Nicole Cantisano

Philippe Menei

Vincent Roualdes

Romuald Seizeur

Didier Philippe Allain

Didier Le Gall

Arnaud Roy

Mickaël Dinomais

Jérémy Besnard

Jérémy Besnard

Jeremy Besnard
email: jeremy.besnard@univ-angers.fr

Patient-reported functional executive challenges and caregiver confirmation in adult brain tumor survivors

Keywords:

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

INTRODUCTION

Since 1970, there has been an increase in the prevalence of primary brain tumors (PBT) in developed countries [START_REF] Mcneill | Epidemiology of Brain Tumors[END_REF][START_REF] Petruzzi | Living with a brain tumor[END_REF]. In France, for example, epidemiological estimates show an incidence rate of 15.5/105 per 100,000 person-years [START_REF] Darlix | Epidemiology for primary brain tumors: a nationwide population-based study[END_REF]. Fortunately, due to significant advances in PBT treatment, there have been considerable improvements in survivorship.

Recent data showed that the overall 5-year survival rate following a diagnosis of a PBT or central nervous system tumor reached 32.1% between 2000 and 2010 in the United States [START_REF] Barnholtz-Sloan | Epidemiology of Brain Tumors[END_REF].

While these results are promising, they should not overshadow the fact that long-term adverse effects-referred to as 'late effect'-can occur in some brain tumor survivors [START_REF] Abu-Hegazy | Neurocognitive Effects of Primary Brain Tumors[END_REF][START_REF] Meyers | Multifaceted end points in brain tumor clinical trials: Cognitive deterioration precedes MRI progression[END_REF]. Several years after treatments end, the effects can persist or emerge-including medical, physical, cognitive and psychosocial sequelae. PBT survivors often experience late effects resulting from a combination of the effects of the tumor itself, and its associated treatments [START_REF] Gehrke | Neurocognitive deficits following primary brain tumor treatment: systematic review of a decade of comparative studies[END_REF]. Among these late effects, cognitive dysfunction is one of the most serious challenges these patients face [START_REF] Godefroy | Dysexecutive syndrome: Diagnostic criteria and validation study[END_REF][START_REF] Lezak | Neuropsychological Assessment[END_REF][START_REF] Ng | Effects of surgery on neurocognitive function in patients with glioma: a meta-analysis of immediate postoperative and long-term follow-up neurocognitive outcomes[END_REF][START_REF] Abrey | The impact of chemotherapy on cognitive outcomes in adults with primary brain tumors[END_REF]. Few studies are devoted to late cognitive effects in adult PBT survivors; these focus on a wide range of identified cognitive functions including attention, language, memory and executive function [e.g., [START_REF] Lezak | The Problem of Assessing Executive Functions[END_REF][START_REF] Gehrke | Neurocognitive deficits following primary brain tumor treatment: systematic review of a decade of comparative studies[END_REF].

One particularly important component of these cognitive domains is executive function (EF). Although there is not a consensus on the definition of these processes, Luria's seminal papers defined functions like planning, inhibition, shifting and action initiation as 'executive' [START_REF] Lezak | The Problem of Assessing Executive Functions[END_REF], which are significantly involved in everyday life. A more contemporary and widely accepted definition of EF characterizes these functions as high-level cognitive processes assumed to underlie behavioral adaptation and regulation in daily life [START_REF] Miyake | The Nature and Organization of Individual Differences in Executive Functions: Four General Conclusions[END_REF]. The main executive processes are inhibition, planning, initiation, rule detection, shifting, working memory and task coordination [e.g., [START_REF] Godefroy | Dysexecutive syndrome: Diagnostic criteria and validation study[END_REF][START_REF] Baddeley | The episodic buffer: a new component of working memory?[END_REF][START_REF] Lieberman | Social: Why Our Brains are Wired to Connect[END_REF]. Most daily life activities depend on the integrity of EF (e.g., shopping, scheduling activities, use of public transport, decision-making, work); therefore, executive difficulties can disrupt PBT patients' everyday lives and can negatively influence caregivers' lives [START_REF] Gregg | Neurobehavioural changes in patients following brain tumour: patients and relatives perspective[END_REF].

For these reasons, EF measurement is of particular interest for clinicians (e.g., neuropsychologists, occupational therapists, rehabilitation doctors). Conventional assessment of EF is based on performance-based tasks, such as the Stroop interference test (inhibition measure) [START_REF] Stroop | Studies of interference in serial verbal reactions[END_REF], the Tower of London (planning measure) [START_REF] Shallice | Specific impairments of planning[END_REF] or the Modified Card Sorting Test (rule detection and set shifting measure) [START_REF] Nelson | A Modified Card Sorting Test Sensitive to Frontal Lobe Defects[END_REF]. These tasks are administered by an examiner in highly standardized conditions, and measures usually involve response time, number of errors or patients' accuracy. Neuropsychological testing embodies the gold standard for assessing cognitive functions [START_REF] Buchanan | Self-report measures of executive function problems correlate with personality, not performance-based executive function measures, in nonclinical samples[END_REF]. However, such performance-based tests sometimes are not feasible in busy clinics, and are not always sensitive to executive difficulties encountered by patients in everyday life [START_REF] Eslinger | Severe disturbance of higher cognition after bilateral frontal lobe ablation: Patient EVR[END_REF][START_REF] Goldstein | Unilateral frontal lobectomy can produce strategy application disorder[END_REF]. One explanation for the discrepancy between EF tasks and daily life executive functioning is that objective EF tasks are very constrained and well-structured, and therefore the ecological validity of actual functioning in life can be reduced [START_REF] Salthouse | Interrelations of Aging, Knowledge, and Cognitive Performance[END_REF][START_REF] Burgess | Mesulam's frontal lobe mystery re-examined[END_REF]. Another hypothesis is that traditional tasks examine component by component, thereby separating integrated executive processes without testing the multidimensional functional aspect implied in these processes [START_REF] Burgess | The ecological validity of tests of executive function[END_REF]. It is also noteworthy that most EF performance-based tests were initially designed for assessment in non-patient populations [START_REF] Burgess | The case for the development and use of "ecologically valid" measures of executive function in experimental and clinical neuropsychology[END_REF].

To sum up, although these tasks are useful, they are insufficient to understand patients' executive difficulties in everyday life with adequate precision [START_REF] Roth | BRIEF-A: Behavior rating inventory of executive function-adult version: Professional Manual[END_REF].

An alternative process for operationalizing and measuring EF involves rating scales, which include self and/or informant reports about executive difficulties when actually carrying out daily life activities. EF rating scales were created for the purpose of providing an ecologically valid measure of competence in complex everyday situations that involve executive functioning. Several measures for rating EF have been developed, such as the Dysexecutive Questionnaire [START_REF] Wilson | Behavioural Assessment of the Dysexecutive Syndrome. Methodology of Frontal And Executive Function[END_REF], the Frontal Systems Behavior Scale [START_REF] Grace | Frontal systems behavior scale: Professional manual[END_REF], or the Behavioral Dysexecutive Syndrome Inventory [START_REF] Godefroy | Dysexecutive syndrome: Diagnostic criteria and validation study[END_REF]. As reported by Randolph and Chaytor [START_REF] Randolph | Promoting the Executive Functions: Core Foundations, Assessment Considerations, and Practical Applications[END_REF], the most commonly used rating scale in adult populations is the Behavior Rating Inventory of Executive Function (BRIEF-A) [START_REF] Roth | BRIEF-A: Behavior rating inventory of executive function-adult version: Professional Manual[END_REF]. Through performance-based tasks, these rating scales measure behaviors of underlying executive processes; however, a mismatch is frequently reported between formal neuropsychological testing and patient-reported measures. As pointed out by Toplak, West, and Stanovich [START_REF] Toplak | Practitioner Review: Do performance-based measures and ratings of executive function assess the same construct?[END_REF], the performance-based tasks evaluate different processes than the rating EF measures, and therefore cannot be used interchangeably. While the objective of performance-based tasks is to provide useful information about the efficiency of processing, rating scales of EF are mostly associated with the ability to achieve goals, and therefore are linked to daily life activities [START_REF] Barkley | The Nature of Executive Function (EF) Deficits in Daily Life Activities in Adults with ADHD and Their Relationship to Performance on EF Tests[END_REF][START_REF] Chevignard | Ecological assessment of cognitive functions in children with acquired brain injury: A systematic review[END_REF].

Few studies have examined EF in adult PBT survivors, and most of them used neuropsychological performance-based tasks [START_REF] Gehrke | Neurocognitive deficits following primary brain tumor treatment: systematic review of a decade of comparative studies[END_REF][START_REF] Ng | Effects of surgery on neurocognitive function in patients with glioma: a meta-analysis of immediate postoperative and long-term follow-up neurocognitive outcomes[END_REF]. Actually, data concerning the functional impact of EF challenges in this population are very scarce. This research gap is regrettable

given the efficient and useful information that rating scales can provide when planning longterm care. In a few rare occasions, some studies have demonstrated executive limitations in adult PBT survivors by implementing self-reports, demonstrating that these difficulties may negatively impact the resumption of activities, including work [START_REF] Collins | Cognitive Tasks Challenging Brain Tumor Survivors at Work[END_REF][START_REF] Loughan | Executive dysfunction in neuro-oncology: Behavior Rating Inventory of Executive Function in adult primary brain tumor patients[END_REF]. Interestingly, EF selfreports seem more predictive of work output than traditional performance-based measures (in breast cancer survivors; [START_REF] Calvio | Measures of Cognitive Function and Work in Occupationally Active Breast Cancer Survivors[END_REF]).

Contrary to cancer survivors who do not suffer from cerebral lesions, PBT survivors often exhibit cognitive difficulties and/or alterations in their awareness of deficits. It has been argued that this lack of awareness can affect the validity of self-report based assessments [START_REF] Molinari | Opportunities and challenges of incorporating clinical outcome assessments in brain tumor clinical trials[END_REF].

In order to test this assumption for the present study, the investigators sought informant reports from close relatives who are frequently engaged with PBT survivors, along with the self-reports completed by the PBT survivors themselves. The level of concordance between patients and caregivers (relatives) was presumed to provide evidence for the consistency of ratings and the presence of executive dysfunction in everyday life.

Since patients and their families are increasingly involved in personal health care decisions, it seems especially important for clinicians [START_REF] Mcneill | Epidemiology of Brain Tumors[END_REF] to know how adult PBT survivors perceive their difficulties and whether there is any inconsistency in caregivers' perception of such difficulties, and (2) to better comprehend the long-term impact of functional challenges in PBT survivors. The primary objective of this study was to evaluate the functional executive challenges reported by PBT survivors and caregivers using the BRIEF-A. To reach this objective, self and informant EF reports were compared and evaluated in a PBT patient group (survivors), as well as a control group without any history of brain tumor or cancer (Non-Cancer Comparison Group; NCCG). First, it was hypothesized that in contrast to non-cancer controls, adult PBT survivors would report functional executive disorders. Second, patients' caregivers would confirm survivors' self-reports by indicating more executive disorders in daily-life activities than family members of the NCCG (informant reports). Third, according to previous data, a high level of agreement between self and caregiver reports in the patient group was expected [START_REF] Ediebah | Impact of neurocognitive deficits on patient-proxy agreement regarding health-related quality of life in low-grade glioma patients[END_REF][START_REF] Jacobs | Concordance of patient and caregiver reports in evaluating quality of life in patients with malignant gliomas and an assessment of caregiver burden[END_REF]. It should be noted that there are only two very recent studies devoted to this topic in adult PBT patients. One of them examined executive self-assessment exclusively [START_REF] Loughan | Executive dysfunction in neuro-oncology: Behavior Rating Inventory of Executive Function in adult primary brain tumor patients[END_REF], while the other did not include the study of survivors [START_REF] Van Der Linden | Assessment of Executive Functioning in Patients with Meningioma and Low-Grade Glioma: A Comparison of Self-Report, Proxy-Report, and Test Performance[END_REF]. by NC and JB. Patients' illness information is presented on Table 1.

METHODS

Non-cancer controls (NCCG; n=40) were recruited via a research center affiliated to one of the university hospitals. Inclusion criteria were identical to that of PBT patients in order to enable group matching (matching criteria: age, gender and educational level).

Participants had to be aged between 20 and 59 years and French had to be their native language. In addition, based on participants' statements, they should not have had any history of severe psychiatric/neurological disease. Plausible participants were identified within the center's volunteer database and contacted by a nurse. If qualifying participants accepted, they were enrolled in the study.

INSERT TABLE 1 ABOUT HERE

After enrollment, within the hospital's premises and in a quiet room, patients and noncancer controls completed the self-report of the BRIEF-A and the brief neuropsychological evaluation. They also completed a questionnaire gathering socio-demographic data such as age, sex, educational level, professional status and laterality (Table 1). Before they returned to their homes, the informant version of the BRIEF-A was given to patients and to non-cancer controls, who were asked to transmit it to a proxy (e.g., family member, spouse). For the PBT patient group exclusively, the study's investigator called the caregiver a few days later to ask if he/she had any questions concerning the informant report. This also ensured that he/she had effectively completed the questionnaire. For each group, proxies had to mail back the questionnaire to the study's investigator after completion. No exclusion criteria were applied to the proxies.

Measures

Cognitive efficiency brief neuropsychological assessment

The level of global cognitive efficiency and understanding was checked with the Mini-Mental State Examination (MMSE; [START_REF] Folstein | A practical method for grading the cognitive state of patients for the children[END_REF]) for both the PBT patient group and the NCCG.

MMSE scores ranging between 25 and 30 (total range 0-30) are considered with normal limits.

Daily-life executive functioning

Self-report. To evaluate daily-life executive functioning, the self-rated version of the BRIEF-A was used ( [START_REF] Roth | BRIEF-A: Behavior rating inventory of executive function-adult version: Professional Manual[END_REF]; French version: [START_REF] Roy | Adaptation and validation in French of the Behavior Rating Inventory of Executive Function-Adult Version (BRIEF-A)[END_REF]). The BRIEF-A is the questionnaire most widely used to highlight executive behavior problems relating to the performance of daily-life activities [START_REF] Randolph | Promoting the Executive Functions: Core Foundations, Assessment Considerations, and Practical Applications[END_REF]. The self-reported questionnaire comprises 75 items that assess the frequency Furthermore, the BRIEF-A [START_REF] Roth | BRIEF-A: Behavior rating inventory of executive function-adult version: Professional Manual[END_REF][START_REF] Roy | Adaptation and validation in French of the Behavior Rating Inventory of Executive Function-Adult Version (BRIEF-A)[END_REF] assures the instrument's validity through three scales: Negativity (the extent to which the respondent answered items in an unusually negative manner), Unusual Answers (the extent to which the respondent endorsed items in an atypical manner) and Inconsistency (the extent to which the respondent erratically answered similar items). Concerning internal consistency, most BRIEF-A self-reported subscales presented satisfying Cronbach's alpha coefficientsexcept for Inhibition, Shift and Task-monitor scales which presented lower coefficients. All Cronbach's alpha coefficients are displayed on Table 2.

Informant-rated reports. As for the self-rated version, the informant report of the BRIEF-A comprises the same 75 items, which assess the frequency ('often,' 'sometimes,' or 'never') of different aspects of patients' executive functioning related problems within a 4week time frame. All BRIEF-A informant-reported validity scales Negativity, Unusual

Answers and Inconsistency presented acceptable scores. Internal consistency for most BRIEF-A informant reported subscales was likewise satisfying. The Inhibition and Self-monitor scales presented lower Cronbach's alpha coefficients (see Table 2).

Data analysis

Raw data were analyzed using STATISTICA Version 13.3 software (Tibco Software Inc, Palo Alto, CA, USA). First, one-tailed t-tests were conducted to compare each BRIEF-A self-report scale between the PTB patient group and the NCCG. Next, one-tailed t-tests were conducted to compare each BRIEF-A informant report scale between the PTB patient group and the NCCG. Finally, mean-split statistics were conducted within the PBT patient group to determine low and high scores, for both self-reports and informant-reports regarding the BRIEF-A. Pearson Chi Square tests were conducted to compare the frequency of low and high scores of these measures. The alpha level was retained at p < .05. For these explorative analyses, we did not correct for multiple testing.

RESULTS

Comparisons of self and informant reports of daily-life EF in patients and NCCG

In the PBT patient group, most caregivers were patients' spouses (81.08%) or a close family member living with the patient (18.92%). The informant reports for the NCCG were completed mainly by participants' spouses (69.23%) or a close family member (30.77%) living with the non-cancer control participant. Some informant report data was lost, especially in the NCCG (several participants' proxies did not return their questionnaires by mail). In the For the NCCG, the study's investigator did not call proxies; however, the caregivers for the PBT patient group were called by the investigator. Consequently, nine informant-reports were not mailed back for the NCCG (77.5% response rate) and three were not mailed back for the PBT patient group (92.5% response rate). Therefore, informant-report comparisons were conducted for 37 PBT patients and 31 non-cancer control participants for which all data sets were available.

The BRIEF-A self-reports displayed significant differences were found between the PBT patient group and the NCCG for the following subscales and indexes: Emotional Control, Working Memory, Task Monitor, Metacognition Index and Global Executive

Composite. For all these subscales and indexes, the PBT patient group obtained higher scores, indicating more difficulties than non-cancer participants. Intergroup comparisons for the Behavioral Regulation Index did approach statistical significance (see Table 2).

In regards to the informant version of the BRIEF-A, significant differences were found between the PBT patient group and the NCCG for the following subscales and indexes:

Emotional Control, Self-Monitor, Working Memory, Behavioral Regulation Index and Global Executive Composite. For all these subscales and indexes, the PBT patient group obtained higher scores than the non-cancer participants. The differences between the PBT patient group and the NCCG for the Metacognition Index and Plan/Organize did approach statistical significance (see Table 2).

As for the global cognitive/understanding efficiency, the PBT patient group's mean MMSE score was 27.91 (SD = 2.18) and the NCCG's was 28.58 (SD = 1.81). No significant difference was found between groups (t (69) = -1.41; p = .16).

INSERT TABLE 2 ABOUT HERE

Overall ratings for both survivors and caregivers (PBT patient group)

Pearson Chi-Square tests using a median split (Table 3) showed no significant difference between low and high BRIEF-A categories (n=40) and informant reports (n=37).

Low and high score frequencies for all BRIEF-A subscales and categories (self-reports and informant reports) are presented in Table 3.

INSERT TABLE 3 ABOUT HERE

DISCUSSION

The present study aimed to evaluate functional challenges associated with the cognitive late effects of PBT, focusing on executive difficulties. Compared to matched participants in the NCCG, PBT survivors suffer from a negative impact of these functional executive challenges when carrying out activities of daily living. Correspondingly, a high level of concordance was observed between self-and caregiver-ratings of EF difficulties.

Patient-rated measures showed several impairments in daily EF domains: the Global Executive Composite and the Metacognitive index are significantly impaired compared to non-cancer controls' self-reports. Emotional control, working memory and task monitor constitute the three domains endorsed by PBT survivors as significant difficulties. These results are in line with those of a recent study, that employed BRIEF-A self-reports which demonstrated that working memory and task monitoring processes were among the most frequently reported as impaired by PBT patients [START_REF] Loughan | Executive dysfunction in neuro-oncology: Behavior Rating Inventory of Executive Function in adult primary brain tumor patients[END_REF]. The present study also included informant reports, which enabled the comparison between patients' and caregivers' perceptions regarding the impact of EF difficulties in daily life. Informants' reports universally support the results of PBT survivors' self-reports. It was observed that the Global Executive Composite and the Behavioral Regulation Index differed significantly between PBT survivors and non-cancer control participants. Particularly in concordance with their relatives, PBT survivors show increased difficulties in emotional control, working memory and self-monitoring compared to non-cancer controls.

To reiterate, self-and task-monitoring, working memory and emotional control seem the most affected domains by cognitive late effects of PBT. Moreover, working memory and emotional control are acknowledged as impaired by both patients and their caregivers. These results are consistent with those of previous studies using performance-based tasks, which report that executive dysfunction-including working memory impairment-comprises one of the most devastating cognitive sequelae of PBT [START_REF] Gehrke | Neurocognitive deficits following primary brain tumor treatment: systematic review of a decade of comparative studies[END_REF]. The present results enhance these findings by demonstrating that executive disorders are likely to affect the ability to regulate and adapt behaviors in daily life. As to the difficulty to modulate and/or control emotional responses, it can be considered as new data in adult PBT survivors, and already observed in childhood PBT survivors [START_REF] Ellenberg | Neurocognitive status in long-term survivors of childhood CNS malignancies: A report from the Childhood Cancer Survivor Study[END_REF]. It is promising to note that no deficit was reported for several subscales of the BRIEF-A. According to patients (self-reports) and caregivers (informant reports), inhibition, shifting, initiation, planning and organization skills seem preserved in the study's sample when compared to non-cancer controls. This means that executive competences may be selectively altered in PBT survivors, and that the late effects of the disease can be circumscribed to particular domains. This encouraging result contrasts conclusions brought forth by most previous publications devoted to the study of EF in adult PBT survivors, in which only difficulties on performance-based tasks are reported [e.g.,

The third hypothesis was confirmed: comparisons between patients' self-reported measures and caregivers' reports showed a high level of concordance between both types of ratings, for all executive domains considered. This result fits well with that of a recent study published by Van der Linden et al. [START_REF] Van Der Linden | Assessment of Executive Functioning in Patients with Meningioma and Low-Grade Glioma: A Comparison of Self-Report, Proxy-Report, and Test Performance[END_REF], which highlights a moderate level of agreement between patients with PBT and their proxies with the BRIEF-A. This study appears complementary to that study, which did not include survivors (on average, patients were recruited three months after surgery). The authors conclude that the proxy reports seem to constitute an adequate estimation of patients' reports. The present study's observations are fully in line with this conclusion, indicating that the everyday life functional difficulties reported by survivors' close relatives reflect what the PBT survivors themselves report.

Regarding implications for cancer survivors, it is recommended that specialists should pay serious attention to PBT survivors' perception of their difficulties, and therefore guarantee optimal post-cancer care. This assertion fits well with recent models and interventions devoted to survivorship care, which underscore the need to focus on patients' experience. Certainly, in a biopsychosocial perspective, it is necessary to separate the concept of disease-which refers to pathological biological processes-from that of illness, which refers to the patient's experience and representation of the disease. As stated recently by Leeper and Milbury [46], care processes should integrate the concept of illness (i.e., patient's representation) in order to be appropriately designed. They propose framework for survivorship care in neuro-oncology that lists several patient-reported outcomes to be considered, including cognitive status, as is measured in the present study.

Several potential limitations must be acknowledged. The first is that the PBT patient group is heterogeneous in regards to etiology, treatment, and lesion location. Although it can be considered that this heterogeneity is representative of what is found in clinical practice, the small sample size did not allow a valid statistical examination of the potential influence of these variables. Although this limitation is not specific to our study [e.g., [START_REF] Loughan | Executive dysfunction in neuro-oncology: Behavior Rating Inventory of Executive Function in adult primary brain tumor patients[END_REF][START_REF] Van Der Linden | Assessment of Executive Functioning in Patients with Meningioma and Low-Grade Glioma: A Comparison of Self-Report, Proxy-Report, and Test Performance[END_REF], a more homogenous and larger sample would enable more refined statistical analyses. It should be noted, however, that some studies have shown that the effect of chemotherapy or radiotherapy on cognitive functioning did not differ in patients with PBT, and that the addition of chemotherapy to radiotherapy has no influence on these variables [START_REF] Reijneveld | Health-related quality of life in patients with high-risk low-grade glioma (EORTC 22033-26033): a randomised, open-label, phase 3 intergroup study[END_REF]. Additionally, although the prevalence of cognitive deficits can vary according to tumor location or tumor type, most studies find that patients with PBT frequently encounter executive difficulties [START_REF] Meyers | Multifaceted end points in brain tumor clinical trials: Cognitive deterioration precedes MRI progression[END_REF][START_REF] Richard | Cognitive rehabilitation for executive dysfunction in brain tumor patients: a pilot randomized controlled trial[END_REF][START_REF] Tringale | Identifying early diffusion imaging biomarkers of regional white matter injury as indicators of executive function decline following brain radiotherapy: A prospective clinical trial in primary brain tumor patients[END_REF]. The present study was predominantly interested in the long-term impact of functional executive difficulties in PBT survivors, regardless of etiology and lesion location. Moreover, this study is one of the few in the field to include a group of patients who had completed their treatment for at least two years.

Several studies have demonstrated that self-cognitive assessments could be impacted by psychological distress, such as anxiety and depression [START_REF] Nicol | Subjective cognitive functioning and associations with psychological distress in adult brain tumour survivors[END_REF]. Hence, another important limitation of this study is that psychological distress was not assessed; yet it is plausible that the high level of agreement between patients and informants' ratings helps to mitigate this flaw. Another variable that may modulate the level of agreement between patient and caregivers is patients' cognitive status. A low level of concordance was shown in patients with cognitive deficits [START_REF] Ediebah | Impact of neurocognitive deficits on patient-proxy agreement regarding health-related quality of life in low-grade glioma patients[END_REF]. In the present study, patients do not suffer from general cognitive deficits, as measured by the MMSE. These conclusions might have been different if patients had presented an alteration of cognitive efficiency, an assumption that needs further investigation. It should be noted however that Van der Linden et al. [START_REF] Van Der Linden | Assessment of Executive Functioning in Patients with Meningioma and Low-Grade Glioma: A Comparison of Self-Report, Proxy-Report, and Test Performance[END_REF] recently contradict the conclusions of Ediebah et al. [START_REF] Ediebah | Impact of neurocognitive deficits on patient-proxy agreement regarding health-related quality of life in low-grade glioma patients[END_REF] by reporting no relationship between performance-based tests and patient-proxy agreement.

Despite the aforementioned limitations, this study provides new evidence regarding the impact of EF difficulties on PBT patients' daily functioning, which can be elicited several years after treatment ends. The present findings justify larger-scale prospective studies, in

France and other countries, to confirm these preliminary results, and identify risk factors for EF challenges. This type of study should also be conducted with survivors of non-central nervous system cancers; while less severe than in patients with brain tumors, cognitive problems can exist and might affect significantly activities of daily life [START_REF] Calvio | Measures of Cognitive Function and Work in Occupationally Active Breast Cancer Survivors[END_REF]. The high level of agreement between patients' and informants' ratings also validates the interest to consider patients' perception of their functional executive challenges, suggesting that survivors' ratings reflect what is observed by caregivers. In addition, the present results provide preliminary information necessary for the conception of cognitive interventions aimed at EF improvement.

Participants

  This study received French regulatory ethical approval (Comité de Protection des Personnes Ouest II, n°2015/27, ID-RCB n°2015-A01192-47) which includes the International Review Board Authorization (N°NCT02693405). The Angers University Hospital was the study's promoter. Each participant signed an informed consent regarding their enrollment and future publishing of their anonymous data. Data were collected between June 2016 and December 2018. Patients (PBT patient group; n = 40) were recruited among neurosurgery departments in three university hospitals located in northwestern France. Eligibility regarding study criteria was verified by experienced physicians (PM, VR, RS and MD) before follow-up consultations. If they were eligible, patients were invited to participate in the study during these consultations, which included a neuropsychological assessment. Inclusion criteria were as follows: (1) patients had to be aged between 20 and 59 years to avoid the effect of normal aging on neurocognitive functioning; (2) they must have been treated (chemotherapy, radiation therapy or surgery) for a PBT. Patients with brain metastases of non-central nervous system tumors were not included. (3) As this study is devoted to functional late effects of PBT, patients should have completed their treatment at least two years prior to enrollment in the study. In addition, since the study's methodology (self-reports) requires a satisfactory level of understanding, (4) patients should not have presented major cognitive/understanding impairments, and (5) they had to be French native speakers. Medical data (tumor type and location, type of treatment, time since end of treatments) were collected and double-checked

(

  'often,' 'sometimes,' or 'never') of different aspects of patients' executive functioning related problems within a 4-week time frame. Based on its factorial structure, the BRIEF-A provides both global and specific subdomains indices. A Global Executive Composite (GEC) can be obtained and broken down into two index scores: (1) the Behavioral Regulation Index (BRI) and (2) the Metacognition Index (MCI). The BRI includes four clinical subscales. The Inhibit scale measures both inhibitory control and impulsivity -individuals' ability to resist impulses and to stop one's behavior at the appropriate time. The Shift scale assesses the capacity to move easily from one situation to another as required by circumstances. The Emotional Control scale assesses an individual's ability to modulate or control his or her emotional responses. The Self-Monitor scale assesses aspects of social or interpersonal awareness. The MCI consists of five clinical subscales. The Initiate scale reflects an individual's ability to begin a task/activity and to independently produce ideas, responses, or problem-solving strategies. The Working Memory scale measures the ability to hold information in mind to accomplish a task. The Plan/Organize scale measures an individual's ability to manage current and future-oriented task demands. The Task Monitor scale reflects the ability to keep track of one's problem-solving success or failure and to identify and correct mistakes during behaviors. Finally, the Organization of Materials scale measures orderliness of work, living, and storage spaces.

  At a clinical level, these findings support the usefulness of questionnaires such as BRIEF-A to orient survivorship care. While research devoted to PBT survivors is still at an early stage, rehabilitation of executive processes is feasible and well demonstrated in patients with acquired brain injury[START_REF] Tornås | Rehabilitation of Executive Functions in Patients with Chronic Acquired Brain Injury with Goal Management Training, External Cuing, and Emotional Regulation: A Randomized Controlled Trial[END_REF]. That being said, most cognitive interventions focus on the improvement of scores on performance-based tasks, hypothesizing that reducing cognitive impairments will reduce problems in everyday life. As recommended by the Society for Cognitive Rehabilitation[START_REF]What Is Cognitive Rehabilitation?[END_REF], cognitive interventions must focus primarily on functional competence in real life. In this vein, the current study's results contribute by demonstrating that questionnaires can provide relevant information about the negative impact of cognitive sequelae in patients' daily life, which in turn is necessary for orienting cognitive interventions. They can help to target specific EF deficits endorsed by both patients and caregivers. At a later stage, questionnaires can also provide helpful indicators regarding the expected efficacy of interventions, by demonstrating (or not) a decrease in the difficulties experienced by patients. By highlighting specific daily EF difficulties in adult survivors of PBT, the results from this study contribute to guiding future research devoted to cognitive interventions in this population, which should focus on the real-life efficacy of the rehabilitation.

Table 1 .

 1 Participants' socio-demographic data and clinical characteristics of PBT patients.

		PBT Patients	NCC
	Characteristic	(n= 40)	(n = 40)
		Mean (± SD) / Number (%)	Mean ± SD / Number (%)
	Age (Years)	41.20 (11.06)	40.88 (11.06)
	Gender
	Men	28 (70%)	25 (63%)
	Women	12 (30%)	15 (37%)
	Laterality
	Left handed	5 (12.5%)	2 (5%)
	Right handed	34 (85%)	36 (90%)
	Ambidextrous	1 (2.5%)	2 (5%)
	Bilingual	4 (10%)	8 (20%)
	Visual aid (glasses)	25 (62.5%)	27 (67.5%)
	Education (Years)*	11.83 (3.29)	12.38 (2.87)
	End of treatments (Years)	3.67 (2.31)
	Tumor Diagnosis
	Oligodendroglioma	11 (27.5%)	--
	Oligoastrocytoma	5 (12.5%)	--
	Astrocytoma	6 (15%)	--
	Glioblastoma	3 (7.5%)	--
	Glioma	5 (12.5%)	--
	Other tumors**	10 (25%)
	Treatments
	Chemotherapy	20 (50%)	--
	Radiotherapy	20 (50%)	--
	Tumor Resection	39 (97.5%)
	Note. *Number of years after 1 st grade (elementary school); **Craniopharyngioma,
	Meningioma, Medulloblastoma, Subependymoma, Neurocytoma, Germinoma.

Table 2 .

 2 Group comparisons for BRIEF-A self-reports and informant reports.

	PBT Patients Mean (SD)	NCC Mean (SD)	t	df	p

Table 3 .

 3 Comparisons between PBT patients' and their caregivers' executive function ratings. Low = under median; High = over median

		Self-Report	Informant Report
	BRIEF-A Subscales	n=40	n=37	Pearson Chi-square p
		N (%)	N (%)
	Inhibit			.01	.92
	Low	19 (47.50)	18 (48.65)
	High	21 (52.50)	19 (51.35)
	Shift			.02	.89
	Low	19 (47.50)	17 (45.95)
	High	21 (52.50)	20 (54.05)
	Emotional Control			.13	.72
	Low	20 (50)	17 (45.95)
	High	20 (50)	20 (54.05)
	Self Monitor			.35	.55
	Low	20 (50)	16 (43.24)
	High	20 (50)	21 (56.76)
	Initiate			.25	.62
	Low	14 (35)	15 (40.54)
	High	26 (65)	22 (59.46)
	Working Memory			.04	.91
	Low	20 (50)	18 (48.65)
	High	20 (50)	19 (51.35)
	Plan/Organize			.28	.60
	Low	16 (40)	17 (45.95)
	High	24 (60)	20 (54.05)
	Task Monitor			.78	.38
	Low	18 (45)	13 (35.14)
	High	22 (55)	24 (64.86)
	Organization of Materials			.16	.69
	Low	18 (45)	15 (40.54)
	High	22 (55)	22 (59.46)
	Behavioral Regulation Index (BRI)			.13	.72
	Low	20 (50)	17 (45.95)
	High	20 (50)	20 (54.05)
	Metacognition Index (MCI)			.01	.92
	Low	21 (52.21)	19 (51.35)
	High	19 (47.50)	18 (48.65)
	Global Executive Composite (GEC)			.35	.55
	Low	20 (50)	16 (43.24)
	High	20 (50)	21 (56.76)
	Note. Median split :

Acknowledgments

We wish to thank participants, families, and informants for their useful implication in this study.

We thank Hector Salcedo for his help in proofreading the current manuscript.

Ethics approval

The present study received French regulatory ethical approval (Comité de Protection des Personnes Ouest II, n°2015/27, ID-RCB n°2015-A01192-47) which includes the International Review Board Authorization (N°NCT02693405).

Fundings

This study was supported by the French National Cancer Institute (INCa) (grant number SHSESP14-14-041).

Consent to participate

Informed consent was obtained from all individual participants included in the study.

Consent to publish

Participants signed informed consent regarding publishing their data.

Conflicts of interest

The authors have no conflict of interest to declare.

Authors' contribution

JB, PA and DLG contributed to the study conception and design. PM, VR, RS and MD recruited participants. Material preparation, data collection and analysis were performed by NC, AR and JB. JB supervised the project. NC and JB wrote the first version of the manuscript. All authors commented on the manuscript.


M Fiacchini
email: mirko.fiacchini@gipsa-lab.fr

G Millérioux
email: gilles.millerioux@univ-lorraine.fr

Dead-beat stabilizability of autonomous switched linear discrete-time systems

Keywords:

The paper deals with dead-beat stabilizability of autonomous switched linear discrete-time systems. More precisely, it is investigated with the problem of finding a condition on the sequences of switches which guarantee that the state of the system reaches the origin in finite steps. A literature overview highlights the fact that such a problem is an open problem in the general case. First, we give necessary and sufficient conditions under which such an autonomous switched linear discrete-time system is dead-beat stabilizable. Then, an algorithm is proposed for deciding when such a sequence exists and for providing the sequence. It is shown that the complexity of the test can be much lower than the exhaustive search.

INTRODUCTION

Switching systems are dynamical systems for which the state dynamics vary between different operating modes according to a switching sequence [START_REF] Liberzon | Switching in Systems and Control. Systems and Control: Foundations and Applica-tions[END_REF]. Such systems are found in many practical and theoretical domains. For example they appear in the study of networked control systems [START_REF] Alur | Compositional modeling and analysis of multi-hop control networks[END_REF]; [START_REF] Jungers | Feedback stabilization of dynamical systems with switched delays[END_REF], in congestion control for computer networks [START_REF] Shorten | A positive systems model of tcp-like congestion control: asymptotic results[END_REF], in viral mitigation [START_REF] Hernandez-Vargas | Optimal and mpc switching strategies for mitigating viral mutation and escape[END_REF], as abstractions of more complex hybrid systems [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF], and other fields (see e.g. [START_REF] Jungers | The joint spectral radius[END_REF]; [START_REF] Lin | Stability and stabilizability of switched linear systems: a survey of recent results[END_REF]; [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF] and references therein).

The problem considered here is to find a condition on the sequences under which a switched system reaches the origin in finite steps. As somehow related problems, the following literature deserves a special attention.

In the survey [START_REF] Sun | Analysis and synthesis of switched linear control systems[END_REF], conditions for controllability and observability of switched linear systems based on the concept of A-invariant are provided. Special results concerning the discrete-time case given by Conner andStanford (1984, 1987) are recalled. These papers provide bounds on the minimal length of a controlling sequence for completely controllable systems. In particular, a bound on the length of the switching sequence guaranteeing that a state is reached from the origin is given. However, the results are given for switched systems with nonsingular transition matrices (defined as reversible in [START_REF] Sun | Analysis and synthesis of switched linear control systems[END_REF]) and with control inputs.

The other, and more recent, survey on the topic is [START_REF] Lin | Stability and stabilizability of switched linear systems: a survey of recent results[END_REF], where the autonomous cases in the This work was supported by Research Grants ANR-13-INSE-0005-01 from the Agence Nationale de la Recherche continuous and discrete contexts are examined. There, after dealing with the conditions for stabilizability of continuous-time autonomous switched linear systems, it is claimed that the extension of the synthesis method to discrete-time counterpart is not obvious. A sufficient condition, provided by the authors themselves, is presented. Another criterion, probably less conservative, is given in [START_REF] Geromel | Stability and stabilization of discrete-time switched systems[END_REF] but is also sufficient only. Necessary and sufficient conditions are provided in [START_REF] Fiacchini | Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: A set-theory approach[END_REF] but they are computationally complex. The difficulty in obtaining some relevant results stems from the fact that the question of deciding the stabilizability of a switching system is known to be hard in general (see [START_REF] Jungers | The joint spectral radius[END_REF], Section 2.2, for hardness results). Finite-time stabilizability, that is the problem of checking whether there exists a finite sequence of switches that gives the null matrix, is even more challenging.

Two papers deal specifically on the dead-beat stability issue for autonomous switched linear discrete-time systems, namely on the problem of checking if every sufficiently long sequence of switches leads to a null matrix. In [START_REF] Parriaux | Nilpotent semigroups for the characterization of flat outputs of switched linear and LPV discrete-time systems[END_REF], dead-beat stability has been used to characterize flatness of controlled switched linear discretetime systems. Actually, it has been shown that a switched linear system is flat if a so-called auxiliary system, taking the form of an autonomous switched linear system, is deadbeat stable. In that typical case, the dynamical matrices of the auxiliary system may be not invertible. The issue has been tackled with the notion of nilpotent semigroups. Dead-beat stability has also been examined in [START_REF] Philippe | Deciding the boundedness and dead-beat stability of constrained switching systems[END_REF]. The specificity is that the switching sequence is constrained. The dead-beat stability is expressed in terms of joint spectral radius. A polynomial time algorithm to decide on the dead-beat stability is provided. Concerning the problem of dead-beat stabilizability, we must quote the papers [START_REF] Paterson | Unsolvability in 3 × 3 matrices[END_REF]; [START_REF] Blondel | When is a pair of matrices mortal?[END_REF]; [START_REF] Bournez | The mortality problem for matrices of low dimensions[END_REF] which deal with the problem of mortality of a set of matrices. A set of matrices is mortal if the zero matrix can be expressed as the product of finite length of matrices. Clearly, this issue is closely related to the concern under investigation in the present paper. The decidability on the mortality of a set of matrices is discussed in the cited papers, which prove that, except some particular cases whose mortality problem results NPcomplete, e.g. pairs of integer matrices, the problem is unsolvable. For instance [START_REF] Paterson | Unsolvability in 3 × 3 matrices[END_REF] proves that the problem is unsolvable for finite sets of 3 × 3 matrices over the integers. Thus, in general, the problem of checking mortality of a set of matrices is unsolvable.

From this literature overview, it can be concluded that dead-beat stabilizability of switched linear discrete-time systems is still an open problem. This papers is an attempt to go further regarding this issue. In particular we are interested in providing a constructive method to check if a finite sequence of switches gives the null matrix and to compute it, if it exists. To show the benefit of the proposed method, it will be compared with the bruteforce approach based on the exhaustive search over all the possible sequences of bounded length.

The paper is organized as follows. In Section 2, a necessary and sufficient condition ensuring the existence of a sequence of finite length such that the state reaches the origin from any arbitrary initial condition, is provided. Then, in Section 3, an algorithm for testing dead-beat stabilizability is provided. The complexity issue is discussed. In Section 4, numeric simulations highlight the efficiency of the approach.

Notation Given n ∈ N define N n = {j ∈ N : 1 ≤ j ≤ n}, the set of integers ranging between 1 and n. The set of q switching modes is I = N q and the related matrices forms a finite collection A ⊆ R n×n , whose i-th element is denoted with A i , i.e. A = {A i } i∈I , with A i ∈ R n×n for all i ∈ I. All the possible sequences of modes of length N is

I N = N j=1 I and |σ| = N if σ ∈ I N . Given σ ∈ I N and i, j ∈ N N with i ≤ j, σ i is the i-th element of σ and σ [i:j]
the subsequence of σ given by the elements from the i-th to the j-th, with σ [i:j] the empty sequence if i > j. Given σ, δ sequences of modes, (σ, δ) is their concatenation. Given σ ∈ I N , define:

A σ = N j=1 A σj = A σ N • • • A σ1 ,
and n j=m A σj = I if m > n where I stands for the identity matrix. Given a ∈ R, a is the smallest integer greater than or equal to a.

NECESSARY AND SUFFICIENT CONDITION

FOR DEAD-BEAT STABILIZABILITY

Problem statement

Consider the switched linear discrete-time system

x k+1 = A σ(k) x k , (1)
where x k ∈ R n is the state at time k ∈ N and σ : N → I is the switching law.

We aim at giving a necessary and sufficient condition for the existence of an integer K ∈ N and a sequence σ ∈ I K such that for any initial condition x 0 ∈ R n , the state at time K reaches zero. That is

∃K ∈ N, ∃σ ∈ I K s.t. x K = 0, ∀x 0 ∈ R n , that is equivalent to ∃K ∈ N, ∃σ ∈ I K s.t. A σ(K) • • • A σ(0) = 0 (2)
We also aim at using the necessary and sufficient condition for computing such a sequence, when numerically possible. Remark 1. According to the considerations made in the introduction on the notion of mortality, the existence of such a K cannot, in general, be performed in finite time, resulting in an unsolvable problem. In this paper we are interested in conditions to determine K and σ, whenever they exist.

Main result

Denote with

I s = {i ∈ I : det A i = 0}, I ns = {i ∈ I : det A i = 0},
(3) which means that I s and I ns are the sets of singular and nonsingular matrices in I, respectively. The following lemma will be instrumental for the sequel. Lemma 1. [START_REF] Meyer | Matrix analysis and applied linear algebra[END_REF]) For every pair of matrices A ∈ R m×n and B ∈ R n×p , we have that dim ker(AB) = dim ker(B) + dim im(B) ∩ ker(A) .

The following proposition holds. Proposition 1. The system is dead-beat stabilizable if and only if there exist m ∈ N n finite mode sequences σ j ∈ I n j with n j ∈ N, where j ∈ N m , such that

m k=1 dim   im   k-1 j=1 A σ j   ∩ ker (A σ k )   = n, (4)
and with σ j n j ∈ I s for every j ∈ N m . Proof 1. The system is dead-beat stabilizable if and only if there exists a finite sequence γ ∈ I N such that A γ = 0, which is equivalent to ker(A γ ) = R n and also to dim ker(A γ ) = n. Hence, from Lemma 1, the system is dead-beat stabilizable if and only if there exist m ∈ N n sequences δ j ∈ I r j with r j ∈ N for all j ∈ N s such that

A γ = m j=1 A δ j and n = dim ker(A γ ) = dim ker(A δ m-1 A δ m-2 . . . A δ 1 ) + dim im(A δ m-1 A δ m-2 . . . A δ 1 ) ∩ ker(A δ m ) = dim ker(A δ m-2 A δ m-3 . . . A δ 1 ) + dim im(A δ m-2 A δ m-3 . . . A δ 1 ) ∩ ker(A δ m-1 ) + dim im(A δ m-1 A δ m-2 . . . A δ 1 ) ∩ ker(A δ m ) = m k=1 dim   im   k-1 j=1 A δ j   ∩ ker (A δ k )   .
(5)

Clearly, the number dim ker (A δ k ) can be assumed to be greater than 0 for all k ∈ N m . Since dim ker(AB) = dim ker(A) if dim ker(B) = 0, then, for every j ∈ N m , there must exist i ∈ N r j such that δ j i ∈ I s , i.e. det(A δ j i ) = 0 and then det(A δ j ) = 0. Now, what remains to be proved is that, if there exists δ j ∈ I r j such that (5) holds, then the condition can be satisfied also for σ j ∈ I n j whose last element is related to a singular matrix, i.e. such that σ j n j ∈ I s . For every j ∈ N m , define

i j = max i∈N r j {i : δ j i ∈ I s }, (6)
that is the higher index value related to a singular matrix among those involved in A δj . Such an index exists since det(A δ j ) = 0. For every j ∈ N m , define

δ j = δ j [1:i j ] , δ j = δ j [i j +1:n j ] , (7)
which means that the sequence δ j is the prefix of δ j of length i j and δ j is the sequence of length

|δ j | -i j such that δ j = (δ j , δ j ). Notice that δ j is empty if i j = |δ j |.
Define also

σ j = δ 1 , if j = 1, (δ j-1 , δ j ), if 2 ≤ j ≤ m,
and n j = |σ j |. By construction, σ j n j ∈ I s and δ j exclusively involves modes in I ns , which implies that A δ j is nonsingular. We have to prove that

d k = dim   im   k-1 j=1 A δ j   ∩ ker (A δ k )   = dim   im   k-1 j=1 A σ j   ∩ ker (A σ k )   (8)
for all k ∈ N m . Consider first k ≥ 2. From the first equality in (8), there exist d k linearly independent vectors v l ∈ R n , with l ∈ N d k , that are both in the image of k-1 j=1 A δ j and in the kernel of A σ k , which means that there exists x l ∈ R n such that

v l = k-1 j=1 A δ j x l = A δ k-1 . . . A δ 1 x l = A δ k-1 A δ k-1 A δ k-2 . . . A δ 2 A δ 1 A δ 1 x l = A δ k-1 A σ k-1 . . . A σ 2 A σ 1 x l , 0 = A δ k v l = A δ k A δ k v l .
Recalling that, by construction, A δ j is nonsingular, (see ( 6)), then d k linearly independent vectors can be defined as

u l = A -1 δ k-1 v l , that are such that u l = A -1 δ k-1 v l = A σ k-1 . . . A σ 2 A σ 1 x l , 0 = A δ k A δ k A δ k-1 u l = A δ k A σ k u l . (9)
Then, u l belongs to the image of k-1 j=1 A σ j and also to the kernel of A σ k , since ker(AB) = ker(B) if A is nonsingular. Then, for k ≥ 2, condition (8) is satisfied. For k = 1, condition (8) reduces to d 1 = dim ker(A δ 1 ) = dim ker(A σ 1 ) that holds since ker(A δ 1 ) = ker(A δ 1 A δ 1 ) = ker(A δ 1 A σ 1 ), being A δ 1 nonsingular. This completes the proof.

In other words, Proposition 1 asserts that the system (1) is dead-beat stabilizable if and only if there exists a set of m switching sequences σ j of finite length n j , with j ∈ N m and 1 ≤ m ≤ n, whose last element is related to a singular matrix, i.e. σ j n j ∈ I s , and such that the intersection of the kernel of A σ j and the image of the matrices preceding it has positive dimension. Moreover, the dimension of those intersections is equal to that of the state space R n . This result provides a necessary and sufficient condition for dead-beat stabilizability that would lead to an approach for searching one switching sequence that leads to (2). This approach acts as an alternative to an exhaustive search, that is the brute-force test of all the possible switching sequences that leads to (2).

ALGORITHM

Preliminary considerations

Before detailing a method for testing dead-beat stabilizability, we provide some preliminary considerations. First, one may wonder if, assuming that condition (5) holds, the subspaces given by the intersections provide a basis of the R n . The answer is no, as illustrated by the following simple counterexample. Example 1. Consider the system with q = 1 and

A 1 = 0 1 0 0 then ker(A 1 ) = im(A 1 ) = span{[1, 0] T } and then dim ker(A 1 A 1 ) = dim ker(A 1 )+dim (im(A 1 ) ∩ ker(A 1 )) = 2,
but the one-dimensional subspaces ker(A 1 ) and im(A 1 ) ∩ ker(A 1 ) have the same basis, i.e. [1, 0] T .

From Example 1, another useful consideration, detailed in Remark 2 below, can be done. Remark 2. It seems reasonable to consider, instead of the singular matrices A i with i ∈ I s , their power of order k i , that is the smallest non-negative integer such that ker(A ki i ) = ker(A ki+1 i

). For the trivial case in Example 1, indeed k 1 = 2 and dim ker(A 2 1 ) = 2 and dead-beat stabilizability results directly, no image and intersection computation were required.

Finally, it can be noticed that the first element of the set of sequences σ j can be simplified, as specified in the following corollary. Corollary 2. If the set of sequences σ j ∈ N n j with j ∈ N m satisfies Proposition 1, then the set of sequence ν j with j ∈ N m given by

ν j = σ 1 [r:n 1 ] , if j = 1, σ j , if otherwise, (10)
with r = min

i∈N n 1 {i : σ 1 i ∈ I s }, also fulfills Proposition 1.
Proof 2. Since im(AB) = im(A) if B is nonsingular, then all the images in (4) are unchanged when replacing A σ 1 by A ν 1 . In fact, we have that

A σ 1 = A σ 1 [r:n 1 ] A σ 1 [1:r-1] = A ν 1 A σ 1 [1:r-1] and A σ 1 [1:r-1]
is nonsingular by definition of r. Moreover, since dim ker

(AB) = dim ker(A) if B is nonsingular, for k = 1 in (4) one has dim   im   k-1 j=1 A σ j   ∩ ker (A σ k )   = dim ker(A σ 1 ) = dim ker(A ν 1 ), being A σ 1 [1:r-1] nonsingular.
Corollary 2 substantially claims that if the dead-beat stabilizability condition is satisfied by the sequences σ j , it also holds if the prefix of σ 1 involving nonsingular matrices is removed. Alternatively speaking, one can focus only on stabilizing sequence candidates starting and terminating with a singular matrix.

Computation and complexity

In this subsection, we provide Algorithm 1 to find a dead-beat stabilizing sequence. From Proposition 1 and Corollary 2, every sequence γ ∈ I N such that A γ = 0 is composed by m ∈ N subsequences σ j terminating with an element of I s . Clearly, there might be an infinite number of such γ. The objective is to find such a γ, whose length is minimal.

First, let us assume for simplicity that the sequence of elements terminating the subsequences σ j , with j ∈ N m , is known (this assumption can be removed as explained in Section 3.3 below). Denote such a sequence by s ∈ I m s . Algorithm 1, based on Lemma 3 given below, allows to determine σ j , with j ∈ N m , and the related γ s = (σ 1 , . . . , σ m ). It is worth stressing that it is not really an algorithm but a semi-algorithm since it might not terminate in finite time, an additional termination condition should be added in practice to stop its execution.

Algorithm 1 Define the dead-beat stabilizing sequence γ s of the system (1), given the sequence s ∈ I m s . Input: matrices A = {A i } i∈I , indexes I and I s and sequence s ∈ I m s . 1: p ← 1;

Initialize the sequence counter 2: σ 1 ← s 1 ;

Define the first sequence σ 1 3: n 1 ← 1;

Length of sequence σ 1 4: D ← dim ker(A s1 );

Initialize dimension counter 5: while D < n do 6:

p ← p + 1;
Update the sequence counter 7:

i ← s p ; Singular matrix closing σ p 8:

compute a basis matrix X of im

  p-1 j=1 A σ j   ; 9: find h ∈ N and δ ∈ I h such that dim ker (A i A δ X) = d p > 0;
Equivalent to (11) 10:

σ p ← (δ, i);
Define the p-th sequence σ p 11:

n p ← h + 1; Lenght of sequence σ p 12: D ← D + d p ;
Update the dimension counter 13: end while Output: γ s ← (σ 1 , . . . , σ m ).

Steps 8 and 9 are based on the following lemma. Lemma 3. [START_REF] Meyer | Matrix analysis and applied linear algebra[END_REF]) Given A ∈ R m×n and B ∈ R n×p the equality dim ker (AX) = dim (im (B) ∩ ker(A)) holds, where X is a basis matrix of im (B).

Lemma 3 permits, given i ∈ I, h ∈ N and δ ∈ I h , to check dim   im   p-1 j=1 A σ j   ∩ ker(A i A δ )   = d p > 0; (11) by testing dim ker (A i A δ X) = d p > 0, where X is a basis of im p-1 j=1 A σ j .
In practice, Algorithm 1 consists in building the sequences σ j , each terminating with s j by assumption, such that dim im

  p j=1 A σ j   < dim im   p-1 j=1 A σ j   .
When the image dimension dropping D reaches n, the overall image is the origin and then A γs = 0. Algorithm 1 can be modified to take into account the considerations made in Remark 2. Indeed, if the index k i of a matrix A i ∈ A is larger than one, considering A ki i instead of A i would reduce the number of iterations.

Considerations on complexity

Algorithm 1 applies once the sequence s ∈ I m s is given. Then, the sequence s that corresponds to the last elements of the subsequences σ j composing the sequence γ s such that A γs = 0 should be known a priori. This is not the case in general, unless |I s | = 1. Moreover, even if s were known, Algorithm 1 does not necessarily find such a γ s . In fact, it could happen that a sequence δ, found at Step 9, satisfies dim ker (A i A δ X) = d p > 0 but is not the right subsequence of γ s . Both problems can be overcome by modifying Algorithm 1 such that all the i ∈ I s and the δ ∈ I h for h ∈ Nh, with h ∈ N design parameter, are evaluated at Step 9 and all those satisfying dim ker (A i A δ X) = d p > 0 are stored and used to compute the basis at Step 8. Notice that an exhaustive search is necessary to generate the sequences δ in Step 9, a search that seems unavoidable due to the nature of the problem. In practice, the resulting algorithm is able to compute all the mortal sequences whose subsequences are not longer than h + 1. We preferred not to give here the modified version of Algorithm 1, to avoid unnecessary further complexity in its representation, although it is considered in the following complexity considerations and is used in the numerical section below.

To compare our algorithm with the brute-force approach, we should reasonably assume that an exhaustive search over the sequences of, at most, length p, that are p s=1 q s , has computational complexity O(q p+1 ). Denote q s the number of singular matrices.

Suppose first, for instance, that the subsequences σ j have average length of r, which implies that γ s has length (m -1)r + 1. Then our method requires to perform m -1 searches over q s r-1 s=1 q s sequences (and one over the q s modes for the first subsequence), leading to O((m -1)q s q r + q s ), while the brute-force approach needs to enumerate the (m-1)r+1 s=1 q s sequences, yielding to O(q (m-1)r+2 ). Another way to analyse the benefit is denoting with l ∈ N the length of the shortest mortal sequence and r ∈ N the length of its longer subsequence. Then r is contained between l-m+1 and (l-1)/(m-1) and hence our method complexity is bounded between O((m -1)q s q l-m+1 + q s ) and O((m -1)q s q (l-1)/(m-1) + q s ), whereas the exhaustive search has a complexity of O(q l+1 ). Of course both the modified algorithm and the brute force one should be applied for increasing values of h until a mortal sequence is found or a maximal value of h is attained.

Summarizing, with our approach the exhaustive search is on the subsequences σ j instead of the whole sequence γ s = (σ 1 , . . . , σ m ), where m ≤ n. This means that the exhaustive search is performed in many shorter subsequences instead of a single longer one and the benefit grows with the state dimension.

NUMERIC SIMULATIONS

In this section, we compare the results of the proposed algorithm with the brute-force approach consisting in the enumeration of every bounded length sequence of modes. The test consists in searching for a finite length sequence γ s such that A γs = 0 by means of both methods: Algorithm 1 and the exhaustive search and to compare the respective execution times.

Example 1

As a first test, we consider 5 matrices of dimension 3 × 3, with |I s | = 1 and |I ns | = 4. First, we generate the set of 5 matrices corresponding to the sequence γ s of length l such that A γs = 0. The singular matrix is generated randomly and its kernel dimension is 1. Some of the nonsingular matrices are generated randomly, while others are computed to obtain A γs = 0. The length of σ 2 and σ 3 (recall that |σ 1 | = 1) is the parameter r ∈ N. For different values of r, the sequence γ s is generated, then |γ s | = l = 2r + 1. Algorithm 1 and the exhaustive search methods are applied to compute γ s . Both methods are coded as MEX functions of Matlab. The time spent by the exhaustive search method and Algorithm 1 are respectively denoted with t enu and t alg . They are reported in Table 1. The exhaustive search method has not been applied for r > 5, since too much time would be necessary. Notice that, even for relatively short sequences γ s , Algorithm 1 gets the solution more than 100 time faster, for r = 4, and more than 2000 faster, for r = 5. Notice also that only 4.6s are sufficient to get the solution for r = 9, among 1.9 • 10 13 possible sequences.

H H H H r l = 2r + 1 Q = q l tenu t alg 4

Example 2

As a second test, we consider the state dimension n = 4 and the modes q = 6, with one singular matrix whose kernel is of dimension one. Some nonsingular matrices are generated randomly, others are chosen to get A γs = 0. The length of the three subsequences σ i with i = 2, 3, 4 are randomly chosen. The total length of the sequence γ s is imposed to be 9. Thus σ i with i = 2, 3, 4 can be of length 2, 3 or 4 (the length of σ 1 is 1). Different sets of matrices are generated and both methods are applied to retrieve γ s . The test is repeated for 100 different sets of matrices. The average, the minimal and the maximal computation time is reported in Table 2. Notice that Algorithm 1 gives the solution around 1000 times faster than the exhaustive search approach.

minimal average maximal t alg 0.0101s 0.0125s 0.0294s tenu 9.86s 15.62s 17.6s

Table 2.

CONCLUSION

In this paper, we have considered the problem of characterizing the dead-beat stabilizability for discrete-time switched systems. We have provided a constructive necessary and sufficient condition for dead-beat stabilizability.

Then we have derived from the condition a semi-algorithm for computing stabilizing switching sequences, whenever they exist, with substantially lower complexity than the brute-force exhaustive search. One of the most likely interest of this result is dead-beat control of switched linear systems. Another research line, which should deserve a special attention, is on the use of the condition for constructing dynamics admitting mortal sequences. The resulting dynamical systems would take the form of a new class of automata with finite memory which are typically involved in symmetric cryptography (statistical self-synchronizing stream ciphers as initiated by [START_REF] Daemen | On the design of high speed self-synchronizing stream ciphers[END_REF]).

Table 1 .

 1

		9	1.95 • 10 6	2.25s	0.014s
	5	11	4.48 • 10 7	83s	0.04s
	6	13	1.22 • 10 9		0.08s
	7	15	3.05 • 10 10		0.24s
	8	17	7.63 • 10 11		0.97s
	9	19	1.9 • 10 13		4.6s


E Knappe

M Manga
email: manga@seismo.berkeley.edu

A Le Friant

Rheology of natural sediments and its influence on the settling of dropstones in marine hemipelagic sediment

We obtain a new dataset of rheology measurements from naturally occurring hemipelagic sediment and a terrestrial mud volcano Comparison with existing measurements indicates that particle composition influences yield stress and consistency.

Most pumice clasts recovered from IODP 340 are stratigraphically in place and thus date the occurrence of eruptions.

Introduction

The rheological properties of sediment are influenced by multiple factors including particle concentration, size, size distribution, composition and pore-water salinity [START_REF] Imran | 1D Numerical model of muddy subaqueous and subaerial debris flows[END_REF]. Models for sediment rheology are often based on laboratory characterization of idealized kaolinite and montmorillonite mixtures (e.g. Maar et al., 2001;[START_REF] Baas | Turbulence modulation in clay-rich sediment-laden flows and some implications for sediment deposition[END_REF][START_REF] Talling | Hybrid submarine flows comprising turbidity current and cohesive debris flow: Deposits, theoretical and experimental analyses and generalized models[END_REF]. These idealized mixtures differ from naturally occurring submarine sediment in particle size and distribution as well as composition. However, there are few studies of the rheology of natural submarine deposits (e.g., [START_REF] Imran | 1D Numerical model of muddy subaqueous and subaerial debris flows[END_REF][START_REF] Locat | Numerical analysis of the mobility of the Palos Verdes debris avalanche, California, and its implication for the generation of tsunamis[END_REF][START_REF] Jeong | Determining the viscosity and yield surface of marine sediments using modified Bingham models[END_REF][START_REF] Menapace | Rheology and biostratigraphy of the Maraian serpentine muds unravel mud volcano evolution[END_REF].

Confidential manuscript submitted to Earth and Space Science Technical Reports: Data

The composition of the sediment can affect its rheology and cohesive sediments exhibit complex behaviors that are often difficult to characterize [START_REF] Yang | Rheological properties of dense natural cohesive sediments subject to shear loadings[END_REF]. When materials contain high clay fractions, strong attractive colloidal forces can lead to thixotropic behavior, that is, the viscosity of the material depends on the shear history [START_REF] Santolo | Some observations on the prediction of the dynamic parameters of debris flows in pyroclastic deposits in the Campania region of Italy[END_REF].

Thixotropy is caused by a time dependent microstructural evolution of a material. When a shear stress or strain is applied to the material the viscosity decreases [START_REF] Barnes | Thixotropya review[END_REF]. In sediment, this decrease in viscosity could be caused by the particles developing a preferred orientation or by the breakdown of flocs and aggregates which release entrapped fluid and increases the lubrication between the particles [START_REF] Barnes | Thixotropya review[END_REF][START_REF] Tregger | Influence of clays on the rheology of cement pastes[END_REF]. However, this evolution is reversible: the longer a material is at rest, the more the microstructure rebuilds until it reaches its initial state again.

In addition to the clay fraction, the type of clay can also influence rheology due to differences in cohesion. Smectite and montmorillonite have high cohesion compared to kaolinite and illite. The difference in cohesion affects how easily the microstructure can be broken down.

Naturally occurring sediments differ from idealized mixtures because they contain a range of sediment sizes, fossils, minerals and clays.

Thixotropy is also influenced by salinity and particle concentration. At high salinity, greater than 0.35 g/l to 0.55 g/l, the inter-particle forces are strong enough to favor fast reconstruction of the microstructure, thus making the material strongly thixotropic [START_REF] Perret | Thixotropic behavior during shear of a fine-grained mud from Eastern Canada[END_REF]. Particle concentration also affects the thixotropic behavior of a material. At low particle concentrations, there are fewer interactions between the particles and thixotropic effects are less significant [START_REF] Santolo | Rheologic behavior of reconstituted pyroclastic debris flow[END_REF].

Confidential manuscript submitted to Earth and Space Science Technical Reports: Data

The rheological behavior of sediment has implications for the initiation and flow of submarine sediment-laden density currents. Additionally, seafloor sediment preserves a record of volcanic eruptions in tephra and crypto-tephra layers including isolated cm-sized pumice clasts that record the occurrence of additional volcanic eruptions (le Friant et al., 2013[START_REF] Friant | Submarine record of volcanic island construction and collapse in the Lesser Antilles arc: First scientific drilling of submarine volcanic island landslides by IODP Expedition 340[END_REF][START_REF] Palmer | Discovery of a large 2.4 Ma Plinian eruption of Basse-Terre, Guadeloupe, from the marine sediment record[END_REF][START_REF] Jutzeler | Submarine deposits from pumice-rich pyroclastic density currents dispersing over water: an outstanding example from offshore Montserrat[END_REF]. These eruptions can be dated using the surrounding sediments but this assumes that the clasts are at the same stratigraphic location and did not settle through underlying sediment upon deposition.

In this study we analyze naturally occurring hemipelagic sediment collected from the seafloor adjacent to Lesser Antilles Volcanic Arc. To determine yield stress and consistency as a function of particle concentration a suite of rheometric tests are utilized. The yield stress and consistency of a material depend strongly on the particle concentration. As particle concentration increases, there is an increase in particle-particle interactions, which causes an increase in the friction and thus increases the yield stress and consistency [START_REF] Coussot | On the behavior of fine mud suspensions[END_REF]. We thus use a range of particle concentrations to develop a better understanding of how these natural sediments flow. The same tests are performed on an additional naturally occurring sediment from the Davis-Schrimpf mud volcanoes for further comparison. Additionally, our results are compared to previously published measurements on other materials in order to better identify the factors that influence rheology. For illustrative purposes, we use our results to determine the size of particles that would settle through the hemipelagic sediment on the seafloor and hence what size particles would be preserved in situ within marine sediments. We use these results to assess whether volcanic dropstones collected by IODP 340 are preserved in situ.

Methods

Confidential manuscript submitted to Earth and Space Science Technical Reports: Data

The hemipelagic sediment was collected offshore of the Lesser Antilles volcanic arc from a water depth of 2745 meters. The Lesser Antilles volcanic arc is located at the eastern border of the Caribbean plate. The proximity of our sampling location to volcanic islands, specifically the active volcano Montagne Pelée, causes our sample to contain volcaniclastic sediment as well as marine fossils (le [START_REF] Friant | Late Pleistocene tephrochronology of marine sediments adjacent to Montserrat, Lesser Antilles volcanic arc[END_REF]. The sample, 340 U1400C 3H 4, was collected during IODP 340 from 26 meters below the sea floor from a sedimentary unit that includes multiple landslide deposits (le Friant et al., 2013;[START_REF] Friant | Submarine record of volcanic island construction and collapse in the Lesser Antilles arc: First scientific drilling of submarine volcanic island landslides by IODP Expedition 340[END_REF][START_REF] Brunet | Composition, geometry, and emplacement dynamics of a large volcanic island landslide offshore Martinique: From volcano flank-collapse to seafloor sediment failure?[END_REF]; see supplementary Figure S1a for a location map. For comparison, another natural mud from the Davis-Schrimpf mud volcanoes located next to the Salton Sea in the Imperial Valley, California is analyzed (map in supplementary Figure S1b). The Salton Sea mud volcanoes are driven by the ascent of CO2 from underlying geothermal systems and the mud recirculates within the subsurface plumbing system (Svenson et al., 2009).

Sample Properties

Size distribution is measured using a Sequoia Portable XR Laser Diffraction Particle Size Analyzer. The cumulative size distribution for both samples is shown in Figure 1a. The median grain size for the hemipelagic sediment is 5.5 microns. Grain density was measured as 2700 kg/m 3 on the IODP ship using helium pycnometry (le Friant et al., 2013).

Particle concentration, the amount of solids in the total mixture, is given by (1) where Vs the volume of solids, and Vt is the total volume. We measure by weighing a known volume of material. Uncertainty in is dominated by uncertainty in measured volume. The density of the hemipelagic sediment was confirmed to be 2700 kg/m 3 . Laboratory, with a resolution of 1.3 microns/pixel. The hemipelagic marine sediment is poorly sorted and contains large voids (Figure 1b). Additionally, the sediment also contains foraminifera fossils and crystal fragments, such as hornblende, quartz and muscovite, that are large compared to particles in the surrounding matrix. The Salton Sea sediment has a more uniform size distribution by comparison (Figure 1c). Using x-ray diffraction, we determine that the hemipelagic sediment is composed of 20% feldspar and plagioclase, 19% calcite, 16% quartz, 7% aragonite, and 31% clay. The Salton Sea sediment is composed of 41% quartz, 3% dolomite, 16% plagioclase, 8% orthoclase and 26% clay. The clay in both samples contains kaolinite, illite, and montmorillonite while the hemipelagic sediment also contains smectite.

Rheology Measurement Methods

We measure the rheology using a HAAKE Rheoscope 1. To ensure uniform strain-rate through the sample a cone and plate geometry, with a 4° cone angle and 60 mm diameter plate, is used. We apply either a controlled angular velocity, and hence a shear rate, and then measure the torque required to maintain this rate, or alternatively apply a torque and measure the resulting deformation.

In order to prevent slippage, both the cone and the plate are covered in ANSI 150 grit sand paper (average particle size of 100 microns). The gap size between the plate and the center of the cone was 0.142 mm after adhesion of sand paper. Each test utilizes 8ml of sample. To prevent jamming of particles within the gap, the mud is sieved to remove particles larger than Confidential manuscript submitted to Earth and Space Science Technical Reports: Data 100 microns, which comprise less than a few percent of the volume of the sediment. The volumetrically dominant fine particles have a larger controlling influence on the yield stress, so we do not expect that the small number of fossils removed from the sample would alter the rheology [START_REF] Yu | Experimental study on the influence of clay mineral on the yield stress of debris flows[END_REF]. In order to prevent temperature fluctuations from affecting the measurements, the temperature of the mud is kept constant at 20°C using a Thermo Scientific Haake DC30-K20 Digital Bath. At this temperature water viscosity is 1.05 mPa s compared to 1.57 mPa s at a seafloor temperature of 4 o C.

The hemipelagic sediment is from a marine setting so we control salinity of the water by adding artificial seawater since the salinity of the water affects suspension rheology [START_REF] Jeong | Grain size dependent rheology on the mobility of debris flows[END_REF]. 35 grams of Instant Ocean salt mixture is added to 1 liter of water to create salinity similar to that of the Caribbean Sea.

The thixotropic nature of the sediment led us to focus on the static yield stress, above which the material will start to flow, and the dynamic yield stress, below which the material will no longer flow. Two separate tests are performed: the first to quantify the static yield and the second to determine the dynamic yield and consistency.

Flow Characteristics

In order to characterize rheology we consider an idealized constitutive model. The most frequently used model to describe naturally occurring sediments and muds is the Herschel-Bulkley model (Huang and Garcia, 1998):

(2) where τ is shear stress, τy is the yield stress, is strain rate, K is consistency, and n is the flow index. We adopt the Herschel-Bulkley model, but note that models without a yield stress have Confidential manuscript submitted to Earth and Space Science Technical Reports: Data also been used to model mud rheology especially for low strain rates and high viscosities (e.g., [START_REF] Menapace | Rheology and biostratigraphy of the Maraian serpentine muds unravel mud volcano evolution[END_REF].

During the measurements, it became apparent that both materials did not simply follow the Herschel-Bulkley model (Supplementary Figure S2) as has been documented previously (e.g., [START_REF] Yang | Rheological properties of dense natural cohesive sediments subject to shear loadings[END_REF]. Different stresses are measured when applying increasing and decreasing strain rates, indicating that the properties are history dependent, one signature of thixotropic rheology [START_REF] Moller | An attempt to categorize yield stress fluid behavior[END_REF] (Supplementary Figure 2b). This leads to the static yield stress, which is associated with the start of flow, being higher than the dynamic yield stress, below which the material will stop flowing, due to the static yield stress having to breakdown the microstructure that has formed while the material was at rest. The microstructure breakdown is reversible; if the material is left at rest, the yield stress will increase again [START_REF] Moller | An attempt to categorize yield stress fluid behavior[END_REF].

There are models that quantify the evolving microstructure (e.g., [START_REF] Dullaert | A structural kinetics model for thixotropy[END_REF][START_REF] De Souza Mendes | Modeling the thixotropic behavior of structured fluids[END_REF], however for this study we focus on the static yield stress and the dynamic yield stress. Consistency is analogous to viscosity, and is also quantified prior to the cessation of flow.

Static Yield

Thixotropic fluids have a viscosity that decreases over time as they are sheared. To mitigate the variations in shear history, samples are allowed to rest for ten minutes between each test (Van [START_REF] Kessel | Rheology of cohesive sediments: comparison between anatural and an artificial mud[END_REF]. After running hysteresis tests at varying time intervals between tests, the material recovers most of its strength after 10 minutes. Ten minutes limits the amount of evaporation of water and settling of particles so as to not affect measurements. Static yield is measured using a controlled stress mode while the sample was maintained at a constant Confidential manuscript submitted to Earth and Space Science Technical Reports: Data temperature. For this controlled stress mode, an increasing shear stress, τ, or torque, is applied and the resulting deformation is measured (Supplementary Figure S2a). Shear stress is increased linearly in 70 steps from 0 Pa to 1500 Pa, for 10 seconds each step. Initially, strain increases linearly with stress, however with further increases of stress the strain grows exponentially fast and is immeasurable after a certain shear stress. The location of this exponential increase varied with each test depending on the water content. To calculate static yield stress a Herschel-Bulkley model is fit to the data from increasing portion of the test, providing τ y , K, and n.

Dynamic Yield

In order to determine the dynamic yield stress and the consistency, a shear strain rate, , is applied and the torque, or shear stress, τ, required to maintain this strain rate is measured.

Strain rate is increased in 40 steps uniformly spaced in the log of strain rate from 0.01 s -1 to 1.5 s - 1 , and subsequently decreased in the same 40 steps, with each step lasting 15 seconds (Supplementary Figure S3 shows results for two different particle concentrations). To determine the dynamic yield stress and consistency, the decreasing portion of the test is fit with a Bingham model.

Results

Figure 2a shows the relationship between particle concentration (%) and yield stress (Pa) for our samples compared to other studies. For the hemipelagic sediment and Salton Sea mud, the filled circles show static yield stress and the open circles show dynamic yield stress. Table 1 summarizes the results for our study and Supplementary Table 1 summarizes comparative studies. In our samples the dynamic yield stress is always lower than the static yield stress, with Confidential manuscript submitted to Earth and Space Science Technical Reports: Data the exception of the Salton Sea mud at a concentration of 40.5%. This exception could be the result of sedimentation within the sample that allowed the material to start flowing at a lower static yield stress. In addition to measuring the static yield using a controlled stress mode, we can estimate the static yield from the inflection point of the increasing strain rate portion of the dynamic yield tests [START_REF] Santolo | Rheologic behavior of reconstituted pyroclastic debris flow[END_REF], and the two separately determined yield stresses are similar.

A yield stress becomes apparent in the hemipelagic sediment at a particle concentration of ~10% and increases with increasing particle concentration. Figure 2a also shows the yield stress as a function of particle concentration measured in other studies [START_REF] Coussot | On the behavior of fine mud suspensions[END_REF][START_REF] Coussot | Rheological interpretation of deposits of yield stress Fluids[END_REF]Huang and Garcia, 1998;[START_REF] Remaitre | Flow behavior and runout modeling of a complex debris flow in a clay-shale basin[END_REF][START_REF] Remaitre | Flow behavior and runout modeling of a complex debris flow in a clay-shale basin[END_REF][START_REF] Maciel | Rheological Analysis of water clay compositions in order to investigate mudflows developing in canals[END_REF][START_REF] Santolo | Experimental study on the rheologic behavior of debris flow[END_REF]Blasio et al., 2010;[START_REF] Manga | Large historical eruptions at subaerial mud volcanoes, Italy[END_REF][START_REF] Jeong | Determining the viscosity and yield surface of marine sediments using modified Bingham models[END_REF][START_REF] Yang | Rheological properties of dense natural cohesive sediments subject to shear loadings[END_REF][START_REF] Tran | Bubble mobility in mud and yield stress fluids[END_REF]. In Figure 2, non-natural sediments are represented with squares.

To determine the yield stress as a function of particle concentration, the hemipelagic data is fit with the model developed by [START_REF] Mueller | The rheology of suspensions of solid particles[END_REF]:

(3) where τ * is a fitting parameter, is the particle concentration, is the maximum particle concentration. The fit of the [START_REF] Mueller | The rheology of suspensions of solid particles[END_REF] model to the hemipelagic static yield data is shown in Supplementary Figure 4. We find τ * = 34.9 ± 6.9 Pa and = 0.23 ± 0.004.

Consistency also increases with increasing particle concentration. Following [START_REF] Mader | The rheology of two-phase magmas: A review and analysis[END_REF], the hemipelagic data is fit to the model:

(4)   m  m
Confidential manuscript submitted to Earth and Space Science Technical Reports: Data

where Kr = K/μ0 where μ0 is viscosity of the suspending fluid (Pa s) (Supplementary Figure 5).

We fnd ϕm = 0.25 ± 0.02 Pa and μ0 = 0.14 ± 0.05 Pa s.

Discussion

Both the static and dynamic yield tests indicate that the hemipelagic sediment is thixotropic. In the dynamic tests, there is a significant difference between the increasing strain rate section and the decreasing strain rate section (Supplement 2b). This indicates that the material has a history dependence in its rheology and hence its microstructure. The higher the particle concentration, the more dramatic this hysteresis. In the static yield tests, the material initially behaves like a linear elastic material until a specific stress is overcome and then the strain exponentially increases until the material reaches another critical point and flows too fast for the rheometer to measure (Supplement 2a). [START_REF] Coussot | Rheologic modeling and peculiar properties of some debris flows[END_REF] similarly found that the response of their clay-rich material is essentially elastic until a critical point is reached and the flow becomes unstable and fast. This behavior indicates that the material is initially structured, but when the apparent yield stress is overcome the sediment becomes unstructured and starts to flow at increasing velocities. Alignment of particles with the flow causes the material to flow faster when a shear is applied [START_REF] Barnes | Thixotropya review[END_REF]. Additionally, the static yield stress is significantly larger than the dynamic yield stress, which is consistent with thixotropic behavior (Figure 2).

The scatter within our data could be caused by the history dependence of our samples due to slightly different handling of the material prior to testing. We tried to mitigate this effect by allowing the samples to rest between tests and by loading the samples into the rheometer in the same way. The scatter could additionally be caused by heterogeneities within the natural sample as each data point was obtained from a different aliquot of the mud.

Confidential manuscript submitted to Earth and Space Science Technical Reports: Data

The range of particle concentrations over which a yield stress can be measured for the hemipelagic sediment is approximately between 10-20%. Below a concentration of 10% the particles settle creating a muddy fluid top layer and particle rich layer underneath. Comparing the results with other studies that used sediment suspensions, the hemipelagic marine sediment is able to develop a static stress at comparatively low particle concentrations (Figure 2a).

Additionally, other studies that measured yield stresses at low particle concentrations were kaolinite and water mixtures produced in the lab (Figure 2a laboratory produced mixtures are represented with squares).

The clay content in the hemipelagic sediment is in the higher range compared to the other studies shown in Figure 2a, although not all studies record clay content. The hemipelagic sediment has approximately 31% clay, which could be one of the contributing factors to why there is a yield stresses at such low particle concentrations. Huang and García (1998) measure a yield stress at lower concentration, approximately 7%, but do not record the clay content of the sample they used. The higher clay content increases electrostatic forces between the clay particles. [START_REF] Yu | Experimental study on the influence of clay mineral on the yield stress of debris flows[END_REF] found that increasing the clay content from approximately 10% to 40% clay increased the yield stress by more than an order of magnitude, consistent with [START_REF] Jeong | Determining the viscosity and yield surface of marine sediments using modified Bingham models[END_REF] who reported the highest clay content and similarly large yield stresses. While both the hemipelagic sediment and Salton Sea mud have kaolinite, illite, montmorillonite, the hemipelagic sediment also contains smectite, which is highly cohesive. The addition of this clay type could also contribute to the higher yield stress and more thixotropic behavior compared to the Salton Sea mud. The yield stress of the hemipelagic sediment is also more than an order of magnitude larger than kaolinite suspensions with similar particle concentrations suggesting that Confidential manuscript submitted to Earth and Space Science Technical Reports: Data the presence of smectite is largely responsible for the higher yield stress (in Figure 2, compare Huang and Garcia (1998) and [START_REF] Maciel | Rheological Analysis of water clay compositions in order to investigate mudflows developing in canals[END_REF] data with the present measurements).

The salinity of the hemipelagic marine sediment, at 35 g/l, is well above the salinity that [START_REF] Perret | Thixotropic behavior during shear of a fine-grained mud from Eastern Canada[END_REF] identify as influencing rheology. Above this salinity, there is fast reconstruction of the microstructure, and below the material is likely to favor slower reconstruction. [START_REF] Yang | Rheological properties of dense natural cohesive sediments subject to shear loadings[END_REF], however, found no effect on rheology of salinity in the range of fresh water to 50 g/L salinity. Using the method outlined by [START_REF] Perret | Thixotropic behavior during shear of a fine-grained mud from Eastern Canada[END_REF] to determine the thixotropic nature of a material, the area between the hemipelagic sediment thixotropy test curves is larger than that of the Salton Sea sediment, indicating that the hemipelagic material is slower to recover back to its initial state. This indicates that the clay content has a larger effect than salinity for the two samples we considered.

Materials with high clay contents have a slow rate of reconstruction, increasing their recovery time [START_REF] Barnes | Thixotropya review[END_REF]. This is also the case for the two samples we considered. While the Salton Sea mud has a significant portion of clay, the median particle size of the hemipelagic sediment is much smaller at 5.5 microns, compared to the Salton Sea sediment 35.7 microns, which could account for the difference in recovery times.

Particle Settling through seafloor sediment

Hemipelagic sediment often contains isolated "dropstones" delivered by melting icebergs or volcanic eruptions; in the case of sediments offshore the Lesser Antilles volcanic arc, these are predominately pumice clasts, deposited from rafts of pumice or fall out. Whether these clasts are preserved within their deposit or subsequently sink multiple clast diameters through underlying layers depends on their size and weight. In order to determine whether a larger particle with Confidential manuscript submitted to Earth and Space Science Technical Reports: Data radius can settle within mud, we approximate the clast as a completely submerged sphere.

The force is

(5

)
where ρs is the density of the sphere (kg/m 3 ), ρ is the density of the mud, R0 is the radius of the sphere (m), and g is gravity (m/s 2 ). The density of the sediment depends on the particle concentration (Cv), ρ=1800•Cv+1000 kg/m 3 . The sphere will cause yielding of the mud and sink when the yield stress parameter (6) exceeds a critical value of 0.143 [START_REF] Beris | Creeping motion of a sphere through a Bingham plastic[END_REF]. Pumice is porous, between 60-80% porosity, and at the bottom of the seafloor these pores will be filled with water. The mean density was calculated assuming 60% porosity and using a glass density of 2700 kg/m 3 . We use a dynamic yield stress because we assume the clasts are initially moving when they reach the seafloor. Using equation ( 6) and equation ( 3) to determine the yield stress as a function of particle concentration, we calculate the relationship between the particle concentration and the radius of a clasts that would settle through the sediment (Figure 3). Below this radius, clasts would not settle but instead be preserved in situ. Using an average mud concentration at the seafloor of 10% [START_REF] Hamilton | Variations of density and porosity with depth in deep-sea sediments[END_REF] Table 1 and Supplementary Table 1.

Table 1. The hemipelagic sediment and Salton Sea sediment measured in this study. Includes the parameters used to fit the Herschel-Bulkley model. Data plotted in Figure 2. Clast Radius (cm)



  Figure 1b,c shows three dimensional x-ray microtomography reconstructions of the

Figure 1 .

 1 Figure 1. a) The mean cumulative size distribution of the sediments. The hemipelagic sediment, shown in blue, is the average of four tests, and the Salton Sea sediment, shown in red, is the average of two tests. b) 3-D microtomography reconstructions of the materials. Reconstruction of the hemipelagic sediment, which is poorly sorted and contains large fossils. C) Reconstruction of the Salton Sea sediment, which is much more uniform in size than the hemipelagic sediment.

Figure 2 .

 2 Figure 2. a) Yield stress as a function of particle concentration from multiple different studies, b) consistency as a function of particle concentration from multiple different studies. Clay contents shown in numbers next to the respective data when reported. For our measurements, open symbols show static yield stress and filled symbols show dynamic yield stress. All data in

Figure 3 .

 3 Figure 3. Particle concentration of hemipelagic sediment and the clast radius that would exceed the yield stress parameter defined by[START_REF] Beris | Creeping motion of a sphere through a Bingham plastic[END_REF]. For radii above the blue curve, clasts

Confidential manuscript submitted to Earth and Space Science Technical Reports: Data https://github.com/eknappe/rheology_tech_reports. We thank two reviewers for constructive comments and suggestions.