Junhua Dang
email: dangjunhua@gmail.com

Paul Barker

Anna Baumert

Margriet Bentvelzen

Elliot Berkman

Nita Buchholz

Jacek Buczny

Zhansheng Chen

Valeria De Cristofaro

Lianne De Vries

Siegfried Dewitte

Mauro Giacomantonio

Ran Gong

Maaike Homan

Roland Imhoff

Ismaharif Ismail

Lile Jia

Thomas Kubiak

Florian Lange

Dan-Yang Li

Jordan Livingston

Rita Ludwig

Angelo Panno

Joshua Pearman

Niklas Rassi

Helgi B Schiöth

Manfred Schmitt

A Timur Sevincer

Jiaxin Shi

Angelos Stamos

Yia-Chin Tan

Mario Wenzel

Oulmann Zerhouni

Li-Wei Zhang

Yi-Jia Zhang

Axel Zinkernagel

A Multilab Replication of the Ego Depletion Effect

Keywords: ego depletion, self-control, multilab, preregistration

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

The ego depletion effect refers to the phenomenon that initial exertion of self-control impairs subsequent self-regulatory performance. It has often been tested in between-subject designs using a sequential task paradigm. Participants in such studies complete a task that is thought to require self-control after (a) completing a different self-control task (depletion condition) or (b) completing a control task that does not require self-control (control condition). Participants in the depletion condition generally perform worse than those in the control condition, which is referred to as the ego depletion effect.

According to the strength model of self-control, all acts of self-control draw from a limited mental resource [START_REF] Baumeister | The strength model of self-control[END_REF]. Initial acts of self-control impair subsequent self-control performance because they have consumed the limited resource, leaving insufficient resource for subsequent use. Alternatively, the process model suggests that what underlies ego depletion is not the consumption of some limited resource but the motivated disengagement of work to engage in leisure. That is to say, because effortful control is intrinsically aversive, people generally tend to avoid it. After exerting effortful control, the aversive feeling would accumulate, which leads people not only to more strongly avoid further control but also more strongly value and pursue rewards that can bring gratification [START_REF] Inzlicht | What is ego depletion? Toward a mechanistic revision of the resource model of self-control[END_REF].

Regardless of its explanations, observations of the ego depletion effect have had pervasive influence in social psychology and many related research areas [START_REF] Friese | Is ego depletion real? An analysis of arguments[END_REF]. The effect has been examined in over 600 independent studies [START_REF] Carter | A series of meta-analytic tests of the depletion effect: Self-control does not seem to rely on a limited resource[END_REF] conducted by more than 2,000 researchers [START_REF] Wolff | Self-reports from behind the scenes: Questionable research practices and rates of replication in ego depletion research[END_REF], with various task combinations (for a list of the most frequently used tasks, see [START_REF] Carter | A series of meta-analytic tests of the depletion effect: Self-control does not seem to rely on a limited resource[END_REF][START_REF] Dang | An updated meta-analysis of the ego depletion effect[END_REF]. The paper that reported the ego depletion effect for the first time [START_REF] Baumeister | Ego depletion: Is the active self a limited resource[END_REF] has been cited over 5,000 times in Google Scholar.

In recent years, however, the ego depletion effect has been challenged on multiple grounds (see [START_REF] Friese | Is ego depletion real? An analysis of arguments[END_REF], for a review). Meta-analyses indicate that the effect has been severely overestimated due to small-study biases and that the existing evidence may not be sufficient to support the existence of an ego depletion effect [START_REF] Carter | A series of meta-analytic tests of the depletion effect: Self-control does not seem to rely on a limited resource[END_REF]. Perhaps most notably, a multilab replication effort including 23 laboratories (N = 2,141) did not find an ego depletion effect that was significantly different from zero, d = 0.04, 95% confidence interval (CI) [-0.07, 0.15] (Hagger et al., 2016).

Large-scale replication efforts like the one Hagger and colleagues (2016) have done cannot easily reflect the methodological heterogeneity of ego depletion research [START_REF] Carter | A simple, principled approach to combining evidence from meta-analysis and high-quality replications[END_REF]. Rather than being able to operationalize ego depletion in every possible way, they need to select a specific combination of self-control tasks for their study. Based on both conceptual and practical reasons, Hagger and colleagues (2016) chose a letter-crossing task (that differed in self-control demands between the depletion and control condition) followed by a multisource interference task for their replication project. The appropriateness of this choice has been debated [START_REF] Baumeister | Misguided effort with elusive implications[END_REF][START_REF] Dang | Commentary: A multilab preregistered replication of the ego-depletion effect[END_REF]Hagger & Chatzisarantis, 2016) and it remains possible that the ego depletion effect can be observed using task combinations other than the one chosen by Hagger and colleagues. Evidence supporting the existence of an operationalization-specific ego depletion might be difficult to reconcile with the domain-general assumptions of the strength model of self-control. It might, however, help to account for the heterogeneity of previous results and open new avenues for research on self-control processes.

For this reason, we set out to complement the project by Hagger and colleagues with a second multilab preregistered replication study on the ego depletion effect. For this project, we selected a procedure that has been reported to produce a substantial ego depletion effect in a recent preregistered study [START_REF] Dang | The ego could be depleted, providing initial exertion is depleting: A pre-registered experiment of ego depletion[END_REF]. Inspired by an up-to-date meta-analysis [START_REF] Dang | An updated meta-analysis of the ego depletion effect[END_REF], those authors selected a Stroop task for the first part of the sequential task paradigm, because this task might be particularly effective in affecting subsequent self-control performance. On the subsequent antisaccade task, participants who completed the Stroop task with incongruent stimuli performed significantly worse than participants who completed the Stroop task with congruent stimuli, Hedge's g = 0.48. 95% CI [0.18,0.78]. This finding indicates that the specific task combination used by [START_REF] Dang | The ego could be depleted, providing initial exertion is depleting: A pre-registered experiment of ego depletion[END_REF] can produce an ego depletion effect. However, on the basis of a single study, it is difficult to conclude whether it does so in a reliable way. Therefore, we preregistered and conducted a high-powered multilab project to replicate the experiment by Dang and colleagues in a standardized way.

Method Project Organization

This project was initiated by the first author in May 2017. He wrote e-mails to invite researchers who have conducted ego depletion studies. Thirteen labs across the world (4 from Germany, 1 each from the mainland of China, the United States, Belgium, Italy, France, the Netherlands, Austria, Hong Kong, and Singapore, respectively) agreed to participate but the lab from Austria quit at the end of 2017 without collecting any data due to lacking relevant resources. The remaining 12 labs followed the same protocol to collect data independently. All labs have experience of doing ego depletion studies and some of them also participated Hagger et al.'s (2016) replication project.

Preregistered Protocol

The protocol was preregistered in a transparent way in October 2017 (https://osf.io/4mcnf/). At the same time, materials including questionnaires measuring individual differences, eprime scripts with Chinese and English instructions, and description and templates for data extraction have been uploaded. Each non-English and non-Chinese lab translated the questionnaire and e-prime instructions into their local language. Translation efforts were coordinated in the two labs testing Dutch-speaking participants and in the four labs testing German-speaking participants. All data and analytical scripts are available for replicating the results reported here (https://osf.io/3txav/).

Experimental procedure-The experimental procedure was the same as that in [START_REF] Dang | The ego could be depleted, providing initial exertion is depleting: A pre-registered experiment of ego depletion[END_REF]. Participants first completed a short questionnaire measuring three individual difference variables that might moderate the ego depletion effect: action orientation, lay theory about willpower, and trait self-control. Next, participants engaged in a Stroop task and were randomly assigned to the depletion or the control condition. After the Stroop task, they answered four manipulation check questions before working on the antisaccade task as the outcome measure. Note that three labs measured the individual differences at the end rather than at the beginning of the experiment. The timing of the individual difference measurement was considered as a potential moderator in our auxiliary analyses.

Individual difference measures-Action orientation was measured by the Demand-Related Action Orientation (AOD)subscale of the Action Control Scale [START_REF] Jostmann | On the regulation of cognitive control: Action orientation moderates the impact of high demands in Stroop interference tasks[END_REF]. The AOD scale consists of 12 items. Each item describes a demanding situation and an action-oriented versus a state-oriented coping way. Participants were asked to indicate the way that best describes their own reaction to that situation. Action-oriented responses were coded as 1 whereas state-oriented responses as 0. Scores summed for the entire scale could range from 0 to 12. Lay theory of willpower was measured by 6 items developed by [START_REF] Job | Ego depletion-Is it all in your head? Implicit theories about willpower affect selfregulation[END_REF]. Participants responded on a 6-point scale (1 = strongly disagree, 6 = strongly agree).

Items were scored so that higher values indicate greater agreement with the unlimitedresource theory.

Trait self-control was measured by the 13-item Brief Self-Control Scale [START_REF] Tangney | High self-control predicts good adjustment, less pathology, better grades, and interpersonal success[END_REF]. Participants indicated the extent to which they agree with each statement on a scale from 1 (strongly disagree) to 5 (strongly agree). Higher scores represent better self-control.

Depletion manipulation-Participants assigned to the depletion condition completed a Stroop task in which most trials were incongruent (four different colors, 256 trials, 75% incongruent, such as"BLUE" with red font and "YELLOW" with blue font). In the control condition, all trials were congruent such as BLUE with blue font and YELLOW with yellow font. In each trial, after a 200 ms fixation, the stimulus (i.e., a color word with a specific font color) appeared on the screen until the participant pressed the spacebar, which was then followed by a 500 ms blank screen. Participants were required to read the color of the word aloud and then press the spacebar to proceed toward the next trial.

Manipulation check measures-After the Stroop task, participants answered four manipulation check questions regarding effort ("How much effort did you put into the colornaming task?"), difficulty ("How difficult did you find the color-naming task?"), fatigue ("How tired do you feel after doing the color-naming task?"), and frustration ("Did you feel frustrated while you were doing the color-naming task?") on a 7-point scale (Hagger et al., 2016).

Outcome measure-Following the manipulation check measures, participants were asked to finish an antisaccade task [START_REF] Dang | Individual differences in dopamine level modulate the ego depletion effect[END_REF][START_REF] Unsworth | Attention control and the antisaccade task: A response time distribution analysis[END_REF]. Their task was to identify three target letters (B, P, and R) by pressing a corresponding key (the Keys 1, 2, and 3, respectively) as quickly and accurately as possible. At the beginning of each trial, a fixation cross appeared for 200 ms on the screen with a black background. A white "=" was then flashed either to the left or right of the fixation cross for 100 ms, followed by a 50 ms blank screen and a second appearance of the sign "=" for 100 ms at the same location as the first one. This procedure made it appear as though the sign "=" flashed on-screen, which would easily grasp participants' attention. Following another 50 ms blank screen, the target stimulus (a letter B, P, or R) appeared in the opposite location of the flashing sign for 100 ms, followed by a mask (the letter "H") for 50 ms and the number "8," which remained on-screen at the same location as the target stimulus until a response was given. Participants received 30 practice trials (12 practice trials for learning the response mapping and 18 practice trials for doing the formal test) and 120 real trials.

Participants-The sample size was determined to be large enough to detect an effect as large as in the original study (g = 0.48) with 80% power (given α= .05, two-tailed) in each of the participating labs. Power analysis (G*Power Version 3.1.9.2; Faul, Erdfelder, Lang, & Buchner, 2007) rendered a target sample size of 140 participants (70 in each of the two conditions). Therefore, each lab was required to recruit at least 140 participants who should be between 18 and 30 years, native speakers of the language in which the Stroop color words were presented and not color-blind. They received course credits, money or a gift, or a mix of credits and money or a gift as compensation. The compensation level was determined by local conventions for an experiment lasting for 30-40 min.

There were 1,841 participants in total, of which 1,834 completed the sequential-task paradigm (i.e., the e-prime script). Those whose age was below 18 or above 30 were excluded according to the preregistration protocol, which led to 1,775 participants for final analyses.

Test environment and experimenters-Each participant tested individually in a behavioral lab. The experimenter had to complete the script by himself or herself at least once to get familiar with the script. The experimenter needed to make sure the participant followed the instruction by checking whether the participant's voice response was consistent with the font color rather than the meaning of the color word during practice trials in the Stroop task in the depletion condition, and whether most feedback provided by the computer were correct during practice trials in the antisaccade task.

Preregistered analyses-According to the preregistration, the data from each lab were first analyzed separately. The primary dependent variable was the error rate of the antisaccade task. The reaction times (RT) were also be examined after trimming (i.e., longer than 2,000 ms and shorter than 200 ms; [START_REF] Unsworth | Attention control and the antisaccade task: A response time distribution analysis[END_REF]. The t test was used to examine whether there were differences between the two conditions. Also, regression and simple slope analyses were used to test the interaction between the depletion manipulation and each of the three individual difference variables (one by one).

Next, we used meta-analysis to test the overall effect size of the depletion manipulation. We chose the random effects model that allowed to test the homogeneity of the effect sizes across studies by means of the Q statistic [START_REF] Borenstein | Introduction to meta-analysis[END_REF]. Because the Q statistic cannot be compared across meta-analyses, we also calculated the I 2 , which is expressed as a percentage and is therefore easy to compare, with levels of 25%, 50%, and 75% representing low, medium, and high levels of heterogeneity, respectively. A high level of heterogeneity suggests the majority of the differences observed between individual studies are due to real differences in effect sizes rather than to random sampling errors.

In order to examine the funnel plot asymmetry, we computed two indices: [START_REF] Begg | Operating characteristics of a rank correlation test for publication bias[END_REF] rank correlation (Kendall's τ-b, one-tailed) and Egger's regression test [START_REF] Egger | Bias in meta-analysis detected by a simple, graphical test[END_REF]. The analyses were performed by using the metafor package of R [START_REF] Viechtbauer | Conducting meta-analyses in R with the metafor package[END_REF]).

To meta-analyze the moderating effects of the individual difference variables, the Liptak-Stouffer Z-score method was used to combine results across labs, weighted by sample size [START_REF] Karg | The serotonin transporter promoter variant (5-HTTLPR), stress, and depression meta-analysis revisited: Evidence of genetic moderation[END_REF][START_REF] Whitlock | Combining probability from independent tests: The weighted Z-method is superior to Fisher's approach[END_REF]. First, the p values of the interaction terms were converted to one-tailed p values, with p values less than .50 corresponding to the fact that individuals with action orientation, belief about unlimited resource, and high trait self-control are influenced less by the depletion manipulation, and p values more than .50 corresponding to the fact that individuals with state orientation, belief about limited resource, and lower trait self-control are influenced less by depletion manipulation. Next, these p values were converted to Z-scores using a standard normal curve such that p values less than .50 were assigned positive Z-scores and p values more than .50 were assigned negative Z-scores. Subsequently, these Zscores were combined by calculating:

Z w = ∑ i = 1 k w i Z i ∑ i = 1 k w 2 ,
where the weighting factor w i corresponds to the sample size of each lab, k corresponds to the number of total studies, and Z i corresponds to the Z-score of each lab. The outcome of this test, Z w , follows a standard normal distribution and the corresponding probability can be obtained from a standard normal distribution table.

Auxiliary analyses-We also conducted a series of auxiliary analyses to further explore the data. First, we tested whether the estimated effect size of ego depletion was moderated by factors that pertain to participant characteristics and experimental settings. Second, we repeated the preregistered meta-analyses and abovementioned auxiliary analyses after excluding participants who might have responded randomly during the second task. This exclusion criterion was not preregistered, but we regard the corresponding analyses as an informative robustness check.

Results

Preregistered Analyses

Single lab analyses-For the primary dependent variable, the error rate on the antisaccade task, t tests showed significantly impaired performance in the depletion condition compared to the control condition in one lab (Baumert, Buchholz, Schmitt, and Zinkernagel), and marginally significant differences in the same direction in two other labs (Bentvelzen, Buczny, de Vries, & Homan; Ismail, Jia, and Tan). Lay theory about willpower interacted with the depletion manipulation to predict the error rate in only one lab (De Cristofaro, Giacomantonio, and Panno), ΔR 2 = .04, p = .014. Action orientation and trait selfcontrol did not significantly interact with the depletion manipulation in any lab. The descriptive statistics are shown in Table 1.

For the RT, a marginally significant group difference was observed in only one lab (Ismail, Jia, and Tan). Lay theory about willpower interacted with the depletion manipulation to predict the RT in one lab (Kubiak and Wenzel), ΔR 2 = .03, p = .034. Action orientation and trait self-control did not significantly interact with the depletion manipulation in any lab. The descriptive statistics are shown in Table 2.

Meta-analyses-As shown in Table 3, for the error rate, the weighted average standardized mean difference between the depletion condition and the control condition was significant with a small effect size, d = 0.10, 95% CI [0.01, 0.19], Z = 2.10, p = .036. The Q was not significant, Q(11) = 10.40, p = .495, and the I 2 was 0, which indicates that the effect sizes were rather homogeneous across the participating labs. Neither funnel plot asymmetry index was significant (Kendall's τ -b =-0.33, p = .153; Egger's regression coefficient, t = -0.43, p = .680). The forest plot and the funnel plot are shown in Figures 1 and2,

respectively.

An analogous pattern emerged for the RT. The weighted average standardized mean difference between the depletion condition and the control condition was small and significant, d = 0.10, 95% CI [0.00, 0.19], Z = 2.01, p = .044. The Q was not significant, Q(11) = 5.72, p = .891, and the I 2 was 0. Neither funnel plot asymmetry index was significant (Kendall's τ -b =-0.06, p = .841; Egger's regression coefficient, t =-0.15, p = .884). The forest plot and the funnel plot are shown in Figures 3 and4, respectively.

With regard to the manipulation check measures, as shown in Table 4, the Stroop version used in the depletion condition was perceived to be more effortful, difficult, and tiring. The effect sizes were homogeneous as indexed by the nonsignificant Q and low I 2 . However, on the item regarding frustration, there was no significant difference between the two conditions. Effect sizes were not homogeneous for this measure as indicated by the significant Q and medium I 2 .

Regarding the moderating effects of individual difference variables, the Liptak-Stouffer Zscore was calculated for each of the three variables (i.e., action orientation, lay theory about willpower, and trait self-control) on both the error rate and the RT. As shown in Table 5, the Liptak-Stouffer Z-score method did not reveal any significant moderation of ego depletion except for lay theory about willpower on the RT, Z = 2.06, p = .039, suggesting individuals with an unlimited-resource theory were influenced less by the depletion manipulation.

Auxiliary Analyses

Lab-level moderators-We tested whether the estimated effect size from the metaanalysis was moderated by lab-level factors that pertain to participant characteristics and experimental settings. There were eight factors and we tested them one by one on both the error rate and the RT with meta-regression: (1) participants' average age in each lab (continuous variable), ( 2) the percentage of male participants in each lab (continuous variable), (3) whether the experimenter used a chin fixer (categorical variable: 0 = no, 1 = yes), ( 4) whether the experimenter stayed with the participant in the lab (categorical variable: 0 = no, 1 = yes), ( 5) the number of experimenters (continuous variable), ( 6) the percentage of male experimenters (continuous variable), ( 7) compensation type (categorical variable: 0 = course credits, 1 = money, a gift, or mix of credits and money), and (8) when the individual difference variables were measured (categorical variable: 0 = at the beginning, 1 = at the end).

As shown in Table 6, meta-regression showed that there was only one significant moderating effect. The percentage of male participants moderated the ego depletion effect observed on the error rate, Q M = 4.00, p = .045, indicating that labs that recruited more male participants tended to find smaller effect sizes.

Exclusion of random responses-Given three possible responses in the antisaccade task, guessing probability was 33%. A Binomial test indicates that, when the error rate is higher than 58.33% (70 errors in 120 trials), the null hypothesis that the participant was responding randomly cannot be rejected at the significance level of 0.05. That is to say, we might be confident to say that a participant performed significantly better than random if he or she had an error rate lower than or equal to 58.33%. Therefore, we repeated the preregistered meta-analyses and the auxiliary analyses (i.e., metaregressions) after excluding participants who completed the antisaccade task at levels indistinguishable from chance (i.e., who were likely to have guessed when required to identify the target letter).

After excluding participants whose error rate was higher than 58.33% (n = 175 in the depletion condition and n = 165 in the control condition), for the error rate, the weighted average standardized mean difference between the depletion condition and the control condition was still significant, d = 0.10, 95% CI [0.00, 0.21], Z = 1.96, p = .049, as shown in Table 3. The Q was not significant, Q(11) = 2.89, p = .992, and the I 2 was 0. For the RT, the weighted average standardized mean difference was also significant after exclusion, d = 0.16, 95% CI [0.05, 0.26], Z = 2.93, p = .003. The Q was not significant, Q(11) = 7.67, p = .742, and the I 2 was 0. After exclusion, the Liptak-Stouffer Z-score method did not reveal any significant moderating effect of the individual difference variables on the ego depletion effect as shown in Table 5. In addition, as shown in Table 6, meta-regression revealed no significant lab-level moderators after excluding chance-level performers.

Discussion

In the current multilab collaborating project, we replicated the ego depletion experiment by Dang and colleagues (2017). Preregistered meta-analyses revealed small and significant ego depletion effects on both response accuracy and latency (d = 0.10). After excluding participants who might have responded randomly on the outcome task, the effect size of the accuracy remained the same, but the effect size of the reaction time increased to d = 0.16.

This effect size estimate was comparable to another recent endeavor in which [START_REF] Garrison | Ego depletion reduces attention control: Evidence from two high-powered preregistered experiments[END_REF] verified the effectiveness of another depleting task implied by [START_REF] Dang | An updated meta-analysis of the ego depletion effect[END_REF] meta-analysis (i.e., the attention essay task, paired with the Stroop task as the outcome measure in Study 1 and with the attention network test as the outcome measure in Study 2) in two preregistered experiments with over 1,000 participants and reported an effect size d = 0.20. Although these results suggest that the ego depletion effect should be real, the effect sizes might be smaller than previously thought. In order to detect an effect size of 0.10, 0.16, and 0.20 with 80% power (given α= .05, two-tailed), we will need 1,571, 615, and 394 participants per condition, respectively.

Does this indicate the true effect size of ego depletion is between 0.10 and 0.20? We suggest the answer is either "yes" or "no." On the one hand, in the current ego depletion literature (as well as Hagger et al.'s and our replications), most studies adopted very brief depletion manipulations that were generally less than 10 min. Participants' responses to these weak manipulations could vary to a great extent, with some participants feeling exhausted while others feeling indifferent or even excited as such tasks may serve to "warm-up" their selfcontrol (e.g., [START_REF] Lopez | Recruitment of cognitive control regions during effortful self-control is associated with altered brain activity in control and reward systems in dieters during subsequent exposure to food commercials[END_REF][START_REF] Wenzel | Let there be variance: Individual differences in consecutive self-control in a laboratory setting and daily life[END_REF]. Therefore, there is a substantive heterogeneity in the size of ego depletion in the literature and the average effect size is small (e.g., between 0.10 and 0.20). If we continue with these weak manipulations, the answer is more likely to be yes. On the other hand, if stronger manipulations were implemented, the answer should be more likely to be no and the effect size should be increased as the strength of the manipulation increases. Consistently with this proposition, recent evidence showed that the depletion intensity is positively correlated with subsequent fatigue perception [START_REF] Tsai | Depletion manipulations decrease openness to dissent via increased anger[END_REF]. When the manipulation lasted for 1 hr or more, the effect size increased from medium to large [START_REF] Radel | Testing the ego-depletion effect in optimized conditions[END_REF]Såstad & Baumeister, 2018). Future studies need to systematically test the dose-dependent feature of ego depletion.

In addition, in line with [START_REF] Job | Ego depletion-Is it all in your head? Implicit theories about willpower affect selfregulation[END_REF], in the full sample, the current project found supporting evidence on the RT for the moderating effect of lay theory about willpower such that people with an unlimited-resource theory were influenced less by the depletion manipulation. Relatedly, a group of studies recently reported reversed ego depletion in individuals who believed that initial exertion is energizing [START_REF] Savani | Reverse ego-depletion: Acts of self-control can improve subsequent performance in Indian cultural contexts[END_REF]. However, after excluding participants who might have responded randomly, which might lead to a more accurate measure of the RT, the moderating effect of lay theory about willpower disappeared. The lack of robustness, in combination with the fact that lay theories did not moderate the effect on error rates, indicates that our results do not provide unambiguous support for a moderating role of lay theories of willpower.

Finally, since our main purpose was to test whether the ego depletion effect is replicable, the current research provides little evidence regarding the mechanisms underlying ego depletion. The observed performance decline after initial exertion may result from insufficient resource for subsequent acts of self-control [START_REF] Baumeister | The strength model of self-control[END_REF] or from reduced motivation to exert further control [START_REF] Inzlicht | What is ego depletion? Toward a mechanistic revision of the resource model of self-control[END_REF]. However, our results pave the way for further theoretical analyses and related empirical examinations because after all it will be meaningless to debate why there is such a phenomenon if the phenomenon itself cannot be observed. Forest plot for the error rate. Funnel plot for the error rate. Forest plot for reaction time. Funnel plot for reaction time.

  Figure 1.

  Figure 2.

  Figure 3.

  Figure 4.

Table 1 .

 1 The Descriptive Statistics for the Error Rate.

a Author

M depletion M control SD depletion SD control t p

	.651	.672	.708	.902	.910	.813	.746	.349	.144	.099	.086	.021 *
	-0.45	-0.42	-0.37	-0.12	-0.11	0.24	0.32	0.94	1.47	1.66	1.73	2.33
	.11	.19	.19	.16	.18	.19	.19	.16	.17	.17	.20	.16
	.13	.18	.16	.16	.16	.20	.20	.18	.17	.16	.20	.17
	.56	.41	.40	.23	.44	.35	.39	.32	.36	.29	.39	.42
	.55	.40	.39	.23	.44	.36	.40	.35	.40	.34	.45	.48
	Barker and Imhoff	Chen and Shi	Gong, Li, Zhang, and Zhang	Berkman, Livingston, Ludwig, and Pearman	Rassi and Sevincer	Dewitte, Lange, and Stamos	Zerhouni	De Cristofaro, Giacomantonio, and Panno	Kubiak and Wenzel	Ismail, Jia, and Tan	Bentvelzen, Buczny, de Vries, and Homan	Baumert, Buchholz, Schmitt, and Zinkernagel	a Labs are ordered by the t value.

Table 2 .

 2 The Descriptive Statistics for the Reaction Times.

a Author

M depletion M control SD depletion SD control t p

	-0.40 .690	-0.23 .816	-0.01 .995	0.02 .983	0.11 .915	0.43 .668	0.55 .584	0.76 .449	1.23 .221	1.27 .207	1.29 .199	1.94 .054
	202.81	192.50	205.01	176.73	174.75	156.44	201.65	162.46	194.23	161.87	167.55	153.90
	135.22	206.44	193.14	209.73	189.88	184.38	188.42	170.29	198.54	188.24	198.37	149.98
	694.12	803.70	782.94	780.20	798.48	751.76	843.27	722.40	787.80	602.11	753.01	647.74
	683.16	795.93	782.71	780.93	801.82	764.24	860.71	740.53	826.03	639.87	793.08	696.73
	Chen and Shi	Bentvelzen, Buczny, de Vries, and Homan	Gong, Li, Zhang, and Zhang	Barker and Imhoff	Rassi and Sevincer	Dewitte, Lange, and Stamos	Kubiak and Wenzel	Zerhouni	Baumert, Buchholz, Schmitt, and Zinkernagel	Berkman, Livingston, Ludwig, and Pearman	De Cristofaro, Giacomantonio, and Panno	Ismail, Jia, and Tan	t value. a Labs are ordered by the	* p < .05.	** p < .01.	*** p < .001.

Table 3 .

 3 Meta-Analytical Effect Size Estimates for the Between-Group Difference in Performance on the Antisaccade Task.

			Full Sample	After Exclusion a
		d	95% CI	d	95% CI
	Error rate	.10 [0.01, 0.19] .10 [0.00, 0.21]
	Reaction times .10 [0.00, 0.19] .16 [0.05, 0.26]
	a Results obtained after excluding participants who might have responded randomly.
	* p < .05.
	** p < .01.
	*** p < .001.

Soc Psychol Personal Sci. Author manuscript; available in PMC 2022 January 01.

Table 6 .

 6 The Q M Values of Meta-Regression.

		Full Sample	After Exclusion	a
		Error Rate RT	Error Rate RT
	Age	0.25	1.05	0.01	0.00
	Percentage of male participants	4.00 *	1.01	0.01	0.12
	Chin fixer use	1.34	0.01	0.04	0.49
	Experimenter presence	0.09	0.01	0.06	0.07
	Number of experimenters	2.01	1.60	0.96	0.62
	Percentage of male experimenter	1.22	0.551 0.12	0.15
	Compensation type	0.43	2.63	0.06	0.71
	Timing of individual difference measures 0.01	0.03	0.00	0.72
	Note. RT = reaction times.
	a Results obtained after excluding participants who might have responded randomly.
	* p < .05.
	** p < .01.

*** p < .001. Soc Psychol Personal Sci. Author manuscript; available in PMC 2022 January 01.

Soc Psychol Personal Sci. Author manuscript; available in PMC 2022 January 01.

Acknowledgments

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Vetenskapsrådet (2018-06664).

Author Manuscript


Athanase Papadopoulos

SURVEYS IN GEOMETRY, II CHAPTER 1: INTRODUCTION

Keywords: 12D10, 14H55, 20B25, 26C10, 30F10, 30F20, 20F55, 30F30, 30F60, 32G15, 51F20, 51F15, 51M35, 53A35, 53D30, 53C70, 57K10 Conformal geometry, metric geometry, Teichmüller spaces, surfaces immersed in 3-manifolds, symplectic geometry, Grassmann spaces, finite homogeneous metric spaces, polytopes, the Gauss-Bonnet formula, isoperimetry, Coxeter groups

published or not. The documents may come   L'archive ouverte pluridisciplinaire

This is the second volume of a collection of multi-authored surveys in geometry. The project of publishing these volumes arose from the conviction that the mathematical community is in need of good surveys of topics that are at the heart of current research. Thus I have asked some colleagues and friends to write an expository article on a subject they have been working on, which I find particularly important or interesting. I hope the result will be of use to mathematicians, both beginners and experienced.

The topics surveyed in the present volume include the conformal and the metric geometry of surfaces, Teichmüller spaces, surfaces immersed with prescribed extrinsic curvature in 3-dimensional manifolds, symplectic geometry, the metric theory of Grassmann spaces, finite homogeneous metric spaces, regular and semiregular polytopes, the Gauss-Bonnet formula and its higher-dimensional versions, isoperimetry in finitely generated groups, and Coxeter groups. Let me review now each chapter in some detail.

Chapter 2, by Norbert A'Campo and me, is titled Geometry on surfaces, a source for mathematical developments. It is an overview of the theory of surfaces equipped with various structures: volume forms, almost complex structures, Riemannian metrics of constant curvature, quasiconformal structures, and others. Several topics discussed originate in the Riemann mapping theorem and its generalization, the uniformization theorem. The theory of Riemann surfaces is intertwined with topology and combinatorics. Higher-dimensional structures are also discussed. We also review several applications of a colored graph associated with a branched covering of the sphere, which we call a Speiser graph. Nevanlinna, Teichmüller and others obtained criteria, using this graph, for the determination of the type of a simply connected surface, that is, to know whether the surface is conformally equivalent to the Euclidean plane or to the unit disc. The same graph appears in the theory of dessins d'enfants as well as in a realization theorem of Thurston concerning the characterization of some branched coverings of the sphere. More precisely, Thurston's theorem gives a necessary and sufficient condition for a 4-valent graph on the 2sphere to be obtained as a pullback by a branched cover of the sphere of a closed curve passing through all the singular values. Combinatorial characterizations of polynomials and rational maps among a class of branched covers of the sphere are discussed. We also survey the notion of rooted colored trees associated with slalom polynomials, and certain graphs that are used in the stratification of the space of monic polynomials. New models of the Riemann sphere and of hyperbolic 2-and 3-spaces appear at several places of the survey, some of them in an expected manner, using algebra (the ring of polynomials). By analogy, some constructions based on the field of complex numbers are extended to arbitrary fields.

Chapter 3, by Ken'ichi Ohshika, is titled Teichmüller spaces and their various metrics. This chapter is a survey of three different Finsler metrics on Teichmüller space: the Teichmüller metric, Thurston's asymmetric metric and the earthquake metric. In particular, the author presents some recent results he obtained with Y. Huang, H. Pan and A. Papadopoulos on the so-called earthquake metric, also introduced by Thurston. He reviews a duality established in that paper between the tangent space at an arbitrary point of Teichmüller space equipped with the earthquake norm and the cotangent space at the same point equipped with Thurston's co-norm. He also reports on a result, also obtained in that paper, saying that the earthquake metric is not complete, and he provides a description of its completion.

Chapter 4 by Marc Troyanov is titled Double forms, curvature integrals and the Gauss-Bonnet formula. We recall that the Gauss-Bonnet formula for surfaces is a major achievement of nineteenth century differential geometry, and is one of the very good examples of how topology is closely related to geometry. In its simplest form, the formula says that for a closed surface S equipped with a Riemannian metric, the integral of the Gaussian curvature is equal (up to a universal constant) to the Euler characteristic of the surface. There are much more evolved forms of the formula. Troyanov, in Chapter 4, recalls that the Gauss-Bonnet formula is due to the effort of several mathematicians, namely, K. F. Gauss, P. Bonnet, J. Binet, and W. von Dyck, and that extensions of this formula to higher-dimensional Riemannian manifolds were obtained in the twentieth century by H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern. The extended formula establishes relations between the Euler characteristic of a smooth, compact Riemannian manifold with (possibly empty) boundary and a curvature integral over the manifold plus a boundary term that involves a combination of curvature and the second fundamental form over the boundary. In this chapter, the author revisits the higherdimensional formula using the formalism of double forms, a tool introduced by G. de Rham and further developed by R. Kulkarni, J. Thorpe and A. Gray in the 1960s and 1970s. In particular, he surveys the history and the techniques around Chern's version of the formula, which uses É. Cartan's moving frame formalism. He explores the geometric nature of the boundary term and he provides examples and applications. This chapter is also an occasion for Troyanov to make a detailed account of the various tools that were used by various authors in the proof of the higher-dimensional Gauss-Bonnet formula, starting with H. Hopf's problem on the Curvatura Intega, including the theories of double forms with their geometric applications, the Pfaffian, the moving frame, the Gauss-Kronecker curvature of hypersurfaces, the Lipschitz-Killing curvature, and several other notions. INTRODUCTION 3 Chapter 5, by Graham Smith, is titled Quaternions, Monge-Ampère structures and κ-surfaces. This chapter combines Riemannian geometry, conformal geometry, symplectic geometry and quaternionic geometry (an analogue of complex geometry where the field of complex numbers is replaced by that of quaternions). The study is based on a work of F. Labourie. The latter developed, in his paper Problèmes de Monge-Ampère, courbes pseudo-holomorphes et laminations, published in 1997, a theory of surfaces immersed in three-manifolds with prescribed extrinsic curvature. This theory has found applications in hyperbolic geometry, general relativity, Teichmüller theory and other domains. Labourie's key insight is that if the second fundamental form of an immersed surface of prescribed extrinsic curvature is positive definite (the surface is then called infinitesimally strictly convex), then its Gauss lift is a pseudo-holomorphic curve for some suitable almost complex structure. This allows the application of Gromov's theory developed in his paper Pseudo-holomorphic curves in symplectic manifolds (1987), and in particular, his compactness results for families of immersed surfaces of prescribed extrinsic curvature in 3-dimensional Riemannian manifolds. Smith, in this chapter, builds on Labourie's work, of which he gives a quaternionic reformulation. This leads him to simpler proofs of Labourie's results, and at the same time, to generalisations to higher-dimensions. Two theorems of Labourie are in the background: a compactness result for sequences of quasicomplete pointed infinitesimally strictly convex immersed surfaces in a complete, oriented, 3-dimensional Riemannian manifold with prescribed curvature, and a description of the accumulation points of such a sequence. A key result of this chapter establishes a relation between the solutions of the 2-dimensional Monge-Ampère equation and pseudo-holomorphic curves.

Chapter 6, by Peter Kristel and Eric Schippers, is titled Lagrangian Grassmannians of polarizations. This chapter is an introduction to polarization theory in symplectic and orthogonal geometries. In this setting, one starts with a triple of structures on a real vector space, namely, an inner product, a symplectic form and a complex structure, the triple satisfying a compatibility condition that ensures that when we are provided with two out of these three structures, we can reconstruct the third one, if it exists (which is not always the case). The specification of such a compatible triple is equivalent to a decomposition of the complexified ambient vector space into the eigenspaces of the complex structure. A familiar example of such a triple of structures is that of a Kähler manifold, where the manifold is equipped with three compatible structures: a Riemannian metric, an integrable complex structure and a symplectic form. Kristel and Schippers adopt the natural point of view of fixing one of these structures, and studying the space of structures of a second type allowing the reconstruction of a compatible structure of the third type. For example, we can fix a symplectic manifold and study the space of integrable complex structures on the manifold such that this manifold admits a compatible Riemannian metric. In the case where either the symplectic form or the inner product is fixed, we get a Grassmannian of polarizations. Polarizations appear in several contexts of algebraic and complex geometry: in the study of moduli spaces, in the theory of metaplectic and spin representations, and in conformal field theory. In Chapter 6, Kristel and Schippers survey this circle of ideas, in which the underlying vector spaces are allowed to be infinite-dimensional, emphasizing the symmetry of the symplectic and orthogonal settings. The authors consider two particular situations: the Riemannian one, in which the polarization is an orthogonal decomposition, and the symplectic one, in which the polarization is a symplectic decomposition. The potential fields of applications of this theory of polarizations include representation theory, loop groups, complex geometry, moduli spaces, quantization, and conformal field theory.

The next three chapters are concerned with metric geometry. Chapter 7, by Árpád Kurusa, is titled Metric characterizations of projectivemetric spaces. The author starts by recalling Hilbert's Problem IV of the list of problems he proposed in 1900 at the Paris International Congress of Mathematicians. The problem asks for the construction of all the projective metrics on an open convex subset of projective space, that is, the metrics whose geodesics are the intersections of this open subset with the projective lines of the ambient space. Kurusa addresses the general question of characterizing such spaces under the effect of adding some additional conditions on the metric. On the same occasion, he surveys notions like Hilbert and Minkowski metrics, projective center, Ptolemaic metric, Erdös ratio, conics in metric spaces, general Finsler projective metrics, the Ceva and Menelaus properties, and other properties of triangles in a projective-metric setting. Notions such as plane and line perpendicularity, equidistance of lines, bisectors, medians, bounded curvature and others, that hold in a general metric space, are used. These properties were introduced in such a general setting by Herbert Busemann. The questions of characterizing Minkowski planes, of Hilbert geometries and of the three classical geometries, which were extensively investigated by Busemann, are addressed by Kurusa, who also mentions several open problems on this topic.

Chapter 8, by the same author, titled Supplement to "Metric Characterization of projective-metric spaces", is a supplement to the previous chapter whose goal is to provide proofs of two theorems. The first one, due to B. B. Phadke, says that a projective-metric space is a Minkowski plane if all equidistants to geodesics are geodesics. The proof that the author provides is different from Phadke's original proof. The second theorem, due to the author, says that a projective-metric plane has the Ceva property (a generalization of the classical Ceva property) if and only if it is either a Minkowski plane or a model of hyperbolic or elliptic geometry. The proofs are long, which is the reason why these theorems are included in a separate chapter.

Chapter 9, by Boumediene Et-Taoui, is titled Metric problems in projective and Grassmann spaces. In this chapter, the author studies metric problems in real, complex and quaternionic spaces concerning equiangular lines and equi-isoclinic n-subspaces. Let us recall the setting. We take F to be the real, complex or quaternionic field. Let p ≥ 3 and r ≥ 2 be two integers. A collection of lines in F r is said to be equiangular if any two lines in this collection make the same nonzero angle. A set of p n-subspaces in F r is said to be equi-isoclinic if this set spans F r and if any two lines in this set make the same non-zero angle. As the author notes, such structures appear, under various names, in fields such as discrete geometry, combinatorics, harmonic analysis, frame theory, coding theory and quantum information theory. The author addresses two natural questions, namely, (i) How many equiangular lines can be placed in F r ? (ii) How many equiangular equi-isoclinic n-subspaces can be placed in F r ? The chapter is a survey of the work done and the developments due to various authors in the last 70 years on these and related questions, interpreted in the setting of the metric geometry INTRODUCTION 5 of projective and Grassmann spaces. This metric setting involves classical and fundamental works of Menger, Blumenthal, Lemmens, Seidel and others, as well as works of the author himself with co-authors.

Chapter 10, by Valeriȋ Berestovskiȋ and Yuriȋ Nikonorov, is titled On the geometry of finite homogeneous subsets of Euclidean spaces. The authors review recent results on finite homogeneous metric spaces, that is, spaces on which their isometry group acts transitively. They are especially interested in finite homogeneous metric subspaces of a Euclidean space that represent vertex sets (assumed to lie on a sphere) of compact convex polytopes whose isometry groups are transitive on the vertex set. In particular, the authors are led to the classification of regular and semiregular polytopes in Euclidean spaces, according to whether or not they satisfy the normal homogeneity property or the Clifford-Wolf homogeneity property on their vertex sets. The metric spaces that satisfy these two properties constitute a remarkable subclass of the class of homogeneous metric spaces. These properties are stronger than the usual homogeneity properties used for homogeneous metric spaces. The definitions of normal and Clifford-Wolf homogeneity involve a property satisfied by the isometry taking one point to the other, in the definition of homogeneity. The fact that such a study is closely related to the theory of convex polytopes in Euclidean spaces makes it natural to first check the presence of these properties for the vertex sets of regular and semiregular polytopes. Berestovskiȋ and Nikonorov are then led to the study of the m-point homogeneity property and to the notion of point homogeneity degree for finite metric spaces. They discuss several recent results, in particular, the classification of polyhedra with all edges of equal length and with 2-point homogeneous vertex sets, and they present results on the point homogeneity degree for some important classes of polytopes. While discussing these classification results, the authors explain in detail the main tools used for the study of the relevant objects, and they discuss prospects for future results, presenting several open problems.

Chapter 11 by Gue-Seon Lee and Ludovic Marquis, is titled Discrete Coxeter groups. It is an introduction to Coxeter groups, with a focus on how these groups can be used for the construction of discrete subgroups of Lie groups. Coxeter groups are geometrically defined groups. They were introduced in 1934 by H. S. M. Coxeter. They are generated by reflections in some spaces. They generalize the Euclidean reflection groups and the symmetry groups of regular polyhedra. They appear in several areas of mathematics, in particular in the theory of representations of discrete groups in Lie groups. The topics discussed in this chapter include, besides the general Coxeter groups, the theories of reflection groups in hyperbolic space, convex cocompact projective reflection groups, projective reflection groups, divisible and quasi-divisible domains in Hilbert geometry, and Anosov representations. The latter constitute a generalization of the class of discrete convex cocompact representations of hyperbolic groups into rank one Lie groups to the setting of representations of hyperbolic groups into semi-simple Lie groups. Anosov representations were introduced by Labourie in his paper Anosov flows, surface groups and curves in projective space (2006), in which he studies Hitchin representations.

Chapter 12, by Bruno Luiz Santos et Marc Troyanov, is titled Isoperimetry in finitely generated groups. The setting is that of infinite finitely generated groups equipped with word metrics associated with finite symmetric sets of generators. An isoperimetric inequality is an inequality between the size of an arbitrary finite set and the size of its boundary (for an appropriate definition of boundary). In this chapter, the authors revisit the work done in the 1980s-1990s by N. Varopoulos, T. Coulhon and L. Saloff-Coste on isoperimetric inequalities in finitely generated groups, adding new results and establishing relations with other topics. In particular, they obtain lower bounds on the isoperimetric quotient that appears in the isoperimetric inequality in terms of the U-transform, which is a variant of the classical Legendre transform of a function, or its generalization, the Legendre-Fenchel transform, which is used in physics. The chapter also includes a review of some basic elements from geometric group theory (growth functions, amenability, the Cheeger constant, etc.) as well as some basic elements from the theory of U-transform, including some computational techniques, and the relation between the U-transform and the Legendre transform.

The first two chapters of this volume are based on lectures given by the authors at two thematic programs at Banaras Hindu University, in December 2019 and December 2022, which I organized with Bankteshwar Tiwari. The programs were funded by CIMPA (Centre International de Mathématiques Pures et Appliquées), IMU (International Mathematical Union), SERB (Science and Engineering Research Board of the Government of India), NBHM (National Board of Higher Mathematics of the Government of India), and SRICC (Sponsored Research and Industrial Consultancy Cell) and I.SC (the Institute of Science) of Banaras Hindu University.

Athanase Papadopoulos, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France

Email address: papadop@math.unistra.fr


Evelyne Benoit

A non-invasive method to appraise time-dependent effects of venom toxins on the mouse neuromuscular excitability in vivo

The conventional methods, usually performed in vivo, provide limited information regarding sensory and neuromuscular excitability properties. In the present work, we review the new minimally invasive methods that have been improved to supplement the conventional electrophysiological methods. In particular, the automated sequence of multiple excitability tests developed by Professor Bostock (Qtrac©, Institute of Neurology, London, UK) to assess sensory and neuromuscular excitability in clinical neurophysiology, has been adapted to animal models in vivo, initially in rats and, more recently, in mice. This method is particularly appropriate for venom toxin injections and repeated recordings from the same animal cohort, making it a promising tool to appraise the progression of toxin effects, and that of a given treatment, on the sensory and neuromuscular excitability of animal models in vivo. The in vivo efficiency in mice and long-term follow-up of the effects of sub-lethal doses of spider and conus venom toxins will be emphasised. It is indubitable that this method brings specific information regarding changes in ion transfer and membrane properties induced by a given venom toxin which, thereby, will help to a better characterization of toxin action on the sensory and neuromuscular excitability properties, in vivo.

Biography

Evelyne BENOIT has completed her PhD in Neuroscience at the age of 26 years and her habilitation to lead research in 2010 from Paris-XI-Orsay University. She is researcher at the CNRS and responsible for "Ion transfers and pathophysiology of neuromuscular conduction" studies. To gain a better understanding of these processes, she studies the mechanism of action of emerging neurotoxins and uses some of these bioactive agents because of their specificity of action. Since 2016, she is President of the French Society of Toxinology (http://sfet.asso.fr/international/). She has published more than 100 papers in reputed journals and more than 40 book chapters.

Presenting author details

A

  non-invasive method to appraise time-dependent effects of venom toxins on the mouse neuromuscular excitability in vivo Evelyne Benoit CEA, Institut des Sciences du Vivant Frédéric Joliot, SIMOPRO, and Institut des Neurosciences Paris-Saclay, UMR CNRS/Université Paris-Sud 9197, Université Paris-Saclay, 91190 Gif-sur-Yvette, France

  Full name: Evelyne BENOIT Contact number: 33.6.6375.1939 Twitter account: None Linked In account: None Session name/number: Toxicology research Category: Oral presentation


est destinée au dépôt

Over the last decades, animal venom toxins have been explored as an original source of new antinociceptive drugs. Our target of interest, NaV1.7, has been validated by human genetic evidence and is mainly expressed in dorsal root ganglia (DRG) neurons. High throughput screening of Sanofi's collection of venom toxins was performed on HEK cells overexpressing human Na V subtypes. The results highlight a new toxin, cyriotoxin-1a (CyrTx-1a), sharing the same inhibitory cysteine knot (ICK) motif with toxins originating from the same theraphosids spider family, such as huwentoxin-IV (HwTx-IV), a potential antalgic targeting Na V1.7. However, when tested in mouse isolated neuromuscular preparations at low concentrations (that did not affect the Na V1.4 muscle subtype), HwTx-IV inhibited completely the muscle contraction, thus incriminating the Na V1.6 neuronal subtype. CyrTx-1a and HwTx-IV revealed nanomolar range affinity for NaV1.7, 1.6, 1.1, 1.2 and 1.3 and micromolar range affinity for Na V1.5, 1.4 and 1.8. Moreover, in mouse DRG neurons, the peptides inhibited preferentially tetrodotoxin (TTX)-sensitive Na current flowing mainly through NaV1.7, compared with TTX-resistant Na current flowing through Na V1.8 and 1.9. Within a therapeutic window of concentrations sparing skeletal muscular system, the peptides exhibited a similar analgesic-like effect in mouse pain models (hot plate and von Frey essays). However, compared with HwTx-IV, CyrTx-1a had less marked in vivo sideeffects, as detected by analyzing in vivo the excitability properties of the mouse neuromuscular system, mediated by NaV1.4 and 1.6. In conclusion, the better in vivo pharmacological profile of CyrTx-1a paves the way for further engineering studies aimed to optimize its potential antinociceptive capability.

  Antinociceptive evaluation of cyriotoxin-1a and huwentoxin-IV, two spider toxins that block with high affinity the Nav1.7 subtype of voltage-gated sodium channels Tânia C. Gonçalves, Evelyne Benoit, Denis Servent, Michel Partiseti To cite this version: Tânia C. Gonçalves, Evelyne Benoit, Denis Servent, Michel Partiseti. Antinociceptive evaluation of cyriotoxin-1a and huwentoxin-IV, two spider toxins that block with high affinity the Nav1.7 subtype of voltage-gated sodium channels. 29th Ion Channel Meeting, Association Canaux Ioniques, Sep 2018, Sète, France. hal-04464122

Sanofi R & D, France.

Service d'Ingénierie Moléculaire des Protéines (SIMOPRO), CEA, Université Paris-Saclay, F-91191 Gif sur Yvette, France.


Jean-Michel Courtault
email: jean-michel.courtault@univ-fcomte.fr.

Bertrand Crettez
email: bertrand.crettez@univ-fcomte.fr

Naila Hayek
email: naila.hayek@univ-fcomte.fr.

Characterization of Stochastic Dominance for Discrete Random Variables

Keywords: J.E.L Classification Numbers: D80, D81, G11 Stochastic Dominance, Discrete Random Variables

Available characterizations of the various notions of stochastic dominance concern continuous random variables. Yet, discrete random variables are often used either in pedagogical presentations of stochastic dominance or in experimental tests of this notion. This note provides complete characterizations of the various notions of stochastic dominance for discrete random variables.

Introduction

The notion of stochastic dominance is a key concept in social sciences. It is especially relevant in the economic theory of risk bearing as well as in the analysis of inequalities. To the best of our knowledge, while the definition of stochastic dominance applies to general random variables, available characterizations make the assumptions of a differentiable utility function as well as a continuous density function. There are two reasons why a characterization would be useful for the case of a discrete random variable. First, pedagogical presentations of stochastic dominance often rely on an informal discretization argument whereas a rigorous proof would be more satisfying. For a remarkable pedagogical presentation of first-and second-order stochastic dominance, see for instance [START_REF] Elton | Modern Portfolio, Theory and Investment Analysis[END_REF]. Their presentation uses a discrete distribution (pages 241-247) while the proofs use a continuous distribution (page 254-255). [START_REF] Chavas | Risk Analysis in Theory and Practice[END_REF] is also a very good source on this issue. Chavas even provides expressions in the case of a discrete distribution from which one could characterize first-and second-order stochastic 1 dominance (see [START_REF] Chavas | Risk Analysis in Theory and Practice[END_REF], pages 56-60 and page 65). However, for the case of thirdorder stochastic dominance he gives an expression of the difference of expected utilities which seems to be only an approximation of the continuous case (see [START_REF] Chavas | Risk Analysis in Theory and Practice[END_REF], page 60, "discrete implementation")). This prevents one from having a characterization of thirdorder stochastic dominance in the discrete case. Hence, a general argument seems to be needed 1 . Second, experimental tests of stochastic dominance make use of discrete random variables and discrete variants of characterizations available in the continuous case (see, e.g., [START_REF] Batalio | Animals' Choices over Uncertain Outcomes: Some Initial Experimental Results[END_REF] 2 , [START_REF] Birnbaum | A Comparison of Five Models that Predict Violations of First-Order Stochastic Dominance in Risky Decision Making[END_REF] andBirnbaum andNavarette (1998)). Here again, a proof that the analogy is indeed relevant seems to be missing. This note presents characterizations of stochastic dominance for discrete random variables and provides a rigorous support for the usual practice of using a discrete random variables setting.

This note is organized as follows. Some definitions and notations are presented in section 2. We present a characterization of first-and second-order stochastic dominance (these are the notions which are usually considered) in section 3. A characterization of the k th -order stochastic dominance is offered in the appendix. The proof uses an induction argument.

Definitions and Notations

Let u : R → R be a utility function. Consider a list of real numbers such that: x 1 < x 2 < ... < x n . We set u i = u(x i ), i=1,. . . , n, and u = (u 1 , ..., u n ). Let U 1 denote the class of increasing utility functions on {x 1 , x 2 , ..., x n }, i.e. functions u satisfying

(u i+1 -u i ) (x i+1 -x i ) ≥ 0 for all i, i = 1, .., n -1. Set u 1 i = u i+1 -u i x i+1 -x i ; i = 1, ..., n -1,
and u 1 = (u 1 1 , ..., u 1 n-1 ). u 1 = 0 means that there exists i such that u 1 i = 0. Let U 2 denote the class of utility functions u which are in U 1 and safisfy3 :

(u i+2 -u i+1 ) (x i+2 -x i+1 ) - (u i+1 -u i ) (x i+1 -x i )
≤ 0 for all i, i = 1, .., n -2.

1 Chambers and Quiggin (2004), page 113, have already stressed the interest of studying simple mean preserving spreads in the discrete case. As they put it, "..., most discussion of stochastic dominance revolves around continuous probability distributions. We, therefore, felt it important to develop these ideas in some depth for the discrete case...". Nevertheless, these authors do not address the case of stochastic dominance per se. The notion of increasing risk and mean preserving spread have also been presented for discrete random variables by [START_REF] Rothschild | Increasing risk: I. A definition[END_REF] (see also, [START_REF] Leshno | A Comment on Rothschild and Stiglitz"s "Increasing Risk: I. A definition[END_REF]).

We consider two probability functions f (x) and g(x) where the random variable x can take the n possible values x 1 , ..., x n . We define f i = f (x i ) and g i = g(x i ). We assume that f = g.

Let F and G be their distribution functions respectively. We set

F i = F (x i ) = i j=1 f (x i ) and G i = G(x i ) = i j=1 g(x i ), i = 1, ..., n. Notice that F 1 = f 1 , F n = 1, G 1 = g 1 , G n = 1
. Finally, we let E f and E g be the expectation operators based on f and g respectively.

3 Characterization of first-and second-order stochastic dominance

We first recall Abel's Lemma4 , which is a discrete version of the summation by parts formula.

Lemma 1 (Abel's Lemma) Let a 1 , ..., a n , b 1 , ..., b n be real numbers. Set A i = i j=1 a j and B i = i j=1 b j . Then n i=1 a i b i = n-1 i=1 A i (b i -b i+1 ) + A n b n .
We now give the definition of stochastic dominance.

Definition 1 We say that f first-order stochastically dominates g (and we write f

≥ F SD g) if E f u(x) ≥ E g u(x) for all u ∈ U 1 . Let D 1 = G -F , D 1 i = D 1 (x i ) = G(x i ) -F (x i ) = G i -F i .
The following Proposition characterizes the first-order stochastic dominance property.

Proposition 1 (First-order stochastic dominance). f ≥ F SD g if and only if D 1 i ≥ 0 for all i = 1, ..., n -1.

Proof. By definition, E f u(x) -E g u(x) = n i=1 (f i -g i )u i . Set f i -g i = a i , u i = b i , F i -G i = A i and apply Lemma 1. So: E f u(x) -E g u(x) = n-1 i=1 (F i -G i )(u i -u i+1 ) + (F n -G n )u n = n-1 i=1 (-D 1 i )(u i -u i+1 ) = n-1 i=1 D 1 i (u i+1 -u i ) (1) since F n -G n = 0. It is clear that if D 1 i ≥ 0 for all i = 1, ..., n -1 then f ≥ F SD g. Let us show the converse. Suppose that f ≥ F SD g so n-1 i=1 D 1 i (u i+1 -u i ) = E f u(x) -E g u(x)
≥ 0 for all u ∈ U 1 . If there exists j such that D 1 j < 0, then consider u where u i = u i+1 , ∀i = j, and u j+1 > u j . We have u ∈ U 1 and E f u(x) -E g u(x) = D 1 j (u j+1u j ) < 0 which contradicts the hypothesis. Q.E.D.

We now turn to the case of second order stochastic dominance.

Definition 2 We say that f second-order stochastically dominates g (and we write f

≥ SSD g) if E f u(x) ≥ E g u(x) for all u ∈ U 2 .
The following Proposition characterizes the second order stochastic dominance property.

Proposition 2 (Second-order stochastic dominance) f ≥ SSD g if and only if

D 2 i = i j=1 D 1 j (x j+1 -x j ) ≥ 0 for all i = 1, ..., n -1.
Proof. From Proposition 1,

E f u(x) -E g u(x) = n-1 i=1 D 1 i (u i+1 -u i ) = n-1 i=1 D 1 i (x i+1 -x i ) (u i+1 -u i ) (x i+1 -x i ) (2)
We shall use again Lemma 1 by setting

D 1 i (x i+1 -x i ) = a i , (u i+1 -u i ) (x i+1 -x i ) = b i so that A i = i j=1 a j = i j=1 D 1 j (x j+1 -x j ).
Applying now Lemma 1 gives:

E f u(x) -E g u(x) = n-2 i=1 - i j=1 D 1 j (x j+1 -x j ) (u i+2 -u i+1 ) (x i+2 -x i+1 ) - (u i+1 -u i ) (x i+1 -x i ) + n-1 j=1 D 1 j (x j+1 -x j ) (u n -u n-1 ) (x n -x n-1 ) = n-2 i=1 D 2 i (u i+2 -u i+1 ) (x i+2 -x i+1 ) - (u i+1 -u i ) (x i+1 -x i ) + D 2 n-1 (u n -u n-1 ) (x n -x n-1 ) (3)
It is clear that if D 2 i ≥ 0 for all i = 1, ..., n -1 then f ≥ SSD g. Let us show the converse. Suppose that f ≥ SSD g that is E f u(x)-E g u(x) ≥ 0 for all u ∈ U 2 . If there exists j such that D 2 j < 0, then consider u where

u i = u i+1 , ∀i ≥ j + 1, u j+1 > u j and u 1 i+1 = u 1 i , ∀i ≤ j -1 that is (u i+2 -u i+1 ) (x i+2 -x i+1 ) = (u i+1 -u i ) (x i+1 -x i ) , ∀i ≤ j -1. We have u ∈ U 2 and E f u(x) -E g u(x) = -D 2 j ( -(u j+1 -u j ) (x j+1 -x j )
) < 0 which contradicts the hypothesis. Q.E.D.

A characterization of the k th -order stochastic dominance is offered in the appendix.

Concluding Remarks

In this section we shall give two remarks concerning characterization of stochastic dominance with alternative definitions.

Let U s 1 denote the class of strictly increasing utility functions i.e. functions u satisfying

u 1 i = (u i+1 -u i ) (x i+1 -x i ) > 0 for all i, i = 1, .., n -1.
Remark 1. Some authors (for instance, [START_REF] Chavas | Risk Analysis in Theory and Practice[END_REF], [START_REF] Elton | Modern Portfolio, Theory and Investment Analysis[END_REF]) use the following definition:

Definition 3 We say that f first-order-s-stochastically dominates g (and we write f ≥ F sSD g) if and only if E f u(x) ≥ E g u(x) for all u ∈ U s 1 . The following Lemma, whose proof is in this appendix, shows that definitions 1 and 3 are essentially similar.

Lemma 2 f ≥ F sSD g if and only if f ≥ F SD g.
The next Proposition characterizes definition 3 of stochastic dominance.

Proposition 3 (First-order-s stochastic dominance). f ≥ F sSD g if and only if D 1 i ≥ 0 for all i = 1, ..., n -1.

Proof. This is an immediate consequence of Lemma 2 and Proposition 1. Q.E.D.

The main reason for using definition 1 instead of definition 3 is that the proofs of the characterizations in the high-order case are simpler5 .

Remark 2. One could also propose to strengthen the definition of stochastic dominance in requiring that the inequality be strict. More precisely, we could use the following definition:

Definition 4 We say that f strictly first-order stochastically dominates g (and we write

f > F SD g) if and only if E f u(x) > E g u(x) for all u ∈ U s 1 .
Using an argument similar to that of Proposition 3 we can show that f > F SD g if and only if D 1 i ≥ 0 for all i = 1, ..., n -1 with D 1 j > 0 for some j ∈ i, ..., n -1.

We shall need the following Lemma.

Lemma 3 For every k = 1, ..., n -1, we have:

E f u(x) -E g u(x) = n-k i=1 (-1) k+1 D k i (x i+1 -x i )u k i + k r=2 (-1) r D r n-r+1 u r-1 n-r+1 .
Proof. The proof is by induction. We know from the previous propositions that it is true till k = 2.(Recall also that D 1 n = 0, this is why r starts at 2.) Suppose it is true for k and let us show it is true for k + 1 provided k ≤ n -2:

E f u(x) -E g u(x) = n-k i=1 (-1) k+1 D k i (x i+1 -x i )u k i + k r=2 (-1) r D r n-r+1 u r-1 n-r+1
Apply Lemma 1 by setting:

(-1) k+1 D k i (x i+1 -x i ) = a i , u k i = b i , so A i = i j=1 a j = i j=1 (-1) k+1 D k j (x j+1 -x j ) = (-1) k+1 D k+1 i .
This gives:

n-k i=1 a i b i = n-k-1 i=1 A i (b i -b i+1 ) + A n-k b n-k ,
so:

E f u(x)-E g u(x) = n-k-1 i=1 (-1) k+1 D k+1 i (u k i -u k i+1 )+(-1) k+1 D k+1 n-k u k n-k + k r=2 (-1) r D r n-r+1 u r-1 n-r+1
.

and

E f u(x) -E g u(x) = n-k-1 i=1 (-1) k+1 D k+1 i (x i+1 -x i ) (u k i -u k i+1 ) (x i+1 -x i ) + k+1 r=2
(-1) r D r n-r+1 u r-1 n-r+1 .

Therefore:

E f u(x) -E g u(x) = n-k-1 i=1 (-1) k+2 D k+1 i (x i+1 -x i )u k+1 i + k+1 r=2 (-1) r D r n-r+1 u r-1 n-r+1 .
and the proof is complete. Q.E.D.

The characterization of the k-th-order stochastic dominance property relied on the following remark. From now on, we assume that k ≥ 2.

Remark. Let z ≥ 2 and j be fixed. Consider a function u : R → R such that

u i+1 = u i for i ≥ j + 1 u j+1 > u j u z-1 i = u z-1 i+1 for i ≤ j -1 .
We claim that this function is in U z . Indeed we have u m i = 0 for 1 ≤ m ≤ z, j + 1 ≤ i ≤ nm. We also have:

u 1 j > 0, u 2 j = u 1 j+1 -u 1 j x j+1 -x j = -u 1 j x j+1 -x j < 0 and for m ≤ z, u m j = -u m-1 j x j+1 -x j = (-1) m-1 (u j+1 -u j ) (x j+1 -x j ) m so (-1) m+1 u m j = (-1) m+1 (-1) m-1 (u j+1 -u j ) (x j+1 -x j ) m > 0. Moreover for i ≤ j -1, (-1) z u z-1 i > 0 since u z-1 i = u z-1 j for i ≤ j -1. Now we can show that for 1 ≤ i ≤ j -1 , (-1) z-1 u z-2 i > 0 . Since u z-2 j-1 = u z-2 j -u z-1 j (x j - x j-1 ), we have (-1) z-1 u z-2 j-1 = (-1) z-1 u z-2 j -(-1) z-1 u z-1 j (x j -x j-1
) > 0 and we can compute recursively u z-2 i for 1 ≤ i ≤ j -2. In a similar way we can compute recursively u m i for i ≤ j -1, 2 ≤ m ≤ z -3 to show that (-1) m+1 u m i > 0 .

Proposition 4 (k-th-order stochastic dominance) f ≥ kSD g if and only if

D k i = i j=1 D k-1 j (x j+1 -x j ) ≥ 0 for all i = 1, ..., n -k + 1 and D r n-r+1 ≥ 0 for all r = 2, ..., k -1.
Proof. From the previous Lemma we have:

E f u(x) -E g u(x) = n-k i=1 (-1) k+1 D k i (x i+1 -x i )u k i + k r=2 (-1) r D r n-r+1 u r-1 n-r+1 .
It is clear that if D k i ≥ 0 for all i = 1, ..., nk + 1 and D r n-r+1 ≥ 0, for all r = 2, • • • , k -1, then f ≥ F SD g. Let us show the converse. Assume then that f ≥ kSD g that is E f u(x) -E g u(x) ≥ 0 for all u ∈ U k . Let us first show that D k i ≥ 0, ∀i ≤ nk. If this is false, there exists j, with D k j < 0, 1 ≤ j ≤ nk. We now use the function u presented in the above remark with z = k. We have: E f u(x) -E g u(x) = D k j (-1) k+1 (x j+1x j )u k j < 0 since (-1) k+1 u k j > 0 (u being in U k ). But this contradicts the assumption. Now, let us show that D r n-r+1 ≥ 0, ∀r = 2, • • • , k. If not, there exists p such that D p n-p+1 < 0, 2 ≤ p ≤ k . We use once again the function presented in the above remark with z = p, j = np + 1. So u i+1 = u i for i ≥ np + 2, u n-p+2 > u n-p+1 and u p-1 i = u p-1 i+1 , 1 ≤ i ≤ np. Thus we obtain u p i = 0, for i ≤ np, (-1) p u p-1 n-p+1 > 0, and u m i = 0, 1 ≤ m ≤ p -2, np + 2 ≤ i ≤ nm (only if p > 2).So we have: E f u(x) -E g u(x) = (-1) p D p n-p+1 u p-1 n-p+1 < 0 since (-1) p u p-1 n-p+1 > 0 (u being in U k ). But this contradicts the assumption. Q.E.D.

[START_REF] Batalio | Animals' Choices over Uncertain Outcomes: Some Initial Experimental Results[END_REF] contains an early test of second-order stochastic dominance.

The functions defined on an interval of R satisfying this property are the concave ones. See e.g,[START_REF] Aliprantis | Infinite Dimensional Analysis[END_REF], page 179.

See, e.g., http://planetmath.org/encyclopedia/AbelsLemma.html.

Moreover, definition 1 is implicit in some classic textbooks, e.g.[START_REF] Huang | Foundations for Financial Economics[END_REF] (page 40), and[START_REF] Ingersoll | Theory of Financial Decision Making[END_REF] (page 137). For instance, in order to show the only if part of the characterization in Proposition 3, these authors make use of a non-decreasing function.

Appendix

A1) Characterization of k th -order stochastic dominance.

Set

We let U k denote the class of functions u such that (-1) r+1 u r i ≥ 0 , r = 1, ..., k, i = 1, ..., n-k. let

Definition 5 We say that f k-th-order stochastically dominates g (and we write f

Set m = n -1 and for every i = 1, ..., m, y i = u i+1u i and z i = D 1 i . Let y = (y 1 , ..., y m ) ∈ R m , z = (z 1 , ..., z m ) ∈ R m . y > 0 means ∀i = 1, ..., m, y i > 0. With these notations it suffices to show that the following propositions are equivalent:

We can write (i) and (ii) in the following way:


Rémy Guyonneau

Franck Mercier
email: franck.mercier@univ-angers.fr

Vincent Boucher
email: vincent.boucher@cerema.fr

Rémy Guyonneau

Remy Guyonneau Laris

Remy Guyonneau
email: remy.guyonneau@univ-angers.fr

Robotic system for indoor illuminance map generation

Keywords:

The paper describes Romulux, a mobile robot equipped with 3D LiDAR carrying an illuminance meter and dedicated to measure indoor lighting quantity. The aim is to check compliance with lighting standards in terms of level and uniformity. The general architecture of the robot using the Robot Operating System (ROS) is presented, and technical choices are explained according to the different constraints. The space localization of illuminance data is computed with Synchronous Localization And Mapping (SLAM) algorithms. Then measurements are compared to standards requirements and interpolated into a user-defined grid to generate a dense illuminance map. Two experiments, one in a sport hall and one in an ice rink, are presented and results are exposed. The paper will follow the workflow from designing a robot to a functional system able to generate illuminance maps, compare with standards and save energy in case of over lighting.

Introduction

Lighting requirements are determined by the satisfaction of three basic human needs: visual comfort, visual performance and safety. The quantifiable parameters subject to the requirements CEN (2021a) CEN (2021b) are the illuminance (amount of light arriving on a surface), its uniformity, the discomfort glare and the color rendering index. When constructing a building, the type and number of lighting elements are sized according to those lighting standards, depending on use of the site if it concerns a hospital, a warehouse, a home, etc. European standards also exist for the lighting of sports facilities [START_REF] Cen | Light and lighting -sports lighting . CEN, 2021a. Light and lighting -lighting of work places -part 1: Indoor work places[END_REF], which ensures good visual conditions for players, athletes, referees, spectators and for color TV broadcasting. The objective is to specify recommendations and requirements to produce good quality lighting for sports facilities ensuring the needs of all users. This is done by optimizing the perception of visual information used during sports events, maintaining the level of visual performance, providing acceptable visual comfort and limiting obtrusive light. Standards provide values for lighting control in terms of illuminance, uniformity, glare control and color properties of light sources. It also provides mesh-grids for measuring these values [START_REF] Cen | Light and lighting -sports lighting . CEN, 2021a. Light and lighting -lighting of work places -part 1: Indoor work places[END_REF].

To meet the standards, it is usually appropriate to carry out a control operation by measuring the illuminance. This operation is done by an operator equipped with an illuminance meter, who must perform spot measurements. The steps are

• Referencing the location of the measurement within a normative mesh-grid;

• Moving away from the illuminance meter to perform the measurement;

• Recording the measurement.

The operation must be done many times in order to achieve a relevant mesh of the area of interest [START_REF] Akashi | A field study of illuminance reduction[END_REF]. This operation is tedious and time-consuming. In practice, this operation is rarely done. Instead, installers tend to over-light to ensure compliance with standards.

The automation of controls is interesting in many ways because it could provide better suitability with the standards: the control will be actually done, the quality of the measurements will be constant and not subject to the human operations uncertainties. A better compliance with the standard could lead to:

• Improve user comfort since the standards are studied in particular with regard to this criterion;

• Save energy thanks to the readjustment of the installation lighting power. This can be easily done, for instance, with dimmed LED lights.

Evaluating lighting quality is of interest to reduce energy consumption Li (2010) Muzet, Colomb, Toinette, Gandon-Leger and Christory (2019a) Muzet, Greffier, Nicolaï, Taron and P.Verny (2019b), to predict building integrated performance [START_REF] Todorovic | Beyond the science and art of the healthy buildings daylighting dynamic control's performance prediction and validation[END_REF], to implement lighting control algorithms Mohammad, Hassan, Abdullah, Rahman, Abdullah, Hussin and Said (2014), [START_REF] Guillemin | An innovative lighting controller integrated in a self-adaptive building control system[END_REF], to evaluate visual comfort [START_REF] Nabil | Useful daylight illuminance: a new paradigm for assessing daylight in buildings[END_REF], [START_REF] Akashi | A field study of illuminance reduction[END_REF] or also to manage road lighting [START_REF] Kumar | Energy efficient smart outdoor lightning system using scada control system[END_REF]; [START_REF] Greffier | A grid in perspective for road lighting calculations[END_REF].

Lighting quality evaluation is then based on illuminance measurements related to spatial locations [START_REF] Carter | Field measurement of illuminance[END_REF]. For outdoor applications, those locations are generally recorded thanks to a Global Navigation Satellite System (GPS for instance) J. and Oliver (2017). But for indoor applications, this technology is no more available and an alternative has to be found. The LiDAR technology has been chosen here for its performance regarding the detection range [START_REF] Warren | Automotive lidar technology[END_REF]. To deal with the standards requirements and to acquire quickly and easily data along a specified mesh-grid, an automated illuminance acquisition system has been developed aiming to construct illuminance maps of a room.

A robot has been designed for an indoor environment. The problem of automated lighting measurement is an active research domain. Attempts are made to perform this task, but encounter localization problems Tang, Inoue, Oshio, Tsujimoto, Taniguchi and Kubota (2020) He, Wang, Zhao, Hao, Hu, Tong, Jiang and Shao (2021) [START_REF] Oshio | 2dslam of illuminance measurement robot using 3d-lidar and imu on slopes[END_REF], or demonstrate the ability to perform localized measurements in extremely controlled environments [START_REF] Drábek | Statistical-based control points selection for indoor illuminance measurement[END_REF]. The contribution of this paper is twofold:

• On the design of the platform which is very accomplished and robust in face of localization issues;

• On the expertise of the illuminance measurements by going to the end of the process with illuminance maps.

The paper is organized as follows: first the mechanical and technical specifications of the system are described, then some experiments and results are presented, and the final section concludes and draws perspectives for this approach.

Requirements, solutions and platform presentation

Objectives, constraints and environment

The objective for developing a robot is to ease the illuminance mapping of indoor environments. To that purpose the robot must:

• Acquire illuminance measurements;

• Locate those measurements in the environment.

However, indoor environments have characteristics that should be considered for robot design. Admittedly, the ground is flat and presents little disturbances (door threshold, concrete threshold bar, electric cables, etc.), but more insidious difficulties may appear (tables, chairs, structural posts, etc.) and lead to both navigation avoidance, and more complex mapping. These constraints must guide the form factor of the robot and its kinematic model for the maneuverability of the system.

The illuminance measurements are governed by standards. In order to comply with EN 12193 CEN (2018) and EN 12464 CEN (2021a[START_REF] Cen | Light and lighting -lighting of work places -part 2: Outdoor work places[END_REF] standards, the illuminance is measured by a calibrated illuminance meter. The spectral characteristics of the photo-diode make it conform to the human eye sensitivity and constitute a key point in measuring illuminance (Figure 1). Here, our illuminance meter is made of a commercial luxmeter OPT3001 associated with a diffuser dome and has been calibrated thanks to an integrating sphere. This sensor has a measurement range from 0.01 lux to 83 k lux. The illuminance meter is carried by the robot and it has been decided to place it at a maximum height of 25 cm above the floor to keep the results comparable to those made on the floor. The illuminance meter owns two integration times of 100 ms or 800 ms depending on the noise sensibility mode. Regarding to the robot velocity, the shorter integration time has been selected. According to a robot speed around 1 m/s, an illuminance measurement is integrated over 10 cm. Moreover, the sensor must not be in the shadow of any other component. The placement height in addition to shadow avoidance forces the integration of the illuminance sensor at the top of robot.

Another constraint is to leave clearance for the localization sensor (360°LiDAR). Both sensors (LiDAR and illuminance sensor) therefore have the same placement requirement: to be at the top of the robot. The proposed solution is to place the illuminance meter on top of the LiDAR at its centre. It allows an agreement at the cost of a loss of vision in a 20°cone behind the LiDAR sensor. This solution has the advantage to avoid a transformation between illuminance sensor and LiDAR localization.

Kinematic study

Kinematic with high maneuverability is useful for indoor mobility. The choice has been oriented on a 4 wheels robot, including 2 wheels with stabilizer bar (or anti-roll bar). This prevents ground irregularities and increases the stability of the robot. Moreover, due to the requirement of high maneuverability, differential kinematic appears to be a good solution. Thus, the chosen kinematic is 2 differential wheels with 2 castor wheels on stabilizer bar. This model allows higher mobility than Ackermann steering, up to rotation only [START_REF] Salem | Dynamic and kinematic models and control for differential drive mobile robots[END_REF]. To avoid additional transformation between the several frames of the system (sensors and robot's chassis), the Li-DAR and illuminance sensors are aligned with the center of rotation (CoR) of the robot.

Robotic system for indoor illuminance map generation

The sensors placement is illustrated in Figure 2 while Figure 3 shows the different variables related to robot kinematics.

In the world frame, the speed of a differential robot can be represented by:

⎡ ⎢ ⎢ ⎣ ẋ ẏ θ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ 𝑉 cos 𝜃 𝑉 sin 𝜃 𝜔 ⎤ ⎥ ⎥ ⎦ , (1)
with (𝑥, 𝑦) 𝑇 the position of the robot, 𝜃 its orientation, 𝑉 the linear speed and 𝜔 the angular speed. In a context of differential architecture, these two speeds can be expressed as a function of wheel speed of the form:

[ 𝑉 𝜔 ] = [ 𝑅∕2 𝑅∕2 𝑅∕𝐿 -𝑅∕𝐿 ] [ 𝑣 𝑟 𝑣 𝑙 ] , ( 2
)
with 𝑅 the radius of the wheels, 𝐿 the distance between the wheels and 𝑣 𝑟 and 𝑣 𝑙 respectively the speed of the right and left wheels in rpm.

Doing so, the Jacobian 𝐉 of the system is:

𝐉 = 𝑅 2𝐿 [ 𝐿 𝐿 2 -2 ] . ( 3
)
Depending on a linear 𝑉 and angular 𝜔 speed set-point, it is possible to compute the corresponding wheel speed using Equations 2 and 3:

[ 𝑉 𝜔 ] = 𝐉 [ 𝑣 𝑟 𝑣 𝑙 ] , ( 4
) [ 𝑣 𝑟 𝑣 𝑙 ] = 𝐉 -1 [ 𝑉 𝜔 ] = 1 2𝑅 [ 2 𝐿 2 -𝐿 ] [ 𝑉 𝜔 ] . ( 5
)
It is the relation of Equation 5that is implemented in the low level functions of the robot.

General Architecture

Figure 4 depicts the general architecture of the robot, named Romulux, the robot which measures lux. The CPU unit centralizes sensors acquisition, interface commands and controls the motors while communicating and recording all Robot Operating System (ROS) topics 1 .

For general requirement, it is wanted an autonomy of operation of one hour, which defines the size of the batteries. Indirectly the DC/DC converters take into account consumption and capability of upstream and downstream systems. Two 24V brushed DC motors are used to motorize the robot. Those have a 28:1 metal gearbox, provided by Maxon. No coder are embedded as it is assumed that the LiDAR will be used for trajectory following and steering. The couple motor/reducer is able to produce 1Nm torque and a rotation speed above 220𝑟𝑝𝑚, corresponding to a linear full speed above 3𝑚∕𝑠. The motors are controlled by Sabertooth card 2 , and communicate through serial communication with the processor, a Nvidia Jetson TX2 3 . The general architecture of the proposed solution is presented in Figure 4. The LiDAR sensor is a Velodyn Puck 4 @15Hz scanning at 360°with vertical slices between -15°and +15°and a 100m range with a typical ±3 cm accuracy. The chassis depicted in Figure 5 has been entirely designed and built to respect the maximum height of 25cm for the illuminance sensor. The maximum speed is therefore set at 3m/s with a working speed of around 1m/s.

Romulux software architecture is based on ROS Quigley, Conley, Gerkey, Faust, Foote, Leibs, Wheeler, Ng et al. (2009). ROS is based on a communication model based on nodes and topics. Nodes are autonomous processes that perform specific tasks, such as controlling a robot's motors or processing data from sensors. These nodes can communicate with each other by publishing or subscribing to topics. Topics are virtual communication channels that allow nodes to share messages asynchronously. A node can publish messages on a specific topic, while other nodes can subscribe to this topic and receive these messages. This flexible and decentralized communication architecture allows different components of a robotic system to operate independently and easily interface with each other. Thanks to the ROS architecture and specifically the twist5 data format, it is quite easy to change the remote device. For the presented experiments, a Dualsense Playstation controller has been considered. The connection between the remote controller and the robot is a Bluetooth link. The working range for this remote has been measured up to 30 m, which allows a fine control in visual range of operator without being too close from the robot and thus corrupting the illuminance data. While the robot is being controlled, all the data (LiDAR, illuminance...) are recorded into rosbag files6 . Those allow to replay the data afterward as if the robot was moving in real time. In this scenario, the illuminance map is built off-line. One of the primary experiment is depicted on the following video7 where the SLAM result has been added afterward.

According to the block diagram presented in Figure 4, each wired input/output of the CPU Unit corresponds to a node which interacts with the main node for control or acquisition. A graphical representation is given in Figure 6. Thanks to this architecture, it is quite easy to add or update sensors. The following section presents experiments and results based on that hardware/software architecture.

Experiments and results

Romulux experiment 1

The first scenario took place in a sports hall (30m long x 20m wide), where illuminance constraints are specific to this type of infrastructure. Figure 7 depicts Romulux in the experimental conditions. Different SLAM algorithms (HDL Cross comparison between algorithms. In each are the differences between the paths of the algorithms: from top to bottom the mean, the standard deviation and the maximum (in meters).

paths. In the first case, an external calibrated and accurate measurement system is required. Here we chose the second method consisting in cross comparison of different algorithms.

Results described in Figure 8 depict a maximal deviation between Lego and loam algorithms of 1.11m (at the end of the travel) with an average value of 0.71m (see Table 1). Comparing to the width (20 m) of the hall, it represents a relative error of 3.5% which is quite acceptable and gives relative confidence in the illuminance map. This latter is constructed from the positions measured by Lego algorithm in Figure 9(a), then interpolated into a user-defined grid (every 10 cm in Figure 9)(b). For comparison, the recommended grid in the standards is made of points every 2 m. In the computed illuminance map (b), it can be seen the four lighting areas, which are consistent with on-site observations. The lights are adjustable halogens, of which this orientation was observed. The lighting level shall be at least the Class III level, that is the average horizontal illuminance (Emean) shall be over 300 lx for a uniformity (Emin/Emean) of 0.7. We found an average illuminance of 854 lx and a uniformity of 0.66. It should be noted that this uniformity was altered by obtrusive light coming from windows located on one side of the hall. A fifty percent reduction in luminous flux would help to comply with the standard thresholds and save energy.

Romulux scenario 2

This scenario took place in an international standard ice rink (see Figure 10(a)). Due to event broadcasting, it has to comply with lighting standards for television. The robot's motor skills were not affected by the ice, even though this was a concern at the outset. Figure 10(b) presents an example of recorded point-cloud. The lighting is intense and it is interesting to obtain a map of illumination to assess its homogeneity. For this ice hockey rink (namely Ice Park) at the international or national level of competition (class I), the standards recommend an average horizontal illuminance (Emean) of 750 lx and a 0.7 uniformity (Emin/Emean) . The playing area measures 60m long by 30m wide. The recommended mesh-grid defined to achieve all the measurements contain 17 points in length and 9 points in width either 153 measurement points each separated by 3.5m. These measurements are made classically with an illuminance meter placed on the ground and moved successively on the 153 points of the mesh-grid. The advantage of using an automatic measurement system is then obvious. For this experiment, an IMU has been added to the robot to improve the SLAM algorithm robustification. The IMU is a MTi-610, which is the highest standard among Inertial Measurement Units8

The robot pathway (see Figure 11(a)) has been defined to maximize the measurement points into the mesh-grid and it took around 30 minutes to acquire (half a day would have been needed if doing the measurements manually). The Figure 11(b) shows the result of the generated illuminance map. It presents a maximum of illuminance at a corner corresponding to an entry on the ice rink.The average illuminance is found to be 3607 lx (for an objective of 750 lx, demonstrating the huge over-lighting). The minimal illuminance value is 2754 lx leading to an homogeneity of 0.76 (for an objective of 0.7). Again, a 75% reduction in luminous flux would still comply with standards and save energy.

Stregths and limitations

Regarding those results, it appears that the presented robotic system can be a useful tool for illuminance mapping. It shows several advantages compared to classical processes:

• Considerable time savings comparing to manual measurement. It takes about 30 minutes to cover 1000 m² with the robotic system where at least half a day is needed manually; • As the robot performs data acquisition when moving, it provides much more measurement than doing it by hand leading to a better space discretization;

• As the robot is tele-operated (up to 30 m) it removes illuminance measurement disturbance that can raise when an operator directly manipulates the sensor.

On the other hand, some limitations can be noticed at this state of the project:

• As the SLAM is not currently performed online, the robot is not fully autonomous yet;

• As the illuminance sensor has been calibrated, the light measurement uncertainties are precisely known however localization uncertainties are complex to estimate and are not evaluated so far, but will be the subject of a further work;

• The system is not user-friendly yet and must be operated by a specialist or at least after some training courses where manual measurements are more straightforward.

It should be noted that the presented system is still a prototype and has not been industrialized. It is custom-made and the hardware costs about 20 k€ including 7 k€ for the LiDAR and 4 k€ for the frame. Furthermore, maintenance costs that will raise when using the robot daily are not evaluated so far as it has only been considered for academic researches.

Conclusion and perspectives

This paper describes Romulux, a mobile robot equipped with a 3D LiDAR carrying an illuminance meter. A SLAM algorithm is used to estimate the robot location in an indoor environment in order to construct an illuminance map. Illuminance values can then be compared to the standards. As places are often over-lit, the comparison can lead to downward adjustment in lighting power, thus saving energy while maintaining a sufficient level of safety and comfort. The results are significant and can be replicated in different indoor environments. The first evolution will be to consider outdoor environment: in particular because requests have appeared for measurement campaigns on tramway lines and on outdoor sport facilities. The use of Romulux robot for outdoor applications pushes the limits of its design framework. Particular attention will be paid to the behavior of the robot and the results. To that purpose the robot will be equipped with a GNSS sensor. A second axis of study will be to evaluate the precision of the approach in order to guarantee the measurements and thus the illuminance map. It is needed to take into account the uncertainty of location, resulting from LiDAR precision and SLAM algorithm result. The problem is complex and interval analysis tools will be explored for solving it. A third axis of study will be the complete automation of the system, by integrating algorithms of explorations, and by automating the outputs, ideally by generating them in real time. Finally it can be noted that this approach could be extended with several sensor maps such as temperature, air quality.

Figure 1 :

 1 Figure 1: The OPT3001 and Human Eye sensitivity.

  Figure 2 illustrates our implementation of the illuminance sensor above the LiDAR sensor. The location of illuminance measurements will be provided by Synchronous Localization And Mapping (SLAM) algorithms Birk and Pfingsthorn (2016); Bautin, Lucidarme, Guyonneau, Simonin, Lagrange, Delanoue and Charpillet (2013).

Figure 2 :Figure 3 :

 23 Figure 2: Sensor alignment: the illuminance sensor (little white dome) is above the LiDAR sensor (black tower). Both are aligned over the rotation center of the robot.

Figure 4 :

 4 Figure 4: Romulux architecture.

Figure 5 :

 5 Figure 5: Views of Romulux Robot.

  a) Sports hall point-cloud. b) SLAM paths.

Figure 8 :

 8 Figure 8: Cross comparison of SLAM path generated.

Figure 9 :

 9 Figure 9: Generation of illuminance map.

  a) The robot on the ice rink. b) Point cloud of Ice Park.

Figure 10 :

 10 Figure 10: Romulux in Ice Park.

  Robotic system for indoor illuminance map generation a) robot pathway. b) Ice Park illuminance map.

Figure 11 :

 11 Figure 11: Generation of illuminance map.

Table 1

 1

			Robotic system for indoor illuminance map generation
	/velodyne_nodelet_manager
	/pa_slam /scan	/paslam/probmap /visualization_marker /obstacle /cmd_vel /rviz_to_twist /aru /move_base_simple/goal /aru /clear /rviz /clicked_point /cmd_vel /aru /custom_ui /visualization_marker /rviz_maillage /maillage_to_twist /clear /rviz_maillage /rviz_maillage /move_base_simple/goal /clear /aru
			/twist_controller
		/teleop	/cmd_vel	/controlled_cmd_vel	/motor

Figure 6: ROS Architecture with nodes and topics Figure 7: Romulux in the sport hall.

http://wiki.ros.org/geometry_msgs

http://wiki.ros.org/rosbag

https://youtu.be/0PwFAdcqaLE

https://mtidocs.xsens.com/mti-600-series-user-manual R.Guyonneau et al.: Preprint submitted to Elsevier

Acknowledgment

The authors thank Alain Thomas for the project management, Anthony Colin for the mechanical study, Yannick Besiau for the electrical study, Bruno Nourrisson, Jocelyn Onillon, Jean-François Besson for the mechanical construction, Florian Greffier for the original idea.

CRediT authorship contribution statement

Rémy Guyonneau: ROS programming and SLAM integration, experiments, redaction of the paper. Franck Mercier: mecatronic design and theoretical formalism, experiments, redaction of the paper. Vincent Boucher: illuminance map generation and standards comparisons, experiments, redaction of the paper.

Authors statement

Rémy Guyonneau