Paul Hamberg
email: a.hamberg@erasmusmc.nl

Margaret M Woo

Lin-Chi Chen

Jaap Verweij

Maria Grazia Porro

Lily Zhao

Wenkui Li

Diane Biessen

Sunil Sharma

Thomas Hengelage

Diane Van Der Biessen

Maja De Jonge

Effect of ketoconazole-mediated CYP3A4 inhibition on clinical pharmacokinetics of

Keywords: Panobinostat, LBH589, histone deacetylase inhibitor, CYP3A

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Histone deacetylases (HDACs) are enzymes controlling the acetylation state of lysine protein residues, notably those contained in the N-terminal portions of the core histones, and are functionally counteracted by histone acetylases. The balance between these two enzyme groups controls cell cycle progression, cell survival and differentiation. In neoplastic cells, the abundance of deacetylated histones is usually associated with DNA hypermethylation and gene silencing [START_REF] Khan | Drug insight: histone deacetylase inhibitor-based therapies for cutaneous T-cell lymphomas[END_REF][START_REF] Witt | HDAC family: what are the cancer relevant targets?[END_REF]. As a new class of chemotherapeutic agents, HDAC-inhibitors (HDACi) have demonstrated potent anticancer activities in preclinical studies and are currently in various stages of clinical development. The molecular basis of the anticancer effects of HDACi goes beyond the inhibition of histone acetylation alone. Over the last several years, a growing list of non-histone targets important in the regulation of cell proliferation, cell death and cell migration have been identified, including p53, NF-κB, Ku70, α-tubulin and Hsp90. [START_REF] Bolden | Anticancer activities of histone deacetylase inhibitors[END_REF] HDACs are grouped based on their catalytic mechanism: Class 1 (HDACs 1,2,3 and 8), Class II a/b (HDACs 4,5,6,7,9 and 10), Class III (sirtuin enzymes) and Class IV (HDAC 11) [START_REF] Witt | HDAC family: what are the cancer relevant targets?[END_REF].

Inhibitors of HDAC Class I as well as Class II are arbitrarily called pan-deacetylase inhibitors (pan-DACi).

Panobinostat (LBH589) is a pan-DACi which belongs to a structurally novel cinnamic hydroxamic acid class of compounds [START_REF] Atadja | Development of the pan-DAC inhibitor panobinostat (LBH589): successes and challenges[END_REF].

It is one of the most potent class I/II pan-DACi in vitro and has shown anti-tumor activity in a wide variety of tumor xenograft models. Reports from phase I-II studies have demonstrated signs of clinical activity of panobinostat in cancer patients, e.g. in cutaneous T-cell lymphoma, Hodgkin's or non-Hodgkin's lymphoma, prostate cancer, acute myeloid leukemia, multiple myeloma, and breast cancer [START_REF] Ellis | Histone deacetylase inhibitor panobinostat induces clinical responses with associated alterations in gene expression profiles in cutaneous T-cell lymphoma[END_REF][START_REF] Ottman | Phase IA/II study of oral panobinostat (LBH589), a novel pan-deacetylase inhibitor demonstrating efficacy in patients with advanced hematologic malignancies[END_REF][START_REF] Clive | Elucidation of the metabolic and elimination pathways of panobinostat (LBH589) using [ 14 C]-panobinostat[END_REF]. Phase II studies are ongoing in patients with hematological malignancies and solid tumors. Thrombocytopenia and fatigue were the doselimiting toxicities in the phase I study after oral administration of panobinostat. Other major toxicities were nausea and vomiting.

Panobinostat is metabolized via multiple pathways including reduction, hydrolysis, oxidation, and glucuronidation. The oxidative metabolism of panobinostat is mediated mainly by human cytochrome P450 CYP3A4 (70-98%) with minor involvement of CYP2D6 and CYP2C19.

CYP3A is a predominant family of CYP enzymes, which makes it one of the most important drug-metabolizing enzymes. CYP3A is expressed primarily in liver and small intestine [START_REF] Shimada | Interindividual variations in human liver cytochrome P-450 enzymes involved in the oxidation of drugs, carcinogens and toxic chemicals: studies with liver microsomes of 30 Japanese and 30 Caucasians[END_REF].

Since hepatic CYP3A4 has been estimated to metabolize close to 50% of currently used drugs, as well as endogenous and exogenous steroids, it is evident that anticancer agents including panobinostat given in combination with drugs that are CYP3A4 inhibitors are likely to result in clinical toxicity. Indeed, based on in silico drug-drug interaction (DDI) modeling and simulation, ketoconazole, a strong CYP3A4 inhibitor, was predicted to increase panobinostat AUC by 1.3-2 fold. Thus, a decision was made to study panobinostat at the lowest efficacious dose at 20 mg/day to assess the clinical drug-drug interaction potential. The study was designed to determine the effect of a strong CYP3A-inhibtor on the systemic exposure of panobinostat, in order to assess whether or not co-administration, with a CYP3A inhibitor can be allowed in patients if medically necessary, during treatment with panobinostat.

The primary and secondary objectives of this study were to investigate the effect of multiple doses of ketoconazole on the pharmacokinetics of a single dose of panobinostat and safety, respectively.

Results

Interaction study panobinostat and ketoconazole Page 6/26

The interim analysis resulted in early stop of the study. Fourteen patients were enrolled, all of them evaluable for pharmacokinetic and safety assessment. Thirteen patients continued in the extension phase. One patient discontinued the study in the core phase due to mild increase in serum creatinine. Patient characteristics are shown in Table 1. All fourteen patients have discontinued the study (ten for disease progression, three for adverse events, one for ontreatment death unrelated to study drug due to myocardial infarction), The mean duration of exposure to panobinostat was 140 days (median [range]: 59 [START_REF] Shimada | Interindividual variations in human liver cytochrome P-450 enzymes involved in the oxidation of drugs, carcinogens and toxic chemicals: studies with liver microsomes of 30 Japanese and 30 Caucasians[END_REF]899] days).

Pharmacokinetic evaluation

Pharmacokinetic parameters are shown in table 2. No residual panobinostat was detectable prior to the day 8 dose (the start of the combination panobinostat plus ketoconazole). In the presence of ketoconazole, peak plasma concentration of panobinostat (C max ) increased from 18.52 to 29.98 ng/mL, and the area-under-the-curve 0-inf (AUC 0-inf ) rose from 133 to 220.7 ng.h/mL, generating ratios of 1.6 (90% CI 1.2-2.2) and 1.8 (90% CI 1.5-2.2), respectively. No substantial change in time to maximum concentration (T max ) or half-life was observed.

Individual geometric mean ratios of panobinostat C max and AUC are shown in figure 2. Of note, in a limited number of patients a higher than 2-fold increase in C max and AUC of panobinostat was observed in the presence of ketoconazole.

Pharmacogenetic evaluation

Genotype analysis of CYP3A4*1B, CYP3A5*2, *3, *6 and *7 was performed and available at baseline from all 14 patients. All patients in this study were Caucasians. Fourteen patients were homozygous for the wild-type CYP3A4*1A genotype. Eleven patients were homozygous for the CYP3A5*3 genotype and 3 patients were heterozygous for the CYP3A5*1/*3 genotype.

Interaction study panobinostat and ketoconazole Page 7/26

There is no apparent difference in panobinostat C max or AUC values between patients carrying CYP3A5*1/*3 and CYP3A5*3/*3 alleles.

Safety results

In order to determine whether the combination of panobinostat plus ketoconazole resulted in additional toxicity, the core-phase of the study was subdivided in 3 parts: period 1 from day 1 until the first dose of ketoconazole (on day 5); period 2 from first dose of ketoconazole until concomitant administration of panobinostat with ketoconazole (on day 8) and period 3 from the time-point of this combination until day 15.

Most patients (12/14 patients, 85.5%) had at least one adverse event during the core phase of the study. Overall, the most frequently reported AEs regardless of relationship to study drug were diarrhea (reported in 5 patients), anorexia (4) vomiting (4), nausea (3), hypophosphatemia (3), myalgia (3), and fatigue [START_REF] Witt | HDAC family: what are the cancer relevant targets?[END_REF]. The majority of the AEs were categorized as Grade 1 or Grade 2 (42.9% and 35.7% of all events, respectively).

No relevant difference was seen in ECG readings across the three periods in the core phase.

The highest incidence of QTc-prolongation ≥ 30 milliseconds (ms) as compared to baseline was observed in 5 patients (36%) in period 3 (concomitant panobinostat and ketoconazole), while this increase was seen in 4 patients (29%) and 3 patients (21%) in periods 1 (panobinostat only) and 2 (ketoconazole only), respectively. No increases in QTc ≥ 60 ms compared to baseline were observed. In six patients (43%) other transient ECG abnormalities were detected during period 3, while this was seen in 3 (21%) patients in either period 1 and 2 (table 3).

One possibly study drug related serious adverse event occurred during the core phase of the study. On day 7 after commencing study drug a patient developed a mild increase in his serum

Interaction study panobinostat and ketoconazole

Page 8/26 creatinine level from 1.09 mg/dL at baseline, to a maximum of 1.72 mg/dL. His concomitant medications were doxazosin, irbesartan and hydrochlorothiazide for pre-existent hypertension. His creatinine level returned to baseline after discontinuing the diuretic. Study treatment was permanently discontinued due to this adverse event. Overall, no differences were observed among periods 1, 2 and 3 of the core phase of the trial, Adverse events were reported in all 13 patients continuing in the extension phase. Similar safety profile was seen in the extension phase of the study, the most frequently reported AEs regardless of relationship to study drug were nausea (reported in 10 patients), hypophosphatemia [START_REF] Julious | Sample size of 12 per group rule of thumb for a pilot study[END_REF], diarrhea [START_REF] Shimada | Interindividual variations in human liver cytochrome P-450 enzymes involved in the oxidation of drugs, carcinogens and toxic chemicals: studies with liver microsomes of 30 Japanese and 30 Caucasians[END_REF], vomiting [START_REF] Shimada | Interindividual variations in human liver cytochrome P-450 enzymes involved in the oxidation of drugs, carcinogens and toxic chemicals: studies with liver microsomes of 30 Japanese and 30 Caucasians[END_REF], fatigue [START_REF] Clive | Elucidation of the metabolic and elimination pathways of panobinostat (LBH589) using [ 14 C]-panobinostat[END_REF], anorexia [START_REF] Clive | Elucidation of the metabolic and elimination pathways of panobinostat (LBH589) using [ 14 C]-panobinostat[END_REF], whereas thrombocytopenia was only noted in the extension phase in four patients, most likely due to longer treatment duration and corresponding higher exposure in the extension phase. The spectrum of adverse events reported in this trial was comparable to the toxicity pattern from previous trials with panobinostat [START_REF] Ellis | Histone deacetylase inhibitor panobinostat induces clinical responses with associated alterations in gene expression profiles in cutaneous T-cell lymphoma[END_REF][START_REF] Ottman | Phase IA/II study of oral panobinostat (LBH589), a novel pan-deacetylase inhibitor demonstrating efficacy in patients with advanced hematologic malignancies[END_REF][START_REF] Prince | Panobinostat (LBH589): a potent pan-deaceylase inhibitor with promising activity against hematologic and solid tumors[END_REF][START_REF] Dickinson | Preliminary evidence of the pan deacetylase inhibitor panobinostat (LBH589) in refractory Hodgkin Lymphoma[END_REF]. Asymptomatic hypophosphatemia of Grade ≥ 2 occurred in a total of nine out of fourteen patients (64%) during the trial. In two of these patients hypophosphatemia Grade 2 was present before start of treatment; two other patients showed it on Day 5 in the core phase, before starting ketoconazole. To further explore this finding, ad hoc urine analysis to measure fractional excretion of phosphate was performed in three out of the seven patients with treatment-emergent hypophosphatemia (6 samples) during the extension phase.

The fractional excretion of phosphate ranged from 2.7-19.4%, while the fractional excretion of phosphate exceeded 5% in all these patients in at least one sample (in total 4 out of 6 samples) [Table 4]. No baseline samples were available.

Antitumor activity:

Interaction study panobinostat and ketoconazole Page 9/26 Disease stabilization for at least 3 months was achieved in 37.5% of the patients with two patients (adenoid cystic carcinoma of head and neck, and salivary gland) staying on study treatment for 333 and 899 days, respectively.

Discussion

The prevalence of polypharmacy is increasing and is an important issue with regard to potential drug-drug interactions with untoward consequences. This study aimed to determine whether co-administration of a strong CYP3A inhibitor in patients treated with panobinostat, would result in a clinically relevant change in the exposure after a single dose of panobinostat.

Emerging results from this DDI study with ketoconazole and a radiolabeled mass balance study [START_REF] Clive | Elucidation of the metabolic and elimination pathways of panobinostat (LBH589) using [ 14 C]-panobinostat[END_REF] have shed light on potential drug-drug interaction risk with respect to panobinostat as a CYP3A substrate. Only a limited increase in C max and AUC of panobinostat was seen in most of the patients, with a ratio of 1.6 and 1.8 in the presence of ketoconazole, respectively, without a change in T max or half-life. In the two dose-escalation trials, these two pharmacokinetic parameters have shown to increase linearly with the dose between 20 mg and less than 60 mg. [START_REF] Ottman | Phase IA/II study of oral panobinostat (LBH589), a novel pan-deacetylase inhibitor demonstrating efficacy in patients with advanced hematologic malignancies[END_REF][START_REF] Prince | Phase I study of oral LBH589, a novel deacetylase (DAC) inhibitor in advanced solid tumors and non-hodgkin's lymphoma[END_REF] A comparison of the metabolites observed in vivo with those formed in human liver in vitro and by recombinant cytochrome P450 (CYP) enzymes indicated that the contribution of oxidative metabolism pathways mediated by CYP enzymes in patients was less than the contributions by other panobinostat metabolic pathways (e.g. reduction, hydrolysis, 1-and 2carbon chain shortening, and glucuronidation) (personal communications with Heidi Einolf, Novartis). Therefore, the impact of clinical drug-drug interactions via CYP enzymes was predicted to be minor. This was confirmed in the present study showing a less than 2-fold increase in panobinostat AUC following consecutive doses of ketoconazole at 400 mg. As the

Interaction study panobinostat and ketoconazole

Page 10/26 AUC inhibition /AUC ratio was 1.8, the fm CYP3A (fraction of panobinostat cleared by CYP3A) was estimated to be 0.4 using the following formula [AUC inhibition /AUC = 1/(1-fm CYP3A )] [START_REF] Shou | Modeling, prediction, and in vitro in vivo correlation of CYP3A4 induction[END_REF] indicating CYP pathways contributed < 50% to the overall metabolism of panobinostat.

In this study, the genotype status of CYP3A does not seem to be of relevance as there was no apparent difference in panobinostat levels in patients harbouring CYP3A5*1/*3 compared to those with CYP3A5*3/*3 alleles. However, no conclusions can be drawn with respect to CYP3A4 as only patients with homozygous wild-type CYP3A4*1A genotype were enrolled in this trial.

The impact of inter-individual (60%) and intra-individual (30%) variability (personal communication Margaret M Woo, Novartis), in the pharmacokinetics of panobinostat is larger than the effect of CYP3A inhibitors (table 2), rendering individual titration of the panobinostat dose, based on the observed toxicity during treatment, of far greater relevance than the concomitant use of a CYP3A inhibitor. Close monitoring of patients would be recommended whenever chronic co-administration of a potent CYP3A4 inhibitor is unavoidable during treatment with panobinostat.

It is worth noting that although the recommended phase II dose in patients with solid tumors is set at 20 mg MWF each week, the increase in C max and AUC of panobinostat in the presence of ketoconazole at this dose should be put in perspective as higher doses of 40mg and 60 mg are given MWF every week in patients with Hodgkin Lymphoma and acute myeloid leukaemia, respectively [START_REF] Ottman | Phase IA/II study of oral panobinostat (LBH589), a novel pan-deacetylase inhibitor demonstrating efficacy in patients with advanced hematologic malignancies[END_REF][START_REF] Dickinson | Preliminary evidence of the pan deacetylase inhibitor panobinostat (LBH589) in refractory Hodgkin Lymphoma[END_REF]. However, we should be cognizant that severe thrombocytopenia in patients with solid tumors is regarded as a DLT, whereas this is not necessarily the case in patients with acute myeloid leukaemia due to the underlying disease.

Thrombocytopenia was the most common grade 3/4 toxicity, occurring in 10 out of 13 patients during dose-escalation, in patients with Hodgkin's Lymphoma and was dose-limiting in four of them [START_REF] Dickinson | Preliminary evidence of the pan deacetylase inhibitor panobinostat (LBH589) in refractory Hodgkin Lymphoma[END_REF].

Clinical evaluation in the core-phase of this study was short (15 days), but no unexpected toxicity was observed due to a potential interaction between panobinostat and ketoconazole.

Thrombocytopenia was present in the extension phase, when repeated doses of panobinostat 20 mg MWF were administered, but overall the safety profile remained as expected.

QTc prolongation has drawn attention during a phase I study with continuous intravenous administration of panobinostat [START_REF] Dickinson | Preliminary evidence of the pan deacetylase inhibitor panobinostat (LBH589) in refractory Hodgkin Lymphoma[END_REF]; however, in our current study as well as in the other studies using oral panobinostat this does not seem to be a major issue [START_REF] Ellis | Histone deacetylase inhibitor panobinostat induces clinical responses with associated alterations in gene expression profiles in cutaneous T-cell lymphoma[END_REF][START_REF] Ottman | Phase IA/II study of oral panobinostat (LBH589), a novel pan-deacetylase inhibitor demonstrating efficacy in patients with advanced hematologic malignancies[END_REF][START_REF] Prince | Panobinostat (LBH589): a potent pan-deaceylase inhibitor with promising activity against hematologic and solid tumors[END_REF][START_REF] Dickinson | Preliminary evidence of the pan deacetylase inhibitor panobinostat (LBH589) in refractory Hodgkin Lymphoma[END_REF]. At least at 14 time-points during the 15 days of the core phase of this study, ECGs were procured in order to obtain more data on a potential QTc prolongation. The mean increase at any moment during the core phase compared to baseline was below 5%. Also during the extension phase, the only observed QTc abnormalities were increases in QTc larger than 30 and less or equal to 60 ms .

The occurrence of hypophosphatemia during treatment with HDAC inhibitors has been previously described [START_REF] Mann | FDA approval summary: vorinostat for treatment of advanced primary cutaneous T-cel lymphoma[END_REF][START_REF] Subramanian | Clinical toxicities of Histone Deacetylase Inhibitors[END_REF][START_REF] Gore | A Phase I and Pharmocokinetic Study of the Oral Histone Deacetylase Inhibitor, MS-275, in Patients with Refractory Solid Tumors and Lymphomas[END_REF], and a recent review of toxicities reported with HDAC inhibitors mentions asymptomatic hypophosphatemia as one of the common AEs reported in single-agent vorinostat (5%), romidepsin (8%), and entinostat (29%) [START_REF] Subramanian | Clinical toxicities of Histone Deacetylase Inhibitors[END_REF].

As mild asymptomatic hypophosphatemia was also noted in this study, and in order to evaluate renal phosphate wasting, ad hoc urine analysis was performed during the extension phase and the findings were suggestive of renal phosphate wasting in all three tested patients as defined by a fractional excretion of phosphate > 5%. Due to lack of predefined baseline and serial assessments of phosphate in urine, firm conclusions can not be drawn from this unplanned and explorative analysis which should be regarded as hypothesis generating. Given the occurrence of hypophosphatemia was observed during panobinostat treatment before the initiation as well as after discontinuing of ketoconazole treatment, it seems unlikely that hypophosphatemia is caused by co-administration of a CYP3A-inhibitor. At present the exact mechanism of this side effect is not known and should deserve further studies focusing on potential tubular toxicity as a result of DAC inhibition, as hypophosphatemia might be a class-effect of DAC-inhibitors.

In our study patients were monitored for electrolyte serum levels and received electrolyte supplements as soon as an electrolyte deficit was noted. Similar management of asymptomatic hypophosphatemia was successfully reported in a study with MS-275, a benzamide derivative, in patients with solid tumors and lymphoma [START_REF] Gore | A Phase I and Pharmocokinetic Study of the Oral Histone Deacetylase Inhibitor, MS-275, in Patients with Refractory Solid Tumors and Lymphomas[END_REF].

As drug-drug interactions become more and more important in the era of personalised medicine, the extent of potential interactions should be explored in the early phase of drug development. In this study, in which each patient served as their own control, a weak interaction was found between the pan-DAC-inhibitor panobinostat and the CYP3A inhibitor ketoconazole. Because the interaction is weak it may not be clinically relevant. In our opinion, the observed toxicity is not attributable specifically due to concurrent CYP3A-inhibtion because a similar safety pattern was reported for the drug interaction evaluation phase and as a single agent in the extension phases of the study. Most patients reported gastrointestinal related events as well as metabolism and nutrition disorders, and most adverse events were of either grade 1 or 2. Thrombocytopenia was observed in 4 patients and neutropenia in 2 patients, when panobinostat was administered for a prolonged period of time, but not when administered with ketoconazole. However, the interpretation of the results should be taken with caution due to the variable treatment in each individual and observation duration in the extension phase..

Conclusion

Concomitant treatment with a strong CYP3A inhibitor increased C max and AUC of panobinostat by 1.6 and 1.8 fold, respectively. Close monitoring of patients would be recommended whenever chronic co-administration of a strong CYP3A4 inhibitor is unavoidable during treatment with panobinostat. Asymptomatic hypophosphatemia has been observed and might be a result of renal phosphate wasting; whether this is a HDACi class effect should be further explored.

Patients and Methods

Patient selection

Patient selection was performed at the phase I outpatient department. Patients with histologically or cytologically confirmed advanced or metastatic solid tumors for whom no standard therapy was available and with an Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 were eligible. Since clinical responses were observed in Phase I/II trials with panobinostat, the selection of this phase I patients population with no other therapeutic options was considered appropriate for this study where a prolonged treatment with a potentially active drug was offered to these patients, although based on tumor characteristics no potential benefit was anticipated beforehand. The serum potassium, total corrected calcium, magnesium and phosphorus had to be above the lower limit of normal.

QTc had to be ≤ 450 msec. Concomitant treatment with CYP3A inhibitors and medications which increase the risk of Torsades de Point was not allowed. Significant cardiac disease or other concurrent uncontrolled medical conditions were exclusionary criteria. The study was designed and conducted under the appropriate institutional review boards' approvals and in accordance with the principles embodied in the Declaration of Helsinki. Written informed consent was obtained from each participant. This study is registered as NCT00503451 (ClinicalTrials.gov).

Study design

Patients received single agent panobinostat at 20 mg orally after at least 2 hours fasting on day 1 single agent ketoconazole once daily at 400 mg on days 5-9 (Figure 1). On day 8 panobinostat was administered 1 hour after ketoconazole. If at interim analysis after evaluation of 12 patients any of the four parameters (AUC 0-24 , AUC 0-tlast , AUC 0-∞ , and C max ) had the paired-t statistic (calculated based on log-transformed data) falling outside of (-1.98405, 1.98405) then the recruitment was to be stopped. If not, the trial was to proceed until 16 patients completed the study. The study design also stipulated that, while completer status for 12 patients was being confirmed, enrollment was not halted as long as there were no safety concerns.

All patients enrolled before this interim analysis, were to proceed to the extension phase regardless of the interim result. From day 15 onward, the extension phase of the study started in which patients received single agent 20 mg oral panobinostat each Monday, Wednesday and Friday (MWF) until disease progression, patient withdrawal of consent or unacceptable toxicity. Toxicity was evaluated using the Common Terminology Criteria for Adverse Events (CTCAE) v 3.0.

Safety assessments

Close monitoring by evaluating signs and symptoms, physical examination, laboratory assessments and ECGs was performed during the trial. Patients had scheduled visits on days

Interaction study panobinostat and ketoconazole

Page 15/26 1, 2, 3, 5, 8, 9, 10 and 15. Laboratory assessments were scheduled at day 1, 5, 8, 9, 10 and 15.

Twelve-lead ECGs were scheduled: pre-dose day 1 and at 1.5, 3, 6, 24 and 48h post-dose; pre-ketoconazole dose on day 5 and at 1.5 and 3h post-dose; pre-ketoconazole on day 8 and 1.5, 3, 6, 24 and 48h post-dose panobinostat; and predose on day 15. All ECGs were taken at least in triplicate separated by 2-5 minutes and were reviewed independently at a central laboratory (eRT, Bridgewater, NJ). During the extension phase, single ECGs were performed weekly as were laboratory assessments. Bazett's formula was used to correct QT intervals for heart rate.

Pharmacokinetic evaluation

During cycle 1 days 1 and 8, patients had serial blood sampling collections for pharmacokinetic evaluation of panobinostat. Blood samples (3 ml/sample) were collected in tubes containing sodium heparin prior to panobinostat dose, and 0.5, 1, 1.5, 2, 2.5, 3, 4, 6, 8, 24, and 48 hours post-dose. Samples were centrifuged at 800 g for 15 minutes at 4 C to yield plasma within 60 minutes after blood collection and stored below -60 C until analyses.

Blood samples (3 mL) for ketoconazole concentration determinations were collected pre-dose and 2 h post dose of ketoconazole on days 5, 8 and 9 (Figure 1). Panobinostat is analyzed in human plasma samples using a specific LC-MS/MS method with the lower limit of quantification (LLOQ) of 0.5 ng/mL and a dynamic range of 0.5 to 500 ng/mL of the analyte using 0.1 mL sample volume. A semi-automated protein precipitation extraction of human plasma samples was used to separate panobinostat from human plasma protein. The obtained sample extracts were evaporated to dryness, reconstituted with 10% aqueous acetonitile (containing 0.2% formic acid) and analyzed by LC-MS/MS on a Sciex API3000 or a Sciex API4000 tandem mass spectrometer. A Waters Xbridge C8 column (2.5 µm particle size, 50 x 2.1 mm) was employed in reversed-phase chromatography with gradient elution using 10% acetonitrile in water (containing 0.1% formic acid and 0.1% acetic acid, mobile phase A) and 90% acetonitrile in water (containing 0.25% formic acid and 0.25% acetic acid, mobile phase B) at a flow rate of 0.3 mL:/min. The assay was linear from 0.5 to 500 ng/mL with the bias (%) and CV(%) values of the QC sample results ranging from -0.7 to 0.7% and 2.3 to 11.6%, respectively, based on the within-study validation during the sample analysis.

For determination of ketoconazole in human plasma, a 100 μL aliquot of each sample (Standards, QC samples, study samples, blanks, etc) was pipetted into the appropriate well in a 96-deep well plate. A 100 μL aliquot of the internal standard working solution (250 ng/mL in 50% acetonitrile, v/v) was added to all wells except for the blanks, to which a 100 μL aliquot of 50% acetonitrile was added. A 400 μL aliquot of acetonitrile was added to each well, the plate covered, vortexed for approximately 10 min, and centrifuged at approximately 4000 rpm for approximately 10 minutes at room temperature. Using a TomTec device, a 50 μL aliquot of sample extract was transferred to a clean 96 deep-well plate, followed by an addition of 450 μL of 30% acetonitrile in water (v/v). After a brief vortex mixing, a 10 μL aliquot was injected onto the HPLC system connected to a Sciex API5000 mass spectrometer.

The assay was linear from 50.0 to 15000 ng/mL with the bias (%) and CV(%) values of the QC sample results ranging from -4.2 to 3.3% and 6.0 to 8.6%, respectively, based on the within-study validation during the sample analysis.

Panobinostat plasma concentration-time curve following each dose from individual patient was used to calculate pharmacokinetic parameters which included C max , T max , and AUC 0-∞ . PK parameters were obtained using non-compartmental methods implemented in WinNonlin Pro software (Version 5.01, Pharsight Corporation, Mountain View, CA, USA). C max and T max were obtained by visual inspection of the concentration-time curve. AUC 0-∞ was calculated using the linear up/log down method up to the last measured concentrations and the additional area estimated from that concentration and the value of apparent terminal half-life estimated for the targeted administration. The apparent terminal half-life was estimated using the bestfit variables of a single exponential to the log-linear portion of the plasma concentration-time curve using non-weighted linear regression. Ratio were calculated using the following formula: [PK-parameter] panobinostat +ketoconazole /[PKparameter] panobinostat ).

Pharmacogenetic evaluation

Blood samples (1.5mL in EDTA-containing tubes) were obtained at baseline for genotyping analysis of CYP3A4*1B, CYP3A5*2, *3, *6 and *7 and performed by using AmpliChip CYP450 at Epidauros, Bernried, Germany. DNA preparation and sequencing followed the method described by Sanger et al. [START_REF] Subramanian | Clinical toxicities of Histone Deacetylase Inhibitors[END_REF], using commercial kits provided by Applied Biosystems. Wild type allele CYP3A4*1A is defined as the exclusion of allele CYP3A4*1B.

Wild type allele CYP3A5*1 is defined as the exclusion of alleles *2, *3, *6 and *7.

Sample size calculation and statistical considerations

Pharmacokinetic evaluations to determine the influence of the potent CYP3A inhibitor ketoconazole on the pharmacokinetics of panobinostat were conducted during cycle 1 days 1-14. A sample size of 16 would have provided enough power (82.4%) to detect a minimum increase of 30% (ketoconazole plus panobinostat vs panobinostat alone) with an alpha level of 0.1. Since strong interaction between ketoconazole and panobinostat is possible, a ratio higher than 1.3 (ketoconazole plus panobinostat vs panobinostat alone) might be observed. Thus, a drug-drug interaction (DDI) could become apparent with fewer patients, and stopping if this occurs avoids unnecessary exposure of patients to study medications. An interim analysis of PK data was carried out when 75% of anticipated patients (12 completed patients out of 16 planned completers) were enrolled and had completed the core phase.

The number of 12 patients was chosen with two considerations. A sample size of 12 is generally accepted to evaluate safety [START_REF] Julious | Sample size of 12 per group rule of thumb for a pilot study[END_REF], and secondly it provided reasonable power for stopping at interim. The O'Brien-Fleming method was used to calculate the boundaries at interim and final analysis [START_REF] O'brien | A multiple testing procedure for clinical trials[END_REF].

A linear mixed effects model was fit to the log-transformed PK parameters (C max and AUC 0-24 , AUC 0-t , and AUC 0-inf ). Included in the model was treatment (ketoconazole + panobinostat or panobinostat alone) as fixed effects, and patient as a random effect. For the DDI analysis, the combination treatment (ketoconazole + panobinostat) was the test and panobinostat alone the reference. Proper contrasts were used to estimate the treatment difference (i.e., ketoconazole + panobinostat vs. panobinostat alone). The point estimate of the treatment difference and the corresponding 90% confidence intervals were calculated and back-transformed to obtain the point estimate and CI on the linear scale for the ratio of geometric means of the test as compared with the reference. The median, minimum and maximum of T max difference between test and reference were also presented.

Table 4

 4 Fractional excretion of phosphate in patients reporting hypophosphatemia in the extension phase based on ad hoc urine analysis

	Gender, age, tumor	Fractional excretion of	Fractional excretion of
	diagnosis	phosphate in % (study	phosphate in % (study
		day)	day)
	Subj nr 00010 Female, 55 y, invasive	2.7 (day 36)	8.9 (day 43)
	ductal carcinoma of the
	breast
	Subj nr 00005 Female, 50 y, rectal	5.3 (day 54)	3.2 (day 61)
	adenocarcinoma
	Subj nr 00007 Male, 71 y, cystic	19.4 (day 70)	19.0 (day 77)
	adenoid head and neck
	carcinoma
	Fractional excretion of phosphate:
	{[Phosphate]

urine * [Creatinine] plasma *100}/[Phosphate] plasma *[Creatinine] urine }

Acknowledgements

The assistance of Judith ter Steeg is kindly acknowledged.

Statement of conflict of interest

Margaret M. Woo, Maria Grazia Porro, Lily Zhao, Wenkui Li and Thomas Hengelage are employees of Novartis. None of the other authors have a conflict of interest.

Interaction study panobinostat and ketoconazole

In all rows, the number of patients is given (the percentage of patients between brackets;

N=14 in all periods)

# Period 1 from day 1 until the first dose of ketoconazole; period 2 from first dose of ketoconazole until concomitant administration of panobinostat with ketoconazole (on day 8) and period 3 from the time-point of this combination until day 15.


J Wu
email: jie.wu@univ-lorraine.fr

THE AVERAGE EXPONENT OF ELLIPTIC CURVES MODULO p

Keywords: Mathematics Subject Classification. 11G05, 11R45 Elliptic curves over global fields, density theorems

published or not. The documents may come

The average exponent of elliptic curves modulo

pDOUBLON DE HAL-01094035

Introduction

Let E be an elliptic curve defined over Q. For a prime p of good reduction for E the reduction of E modulo p is an elliptic curve E p defined over the finite field F p with p elements. Denote by E p (F p ) the group of F p -rational points of E p . Its structure as a group, for example, the existence of large cyclic subgroups, especially of prime order, is of interest because of applications to elliptic curve cryptography [START_REF] Koblitz | Elliptic curve cryptosystems[END_REF][START_REF] Miller | Use of elliptic curves in cryptography[END_REF]. It is well known that the finite abelian group E p (F p ) has structure (1.1) E p (F p ) ≃ (Z/d p Z) ⊕ (Z/e p Z)

for uniquely determined positive integers d p and e p with d p | e p . Here e p is the size of the maximal cyclic subgroup of E p (F p ), called the exponent of E p (F p ). The study about e p as a function of p has received considerable attention [START_REF] Schoof | The exponent of the group of points on the reductions of an elliptic curve[END_REF][START_REF] Duke | Almost all reductions modulo p of an elliptic curve have a large exponent[END_REF][START_REF] Cojocaru | Cyclicity of CM elliptic curves modulo p[END_REF][START_REF] Cojocaru | Cyclicity of elliptic curves modulo p and elliptic curve analogue of Linnik's problem[END_REF], where the following problems were considered:

• lower bounds for the maximal values of e p ,

• the frequency of e p taking its maximal value, i.e., the density of the primes p for which E p (F p ) is a cyclic group, • the smallest prime p for which the group E p (F p ) is cyclic (elliptic curve analogue of Linnik's problem). Very recently motivated by a question of Silverman, Freiberg and Kurlberg [START_REF] Freiberg | On the average exponent of elliptic curves modulo p[END_REF] investigated the average order of e p . Before stating their results, let us fixe some notation. Given a positive integer k, let E[k] denote the group of k-torsion points of E (called the k-division group of E) and let L k := Q(E[k]) be the field obtained by adjoining to Q the coordinates of the points of

E[k] (called the k-division field of E). Write (1.2) n L k := [L k : Q].
Denote for all x 2, where

(1.4) C E := ∞ k=1 1 n L k dm=k µ(d) m = p 1 - ∞ ν=1 p -1 p ν n L p ν .
The implied constant depends on E at most. When E has complex multiplication (CM), they [4, Theorem 1.2] also proved that (1.3) holds unconditionally with a weaker error term

(1.5) O E x log 3 x log 2 x ,
where log ℓ denotes the ℓ-fold iterated logarithm.

The aim of this short note is to propose more precise result than (1.3) and (1.5).

Theorem 1.1. Let E be an elliptic curve over Q.

(a) Assuming GRH for the Dedekind zeta function ζ L k for all positive integers k, we have

(1.6) 1 π(x) p x e p = 1 2 C E x + O E x 5/6 (log x) 4/3 .
(b) If E has CM, then we have unconditionally

(1.7) 1 π(x) p x e p = 1 2 C E x + O E x (log x) 1/14 .
Here C E is given as in (1.4) and the implied constants depend on E at most.

Remark. (a) Our proof of Theorem 1.1 is a refinement of Freiberg and Kurlberg's method [START_REF] Freiberg | On the average exponent of elliptic curves modulo p[END_REF] with some simplification.

(b) For comparison of (1.3) and (1.6), we have 9 10 = 0.9 and 5 6 = 0.833 • • • . (c) The quality of (1.7) can be compared with the following result of Kurlberg and Pomerance [6, Theorem 1.2] concernng the multiplicative order of a number modulo p : Given a rational number g = 0, ±1 and prime p not dividing the numerator of g, let ℓ g (p) denote the multiplicative order of g modulo p. Assuming GRH for ζ Q(g 1/k ,e 2πi/k ) (s) for all positive integers k, one has

1 π(x) p x ℓ g (p) = 1 2 C g x + O x (log x) 1/2-1/ log 3 x ,
where C g is a positive constant depending on g.

Preliminary

Let E be an elliptic curve over Q with conductor N E and let k 1 be an integer. For x 1, define

π E (x; k) := p x p∤N E , k|dp 1.
The evaluation of this function will play a key role in the proof of Theorem 1.1. Using the Hasse inequality (see (

p x p splits completely in L k 1 = π E (x; k) + O(log(N E x)).
In order to evaluate the sum on the left-hand side, we need effective versions of the Chebotarev density theorem. They were first derived by Lagarias and Odlyzko [START_REF] Lagarias | Effective versions of the Chebotarev Density Theorem[END_REF], refined by Serre [START_REF] Serre | Quelques applications du théorème de densité de Chebotarev[END_REF], and subsequently improved by M. Murty, V. Murty and Saradha [START_REF] Murty | Modular forms and the Chebotarev density theorem[END_REF]. With the help of these results, one can deduce the following lemma (cf. [4, Lemma 3.3]).

Lemma 2.1. Let E be an elliptic curve over Q with conductor N E .

(a) Assuming GRH for the Dedekind zeta function ζ L k (s), we have

(2.1) π E (x; k) = Li(x) n L k + O x 1/2 log(N E x)
uniformly for x 2 and k 1, where the implied constant is absolute.

(b) There exist two absolute constants B > 0 and C > 0 such that

(2.2) π E (x; k) = Li(x) n L k + O xe -B(log x) 5/14
unformly for x 2 and CN 2 E k 14 log x, where the implied constant is absolute. The next lemma (cf. [4, Proposition 3.2] or [2, Propositions 3.5 and 3.6]) gathers some properties of the division fields L k of E and estimates for n L k , which will be useful later. Denote by ϕ(k) the Euler function.

Lemma 2.2. (a)

The field L k contains Q(e 2πi/k ). Therefore ϕ(k) | n L k and a rational prime p which splits completely in In order to evaluate the last sum, we first notice that the Hasse inequality (3.1) implies d p 2

L k satisfies p ≡ 1(mod k). (b) n L k divides |GL 2 (Z/kZ)| = k 3 ϕ(k) p|k (1 -p -2 ). (c) If E is a non-CM curve, then there exists a constant B E 1 (depending only on E) such that |GL 2 (Z/kZ)| B E n L k for each k 1. Moreover, we have |GL 2 (Z/kZ)| = n L k whenover (k, M E ) = 1 (where M E is Serre's constant ). (d) If E has CM, then ϕ(k) 2 ≪ n L k k 2 .

Proof of

√ p. Thus we can use the formula

1 k = dm|k µ(d) m to write (3.3) p x p∤N E p d p = p x p∤N E p dm|dp µ(d) m = k 2 √ x dm=k µ(d) m p x p∤N E , k|dp p.

Let y 2

√ x be a parameter to be choosen later and define

S 1 := k y dm=k µ(d) m p x p∤N E , k|dp p, S 2 := y<k 2 √ x dm=k µ(d) m p x p∤N E , k|dp p.
With the help of Lemma 2.1(a), a simple partial integration allows us to deduce (under GRH) (3.4)

p x p∤N E , k|dp p = x 2- t dπ E (t; k) = xπ E (x; k) - x 2 π E (t; k) dt = xLi(x) n L k - 1 n L k x 2 Li(t) dt + O E x 3/2 log x = Li(x 2 ) n L k + O E x 3/2 log x .
On the other hand, by Lemma 2.2 we infer that

(3.5) k y 1 n L k dm=k µ(d) m = C E + O(y -1 ).
Thus combining (3.4) with (3.5) and using the following trivial inequality (

S 1 = Li(x 2 ) k y 1 n L k dm=k µ(d) m + O E x 3/2 log x k y dm=k µ(d) m = C E Li(x 2 ) + O E x 2 y log x + x 3/2 y log x . 3.6) dm=k µ(d) m ϕ(k) k 1, we find (3.7)
Next we treat S 2 . By [4, Lemma 3.1 and Proposition 3.2(a)], we see that

k | d p implies that k 2 | (p + 1 -a E (p)) and also k | (p -1), hence k | (a E (p) -2).
With the aid of this and the Brun-Titchmarsh inequality, we can deduce that

S 2 ≪ x y<k 2 √ x |a| 2 √ x,a =2 a≡2(modk) p x,a E (p)=a k 2 |p+1-a 1 + p x,a E (p)=2 k 2 |p-1 1 ≪ x y<k 2 √ x √ x k • x kϕ(k) log(8x/k 2 ) + x k 2 .
By virtue of the elementary estimate

n t 1 ϕ(k) = D log t + O(1) (t 1)
with some positive constant D, a simple integration by parts leads to where we have used the fact that the term x 2 y -1 (log x) -1 can be absorded by x 5/2 y -2 (log(8x/y 2 )) -1 since y 2 √ x. Now the asymptotic formula (1.6) follows from (3.2) and (3.9) with the choice of y = x 1/3 (log x) -2/3 . The proof of (1.7) is very similar to that of (1.6). Next we shall only point out some important differences.

(
Similar to (3.4), we can apply Lemma 2.1(b) to prove (unconditionally)

p x p∤N E , k|dp p = Li(x 2 ) n L k + O E x 2 exp{-B(log x) 5/14 } for k (C -1 N -2 E log x)
S 1 = C E Li(x 2 ) + O E x 2 y -1 (log x) -1 + x 2 ye -B(log x) 5/14
for y (C -1 N -2 E log x) 1/14 . The treatment of S 2 is different. First we divide the sum over k in S 2 into two parts accroding to y < k x 1/4 (log x) 3/4 or x 1/4 (log x) 3/4 < k 2 √ x.

When E has CM, we have (see [3, page 692])

p x p∤N E , k|dp 1 ≪
x ϕ(k) 2 log x for k x 1/4 (log x) 3/4 . Thus the contribution from y < k x 1/4 (log x) 3/4 to S 2 is ≪

x 2 log x y<k x 1/4 (log x) = C E Li(x 2 ) + O E x 2 y log x + x 2 (log x) 5/2 + x 2 ye -B(log x) 5/14 for y (C -1 N -2 E log x) 1/14 . Now the asymptotic formula (1.7) follows from (3.2) and (3.12) with the choice of y = (C -1 N -2 E log x) 1/14 .

  by µ(n) the Möbius function, by π(x) the prime-counting function and by ζ L k (s) the Dedekind zeta function associated with L k , respectively. Assuming the Generalized Riemann Hypothesis (GRH) for ζ L k (s) for all positive integers k,

	Freiberg and Kurlberg [4, Theorem 1.1] shew that
	(1.3)	1 π(x) p x	e p =	1 2	C E x + O E x 9/10 (log x) 11/5

  3.1) below), it is not difficult to check that p ∤ d p for p ∤ N E . Thus the conditions p ∤ N E and k | d p are equivalent to p ∤ kN E and k | d p , that is p ∤ kN E and E p (F p ) contains a subgroup isomorphic to Z/kZ × Z/kZ. Hence by [9, Lemma 1], we have

  Theorem 1.1 Let a E (p) := p + 1 -|E p (F p )|, then e p = (p + 1 -a E (p))/d p if p ∤ N E ,

			0					otherwise.
	By using Hasse's inequality
	(3.1) for all primes p ∤ N E , it is easy to see that |a E (p)| < 2	√ p
	(3.2)	p x	e p =	p x, p∤N E	p d p	+ O	x 3/2 log x	.

  Clearly the inequality (3.8) (taking y = x 1/4 (log x)3/4 ) implies that the contribution fromx 1/4 (log x) 3/4 < k 2 √ x to S 2 is

						3/4	1 ϕ(k) 2 ≪	x 2 y log x	•
			≪	x 1/4 (log x) 3/4 <k 2 √ x	p∤N E , k|dp p x	p ≪	x 2 (log x) 5/2 •
	By combining these two estimates, we obtain
	(3.11)			S 2 ≪	x 2 y log x	+	x 2 (log x) 5/2 •
	Inserting (3.10) and (3.11) into (3.3), we find
	(3.12)	p x, p∤N E	p d p


Elodie Moulin
email: elodie.moulin@leesu.enpc.fr

Live with floods in the Greater Paris. Flood risk integration in the landuse projects

Keywords: Floods, landuse, urban project, management

My thesis is the result of a European research conducted since 2009 about the implementation of flood resilience. This research is multidisciplinary; it involves hard sciences and human sciences researchers. This experience was a first step to develop my PhD subject. Indeed, the interaction between hard and human sciences is a key to a better understanding of how the issue of floods could be appropriate by non technical specialists.

This research has to take into account the special context of the landuse in flooding areas in the Greater Paris, between a strong regulation and the real estate pressure.

Behind the technical aspects of flood risk, the risk integration by the stakeholders is a relevant aspect of the landuse. We will search if the flood risk is considered as a main issue in the execution of an urban project and in this way observe the integration of the risk throughout the project.

INTRODUCTION

Despite the strong regulation concerning the building in flooding areas (Fig. 1), 80% of these areas are built and 828 100 inhabitants are exposed to floods in the Ile-de-France region (IAU, 2011), mainly because the buildings precede the regulation. The land use in flooding area is presented as a solution to solve the real estate pressure. For instance some of the industrial wastelands are along the river and are currently in redevelopment and residential buildings are planned. Moreover, in the Ile-de-France master plan1 it is provided to build 60,000 accommodations pro year, in order to obtain 1.5 millions additional accommodations on the horizon 2030, we can deduce that the phenomenon of landuse in flooding areas will increase. This research fit into the special greater Paris framework (very dense and a strong lack of accommodation) and we have to take into account the metropolitan context as a matter of fact. In fact, we may ask ourselves if there is a place for flood integration in the metropolitan context. The regulatory context is mostly based on risk prevention plan, whereas there are more flood management tools, like the action plans for floods (PAPI). However with the implementation of the Flood directive since 2010, the objective is to homogenize the flood management policy and to make clearer the flood risk prevention and protection policy.

How human sciences (urbanism, geography, social sciences) could deal with flood risk. And how answer to the question: how the integration of the flood risk enables the implementation of an urban project. This paper will first present the context of the development of the PhD project. It will then introduce the methods used to answer to the core question. At last, we will show how first results can be turned into research hypothesis and lead to a new research question. This European project is the first step of our research. This project began in 2009 and will end in December 2012. It involves seven countries Germany, United Kingdom, France, Greece, Cyprus, Netherlands and Spain. The purpose of the project is to find the "road to market" for flood resilient2 tools, as barriers, non-return valves, demountables gates. In order to meet this objective, the project integrates hard sciences and human sciences to develop a flood resilient system. It's structured in five work packages, the first one is the management of the project, the second concerns the test of the flood resilient tools, the third is designing a flood resilient system, the forth is developing technology implementation tools and the fifth includes all the work packages to find the improvement of the flood resilient tools integration. I'm especially involved in the fifth, the one which including mostly human sciences researchers, with the description of stakeholders interactions and the development of workshops integrating stakeholders in order to find the best scenarios of flood resilient systems.

Each country chose a city to develop its research. In France, we chose the Greater Paris and especially the Bièvre River, the urban project Seine Rive Gauche in Paris and the "Orly-Rungis Seine Amont" area (in the west of the Val de Marne County). As human sciences researcher in the framework of the project, we have to describe the flood risk management in France, to conduct interviews with local authorities, associations, cities services and to organize workshops about flood resilience integration. Along the project hard sciences and human sciences researchers have to be in contact to describe as well what is a flood resilient system including hydraulic data and stakeholders interactions. For instance, the workshops are conducted in three parts and focused on the Bièvre River basin; the territorial diagnosis, including flood protection and prevention measures and tools; strategies elaboration, that is to say the elaboration of combination of structural and individual solutions of flood protection and/or vulnerability reduction depending on the foreseen projects of urban development; and the assessment and rating of the scenarios. These workshops are an example of the combination of hard sciences methodology with the territorial diagnosis and the use of models to develop the scenarios and at the same time human sciences methodology to organize into a hierarchy the scenarios and to integrate all stakeholders including inhabitants in the process. Through this project I find how I can investigate the landuse in flooding areas and what question has not been explored by researchers.

The resulting topic

The landuse in urban area has to be linked with the regulatory policy. Indeed, the risk prevention plans are the tools used by the central state services to control the urbanism in these vulnerable areas. These documents establish the constructible areas (with some instructions to take into account) and the non-constructible areas. The risk prevention plan is included in the city master plan, so the mayor has to take it into account when he issues planning permissions. The management of flood risk is constituted by others documents concerning prevention, management of the crisis, etc. The first question is: why, despite regulation, there is still building floodplains. To us, it's maybe in the application of the regulation by stakeholders, that we will find some of the flaw of the flood risk management. We want to study the amount of risk taken by those involved in planning.

At the beginning of my PhD project, we developed some questions, which appeared to us as key points in our reflection. We wanted to identify which step of the development of a building project is the key moment to integrate the flood risk issue, that is to say at which time the flood risk appeared in the debate around the development of the project. What is the good governance around the project, which enables the risk integration? Who are the stakeholders' driving this process? How is built the speech about the improvement of the resilience and the reduction of the vulnerability. Are these concepts openly mentioned or are they underlying? To answer to these questions we planned to take into account different scales of landuse project: the building, the plot, the district. This approach enables us to observe if the integration of the risk rises crescendo along the project or if it progressively disappears of the stakeholders' speeches and actions.

The progress of the SMARTeST project allows us to enrich this PhD project and to develop a method applied to human sciences and especially to urbanism and landuse.

MATERIAL & METHODS: THE URBAN PROJECT, HOW TO APPROACH THE INTEGRATION OF THE RISK

Here we have to define what we hear with the concept of urban project, the central point of this PhD research.

The flood context of the Greater Paris

In this article we will mostly focus on the Seine River, it is characterized by slow rise of the river; and as a matter of fact the river may take several days or even weeks to recover its initial level. Most of the rivers have the same hydraulic context as the Seine River; however some as the Bièvre River are characterized by flash floods. The table 1 below presents the eight biggest floods since 1910; we notice that any floods occur in recent years concerning the Seine River. The urban project: a way to understand the stakeholders' interactions

The urban project is born with the delegation of responsibilities from the global (central state) to the local (city) [START_REF] Pinson | Gouverner la ville par projet, Urbanisme et gouvernance des villes européennes[END_REF]. The central services used to proceed with a long term planning, whereas the elected representatives and town planners tend to act with uncertainties and pragmatic planning. The state urbanism was focus on central services (technical and administrative), whereas in the conduct of urban project the stakeholders networks involve public and private stakeholders, political and technical, expert and layman [START_REF] Pinson | Gouverner la ville par projet, Urbanisme et gouvernance des villes européennes[END_REF]. Most often an urban project involves working with a larger scale, e.g. the neighborhood.

in France and especially in the discussions about the Greater Paris development, a new concept appears: the project urbanism (l'urbanisme de projet). This concept means that the development of a city has to be thought through the urban project, which has to be free from the city and the regional (landuse) planning. This includes a special dispensation that is to say that the urban project doesn't have to take into account the city and the regional master plan.

In the case of flood risk awareness, this concept of project urbanism could be a good or a bad thing. Around the Greater Paris there are a lot of discussions and one of them it's: "We have to build in flooding areas because there are still free from buildings". So with the project urbanism, city planners could use the special dispensation included, to build in flooding areas without taking into account the risk in the development of a project. But on the contrary the concept could be a way to take the risk into account differently at a larger scale. Indeed, risk prevention plan deals with flood risk at the building scale, project urbanism could deals with flood risk at the district/neighbourhood scale.

To answer questions raised above, we have chosen the urban project approach and so we have chosen several urban projects and through its descriptions we will develop how the risk is taken into account. We will interview all stakeholders taking part in the building of an urban project: contracting owner (elected representatives…) project manager (architects, town planners…), property developer, social landlords, central state services etc.

Urban projects selected

We have chosen the Greater Paris as our area of study, in order to continue the work begun with the SMARTeST project. Here we are going to describe the urban projects we selected. This list will perhaps change during our project, functions of information which information we can access. Indeed, some city planners or cities do not always agree with the diffusion of some information about the development of the projects. So the access of information (e.g. archives, plan, meeting report…) will shape the list of the selected projects.

The main project we are studying is a project called the Ardoines Ardoines") located along the Seine River in Vitry-sur-Seine in the Val de Marne County (Fig. 2). The Ardoines are integrated in an operation of national interest (Opération d'Intérêt National, OIN), which included 12 cities. It used to be an industrial wasteland, and now this area is in redevelopment and a public development authority (Etablissement Public d'Aménagement, EPA) is in charge of this redevelopment. This area is interesting because the public development authority brought together several experts to elaborate some guidelines to know how build in this flooding area and how take into account vulnerability, resilience and the management of the crisis. An architect drew up a master plan taking into account the risk and the vulnerability. It's a planning with three steps: the first one could be flooded by a 20 years flood, so it is occupied by a park; the second floor could be flooded by a 50 years flood, here there are some accommodation and activities, but the ground floor of buildings has to be unoccupied; and then the third floor could be flooded by a 100 years flood, it's occupied by the most vulnerable activity e.g. transport infrastructures. This master plan (Fig. 2) has to be tested by hydraulic models. The second selected project is an eco-neighbourhood in the city of Ile saint Denis, in the Seine-Saint-Denis County. It's a farmer warehouse area and now the local authority has a redevelopment project in the city. The territory has got a lot of constraints because it's island, but they decided to give more space to water in the neighbourhood. Here the question of the crisis management is relevant, but at this time this question it's not evocated in the project discussions.

The third selected project is Paris Seine Rive Gauche, a neighbourhood located between the Austerlitz station and the National library. It's a flooded area. During the construction of the building they did not take into account the risk. But yet they weatherproof some basements, for instance to protect the National Library books and archives.

We have selected others projects, we considered them as secondary in our study and as element of comparison. We have chosen an eco-neighbourhood in Mantes-la Jolie, a project of a lake dwelling in Neuilly-sur-Marne and an urban project called Ivry-Confluence. There are all urban projects in flooding areas and are in the study phase. Our investigations concerning these projects are not very advanced in terms of interviews and analyses. After around twenty interviews, our approach evolved but the urban projects stayed the middle of our reflection.

RESULTS AND DISCUSSION

The flood risk isn't a major issue in an urban project Thanks to the interviews, the first result is that the risk is not taken into account by the stakeholders as a major issue in the discussions around the urban project. They apply the risk prevention plans but the flood risk is integrated as a constraint of the urban project. The elected representatives are afraid to communicate about the flood risk to the inhabitants. For instance, in the consultation about the development of the Ardoines project, there is little information about the risk concerning the area. Indeed, the elected representatives don't agree with the master plan, because a part of the area could be flooded by a 20 years flood and they don't accept that the city appears as a flooding city.

On the contrary, the river as the natural element is actually a rise in value element. The discussions around the Greater Paris promote the return of the city towards the river. A member of the Paris city hall says that high rise buildings have to be built along the Seine River, because more people could have a river view.

There is a gap between the landuse policy and the flood risk management policy. This gap could be reduced if the urban project is integrated in a territory and so is thought at a larger scale. Currently, the management of flood risk is limited to the application of the risk prevention plan.

Civil engineering: a way to integrate the flood risk

We observe that the non-integration of the flood risk issue is not because a technical problem; a lack of knowledge about the hazard or a lack of knowledge in the way to build taking into account the vulnerability and the resilience, but more because a non-integration by stakeholders. The risk prevention plan is not very efficient in dense area because this document concerns only new buildings. But despite twenty years of risk prevention policy in the urban project only the risk prevention plan is applied and comes down to civil engineering; e.g. stilts, ground reshaping, build out of the high water mark, etc. It's a failure in the policy of flood risk management regarding all the documents concerning prevention and management of the crisis. The stakeholders taking part in urban project take the risk into account only through constructive measures and don't care the management of the crisis.

CONCLUSIONS

Because of the real estate pressure, the concept of densification through the sustainable development and the fact that the river appears as an element of an urban project promotion, the risk is not considered as a major aspect of an urban project. Currently, we observe that the management of the flood risk is dissected in accordance with the sustainable development and the concept of densification. Indeed, the flood risk management is summarized as constructive measures, without thinking in terms of prevention and crisis management. But in case of flood, who will be responsible for the damages?

Nowadays the human sciences try to "do science" to the side of hard sciences. They appropriate technical subject related to environmental sciences as the flood risk. But they chose others entry points as the urban project others subject as the interactions of the stakeholders. To the scientific community the human scientists' methodology isn't real sciences and could be compared with the work of the journalists. Through projects bringing together human and hard scientists, the understanding and the exchange of way of "do sciences" may be rewarding for each.

Figure 1 :

 1 Figure 1 : Flooding areas in Greater Paris. Source: IAU Ile-de-France

Fig. 2 :

 2 Fig. 2: The Ardoines master plan. Source: SEURA, 2009.

Table 1 : The eight biggest flood events since 1910. Source: Les Dossiers de La Seine en Partage, "Le risque d'inondation en Ile-de-France", 2005. Crisis point SEINE RIVER : THE EIGHT BIGGEST FLOODS SINCE 1910

 1

in the draft version of

12thWWW YES, Arcueil, France -20-25 Mai 2012

Here we use the most common definition of the resilience: the ability of a system/community/society/defence to react to and recover from the damaging effect of realised hazards. www.floodsite.net

12thWWW YES, Arcueil, France -20-25 Mai 2012


B Trouette
email: benoit.trouette@limsi.fr

C Delcarte

G Labrosse
email: gerard.labrosse@u-psud.fr

About the ellipticity of the discrete Laplacian in polar coordinate with Neumann condition

Keywords: polar laplacian, spectral collocation method, ellipticity, mapping

The Chebyshev Gauss-Radau discrete version of the polar-diffusion operator,

) , k being the azimuthal wave number, presents complex conjugate eigenvalues when it is associated with Neumann boundary condition imposed at r = 1. It is shown that this ellipticity violation of the original continuous problem is genuine and not due to some round-off error.

A way to avoid these complex conjugate eigenvalues is proposed, at the expense of some loss of accuracy. An evaluation is performed of the impact this approach has on the spectral accuracy of the solution.

Introduction

Temporal discretization of the balance equations which govern the flow dynamics often leads to elliptic Helmholtz equations. When these multidimensional elliptic problems are separable, and posed in orthogonal geometries, their numerical solution can be efficiently obtained by using the Successive Diagonalization Technique (SDT), ( [START_REF] Haldenwang | Chebyshev 3D Spectral and 2D Pseudospectral Solvers for the Helmholtz Equation[END_REF][START_REF] Lynch | Direct Solution of Partial Differential Equations by Tensor Product Methods[END_REF]). This method amounts to work in the Helmholtz operator numerical eigenspace, itself being tensiorally constructed from the eigenspaces of the second derivatives (see [START_REF] Haldenwang | Chebyshev 3D Spectral and 2D Pseudospectral Solvers for the Helmholtz Equation[END_REF]). Making use of a spectral collocation method leads then to cheap accurate numerical solutions of a large variety of diffusion problems, with any kind of boundary conditions, Dirichlet, Neumann or Robin, with constant coefficients. This approach has been extended in [START_REF] Nguyen | A spectral collocation method to solve Helmoltz problems with boundary conditions involving mixed tangential and normal derivatives[END_REF] to the case where the boundary conditions involve mixed tangential and normal derivatives.

The key-point which makes this SDT attractive, for easy to implement, is the ellipticity of the original continuous problem, provided this ellipticity be preserved at the discrete step. This occurs in most of the cases, for instance in the Cartesian configuration, viz. with ∂ 2 ∂z 2 completed with any boundary condition imposed at z = ±1. In cylindrical coordinates, the easy implementing of the SDT depends therefore upon the ellipticity of the companion operator of ∂ 2 ∂z 2 , namely the polar operator

( 1 r ∂ ∂r ( r ∂ ∂r ) -k 2 r 2
) , k being the azimuthal wave number. The polar ellipticity is preserved in its Chebyshev Gauss-Radau discrete version when a Dirichlet boundary condition is imposed at r = 1, say. When a Neumann condition is imposed at r = 1, a few couples of complex conjugated eigenvalues show up for given values of k when the radial cut-off frequency N gets larger than 12. This paper will show that the presence of these complex eigenvalues, which violates the ellipticity of the original polar problem, is genuine and not due to some round-off error amplification associated itself with the bad conditioning of the Gauss-Radau matrix. A way to avoid these complex conjugated eigenvalues, in implementing the SDT, is proposed, at the expense of some loss of accuracy.

An evaluation is made of the impact this approach has on the spectral accuracy of the solution.

Continuous problems

The inhomogeneous problem

Let us consider the diffusion equation in polar coordinates (e r ,e φ ),

( 1 r ∂ ∂r ( r ∂ ∂r ) + 1 r 2 ∂ 2 ∂φ 2 ) u = f (r, φ) with r ∈]0, 1[ and φ ∈ [0, 2π[. ( 1
)
where u(r, φ) and f (r, φ) are real fields. To be quite general a Robin boundary condition is

imposed at r = 1, viz. a u(r = 1, φ) + b ∂u ∂r r=1 = c(φ) with φ ∈ [0, 2π[, (2)
where the real coefficients a and b are assumed to satisfy the ellipticity sufficient condition, a b > 0. Dirichlet or Neumann conditions respectively correspond to fixing (b = 0, a = 1) or (a = 0, b = 1) in [START_REF] Lynch | Direct Solution of Partial Differential Equations by Tensor Product Methods[END_REF]. All the fields are necessarily periodic in the azimuthal direction. They can therefore be expanded in Fourier series, with

     u(r, φ) f (r, φ) c(φ)      = ∞ ∑ |k|=0      ūk (r) fk (r) ck      e ikφ , integer k. (3)
Since the left-hand-side fields are real their k = 0 Fourier components are complex conju-

gated, •k (r) = • * -k (r), • = u, f, c
, for k = 0. Thus, the unknown fields of the expansion (3) are those which correspond to k ≥ 0 for example. Plugging (3) into ( 1) and ( 2) leads to the following set of mono-dimensional problems,

( 1 r ∂ ∂r ( r ∂ ∂r ) - k 2 r 2 ) ūk = fk (r) with r ∈]0, 1[ and for k = 0, • • • , ∞, (4)
each completed with a boundary condition, viz.

a ūk (r = 1) + b dū k dr r=1 = ck for k = 0, • • • , ∞. (5)

The associated homogeneous problems

The associated homogeneous problems read

( 1 r d dr ( r d dr ) - k 2 r 2 ) u k = -λ 2 u k ; r ∈]0, 1] , k = 0, • • • , ∞, (6)
completed with the homogeneous boundary conditions

a u k (r = 1) + b du k dr r=1 = 0 for k = 0, • • • , ∞. (7)
The analytical solutions to ( 6)-( 7) express in terms of the Bessel functions of the first kind,

u k (r) = J k (λ r),
where λ is anyone of the roots, in infinite number, of the following equations,

for k = 0 : a J 0 (λ) = b λ J 1 (λ), for k ≥ 1 : a J k (λ) + b 2 λ (J k-1 (λ) -J k+1 (λ)) = 0.
All the λ's are real.

Discretized problems

The inhomogeneous problems

Let N be the radial cut-off frequency and r p , with p = 0, • • • , N , be the radial location of the Chebyshev Gauss-Radau collocation points, where

r p = 1 2 ( 1 -cos ( (2p + 1)π 2N + 1 )) , p = 0, • • • , N. (8)
Chosing the Gauss-Radau nodes allows us to avoid the r = 0 singular position which occurs in the operator (4), the closest node to r = 0 being at

r 0 = 1 2 ( 1 -cos ( π 2N +1
)) ≃

( π 2(2N +1) ) 2 .
The discrete version of ( 4) and ( 5) is obtained by introducing the polynomial approximation of the ūk (r)'s,

ū(N) k (r) = N ∑ p=0 (ū k ) p l (N ) p (r) with (ū k ) p ≡ ū(N) k (r p ) , the l (N )
p (r)'s being the Lagrange polynomials based over the Chebyshev Gauss-Radau nodes [START_REF] Chénier | Sensitivity of the liquid bridge hydrodynamics to local capillary contributions[END_REF]. Let D and D (2) be the respective Gauss-Radau matrix representations of d dr and

( 1 r ∂ ∂r ( r ∂ ∂r )
)

, the superscript "(2)" indicating that D (2) is not the square of D. The discretized version of ( 4) and ( 5) reads

N ∑ q=0 D (2) pq (ū k ) q - k 2 r 2 p (ū k ) p = ( fk ) p , p = 0, • • • , N -1 , for k = 0, • • • , ∞, (9)
and

a (ū k ) N + b N ∑ q=0 D N q (ū k ) q = ck for k = 0, • • • , ∞, (10)
where ( fk ) p ≡ fk (r p ). The matrix system (9) is rectangular. It is made square upon eliminating the (ū k ) N 's through [START_REF] Chénier | Thermocapillary flows and vorticity singularity, Interfacial Fluid Dynamics and Transport Process[END_REF], with

(ū k ) N = ck -b ∑ N -1 q=0 D N q (ū k ) q a + b D N N for k = 0, • • • , ∞.
The resulting discrete system reads

N -1 ∑ q=0 ( D (2) R ) pq (ū k ) q - k 2 r 2 p (ū k ) p = ( fk ) p - ck D (2) pN a + b D N N , p = 0, • • • , N -1, (11)
where

( D (2) R ) pq = D (2) pq - b D (2) pN D N q a + b D N N . ( 12
)

The homogeneous problems

Implementing the SDT is then based on the eigenvalues, and eigenvectors, of the matrices

( D (2) R -k 2 r 2 )
which come from the l.h.s. of [START_REF] Leriche | High-order direct Stokes solvers with or without temporal splitting : numerical investigations of their comparative properties[END_REF], wherein k 2 r 2 stands for the diagonal matrix of entries k 2 r 2 p , with p = 0, • • • , N -1. Let us therefore introduce the discrete eigenvalue problem,

N -1 ∑ q=0 ( D (2) R ) pq ( Ūk ) q - k 2 r 2 p ( Ūk ) p = ξ ( Ūk ) p , p = 0, • • • , N -1 for k = 0, • • • , ∞.
It leads to N numerical eigenvalues, ξ n , for n = 1, • • • , N , ordered with increasing absolute value of their real parts.

The numerical eigenvalues of the polar-diffusion problem

In the Neumann case

For the Cartesian operator, d 2 dx 2 + b.c., the numerical eigenvalues can be expressed analytically, ( [START_REF] Vandeven | On the eigenvalues of second-order spectral differentiation operators[END_REF]), whereas, so far, those of

( D (2) R -k 2 r 2 )
can only be obtained by numerical experiments. These latter were performed, for the sake of this analysis, for N ≤ 199 and k ≤ 499. In the Dirichlet case, viz. fixing a = 1 and b = 0 in ( 7) and ( 12), the ξ n 's were found as being all real and negative, ξ n = -λ 2 n with real λ n 's, for any N and k values. The situation is fairly different in the Neumann case, a = 0 and b = 1 in ( 7) and (12).

( D (2) R -k 2 r 2 )
has been made. The result is given in Fig. 3 : there is one particular value of k for each value of N , for N ≥ 13, where complex eigenvalues occur in the Neumann spectrum. Should these complex eigenvalues be considered as spurious, and/or simply due to the bad conditioning of the matrix

( D (2) R -k 2 r 2
)

? The answer is supplied by two numerical tests, both performed with k = 39. As indicated by Fig. 3 complex eigenvalues occur

R -k 2 r 2 (2)
) .

with N = 19, 20, 21 for example. First, computing them for N = 20 while increasing the number of significant digits, as it is made possible by the Mathematica software, shows that the complex eigenvalues are not sensitive to round-off errors. They definitely converge to more and more accurately determined values, -11879.8 ± 17.3509 i for example. Second, computing the eigenvalues for N = 22, now, leads to a purely real spectrum, despite the fact that the matrix conditioning should be worst than with N = 20. It can thus be concluded that the complex eigenvalues are genuine eigenvalues of the discrete problem.

In the Robin case

When the Robin conditions (5) are adopted, and discretized using a Gauss-Radau grid with N +1 nodes, the order of magnitude of the Neumann contribution is b N 2 . It is therefore expected that complex eigenvalues will occur in the polar-diffusion problem if b N 2 ≫ a.

4. Is there a way to avoid the complex eigenvalues ?

The existence of these complex eigenvalues does not prevent from using the Successive Diagonalization Technique, but it makes it a bit more fastidious to code. Is there an easy way to suppress them ? It will be shown that a particular mapping brings a partial answer.

The mapping

In [START_REF] Kosloff | A modified Chebyshev pseudospectral method with an O(N -1 ) time step restriction[END_REF], D. Kosloff and H. Tal-Ezer proposed a mapping from the internal Gauss-Lobatto points to a new set of points in order to reduce, from an O (N -2 ) to an O (N -1 ) criterion, the limitation of the time step for solving the 1D hyperbolic equation. This is obtained by moving the Gauss-Lobatto nodes of abscissae y i to other locations of abscissae x i , according to the α-mapping relation

x i = g(y i ; α) = arcsin(α y i ) arcsin(α) ∈ [-1, 1] for i = 0, • • • , N with 0 ≤ α < 1.
For α much smaller than 1 the mapping just slightly moves the Gauss-Lobatto nodes, while values of α close to 1 lead to almost evenly distributed nodes over [-1, 1]. We have applied this transformation on the Gauss-Radau collocation points of abscissae 0 < r i ≤ 1, with i = 0, • • • , N , for solving the radial diffusion equation [START_REF] Leriche | High-order direct Stokes solvers with or without temporal splitting : numerical investigations of their comparative properties[END_REF]. Any function f (r) is then transformed into h(x) whose first derivative is obtained from

dh dx = 1 g ′ (r; α) df dr with g ′ (r; α) = ∂g ∂r .
The radial discrete first derivative D is transformed as

D → A • D
where A is a diagonal matrix whose entries are given by

A ii = 1 g′(r i ; α) = α arcsin(α) √ 1 -(α r i ) 2 , i = 0, • • • , N.
In the same way, the matrix D 2 which represents d 2 dr 2 is modified according to

D 2 → A 2 • D 2 + B • D,
the diagonal matrix B being defined by

B ii = - g′′(r i ; α) [g′(r i ; α)] 3 , i = 0, • • • , N,
where

g′′(r i ; α) = α 3 arcsin(α) • r i [1 -(α r i ) 2 ] 3/2 .
( D (2) R -k 2 r 2 )
, with (α = 0.99) and without mapping.

Its impact on the polar-diffusion spectrum

Mapping and accuracy

Choosing the Gauss-Radau (or Gauss-Lobatto) nodes for discretizing the differential equations is not made by convenience. It is indeed well known ( [START_REF] Canuto | Spectral Methods in Fluid Springer Series in Computational Physics[END_REF]) that using these nodes is at the heart of the high level of numerical accuracy the spectral methods are able to achieve. It is therefore expected that the mapping should degrade the overall accuracy of the Gauss-Radau scheme, more and more with increasing α.

(α) n | ≃ O ( N 2
) .

Comparison with an analytical solution of the polar-diffusion problem

Consider the equation

( 1 r ∂ ∂r ( r ∂ ∂r ) - k 2 r 2 ) u = f (r) , r ∈]0, 1[, (13)
where the source term is chosen as being f (r) = (N 2 -k 2 ) r N . The analytical solution is a polynomial, u(r) = r N , which will coincide (within the zero machine accuracy) with the numerical solution provided by the Gauss-Radau Chebyshev solver of (13). This solution of (13) can be considered with the Neumann boundary condition du dr r=1 = N . It is then very easy with this analytical solution to compute the error coming from the α-mapping solver, wheresoever are located the associated nodes. Let u (N ) (r p ), for p = 0, • • • , N , be the nodal values obtained from this latter solver, and

E = max p=0,••• ,N -1 u (N ) (rp)-r N p r N p
be the relative error in absolute norm between the analytical and numerical solutions. Figure 7 shows the error E as a function of the mapping parameter α for various N values. For small α values or large N the error is very small.

Comparison on a physical configuration

We now consider the axi-symmetric floating-zone problem in microgravity environment which is described in detail in [START_REF] Chénier | Stability of the axisymmetric buoyant-capillary flows in a laterally heated liquid bridge[END_REF] and [START_REF] Chénier | Sensitivity of the liquid bridge hydrodynamics to local capillary contributions[END_REF]. A cylindrical liquid bridge, of height 2H and radius R, is maintained by surface tension between two horizontal isothermal solid disks. Its lateral free surface is submitted to a steady and uniform heat flux. This triggers and maintains a thermo-capillary flow : the surface tension variations, due to the thermal gradients on the free surface, generate tangential stresses and then viscous driving of the liquid. The flow is generally studied in a two-parameter space : the Prandtl number (P r), ratio between momentum and thermal diffusivities, and the Marangoni number (M a), ratio between thermocapillary and thermal diffusion velocities. The axi-symmetric velocity and temperature fields of the flow were deeply studied on a large domain of the parameters space in [START_REF] Chénier | Stability of the axisymmetric buoyant-capillary flows in a laterally heated liquid bridge[END_REF] and [START_REF] Chénier | Sensitivity of the liquid bridge hydrodynamics to local capillary contributions[END_REF]. The impact of the numerical treatment of the vorticity singularity which occurs at the junction of the free surface with the rigid disks was explored in [START_REF] Kasperski | On the numerical treatment of viscous singularity in wall-confined thermocapillary convection[END_REF] and [START_REF] Chénier | Thermocapillary flows and vorticity singularity, Interfacial Fluid Dynamics and Transport Process[END_REF].

The axi-symmetric physical problem is governed by the following non-dimensional Navier-Stokes and energy equations, in the Boussinesq approximation framework,

∂u ∂t + (u.∇) u = -∇p + P r ( ∇ 2 - e r r 2 ) u, (14)
∂θ ∂t + (u.∇) θ = ∇ 2 θ, (15)
∇.u = 0,

where u, θ and p respectively are the non-dimensionnal velocity, temperature and pressure.

The unit vectors e r and e z define respectively the radial and axial directions, their origin being located at the center of the liquid bridge. The operators are defined as follows: To complete the set ( 14)-( 16), boundary conditions are specified:

∇ = e
• z = ±1    u = 0 (no-slip conditions), θ = 0 (imposed temperature), • r = 1            v = 0 (non-deformable free surface), ∂w ∂r = -M a ∂θ ∂z f (z) (stress condition), ∂θ ∂r = q(z) (heat flux). (17)
with q(z) = (1 -z 2 ) 2 the heat flux. The parameter values are P r = 0.01 and M a = 106.

The function f (z) = (1 -z 2n ) 2 is introduced for regularizing the vorticity singularity, n being here fixed to 13 according to the results of [START_REF] Kasperski | On the numerical treatment of viscous singularity in wall-confined thermocapillary convection[END_REF] and [START_REF] Chénier | Thermocapillary flows and vorticity singularity, Interfacial Fluid Dynamics and Transport Process[END_REF].

The system ( 14) -( 17) is space-discretized with a Chebyshev collocation method based on radial Gauss-Radau and axial Gauss-Lobatto grids. Uncoupling the velocity and pressure fields is made with the Projection-Diffusion method, ( [START_REF] Leriche | High-order direct Stokes solvers with or without temporal splitting : numerical investigations of their comparative properties[END_REF]). The time integration is performed with an usual second order finite difference scheme. Let w ij be the set of the axial velocity nodal values obtained in this way, and w map ij the corresponding set obtained from a mapping applied on the radial grid, the α parameter being fixed to 0.99. obtained with 70 and 100 grid points in the radial and axial directions respectively. In this figure are superimposed the iso-w lines supplied by the mapped (α = 0.99) Gauss-Radau grid. As expected from the comment made from Fig. 5 the maximum of the error is located in the regions of steep vorticity gradient. Yet this relative error is small. The grid transformation can thus be used in order to avoid complex conjugate eigenmodes.

Conclusion

The Chebychev Gauss-Radau discrete version of the polar-diffusion operator, L =

Figure 1 :

 1 Figure 1: log 10 (|ξ n |) as a function of log 10 (n), for the Dirichlet and Neumann cases, obtained with various N 's and k = 4.

Figure 1

 1 Figure 1 displays the Dirichlet and Neumann numerical spectra obtained with various values of N , all with k = 4. The eigenvalues are all real and negative. They converge towards the analytical ones lying along the envelope which is common to all these plots. But taking now N = 128 and k = 11 for computing the Neumann spectrum leads to (N -2) real negative eigenvalues, plus a pair of conjugate complex eigenvalues, while the Dirichlet ξ n 's remain all real and negative. This is shown in Fig. 2 where log 10 (|Re(ξ n )|) and log 10 (|Im(ξ n )|) are plotted as functions of log 10 (n). Both (equal) imaginary parts are small, but definitely not compatible with the zero machine. A complete scanning of the k(N ) values which lead to complex eigenvalues of

Figure 2 :

 2 Figure 2: log 10 (|Re(ξ n )|) and log 10 (|Im(ξ n )|) as functions of log 10 (n), for the Dirichlet and Neumann cases, obtained with N = 128 and k = 11.

Figure 3 :

 3 Figure 3: The set of k(N ) values which lead to complex eigenvalues of ( D

Figure 4 :

 4 Figure 4: The set of k(N ) values leading to complex eigenvalues of

Figure 4

 4 Figure4shows the way the k(N ) configuration which leads to complex eigenvalue has been modified with the α = 0.99 mapping. Complex eigenvalues are still occurring, but for higher values of N , viz. N > 80, and also for much larger values of k.Thus, thanks to the mapping, spectra which were previously "polluted" by complex eigenvalues are now purely real. But does this mapping affect, and to what extent, the spectrum itself of the polar-diffusion operator ? Let us denote ξ (α) the set of the numerical eigenvalues obtained from an α-mapping and compare them to those, ξ, obtained without mapping. One of the Neumann cases presented in Fig.1is chosen, the one which corresponds to N = 128, k = 4. In Fig.5are plotted the relative differences ξ (α) n -ξn ξn obtained for several values of α. Two regions clearly show up. The first region is the part of the spectra where the numerical eigenvalues are in good agreement with the analytical ones, the common envelope in Fig.1. The mapping significantly affects the accuracy of the numerical eigenvalues, but at a level which is without any practical importance. The second region lies in the purely numerical part of Fig.1, wherein the eigenvalues strongly depart from the analytical ones. This is the region of Fig.5which exhibits a very steep increasing of the

Figure 5 :

 5 Figure 5: Relative differences ξ (α) n -ξn ξn , for the Neumann cases, obtained with various N 's and k = 4 with

Figure 7 :

 7 Figure 7: Maximum relative error E for different N as functions of α, for k = 4.

  r (∂/∂r) + e z (∂/∂z), ∇ 2 = (1/r)(∂/∂r)[(r(∂/∂r))] + ∂ 2 /∂z 2 . Noting u = v e r + w e z , one has also ∇.u = (1/r)[∂(rv)/∂r] + (∂w/∂z) and u.∇ = v(∂/∂r) + w(∂/∂z).

Figure 8 :

 8 Figure 8: Maximum relative error E on the axial velocity field w and iso-w lines for M a = 106 and P r = 0.01.

Figure 8

 8 Figure8shows the relative errorE = max i,j |w map ij -w ij | max i,j |w map ij |

  Neumann boundary condition at r = 1, does not preserve the ellipticity which is expected for L in the continuous realm. Numerical complex conjugate eigenvalues are indeed obtained for a set of values of both the azimuthal wave-number k and the

  |, for the Neumann cases, obtained with α = 0.9999999999. They suggest the asymptotic laws max n |ξ n | ≃ O ( N 4 ) and max n |ξ

	11
			without mapping α=0.9999999999 slope=3.87241
	10			slope=1.93921
	9
	8
	7
	6
	5
	4
	3
	0.6		0.8	1	1.2	1.4	1.6	1.8	2	2.2
						log 10 (N)
	Figure 6: Scaling laws in O	(	N β )	of the largest |ξ n | and |ξ	(α)

n