MD Xavier Carcopino

MD Julien Mancini

MD Jean Gondry

MD Julien Chevreau

MD Gery Lamblin

MD Anthony Atallah

MD Vincent Lavoue

MD Claire Caradec

MD Jean-Jacques Baldauf

MD Angélique Bryand

MD Sebastien Henno

MD Aubert Agostini

MD Serge Douvier

MD Adelaïde Jarniat

MD Didier Riethmuller

MD Anne Mendel

MD Jean-Luc Brun

MD Hajanirina Rakotomahenina

MD Lise Preaubert

Risk Factors of Inadequate Colposcopy After Large Loop Excision of the Transformation Zone: A Prospective Cohort Study

Keywords: LLETZ, colposcopy, follow up, risk, high grade intraepithelial lesion, stenosis (

Objective:

The aim of the study was to identify the risk factors of post-large loop excision of the transformation zone (LLETZ) inadequate colposcopy.

Materials and Methods: From December 2013 to July 2014, a total of 157 patients who had a LLETZ performed for the treatment of high-grade intraepithelial lesion with fully visible cervical squamocolumnar junction were included. All procedures were performed using semicircular loops. The use of colposcopy made during each procedure was systematically documented. Dimensions and volume of LLETZ specimens were measured at the time of procedure, before formaldehyde fixation. All participants were invited for a follow-up colposcopy 3 to 6 months after LLETZ. Primary end point was the diagnosis of post-LLETZ inadequate colposcopy, defined by a not fully visible cervical squamocolumnar junction and/or cervical stenosis.

Results: Colposcopies were performed in a mean (SD) delay of 136 (88) days and were inadequate in 22 (14%) cases. Factors found to significantly increase the probability of post-LLETZ inadequate colposcopy were a history of previous excisional cervical therapy [adjusted odds ratio (aOR) = 4.29, 95% CI = 1.12-16.37, p = .033] and the thickness of the specimen (aOR = 3.12, 95% CI = 1.02-9.60, p = .047). The use of colposcopy for the guidance of LLETZ was statistically associated with a decrease in the risk of post-LLETZ inadequate colposcopy (aOR = 0.19, 95% CI = 0.04-0.80, p = .024) as the achievement of negative endocervical margins (aOR = 0.26, 95% CI = 0.08-0.86, p = .027).

Conclusions: Although the risk of post-LLETZ inadequate colposcopy is increased in patients with history of excisional therapy and with the thickness of the excised specimen, it could be reduced with the use of colposcopic guidance and the achievement of negative endocervical margins.

L arge loop excision of the transformation zone (LLETZ) is a routine procedure worldwide, because it is the first-line treatment of high-grade intraepithelial lesion (HSIL) of the cervix. Quality criteria for optimal LLETZ include the completeness of excision with the achievement of negative margins, while producing the minimal excised volume and depth of excision to minimize subsequent obstetrical and neonatal morbidity. [START_REF] Kyrgiou | Obstetric outcomes after conservative treatment for intraepithelial or early invasive cervical lesions: systematic review and meta-analysis[END_REF][START_REF] Kyrgiou | Adverse obstetric outcomes after local treatment for cervical preinvasive and early invasive disease according to cone depth: systematic review and meta-analysis[END_REF] Obtaining negative margins is important, because incomplete excision exposes women to a significant risk of posttreatment residual and/or recurrent disease, particularly when the lesion involves the endocervical canal. [START_REF] Ghaem-Maghami | Determinants of success in treating cervical intraepithelial neoplasia[END_REF][START_REF] Ghaem-Maghami | Incomplete excision of cervical intraepithelial neoplasia and risk of treatment failure: a meta-analysis[END_REF] However, this risk remains higher to the general female population, even when negative margins are achieved. Women who had had a LLETZ remain therefore exposed to a 3-to 4-fold increased risk of developing subsequent cervical cancer at least for 20 years. [START_REF] Rebolj | Risk of cervical cancer after completed post-treatment follow-up of cervical intraepithelial neoplasia: population based cohort study[END_REF][START_REF] Strander | Long term risk of invasive cancer after treatment for cervical intraepithelial neoplasia grade 3: population based cohort study[END_REF][START_REF] Soutter | Invasive cervical cancer after conservative therapy for cervical intraepithelial neoplasia[END_REF][START_REF] Melnikow | Cervical intraepithelial neoplasia outcomes after treatment: long-term follow-up from the British Columbia Cohort Study[END_REF] Thus, prolonged and careful post-LLETZ follow-up is mandatory, whatsoever the margins status. For the last decade, the value of human papillomavirus testing has been demonstrated in this indication. Although a negative human papillomavirus test has now been admitted as the best test of cure for patients, colposcopy remains needed when this test is found to be positive. [START_REF] Verguts | Prediction of recurrence after treatment for high-grade cervical intraepithelial neoplasia: the role of human papillomavirus testing and age at conisation[END_REF][START_REF] Paraskevaidis | The role of HPV DNA testing in the follow-up period after treatment for CIN: a systematic review of the literature[END_REF][START_REF] Kocken | High-risk human papillomavirus testing versus cytology in predicting post-treatment disease in women treated for high-grade cervical disease: a systematic review and meta-analysis[END_REF][START_REF] Arbyn | Evidence regarding human papillomavirus testing in secondary prevention of cervical cancer[END_REF] Although being the key examination in this indication, the accuracy of colposcopy performed after previous excisional therapy of HSIL is however questionable because the healing process might result in changes in the appearance of the transformation zone (TZ). However, the main limitation of post-LLETZ colposcopic examination is the possibility of inadequate colposcopy due to the inability to visualize the entire TZ. Known risk factors for inadequate colposcopy include age, severity of lesion, and estrogen status of the patient. [START_REF] Carcopino | Specific timing for colposcopy: is it worthwhile?[END_REF] However, inadequate colposcopy is also one of the main adverse effects of excisional therapies of the cervix, including LLETZ. [START_REF] Carcopino | Specific timing for colposcopy: is it worthwhile?[END_REF] However, data on the precise risk factors for inadequate colposcopy after LLETZ are limited because most studies have focused on the sole risk of cervical stenosis without considering the position and visibility of the squamocolumnar junction. [START_REF] Penna | Laser CO2 conization in postmenopausal age: risk of cervical stenosis and unsatisfactory follow-up[END_REF][START_REF] Baldauf | Risk of cervical stenosis after large loop excision or laser conization[END_REF][START_REF] Houlard | Risk factors for cervical stenosis after laser cone biopsy[END_REF][START_REF] Kliemann | Minimal cold knife conization height for high-grade cervical squamous intraepithelial lesion treatment[END_REF] This point is however crucial because it is clinically essential to identify how post-LLETZ inadequate colposcopy could be avoided, thus preserving the possibility for the follow-up of these women.

The aim of this study was to identify the risk factors of inadequate colposcopy after LLETZ. We also aimed to identify the risk factors for post-LLETZ cervical stenosis.

MATERIALS AND METHODS

This study is a secondary follow-up analysis of the ICORAD prospective multicenter observational study, performed in nine teaching hospitals in France, that aimed to investigate the benefit of the use of direct colposcopic vision for the guidance of LLETZ. [START_REF] Preaubert | Benefits of direct colposcopic vision for optimal LLETZ procedure: a prospective multicenter study[END_REF] ICORAD is the French acronym of "Impact de la Colposcopie sur la Résection à l'Anse Diathermique," which translates to "Impact of Colposcopy on LLETZ." From December 2013 to July 2014, a total of 249 all patients who had a LLETZ performed for the treatment of colposcopically and histologically proven HSIL were recruited to participate in this study. Only patients having a pretherapeutic fully visible cervical squamocolumnar junction were included; women with inadequate pretherapeutic colposcopy were not included in the current study. Written informed consents were obtained from all participants before excisional therapy was performed. Histological analysis of the LLETZ specimen showed invasive carcinoma in 13 cases and the absence of any intraepithelial lesion in 20 cases; all of these 33 cases were secondarily excluded from the study. Thus, a total of 216 patients were finally included. The study received approval from the ethics committee of the French College of Obstetricians and Gynecologists, Paris, France, in October 2013 (CEROG 2013-GYN-1001). Patients' characteristics such as age, parity, history of previous excisional procedure, indication for excision, and characteristics of the cervix were systematically recorded.

Large Loop Excision of the Transformation Zone Procedures

All procedures were performed under anesthesia, by a total of 45 gynecologists surgeons, according to their routine practice. Precise description of the characteristics of the surgeons was made in the initial ICORAD study publication. [START_REF] Preaubert | Benefits of direct colposcopic vision for optimal LLETZ procedure: a prospective multicenter study[END_REF] The use of colposcopy made during each LLETZ procedure was systematically documented. Three groups were considered: LLETZ achieved without any colposcopic examination at the time of the procedure and solely guided by the previous colposcopic report, LLETZ achieved under naked eyes vision immediately after a colposcopic examination had been performed, and LLETZ fully achieved under direct colposcopic vision (DCV). All LLETZ procedures were performed using a semicircular loop only. Dimensions of the loops were left to the appreciation of gynecologists, depending on the size of the cervix and of the type and size of the abnormal TZ. The abnormal TZ was considered to be large if at least three quadrants of the cervix were involved. The possible use of additional destructive therapy, using either ball diathermy or laser ablation, was systematically recorded. Specific attention was also given to how local hemostasis was achieved immediately after LLETZ. Thus, the use and exact duration of electrocauterization, cervical stitches, hemostatic compress, or vaginal clothing were systematically recorded. Finally, the occurrence of intraoperative complications was registered if any.

Dimensions and Volume of LLETZ Specimens

Gynecologists systematically measured exact dimensions and volume of opened LLETZ specimens at the time of procedure, just before formaldehyde fixation as previously reported. Before the study began, all gynecologists participating in the study received specific training regarding how the specimens' dimensions and volume were to be measured. Briefly, the volume was measured using the displacement fluid technique using a 25-mL graduated tube. Dimensions recorded were the thickness, the length, and the circumference of each specimen, as illustrated in Figure 1. In case of multiple specimens were obtained, all the fragments were placed at the same time in the 25-mL graduated tube for global volume assessment. In addition, although the assessment of the circumference of the excised tissue resulted in the addition of the circumference of each fragment, only the length and thickness from the largest fragment were considered.

Follow-up Colposcopic Assessment

All participants were invited to have a systematic follow-up colposcopy 3 to 6 months after LLETZ. Evaluation of the TZ was performed according to the classification of the Internal Federation of Cervical Pathology and Colposcopy. [START_REF] Bornstein | colposcopic terminology of the International Federation for Cervical Pathology and Colposcopy[END_REF] Briefly, the TZ was classified as type 1 when it was entirely ectocervical and visible. Type 2 and 3 TZ both extend into the cervical canal, but whereas TZ was referred as a type 2 if the border between squamous and columnar epithelium was completely visible, it was referred as a type 3 if not. Primary end point of the study was the diagnosis of post-LLETZ inadequate colposcopy, defined by a type 3 TZ and/or a cervical stenosis, resulting in a not fully visible cervical squamocolumnar junction. [START_REF] Bornstein | colposcopic terminology of the International Federation for Cervical Pathology and Colposcopy[END_REF][START_REF] Tatti | Colposcopy: a global perspective: introduction of the new IFCPC colposcopy terminology[END_REF] Secondary end point of the study was the post-LLETZ diagnosis cervical stenosis.

Statistical Analysis

Data were presented as the mean plus the standard error for continuous variables and as counts (percentages) for categorical data. Analysis of variance and Student t tests were used to compare continuous data, and χ 2 tests were used to compare categorical data. A binary logistic regression model was used to study the factors independently associated with posttreatment inadequate colposcopy and cervical stenosis. Only variables associated with a p value of less than .10 in univariate analysis were considered for inclusion in the multivariate model. Circumference and length of the excised specimen were removed because they strongly correlated to thickness of the specimen and were not significant after accounting for it. In addition, the use of local analgesia was also not included in the multivariate model because it was strongly associated with the use of DCV for the guidance of LLETZ. Statistical analyses were performed with IBM SPSS Statistics 20.0 (IBM Inc, New York, NY). A two-sided p value of less than .05 was considered statistically significant.

RESULTS

Among the 216 patients included in the study, 59 (27.3%) did not attend to the follow-up colposcopic assessment they were given 3 to 6 months after LLETZ therapy and were secondarily excluded. No significant difference between characteristics of patients who received follow-up compared with those who did not was observed. Finally, a total of 157 women were included in this study.

Patients' characteristics are summarized in Table 1. Colposcopic examinations were performed in a mean (SD) delay of 4.5 (2.9) months after LLETZ. Colposcopy was documented as inadequate in 22 cases (14%). Women having post-LLETZ inadequate colposcopy were significantly older ( p = .015) and significantly less likely to be nulliparous ( p = .049). Although the volume, circumference, and length of the specimens were not associated with a significant change in the probability of inadequate post-LLETZ colposcopic evaluation, the thickness of the specimens was found to significantly increase that risk. Thus, mean (SD) thickness of patients with inadequate post-LLETZ colposcopy were significantly more important than others [10.5 (4.7) mm vs 7.8 (3.3) mm, respectively, p = .005]. The best prediction for inadequate post-LLETZ colposcopic examination was found for a 9.5-mm thickness cut-off (63.6% sensitivity and 66.7% specificity). Although the final histological diagnosis and achievement of clear margins were not statistically different between patients having inadequate colposcopy and others, patients with inadequate colposcopy were significantly less likely to have negative endocervical margins [13 (61.9%) vs 13 113 (84.3%), respectively, p = .03]. Finally, the use of colposcopy for the guidance of LLETZ was significantly associated with a reduced risk of inadequate colposcopy. Thus, compared with patients who had had a LLETZ performed without any use of colposcopy or just after colposcopic examination, the lowest proportion of inadequate colposcopy was observed among patients who had had LLETZ fully performed under DCV [19 (86.4%) vs 3 (13.6%), respectively, p = .005].

Factors Independently Associated With Post-LLETZ Inadequate Colposcopy

In multivariate analysis, neither the age nor the nulliparity was found to have a significant impact on the probability of post-LLETZ inadequate colposcopic examination (see Table 2). Factors found to significantly increase the probability of post-LLETZ inadequate colposcopy were a history of previous excisional cervical therapy [adjusted odds ratio (aOR) = 4.29, 95% CI = 1.12-16.37, p = .033] and the thickness of the specimen (aOR = 3.12, 95% CI = 1.02-9.60, p = .047). The use of DCV for the guidance of LLETZ was statistically associated with a decrease in the risk of posttreatment inadequate colposcopy (aOR = 0.19, 95% CI = 0.04-0.80, p = .024). Finally, the achievement of negative endocervical margins was found to significantly reduce the probability of post-LLETZ inadequate colposcopy (aOR = 0.26, 95% CI = 0.08-0.86, p = .027).

Risk Factors for Post-LLETZ Cervical Stenosis

A post-LLETZ cervical stenosis was diagnosed in five cases (3.2%), representing 22.7% of the cases of inadequate colposcopic examinations (see Table 3). Compared with others, patients diagnosed with post-LLETZ cervical stenosis were significantly older (p = .008). In addition, the thickness of the specimens was associated with a significant change in the probability of post-LLETZ cervical stenosis. Thus, mean (SD) thickness of specimen from patients with post-LLETZ stenosis was significantly more important than others [12.8 (5.1) mm vs 8 (3.5) mm, respectively, p = .004]. Finally, patients with post-LLETZ stenosis had a higher mean volume of cervix excised. However, the difference was not significant [3.4 (0.5) mL vs 2.2 (1.3) mL, respectively, p = .057]. In multivariate analysis, both the age of patients and the thickness of specimens were found to have a significant impact on the risk of post-LLETZ cervical stenosis (aOR = 1.11, 95% CI = 1.01-0.22, p = .029, and aOR = 1.21, 95% CI = 1.03-0.1.42, p = .023, respectively) (see Table 4).

DISCUSSION

Main Findings

With a proportion of 14% of patients diagnosed with inadequate post-LLETZ colposcopic examination, this study confirms that patients who had a LLETZ for the therapy of HSIL are exposed to a genuine risk of inadequate colposcopy, thus compromising follow-up. According to results from previous study, our results also show the risk of post-LLETZ inadequate colposcopy to depend on the thickness of the excised specimen. [START_REF] Houlard | Risk factors for cervical stenosis after laser cone biopsy[END_REF] However, this study is the first to report this finding after LLETZ because it has been previously solely shown in patients who had had a laser conization. In addition, this study is the first to report this risk to depend on the use of DCV for the guidance of LLETZ and on the achievement of negative endocervical margins. Finally, this study also shows inadequate post-LLETZ colposcopy to partially result from cervical stenosis, with this risk depending on the age of patients and thickness of the excised specimen.

Strength and Limitations

Although the prospective nature of this study brings the possibility for optimal evaluation of the impact of the volume and dimensions of LLETZ specimen on the risk of further inadequate colposcopic examination, there are possible biases that need to be carefully considered when interpreting our results. In particular, colposcopic examination is known to be subjective, so is the characterization of the precise TZ type and the diagnosis of inadequate examination. We believe this point to be important because the range of inadequate colposcopic examination in literature varies from 3% to 20% depending on the population studied and the colposcopist. [START_REF] Carcopino | Specific timing for colposcopy: is it worthwhile?[END_REF][START_REF] Makkar | Vaginal misoprostol versus vaginal estradiol in overcoming unsatisfactory colposcopy[END_REF][START_REF] Pretorius | Which colposcopies should include endocervical curettage?[END_REF] One should therefore consider the lack for additional post-LLETZ colposcopic examination by a second examiner as a limitation in the interpretation of our results. It is however noticeable that with a proportion of 14% post-LLETZ inadequate colposcopy, our results are consistent with those from previous literature that reported this risk to range from 6% to 30%. [START_REF] Thompson | Is colposcopy necessary at twelve months after large loop excision of the transformation zone? A clinical audit[END_REF][START_REF] Suh-Burgmann | Risk factors for cervical stenosis after loop electrocautery excision procedure[END_REF] Because this diagnosis is far less subjective, we also aimed to identify the cases of post-LLETZ stenosis. However, the small size of our study combined with the low proportion of cases of post-LLETZ cervical stenosis limits the interpretation of our results. Another point to be considered is that estimating the circumference of fragmented specimens by summing the circumference of the fragments could significantly have overestimated the circumference of the original specimen. However, multiple fragments were obtained in few cases (18.5%), with no impact on the risk of inadequate post-LLETZ colposcopy (see Table 1). Even if a bias cannot be excluded, we believe the impact to be minimal. Finally, one should also consider that no sample size calculation was made because this study is a secondary follow-up analysis of a prospective project. [START_REF] Preaubert | Benefits of direct colposcopic vision for optimal LLETZ procedure: a prospective multicenter study[END_REF]

Interpretation

In addition to the potential obstetrical impact and consequences on subsequent pregnancies, our results highlight the importance of the risk of inadequate colposcopic examination after LLETZ therapy. Although crucial, this point is however often under considered when making the decision of excisional therapy. In case when excisional therapy is considered, our results suggest to carefully put in balance the risk of compromised follow-up due to inadequate colposcopic examination with the benefits of LLETZ. Interestingly, as previously reported, our results demonstrate the age of patients as a key factor for the risk of post-LLETZ inadequate colposcopic examination due to cervical stenosis. [START_REF] Houlard | Risk factors for cervical stenosis after laser cone biopsy[END_REF] This finding is consistent with those from previous studies, and it is noticeable that in the absence of any excisional therapy, age itself has already been identified as a determinant factor in the position and visibility of the TZ. [START_REF] Carcopino | Specific timing for colposcopy: is it worthwhile?[END_REF] Although, compared with other identified risk factors, the impact of the age seems to be minimal, this finding should be carefully considered when making the decision for excisional therapy in older patients. The two other factors that were identified to significantly increase the risk of inadequate post-LLETZ colposcopy were a history of excisional therapy and the thickness of the specimen excised. Considering those two factors, they both suggest the depth of excision to play a key role in the risk of subsequent inadequate colposcopic examination. Both the volume and the depth of the excision have been demonstrated to significantly impact the risk of subsequent obstetrical morbidity, with the risk of premature delivery being directly related to these two parameters. [START_REF] Kyrgiou | Obstetric outcomes after conservative treatment for intraepithelial or early invasive cervical lesions: systematic review and meta-analysis[END_REF][START_REF] Kyrgiou | Adverse obstetric outcomes after local treatment for cervical preinvasive and early invasive disease according to cone depth: systematic review and meta-analysis[END_REF] The depth and volume of excision are also likely to induce cervical morphological changes such as a reduction in the length of the cervix after LLETZ. [START_REF] Carcopino | Image assessment of cervical dimensions after LLETZ: a prospective observational study[END_REF] Although one could hypothesize that the volume excised would significantly impact the risk of inadequate post-LLETZ colposcopic examination, our results do not support this hypothesis. Thus, although the volume directly depends on the thickness of the excision performed, it also depends on all the other dimensions of the excised specimen. However, because the length and the circumference of the excised specimen were not found to increase the risk of inadequate post-LLETZ colposcopic examination, our results suggest that a large but not deep excision is not likely to end in making the post-LLETZ colposcopic follow-up impossible. This result therefore brings additional argument to minimize the thickness of the excised specimen, ideally in precisely adapting the LLETZ to the precise characteristics of the abnormal TZ and to the exact position of the squamocolumnar junction.

It is important to note that the use of colposcopy during procedures was left to the appreciation of surgeons according to their routine practice, and our study therefore illustrates the wide range of variations in how LLETZ is performed. In previous studies, the use of DCV was shown to help in minimizing the dimensions of the excised specimen, including the thickness, without compromising the margins status. [START_REF] Preaubert | Benefits of direct colposcopic vision for optimal LLETZ procedure: a prospective multicenter study[END_REF][START_REF] Grisot | How to optimize excisional procedures for the treatment of CIN? The role of colposcopy[END_REF][START_REF] Carcopino | Direct colposcopic vision used with the LLETZ procedure for optimal treatment of CIN: results of joint cohort studies[END_REF] In addition to the potential subsequent obstetrical benefit, the systematic use of DCV should therefore also be recommended to reduce the risk of compromised post-LLETZ follow-up as a result of the reduction of the thickness of the excised tissue. However, independently to the thickness of the excised specimen, our study shows the use of DCV to significantly and massively reduce the risk of inadequate colposcopic examination. It is noticeable that this effect was not observed when solely considering the risk of post-LLETZ cervical stenosis. Although this finding suggests that guidance of LLETZ by DCV mostly ends in a reduction of the probability of a type 3 TZ, we cannot exclude the possibility that colposcopists from this group were more likely to identify the squamocolumnar junction, resulting in a lower proportion of type 3 TZs. At this stage, we acknowledge that this result should be considered with caution and that other prospective studies with double post-LLETZ colposcopic assessment are needed to properly evaluate this parameter.

Finally, we found the achievement of negative margins to significantly, independently, and strongly reduce the risk of post-LLETZ inadequate colposcopic evaluation. It is noticeable that this finding was not observed for the ectocervical margins.

Although we feel difficult to properly explain this result, it might suggest that when not achieving clear endocervical margins and therefore not fully excising the squamocolumnar junction, the healing process may result in the burying of the junction deep inside the cervical canal. We acknowledge this finding to be surprising and that more prospective studies are needed to properly investigate this parameter. At this stage, we believe our results to provide genuine arguments to promote that all efforts should be made to achieve ideal balance between minimal excision and negative margins. This point seems crucial, not only to minimize the risk of subsequent obstetrical morbidity but also to minimize the risk of compromised follow-up in women with identified increased risk to develop recurrence and subsequent invasive cancer.

CONCLUSIONS

After LLETZ, patients with initial type 1 or 2 TZ are exposed to the risk of inadequate colposcopic examination, thus compromising follow-up. This risk partially results from cervical stenosis, mostly depending on the age of patients, and should be put in balance when making the decision for excisional therapy, particularly in older women. With this risk depending on the thickness of the excised specimen as well as on the use of DCV for the guidance of LLETZ and on the achievement of negative endocervical margins, this study brings additional arguments to recommend the systematic use of DCV for the guidance of LLETZ and optimal excision. More prospective studies remain needed to confirm our findings.

a

  Defined by the involvement of ≥3 quadrants of the cervix. b 2 or 3 fragments obtained.

TABLE 2 .

 2 Factors Associated With the Occurrence of Post-LLETZ

	Unsatisfactory Colposcopic Examination (Binary Logistic
	Regression Model, n = 155)
		aOR (95% CI) p
	Age (per 1-y increase)	1.04 (0.98-1.11) .180
	Nulliparity	0.36 (0.09-1.48) .154
	History of previous excisional therapy	4.29 (1.12-16.37) .033
	Use of direct colposcopic vision	0.19 (0.04-0.80) .024
	Thickness of the excised specimen ≥10 mm 3.12 (1.02-9.60) .047
	Negative endocervical margins	0.26 (0.08-0.86) .027

aOR indicates adjusted odds ratio.

TABLE 1 .

 1 Patients' Characteristics According to the Diagnostic of Post-LLETZ Inadequate Colposcopy

		Inadequate colposcopic examination
	N = 157	Yes (n = 22)	No (n = 135)	p

All values are expressed as n (%), unless otherwise specified. TZ indicates transformation zone; LLETZ, large loop excision of the transformation zone; DCV, direct colposcopic vision; HSIL, high-grade squamous intraepithelial lesion.

TABLE 3 .

 3 Patients' Characteristics According to the Diagnostic of Post-LLETZ Stenosis of the Cervical Os

		Stenosis
	N = 157	Yes (n = 5)	No (n = 152)	p

  Defined by the involvement of ≥3 quadrants of the cervix.

					.00
	Dimensions of the excised specimen mean (SD), mm
	Thickness	8.1 (3.7)	12.8 (5.1)	8 (3.5)	.004
	Length	12.1 (5.6)	13.8 (3.9)	12.1 (5.7)	.503
	Circumference	45.4 (15.7)	56.8 (12.3)	45.1 (15.6)	.083
	Volume of the excised specimen, mean (SD), mL	2.3 (1.3)	3.4 (0.5)	2.2 (1.3)	.057
	Final histological diagnosis
	HSIL	149 (94.9)	5 (100)	144 (94.7)	1.00
	Negative margins	101 (65.2)	4 (80)	97 (64.7)	.659
	Negative endomargins	126 (81.3)	4 (80)	122 (81.3)	1.00
	Negative ectomargins	126 (80.8)	5 (100)	121 (80.1)	.584
	Delay LLETZ-colposcopy, mean (SD), mo	4.0 (0.2)	3.8 (1.3)	4.0 (0.2)	.880
	All values are expressed as n (%), unless otherwise specified.
	TZ indicates transformation zone; LLETZ, large loop excision of the transformation zone; DCV, direct colposcopic vision; HSIL, high-grade squamous
	intraepithelial lesion.
	b 2 or 3 fragments obtained.

a

TABLE 4 .

 4 Factors Associated With Post-LLETZ Stenosis of the Cervical oz (Binary Logistic Regression Model, N = 157)

	aOR (95% CI)	p

© 2017, American Society for Colposcopy and Cervical Pathology Copyright © 2017 American Society for Colposcopy and Cervical Pathology. Unauthorized reproduction of this article is prohibited.

ACKNOWLEDGMENTS

The authors thank all the women who participated in this study.


Compte

Jean Jaurès

On November

Sylvie Chaperon

Marine Rouch

Justine Zeller

Sou Linne Baik

Beauvoirian disturbances : Review of a French workshop

come   L'archive ouverte pluridisciplinaire

Indeed, Beauvoir's life and work could feed the history of political and social movements (Algerian war, Vietnam and Russel Tribunal, and, of course, the Second Wave feminist movements), sexual history (The Second Sex, Beauvoir's private life), and even the history of tourism as journeys had a huge place in her life. On this subject, a doctoral dissertation has been defended in 2012 by Tiphaine Martin who focused on travel writings within the Memoirs.

that Beauvoir studies, in France and other countries are destined for a bright future. Two vast conferences have reinvigorated them. The first one was held in 1999 upon the fiftieth birthday of The Second Sex (co-organized by Sylvie Chaperon and Christine Delphy), the second in 2008 on the occasion of Simone de Beauvoir's hundredth birthday (organized by Julia Kristeva). One might take place in Paris in 2018 along with various events to celebrate her 110th birthday. Regarding doctoral researches: since 7 were defended and 10 are being pursued.

But we are facing a cloud on the horizon for if French Beauvoir studies are flourishing, they lack visibility, maybe due to a lack of unity and legitimacy. The transversal aspect of Beauvoir's work has been encountering the strict separation of French academic disciplines, whereas the United States have their Simone de Beauvoir Society which gathers researchers from various disciplines since the 80's. A glimmer of hope came in 2012 with a latest attempt of an interdisciplinary book on Beauvoir, which contains a lot of until then unpublished documents that could bring new food for thought 6 .

We tried to join this movement and gathered philosophers, historians and one linguist to investigate some of the « perturbations » that came from Beauvoir 7 .

« Perturbation » #1: Lighting domestic work

Long before French materialist feminist Christine Delphy's work, Simone de Beauvoir perceived the specificity of domestic work for women. María Luisa Femenías, Argentinian philosopher, chose to explore the work question to show that in 1949, Beauvoir's conception of work is close to Engels' philosophy and a socialist interpretation of society.

Beauvoir refers to work with a specific lexical field: monotony, oppression and subordination. In The Second Sex, she applies these ideas to domestic work and analyzes the way it is lived by women according to their situation (women workers [in farms or factories] or bourgeois housewives for example). But first, she stresses the education of girls that focuses on the house and keeps them away from productive work (as opposed to reproductive work). « Ces dialectiques peuvent donner au travail ménager l'attrait d'un jeu : la fillette s'amuse volontiers à faire briller l'argenterie, à astiquer les boutons de porte. » (TSS,2,p. 263.) This situation also comes from a socialization of the feminine strength as Beauvoir writes in the introduction of her essay, which assigns production to men and reproduction and primary tasks to women.

From then on, Beauvoir considers that women's defeat lies on this sexual work division. Indeed, this matter of fact makes them dependent on men since they live among and with them and can hardly form a group. They do not have an egalitarian access to work and many social structures, such as maternity leave or the banning of extended work, are confirmation that women are relegated to the household and that their salary is just a contribution.

Still, they work -inside and outside their household. The last pages of The Second Sex's "History" part explores different situations. Inside the household, it is the "supplice de Sisyphe":

"Jour après jour, il faut laver les plats, épousseter les meubles, repriser le linge qui seront à nouveau demain salis, poussiéreux, déchirés. La ménagère s'use à piétiner sur place ; elle ne fait rien ; elle perpétue seulement le présent ; elle n'a pas l'impression de conquérir un Bien positif mais de lutter indéfiniment contre le Mal. » (TSS, 2, p. 264.) ; « Laver, repasser, balayer, dépister les moutons tapis sous la nuit des armoires, c'est arrêtant la mort refuser aussi la vie : car d'un seul mouvement le temps crée et détruit. » (p. 265).

Women from bourgeoisie backgrounds often do not have a salaried work. Sometimes, they have maids to assist them. But others cannot and have to reconcile those two lives: "Il en va tout autrement pour l'ouvrière, l'employée, la secrétaire, la vendeuse, qui travaille audehors. Il leur est beaucoup plus difficile de concilier leur métier avec le soin du ménage (courses, préparation des repas, nettoyage, entretien des vêtements demandent au moins trois heures et demie de travail quotidien et six heures le dimanche ; c'est un chiffre considérable quand il s'additionne à celui des heures d'usine ou de bureau). » (TSS, 1, p. 231.) Discussions during this session pointed out the fact that, despite those negative interpretations of work, Beauvoir, especially in the 70's -argues that human beings need work to realize themselves. For that to be possible, capitalist society must end and move on to a socialist and more egalitarian work organization. Indeed, in 1949, Beauvoir thinks that the advent of socialism will improve women's condition. But in the 1970's, she concedes that sex struggle needs to prevail over class struggle and that a socialist society won't mean women's emancipation. Only a financial independence will help women to emancipate themselves. But sacrifices will have to be made, she warned several times.

For María Luisa Femenías, it is because Beauvoir considers that power relations exist between men and women that she can perceive the specific women's relationships to work and thus, understand how the Woman was historically built as the Other. From a macropolitical point of view, Beauvoir's work allows one to consider a larger structure of men and women's power relations.

« Perturbation » #2: Theorizing the ambiguity

Beauvoir's theorization of ambiguity could shake our thought and social structures up.

That is what Françoise Rétif proposes to explore from the first beauvoirian philosophical essays, Pyrrhus and Cineas, The Ethics of Ambiguity, and her youth diary. Ambiguity implies an idea of duality which is a characteristic of Beauvoir: she is as complex as her entire work.

Françoise Rétif introduces the notion of androgyny, which she developed in her essay Simone de Beauvoir: l'autre en miroir (1998). Indeed, the youth diary shows how Beauvoir defines herself in ambiguity -between "masculine and feminine". Her bourgeois family circle imposed on her a model according to which there is no symmetry between genders. Masculine incarnates superiority through knowledge and intellectual life -to which F. Rétif refers to as the cultural side -, whereas feminine is relegated to spiritual life -referred to as the natural side. But the young Beauvoir cannot adapt to those principles and tries to philosophically handle her internal conflicts. She early decides that she won't give up on anything : « Et puis, mes chers amis, vous n'aimez pas les jeunes filles, mais songez que non seulement elles ont une raison à satisfaire, mais un coeur lourd à comprimer. Et en cela je veux rester femme, plus masculine encore par le cerveau, plus féminine par la sensibilité » (Cahiers de jeunesse, p. 374).

Or later:

"je suis terriblement avide, aussi, je veux tout de la vie, être une femme et aussi un homme » (Letter to Nelson Algren, July 3rd, 1947). Simone de Beauvoir adds value to ambiguity in a process of correction of Philosophy in general and sartrean existentialism particularly, which do not consider human beings as consciousness incarnated in bodies. Plus, Sartre, in the 30's and the 40's, insists on human failure: « Et il est vrai aussi que dans l'Être et le Néant, Sartre a surtout insisté sur le côté manqué de l'existence humaine » (Pour une morale de l'ambiguïté, p. 17). For Beauvoir, bodies and passion can reconcile them through (physical and psychological) pleasure during which one is subject and object at the same time. The Other -one's existence, one's freedom -becomes necessary.

That is how Beauvoir reintroduces nature and body in order to create a balance. The philosopher was one of the first -if not the first -to invent gender in its current acceptation but her ethics of ambiguity invites to think over the traditional boundary between men and women, between culture and nature. Sylvie Chaperon's talk echoes Françoise Rétif's considerations for she shows that Beauvoir's double gender identification is perceptible in her sexual life. The historian advances the hypothesis according to which Beauvoir was a physical and psychological bisexual before the word even took the meaning it has today. John Gagnon, which considers sexuality as both sexual and social phenomenon, uses what he calls "sexual script" in order to understand sexual conducts that take place out of norms. Sylvie Chaperon proposes to interpret Beauvoir's ambiguous sexual life in light of Gagnon's theory. Beauvoir's sexual script is characterized by the flame theme: her passionate friendships with young girls sometimes become sexual, or, some other times, stay platonic. Those "flames" are only accepted -although they are forbidden by law -because they are regarded as transitory, and consequently do not challenge norms. In The Second Sex, Beauvoir treats this kind of conducts as so: Is not "The lesbian" chapter in the "Formation" part? « Perturbation » #3: leading women towards emancipation Simone de Beauvoir led millions of women on the path of emancipation, inviting them to shake not only their daily life but also society, and to assert themselves. Simone de Beauvoir was a tireless letters writer. Her published correspondences (To Sartre, Algren and Bost) show this but the letters she received from her readers confirm that more. Those letters, which she used to answer almost systematically, are kept at the Bibliothèque nationale de Thus, from the first volume of the autobiography, Beauvoir has become a privileged confidante and has been receiving testimonies on women's private lives: pregnancies, abortions, loneliness, domestic violence etc. In her replies (not kept, but Marine Rouch found some of them) Beauvoir appears as an intellectual mentor and personal counselor.

Since letters to writer allow the readership's concretization to the writer's eye, Marine Rouch proposes the hypothesis of an influence in return of these women letters on Simone de Beauvoir's feminist, intellectual and militant path. Indeed, the radicalization in Beauvoir's militancy corresponds to the peak in letters written by women in which they confide themselves and appropriate Beauvoir's work. This hypothesis is being pursued.

As a linguist, Alice Caffarel-Cayron -like Beauvoir and Halliday -considers language as a way to act upon the world. Halliday writes: « Language is not only a way of thinking about the world; it is also, at one and the same time, a way of acting on the worldwhich means, of course, acting on the other people in it. » (2009:4 10 ). From this theory, Alice Caffarel-Cayron tries to understand how Beauvoir's language and conception of literature act upon her readers through a transcendental force. She bases her talk on a particular correspondence: her mother's. On July 11st 1964, Claire Cayron wrote Simone de Beauvoir for the first time, after reading Force of Circumstance: « L'année dernière, c'est après avoir lu la Force des Choses que m'est venue l'envie de vous écrire et de solliciter votre avis sur ma première tentative d'expression. C'est encore le cas aujourd'hui. Vous avez mis dans ce livre un tel pouvoir de communication, un tel besoin même (me semble-t-il), qu'il paraît normal, l'ayant clos, de venir bavarder avec vous. »

The two women engage in a regular correspondence that shows the transformations of the reader. When she first wrote Beauvoir, Claire Cayron was a single and unemployed 8 NAF 28501. 9 Lettre du 20 juin 1959. 10 Halliday, M.A.K. (2009). The Essential Halliday, edited by Jonathan Webster. Continuum mother. Over the years, she is pushed by Beauvoir to write and publish books: La nature chez Simone de Beauvoir (1972), Divorce en France (1974). To Alice Caffarel-Cayron, Beauvoir's language is dialogical. In her work and letters, she invites people to respond, to talk to her. By analyzing this, the linguist seeks to understand Beauvoir's influence on "millions of hearts".

Focusing on "perturbations" has allowed scholars to see more clearly the complexity and the ambiguity of Beauvoir's work's but also its ability to interrogate our current representations. At this birthday year, let's venture that more and new reflections will challenge Beauvoir studies. Marine Rouch. January 2018.

France

  since 1995 8 . Marine Rouch is pursuing a doctoral research on them. They allow the evaluation of Beauvoir's work influence and reception by anonymous people. From 1958, letters written by women are majority: 60% in 1958, more than 80% in 1959 and between 68% and 72% in the next years. The autobiography seems to have been what pushed women to write the author: Beauvoir seems closer while The Second Sex, although revolutionary, was a dry reading. « Après vos mémoires, vous êtes descendue d'un piédestal, dans le bon sens, vous êtes devenue plus humaine et votre supériorité culturelle et intellectuelle ne vous rend plus si lointaine. » 9(Letter, 20, june 1959)

Écriture et deuil : la mort chez Marguerite Yourcenar, Simone de Beauvoir et Annie Ernaux

La mélancolie entre philosophie et littérature : lecture de l'oeuvre autobiographique de Simone de Beauvoir par Jing Zhao, 2014.

Fictions de soi : l'écriture du deuil dans les oeuvres romanesques de Simone de Beauvoir

www.theses.fr

Si j'en suis arrivée là, c'est grâce à vous". Ecritures des femmes et des hommes "ordinaire" : le lectorat de Simone de Beauvoir.


Andrey Polyakov

On global existence of strong and classical solutions of Navier-Stokes equations

Keywords: Navier-Stokes-Equations, Strong Solutions, Dilation Symetry

This short note studies the problem of a global expansion of local results on existence of strong and classical solutions of Navier-Stokes equations in R 3 .

Introduction

In this short note we exploit a very simple idea of global expansion of regularity by means of dilation symmetry, which is well known for systems in R n .

Let a vector field f : R n → R n be symmetric with respect to a dilation of the argument, i.e. ∃α ∈ R such that f (e s u) = e (α+1)s f (u), ∀u ∈ R n , ∀s, ∈ R If u(•) : [0, +∞) → R n is a classical solution of du dt = f (u), t > 0 with the initial condition u(0) = u 0 then u s (t) := e s u(e αs t)

is defined on [0, +∞) and, due to symmetry, we derive du s dt = e (α+1)s f (u(e αs t)) = f (u s (t)), t > 0,

A. Polyakov Inria Lille, Univ. Lille, CNRS, UMR 9189 -CRIStAL, (F-59000 Lille, France), Tel.: +33-359577802 E-mail: andrey.polykov@inria.fr i.e. u s is a classical solution of the same differential equation with the initial condition u s (0) = e s x 0 , where s ∈ R.

Let ∃ε > 0 such that a classical solution of the differential equation exists on [0, +∞) for any initial value u(0) = u 0 ∈ B ε := {u ∈ R n : |u| < ε}. To construct a solution for u 0 / ∈ B ε we first need to scale u 0 → e s0 u 0 , where s 0 ∈ R is such that

|e s0 u 0 | < ε.
If u(, e s0 u 0 ) is a solution with the initial condition u(0) = λ 0 u 0 then ũ(t) = e -s0 u(e -αs0 , e s0 u 0 ) is also a solution of the considered system and, obviously, ũ(0) = e s0 u(0, e s0 u 0 ) = e -s0 e s0 u 0 = u 0 .

In this paper we use the dilation symmetry (see, [1, formula (1.5)]) of the Navier-Stokes equations in R 3

∂ t u = ν∆u -(u • ∇)u -∇p, 0 = divu
where u denotes the velocity of a fluid, p denotes the scalar pressure and ν > 0 denotes viscosity of the fluid, in order to understand when a global-intime existence of strong or classical solutions of the Navier-Stokes equations for small initial data implies the existence of global-in-time strong or classical solutions for large initial data. We refer the reader, for example, to [START_REF] Leray | Sur le mouvement d'un liquide visqueux que limitent des parois[END_REF] and [START_REF] Ozanski | Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF],

for more details about global-in-time existence of strong solutions for small initial data. Mainly, the standard notation is utilized through the paper, e.g. R is the field of real numbers; L 1 loc ((0, T )×R n , R) denotes the space of locally integrable functions (0,

T ) × R n → R; L p (R n , R m ), 1 ≤ p ≤ +∞ is a Lebesgue space of function R n → R m with the norm • p ; C ∞ c ((0, T ) × R n , R m ) is a space of smooth functions (0, T ) × R n → R m with compact support and C ∞ 0 ([0, T ) × R n , R m
) is a space of smooth functions which vanish at infinity, where 0 < T ≤ ∞. For composition of operators A, B we also use the notation A • B.

Let L p µ (R n , R m ) denotes the following normed vector space of functions

R n → R m L p µ (R n , R m ) := {u : u p,µ < +∞} , µ ∈ R u p,µ := R n |x| µp |u(x)| p dx 1/p , 0 < p < ∞ u ∞,µ := ess sup(|x| µ u(x)), p = ∞,
which can be treated as a wighted L p .

Preliminaries: Dilations in functional spaces

The following lemmas deal with the most common dilation groups in functional spaces.

Lemma 1 The operator d(s) given by (d(s)z)(x) = e αs z(e γs t, e βs x),

where s ∈ R, z is a function (0, T ) × R n → R m , 0 < T ≤ +∞, x ∈ R n and α, β, γ ∈ R are constant parameters, -maps C ∞ c ((0, T ) × R n , R m ) onto C ∞ c ((0, e -γs T ) × R n , R m ); -maps C ∞ 0 ([0, T ) × R n , R m ) onto C ∞ 0 ([0, e -γs T ) × R n , R m ). The inverse operator is given by [d(s)] -1 = d(-s). (1)
Proof 1) Since the linear function (t, x) → (e γs t, e βs x) maps a compact in (0, T ) × R n to a compact in (0, e -γs T ) × R n then d(s) defined on whole

C ∞ c ((0, T ), R n ) and if z ∈ C ∞ c ((0, T ), R n ) (i.e. z is smooth and has a com- pact support in (0, T ) × R n ) then d(s)z ∈ C ∞ c ((0, e -γs T ), R n ) (i.e. d(s)z is also smooth, but it has a compact support in (0, e -γs T ) × R n ). Obviously, (d(s) • d(-s))z = (d(-s) • d(s))z = z for any z ∈ C ∞ c ((0, T ), R n ) and any s ∈ R. Let us show that d(s) maps C ∞ c ((0, T ), R n ) onto C ∞ c ((0, e -γs ), R n ). Sup- pose the opposite: ∃z * ∈ C ∞ c ((0, e -γs T ), R n ) such that z * = d(s)y, ∀y ∈ C ∞ c ((0, T ), R n ). This is impossible, since d(-s)z * ∈ C ∞ c ((0, T ), R n ) and z * = (d(s) • d(-s))z * ∈ d(s)C ∞ c ((0, T ), R n ). 2) The proof for C ∞ 0 is almost identical. Obviously, if z ∈ C ∞ 0 ([0, T ), R n ) (i.e.
z is smooth and vanishing at infinity) then d(s)z ∈ C ∞ 0 ([0, e -γs T ), R n ) (i.e. d(s)z is also smooth and vanishing at infinity) and d(s

) • d(-s)z = d(-s) • d(s)z = z for any z ∈ C ∞ 0 ([0, T ), R n ). Let us show that d(s) maps C ∞ 0 ([0, T ), R n ) onto C ∞ 0 ([0, e -γs T ), R n ). Sup- pose the opposite: there exists z * ∈ C ∞ 0 ([0, e -γs T ), R n ) such that d(s)y = z * , ∀y ∈ C ∞ 0 ([0, T ), R n ). This is impossible since d(-s)z * ∈ C ∞ 0 ([0, T ), R n ) and z * = d(s) • d(-s)z * ∈ d(s)C ∞ 0 ([0, T ), R n ).
Lemma 2 The operator d(s) given by

(d(s)z)(x) = e αs z(e βs x), (2)
where s ∈ R, z is a function R n → R m , x ∈ R n and α, β ∈ R are constant parameters, is -a linear bounded invertible operator on L p (R n , R m ), d(s)z p = e (α-nβ/p)s z p , z ∈ L p (R n , R m ), s ∈ R, -a linear bounded invertible operator on L p µ (R n , R m ), d(s)z p = e (α-β(µ+n/p))s z p , z ∈ L p (R n , R m ), s ∈ R, where 0 < p ≤ ∞. The inverse operator is given by [d(s)] -1 = d(-s). Proof Notice that L p = L p 0 . Let 1 ≤ p < ∞. If z ∈ L p µ (R n , R m ) then R n |x| µp |z(x)| p dx < +∞ and R n |x| µp |z(x)| p dx = e nβs R n |e βs x| µp |z(e βs x)| p dx = e ((n+µp)β-αp)s R n |x| µp (d(s)z)(x)| p dx < +∞. Since e ((n+µp)β-αp)s > 0 for any α, β, p, s ∈ R then d(s)z ∈ L p µ (R n , R m ) for any s ∈ R. Obviously, d(s) is a linear operator on L p µ , i.e. d(s)(µ 1 z 1 + µz 2 ) = µ 1 d(s)z 1 + µ 2 d(s)z 2 , for any µ 1 , µ 2 ∈ R and z 1 , z 2 ∈ L p µ (R n , R m ).
Moreover, the latter identities imply that d(s)z p,µ = e (α-(n/p+µ)β)s z p , z p,µ :=

R n |x| µp |z(x)| p dx 1/p .
Hence, the operator d(s) :

L p (R n , R m ) → L p (R n , R m ) is bounded for any s ∈ R . Let p = ∞. If z ∈ L ∞ µ (R n , R m

Global Existence of Strong Solutions of Navier-Stokes Equations

Below for shortness we omit R 3 in the notations of R 3 , L p , C ∞ 0 and C ∞ c if the context is clear. Without loss of generality (see e.g. [4, page 4]) we also assume that ν = 1, where ν is a viscosity coefficient.

Let us consider the weak form of the Navier-Stokes equations

R 3 u(0)•ξ(0)+ T 0 R 3 u•(∂ t ξ+∆ξ)+p divξ = T 0 R 3 u•(u•∇)ξ, ∀ξ ∈ C ∞ 0 ([0, T )×R 3 , R 3 )
(3) with u(t) ∈ V for t ∈ (0, T ), where V is a set of the so-called weakly divergence free velocity fields:

V := u ∈ L 2 : R 3 u • ∇φ = 0, ∀φ ∈ C ∞ 0 (R 3 , R) .
The classical idea of analysis is to prove existence and regularity of weak solutions and next to show that any weak solution is smooth. For Navier-Stokes equations this analysis has been initiated by Jean Leray in 1938 (see [START_REF] Leray | Sur le mouvement d'un liquide visqueux que limitent des parois[END_REF]). We use the recent review [START_REF] Ozanski | Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF] of his results.

, T ) if u(t) ∈ V for t ∈ (0, T ), p ∈ L 1 loc ((0, T ) × R 3 , R), u ∈ C([0, T ), L 2 ) ∩ C((0, T ), L ∞ ), u (
t) ∞ is bounded as t → 0 + and (3) is satisfied.

A possible way for expansion of regularity to a larger set of initial conditions is to use the dilation symmetry as explained in the introduction. Several symmetries for Navier-Stokes equations are known (see e.g. [START_REF] Fushchych | Symmetry reduction and exact solutions of the navierstokes equations[END_REF] and references therein). Below we use just one of them (see [1, formula (1.5)]).

Lemma 3 If (u, p) is a strong solution of the Navier-Stokes equations with the initial data u(0) = u 0 ∈ V ∩ L ∞ defined on [0, T ) then for any s ∈ R the pair (u s , p s ) given by u s (t, x) = e s u(e 2s t, e s x), p s (t, x) = e 2s p(e 2s t, e s x), t ≥ 0,

x ∈ R n , s ∈ R.
is a strong solution of the Navier-Stokes equations defined on [0, e -2s T ) with the initial value u s (0) = d(s)u 0 .

Proof 1) Let d 1 (s) be defined by the formula (2) with α = 1, β = 1.

Since u is a strong solution then the function t → u(t) is continuous in L 2 and in L ∞ . According to Lemma 2, d 1 (s) is a linear bounded invertible operator on L 2 and on L ∞ . Hence, for any fixed s ∈ R the function t → d 1 (s)u(t) is also continuous in L 2 and L ∞ , consequently, u s ∈ C([0, e -2s T ), L 2 ) and u s ∈ C((0, e -2s T ), L ∞ ).

Notice d 1 (s)z ∞ = e s z ∞ for any z ∈ L ∞ and any s ∈ R (see Lemma 2). Since u(t) ∞ is bounded as t → 0 + then u s (t) ∞ = e s u(e s t) ∞ is also bounded as t → 0 + .

If u 0 ∈ V then d 1 (s)u 0 ∈ V for any s ∈ R. Indeed,

u 0 • ∇φ = 0, ∀φ ∈ C ∞ 0
and using the change-of-variable theorem in the Lebesgue integral we derive

0 = R 3 u 0 (x)•∇φ(x)dx = e 3s R 3 u 0 (e s x)•(∇φ)(e s x)dx = e 2s R 3 (d 1 (s)u 0 )•∇ φ, where φ = d 1 (s)φ. Since d 1 (s) maps C ∞ 0 onto C ∞ 0 (see Lemma 1) then 0 = (d 1 (s)u 0 ) • ∇ φ, ∀ φ ∈ C ∞ 0 , i.e. d 1 (s)u 0 ∈ V for any s ∈ R. Hence, the inclusion u(t) ∈ V , t ∈ [0, T ) implies u s (t) ∈ V , t ∈ [0, e -2s T ).
2) Let d 2 (s) be given by the formula (1) with α = 2, β = 1 and γ

= 2. Since p ∈ L 1 loc ((0, T ) × R n , R) then T 0 R 3 |p(t, x)ξ(t, x)|dxdt < +∞, ∀ξ ∈ C ∞ c ((0, T ) × R 3 , R)
hence using the change-of-variable theorem in the Lebesgue integral for the functions t → e 2s t and x → e s x we derive

T 0 R 3 |p(t, x)ξ(t, x)|dxdt = e 5s e -2s T 0 R 3
|p(e 2s t, e s x)ξ(e 2s t, e s x)|dxdt =

e s e -2s T 0 R 3 |p s • d 2 (s)ξ| < +∞, ∀ξ ∈ C ∞ c ((0, T ) × R 3 , R). Since the operator d 2 (s) maps C ∞ c ((0, T )×R 3 , R) onto C ∞ c ((0, e -2s T )×R 3 , R) then e -2s T 0 R 3 |p s • ξ| < +∞, ∀ ξ ∈ C ∞ c ((0, e -2s T ) × R 3 , R).
Therefore, p s ∈ L 1 loc ((0, e -2s T ) × R 3 , R).

3) Let us show that (u s , p s ) satisfies the equation (3). Since (u, p) is a solution defined on [0, +∞) then ∀ξ ∈ C ∞ 0 ([0, T ) × R 3 , R 3 ) we have

R 3 u(0, x)•ξ(0, x)dx+ T 0 R 3 u(t, x)•(∂ t ξ(t, x)+(∆ξ)(t, x))+p(t, x)(divξ)(t, x)dx dt = T 0 R 3 u(t, x) • (u(t, x) • ∇)ξ(t,
(0)•ξ s (0)+e 2s e -2s T 0 u s •(∂ t ξ s +∆ξ s +p s (divξ s ) = e 2s e -2s T 0 R 3 u s •(u s •∇)ξ s , Since by assumption r 1 (1 -µ 1 -3/q 1 ) + r 2 (1 -µ 2 -3/q 2 ) = 0 then for any u 0 ∈ V ∩ L q1 µ1 ∩ L q2 µ2 there exists s 0 ∈ R such that d 1 (s)u 0 r1 q1 d 1 (s)u 0 r2 q2 < ε.
Hence, if a strong solution (u, p) with u(0) = d 1 (s 0 )u 0 exists on [0, +∞) then by Lemma 3 the pair (ũ, p) given by ũ(t, x) = e -s0 u(e -2s0 t, e -s0 x), p(t, x) = e -2s0 p(e -2s0 t, e -s0 x), t ∈ [0, +∞), x ∈ R 3 is also a strong solution of the Navier-Stokes equation. Since u(0) = d 1 (s 0 )u 0 means that u(0, x) = e s0 u 0 (e s0 x), x ∈ R 3 then ũ(0, x) = e -s0 u(0, e -s0 x) = u 0 (x),

i.e. ũ(0) = u 0 ∈ V ∩ L q1 µ1 ∩ L q2 µ2 and the proof is complete. Notice that taking µ 1 = µ 2 = 0 we derive the usual L q1 and L q2 spaces in the latter corollary.

Let us mention the following properties of the strong solutions (see Definition 1) proven before:

-Global-in-time existence for small initial data [4, Corollary 3.13 and Lemma 3.10] There exist ε > 0 and C > 0 such that for any

u 0 ∈ {u ∈ V ∩ L ∞ ∩ L 2 : u 2 2 u ∞ < ε} (4)
or

u 0 ∈ {u ∈ V ∩ L ∞ ∩ L 2 : u 2 ∇u 2 < ε} (5)
or u 0 ∈ {u ∈ V ∩ L ∞ ∩ L 2 : u 2(q-3) 2 u q q < ε}, q > 3 (6) To expand globally the regularity of the Navier-Stokes equation a globalin-time existence of strong solutions for small initial data has to be proven for the norms satisfying Corollary 1.

  ) then ess sup|z(x)| = ess sup(|e βs x| µ |z(e βs x)|) < +∞ for any β, s, µ ∈ R and d(s)z ∞ = e (α-βµ)s z ∞ for any s ∈ R. Therefore, d(s) is also a linear bounded operator on z ∈ L ∞ (R n , R m ). Obviously, (d(s) • d(-s))z = (d(-s) • d(s))z for any z : R n → R n and any s ∈ R and we derive [d(s)] -1 = d(-s).

Definition 1 (

 1 [START_REF] Ozanski | Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF], Definition 3.7) A pair (u, p) is said to be a strong solution of the Navier-Stokes equations on [0

a

  strong solution (u, p) with the initial data u(0) = u 0 exists on [0, +∞) and u(t) ∞ ≤ C u 0 ∞ . -Uniqueness of strong solutions [4, Theorem 3.9] For any u 0 ∈ V ∩ L ∞ a strong solution with u(0) = u 0 is unique. -Smothness [4, Corollary 3.3].If (u, p) is a strong solution of the Navier-Stokes equation then∂ k t ∇ m u, ∂ k t ∇ m p ∈ C((0, T ), L 2 ) ∩ C((0, T ), L ∞ ), ∀m, k ≥ 0 and, in particular, u, p ∈ C ∞ (R 3 × (0, T )) constitute a classical solution of the Navier-Stokes equations on (0, T ) × R 3 .None of conditions (4), (5), (6) satisfy Corollary 1.

  x)dx dtUsing the change-of-variable theorem in the Lebesgue integral for the functions t → e 2s t and x → e s x we derive 2s t, e s x) • (∂ t ξ(e 2s t, e s x)+ (∆ξ)(e 2s t, e s x)) + p(e 2s t, e 2s x)(divξ)(e 2s t, e s x)dx dt = 2s u s (t, x)• ∂ t ξ(e 2s t, e s x)+(∆ξ)(e 2s t, e s x)

				e -2s T
	e 3s	R 3	u(0, e s x) • ξ(0, e s x)dx + e 5s e -2s T u(e e 5s 0 R 3 u(e 2s t, e s x) • (u(e 2s t, e s x) • ∇)ξ(e 2s t, e s x)dx dt
				0	R 3
	or, equivalently,
				e -2s T
	e 2s		u s (0, x)•ξ(0, e s x)dx+e 2s
	R 3		0	R 3
				+e s p s (t, x)(divξ)(e 2s t, e s x)dx dt =
				e -2s T
				e 3s	u s (t, x) • (u s (t, x) • ∇)ξ(e 2s t, e s x)dx dt
				0	R 3
	Let us denote ξ s (t, x) = ξ(e 2st , e s x). Hence,
	(∆ξ and
	3
	e 2s	u s
	R

e s )(t, x) = e 2s (∆ξ)(e 2st , e s x), (∇ξ s )(t, x) = e s (∇ξ)(e 2s t, e s x), (∂ t ξ s )(t, x) = e 2s (∂ t ξ)(e 2s t, e s x), (divξ s )(t, x) = e s (divξ)(e 2s t, e s x),

where ξ s ∈ C ∞ 0 ([0, e -2s T ), ×R 3 , R 3 ). Since for any s > 0 the operator d 3 (s) defined as (d 3 (s)ξ)(t, x) = ξ(e 2s t, e s x), t > 0,

holds for all ξ s ∈ C ∞ 0 ([0, e -µs T ) × R 3 , R 3 ). Therefore, (u s , p s ) is a strong solution of the Navier-Stokes equations on [0, e -2s T ) and u s (0) = d(s)u 0 .

The proven lemma implies the following result, which describes the cases when global-in-time existence of strong solutions for small initial data is equivalent to global-in-time existence of strong solutions for large initial data.

A strong solution of the Navier-Stokes equations with arbitrary the initial data u(0) = u 0 ∈ V ∩ L q1 µ1 ∩ L q2 µ2 exists on [0, +∞) if and only if there exist ε > 0 such that for any

a strong solution (u, p) with the initial data u(0) = u 0 exists on [0, +∞).

Proof Let d 1 be defined as in the proof of Lemma 3. Assume that for any u 0 ∈ {u ∈ V ∩ L q1 µ1 ∩ L q2 µ2 : u r1 q1,µ1 u r2 q2,µ2 < ε} there exists a global in time strong solution with u(0) = u 0 and let us show that for any u 0 ∈ V ∩L q1 µ1 ∩L q2 µ2 : u 0 r1 q1,µ1 u 0 r2 q2,µ2 ≥ ε then the Navier-Stokes equations also have a strong solution on [0, +∞).

In the proof of Lemma 3 we have shown that d 1 (s)u 0 ∈ V for any s ∈ R, by Lemma 2 and d 1 (s)u 0 ∈ L q1 µ1 and for d 1 (s)u 0 ∈ L q2 µ1 any s ∈ R. By Lemma 2 we also derive )+r2(1-µ2-3 q 2

)) u 0 r1 q1,µ1 u 0 r2 q2,µ2 .


Marzieh Gheisari
email: marzieh.gheisari-khorasgani@inria.fr

Teddy Furon
email: teddy.furon@inria.fr

Laurent Amsaleg
email: laurent.amsaleg@irisa.fr

Behrooz Razeghi
email: behrooz.razeghi@unige.ch

Slava Voloshynovskiy

Aggregation and embedding for group membership verification

Keywords: group representation, discrete embedding, aggregation, data privacy, verification

INTRODUCTION

Verifiying that an item/device/individual is a member of a group is needed for many applications granting or refusing access to sensitive resources. Group membership verification is not about identifying first and then checking membership. Rather, being granted with access requires that the members of the group could be distinguished from non-members, but it does not require to distinguish members from one another.

Group membership verification protocols first enroll eligible signatures into a data structure stored at a server. Then, at verification time, the structure is queried by a client signature and the access is granted or not. For security, the data structure must be adequately protected so that a honest but curious server cannot reconstruct the signatures. For privacy, verification should proceed anonymously, not disclosing identities.

A client signature is a noisy version of the enrolled one, e.g. due to changes in lighting conditions. The verification must absorb such variations and cope with the continuous nature of signatures. They must be such that it is unlikely that a noisy version for one user gets similar enough to the enrolled signature of any other user. Continuity, discriminability and statistical independence are inherent properties of signatures.

This paper proposes a group membership verification protocol preventing a curious but honest server from reconstructing the enrolled signatures and inferring the identity of querying (trusted) clients. It combines two building blocks: Block #1: One building block hashes continuous vectors into discrete embeddings. This lossy process limits the ability of the server to reconstruct signatures from the embeddings. Block #2: The other building block aggregates multiple vectors into a unique representative value which will be enrolled at the server. The server can therefore not infer any specific signature from this value. Sufficient information must be preserved through the aggregation process for the server to assert whether or not a querying signature is a member of the group. These two blocks can be assembled according to two configurations: block #1 before block #2, the system acquires and then hashes the signatures before aggregating them. The opposite configuration is where acquired signatures are aggregated before hashing the result of this aggregation. At query time, the newly acquired signature is always hashed before being sent to the server. Weaknesses and strengths of these two configurations are explored in the paper.

RELATED WORK

Group membership verification protocols relying on cryptography exist [START_REF] Schechter | Anonymous authentication of membership in dynamic groups[END_REF] but are more relevant to authentication, identification and secret binding. Other approaches apply homomorphic encryption to signatures, compare [START_REF] Ramón Troncoso-Pastoriza | Fully private noninteractive face verification[END_REF] and threshold them [START_REF] Erkin | Privacy-preserving face recognition[END_REF][START_REF] Sadeghi | Efficient privacy-preserving face recognition[END_REF] in the encrypted domain, and need active participation of clients. Approaches involving cryptography, however, are extremely costly, memory and CPU wise.

Group membership is linked to Bloom filters that are used to test whether an element is a member of a set. When considering security, a server using Bloom filters cannot infer any information on one specific entry [START_REF] Bianchi | better than nothing" privacy with bloom filters: To what extent?[END_REF]. Note that Bloom filters can not deal with continuous high dimensional signatures and that queries must be encrypted to protect the privacy of users [START_REF] Boneh | Public key encryption that allows pir queries[END_REF][START_REF] Beck | Approximate two-party privacy-preserving string matching with linear complexity[END_REF]. Bloom filters, adapted to our setup, however, form a baseline in our experiments (see Sect. 5.3).

Embedding a single high dimensional signature is quite a standard technique. The closest to our work is the privacypreserving identification mechanism based on sparsifying transform [START_REF] Razeghi | Privacy preserving identification using sparse approximation with ambiguization[END_REF][START_REF] Razeghi | Privacypreserving outsourced media search using secure sparse ternary codes[END_REF][START_REF] Razeghi | Privacy-preserving identification via layered sparse code design: Distributed servers and multiple access authorization[END_REF]. It produces an information-preserving sparse ternary embedding, ensuring privacy of the data users and security of the signature.

Aggregating signals into similarity-preserving representations is very common in computer vision [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF][START_REF] Jégou | Aggregating local image descriptors into compact codes[END_REF][START_REF] Perronnin | Fisher kernels on visual vocabularies for image categorization[END_REF]. They do not consider security or privacy. In [START_REF] Iscen | Memory vectors for similarity search in high-dimensional spaces[END_REF], Iscen et al. use the group testing paradigm to pack a random set of image signatures into a unique high-dimensional vector. It is therefore an excellent basis for the aggregation block: the similarities between the original non-aggregated signatures and a query signature is preserved through the aggregation.

NOTATIONS AND DEFINITIONS

Signatures are vectors in R d . If N users/items belong to the group, then the protocol considers N signatures, S = {x 1 , . . . , x N } ⊂ R d . The signature to verify is a query vector y ∈ R d . Group membership verification considers two hypotheses linked to the continuous nature of the signatures:

H 1 : The query is related to one of the N vectors. For instance, it is a noisy version of vector j, y = x j + n, with n to be a noise vector.

H 0 : The query is not related to any vector in the group.

We first design a group aggregation technique s which computes a single representation from all N vectors r := s(S). This is done at the enrollment phase. Variable denotes the size in bits of this representation.

At the verification phase, the query y is hashed by a function h of size in bits. This function might be probabilistic to ensure privacy. The group membership test decides which hypothesis is deemed true by comparing h(y) and r. This is done by first computing a score function c and thresholding its results: t := [c(h(y), r) > τ ].

Verification Performances

The performances of this test are measured by the probabilities of false negative, p fn (τ ) := P(t = 0|H 1 ), and false positive, p fp (τ ) := P(t = 1|H 0 ). As τ varies from -∞ and +∞, these measures are summarized by the AUC (Area Under Curve) performance score. Another figure of merit is p fn (τ ) for τ s.t. p fp (τ ) = , a required false positive level.

Security and Privacy

A curious server can reconstruct a signature x from its embedding (for instance the query): x = rec(h(x)). The mean squared error is a way to assess its accuracy: MSE = E( Xrec(h(X)) 2 )/d. The best reconstruction is known to be the conditional expectation: x = E(X|h(x)).

Reconstructing an enrolled signature from the group representation is even more challenging. Due to the aggregation block, the curious server can only reconstruct a single vector x from the aggregated representation, and this vector serves as an estimation of any signatures in the group:

MSE e = (dN ) -1 N j=1 E( X j -X 2 ).
(1)

VERIFICATION FOR A FEW GROUP MEMBERS

This section discusses the verification protocol when N is small. We study the two different configurations for assembling block #1 and block #2.

Block #1: Embedding. An embedding h : R d → A maps a vector to a sequence of discrete symbols. This quantization shall preserve enough information to tell whether two embeddings are related, but not enough to reconstruct a signature. We use the sparsifying transform coding [START_REF] Razeghi | Privacy preserving identification using sparse approximation with ambiguization[END_REF][START_REF] Razeghi | Privacypreserving outsourced media search using secure sparse ternary codes[END_REF]. It projects x ∈ R d to the range space of a transform matrix W ∈ R ×d . The output alphabet A = {-1, 0, +1} is imposed by quantizing the components of Wx whose amplitude is lower than λ to 0, the others to +1 or -1 according to their sign. In expectation, S = 2dΦ(-λ/σ x ) symbols are non null.

Block #2: Aggregation. Aggregation a processes a set of input vectors to produce a unique output vector. When block #1 is used before, that is, when considering s = a • h , then a : R ×N → R . When block #2 is used before block #1, that is when considering s = h • a, then a : R d×N → R d .

Aggregation strategies

The nature of a highly depends on the type of vector the aggregation function receives. When considering s = h • a, then a gets continuous signatures. In this case it is possible to design two aggregations schemes that are:

a(S) = x∈S x = G1 N or (2)
a(S) = (G † ) 1 N . ( 3
)
where

G is the d × N matrix G := [x 1 , . . . , x N ], 1 N := (1, . . . , 1) ∈ R N , and G † is the pseudo-inverse of G.
Eq. ( 2) is called the sum and (3) the pinv schemes in [START_REF] Iscen | Memory vectors for similarity search in high-dimensional spaces[END_REF].

When considering s = a • h, then a gets the embeddings of the signatures. Two additional aggregation strategies are the sum and sign pooling (4) and the majority vote (5):

r = sign( x∈S h(x)) or (4)
r i = arg max s∈{-1,0,1} |{x ∈ S|h(x) i = s}| (5)

Four resulting schemes

The assemblage of the blocks and the aggregation strategies overall create four variants. We name them:

• HoA-2: this scheme sums the raw signatures into a unique vector before embedding it in order to obtain r. It therefore corresponds to the case where s = h • a, the aggregation a being defined by (2). • HoA-3: here also, aggregation precedes embedding, s = h • a, and a is defined by (3). • AoH-4: this scheme embeds each signature before aggregating with sum and sign pooling as defined by (4). • AoH-5: here also, embedding precedes aggregation, but the majority vote is used as defined by ( 5). The score function c comparing the hashed query with the group representation is always c(h(y), r) = -h(y) -r .

RECONSTRUCTION AND VERIFICATION

This section makes the following assumptions: i) Enrolled signatures are modelled by X ∼ N (0 d , σ 2

x I d ), ii) Square orthogonal matrix W known by the attacker.

Ability to reconstruct from the embedding

Now that W preserves the norm, the MSE on X is the same as the mean square reconstruction error on Z = WX, which is also white Gaussian distributed. Thanks to the independance of the components of Z, the conditional expectation can be computed component-wise. We introduce the density function conditioned on the interval R s ⊂ R: Observing the i-th symbol of h(x) equals s reveals that z i ∈ R s . This component is reconstructed as ẑi (s) := E(Z|R s ). Note that ẑi (0) = 0 because f (z|R 0 ) is symmetric around 0. For s = 1, the reconstruction value equals

f (z|R s ) := φ σx (z).1 Rs (z)/P(Z ∈ R s ), (6)
ẑi (1) = +∞ -∞ z.f (z|R 1 )dz = σy p1 √ 2π e -λ 2 2σ 2
x , where p 1 := P(Z ∈ R 1 ) = Φ(-λ/σ x ). By symmetry, ẑi (-1) = -ẑ i (1), and MSE admits the following close form:

MSE = σ 2 x .MSE(λ) (7)
MSE(λ) := 1 - 1 πΦ(-λ/σ x ) e -λ 2 σ 2 x . (8)
This quantity starts at 1 -2π -1 when λ = 0. The embeddings are then full binary words (p 1 = 1/2). All components are reconstructed by ±ẑ i but with a large variance. As λ increases, this variance decreases but less non-null components are reconstructed. MSE(λ) achieves a minimum of ≈ 0.19 for λ ≈ 0.60, where 55% of the symbols of an embedding are non null. Then, MSE(λ) increases up to 1 for a large λ: the embeddings becomes sparser and sparser. When fully zero, each component is reconstructed by 0, and MSE equals σ 2 x .

Ability to reconstruct the signatures

The curious server tries to reconstruct a unique vector x from r which represents the N enrolled signatures. Note that r is scale invariant: scaling the signatures by any positive factor does not change r. Suppose that the curious server reconstructs x = κu. The best scaling minimizing MSE e (1) is: κ = u -2 u m, with m := N -1 N j=1 x j . The curious server can not compute κ giving birth to a larger distortion:

MSE e ≥ 1 N N j=1 x j 2 - (u m) 2 u 2 . (9)
This lower bound is further minimized by choosing u ∝ m. Therefore, aggregation ( 2) is less secure as the other schemes do not allow the reconstruction of m. In the worst case (2), the curious server estimates m by N -1 rec(h(a(S))):

d.MSE e =E( X j -N -1 rec(h(a(S))) 2 ) (10) =E( X j - a(S) N 2 ) + E( a(S) -rec(h(a(S))) 2 ) N 2 .
The first term is the squared distance between X j and m, whereas the second term corresponds to the error reconstruction for inverting the embedding. In the end:

MSE e = σ 2 x 1 - 1 N (1 -MSE(λ)) . (11)
This figure of merit increases with N because MSE(λ) ≤ 1, ∀λ ≥ 0: Packing more signatures increases security.

Verification performances

We compare to a baseline defined as a Bloom filter optimally tuned for given N and p fp having length B = N | log p fp | log(2) -2 . An embedding h is mandatory to first turn the real signatures into discrete objects. This means that, under H 1 , a false negative happens whenever h(x j + n) = h(x j ).

Fig. 2 shows the AUC vs. MSE [START_REF] Beck | Approximate two-party privacy-preserving string matching with linear complexity[END_REF] for the schemes of Sect. 4.1 for different sparsity S/d. Two schemes performs better. For low privacy (small MSE q ), HoA-3 achieves the largest AUC (with 0.5 ≤ S/d ≤ 0.7) ; for high privacy, AoH-4 is recommended (with S/d ≤ 0.2). In these regimes, the performances are better than the Bloom filter.

Fig. 3 shows how the verification performances decrease as the number N of enrolled signatures increases. As mentioned in [START_REF] Iscen | Memory vectors for similarity search in high-dimensional spaces[END_REF], the behavior of the aggregation scheme depends on the ratio N/d. The longer the signatures, the more of them can be packed into one representation.

VERIFICATION FOR MULTIPLE GROUPS

When N is large, aggregating all the signatures into a unique r performs poorly. Rather, for large N , we propose to partition the enrolled signature into M > 1 groups, and to compute M different representatives, one per partition.

Random assignment:

The signatures are randomly assigned into M groups of size n = N/M . Clustering: Similar signatures are assigned to the same group. The paper uses the k-means algorithm to do so. Yet, the size of the groups is no longer constant. tp ) the operating point of group number k, 1 ≤ k ≤ M . The overall system outputs a positive answer when at least one group test is positive. Denote by (P fp (M ), P tp (M )) the performance of the global system. Under H 0 , the query is not related to any vector. Therefore,

P fp (M ) = 1 - M k=1 (1 -p (k) fp ), (12)
Under H 1 , the query is related to only one vector belonging to one group. A false negative occurs, if this test produces a false negative and the other tests a true negative each:

P fn (M ) = M k=1 n k N p (k) fn l =k (1 -p (l) fp ). (13)
The operating point of a group test is mainly due to the size of the group. The random assignment creates even groups (if M divides N ), so these share the operating point (p fp , p tp ). Fig. 4 shows the experimental AUC and the one predicted by ( 12) and ( 13) when M ranges from 8 to 512. Since clustering makes groups of different sizes, we show the performances versus n min = min 1≤k≤M (n k ), where n k is the size of k-th group. The theoretical formulas are more accurate for random partitioning where the group are even. Estimations of (p (k) fp , p (k) fn ) were less precise with the clustering strategy, and this inaccuracy cumulates in [START_REF] Jégou | Aggregating local image descriptors into compact codes[END_REF] and [START_REF] Perronnin | Fisher kernels on visual vocabularies for image categorization[END_REF].

Clustering improves the verification performances a lot especially for HoA-3. A similar phenomenon was observed in [START_REF] Iscen | Memory vectors for similarity search in high-dimensional spaces[END_REF]. Yet, Fig. 5 shows that it does not endanger the system: MSE e is only slightly smaller than for random assignment, and indeed close to 1 for n min ≥ 100. This is obtained for M = 32 for HoA-3 giving AUC = 0.97. The space is so big that the clusters are gigantic and not revealing much about where the signatures are. However, the anonymity is reduced because the server learns which group provided a positive test. This is measured in term of k-anonymity by the size of the smallest group, i.e. n min . Fig. 4 indeed shows the trade-off between k-anonymity and the verification performances.

CONCLUSION

This paper proposed four schemes for verifying the group membership of continuous high dimensional vectors. The keystones are the aggregation and embedding functions. They prevent accurate reconstruction of the enrolled signatures, while recognizing noisy version. However, the anonymity is slightly revealed when managing many signatures aggregated into several representatives: the server is only able to link each signature to its group number. Yet, the full identity of the user is preserved.

Fig. 1 :

 1 Fig. 1: Block diagram of the proposed model.

Fig. 2 :

 2 Fig. 2: Unique group: AUC vs. MSE q /σ 2 y . N = 128, d = 1024 , σ 2 n = 0.01 for varying S ∈ (0.1 × d, 0.9 × d).

4 Fig. 3 :

 43 Fig.3: Unique group: AUC and p tp vs. N . Solid and dashed lines correspond to AUC and p tp @p fp = 10 -2 .

Fig. 4 :Fig. 5 :) and 1

 451 Fig. 4: Multiple groups: AUC vs. n min . Dotted lines are theoretical AUC. N = 4096, d = 1024, σ 2 n = 10 -2 , S/d = 0.6 for HoA-3, and S/d = 0.85 for AoH-4.


Keywords: Protein-Protein Interactions, 14-3-3 proteins, Drug discovery, NMR resonance assignments

14-3-3 proteins are a group of seven dimeric adapter proteins that exert their biological function by interacting with hundreds of phosphorylated proteins, thus influencing their subcellular localization, activity or stability in the cell. Due to this remarkable interaction network, 14-3-3 proteins have been associated with several pathologies and the proteinprotein interactions established with a number of partners are now considered promising drug targets. The activity of 14-3-3 proteins is often isoform specific and to our knowledge only one out of seven isoforms, 14-3-3ζ, has been assigned. Despite the availability of the crystal structures of all seven isoforms of 14-3-3, the additional NMR assignments of 14-3-3 proteins are important for both biological mechanism studies and chemical biology approaches.

Herein, we present a robust backbone assignment of 14-3-3σ, which will allow advances in the discovery of potential therapeutic compounds. This assignment is now being applied to the discovery of both inhibitors and stabilizers of 14-3-3 protein-protein interactions.

Biological context

The 14-3-3 family includes seven different isoforms (β, γ, ε, η, σ, τ and ζ) expressed in all eukaryotic organisms. 14-3-3 proteins are adapter proteins, expressed in an ubiquitous manner in all tissues and cell compartments. Despite the considerably high structural homology between different isoforms, 14-3-3 activity is often isoform specific [START_REF] Fu | 14-3-3 proteins: structure, function, and regulation[END_REF]. To our knowledge, 14-3-3ζ is the only isoform of 14-3-3 whose assignment is reported [START_REF] Killoran | Structural Analysis of the 14-3-3ζ/Chibby Interaction Involved in Wnt/β-Catenin Signaling[END_REF]. 14-3-3 proteins exert their biological functions through the modulation of the activity of hundreds of phosphorylated proteins. This remarkable interactome makes 14-3-3 proteins influent actors in many cellular events and, by consequence, in several pathologies like cancer, Alzheimer's disease, Parkinson's disease, among others [START_REF] Kaplan | 14-3-3 adaptor protein-protein interactions as therapeutic targets for CNS diseases[END_REF] . With the rise of the interest on the modulation of Protein-Protein Interactions (PPIs) in modern drug discovery, 14-3-3 proteins are being seen as promising targets for several pathologies. Especially over the last 10 years, there has been a remarkable effort towards the discovery of compounds capable of inhibiting or stabilizing 14-3-3 PPIs [START_REF] Stevers | Modulators of 14-3-3 Protein-Protein Interactions[END_REF]. In particular, we have been interested in modulating the Tau/14-3-3 interaction, which is thought to have detrimental effects in neuronal cells in Alzheimer's disease [START_REF] Joo | Involvement of 14-3-3 in tubulin instability and impaired axon development is mediated by Tau[END_REF]. Peptidomimetic compounds have been designed and shown to inhibit this PPI [START_REF] Milroy | Stabilizer-Guided Inhibition of Protein-Protein Interactions[END_REF][START_REF] Andrei | Inhibition of 14-3-3/Tau by Hybrid Small-Molecule Peptides Operating via Two Different Binding Modes[END_REF]. Assignment of 14-3-3σ spectra will further help the development of these promising compounds.

In physiological conditions, 14-3-3 proteins exist as functional dimers, both as hetero and homodimers [START_REF] Fu | 14-3-3 proteins: structure, function, and regulation[END_REF]. They are acidic proteins and each monomer has a molecular weight (MW) of around 30 kDa, being composed by nine α-helices, commonly named αA-I, disposed in an antiparallel fashion. At the C-terminus of the last helix of 14-3-3 proteins (αI), there is a flexible and highly acidic tail, which is the less conserved region of 14-3-3 [START_REF] Obsil | Structural basis of 14-3-3 protein functions[END_REF].

Here we report the backbone assignments ( 13 C α , 13 C β , 13 CO, 15 N and 1 H N ) of 14-3-3σ (a truncated form comprising residues 1-231 out of 248). Due to the large number of partners of 14-3-3σ, the assignment of this protein opens new possibilities for the study of hundreds of PPIs. Our assignment is currently being applied to drug screening and to the study of the effect of small molecules on the modulation of 14-3-3 PPIs. In the future, it may have a role on the development of new 14-3-3 PPI modulators that can be employed as tool compounds and hopefully, be the basis for the development of new therapeutic molecules.

Sample preparation

The plasmids used for protein expression were kindly provided by Dr. Christian Ottmann from the Eindhoven University of Technology, The Netherlands. The cDNA coding for human 14-3-3σ (residues 1-248) or 14-3-3σΔC17 (residues 1-231) was cloned into the pProExHtb vector between the BamHI and NotI sites. A Tobacco Etch Virus (TEV) cleavage site followed by a linker (GAMGS) are present between the regions coding for the N-terminal His 6 -tag and for the protein, yielding a recombinant expression plasmid with the following configuration: His 6 tag-TEV cleavage site-Linker-14-3-3σ/14-3-3σΔC17. Therefore, 14-3-3σ and 14-3-3σΔC17 were both expressed as N-terminally His 6 -tagged proteins. 15 N 13 C 2 H labeled 14-3-3σ or 14-3-3σΔC17 were expressed in E. coli BL21 (DE3) cells. A 20 ml pre-culture in Luria-Bertani (LB) medium containing 100 mg/L ampicillin was grown overnight at 310 K and was used to inoculate 1 L of deuterated M9 minimal medium supplemented with 2 g/L 13 C 6 2 H 7 D-Glucose, 1 g/L 15 N Ammonium Chloride, 0.4 g/L Isogro 15 N 13 C 2 H Powder -Growth Medium (Sigma Aldrich) and 100 mg/L ampicillin. The culture was grown at 310 K to an OD 600 of 0.9 and induced with 0.5 mM Isopropyl-β-D-Thiogalactopyranoside (IPTG). Incubation was continued for 15h at 301 K with vigorous shaking. Cells were harvested by centrifugation, resuspended in a buffer containing 50 mM Tris-HCl, 500 mM NaCl, 5% (v/v) glycerol, 25 mM imidazole, 2 mM β-Mercaptoethanol (BME), pH 8.0 and lysed with a homogenizer. The cellular debris were then eliminated by centrifugation and the supernatant was loaded into a Ni-NTA column (GE Healthcare).

Elution of proteins was performed with a buffer containing 50 mM Tris-HCl, 500 mM NaCl, 5% (v/v) glycerol, 250 mM imidazole, 2 mM BME, pH 8.0. The N-terminal His 6 -tag was then cleaved by the TEV protease for 2 hours at 293 K, followed by 12 hours at 277 K, while being dialyzed against 100 mM sodium phosphate, pH 6.8 and 50 mM NaCl (NMR buffer).

The molar ratio between 14-3-3σ or 14-3-3σΔC17 and TEV protease was 50:1. The protein without His 6 -tag was collected in the flow-through of a Ni-NTA column, using NMR buffer for the elution, while the uncleaved fraction and the His 6 -tagged peptide were retained on the column. The protein was then concentrated to 2 mM, aliquoted, flash frozen and stored at 193 K. Typical yields were in the range of 45-70 mg of protein per liter of culture.

NMR spectroscopy

All NMR experiments were recorded using a 900 MHz Bruker Avance spectrometer, equipped with a cryoprobe. Samples, in a buffer containing 100 mM sodium phosphate, pH 6.8, 50 mM NaCl, 1mM DTT (Dithiothreitol), EDTA-free Protease Inhibitor Cocktail (Roche, Switzerland) and 10% (v/v) D 2 O, were transferred to Shigemi tubes and experiments were acquired at 305 K. The concentration of 15 N 13 C 2 H labeled protein for assignment experiments ranged from 0.8 mM to 1.0 mM. Backbone assignments were obtained from TROSY-HNCACB, TROSY-HN(CO)CACB, BEST-TROSY-HNCO and 1 H-15 N-NOESY-HMQC spectra on 15 N 13 C 2 H labeled 14-3-3σ or 14-3-3σΔC17 samples. The mixing time of the 1 H-15 N-NOESY-HMQC spectrum was 120 ms. Non-Uniform Sampling was employed for the acquisition of all 3D spectra. The reference for the 1 H chemical shift was relative to

Trimethyl silyl propionate. The 15 N and 13 C chemical shifts were referenced indirectly. All spectra were collected and processed with Topspin 3.5 (Bruker Biospin) and analyzed with Sparky 3.12 (T. D. Goddard and D. G. Kneller, SPARKY 3, University of California, San Francisco). Automatic assignments were performed with MARS version 1.2 (Jung and

Zweckstetter 2004) and with PINE-NMR server [START_REF] Bahrami | Probabilistic Interaction Network of Evidence Algorithm and its Application to Complete Labeling of Peak Lists from Protein NMR Spectroscopy[END_REF].

Assignments and data deposition

For sequence numbering purposes, the residues of the linker left after TEV cleavage (GAMGS) were not considered. Accordingly, residue 1 corresponds to the first Methionine residue of 14-3-3σ. Similarly to what has been reported for the assignment of 14-3-3ζ [START_REF] Killoran | Structural Analysis of the 14-3-3ζ/Chibby Interaction Involved in Wnt/β-Catenin Signaling[END_REF], we also observed the presence of highly intense peaks in the 1 H-15 N TROSY-HSQC spectrum of the full-length 14-3-3σ (residues 1-248). Because these intense signals were overlapping with signals corresponding to the ordered core-region, we decided to use a construct of 14-3-3σ without the C-terminal flexible region (14-3-3σΔC17, containing residues 1-231) for the backbone assignment. The elimination of the resonances corresponding to this flexible region indeed allowed to detect more resonances on the 1 H-15 N TROSY-HSQC spectrum and additionally improved the overall quality of the 3D spectra.

Removal of the disordered terminal tail did not affect the structure of the protein in solution as resonances in the 2D spectra of the protein with or without the tail superimposed. The final construct, after cleavage of the N-terminus by the TEV protease, yielded a 14-3-3σΔC17

protein with a MW of 26.51 KDa. Given its association as a dimer and therefore the formation of a biomolecule with a MW higher than 50 KDa, it was necessary to produce a deuterated protein, to use a high-field spectrometer and to extend acquisition times. A good coverage of the protein assignment was reached although it should be noted that not all the amide peaks of the protein were detected on the 1 H-15 N TROSY-HSQC (shown in figure 1 a)

and in some cases, backbone signals in the 13 C dimension were of low intensity. The incomplete 2 H/ 1 H exchange may have been a factor that prevented the detection of some resonances, in particular those corresponding to residues located at the dimer interface.

Overall, we were able to obtain a robust assignment of 14-3-3σΔC17 comprising 177 out of

Secondary structure

The obtained chemical shift assignments were further used for the analysis of the secondary structure of 14-3-3σ by two different methods. On one hand, the ensemble of backbone chemical shifts ( 13 C α , 13 C β , 13 CO, 15 N and 1 H N ) was submitted to CSI 3.0 sever [START_REF] Hafsa | CSI 3.0: a web server for identifying secondary and supersecondary structure in proteins using NMR chemical shifts[END_REF] in order to obtain the Chemical Shift Indexes predictive of the secondary structure. On the other hand, the 13 C α and 13 C β chemical shift values were used to calculate the Secondary Structure Propensity (SSP) scores [START_REF] Marsh | Sensitivity of secondary structure propensities to sequence differences between α-and γ-synuclein: Implications for fibrillation[END_REF]. Prediction by SSP scores allowed comparison with the secondary structure analysis of the previously assigned 14-3-3 isoform [START_REF] Killoran | Structural Analysis of the 14-3-3ζ/Chibby Interaction Involved in Wnt/β-Catenin Signaling[END_REF].

The results of the predictions made by both softwares are summarized in figure 2. Indeed, the results obtained from both methods are coherent and well correlated with what is reported in the crystal structure of 14-3-3σ (PDB ID: 1YZ5). The SSP scores of 14-3-3σ presented here are almost identical to the ones of 14-3-3ζ showing that the secondary structures of both isoforms in solution are very similar. The only remarkable difference is the length of helix αD, which is shorter in 14-3-3σ at its N-terminus due to the replacement of Q 79 of 14-3-3ζ by a proline residue.

Fig. 1 Assignment of 14-3-3σ is robust and covers homogeneously the protein's surface. a) Annotated 1 H-15 N TROSY-HSQC spectrum of 15 N 13 C 2 H labeled 14-3-3σΔC17 0.8 mM acquired at 305 K in a buffer containing 100 mM sodium phosphate, pH 6.8, 50 mM NaCl, 1mM DTT, EDTA-free Protease Inhibitor Cocktail (Roche, Switzerland) and 10% (v/v) D 2 O.

The spectrum was acquired with 3426 complex points in the 1 H dimension and 256 complex points in the 15 N dimension with 64 scans per increment. The spectral window was between 4.72 ppm and 14.64 ppm for the 1 H dimension and between 80.2 ppm and 140.1 ppm for the 15 N dimension. The selected window contains all the assigned backbone resonances. b) Cartoon representation of the crystal structure of 14-3-3σ (PDB ID: 1YZ5, gray cartoon) with the assigned residues colored in red. The structure is shown both as a dimer and as a monomer of 14-3-3σ.

Fig. 2

Secondary Structure analysis of 14-3-3σ by calculation of the SSP scores and by the CSI 3.0 server reveals that 14-3-3σ is composed by nine α-helices, as expected from the crystal structure. a) SSP scores calculation for 14-3-3σ. The y axis of the plot represents the score (~ 1 for α-helix, ~ 0 for random coil, ~ -1 for β-strand) for each residue of 14-3-3σ (represented on the x axis). Yellow bars correspond to non-assigned residues or to residues preceding prolines, which were not considered for the SSP score calculation. Random coil chemical shift values used for the calculation of SSP scores were taken from RefDB database [START_REF] Zhang | RefDB: A database of uniformly referenced protein chemical shifts[END_REF]. The plot is divided in 3 lines. b) Output of the CSI 3.0 server in a residue specific-way (blue cartoon stands for α-helix, red arrow stands for β-strand and black line stands for random coil). The plot is divided in 3 lines. The identification of the helices of 14-3-3σ (αA-αI) is shown above the output of the CSI 3.0 server.

  226 1 H N and 15 N resonances (78%), 200 out of 231 13 C α resonances (87%), 187 out of 219 13 C β resonances (85%) and 176 out of 231 13 CO resonances (77%). The assignment covers homogeneously the protein's sequence (figure 1 b). The assigned chemical shifts were deposited into the Biological Magnetic Resonance Database (http://www.bmrb.wisc.edu/) under the BMRB accession number 27563.

Acknowledgments

We thank MSc. Eline Sijbesma and Dr. Christian Ottmann from the Eindhoven University of Technology for kindly providing us the plasmids of both 14-3-3σ and 14-3-3σΔC17. We also thank Dr. Elian Dupré from Lille University for his help with the automatic assignments software.

The research is supported by funding from the European Union through the TASPPI project (H2020-MSCA-ITN-2015, grant number 675179) and by the LabEx (Laboratory of Excellence) DISTALZ (ANR, ANR-11-LABX-009). The NMR facilities were funded by the Nord Region Council, CNRS, Institut Pasteur de Lille, the European Community (ERDF), the French Ministry of Research and the University of Lille and by the CTRL CPER cofunded by

the European Union with the European Regional Development Fund (ERDF), by the Hauts de France Regional Council (contract n°17003781), Métropole Européenne de Lille (contract n°2016_ESR_05), and French State (contract n°2017-R3-CTRL-Phase 1). We acknowledge support for the NMR facilities from TGE RMN THC (CNRS, FR-3050) and FRABio (Univ.

Lille, CNRS, FR-3688).

Backbone chemical shift assignments of human 14-3-3σ

João Filipe Neves 1 , Isabelle Landrieu 1 *, Hamida Merzougui 1 , Emmanuelle Boll 1 , Xavier Hanoulle 1 , François-Xavier Cantrelle 1 1 UMR 8576 CNRS-Lille University, 59000 Lille, France

Conflict of interest

The authors declare that they have no conflict of interest.