loria.fr

Dominique Fohr
email: dominique.fohr@loria.fr

Generalisability of Topic Models in Cross-corpora Abusive Language Detection

Rapidly changing social media content calls for robust and generalisable abuse detection models. However, the state-of-the-art supervised models display degraded performance when they are evaluated on abusive comments that differ from the training corpus. We investigate if the performance of supervised models for cross-corpora abuse detection can be improved by incorporating additional information from topic models, as the latter can infer the latent topic mixtures from unseen samples. In particular, we combine topical information with representations from a model tuned for classifying abusive comments. Our performance analysis reveals that topic models are able to capture abuse-related topics that can transfer across corpora, and result in improved generalisability.

Introduction

With the exponentially increased use of social networking platforms, concerns on abusive language has increased at an alarming rate. Such language is described as hurtful, toxic, or obscene, and targets individuals or a larger group based on common societal characteristics such as race, religion, ethnicity, gender, etc. The increased spread of such content hampers free speech as it can potentially discourage users from expressing themselves without fear, and intimidate them into leaving the conversation. Considering variations of online abuse, toxicity, hate speech, and offensive language as abusive language, this work addresses the detection of abusive versus non-abusive comments.

Automatic detection of abuse is challenging as there are problems of changing linguistic traits, subtle forms of abuse, amongst others [START_REF] Vidgen | Challenges and frontiers in abusive content detection[END_REF]. Moreover, the performance of models trained for abuse detection are found to degrade considerably, when they encounter abusive comments that differ from the training corpus [START_REF] Wiegand | Detection of Abusive Language: the Problem of Biased Datasets[END_REF][START_REF] Arango | Hate speech detection is not as easy as you may think: A closer look at model validation[END_REF][START_REF] Durairaj Swamy | Studying generalisability across abusive language detection datasets[END_REF][START_REF] Karan | Cross-domain detection of abusive language online[END_REF]. This is due to the varied sampling strategies used to build training corpus, topical and temporal shifts [START_REF] Florio | Time of your hate: The challenge of time in hate speech detection on social media[END_REF], and varied targets of abuse across corpora. Since social media content changes rapidly, abusive language detection models with better generalisation can be more effective [START_REF] Yin | Towards generalisable hate speech detection: a review on obstacles and solutions[END_REF]. To this end, a cross-corpora analysis and evaluation is important.

Topic models have been explored for generic cross-domain text classification [START_REF] Jing | Cross-domain labeled LDA for cross-domain text classification[END_REF][START_REF] Zhuang | Concept learning for crossdomain text classification: A general probabilistic framework[END_REF][START_REF] Li | Topic correlation analysis for cross-domain text classification[END_REF], demonstrating better generalisability. Moreover, they can be learnt in an unsupervised manner and can infer topic mixtures from unseen samples. This inspires us to exploit topic model representations for crosscorpora abuse detection.

Recently, [START_REF] Caselli | Hatebert: Retraining bert for abusive language detection in english[END_REF] have "retrained" BERT [START_REF] Devlin | BERT: Pre-training of deep bidirectional transformers for language understanding[END_REF] over large-scale abusive Reddit comments to provide the HateBERT model which has displayed better generalisability in cross-corpora experiments. Furthermore, [START_REF] Peinelt | tBERT: Topic models and BERT joining forces for semantic similarity detection[END_REF] show that combination of topic model and BERT representations leads to better performance at semantic similarity task. Taking these studies into account, we investigate if combining topic representation with contextualised HateBERT representations can result in better generalisability in cross-corpora abuse detection. Cross corpora evaluation on three common abusive language corpora supports and demonstrates the effectiveness of this approach. Besides, we bring some insights into how the association of unseen comments to abusive topics obtained from original training data can help in cross-corpora abusive language detection.

The rest of the paper is organised as follows: Section 2 describes the architecture of the combination of topic model and HateBERT. Section 3 presents our experimental settings. An analysis of the results obtained is present in Section 4, and Section 5 concludes the paper.

In this work, we leverage the Topically Driven Neural Language Model (TDLM) [START_REF] Jey | Topically driven neural language model[END_REF] to obtain topic representations, as it can employ pre-trained embeddings which are found to be more suitable for short Twitter comments [START_REF] Yi | Topic modeling for short texts via word embedding and document correlation[END_REF]. The original model of TDLM applies a Convolutional Neural Network (CNN) over wordembeddings to generate a comment embedding. This comment embedding is used to learn and extract topic distributions. [START_REF] Cer | Universal sentence encoder[END_REF] show that transfer learning via sentence embeddings performs better than word-embeddings on a variety of tasks. Hence, we modify TDLM to accept the transformer based Universal Sentence Encoder (USE) [START_REF] Cer | Universal sentence encoder[END_REF] embeddings extracted from input comments, instead of the comment embeddings from CNN. The modified model is denoted as U-TDLM hereon. Refer to Appendix A.1 for the architecture of U-TDLM and also to [START_REF] Jey | Topically driven neural language model[END_REF].

U-TDLM is trained on the train set from the source corpus and is used to infer on the test set from a different target corpus. The topic distribution per comment c is given by

T c = [p(t i |c)] i=1:k ,
where k is the number of topics. T c is passed through a Fully Connected (FC) layer to obtain transformed representation T c . Besides, we first perform supervised fine-tuning of HateBERT 1 on the train set of the source corpus. The vector corresponding to the [CLS] token in the final layer of this fine-tuned HateBERT model is chosen as the Hate-BERT representation for a comment. It is transformed through an FC layer to obtain the C vector. Finally, in the combined model (HateBERT+U-TDLM), the concatenated vector [T c ; C] is passed through a final FC and a softmax classification layer. The readers are referred to Appendix A.2 for the architecture of the individual, and the combined models.

3 Evaluation Set-up

Experimental Settings

We perform experiments on three different publicly available abusive tweet corpora, namely, HatEval (Basile et al., 2019), Waseem [START_REF] Waseem | Hateful symbols or hateful people? predictive features for hate speech detection on Twitter[END_REF], and Davidson [START_REF] Davidson | Automated hate speech detection and the problem of offensive language[END_REF]. We target a binary classification task with classes: abusive and non abusive, following the precedent of 1 Pre-trained model from https://osf.io/tbd58/ previous work on cross corpora analysis [START_REF] Wiegand | Detection of Abusive Language: the Problem of Biased Datasets[END_REF][START_REF] Durairaj Swamy | Studying generalisability across abusive language detection datasets[END_REF][START_REF] Karan | Cross-domain detection of abusive language online[END_REF]. For HatEval, we use the standard partition of the shared task, whereas the other two datasets are randomly split into train (80%),development (10%), and test (10%). The statistics of the traintest splits of these datasets are listed in Table 1. We choose a topic number of 15 for our experiments based on the results for in-corpus performance and to maintain a fair comparison. Besides, the best model checkpoints are selected by performing early-stopping of the training using the respective development sets. The FC layers are followed by Rectified Linear Units (ReLU) in the individual as well as the combined models. In the individual models, the FC layers for transforming T c and the HateBERT representation have 10 and 600 hidden units, respectively. The final FC layer in the combined model has 400 hidden units. Classification performance is reported in terms of mean F1 score and standard deviation over five runs, with random initialisations.

Datasets

Data Pre-processing

We remove the URLs from the Twitter comments, but retain Twitter handles as they can contribute to topic representations. 2 Hashtags are split into constituent words using the tool CrazyTokenizer3 , and words are converted into lower-case. U-TDLM involves prediction of words from the comments based on topic representations. In this part, our implementation uses stemmed words and skips stopwords. All models are trained on the train set of the source corpus. The in-corpus performance of the models is obtained on the source corpora test sets, while the cross-corpora performance is obtained on target corpora test sets. It is shown in Table 2 that the cross-corpora performance degrades substantially as compared to the in-corpus performance, except for HatEval which indeed has a low in-corpus performance. HatEval test set is part of a shared task, and similar in-corpus performance have been reported in prior work [START_REF] Caselli | Hatebert: Retraining bert for abusive language detection in english[END_REF]. Overall, comparing the cross-corpora performances of all models, we can observe that the combined model (HateBERT + U-TDLM) either outperforms Hate-BERT or retains its performance. This hints that incorporating topic representations can be useful in cross-corpora abusive language detection. As an ablation study, we replaced U-TDLM features with random vectors to evaluate the combined model. Such a concatenation decreased the performance in the cross-corpora setting, yielding an average macro-F1 score of 59.4. This indicates that the topic representations improve generalisation along with HateBERT.

Results and Analysis

Case-studies to Analyse Improvements from U-TDLM

We investigate the cases in Train on Waseem →Test on Davidson: In this case, U-TDLM shows poor performance due to the large number of False Negatives (#FN for U-TDLM: 1824), and less True Positives (#TP for U-TDLM: 266). The combined model, on the other hand, has higher True Positives compared to those obtained from HateBERT (#TP for HateBERT+U-TDLM: 1556, #TP for HateBERT: 1267). The count of True Negatives with the combined model remains similar to that in HateBERT (#TN for Hate-BERT + U-TDLM: 314, #TN for HateBERT: 340). This indicates that U-TDLM introduces some complementary information in the combined model.

We analyse a few abusive comments in the test set of Davidson (target) weights to the relevant abuse-related topics from Waseem. As indicated in the first example, an abusive comment against women that discusses cooking, in Davidson, is mapped to the topics 4 (sexism) and 12 (cooking show) from Waseem. Similarly, the second comment gets high weight in the three topics 4, 9 and 12 due to its sexist content and use of a profane word. Other pairs of corpora that yield improved performance with the combined model also follow similar trends as above.

Train on HatEval →Test on Davidson: In this case, while U-TDLM performs considerably well, the combined model only provides a slight improvement over HateBERT, as per Table 2. U-TDLM has a higher TP when compared to both HateBERT and the combined model (#TP for U-TDLM: 1924, #TP for HateBERT+U-TDLM: 1106, #TP for Hate-BERT: 1076), with lower TN (#TN for U-TDLM: 130, #TN for HateBERT+U-TDLM: 373, #TN for HateBERT: 374).

Few abusive comments from Davidson that are correctly classified by U-TDLM alone are presented in Table 4. The first comment for this case have high weights for the abuse-related topics 3 and 7 from HatEval due to the presence of the profane word "f*ggot". The second comment only gets a high weight for topic 10, which deals with politics. This is due to the word "leftist", which is associated with a political ideology. As per our analysis, we found that all of these source topics are highly correlated with the abusive labels in the source corpus of HatEval. As such, these comments from the target corpus of Davidson are correctly classified as abusive by U-TDLM.

Discussion and Conclusion

An in-corpus and cross-corpora evaluation of Hate-BERT and U-TDLM has helped us confirm our perspective on generalisation in the abusive language detection task. A contextualised representation model like HateBERT can achieve great levels of performance on the abusive language detection task, only when the evaluation dataset does not differ from the training set. The performance of this model degrades drastically on abusive language comments from unseen contexts. Topic models like U-TDLM, which express comments as a mixture of topics learnt from a corpus, allow unseen comments to trigger abusive language topics. While topic space representations tend to lose the exact context of a comment, combining them with Hate-BERT representations can give modest improvements over HateBERT or at the least, retain the performance of HateBERT. These results should fuel interest and motivate further developments in the generalisation of abusive language detection models.

Table 1 :

 1 Statistics of the datasets used (average comment length is calculated in terms of word numbers).

	Number of	Average	Abuse
	comments	comment	%
		length
	Train Test
	HatEval 9000 3000	21.3	42.1
	Waseem 8720 1090	14.7	26.8
	Davidson 19817 2477	14.1	83.2
	.

Table 2 :

 2 Table2presents the in-corpus and cross-corpora evaluation of the HateBERT and U-TDLM models. Macro average F1 scores (mean±std-dev) for in-corpus and cross-corpora abuse detection. The best in each row for the cross-corpora performance is marked in bold.

	Train set	In-corpus performance HateBERT U-TDLM	Cross-corpus test set	Cross-corpora performance HateBERT U-TDLM HateBERT + U-TDLM
	HatEval	53.9±1.7	41.5±0.6	Waseem Davidson	66.5±2.2 59.2±2.5	55.5±2.6 64.4±2.3	67.8±2.4 60.4±1.4
	Waseem 86.1±0.4	73.7±1.4	HatEval Davidson	55.8±1.4 59.8±3.6	36.7±0.0 28.2±2.4	55.4±0.7 64.8±1.8
	Davidson 93.7±0.2	75.6±0.8	HatEval Waseem	51.8±0.2 66.6±3.0	50.5±1.3 48.7±3.3	51.8±0.3 68.5±2.1
	Average 77.9	63.6		60.0	47.3	61.5

Table 2

 2 HateEval, test on Davidson). Some of the prominent topics from Waseem and HateEval associated with abuse, and the top words corresponding to these topics are provided in Table3 and Table 5, respectively. For better interpretation, topic names are manually assigned based on the top words and the knowledge of the individual corpora. We consider the abusive class as positive, and the non-abusive class as negative in the subsequent discussion.

	which report rel-
	atively large improvements, as compared to Hate-
	BERT, either with HateBERT+U-TDLM (train on
	Waseem, test on Davidson) or only with U-TDLM
	(train on

Table 3 :

 3 U-TDLM trained on Waseem's train set (topic names are assigned manually for interpretation).

Table 4 :

 4 in Table 4, which are wrongly classified by HateBERT, but correctly detected as abusive by the combined model. The topical membership of these abusive comments from Davidson indicates that U-TDLM associates high Abusive comments in the target corpus, correctly classified by HateBERT+U-TDLM (Waseem →Davidson) and U-TDLM (HatEval →Davidson). "Source topics" : topics that are assigned high weights by U-TDLM trained on Source.

	Source			Abusive Comments in Target	Source
	→Target			topics
			When women are so proud that they don't like to cook; clean b*tch stop	4, 12
	Waseem		being lazy..It's not cute.
	→Davidson	ya girl is a slimy ass h*e. get her under control and tell her to stop spraying	4, 9, 12
			bullshit out her mouth all day.
			No. Its wrong to try to change f*ggots; There is no "therapy"....sympathize	3, 7
	HatEval		like they are retards.
	→Davidson	Naturally, when a shitty leftist rag talks trash about another shitty leftist	10
			rag, you better fall in line...
	Topic	Names	Top words
	id
	3	Explicit	men, c*ck, d*ck, woman,
		abuse 1	picture, sl*t, s*ck, guy
	7	Explicit	b*tch, ho*, n*gger, girl-
		abuse 2	friend, f*ck, shit, s*ck,
				dumb
	10	Politics	therickwilson, anncoulter,
		related	c*nt, commies, tr*nny,
				judgejeanine, keitholber-
				mann, donaldjtrumpjr

Table 5 :

 5 U-TDLM trained on HatEval's train set (topic names are assigned manually for interpretation).

Eg., the topic associated with @realDonaldTrump.

https://redditscore.readthedocs.io/ en/master/tokenizing.html

Acknowledgements

This work was supported partly by the french PIA project "Lorraine Université d'Excellence", reference ANR-15-IDEX-04-LUE.

A Appendices

A.1 Topic Model U-TDLM


André É Leroy

Mona Abdi

GRAPHS OF COMMUTATIVELY CLOSED SETS

Keywords: Matrix, Commutatively closed, Closure

Introduction

A subset S of a ring R is commutatively closed if for any a, b ∈ R such that ab ∈ S, we also have ba ∈ S. This notion was introduced and studied in [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF]. The aim of this paper is to investigate further this property and, in particular, to involve graph theory in the subject. Let us first mention some easy examples. Any subset of a commutative ring is commutatively closed. The set {1} is commutatively closed if and only if the ring R is Dedekind finite. Similarly, the set {0} is commutatively closed if and only if the ring R is reversible. The intersection of two commutatively closed subsets is easily seen to be commutatively closed and hence every subset of R is contained in a minimal commutatively closed subset: its commutative closure. The closure of a subset S ⊂ R is denoted by S. In [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF] different characterizations of S were given. It was shown that this notion is related to various kind of subsets such as idempotents elements, nilpotent elements, invertible elements, and various kinds of regular elements. It was also used to characterize, semicommutative rings, 2-primal rings, and clean rings . We will briefly recall some of the notions used in [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF] at the end of this introduction.

In section 1, we describe a graph structure on the sets {a} and define the diameter. Our aim is to compute the diameter of some classical rings, in particular, the ring of matrices over a division ring. In section 2, we give some ways of constructing {a} and study some particular rings such as the free algebras. In the last section, we investigate some properties of the commutatively closed graph of matrix rings and semisimple algebras.

For two elements a, b ∈ R we write a ∼ 1 b if there exists c, d ∈ R such that a = cd and b = dc. We then define by induction a ∼ n+1 b if and only if there exists an element c ∈ R such that a ∼ 1 c and c ∼ n b. For two elements a, b ∈ R, we also define a ∼ b if there exists n ∈ N such that a ∼ n b. For a subset S ⊆ R we denote S i the set {x ∈ R | x ∼ i s, for some s ∈ S}. Since our ring R is unital, the chain S i is ascending and we have S = ∪ i≥0 S i (for details about these constructions we refer the reader to [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF]).

We recall that an element a ∈ R is called von Neumann regular if there is x ∈ R such that a = axa. Similarly, we define a ∈ R to be a π-regular element of R if a n xa n = a n for some x ∈ R and n ≥ 1. An element a ∈ R is called right (left) π-regular, if a n+1 x = a n (xa n+1 = a n ) for some x ∈ R and n ≥ 1. We call a ∈ R strongly π-regular if it is both left and right π-regular.

Throughout this paper R will be a unital ring, U (R) and N (R) will stand for the set of invertible and nilpotent elements of R, respectively. For an element a of a ring R we denote l(a) (resp. r(a)) its left (resp. right) annihilator.

Let us mention some results from [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF]. They give motivation for the subject and basic information about the results.

Theorem 0.1. [1, Theorem 2.7] Let ϕ : R -→ S be a ring homomorphism, then

(1) For any X ⊆ R, ϕ(X) ⊆ ϕ(X).

(2) If ϕ is a ring isomorphism, then for any X ⊆ R, ϕ(X) = ϕ(X).

(

) If T ⊆ S is commutatively closed in S, then ϕ -1 (T ) is commutatively closed in R. (4) If S is reversible, Ker(ϕ) is commutatively closed. (5) If S is Dedekind-finite, then ϕ -1 ({1}) is commutatively closed. Proposition 0.2. [1, Proposition 3.6] (1) If R is Dedekind-finite and a ∈ U (R), then {a} = {uau -1 | u ∈ U (R)}. (2) The set of unit U (R) of a ring R is commutatively closed if and only if R is Dedekind-finite. Proposition 0.3. [1, Proposition 4.7] Let k be a commutative field and n ∈ N, the class of {0} in M n (k) is the set of nilpotent matrices. 3
For notions related to graph theory, we refer the reader to [START_REF] West | Introduction to Graph Theory[END_REF].

Commutatively closed graphs and their diameters

First, we start this section with some examples of commutatively closed sets.

Examples 1.1.

(1) A ring R is reversible if and only {0} is commutatively closed;

(2) A ring R is Dedekind finite if and only if {1} is commutatively closed;

(3) For a right R-module M R the set E = {u ∈ R | ann M (u -1) = 0} is commutatively closed. Indeed, if u = ab ∈ E and 0 = m ∈ M is
such that m(ab -1) = 0, then ma = 0 and ma(ba -1) = m(ab -1)a = 0. This gives that ba ∈ E. (4) The ring R is symmetric if and only if for every a, b, c ∈ R, abc = 0 implies that acb = 0. This can be translated into asking that for any a ∈ R, r(a) := {b ∈ R | ab = 0} is commutatively closed. We refer the reader to [START_REF] Gürgün | On Cline's formula for some certain elements in a ring[END_REF] for more information on this kind of ring. ( 5) U (R) -1 is always commutatively closed. More generally, denoting the set of regular elements by Reg(R) = {r ∈ R | ∃x ∈ R such that r = rxr} we have that Reg(R) -1 is always commutatively closed. A similar result is true for the set of unit regular elements and also for the set of strongly π-regular elements. For proofs of these facts we refer the reader to [START_REF] Lam | Jacobson's lemma for Drazin inverses[END_REF]. [START_REF] Lam | Lectures on modules and rings[END_REF] We consider Z l (R) = {a ∈ R | r(a) = 0} the set of left zero divisors. We define similarly Z r (R) the set of right zero divisors. Similarly as in the item above, Z r (R) + 1 (resp. Z l (R) + 1) is commutatively closed ( [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF]). [START_REF] West | Introduction to Graph Theory[END_REF] The set N (R) of nilpotent elements of a ring R is easily seen to be commutatively closed. (8) Recall that an element r ∈ R is strongly clean if there exist an invertible element u ∈ U (R) and an idempotent e 2 = e ∈ R such that eu = ue and r = e + u. It is proved in Theorem 2.5 of [START_REF] Gürgün | On Cline's formula for some certain elements in a ring[END_REF] that the set of strongly clean elements is commutatively closed. In the same paper the author also shows that the set of Drazin (resp. almost, pseudo) invertible elements is also commutatively closed.

The three first statements of the following results were obtained in [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF]. So we will only prove the last one.

Proposition 1.2.

(1) For any n ≥ 1 and a, b ∈ R, we have a ∼ n b if and only if there exist two sequences of elements in R x 1 , x 2 , . . . , x n and y 1 , y 2 , . . . , y n such that a = x 1 y 1 , y 1 x 1 = x 2 y 2 , y 2 x 2 = x 3 y 3 , . . . , y n x n = b.

(2) If a ∼ n b, then a -b is a sum of n additive commutators.

(3) If a ∼ n b, then there exist x, y ∈ R such that ax = xb and ya = by.

Moreover, for l ∈ N, we also have b n+l = ya l x and a n+l = xb l y. In particular, b n = yx and a n = xy. (4) If elements a, b in a ring R are such that a ∼ m b, then for any 0 < l < m, we have a l ∼ q b l , where q is such that q = m l Proof. (4) We can write m = lq -r for some 0 ≤ r < l and, since 1 R ∈ R, a ∼ m b implies that we also have a ∼ m+r=lq b. We prove that a l ∼ q b l by induction on q. If q = 1, the statement (3) above shows that a m = xy and b m = yx for some x, y ∈ R and hence a m ∼ 1 b m . So let us now assume that q > 1 and put n := m + r = lq, so that we have a ∼ n b. We use the same notations as in statement (1) and assume that there exist two sequences of elements in R x 1 , x 2 , . . . , x n and y 1 , y 2 , . . . , y n such that a = x 1 y 1 , y 1 x 1 = x 2 y 2 , y 2 x 2 = x 3 y 3 , . . . , y n x n = b. We than have that a = x 1 y 1 ∼ l y l x l ∼ l(q-1) b. The case q = 1 and the induction hypothesis lead to a l ∼ 1 (y l x l ) l ∼ q-1 b l . This gives the conclusion.

Remarks 1.3.

(1) With the same notations as in the above proposition 1.2, if a ∼ n b and x, y are such that ax = xb and by = ya, then left multiplication by the elements y and x give rise to maps in Hom R (R/aR, R/bR) and Hom R (R/bR, R/aR). We then have the compositions

L x • L y = L a n and L y • L x = L b n . In particular, if a ∼ b then Hom R (R/aR, R/bR) = {0}.
(2) In section 3, we will study the ring of matrices over a division ring D.

It is interesting to observe that if A, B ∈ M n (D) are such that A ∼ B, then the central spectrum Sp(A) and Sp(B) are equal. Indeed, there exists

m ∈ N such that A ∼ m B. Proposition 1.2 (3) shows that there exist matrices X, Y ∈ M n (D) such that AX = XB and Y A = BY . So, if Av = λv for some 0 = v ∈ D n and λ ∈ k (k is the center of D), then BY v = Y Av = Y λv = λY v. Noting that λ m v = A m v = XY v, we conclude that Y v = 0 and hence λ is indeed a central eigenvalue of B.
Let us remark that in the case D is commutative, the equality between the spectrums is a consequence of the fact that the characteristic polynomials of A and B are the same (cf. [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF]).

In the following, we define the commutatively closed graph and show that the graph C(a) is always connected for every a ∈ R. Also, we analyze some other of his properties.

Definitions 1.4.

(1) Let a be an element in a ring R and let C(a) denote its commutative closure as defined in the introduction. We define a graph structure with the elements of C(a) = {a} as vertices and two distinct vertices x and y of C(a) are said to be adjacent if and only if y ∈ {x} 1 . The commutatively closed graph of a ring R is the union of all the graphs C(a), for a ∈ R. It will be denoted by C(R).

(2) Let a be an element in R. In the class {a} of a, we define a distance as follows: For two elements x, y ∈ {a}, we put d(x, y)

= min{n ∈ N | y ∼ n x}.
One can easily check that d is indeed a distance defined on {a}.

(3) Let R be a ring and a ∈ R, the diameter of a graph C(a) is defined as follows:

diam C(a) = sup d(x, y) | x, y ∈ {a} . Also, we define the diameter of a set S ⊆ R of a ring S as follows: Proof. We leave the easy proof to the reader. Proposition 1.6. Let a be an element in a ring R. If n ∈ N is the smallest integer such that a = {a} n , then n ≤ diam R C(a) ≤ 2n.

diam(S) = sup diam C(a) | a ∈ S .
Proof. We know that the distance between a and every other element of

{a} n is at most n. Now, if b, c ∈ C(a) then d(b, c) ≤ d(b, a) + d(a, c) ≤ 2n. So that diam R C(a) ≤ 2n. The fact that n is minimal such that a = {a} n implies that n ≤ diam R C(a) .
Lemma 1.7. Let R and S be two rings, and

(a, b) ∈ R × S. Then {(a, b)} = {a} × {b}. Proof. Let (c, d) ∈ {(a, b)}. Thus (c, d) ∼ n (a, b), for some n ≥ 0. So there are (c 1 , d 1 ), . . . , (c n , d n ) ∈ R × S such that (c, d) ∼ 1 (c 1 , d 1 ) ∼ 1 • • • ∼ 1 (c n , d n ) = (a, b). One can easily see that c ∼ n a and d ∼ n b. Hence (c, d) ∈ {a} × {b}. Now, let (c, d) ∈ {a} × {b}. Since c ∈ {a} and d ∈ {b}, thus there are m, n ∈ N such that c ∼ m a and d ∼ n b. So we have c ∼ 1 c 1 ∼ 1 • • • ∼ 1 c m = a and d ∼ 1 d 1 ∼ 1 • • • ∼ 1 d n = b, for some c 1 , . . . , c m ∈ R and d 1 , . . . , d n ∈ S. We may assume that m < n. Thus we have (c, d) ∼ 1 (c 1 , d 1 ) ∼ 1 • • • ∼ 1 (c m , d m ) = (a, d m ) ∼ 1 (a, d m+1 ) ∼ 1 • • • ∼ 1 (a, d n ) = (a, b). Hence (c, d) ∼ n (
(S) = d 2 . Then diam(R × S) = max{d 1 , d 2 }. Proof. Assume that (a, b) ∈ R × S. For every (a 1 , b 1 ), (a 2 , b 2 ) ∈ {(a, b)}, we have a 1 , a 2 ∈ {a} and b 1 , b 2 ∈ {b}, by Lemma 1.7. Since diam(R) = d 1 and diam(S) = d 2 , then d(a 1 , a 2 ) ≤ d 1 and d(b 1 , b 2 ) ≤ d 2 . Thus there exist t ≤ d 1 and s ≤ d 2 such that a 1 ∼ t a 2 and b 1 ∼ s b 2 . Then we have a 1 ∼ 1 c 1 ∼ 1 • • • ∼ 1 c t = a 2 and b 1 ∼ 1 v 1 ∼ 1 • • • ∼ 1 v s = b 2 , for some c 1 , . . . , c t ∈ R and v 1 , . . . , v s ∈ S. Let t < s. We have (a 1 , b 1 ) ∼ 1 (c 1 , v 1 ) ∼ 1 • • • ∼ 1 (c t , v t ) = (a 2 , v t ) ∼ 1 (a 2 , v t+1 ) ∼ 1 • • • ∼ 1 (a 2 , v s ) = (a 2 , b 2 ). Hence d (a 1 , b 1 ), (a 2 , b 2 ) ≤ max{d 1 , d 2 }. Therefore diam(R × S) = max{d 1 , d 2 }.
1.9. In the above proposition, we showed that if R and S are two rings with finite diameter, then the diameter of R × S is also finite.

Let us remark that is easy to construct elements a, b, c, d in a ring such that a ∼ 1 b and c ∼ 1 d, but there is no path between ac and bd. For instance, consider the free algebra K < X, Y > where K is a field. Let a = XY, b = Y X, c = d = X 2 Y . We leave to the reader to check that there is no path from ac = XY X 2 Y to bd = Y X 3 Y . Proof. 1) The first statement is a direct consequence of the definition.

2) The second statement is easily obtained from Lemma 1.10.

∈ {0} n-1 . We claim that A / ∈ {0} n-k for 1 < k < n. Let A ∈ {0} n-k . Thus A ∼ n-k 0. By Proposition 1.2 (3), there exist X, Y ∈ R such that A (n-k)+l = X(0) l Y (∀l ∈ N). If we put l = k + 1, then A n-1 = 0, which is a contradiction. Hence d(0, A) = n -1 for every n ∈ N. Therefore diam(R) = ∞.

Constructing the closure of a subset of R

Let us first remark that the commutative closure S of a subset S ⊆ R is the union of the commutative closure {s} of its elements s ∈ S. On the other hand to construct the commutative closure of an element we need to use factorizations of this element. We will use the following tools to build elements of s, for s ∈ R.

• For any a ∈ R, we have a ∼ • Somewhat more general than the above point, let us remark that if xb = 0 we always have xy ∼ 1 (y + b)x (resp. if c, y ∈ R are such that cy = 0 then xy ∼ 1 y(x + c)). We even have, for a = xy n , that (y + r(x)) n x ∈ {a} n (and for b = x n y we have y(x + l(y)) n ∈ {b} n ). • For a, b ∈ R we have a(1 + ba) ∼ 1 a(1 + ab). We leave the short proofs of these statements to the reader and start to apply them to different cases. Proof. Let c be any element in {b} n . According to Proposition 1.2, there are two sequence of elements x 1 , . . . , x n , y 1 , . . . , y n in r such that b = x 1 y 1 , y 1 x 1 = x 2 y 2 , . . . , y n x n = c. we thus have a = u -1 bu = (u -1 x 1 )(y 1 u) ∼ 1 y 1 x 1 ∼ n-1 c. We thus conclude that a ∼ n c. This yields the result. Theorem 2.4. Let k be a field and R = K x, y be the free k-algebra. Then diam(R) is infinite.

Proof. Consider x, y ∈ R. We show that for every l ∈ N, d(x + yx l , x + x l y) = l.

Since x + yx l = (1 + yx l-1 )x ∼ 1 x(1 + yx l-1 ) = (1 + xyx l-2 )x ∼ 1 x(1 + xyx l-2 ) = (1 + x 2 yx l-3 )x ∼ 1 • • • ∼ 1 x(1 + x l-1 y) = x + x l y, so d(x + xyx l , x(1 + x l y)) ≤ l.
The path we just described between the two elements x + xyx l and x + x l y is the only path between these two elements. This is a consequence of the fact that, for r, s ≥ 1, the only factorization in R of x + x s yx r are x + x s yx r = x(1 + x s-1 yx r ) = (1 + x s yx r-1 )x. We leave the arguments to the reader.

k[x 1 , x 2 , . . . , x m , . . . ] such that σ(x i ) = x i+1 . Since the chain x 1 x ∼ 1 xx 1 = x 2 x ∼ 1 xx 2 = x 3 x ∼ 1 .
. . is infinite we get that diam(R) is infinite. Also assume that I = (x). We can easily see that

I is commutatively closed. Since R I ∼ = k[x 1 , x 2 , . . . , x m , . . . ] is commutative, we have diam(k[x 1 , x 2 , . . . , x m , . . . ]) = 0.
Example 2.6. Let R and S be two rings and ϕ : R → S is a morphism. In general we can get any relation between diam(R) and diam(S). For instance, there is homomorphism ϕ : k x, y → k, where k is a field. We know diam(k) = 1 while diam(k x, y ) is infinite.

The next proposition establishes a nice connection of our study with the notion of stably associated elements in a ring. In fact, we will just use a very special case of this notion and hence we do not introduce a formal definition (see PM Cohn [START_REF] Cohn | Skew fields, a general theory of division rings[END_REF] for more information about that). Let us recall that two square matrices A, B ∈ M n (R) are said to be associated if there exist invertible matrices P, Q such that P AQ = B. We shall say that A and B are stably associated if diag(A, I) is associated to diag(B, J) for some unit matrices I, J (not necesserily of same size).

Proposition 2.7. Let x, y be elements in a ring R. The 2 × 2 diagonal matrices diag(1 + xy, 1) and diag(1 + yx, 1) are associated.

Proof. First remark that -y -1 1 0

1 x 0 1 = -y -yx -1 1 x ; 1 0 -y 1 x 1 -1 0 = x 1 -yx -1 -y ,
so that the matrices on the right-hand side of the above equalities are invertible.

We have:

-y -yx -1 1 x 1 + xy 0 0 1 x 1 -yx -1 -y = 1 + yx 0 0 1 .
This proves our proposition.

Remark 2.8. Let us remark that the above result is due to the fact that, in the language used in [START_REF] Cohn | Skew fields, a general theory of division rings[END_REF], the equality (1

+ xy)x = x(1 + yx) is comaximal.
This leads to the following statement:

Proposition 2.9. Let S be a connected subset of a ring R. Then S is commutatively closed if and only if for any two elements a, b ∈ S we have that the diagonal matrices diag(1 -a, 1) and diag(1 -b, 1) are associated in M 2 (R).

Proof. if S is commutatively closed and connected (equivalently S = C(a) for some a ∈ S) subset of R and a, b ∈ S. then there is a path from a to b in S and, as in Proposition 1.2, we have two sequences x 1 , x 2 , . . . , x n and y 1 , y 2 , . . . , y n such that a = x 1 y 1 , y 1 x 1 = x 2 y 2 , y 2 x 2 = x 3 y 3 , . . . , y n x n = b and Proposition 2.7 easily implies that diag(1 -a, 1) and diag(1 -b, 1) are associated. Conversely, if a = xy ∈ S, then Proposition 2.7 gives that the diagonal matrices diag(1 -xy, 1) and diag(1 -yx, 1) are associated and hence yx ∈ S.

In the next corollary, we get some generalizations of classical examples.

Corollary 2.10. The set 1 -S is commutatively closed for any one of the following subsets S of R:

(1) S = U r (R) (resp. S = U l (R) or S = U (R)), the set of right (rep. left, two sided) invertible elements of R. (2) S = {A ∈ M n (K) | Rank(A) = l},
where K is a field and l ≤ n ∈ N.

(3) S = reg(R) the set of regular elements of R. (4) S is the set of strongly π-regular elements.

(5) S is the set of left (right) zero divisors in R.

Proof. We refer the reader to [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF] for the proofs or references for these statements.

On Commutatively closed graph over matrix rings

In this section, we study some properties of the commutatively closed graph over matrix rings.

For a ring R and n ∈ N, we denote N n (R) the set of elements of R that are nilpotent of index n. Proposition 3.1. Let R be a ring. Then (1) For any i ∈ N, we have {0} i ⊆ N i+1 (R). In particular, 0 ⊆ N (R).

(2) For any strictly upper triangular matrix

U ∈ M n (R), U ∈ {0} n-1 ⊆ {0}. (3) Let U n (R) ⊆ M n (R) be the set of all n × n strictly upper triangular matrix over R. Then diam U n (R) ≤ 2(n -1).
Proof.

(1)This is easily proved by induction using Proposition 1.2.

(2) We may assume that U = 0 and we denote the lines of U by L 1 , L 2 , . . . , L n . In fact, the last line L n is zero, and we define r ∈ {1, . . . , n -1} to be minimal such that L i is zero for i > r. We will prove that U ∈ {0} r by induction on r. We write

U = I r,r 0 0 0 U and B := U I r,r 0 0 0 ∈ {U } 1 ,
where I r,r denotes the identity matrix of size r × r.

If r = 1, we get that B = 0 ∈ M n (D) and this yields the thesis.

If r > 1, write B = (R 1 , . . . , R n ) where R i is the i th row of B.
The matrix B is easily seen to be upper triangular and such that the rows R r , . . . , R n are zero. This means that this matrix has at least one more zero row than the matrix U . The induction hypothesis gives that B ∈ {0} r-1 , but then U ∈ {B} 1 ⊆ {0} r ⊆ {0}, as required.

(3) By the above statement (2), we know U n (R) ⊆ {0} n-1 ⊆ {0}. So that for two matrices A, B ∈ U n (R), we have A ∼ n-1 0 ∼ n-1 B. This yields the conclusion.

We will now determine the diameter of the class C(0) ∈ M n (D) where D is a division ring. The following lemma is well known, we give a proof for completeness. Lemma 3.2. Every nilpotent matrix with coefficients in a division ring is similar to a strictly upper triangular matrix.

Proof. The proof is based on the fact that any nonzero column can be the first column of an invertible matrix. So let A ∈ M n (D) be a nilpotent matrix with coefficients in a division ring D and let u ∈ M n,1 (D) be a nonzero column such that Au = 0. Let U ∈ M n (D) be an invertible matrix having u as its first column. We conclude that

U -1 AU = 0 r 0 A 1 ,
for some row r ∈ M 1,n-1 (D). It is easy to check that A 1 is again nilpotent. An easy induction on the size of the nilpotent matrix yields the proof.

The next result was proved in [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF] for matrices with coefficients over fields. Proof. We have seen that, in any ring, {0} i ⊆ N (R) i+1 (cf. Proposition 3.1). Conversely, if A ∈ M n (D) is nilpotent, the above lemma 3.2 shows that there exists an invertible matrix P and a strictly upper triangular matrix U ∈ M n (D) such that P AP -1 = U . Since the class of an element is the same as the class of any of its conjugate, we conclude that {A} = {U }. Proposition 3.1 (1) and ( 2) implies that {U } = {0}.

A Jordan block J l (associated to zero) is a matrix of the form

J l =          0
         ∈ M l (D) (1)
If A ∈ M n (D) is a nilpotent matrix, where D is a division ring, A is similar to a diagonal sum of Jordan blocks. This is classical if D is commutative and for a proof in a noncommutative setting we may refer to Chapter 8 of P.M. Cohn's book ( [START_REF] Cohn | Skew fields, a general theory of division rings[END_REF]) or to the more recent paper [START_REF] Hoopes-Boyd | Representing nilpotent matrices as single commutators[END_REF]. Let us notice that J 1 = 0. Lemma 3.4. Let J l be a matrix block of size l > 1. Then J l ∼ l-1 0 Proof. As above, we write J l for the Jordan matrix presented in [START_REF] Alghazzawi | Commutatively closed sets in rings[END_REF].

We proceed by induction on l. If l = 2, we have

J 2 = 0 1 0 0 = 1 0 0 0 0 1 0 0 ∼ 1 0 1 0 0 1 0 0 0 = 0 0 0 0 .
Similarly, for l > 2, we have

J l = ( l-1 i=1 e ii )J l ∼ 1 J l ( l-1 i=1 e ii ) = J l-1 0 0 0 ∼ l-2 0.
Where we have used the induction hypothesis: J l-1 ∼ l-2 0. This implies that J l ∼ l-1 0, as required.

Let us extract from the above proof the following observation.

Corollary 3.5. If l > 1, we have J l ∼ 1 diag(J l-1 , J 1 ).

We will now look more closely to the class of nilpotent matrices.

Proposition 3.6. Let D be a division ring. Then for A ∈ M n (D), we have

A l+1 = 0 ⇔ A ∼ l 0.
Proof. We suppose that A = 0. We know (cf. Proposition 3.3) that the class C(0) consists of all the nilpotent matrices. Thanks to Lemma 2.2 we know that we can replace A by a conjugate to evaluate the length of path from A to 0. Using a result from [START_REF] Hoopes-Boyd | Representing nilpotent matrices as single commutators[END_REF], we know that there exists an invertible matrix P ∈ GL n (D) such that P AP -1 is of the form diag(J n1 , J n2 , . . . , J ns ) where the square matrices J i ∈ M i (D) are of the form given in (1) above. Moreover the maximal size of the Jordan blocks is l + 1 i.e., for all 1 ≤ i ≤ s, we have n i ≤ l + 1. Since for any integer i ∈ N, we have {0} i ⊆ {0} i+1 , the above Lemma 3.4 implies that, for all 1 ≤ i ≤ s, J ni ∼ l 0. This easily leads to the conclusion that A ∼ l 0. The converse was proved in Lemma 3.1.

Theorem 3.9. Let D be a division ring n ∈ N. Then

diam(M n (D)) = n -1.
Proof. A consequence of the Fitting Lemma is that any matrix A ∈ M n (D) is similar to a block diagonal matrix of the form diag(U, N ) where U is an invertible matrix and N a nilpotent matrix. Lemma 2.2 shows that to compute the distance between two different matrices that are in the same commutative class, we may use similar matrices. Thus we need to compute d(A, B) where A and B are of the form A = diag(U, N ) and B = diag(V, M ). The preceding lemma 3.8 shows that U, V ∈ Gl s (D) and N, M ∈ M n-s (D). Theorem 3.7 and Lemma 1.10 show that we may assume 0 < s < n.

Since the matrices N and M are nilpotent we conclude that their distance is less or equal to n -2. This means that there is a sequence of factorizations of length ≤ n -2 linking M and N . We claim that the matrices U and V are in fact similar. Assume that A ∼ r B, according to Proposition 1.2 we know that there exist matrices X, Y ∈ M n (D) such that AX = XB and Y A = BY , moreover for any l ∈ N, we have that A r+l = XB l Y . Choosing l such that M l = N l = 0, and writing X, Y as blocks matrices with X 1 , Y 1 ∈ M s (D), this last equality shows that

U r+l 0 0 0 = X 1 X 2 X 3 X 4 V l 0 0 0 Y 1 Y 2 Y 3 Y 4 .
Comparing the blocks on the top left corner gives U r+l = X 1 V l Y 1 . Since U is invertible, we conclude that the matrices X 1 and Y 1 are also invertible. Now, comparing again the top left corner blocks of the equality AX = XB, we get U X 1 = X 1 V . This shows that U and V are similar, as claimed. This implies that U ∼ 1 V . Since A = diag(U, N ) and B = diag(V, M ) (with sizes of U and N greater or equal to 1). We conclude, thanks to Theorem 3. Proof. This is a simple consequence of Theorem 3.9 and Proposition 1.8

Definition 3.11. The girth of a graph G, denoted by gr(G), is the length of shortest cycle in G, provided G contains a cycle; otherwise gr(G) = ∞.

Note that if R is a ring, then we define the commutatively closed girth of R as follows:

gr(C(R)) = min{gr C(a) | a ∈ R}.

Theorem 3.12. Let D be a division ring. Then gr Mn(D) C(0) = 3.

Proof. We know the class of {0} in R = M n (D) is the set of nilpotent matrices, where D is a division ring (cf. Proposition 3.3). It is enough, to find three nilpotent matrices that form a cycle. It is easy to see that E 1n , E n1 ∈ {0} 1 (Since E 1n = E 11 (E 1n ), 0 = E 1n (E 11 ), E 1n = E 1n E nn and E nn E 1n = 0) and E 1n ∼ 1 E n1 . Hence E 1,n , E 2,n-1 and 0 form a cycle. Therefore gr C(0) = 3.

As an immediate consequence of Theorem 3.12 and definition of gr(C(R)) , we get the following.

Theorem 1. 5 . ( 1 )

 51 for a, b ∈ R, we have a ∼ b if and only if b ∈ {a}. (2) The relation ∼ on R is an equivalence relation. (3) If b ∈ C(a) then, for any l ∈ N, b l ∈ C(a l ). (4) For a ∈ R, the graph C(a) is connected. (5) A subset S of R is closed and connected if and only if it is the closure of an element of R. (6) If S is the closure of a subset S ⊆ R then diam(S) = diam(S).

  a, b), and so (c, d) ∈ {(a, b)}. Proposition 1.8. Let R, S be two rings, and let diam(R) = d 1 and diam

Lemma 1 .

 1 10. If R is a Dedekind finite ring and a ∈ U (R), then diam R (C(a)) = 1. Proof. Since R is Dedekind finite and a ∈ U (R), thus {a} = {uau -1 | u ∈ U (R)}, by Proposition 0.2. So for every b, c ∈ {a}, there exist u, v ∈ U (R) such that b = uau -1 and c = vav -1 . Hence b = uv -1 cvu -1 . Thus b and c are adjacent. Therefore diam R (C(a)) = 1. Proposition 1.11. 1) R is commutative if and only if diam(R) = 0. 2) Let R be a division ring. Then diam(R) = 1.

Theorem 1 .

 1 12. If R is not Dedekind-finite, then diam(R) = ∞. Proof. Assume that R is not Dedekind-finite. Thus there exist a, b ∈ R such that ab = 1 but ba = 1. So we have nonzero element e ij = b i (1 -ba)a j . Consider A = e 12 + e 23 + • • • + e n-1,n . One can easily check that A n = 0 and A n-1 = e 1n = 0, since ba = 1. Write A = (e 11 + e 22 + . . . e n-1,n-1 )A ∼ 1 A(e 11 + e 22 + . . . e n-1,n-1 ) = e 12 + e 23 + • • • + e n-2,n-1 . Continuing this process, we conclude A

1 a( 1 +

 1 b) (resp. a ∼ 1 (1 + b)a) for any b ∈ l(a) (resp. b ∈ r(a)).

Example 2 . 1 .

 21 Let R = M 2 (k) and A, B ∈ R. Also let A BA / ∈ {1 + AB} = {I} = {I}, as wanted. Lemma 2.2. Let R be any ring, and let a, b ∈ R and u ∈ U (R) such that b = uau -1 . Then, for any n ≥ 1, {a} n = {b} n .

Example 2 . 3 .

 23 Let R = k x, y, z, t /I, where k is a field and I is the ideal generated by the element xy -zt. Then yx, tz ∈ {xy} 1 and d(yx, tz) = 2.

Example 2 . 5 .

 25 Let R be a ring and I a commutatively closed ideal of R. Also,let diam( R I ) is finite. Then the diameter diam(R) is not necessarily finite. Consider R = k[x 1 , x 2 , . . . , x m , . . . ][x; σ],where k is field and σ is an automorphism on

Proposition 3 . 3 .

 33 Let D be a division ring and n ∈ N, the class of {0} in R = M n (D) is the set of nilpotent matrices.

7 ,

 7 that d(A, B) = d(M, N ) ≤ n -2. Hence the class with the biggest distance is the class of nilpotent matrices, so that Theorem 3.7 implies that Diam(M n (D)) = n -1. Theorem 3.10. Let R be a semisimple ring and R = M n1 (D 1 ) × • • • × M n l (D l ) be its Wedderrburn Artin decomposition where D 1 , . . . , D l are division rings. Then diam(R) = max{n i -1 | 1 ≤ i ≤ l}.

Corollary 3 .

 3 13. Let D be a division ring. Then gr C(M n (D)) = 3. Corollary 3.14. if R is a semisimple ring then gr C(R) = 3.

Theorem 3.7. Let R = M n (D), where D is a division ring. Then, for nilpotent matrices A, B ∈ M n (k), with nilpotent indexes n(A), n(B) respectively. We have d(A, B) ≤ max{n(A), n(B)} -1. In particular, diam R C(0) = n -1.

Proof. We know that the class of zero in M n (D) is exactly the set of nilpotent matrices. Hence the matrices A, B ∈ C(0) and the distance between A to B is the shortest path from A, B in the graph defined by C(0). Let us write l = n(A) and s = n(B), by symmetry we may assume that l ≥ s. Let diag(J n1 , J n2 , . . . , J nr ) be the Jordan form of B, where r ≥ 1 and s = n 1 ≥ n 2 , • • • ≥ n r . We will use induction on r. In the proof, to avoid heavy notations, we will write, (c 1 , c 2 , . . . , c t ) for diag(c 1 , c 2 , . . . , c t ) (where c 1 , c 2 , . . . , c t are square matrices). If r = 1, then B = J s and A = (J l , A ′ ) where A ′ is a nilpotent matrix of index ≤ l. Using repeatedly Corollary 3.5, we can write A = (J l , A ′ ) ∼ l-s (J s , A ′′ ) and (A ′′ ) s = 0, so d(A, (J s , A ′′ )) ≤ l -s and

Suppose now that the formula is proved for matrices B having less than r > 1 Jordan blocks and consider a matrix B = (J n1 , . . . , J nr ) = (J s , B ′ ), with (B ′ ) s = 0. As above we have A = (J l , A ′ ) ∼ l-s (J s , A ′′ ) and the induction hypothesis gives also

In particular, since the maximal index of nilpotency for matrices in

We will now show that the diameter of the matrix ring M n (D), over a division ring D, is itself n -1. The following lemma is far from surprising but needs to be proved.

Lemma 3.8. Suppose that D is a division ring and that A, B ∈ M n (D) are of the form

where U ∈ GL r (D) and V ∈ Gl s (D) are invertible matrices and N, M are nilpotent matrices. If A ∼ B then r = s.

Proof. Let us first remark that there exists l ∈ N such that N l = 0 and M l = 0. Theorem 1.5 implies that we may assume M = 0 and N = 0. Now, Taking powers again, Proposition 1.2 shows that we may assume A ∼ 1 B. So let us write A = XY, B = Y X ∈ M n (R) and decompose X and Y as follows

where X 1 ∈ M r×s (D), Y 1 ∈ M s×r (D). This fixes the size of all the other matrices appearing in X and Y . Since M and N are zero, the two equations AX = XB and Y B = BY quickly imply that X 2 , X 3 , Y 2 , Y 3 are all zero. and we get U = X 1 Y 1 and V = Y 1 X 1 . Since U and V are invertible we easily conclude that r = s.