Lucas Fonseca

Ana Claudia Lopes

Claudia Ochoa-Diaz

Christine Azevedo Coste

Emerson Fachin-Martins

Antonio Padilha

Lanari Bo

Lucas O Da Fonseca

Ana Claudia

Garcia Lopes

Christine Azevedo-Coste

Antônio P L Bó

Towards Transfers in Paraplegia Assisted by Electrical Stimulation and Inertial System

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

I. INTRODUCTION

A spinal cord injury (SCI) can deeply impact the individual's life and there are numerous challenges they must face everyday. Transfer from/to wheelchair is a key ability to gain mobility and independence, allowing greater interaction with the environment, besides social participation and improvements of the quality of life for individuals with spinal cord injury [START_REF] Eb Marsolais | Augmentation of transfers for a quadriplegic patient using an implanted FNS system[END_REF], [START_REF] Rj Triolo | Implanted FNS systems for assisted standing and transfers for individuals with cervical spinal cord injuries: clinical case reports[END_REF], [START_REF] Gagnon | Electromyographic patterns of upper extremity muscles during sitting pivot transfers performed by individuals with spinal cord injury[END_REF], [START_REF] Kankipati | Upper limb joint kinetics of three sitting pivot wheelchair transfer techniques in individuals with spinal cord injury[END_REF]. The most commonly performed type of transfer by paraplegic subjects is sitting pivot transfer (SPT) [START_REF] Veterans | Preservation of upper limb function following spinal cord injury: a clinical practice guideline for health-care professionals[END_REF], [START_REF] Gagnon | Biomechanics of Sitting Pivot Transfers Among Individuals with a Spinal Cord Injury: A Review of the Current Knowledge[END_REF]. In most of the cases, they have potential to perform this task independently. The SPT is executed on average 15 to 20 times a day [START_REF] Veterans | Preservation of upper limb function following spinal cord injury: a clinical practice guideline for health-care professionals[END_REF]. This large number of transfers also contributes to the development or perpetuation of secondary upper limbs musculoskeletal impairments over time and it is known that after SCI, individuals have great risks of pain and injury in the upper limb due to joint overloads during activities of daily living [START_REF] Veterans | Preservation of upper limb function following spinal cord injury: a clinical practice guideline for health-care professionals[END_REF], [START_REF] Desroches | Magnitude of forward trunk flexion influences upper limb muscular efforts and dynamic postural stability requirements during sitting pivot transfers in individuals with[END_REF], [START_REF] Gagnon | Trunk and upper extremity kinematics during sitting pivot transfers performed by individuals with spinal cord injury[END_REF]. Functional electrical stimulation (FES) can be used as a potential technological resource to assist these people during transfers [START_REF] Claudia | Electrical stimulation to reduce the overload in upper limbs during sitting pivot transfer in paraplegic: a preliminary study[END_REF], [START_REF] Murphy | Feasibility of closed-loop controller for righting seated posture after spinal cord injury[END_REF], [START_REF] Jovic | Impact of Functional Electrical Stimulation of Lower Limbs during Sitting Pivot Transfer Motion for Paraplegic People[END_REF], [START_REF] Jovic | Coordinating Upper and Lower Body During FES-Assisted Transfers in Persons With Spinal Cord Injury in Order to Reduce Arm Support[END_REF]. However, how the user activates the stimulation poses a problem. Since he uses both upper limbs during the transfer, any strategy that requires manual activating at the time of stimulation is This work was funded by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -CAPES, and Fundação de Apoio à Pesquisa -FAP/DF not practical. The moment of the stimulation onset must be automated in order to free the user's upper limbs.

We hypothesized that kinematic information could be used to trigger the stimulation with minimal user training, as proposed by [START_REF] Jovic | Coordinating Upper and Lower Body During FES-Assisted Transfers in Persons With Spinal Cord Injury in Order to Reduce Arm Support[END_REF]. A possible system for FES-assisted SPT could work as follows: the user has some kind of interface, like a switch, that he uses to enter a "SPT mode" when he is already in the starting position. Then he places his hand wherever he wishes and perform the transfer. The IMU, placed over his C7 vertebra, captures the trunk angle and triggers the lower limbs stimulation according to the relative angle threshold previously set to that user.

We have previously performed similar tests with able-bodied subjects [START_REF] Claudia | Electrical stimulation to reduce the overload in upper limbs during sitting pivot transfer in paraplegic: a preliminary study[END_REF] with manual stimulation triggering. This work was motivated by the following questions: 1) can an inertial measurement unit (IMU) collect kinematic information capable of reliably trigger FES for SPT assist on paraplegic subjects? And 2) can this IMU be placed on the user such as a wearable device? In order to do that we captured kinematic data from SCI subjects performing numerous SPTs with two different systems simultaneously; an IMU and a motion capture system. In addition, we developed a glove that allowed the subject to activate the stimulation as intended during the transfer.

II. METHODS

A. Subjects and setup

Three patients underwent the experiment: 2 men and 1 woman aging between 34 and 49 years old, all with AIS A (American Spinal Injury Association Impairment Scale) lesions between T2 and T11. They signed the Consent Form, which, along with this work, was approved by the Research and Ethics Committee (CAAE: 54748116.9.0000.0022). This work was approved by the Ethical Committee from the SARAH Network Rehabilitation Hospitals.

The patients were involved in two sessions. The first one took place at the hospital, where a physiotherapist applied a physical exam to make sure he/she was able to perform the task of SPT. The second session was the experiment in the laboratory.

Two benches were placed next to each other, with a 10º angle in between. The patient's feet were positioned in front of the benches, and equidistantly to both of them. There were specific places for positioning the hands, which were kept unchanged during the whole work. The experimental set-up can be seen on Fig. 1.

In order to collect accurate kinematic data, a golden standard motion capture system was used. Also, an IMU collected data for the purpose of this work. Finally, pressure sensors on the hands provided readings related to the pressure applied by the hand on the support.

B. Stimulator

The stimulator was a Rehastim (Hasomed, Germany). Two channels were used, one for each leg on the quadriceps muscle group. An emergency stop button was positioned on the metal structure, easily accessible by both patient and supervising physiotherapist. Stimulation was applied at 50 Hz and 450 µs. The current was determined on the day of the experiment, based on each subject's individual response to achieve leg extension for 10s.

C. Hands pressure sensors

A couple of hand pressure sensing gloves were custommade. Each glove embeds three FlexiForce sensors (Tekscan, EUA), which were connected to a microcontroller. The resistive sensors are placed on specific palm areas to maximize the force application. A microcontroller sampled data at 1 kHz and forwarded it to a computer at 20 Hz.

D. Inertial measurement unit

One inertial measurement unit (IMU, Yost Labs, EUA) was positioned over the patient's C7 vertebrae. It communicated wirelessly with the computer with a proprietary protocol and was able to sample orientation data at about 170 Hz. In order to simulate real applications, IMUs readings were performed with respect to the initial position, i.e. angle data was always set to 0 at the start of each trial. Indeed, for everyday use, a helpful feature is possibility of using the device without any initialization, such as positioning in vertical position for providing an absolute reference. In our case, relative reference position could be obtained by high-pass filtering the measured angle.

E. Motion capture system

Qualisys QTM (Qualisys, Sweden) was used to capture motion data, as well as data from the force plates. It was all sampled at 200 Hz and recorded on a local computer. The upper body marker protocol1 was used with 31 markers. The C7 marker, which is usually positioned over the C7 vertebrae, was instead positioned over the IMU.

F. Experimental protocol

At the beginning of each trial, the subject was either on one bench or the other, already prepared to execute the transfer. Their hand would be placed on the predefined places, but it was asked them not to apply weight on it to prevent the stimulation activation ahead of time. The system would start collecting data, and a researcher would tell the subject they could do the transfer whenever they wanted. The subject would do it, and when they supported their weight on their upper limbs, the stimulation would be activated and cause the lower limb muscles to contract, assisting with the transfer. After the stimulation was over, and the transfer was finished, the trial was concluded.

Each subject underwent 12 transfers, 6 in each direction (left and right). Out of the total 12, 6 were performed with stimulation and 6 without it. Before the 12 transfers in which data was collected for this work, there were 6 transfers with the purpose of familiarizing the user with the system, particularly the stimulation activation method.

G. Control strategy

The hand pressure sensors gave the subjects the control over when to activate the stimulation. They could prepare themselves for the transfer and activate the stimulation when they supported their weight on their upper limbs, in a intuitive way. The stimulation then remains activated for 1 second during the lift pivot phase. The lift pivot phase is defined between the moment the subject supports his weight on the upper limbs and the moment he reaches the target seat, and it lasts on average 1 second, according to [START_REF] Gagnon | Electromyographic patterns of upper extremity muscles during sitting pivot transfers performed by individuals with spinal cord injury[END_REF].

Knowing when the subject intended to activate the stimulation, we could evaluate if the wireless IMU provided kinematic data that is consistent intra-subject with the user intended stimulation activation timing. If this is indeed the case, a simpler measurement system (i.e. IMUs) could be used instead of sensorized gloves for controlling stimulation onset in FESaided SPT. This is particularly relevant for the users, who will have their upper limbs free to guide the transfer.

H. Data analysis

After the trunk angles were recorded for both the motion capture system and the IMU, intra-subject trials correlation of trunk angle on the sagittal plane was calculated. Then the trunk angle was analyzed along the moment in which the stimulation was activated by the subject. The same was done with the trunk angle first derivative, second derivative, relative angle and standard deviation (along a moving window of 0.1s). Since this work aims at developing wearable technologies which must work in real time, no offline non causal filters were used.

III. RESULTS

The trunk angle correlation between all trials in each subject presented correlation greater than 0.75, where 1 indicates complete similarity, both with the motion capture system and the IMU data. We may infer that, even though all subjects had just been presented to a new technology and, thus, a new way of performing transfers, there was reasonable degree of similarity between the trials, concerning trunk angle on the sagittal plane. Fig. 2 shows the trunk angle on the sagittal plane on all 12 trials for one subject with the motion capture system (Fig. 2a) and the IMU (Fig. 2b). From the moment the subject was told he could do the transfer, there was no instructions for specific timing on when he should start it. Therefore, for visualization purposes, all trials were centralized by the peak trunk angle and plotted on top of each other. Also, the data was trimmed to show only the 5 seconds around the peak trunk angle. This is why not all trials start at exactly 0º on Fig. 2b, even though the IMU angles are set to 0º at the beginning.

Fig. 3 show the angles on which the stimulation was activated by one subject related to the initial angle position. Since the angles are relative to the initial position, it was expected that the activating angles would be low. Table I summarizes these relative angles means and standard deviations for the IMU data.

The first and second derivatives were calculated, but it seemed there was no simple threshold that could reliably be set to trigger the stimulation because the signal oscillated too much. The same happened with the windowed standard deviation.

IV. DISCUSSION

The differences between the angles captured by the motion capture system, Fig. 2a, and the IMU, Fig. 2b, are due to two reasons. First, the procedure on which the IMU angles are set to 0º on the beginning of each trial. Also, the two systems do not measure the exact same body segment. The trunk angle from the motion capture system is based on the trunk body segment, which was built using the shoulders and hips markers. Therefore, it does not capture the curvature along the trunk. The IMU, however, was placed over the C7 vertebra. As a result, the IMU captures the upper trunk orientation, which may also be affected by the neck, but ignores the lower trunk movement, close to the hips. We can see on Fig. 2a that the absolute angle varies substantially between trials of the same subject, which also makes it unsuitable to be used as trigger for the stimulation. However, the relative angle between the starting position and the moment of activation is more consistent, as can be seen on Fig. 3. Note that, despite the differences aforementioned between the two systems used, the relative angle seems similar between the two.

Having a high trunk angle correlation between trials on each subject is a strong indication that a device to automate the stimulation trigger is feasible. Table I shows low standard variations for the angles in which the stimulation was activated. That suggests that these angles collected by the IMU can be used to set a trigger, which must be calibrated for each individual, to activate the stimulation without the need of hand sensors or any other method.

It is important to point out that the transfers performed on this work were done without prior training, and the system performed reliably with all users. A real life application, however, would profit from some practicing, which we believe would further narrow the angle deviation. In addition, the relative angle would probably pose an easier learning curve for the user than something like the acceleration (second derivative), which would be harder to understand.

V. CONCLUSION

The IMU is a small device that could easily be adapted as a wearable device. The results found on this work indicate its data can be used to trigger the stimulation of lower limbs and assist SPT on paraplegic users. That would free their upper limbs to be used to support their remaining weight and balance, and possibly reduce the load, particularly on the shoulders. That, along with the high trunk angle correlation between trials, indicate that a reliable wearable device for such purpose if feasible.

Figure 1 :

 1 Figure 1: Experimental set-up. The gloves embed pressure sensors. It is possible to see the markers over the subject body, which are captured by the motion capture system. The benches are next to each other, with a 10º angle between them. The legs are constrained to keep them for falling to the sides.

  Trunk angles from the motion capture system. Trunk angles from the IMU.

Figure 2 :

 2 Figure 2: Trunk angles for all trials of subject B. The * marker indicates the moment the simulation was activated by the subject in each trial.

  Relative trunk angle from the motion capture system. Relative trunk angle from the IMU.

Figure 3 :

 3 Figure 3: Trunk angle on the moment the stimulation was activated by the subject B related to the initial angle position. Each point depicts to one trial. The green dashed line is the average of all measurements, and the pink dashed lines represent one standard deviation up and down.

Table I :

 I Mean and standard deviation of relative trunk angle position from the IMU data.

	Subject Mean [º] Standard deviation [º]
	A	7,74	2,06
	B	4,11	3,78
	C	5,28	3,31

Plug-in Gait Full-Body (C-Motion, available in: https://cmotion.com/v3dwiki/index.php?title=Tutorial:_Plug-In_Gait_Full-Body)


Arthur Charpentier

Baptiste Coulmont

We are not alone ! (at least, most of us). Homonymy in large scale social groups

1 First and last-names Homonyms

The Western system of identification is based on a first and a last-name : the first-name is a personal name, the last-name is a transmitted family name, often from the father to his children. According to [START_REF] Scott | The Production of Legal Identities Proper to States: The Case of the Permanent Family Surname[END_REF] this system is first of all a government device, to monitor individuals and to ensure the rights and duties of citizen : it surfaced with the emergence of state governments. Nowadays, the more stable the state, the stronger this system: it gives a legal civil identity to everyone under its scope.

The "first-name + last-name" couple is not, and never was, sufficient to identify someone without any ambiguity. Historians and anthropologists have often remarked that in small European villages, many individuals shared the same identity. In small settings where everyone was known to everyone, there was no "collective interest in the clear and unambiguous individuation of persons through their names" [START_REF] Piña-Cabral | The Functional Fallacy: On the Supposed Dangers of Name Repetition[END_REF]). In small villages, nicknames (Big John), toponyms (John from the lake) and paraphrases (the son of Jake) could be much more efficient to distinguish someone from everyone else.

If this system worked for a long period of time, it was thanks to local agents of the state who could translate a local identity (Big John) into the civil identity needed by the state or the central authorities (John Martin) and reassure the state that John Martin the conscript or John Martin the suspected tax evader was indeed Big John. With additional elements such as the precise date of birth, the place of birth, the names and profession of the parents... the first and last-names could be used to identify someone in a much larger regional or national setting [START_REF] Noiriel | The Identification of the Citizen: The Birth of Republican Civil Status in France[END_REF]).

And today in our "global village" the first and lastnames are still the basis for worldwide identification. But without intimate knowledge or local agents in charge of the disambiguation, the collision of identities becomes problematic and more frequent. Every day in a random airport, someone sharing the identity of a known terrorist may be interrogated by customs * Université de Rennes 1

† Université Paris 8 and INED agents or banned from flying. And every second, bibliographic databases are trying to differentiate John Lee the mathematician from John Lee the biologist in order to compute their scientific outputs [START_REF] Gomide | Name Usage Pattern in the Synonym Ambiguity Problem in Bibliographic Data[END_REF]). Yesterday's homonymy was the shared sign of belonging to the same locality. There may have been hundreds of John Martins around 1700, but if they were not from the same place, they did not know they existed. Today's homonymy is shared between strangers in random places. In our interconnected societies, electronic social networks and multiple registrations enable us to "meet" or to "bump into" people with the same names as ours, often in circumstances when we have to assert a right (to vote, to travel, to buy...) based on our civil identity. From the point of view of the individual, then, homonymy is a random annoyance, a discomfort or a personal catastrophe, depending on the circumstances.

But from the point of view of the manager of any large scale register, today's homonymy seems to be a very common nuisance, if we consider the great numbers of personal identifiers that are meant to distinguish individuals without ambiguity. Personal identification numbers such as the Social Security Number in the United States, or the "numéro d'inscription au répertoire des personnes physiques" (NIR) in France were created to resolve this particular problem [START_REF] Lévy | ) au projet Safari[END_REF]).

These numbers are not used daily by people, who still prefer to be known by their names, and who do not gain anything by using a number instead. In the academic field, the "ORCID" promises to be "a persistent digital identifier that distinguishes you from every other researcher". It is meant to be used widely and the incentive is another promise : it "ensur[es] that your work is recognized".

But we do not know how frequent these identity collisions are. We do not know if, in a large scale society, many people have homonyms, or if only a small percentage does. An because of privacy issues, and of required anonymity of databases, estimating the proportion of homonyms in a large group is rather difficult. For instance in France, if researchers were allowed to access the NIR database, we could easily get the true proportion. But it is not the case.

In this article1 a propose a technique to provide an estimation of the proportion of homonyms in large scale groups based, on the distribution of first-names and last-names in a (smaller) subset of these groups. The main result is that, in societies such as France or the United States, identity collisions (based on first + last-names) are frequent. The large majority of the population has at least one homonym.

A Birthday Paradox Problem

The birthday problem concerns the probability that, in a set of randomly chosen people, some pair of them will have the same birthday. It is usually seen as a paradox since, within a group of n = 23 persons, it is more likely to find than not to find two people sharing the same birthday, which seems counter-intuitive given the large number of possible birthdays and the small size of the group. Here also, we want to compute the probability that people share the same names (so called "homonymy") in a given group. The first difference between these two problems is that the number of possible birth dates is known (365 -if we exclude the 29th of February) while names belong to a much larger set. The second difference, in the computation of probabilities on birthdays, is that we usually assume that births are uniformly distributed over the year. But we cannot assume names to be uniformly distributed (and purely randomly chosen) : as suggested in [START_REF] Li | Analyses of baby name popularity distribution in U.S. for the last 131 years[END_REF] for first-names, a Pareto/Zipf law can be considered, see Figure 1, for first and lastnames distribution, in France. With Zipf's law the probability of occurrence is inversely proportional to the rank. Here we consider a mild version a power function (for instance the probability of occurrence can be inversely proportional to the square root of the rank). It is related to Pareto principle where a small number of names is accounting for a large proportion of the observations. For instance here there are more than 74,000 firstnames in Paris and Marseille (1,757,895 persons on our lists). The top 10 (0.01% of all possible firstnames) account for almost 9.5% of the population. And 740 first-names (1%) are shared 82% of the population. It is an extremely strong version of the Pareto principle (the popular 80/20 rule: 80% of the wealth is owned by 20% of the population). And out of 300,000 last-names, the top 1% is shared by 31% of the population. As a comparison, with a pure Zipf's law (when the probability of occurrence is proportional to the inverse of the rank) the top 1% of the first-names should be shared by 60% of the population. And for financial wealth [START_REF] Bricker | Changes in U.S. Family Finances from 2013 to 2016[END_REF]) claims that in the U.S., in 2016, the richest 1% of families controlled 38.6% of the country's wealth.

Probit and Logit Transforms

Following ideas in demography, [START_REF] Pearl | A further note on the mathematical theory of population growth[END_REF]) and [START_REF] Winsor | A comparison of certain symmetrical growth curves[END_REF]) introduced the logistic function in the 1920's (see [START_REF] Cramer | Logit models from economics and other fields[END_REF]) for more details about the origin of the function and the term). This function yields the popular logistic regression, to model a probability as a (linear) function of some covariates, since it is naturally obtained when the logarithm of an odds-ratio is a linear function of the covariates, where

P = P[Y = 1|X = x] satisfies logit(P ) = log P 1 -P = a+bx or P = e a+bx 1 + e a+bx
At the same time, [START_REF] Bliss | The Method of Probits[END_REF]) suggested an alternative regression model, based on a latent representation. Assume that there is an (unobservable) variable Ỹ , with Ỹ = a + bx + ε, where ε is a Gaussian noise. Assume also that

Y = 0 if Ỹ ≤ 0 1 if Ỹ > 0 Then P = Φ(a + bx) or Φ -1 (P ) = a + bx,
where Φ is the cdf of the centered reduced N (0, 1) distribution. Φ -1 was called the probability unit transform, or shortly the probit transform. Hence, statistician like to represent either the logit or the probit transformation of probabilities, hoping they will have a linear model for the later.

This non-uniformity makes our problem complex (see [START_REF] Munford | A note on the uniformity assumption in the birthday problem[END_REF], DasGupta (2005), [START_REF] Inoue | Methods for studying generalized and coupon collection problem[END_REF], or [START_REF] Nunnikhoven | A birthday problem solution with non-uniform birth frequency[END_REF] for some attempts, and more recently Cortina Borja ( 2013)). Let P n,k denote the proportion of people having an homonym in a group of size n when k names are available. Before using real data, let us run simulation to visualize the evolution of the proportion P n,k as a function of n (or its logarithm). Let us consider first the case where names are uniformly (randomly) chosen among the k possible names (analogous to the birthday problem when k = 365). This case is on the left of Figure 2.

For different k's, different lines represent the evolution of P n,k , which is (naturally) increasing with n : the larger the group, the more homonyms there should be. Since probabilities are constrained to lie in the interval [0, 1], alternatives can be to visualize the odd-ratio or the probit transformation of those probabilities. If the graph on top show the evolution of P n,k , the graph in the middle show the evolution of the logit transformation of that proportion (the logarithm of the odds ratio) in the middle, and the probit transformation below.

On the right of Figure 2 we can visualize the evolution of the proportion of homonyms, P n,k , as a function of the group size n, when names have a Zipf law on the set of possible names (of size k). One can observe that either the logit or the probit transformation of proportion P n,k is linearly increasing in the logarithm of n. We will use that property to estimate the proportion of homonyms when n is too large (or when our sample size is too small).

First and last-names

In [START_REF] Höhle | Why the "most popular" baby names might not be the most popular[END_REF], only the first-name was considered, but to study homonymy, we need to study pairs (firstname, last-name). We now face a bivariate problem, with -potentiallyk 1 first-names and k 2 lastnames. The two components cannot be considered as independent, see among recent references Chalabi & Flower (2014), which starts with the observation that "Michael Smith" might not be the most popular American names, even if "Michael" and "Smith" are probably the most popular first and last-names. Similarly on French data, "Jean"/"Marie" and "Martin" are the most popular first and last-names (for males and females respectively, on Paris and Marseille electoral lists) but the most popular full name is actually "Thi Nguyen", with 350 records (84 people in Paris and Marseille named "Jean Martin", the third highest score). To go one step further in the analysis, a chi-square test can be performed, as in Chalabi & Flower (2014). On Figure 3 are plotted (normalized) residuals from a chi-square test between firstnames of the top 25, and last-names. For instance, "Michel " is both a popular first and last-names. But the proportion of "Michel Michel " is much smaller than was would be expected if the two were independent. One can also observe some strong 'communitarian' effect, with Vietnamese names ("Than" or "Thi" for the first-name and "Nguyen" or "Tran" for the last-name), African names (such as the popular Peul last-name "Diallo" and the first-name "Mamadou") or Jewish names ("Cohen" and "Levy " are positively correlated with "David ", but negatively correlated with "Marie "). Finally there are also negative correlations for names that share similar phonemes (such as "Pierre Robert " both ending with [EK], "Alain Martin " ending with [Ẽ] and "Laurent Durand " [Ã]). As explained in the introduction, our sample size is too small to estimate the proportion of people with an homononym in the country. So we will have to extrapolate P n for n large, based on accurate estimations obtained when n is much smaller. And having a linear model to extrapolate is the most convenient way to do it. As discussed in the previous section, having a linear relationship between the probit transform of P n and log(n) would be interesting because that yields the popular least square problem : we fit a linear trend on observed data, and extrapolate for non-observed ones.

Application on French Data

In order to compute the probability P n in the context of French names, the electoral roll of Paris and Marseille (for the year 2015) has been used. In this dataset, we have the first-name, last-name and date of birth of registered electors in Paris and Marseille (1,757,895 observations). Overall, we kept 1,542,528 observations, because of some typos in the original dataset. There where k 1 = 74, 085 first-names in that dataset, and k 2 = 309, 907 last-names (almost half of those appeared only once). Because of the variety of the first and last-names, our sample size (n = 1.5 million) was too small to estimate the proportion of people with an homonym in the entire French population (65 million). Resampling from pairs (first and last-names together) will over-estimate the proportion of homonyms in a very large group. Nevertheless, as mentioned in the previous section, it is not realistic to draw first and last-names independently, since both are correlated. On Figure 4 we can visualize the proportion of homonyms when drawing from the French population in Paris, either drawing pairs or drawing first and last-names independently. When n is not too large, drawing pairs should yield a good approximation (the red line on Figure 4), but when n is too small we keep sampling from our too small sample, so we should over-estimate the proportion of people with an homonym. On the other hand, drawing independently first and last names will under-estimate the proportion of people with an homonym since it does not account for the dependence that exists between the two. Nevertheless, in that case, we can observe that the assumption of a linear relationship between the probit transform of the proportion and log(n) is valid, whatever the value of n. So, as we can see on Figure 5, we use a linear approximation when n is between 5,000 and 50,000, on approximations obtained drawing pairs, and then we extrapolate that linear approximation for large n's. As we can see in the second box on another dataset in the United States, the approach we use here is rather general, and not only valid in France.

Temporal evolution of that Proportion

On two larger datasets2 , we can observe the evolution of first and last-names in France, see Tables 1 and2 (those datasets contained statistics about first and last-names, respectively, but not paired). The pattern observed in Table 1 for the first-names was already observed in (Li ( 2012)) (on U.S. firstnames). Ninety years ago, most of the first-names were related to Saints or the Bible, so the list was much smaller (with a lot of "Marie", "Jean", "David "), while nowadays there are first-names from various place in the world (not to mention odd names that can now be given to babies nowadays). Old firstnames remain, and new ones appear, but it is not the case for last-names. Last-names are more stable with time, even if one could also object that there were less "Nguyen" in France in 1915 than in 2015. But in the case of Vietnamese names, even in Vietnam there are less names than in France (names are related to clans, and they are about 200 -85% of Vietnamese people share 12 names as mentioned in [START_REF] Krowolski | Nom et appelation au Viêt-Nam[END_REF])). This might explain the (relative) stability observed on last-names.

It is then possible, assuming independence between first and last-names, to visualize the evolution of the proportion of homonyms, approximated using Monte Carlo simulations, on Figure 6, for groups of size 10,000 up to 200,0000 people (from bottom to top).

Application to Ohio Data

It is a dataset with 7.8 million individuals. Using independence between first and last-names to generate names, here also we obtain that the probit transform of the proportion of people with an homonym is linear in the logarithm of the population size. Further, observe that in that group, 50% of people have an homonym in that specific state. It might be interesting to extrapolate to a much higher n. As described in Figure below, in a population of n = 320 million people (the country size), we can estimate that 95.1% Americans have an homonym (in the United States).

Conclusion

As the interconnexion of our world increases and as the realm of interactions widens , we encounter a increasing number of homonyms. These collisions are annoying. But we continue to value the use of a basic identification system. Some contemporary changes reduce the chance of collisions: we increasingly choose rare names for our children, and, at least in Europe, the transmission of the father's last-name is slowly replaced by the possibility to choose to transmit the mother name or to create a combination of both parents' names.

7 Datasets

Figure 1 :

 1 Figure 1: Empirical distribution of first (on the left) and last (on the right) names in France, with the estimate Pareto/Zipf fit.

Figure 2 :

 2 Figure 2: Evolution of P n,k as a function of the group size n (on a log scale) as a function of k, for different distributions p (uniform on the left and Pareto on the right).

Figure 3 :

 3 Figure 3: Pearson's residuals from a chi-square test of independence in the contingency table last vs. firstnames, in France.

Figure 4 :

 4 Figure 4: Proportion of homonyms when drawing from the French population in Paris, either drawing pairs (first and last-names) in blue, or drawing first and last-names independently, in red. Empirical probabilities P n are on the left, and the probit transform of P n is on the right.

Figure 5 :

 5 Figure 5: Proportion of homonyms with a linear extrapolation when pairs were drawn (linear on the probit transform as a function of log n).

Figure 6 :

 6 Figure 6: Evolution of the proportion of homonyms, P ⊥ n , assuming independence between first and lastnames.

Table 1 :

 1 first-names in France.

	time period	size	top 10 top 100
	1916-1940	95,000 25.17% 79.05%
	1941-1965	105,000 20.50% 72.61%
	1966-1990	245,000 12.59% 56.98%
	time period	size top 10 top 100
	1916-1940	638,000 1.83%	8.66%
	1941-1965	669,000 1.76%	8.41%
	1966-1990	814,000 1.57%	7.83%

Table 2 :

 2 last-names in France.

  /www6.sos.state.oh.us/ords/f?p=111:1 • Paris and Marseille Voter Files /www.data.gouv.fr/fr/datasets/fichierdes-prenoms-edition-2016/ • Fichiers des noms de famille -1891-1990 -Édition 1999, INSEE [producteur], ADISP-CMH [diffuseur]

	• Ohio	Voter	Files	available	at
	https:/• Fichier	des	prénoms,	édi-
	tion	2016,	INSEE,	available	at
	https:/

Additional material, including mathematical explanation and R codes used for computations (and to produce graphs) is available on a GitHub repository, https://github.com/freakonometrics/homonym

The first one is the fichier des prénoms, 2016 edition available from https://www.data.gouv.fr/fr/datasets/fichier-desprenoms-edition-2016/ produced by the National Institute of Statistics (INSEE) and the second one is the fichiers des noms de famille -1891-1990 -1999 edition, produced by INSEE, available from ADISP-CMH.


L F Zagonel

C A Figueroa

F Alvarez

In situ photoemission electron spectroscopy study of nitrogen ion implanted AISI-H13 steel

In this paper we report the effect of hydrogen on the structural properties of AISI-H13 steel nitrogen-implanted samples in low oxygen partial pressure atmosphere. The samples were implanted in a high vacuum chamber by using a broad ion beam source. The H 2 + /N 2 + ion composition of the beam was varied and the surface composition studied in situ by photoemission electron spectroscopy (XPS). The samples were also ex situ analyzed by X-ray diffraction and scanning electron microscopy (SEM), including energy-dispersive spectroscopy measurements. It was found that hydrogen has the effect of modifying the amount of retained nitrogen at the surfaces. This result shows that hydrogen plays a role beyond the well-established effect of oxygen etching in industrial machines where vacuum is relatively less well controlled. Finally, an optimum concentration of 20-40% [H 2 ]/[H 2 +N 2 ] ion beam composition was determined to obtain maximum nitrogen incorporation on the metal surface.

Introduction

Plasma nitriding is a well-established technique broadly applied to improve surface properties of steels [1]. Nevertheless, some important physical mechanisms involved in the process are still not well understood. Indeed, the microscopic mechanisms in nitriding processes and their effects on the metal structure continue to be an important subject of study [2][3][4]. The use of hydrogen in plasma nitriding is a standard procedure to increase nitrogen penetration in steel [5]. The presence of stable oxides on the surface blocks nitrogen diffusion and the beneficial effect of hydrogen attributed to etching effects on the oxides compounds. However, the role of hydrogen is sometimes suggested to go beyond the chemical etching effects. Moreover, some authors claim that the presence of hydrogen has effects on the plasma structure, influencing nitrogen incorporation in the bulk of the metal [6,7]. Therefore, to observe specific effects of hydrogen, experiments in an oxygen-free atmosphere are mandatory.

In this paper we report a comprehensive nitriding study of AISI-H13 steel in accurately controlled conditions. The samples were implanted in a high vacuum chamber (partial pressure ~2 x10 5 Pa) by using a Kaufman cell fed with different nitrogen-hydrogen mixtures. The deposition chamber is attached to an ultrahigh vacuum chamber for in situ photoemission electron spectroscopy analysis (XPS) for composition and structural analysis. Also, the structure evolution of the material was studied by ex situ X-ray diffraction. Morphology studies, by ex situ SEM analysis, show the nitrogen diffusion via grain boundaries. Experimental results show that 20-40% hydrogen partial pressure increases nitrogen concentration in the metal bulk. This experimental finding confirms that hydrogen plays a role beyond the well-established oxygen etching.

Experimental

The samples studied in the present work are rectangular (20x15x2 mm) slices from the same AISI-H13 steel lot. Table 1 displays the material composition as determined by XPS analysis in sputtered cleaned raw samples. The nitriding process was performed on a high vacuum system with a base pressure always better than 2x10 5 Pa (P[O 2 ] ~10 6 Pa) using a Kaufman cell. This system is attached to a UHV chamber where in situ X-ray photo-electron spectroscopy (XPS) measurements are performed. More details of the implanting system can be found elsewhere [8]. To carry out this study, all system parameters were maintained constant and the hydrogen and nitrogen flux proportion accurately adjusted. These gases are admitted into the Kaufman cell through independent mass flow controllers.

The hydrogen partial pressure, C H =[H 2 ]/[N 2 +H 2 ], was varied from 0 to 80 at.% maintaining fixed the total chamber pressure (~1.4 10 2 Pa). Other important parameters maintained constant during ion implantation are: ion beam energy, 600 eV; ion beam effective current density: 2 mA/cm 2 (measured with a faraday cup); temperature: 450 8C; implantation time: 2 hours. At this energy, both nitrogen and hydrogen will be dissociated and shallow implanted [2,9,10]. Indeed, SRIM simulations show that at 300 eV, nitrogen and hydrogen will reach 1 and 5 nm depths, respectively [START_REF] Biersack | The simulation used the TRIM software[END_REF]. These characteristic depths show that a thin oxide layer may indeed be an efficient barrier for nitrogen incorporation and diffusion. Moreover, hydrogen passivation on this first 5 nm layer might increase both, nitrogen retention (up to saturation concentrations) and mobility (through alloying elements traps screening) [12]. Finally, after the nitriding process, the samples are transferred to the ultrahigh vacuum chamber (~5 x10 7 Pa) for XPS measurements.

The neutral species contained in the beam are difficult to measure. However, we can make an attempt to estimate their concentrations. Conservation laws forbid neutralization of the ions in the gas phase at the working pressures used in the present study [START_REF] Lieberman | Principle of Plasma Discharges and Materials Processing[END_REF]. Therefore, the origin of the neutral species in the beam could stem from two sources: (1) neutralization due to electrons injected by the filament neutralizer of the Kaufman cell, and (2) secondary electrons emitted due to ion impact on grounded parts of the Kaufman cell. In the experimental conditions reported in this paper the beam neutralizer was not used. Therefore, this contribution to the formation of neutral species is not present. Regarding secondary electron emission, it can be neglected. Indeed, the secondary emission coefficient of 100 eV N 2 + ions hitting molybdenum is c~0.032 [START_REF]Most of the parts of the Kaufman cell used in the experiment are made on molybdenum[END_REF]15]. Considering that c weakly depends on energy, one can estimate that at most ~3.2% of the N 2 + ion will be neutralized in the ion beam. The XPS spectra were obtained by using the 1486.6 eV photons from an Al target (K-line) and a VG-CLAMP-2 electron analyzer. The total apparatus resolution was ~0.85 eV (line-width plus analyzer). The relative atomic composition at the sample surfaces was determined by integrating the core level peaks, properly weighted by the photo-emission cross-section. As is well known, XPS gives information of the outmost atomic material layers (~0.5 nm) [START_REF]Practical Surface Analysis, Auger and X-ray Photoelectron Spectroscopy[END_REF]. Crystallographic information was acquired with the usual h-2h mode X-ray diffraction (XRD) with the Cu Ka-line. Material hardness measurements were performed using a standard Vickers micro-indenter (0.05 kg load, Shimadzu) and the morphological characterization by a scanning electron microscopy (SEM) (Jeol JMS-5900LV), equipped with energy depressives spectrometry (EDS). The samples for SEM measurements were mirrorpolished and 120 seconds etched in 2% nital solution. SEM images and EDS were obtained using 8 keV electron beam energy and 15 keV, respectively.

Results and discussion

3.1. Influence of the H 2 /[H 2 +N 2 ] gaseous mixture on surface composition and hardness Fig. 1(a) shows the nitrogen and iron concentrations obtained from XPS studies as a function of hydrogen partial pressures in the Kaufman cell in the absence of oxygen. This plot shows the nitrogen dependence retention on the surface material by the presence of H 2 + in the ion beam. The curves show that a maximum (minimum) nitrogen (iron) concentration at the surfaces is obtained for 30-40% (20-30%) H 2 /[H 2 +N 2 ] gaseous mixtures. This indicates that even if the plasma nitriding process takes place in a very low oxygen partial pressure chamber, some hydrogen will enhance nitrogen surface retention. The observed diminishing N concentration above around 30-40% gaseous concentration is probably due to chemical etching. Also, the increment of hydrogen partial pressure decreases the effective nitrogen ionic current, contributing to a lower nitrogen concentration on the surface. Fig. 1(b) displays surface hardness vs. gaseous mixture. Within the experimental error, hydrogen slightly reduces hardness to an almost constant value, independent of gas mixture composition. As we shall see below, however, the crystalline structure depends on the gaseous mixture. Finally, it is important to remark that the oxygen concentration measured by XPS on the surfaces was smaller than 3 at.% for all studied samples. This is expected since XPS analyses were performed in situ in a high vacuum chamber. X-ray diffraction also reveals that samples treated with less hydrogen are more expanded (not shown) possibly explaining the greater hardness of sample H 2 -00.

The photoelectron emission spectra (XPS) of the allowing elements, such as Cr2p 3/2 and V2p 3/2 , show the onset of a second peak. For chromium, this second peak, associated with Cr 1+x N (x near to 0), has a chemical shift of ~1.1 eV and is responsible for ~40% of the total signal independently of the [H 2 ]/[N 2 +H 2 ] mixture feeding the ion source [START_REF] Marcus | [END_REF]. For the sake of clarity, only the results obtained with [H 2 ]/[N 2 +H 2 ]=50% are displayed in Fig. 3(a). We remark that the curve asymmetry is due to shakeup process [START_REF]Practical Surface Analysis, Auger and X-ray Photoelectron Spectroscopy[END_REF]. Assuming that, after nitrogen incorporation, two bands associated to chromium environments, where used to fit the curves (see Fig. 3). The first band corresponds to the Cr2p 3/2 core electrons of nitrogen-free material and the second one to the electrons associated to Cr2p 3/2 core electrons in chromium nitrides compounds. The latest contribution is chemically shifted due to the electronegative effect of nitrogen [START_REF]Practical Surface Analysis, Auger and X-ray Photoelectron Spectroscopy[END_REF].

Influence of the H 2 /[H 2 +N 2 ] gaseous mixture on the material structure

The X-ray diffraction patterns of the studied samples are enlightening about the phases present in the material. Fig. 2 shows the evolution of the crystalline material on N incorporation. A pristine sample is also included for comparison purposes. Except for sample H2-80, implanted with 80% H2/[H2+N2] gaseous mixture, the -phase (Fe 2 -3 N) is sizable in all curves [START_REF] Walkowicz | [END_REF]18]. The -phase (bcc) is also apparent as a shoulder in some nitrated samples. The presence of these phases is consistent with the iron/nitrogen stoichiometry obtained from the XPS data after subtracting nitrogen bounded to alloying elements (see Fig. 1b). The constituent alloying elements (Cr, V, Mo) produces two effects: stabilization of the original a-phase and more retention of nitrogen in the structure [19]. In fact, depending on the nitrogen concentration, two phenomena are possible. On the one hand, at relative low N content, the -phase is predominant and N does not exceed the solubility limit of the element in the Fe-N binary system [START_REF]Desk Handbook: Phase Diagrams for Binary Alloys[END_REF]. On the other hand, at higher N content, the q-phase becomes dominant. Thus, the presence of the -phase observed in different samples even at nitrogen concentration above the solubility limit Similar results are obtained for all studied gaseous mixtures. Also, we note that the chemical shift to higher energies observed in the photoemission electron core level binding energies is consistent with the high N electronegativity. The photoelectron emission spectrum associated with the N1s electron is a broad band centered at ~394.4 eV (Fig. 3b). This band (width ~1.9 eV) is constituted by electrons stemming from contributions of several metallic nitrides such as CrN, VN, MoN, and Fe 2- 4 N. We note that this band overlaps with the one associated with the spectrum of the Mo3p 3/2 electron core level. In Fig. 3, a ~10% contribution stemming from the Mo p 3/2 core electron (not shown) has been properly subtracted.

Morphology

Fig. 4 shows the SEM result (backscattered mode) obtained in sample H 2 -50 taken from its transversal cross section. Several features are observed in this picture. First, the dark zones around grain boundaries near the surface indicate the presence of light nuclear density elements such as N. Therefore, grain boundary diffusion through the presence of defects of N is strongly suggested for the picture, explaining the relative thick diffusion layer for this sample (~30 Am). Second, the small bright spots indicate metallic nitrides precipitates. The higher density of these spots at the outmost layers of the samples is due to higher N concentrations near the sample surfaces. At the surface, therefore, the large amount of precipitates suggests a composition bulk grain change by alloying elements depletion. In order to verify this hypothesis, spatially resolved energy-dispersed X-ray spectroscopy (EDS) was performed at the borders and bulk region of sample grains. In particular, the probed zone was located at the outmost sample layers and deeper in the studied samples. At the outmost layers, the EDS results confirm the increasing (decreasing) alloying elements, such as Cr and V, at the grain boundary (bulk) as compared with normal concentration found in deeper material layers. This is understood because N extracts allowing element from the bulk grains to form the precipitates, depleting the bulk of the grains near the material surfaces. Unfortunately, SEM pictures of the studied samples do not show sizable differences in samples treated with diverse H 2 /[H 2 +N 2 ] mixtures. However, the effect observed on N retention is probably due to the formation of hydrides at the grain boundary, playing an important role in the diffusion process [START_REF] Marcus | [END_REF][22][23].

Conclusion

In conclusion, in this work we have shown that, at very low oxygen partial pressure (~10 6 Pa), appropriated nitrogen dilution in hydrogen improves nitrogen retention at the surface of the material. This results shows that hydrogen has a specific role going beyond that of the wellestablished oxygen etching. Depending on the nitrogen concentration, two phenomena can occur. First, for relative low N concentration, the a-phase is stabilized. Second, at higher N surface retention, the q-phase is formed. Scanning electron microscopy confirms that N diffuses by grain boundary. Moreover, metallic nitrite precipitates blocks N diffusion deeper in the bulk material. The presence of H neutralizes allowing components potentially acting as N traps. Finally, the incorporation of H to the ion beam slightly reduces the material hardness as compared the material implanted with a pure N 2 + ion beam and influences the formed crystalline structure.

Fig. 1 .

 1 Fig. 1. (a) Nitrogen and iron surface concentration as function of gaseous nitrogen concentration feeding the ion source. (b) Frontal hardness and iron/ nitrogen ratio as function of gaseous nitrogen concentration feeding the ion source. Lines are guide to the eyes.

Fig. 2 .

 2 Fig. 2. X-ray difractogram of nitrated samples. The characteristic position

Fig. 3 .

 3 Fig. 3. (a) Spectra corresponding to the Cr2p3/2 photoelectrons for untreated and nitrided samples with 50% hydrogen feeding the ion source. It can be observed the onset of a second band, 1.1 eV chemical shifted toward increasing binding energies. (b) N 1s band formed by contribution stemming from all N chemical states.

Fig. 4 .

 4 Fig. 4. SEM image in backscattered configuration (sample H 2 -50). The dark grain contours are evidences of nitrogen presence. The bright smallest spots are metallic precipitates.

Table 1 AISI

 1 H13 steel composition, at.%

	Element	Fe	C	Mn	Si	Cr	Mo	V
	Concentration	90.5	0.6	0.3	0.3	6.7	1	0.7
	(F0.5 at.%)

Acknowledgements

This work is part of the PhD thesis of LFZ. The authors are indebted to D. Ugarte for the SEM measure-ments. This work was partially sponsored by FAPESP, project #97/12069-0. LFZ and CAF are FAPESP fellows. FA is CNPq fellow. The electron microscopy work has been performed with JSM-5900LV microscope of the LME/LNLS, Campinas.