Healing Religions and Well-being

Régis Dericquebourg

Exposé fait dans l'atelier : Religion, Healing, Welfare and Well-being : Policies and Praxis in Comparative Perspective. Organizers : Grace Davie, University of Exeter,and Ian Varga,Queens University,Kingston. Convener : Adam Possamaï,University of Western Sydney. 66th Annual Meeting of The Association for ther Sociology of religion : The Causes and Consequences of Contemporary Moralities, San Francisco, Cal.. August 13-15, 2004. For many years, I have been studying a subset of the religious field that I have named healing religions or religions with therapeutic vocation or also healing Churches. I will give some examples later on.

Generally speaking, it can be noted that many religions show great concern about health. Some religions, like Mormonism, reduce the risk of illness by proposing some life hygiene. Other faiths practice healing. Catholicism is known for its pilgrimages, such as Lourdes, for its masses for sick people, for the anointing of the sick the importance of which was stressed by the Pope in 1998, for devotions to healing saints. In Protestantism, pastors can lay on hands on suffering people. Asian spiritual movements (Buddhism and Hinduism) propose the extinction of sufferings and have invented a medicine adapted to their beliefs, such as the Yang Ji medicine which is at the confluence of biomedicine and three religious traditions: Confucianism, Buddhism and Taoism. In their respective peripheries, one can also fend healing mediums and people invocating goddesses specialized in one type of illness (Bertrand, Harman). Muslims have invented the unani medicine (Kakar, 1997). In his travel stories dating back to 1347, Ibn Battuta describes healings of paralytics near the grave of Ali (Battuta, 1982). In Greek ancient times, healthcare was put under the auspices of the god of medicine, Asclepios, who became Esculape in the Roman Empire. Some New Age practices aim at the conservation and the improvement of health as well as healing, although New Age is not a religious therapeutic movement.

Among smaller religions, Mahikari and Soka Gakkai for example have healing practices (Louis Hourmant, 1999). In the wake of Protestantism and Catholicism, Pentecostals and Charismatics can pray to get healings.

From a global perspective, religions may be a source of some psychological wellbeing and even better health for believers. From Freud's point of view, they appear to be « concernbreakers ». This expression is linked to medicines used by man to face the harshness of destiny and to endure the sufferings and the deprival imposed on him by civilization. Such medicines are expected to make life in community possible even if for the founding father of psychoanalysis, they hardly achieve their goal (S. Freud, L'avenir d'une illusion, Paris, P.U.F., ed. In 1975, p 25).

However, according to Max Weber, the various fruit of salvation (Heilsgüter) that religions promise and propose cannot be considered by empirical researchers as exclusively or even primarily related to the beyond » (Weber, French translation, Grossein, p 345). Indeed, « Apart from Christianity and a small number of specifically ascetic faiths, the fruit of salvation, proposed by an religions, whether primitive or civilized, prophetic or not, mainly concerns this world: health, long life, wealth were the promises of Chinese, Vedic, Zoroastrian, Judaic, Ancient, Islamic religions ; such were the promises made to pious laymen by Hinduism and Buddhism. » (Weber, French translation Grossein, p 346).

From a particular point of view, empirical studies of the Jesus Movement, the Ananda Cooperative Village, the Children of God, Moonism and the Divine Light Mission carried out by researchers like Richardson, Simmonds and Harder, Kuner, Galanter, Nicholi (James Richardson, 1985) have shown that religious movements linked to Asian spiritual movements or Judeo-Christian monotheism can have indirect therapeutic effects and facilitate the rehabilitation and the reinsertion of drug addicts and unstable persons. See also M. Argyle and B. Beith Hallami, 1975. Yet, although many big religions and religious minorities show great concern about health and well-being, their primary objective is ontological salvation (entry of the soul in the light) or collective salvation (final resurrection and entry in the paradise restored on earth). In their theologies and in their practices, their concern about health is secondary. This makes them different from the healing Churches where the will to heal the body, the mind and the difficulties of life is in the heart of faith and publicly announced. For example: Christian Science, Antoinism, Maguy Lebrun's prayer groups, the Christ of Montfavet group (now named Universal Alliance) and Invitation à la Vie (IVI). The last three are based in Europe but there are similar movements elsewhere in the world. I set aside Pentecostal and Charismatic groups because their primary objective is the soul' s salvation and the recognition of Christ as the redeemer whose power is proved by some through the healing charisma.

Ideal type of healing Churches

In my research works, I defined healing religions by outlining their ideal type according to the Weberian typification method, that is to say by stressing, linking and organizing the traits that are common to a specific category of phenomena in an ideal table that does not necessarily exist as such in reality (Weber, 1995:181).

Each case has some significant characteristics that bring it closer to the ideal type but what is essential is to measure the deviation from the ideal type so as to find its limits The presentation of this ideal type of healing religions would require more than just a few minutes. I had the opportunity to describe it in details in my thesis for the position of research director (Dericquebourg, 1999) and in several articles (Dericquebourg, 1995, Lautman), Dericquebourg, 1997 RFP), (Dericquebourg, 2002, Benoist). Here are its characteristics: They publicize their quest for healing and therefore attract sick people or people in psychological and moral distress.

They have accredited religious therapists whose healing charisma was legitimized by the movement. From a Weberian point of view, they can be qualified « religions virtuoso ». They are for example practitioners in Christian Science. Healing Churches grant them a central role and a legitimate authority in the leadership of the movement. They thereby avoid possible tensions between the managers and the owners of a specific charisma.

They have a broad definition of illness: not only physical and psychological ailments but also the material, social, professional, sentimental or family vicissitudes of life. This is to be connected to the concept of infirmitas in the Middle Ages which meant at the same time poverty, illness and the condition of the pilgrim.

They provide a metaphysical explanation for the illness and its spiritual treatment which is based on the belief system of the movement. The treatment is not applied in an epistemological vacuum. It is the prolongation of a theology. When healing is achieved, it authenticates the movement.

They consider the spiritual treatment as a stage on the way to salvation because it offers the sick the opportunity to meet a supernatural reality and to enter the way to salvation. However, some people only use religious therapists when they need them, just as providers of services, but never adhere to the proposed spiritual way. A tension has been noticed between the demand for healthcare and the offer of salvation. Few people follow the proposed way to salvation, even if they were healed. Such religious groups sometimes regret that they are only « client-cults ».

The spiritual treatment is a mystical experience in so far as it creates a direct relationship with god or a supernatural entity.

They are not ceremony-oriented in so far as their main practice is the experience of the spiritual treatment. When they exist, religious offices consist in reading the fundamental doctrines, like the reading of verses from the Bible and of their comments in Science and Health with the Key to the Scriptures in the Church of Christian Science or like the reading of the Revelation of « Father Antoine » in Antoinism.

They know how to manage the failure of the treatment so that it does not question the validity of their doctrine.

They create an epistemological rupture with biomedicine both in the explanation of the illness and in the type of medication they propose. In healing religions, illness in the broad sense has a metaphysical cause and the treatment is spiritual. Although the medical establishment and healing Churches are potentially in conflict, they manage to reach a modus vivendi which allows them to coexist in society without being in confrontation, except when medical doctors can invoke negligence and lack of discernment of the religious therapists in a death case. Generally speaking, they are rarely sued for illegal practice of medicine.

Another characteristic of healing religions is practicing prevention. Their practice is not limited to the peak experience of healing. In their view, it is important to be healed, it is essential not to get ill or not to relapse. That is the reason why they have developed practices aiming at keeping illness away. The daily prayer of the Christian Scientist will protect against « malicious mental poison » or regular harmonization in Invitation à la Viea sort of massage of the body accompanied by prayers will prevent illnesses. Some recommend some food hygiene excluding tobacco, alcohol, drugs and some food items presented as harmful as in the movement of the Christ of Montfavet or as the Adventists or the Mormons.

The healings they achieve cannot be considered miracles which from a theological point of view are exceptional events through winch God reminds men of his presence and his power. Healing religions reject « divine lottery » in which the one gets healed and another does not, as in Lourdes. In healing religions, healing must take place if it is done the right way, even if the healing procedure must be repeated as often as it must.

12) I will also add that healing religions are distinct from magic because contrary to magicians, 1) they have worked out a rational religions thinking under the form of a theology along with an axiology; 2) they claim to be universal and do not use a local symbolical system which would limit their power to a given local area; 3) their therapists are not isolated entrepreneurs like magicians, even if the latter ones can unite in corporations to teach their art ; religious therapists are accredited and supported by a community of believers which communicates and propagates a belief ; 4) healing religions propose salvation based on manipulations (beneficial effects here below), like magicians, but they also have an ontological salvation, the soul's or an immortal principle's salvation while magicians do not care about salvation.

Religion and well-being

The modus operandi of religious therapy always consists in a protocol which is common to ail therapies : a) a demand originating in suffering or distress and addressed to a therapist accredited by his community; b) a relationship with the therapist based on hope (believing expectation); c) a theory about illness that cannot be questioned by a possible failure of the therapy; d) the implementation of the spiritual treatment; e) the installation of new social attitudes and behaviors; f) a debt which can be settled through a payment and sometimes a conversion which can sometimes be considered a limitless transfer (in the Freudian sense) to the healing movement.

Contrary to healers of traditional societies, the religious therapist does not aim at reintegrating the patient into his or her family or community The new social attitudes acquired through the spiritual treatment are either at the micro-relational level or at the moral level. The treatment does not manipulate a local symbolical system which limits its efficiency to a cultural area as in traditional religions (Rosny, 1992). The religious treatment concerns the individual in modem societies with a universal outreach. Healing religions allow the patient to put his or her illness in a theory (Pedinielli, 1989) and to appropriate it while giving it a meaning that goes beyond the framework of life on earth; a medical doctor can say how someone gets ill but he or she cannot say why the illness strikes Mr X. The religious therapist answers the question « Why me? » which is posed in the face of the injustice caused by the illness which strikes some people and not others. At this level, he relieves the existential anguish of the sick or distressed person.

It can also be noted that healing religions do not increase the standard of sufferings. There is no transfiguration of the sufferings as in a part of Christianity. They are mostly viewed as the result of errors of judgment like the belief in the existence of evil in the creation whilst a good god cannot have created it. The only thing that man must do is to get rid of the belief in illness and to live pleasantly. It is a sort of contemporary secular hedonism where man claims happiness and health as a right. The mystical habitus that it produces must manifest itself « through the sweet euphoria of man with god. » The individual is not induced to withdraw from the world but is encouraged to acquire a bigger intensity of enjoying life and of happiness.

The end of suffering as a present-day

Healing religions are in the minority in the world. However, it can be noticed that official medicine, though established, has not replaced the recourse to religious therapists. We have wondered if the plausibility of the religious therapy was rooted in the conscience of the public. To this end, we have carried out a survey among students of the University of Lille (France). We have selected students in psychology because they are rather cautious towards belief phenomena which they link to the « obscure » foundation of the mind.

In a first move (2004), we proposed those students a list of 17 criteria and we asked them to choose three in order of importance to qualify a religion « a true religion. » We received 101 fully filled in questionnaires. The first three criteria selected by the students are : a) « makes life meaningful » (43) b) « allows its faithful to overcome the hardships of life » (41) c) « leaves its faithful the possibility not to believe in god » (33) (This is a curious choice!) After a balancing of the choices (1 = 3 points, 2 = 2 points, 3 = 1 point), we have this range: a) « leaves its faithful the possibility not to believe in god » (97) b) equally: « allows its faithful to overcome the hardships of life » (94) and « makes life meaningful » (94) c) « defends human rights » (66). We can see that guaranteeing the well-being is in good position.

In a second survey ( 2004), we have asked another sample of students (95 responses) to agree or to disagree with 12 proposals concerning religious healing. Items 1, 2, 8, 9, 11 concern the possibility of a « out of track » healing. The average percentages of those who agreed are: Overall: 53.46% Believers: 63.82% Unbelievers: 44.99% Items 4,5,7 concern the recourse to religious therapy along with a classical medical treatment. The average percentages of those who agree are: Overall: 68.76% Believers: 73.75% Unbelievers: 59.71%

These results show us that those students have a positive attitude towards the quest for healing or well-being in a religion, including for a large share of unbelievers. We can find here the idea of a magical transformation of the world which is consistent with other vider and more statistically refined surveys (Michelat). The results also show a positive attitude towards the recourse to a combination of medicine and spiritual treatment of illnesses. We will interpret this in terms of mobilization of resources as the students agree to resort to medicine and to religion to have a maximum of chances to get healed or to feel better.

In a short conclusion, we will say that religions give a global response to the suffering of man but we have shown that some religions are specialized in religions healing, in recovering and in keeping well-being. These religions carry out a function that big religions only partially take up or have left aside although it is one of the expectations of a man in distress. They answer the questions on the meaning of illness and unhappiness that medical doctors neglect. This is the reason why, according to us, healing religions will go on existing, even marginally, despite the growing hegemony of biomedicine.

table 1 .

 1

	items

table 2 (

 2 percentages of students who agree with the items).

	Régis Dericquebourg. ISA conference. 2004.
	items	together believers unbilievers
		agree % agree % agree %
	1. today, one may always heal by the prayer.	33,68 40,42	25
	2. some people have a personal gift of healing.	54,73 55,31	54,16
	3. The healing by the prayers is permanent even though the healing with medicine is temporary.	1,06	2,1.2	0
	4. The cure by religious therapist is not inconsistent with the cure by the medecine.	58,94 80,85	83,33
	5. When the medicine is not able to heal a person, one
	understand that these person seeks the recovery with a faith	84,21 85,10	83,33
	healer.
	6. When a person is healed by a religion, it is normal that she converts to it.	59,57 63,04	54,16
	7. One may be healed by a religion without joining it.	63,15 55,31	66,66


Axel Barrau
email: axel.barrau@mines-paristech.fr

Silvère Bonnabel
email: silvere.bonnabel@mines-paristech.fr

Navigating with highly precise odometry and noisy GPS: a case study *

Navigating with highly precise odometry and noisy

GPS: a case study *

Introduction

The extended Kalman filter (EKF) has been developed by the NASA in the 1960s for the Appolo program. It has since been the state of the art for (industrial) navigation. The engineers appreciate its ease of implementation, its rather easy tuning based on the sensors' noises, and its optimal stochastic properties up to second order terms. Yet it is known to loose its properties when the initial errors become too large, due to the nonlinearities. As a result, over the past decade or so we have witnessed many attempts to derive alternative observers with asymptotic convergence properties for attitude and pose estimation. Those important properties -reserved to the deterministic settingoften use the Lie group structure of the state space, see, e.g., [START_REF] Hua | Implementation of a nonlinear attitude estimator for aerial robotic vehicles[END_REF][START_REF] Wolfe | Bayesian fusion on Lie groups[END_REF][START_REF] Batista | Attitude and earth velocity estimation-part i: Globally exponentially stable observer[END_REF][START_REF] Park | Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map[END_REF][START_REF] Izadi | Rigid body attitude estimation based on the lagrange-d'alembert principle[END_REF][START_REF] Bourmaud | Discrete extended Kalman filter on Lie groups[END_REF][START_REF] Amit | Attitude state estimation with multirate measurements for almost global attitude feedback tracking[END_REF][START_REF] Zamani | On the distance to optimality of the geometric approximate minimum-energy attitude filter[END_REF][START_REF] Bonnabel | Non-linear symmetry-preserving observers on Lie groups[END_REF].

1
The Invariant Extended Kalman Filter (IEKF) is a relatively recent variant of the EKF meant to account for the invariance/symmetries properties of the state space when devising EKFs on Lie groups, see [START_REF] Bonnabel | Left-invariant extended kalman filter and attitude estimation[END_REF][START_REF] Bonnabel | Invariant extended Kalman filter: theory and application to a velocity-aided attitude estimation problem[END_REF][START_REF] Ph | Generalized multiplicative extended kalman filter for aided attitude and heading reference system[END_REF][START_REF] Barrau | Intrinsic filtering on lie groups with applications to attitude estimation[END_REF]. As such, it can be viewed as a variant of the multiplicative extended Kalman filter (MEKF) [START_REF] John L Crassidis | Survey of nonlinear attitude estimation methods[END_REF] for attitude estimation, and as an extension to it for pose estimation. It has the merit to retain all the EKF advantages -being a variant of it -while posessing a nicer geometric structure. In [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF], local guaranteed convergence properties are derived for a large class of systems, and simulations indicate remarkable robustness to large initialization errors for high precision navigation problems. However no global property of the IEKF has ever been proved. The present paper allows to gain insight into the global IEKF properties by considering a simple example.

We study a non-holonomic car with perfect odometry and noisy position (i.e. GPS) measurements, where the initial position is known but not the orientation. This simplified navigation problem can be viewed as a limit case of high-accuracy navigation where inertial sensors allow precise navigation for several minutes in the absence of measurement. The degenerate situation of perfectly known dynamics poses no problem to the linear Kalman filter, the gains being possibly large at the beginning if the prior information is inaccurate, and then decreasing to zero. Yet, it poses great challenges to the EKF as gains can go to zero whereas, due to nonlinearities, the estimation error has not reduced enough, leading to a static asymptotic non-zero error, or divergence, even for unnoisy measurements. Note that, the general local convergence proof of [START_REF] Arthur | The convergence of the extended kalman filter[END_REF] requires full rank process noise covariance, which is violated here. The usual remedy is either to artificially inflate the process noise covariance matrix, known as "robust tuning" (see [START_REF] Garry A Einicke | Robust extended kalman filtering[END_REF][START_REF] Sonnemann Reif | An ekf-based nonlinear observer with a prescribed degree of stability[END_REF]) -but this results in degraded stochastic performance as the filter's tuning does not match with the true sensors' characteristics, or to use optimization methods over the first steps to initialize the EKF with a better guess of the true state. Note also, in passing, that as the state conditional probability does not have a density, even particle filters may face severe issues for the considered problem (the resampling step needs process noise to be useful). Here we show the nice geometric structure of the IEKF allows it to perfectly cope with the absence of process noise.

The main merit of the present paper is thus to show one can devise an EKF variant for a simple yet instructive navigation example, which posesses global properties both in terms of behavior and convergence. To our best knowledge this is the first time an EKF variant with global convergence properties is derived for an example of engineering interest, whereas the standard EKF does not even converge. Of course, the symmetries play a prominent role as they bring a lot of structure into the problem. The results obtained suggest great potential benefits of the IEKF over EKF for high precision navigation.

The contributions and organization of the paper are as follows. In Section 2 we recall the implications of deterministic dynamics in the linear case, and how the linear Kalman filter naturally encodes this information. We also introduce our simplified car model and derive the standard EKF equations for it. In Section 3, we discuss the physical implications of the choice of deterministic dynamics for the simplified car, and prove the IEKF naturally encodes this information, reminding the linear case: This is our main contribution. Finally, in Section 4, we leverage those results to derive some global convergence properties of the IEKF for this problem.

2 Filtering with perfect odometry

The linear case

Consider in R n a deterministic continuous-time linear system ẋt = A t x t with noisy discrete output measurements

Y n = H n x tn + V n at times t 1 < • • • < t n ,
where V n is a Gaussian variable. Assume that initially part of the state is known, that is, for some matrix C ∈ R p×n we have Cx 0 = α with α ∈ R p a known vector. This means that the state is initially known to be in a subspace of dimension n -p. By propagating this subspace through the deterministic dynamics, the state is known to be in a subspace of dimension n -p at any time. Indeed letting C t ∈ R p×n be the solution of Ċt = -C t A t with C 0 = C, we have at all times C t x t = α as proved by differentiation.

A Kalman filter perfectly handles this kind of singular information about the state. Indeed, the fact that C 0 x 0 be initially perfectly known, means in probabilistic terms that there is no dispersion of the distribution of x t orthogonally to the subspace the state lives in, and thus C 0 P 0 C T 0 = 0 where P t denotes the covariance matrix output by the Kalman filter. This equality implies that at all times the estimate of the Kalman filter remains in the subspace the state lives in (if initialized inside it), and P t keeps reflecting the absence of dispersion of the probability distribution of the state orthogonally to this subspace, as shown by the following trivial result. Note that the result in fact directly stems from the fact the filter computes the true probability distribution of the state conditioned on the outputs. Proposition 1. Consider the deterministic dynamics ẋt = A t x t with noisy measurements Y n = H n x tn + V n . If initially we have C 0 x 0 = α, implying C t x t = α ∀t ≥ 0 where Ċt = -A t C t , then the linear Kalman filter is such that C t xt = α ∀t ≥ 0. Moreover, the absence of dispersion orthogonally to the subspace the state lives in is correctly captured by the covariance matrix as C t P t C T t = 0 ∀t ≥ 0. Proof. Between two measurements the Riccati equation d dt P t = A t P t + P t A T t implies C t P C T t = 0, and when there is a measurement the covariance updates writes P + tn = (I -K n H n )P tn , and thus C tn P tn C T tn implies C tn P + tn C T tn . Indeed, as P tn is symmetric, we have necessarily C tn P tn = 0 and thus C tn K n = 0. The latter equality also implies the state is forced to remain in the subspace, as the update writes x+

tn = xtn + K n (Y n -H n xtn ).

Implications of perfect odometry for the nonlinear simplified car

Consider a non-holonomic car with deterministic dynamics (e.g. [START_REF] Guillaume | Observation and control of a simplified car[END_REF])

d dt θ t = ω t , d dt x t = cos(θ t )u t sin(θ t )u t , (1)
where θ t is the heading of the car, x t is the position vector, and ω t , u t are the angular and linear velocities computed through odometry. We also consider noisy position measurements. Those measurements can interpreted as GPS measurements, or some other position measurements that arise in mobile robotics. Our goal is to understand what happens when the odometry becomes infinitely more precise than the other sensors, a situation that may arise in mobile robotics, and which also serves as a simplified problem for high precision inertial navigation. We thus consider noisy position measurements

Y n = x tn + V n , (2)
where V n is an i.i.d. Gaussian noise with covariance matrix R. Furthermore, we assume that the car's inital position is known, but its heading is not. As the state initially belongs to a one-dimensional submanifold of the state space, and the dynamics is prefectly known, it remains at all times in a one-dimensional submanifold one can compute.

Proposition 2. Consider the dynamics (1). Assume the initial position is known, i.e., x 0 = 0 2,1 . The dynamics being deterministic, at all times, the state (θ t , x t ) belongs to the image of the set S 1 × {0} × {0} through the flow of (1). This set being invariant to rotations, it can be parameterized in a more concise way as follows. Let b t be defined by the differential equation

b 0 = 0 0 , d dt b t = - 0 -ω t ω t 0 b t + u t 0 . (3)
Then at all times the state satisfies R(θ t ) T x t = b t where R(θ) denotes the rotation of angle θ.

Proof. we have:

d dt R(θ t ) T x t = R(θ t ) T -ω t Jx t +R(θ t ) u t 0 = -ω t JR(θ t ) T x t + u t 0
where J := 0 -1 1 0 commutes with R(θ t ) T . Thus the quantity R(θ t ) T x t satisfies

(3).

The physical interpretation of the proposition is clear: Assume for the sake of simplicity that ω t ≡ 0. This means the car moves along a straight line, and the condition of Proposition 2 becomes x t = R(θ t )(α(t), 0) T = R(θ 0 )(α(t), 0) T where α(t) = t 0 u s ds is the total traveled distance according to the odometer, that is, the car belongs to a circle centered at the inital position and of radius α(t). Let us see now whether a nonlinear counterpart of Proposition 1 might hold for the extended Kalman filter (EKF).

EKF equations

In the latter problem, the initial position is assumed to be known but the not the heading, so we let initially x0 = 0 2,1 . The initial covariance matrix must reflect dispersion only on the heading angle, so we let it be equal to, e.g.,

P 0 =   π/2 0 0 0 0 0 0 0 0   .
Of course, after a few time steps the position becomes also uncertain due to the initial heading error but Proposition 2 holds. The standard EKF equations for the continuous time with discrete observations system considered in Section 2.2 rely on a propagation step between two observations, and an update step at each measurement time.

Propagation:

d dt θt = ω t , d dt xt = cos θt sin θt u t , t n-1 < t < t n , d dt P t = A t P t + P t A T t , t n-1 < t < t n , (4)
with

A t =   0 0 0 -sin( θt )u t 0 0 cos( θt )u t 0 0   .
Note that Equation (4) contains no process noise (usually denoted by Q t ) due to deterministic dynamics.

Update:

The update consists of the following steps:

• Computation of the gain:

K n = P tn H T HP tn H T + R -1 ,
with H = 0 1 0 0 0 1 .

• Computation of the innovation:

z = Y n -xtn .
• Computation of the new estimate:

θ+ tn x+ tn = θtn xtn + K n z.
• Update of the covariance matrix:

P + tn = (I -K n H) P tn .
Due to the update, we see no reason why the EKF estimates should remain in the submanifold defined at Proposition 2. Simulations below will confirm this indeed. But before that, let us study what happens with the IEKF for the considered problem.

Invariant filtering with perfect odometry

The considered system is left-invariant on the Lie group SE(2) [START_REF] Bonnabel | Non-linear symmetry-preserving observers on Lie groups[END_REF]. A Left-invariant IEKF for our system is based on the error variable:

θ -θ R(θ) T (x t -x t ) , (5)
hence its name: the error variable is invariant to left multplications on SE(2) (that is change of global frame) which write θ → θ + α, x → R(α)x. As a result, the covariance matrix P t output by the IEKF is an approximation to the dispersion of this invariant error.

IEKF equations

The IEKF equations for the considered example has already been derived in [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF] and are based on a propagation and an update step as follows:

Propagation:

d dt θt = ω t , d dt xt = cos θt sin θt u t , t n-1 < t < t n d dt P t = A t P t + P t A T t , t n-1 < t < t n , (6)
with

A t =   0 0 0 0 0 ω t -u t -ω t 0   .
Again, the equation contains no process noise.

Update:

The update consists of the following steps:

• Computation of the gain:

K n = P H T HP t H T + R -1 ,
with H = 0 1 0 0 0 1 .

• Computation of the innovation:

z = R( θtn ) T (Y n -xtn ) .
• Computation of the new estimate:

R( θ+ tn ) x+ tn 0 1×2 1 = R( θtn ) xtn 0 1×2 1 exp(K n z) (7)
where exp : R 3 → R 3 is the exponential map of the Lie group

SE(2) defined for θ ∈ R, x ∈ R 2 by exp θ x = R(θ) B(θ)x 0 1×2 1 , with B(θ) = 1 θ sin θ -(1 -cos θ) (1 -cos θ)
sin θ .

• Update of the covariance matrix:

P + tn = (I -K n H) P tn . (8)
with initial values such as in Section 2.3. The IEKF is a left-invariant observer in the sense of [START_REF] Bonnabel | Non-linear symmetry-preserving observers on Lie groups[END_REF]: the output error and the gains are computed in the frame of the car, making them insensitive to change of global frame (i.e. left group multiplications). However, it has the additional property of being an EKF variant, and as such its tuning is optimal for the noisy problem up to second order terms.

Implications of perfect odometry for the IEKF

It turns out that, quite surprisingly, a counterpart of Proposition 1 can be derived for the IEKF. Indeed, Proposition 2 shows the true state lives in a one-dimensional manifold defined at all times by R(θ t ) T x t = b t . The following result shows that whatever the initial heading error, the IEKF output estimate lives in this very manifold too. The proof has been moved to the Appendix.

Theorem 1. The IEKF estimate as defined just above satisfies at all times R( θt ) T xt = b t with b t defined at eq (3) and thus lives in the same one-dimensional manifold as the state space whatever the initial heading value θ0 .

Numerical illustration of the theorem

To fix ideas, assume to car is moving along a straight line, i.e. ω t ≡ 0. In this case the theorem implies xt = R( θt )(α(t), 0) where α(t) = t 0 u s ds is the distance traveled according to the odometer. This implies the two physically logical to expect properties of the estimate, that are yet never met by the standard EKF:

1. Despite the updates applied by the IEKF to the state, the total distance traveled by the estimate matches exactly the distance given by the odometer 2. The car estimated heading is always parrallel to a ray that passes through the origin Those properties are logical to expect since the odometer (assumed perfect) indicates the car is necessarily travelling along a (unknown) straight line emanating from the origin. Figure 1 displays the trajectory of the car for ω t ≡ 0 and the trajectories of the EKF and IEKF estimates. We see the IEKF achieves property (2) above, as evidenced by the dotted segments that prolongate the trajectory of the car between two updates.

Figure 2 displays the error between the total distance traveled by the estimate of respectively the IEKF and the EKF and the total distance traveled by the true car. Due to perfect odometry, the user can evaluate exactly the distance traveled by merely integrating the odometer measurements. We see the IEKF makes no mistake on this value whereas the EKF totally miscalculates it.

Implications in terms of convergence

In this section, once again for the sake of simplicity, we consider the car moves along a straight line, i.e., ω t ≡ 0.

Numerical evidence

The results of EKF and IEKF in terms of position error are displayed on Figure 3 with identical covariance matrices tuning. To obtain cleaner curves, we fed the filters with unnoisy measurements (which amounts to study their behavior when used as nonlinear observers). We see the IEKF outperforms the EKF as

• the IEKF initialized with a 40 • angle error, beats the EKF initialized with a 5 • angle error (top plot) !

• All EKFs (even with a 5 • angle initial error) totally fail to converge (bottom plot) whereas the IEKF does.

This last feature may be explained by the fact the dynamics being deterministic, the gains tend to 0. If the EKF has failed to reduce the error enough during the transitory phase, it may fail to converge as observed here.

IEKF almost global convergence properties

The following theorem leverages the unexpected "physical" properties underlined by Theorem 1, to prove almost global convergence of the IEKF used as a non-linear observer (i.e. noise being turned off). This is a remarkable result as global convergence properties of EKFs are in general very difficult to derive (in the present case they are impossible to derive as they do not converge). The proof has been moved to the Appendix.

Theorem 2. Consider deterministic dynamics (1) with noisy output (2). Assume the observations occur at a constant rate (t n = n∆T with ∆T > 0) with noise covariance of the form R = rI 2 with r > 0. Assume that the initial position is known and equal to 0 2,1 and the car is moving along a straight line with constant speed i.e., ω t ≡ 0 and u t ≡ 1.

Then, for any initial orientation error such that | θ(0) -θ(0)| = π mod 2π the IEKF used as a nonlinear observer globally converges to the true state. Moreover, we have the following convergence rates : θtn -θ tn ∼ C/n 3 and xtn -x tn ∼ C/n 2 for some C > 0.

Figure 1 :

 1 Figure 1: Illustration of Theorem 1: Car with perfect odometry, with known initial position, and unknown initial orientation. True and estimated trajectories (EKF and IEKF). The odometry indicates ω t ≡ 0, thus both filters move along straight lines between two updates. But the IEKF update is such that the car always moves along rays of a circle centered at the known initial position, as evidenced by the dotted segments, whereas this physical information is totally destroyed by the standard EKF updates.

Figure 2 :

 2 Figure 2: Error between the distance traveled by the estimate for respectively IEKF and EKF (dotted line) and the true distance traveled (according to perfect odometer). The IEKF preserves the information delivered by the odometer whereas the IEKF loses it completely.

Figure 3 :

 3 Figure 3: Influence of singular information: Position error of the EKF for GPS and perfect odometry, with initial position known and initial heading unknown. Top plot: We see the IEKF initialized with a large error (40 • ) even beats the EKF initialized with a small error (5 • ). Bottom plot : Vertical zoom and extended running time (1 million steps). The EKF errors stabilize at a non-zero value whereas the IEKF error goes to 0.

A Proof of Theorem 1

Proposition 2 proved R(θ t ) T x t = b t . Now let He t = R( θt ) T [-Jb t , I 2 ] and let

Our goal is to prove N t ≡ 0. We have M 0 = 0 and N 0 = 0. Assume M t n-1 = 0, N t n-1 = 0. During the propagation step d dt N t = 0 by mimicking the proof of Proposition 2 and thus N tn = 0. Moreover

and thus M tn = 0. As P t is symmetric this means H e tn P tn = 0, proving that [-Jb tn , I 2 ] T defines a basis of the orthogonal space to Im P tn and thus to Im K n . This implies the image of K n is spanned by w := 1 Jb tn . Let us now focus on [START_REF] Bourmaud | Discrete extended Kalman filter on Lie groups[END_REF]. We have just proved

K n z = αw with α ∈ R that we assume equal to 1 without loss of generality. It follows directly from the definition of the Lie exponential map on SE(2) that θ+ , x+ defined by [START_REF] Bourmaud | Discrete extended Kalman filter on Lie groups[END_REF] are the solution at s = 1 to the equation

Let a(s) := R( θ(s)) T x(s). We have then d ds a(s) = -JR( θ(s)) T x(s) + Jb tn = J(-a(s) + b tn ). As a(0) -b tn = 0 this implies a(s) -b tn ≡ 0 and in particular a(1) = b tn .

We have thus proved that N + tn = N tn = 0. To complete the induction over n, we must check also that M + tn = M tn = 0. Note that M t = 0 is equivalent to [-Jb t , I 2 ] T P t = 0. As N t is preserved by the update, so is b t . Thus ( He tn ) + P + tn ( He

B Proof of Theorem 2

The solution of the Riccati equation ( 6), [START_REF] John L Crassidis | Survey of nonlinear attitude estimation methods[END_REF] reads

with a(0) = p 0 , d dt a(t) = 0 during the propagation (t ∈]t n-1 , t n ]) and

at the update steps. Checking this value of P t verifies ( 6),( 8) is straightforward. Now we compute the value of a(t) for any t. Eq. ( 10) can be re-written 1 a(tn) + = 1 a(tn) + ∆T 2 n 2 r , where t n has been replaced with n∆T . As we have a(t + n ) = a(t n+1 ) we can apply the square pyramidal number formula to obtain:

In turn, let us write a recursive equation verified by the heading error θt = θt -θ t . We will need the update function of θtn (extracted from eq. ( 7)):

where we have to compute K n (1, :) (first line of the gain matrix) and z. First, using the expression of P t , we can write the first line of the gain K n :

Then we write the innovation as a function of the heading error θt . As we have ω t = 0 and u t = Cste = u, Proposition 2 boils down to the relation x t = R(θ t )(tu, 0) T . Following Theorem 1 we also have at all time: xt = R( θt )(tu, 0) T , which allows writing the innovation as a function of θt :

Putting together ( 13) and ( 14) we obtain: θt n+1 = θtn -α n sin( θtn ), [START_REF] Park | Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map[END_REF] with α n = t 2 n a(tn) t 2 n a(tn)+r . As a first consequence, | θt | is decreasing. Thus, sin(x) x being decreasing on [0, π], we have sin(θt) θt ≥ sin(θ 0 ) θ 0 , i.e. sin (θ t ) ≥ sin(θ 0 ) θ 0 θ t . Introducing the latter inequality into [START_REF] Park | Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map[END_REF]


Jean-Louis Loday
email: loday@math.unistra.fr

J.-L Loday

Exponential series without denominators

For a commutative algebra which comes from a Zinbiel algebra the exponential series can be written without denominators. When lifted to dendriform algebras this new series satisfies a functional equation analogous to the Baker-Campbell-Hausdorff formula. We make it explicit by showing that the obstruction series is the sum of the brace products. In the multilinear case we show that the role the Eulerian idempotent is played by the iterated pre-Lie product.

Introduction

The classical exponential series exp(x)

= 1 + x + x 2 2! + • • • + x n n! + • • •
can be written without denominators provided one assumes some properties on the commutative algebra. For instance in a Zinbiel algebra the term x n n! can be replaced by the iterated product x ≺n := x ≺ (x ≺n-1 ), with x ≺1 = x. Recall that a Zinbiel algebra is a commutative algebra in which the product splits as xy = x ≺ y + y ≺ x. Matrices with coefficients in a commutative algebra (resp. Zinbiel algebra) are endowed with a structure of associative algebra (resp. dendriform algebra). Recall that a dendriform algebra is an associative algebra whose product is the sum of two binary operations: xy = x ≺ y + x ≻ y, supposed to satisfy some relations. There is a unique way to extend the exponential series from commutative algebras to associative algebras. However there are several ways to extend the exponential series from Zinbiel algebras to dendriform algebras: either take x n n! or x ≺n or x ≻n . These choices are equal in a Zinbiel algebra, but different in a dendriform algebra. We adopt the notation e(x) := 1

+ x + x ≺2 + • • • + x ≺n + • • • , e ′ (x) := 1 + x + x ≻2 + • • • + x ≻n + • • •
for the exponential series without denominators. The classical functional equation exp(x) exp(y) = exp(x + y) holds only when we are in the commutative or in the Zinbiel setting. In this last case it can be written: e(x)e(y) = e(x + y).

In the associative setting there is an error term for exp which is called the Baker-Campbell-Hausdorff formula. We prove that, in the dendriform context, the functional equation for the exponential without denominators is: In the associative framework the BCH-formula is best understood by looking at the multilinearized case. Then, the multilinear polynomial involved in the functional equation of the exponential turns out to be the Eulerian idempotent. In the dendriform case, we show that the analogue of the Eulerian idempotent for the exponential without denominators is played by the iterated pre-Lie product

e(x)
h(x 1 , . . . , x n ) = {x 1 , {x 2 , {• • • {x n-1 , x n }•}}},
where the pre-Lie product is {x, y} := x ≻ yy ≺ x.

Here is the content of this paper. In the first section we recall the notions of Zinbiel algebras, dendriform algebras and some of their properties. In the second section we introduce and study the exponential without denominators. In particular we compute its inverse, the logarithm without denominators, which is closely related to the 1 1 2 -logarithm of Kontsevich. In section 3 we prove the BCH-type formula for the exponential series without denominators. As a corollary we show that the multilinear obstruction to the additivity of the exponential is the iterated pre-Lie product.

In this paper K denotes a unital commutative ring (for instance Z) which is the ground ring. We sometimes need to suppose that K contains Q. The tensor product over K is simply denoted by ⊗.

Zinbiel and dendriform algebras

We introduce the notion of Zinbiel algebra (called dual Leibniz algebra in [START_REF] Loday | Cup-product for Leibniz cohomology and dual Leibniz algebras[END_REF]) and its relationship with commutative algebras. We also introduce the notion of dendriform algebras because matrices over a Zinbiel algebra bear the structure of a dendriform algebra.

Definition

A Zinbiel algebra is a module A over K equipped with a binary operation x ≺ y such that

(x ≺ y) ≺ z = x ≺ (y ≺ z + z ≺ y).
Symmetrizing the Zinbiel operation, that is defining xy := x ≺ y + y ≺ x, we get a product on A which is commutative and associative. Indeed, one gets

(xy)z = (x ≺ y + y ≺ x) ≺ z + z ≺ (xy) = x ≺ (yz) + y ≺ (xz) + z ≺ (xy) = x ≺ (yz) + y ≺ (zx) + z ≺ (yx) = x ≺ (yz) + (y ≺ z) ≺ x + (z ≺ y) ≺ x = x(yz).
Hence there is a forgetful functor between categories of algebras:

Zinb-alg → Com-alg.

Free Zinbiel algebra

It is shown in [START_REF] Loday | Dialgebras and related operads[END_REF] that the free Zinbiel algebra over the vector space V is the reduced tensor module

T (V ) := V ⊕ • • • ⊕ V ⊗n ⊕ • • • where the generic element v 1 • • • v n ∈ V ⊗n corresponds to the product v 1 ≺ (v 2 ≺ (• • • ≺ v n )).
As a consequence the space of n-ary operations of the operad Zinb is the S n -module Zinb(n) = K[S n ]. More precisely we have:

Theorem 1.3 Let V be a vector space and let T (V ) be the reduced tensor module over V . The half-shuffle:

v 1 • • • v p ≺ v p+1 • • • v p+q := ∑ σ v 1 v σ -1 (2) • • • v σ -1 (p+q)
where σ is a (p -1, q)-shuffle acting on {2, . . . , p + q}, makes T (V ) into the free Zinbiel algebra on V . The associated commutative algebra is the (nonunital) shuffle algebra, denoted by T sh (V ). The Zinbiel algebra ( T (V ), ≺) is in fact the free Zinbiel algebra over V , denoted by Zinb(V ).

Proof. See [START_REF] Loday | Dialgebras and related operads[END_REF].

Example

If V is one-dimensional spanned by x, then Zinb(K x) is spanned by x ≺n for n ≥ 1.

The operations are given by

x ≺p ≺ x ≺q = p + q -1 q -1
x ≺p+q and x ≺p x ≺q = p + q q x ≺p+q .

Dendriform algebra

A dendriform algebra is a K-module A equipped with two linear maps (binary operations)

≺: A ⊗ A → A and ≻: A ⊗ A → A
called the left operation and the right operation respectively, satisfying the following three relations

   (x ≺ y) ≺ z = x ≺ (y ≺ z) + x ≺ (y ≻ z), (x ≻ y) ≺ z = x ≻ (y ≺ z), (x ≺ y) ≻ z + (x ≻ y) ≻ z = x ≻ (y ≻ z).
From these axioms it follows readily that the binary operation

xy := x ≺ y + x ≻ y .
is associative. Under this notation the first relation becomes

(x ≺ y) ≺ z = x ≺ (yz).
In the proof of the main Theorem we allow ourselves to write x ≺ yz in place of x ≺ (yz).

Let us mention that numerous combinatorial Hopf algebras come with a dendriform structure, cf. for instance [START_REF] Aguiar | Infinitesimal bialgebras, pre-Lie and dendriform algebras[END_REF][START_REF] Burgunder | Tridendriform structure on combinatorial Hopf algebras[END_REF][START_REF] Loday | Combinatorial Hopf algebras[END_REF][START_REF] Novelli | Hopf algebras and dendriform structures arising from parking functions[END_REF].

Commutative dendriform algebra

By definition a commutative dendriform algebra is a dendriform algebra whose left and right operations are related by the following symmetry relation

x ≺ y = y ≻ x, for any x and y.

By direct inspection we see that a Zinbiel algebra is a commutative dendriform algebra and vice-versa (cf. [START_REF] Loday | Dialgebras and related operads[END_REF]).

Proposition 1. [START_REF] Loday | Cup-product for Leibniz cohomology and dual Leibniz algebras[END_REF] The module of n × n-matrices M n (A) with coefficients in the Zinbiel algebra A is a dendriform algebra.

Proof. It is straightforward to check that the matrices over a dendriform algebra is still a dendriform algebra. The formulas are like in the classical case. Since a Zinbiel algebra is a particular case of a dendriform algebra, we are done.

Dendriform calculus

We recall some results from [START_REF] Loday | Dialgebras and related operads[END_REF] about computation in a dendriform algebra A.

It is often helpful to suppose that A is equipped with a unit 1. So A = K 1 ⊕ I where I is a dendriform algebra, and the relationship between 1 and the elements of I are given by:

x ≺ 1 = x, 1 ≺ x = 0, x ≻ 1 = 0, 1 ≻ x = x.
Observe that 1 ≺ 1 and 1 ≻ 1 are not defined, so, in order for an expression like (1 + u)(1 + v) to make sense, we have to write it as

(1 + u)(1 + v) = 1 + u + v + uv = 1 + u + v + u ≺ v + u ≻ v .
The free unital dendriform algebra on one generator is spanned by the planar rooted binary trees. We denote by PBT n the set of planar rooted binary trees t with n leaves. In low dimension we have:

PBT 1 = { | } , PBT 2 = ⑧ ⑧ ❄ ❄ , PBT 3 = ❄ ❄ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ , ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ , PBT 4 = ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ , ❄ ❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ , ⑧ ⑧ ❄ ❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ , ⑧ ⑧ ❄ ❄ ❄ ❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ , ❄ ❄ ❄ ❄ ❄ ❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ .
When t has n + 1 leaves (i.e. t ∈ PBT n+1 ), it determines an n-ary operation. Applied to the generic n-tuple

x 1 • • • x n we denote the result by t(x 1 • • • x n ). For instance: |() = 1, ⑧ ⑧ ❄ ❄ (x) = x, ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ (xy) = x ≻ y, ❄ ❄ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ (xy) = x ≺ y.
More generally, if the tree t is the grafting of t l and t r , denoted by t = t l ∨ t r , then

t(x 1 • • • x n ) = t l (x 1 • • • x i-1 ) ≻ x i ≺ t r (x i+1 • • • x n ).
The free dendriform algebra over the vector space V is Dend(V ) = n Dend n ⊗ V ⊗n where Dend n = K[PBT n+1 ]. In order to check an equality of multilinear elements in the free dendriform algebra over the set {x 1 , . . . , x n } it is equivalent to check the equality for each permutation of the variables individually (in other terms the operad Dend is a nonsymmetric operad).

It is sometimes helpful to adopt the notation x t := t(x • • • x) or simply t when there is only one variable into play. For instance we have:

x t = x ≺n = x ≺ (x ≺ (• • • ≺ x))
for t the right comb with n + 1 leaves. It can be shown that

x n = ∑ t∈PBT n+1
x t in any dendriform algebra.

We already introduced the notation x ≺n . Similarly we define x ≻n := (x ≻n-1 ) ≻ x, with x ≻1 := x.

If we now work in a Zinbiel algebra, then it can be shown that x t = #ϕ -1 (t) x ≺n where ϕ : S n → PBT n+1 is the surjective map constructed in loc.cit. As a consequence we have

x n = ∑ t∈PBT n+1 x t = ∑ t∈PBT n+1 #ϕ -1 (t) x ≺n = ∑ σ ∈S n x ≺n = n! x ≺n ,
as we already know.

Pre-Lie product and brace products

Recall that a binary operation is said to be left pre-Lie if its associator is symmetric in the first two variables. In a dendriform algebra the binary operation {x, y} := x ≻ yy ≺ x is a left pre-Lie product. It is a direct consequence of the dendriform axioms. More generally, one can form the following (n + 1)-ary operation:

{x 1 , . . . , x n ; y} := n ∑ i=0 (-1) n-i (x 1 ≺ (x 2 ≺ (• • • ≺ x i ))) ≻ y ≺ (((x i+1 ≻ • • • ) ≻ x n-1 ) ≻ x n ) .
In [START_REF] Ronco | Primitive elements in a free dendriform algebra[END_REF] M. Ronco showed that these operations are primitive for the Hopf structure of the free dendriform algebra and that they satisfy the axioms of a brace algebra. They are called brace products. Observe that {x; y} = {x, y} is the left pre-Lie product.

A commutative diagram of algebras

The relationship between all these types of algebras can be summarized by the existence of a commutative diagram of categories of algebras [START_REF] Loday | Dialgebras and related operads[END_REF][START_REF] Ronco | Primitive elements in a free dendriform algebra[END_REF]:

Zinb-alg / / Dend-
K[x] n K[PBT n+1 ] o o o o n! x ≺n ∑ t x t ✤ o o K[x] O O O O K[x] ∼ = o o O O O O x n ❴ O O x n ❴ O O ✤ o o
Observe that there are various possibilities to lift the element x n n! to the free dendrifrom algebra, for instance x t #ϕ -1 (t) for any t ∈ PBT n+1 .

Exponential series

We introduce the exponential series without denominators and we compute their inverse for the associative product and for composition (logarithm without denominators).

Exponential series without denominators

Let V be a module over K and let Dend(V ) ∧ be the infinite product

Dend(V ) ∧ := ∏ n Dend n ⊗ V ⊗n .
For any x ∈ V a series in x is an element of Dend(V ) ∧ made out of products of the only variable x ∈ V . There are several different ways to extend the exponential series exp(x) = e(x) = e ′ (x) on Zinbiel algebras to dendriform algebras. For instance we have

1. exp(x) = 1 + x + x 2 2! + • • • + x n n! + • • • , the classical exponential series when Q ⊂ K,
2. e(x) := 1 + ∑ n≥1 x ≺n , e ′ (x) := 1 + ∑ n≥1 x ≻n , the exponential series without denominators, 3. ee(x) := 1 + ∑ n≥1 1 2 (x ≺n + x ≻n ), when 2 is invertible in K. Observe that we have e(x) = 1 + x ≺ e(x). Proof. Let us first prove that x ≻i ≻ x ≺ j = x ≻i+1 ≺ x ≺ j-1 . We have

x ≻i ≻ x ≺ j = x ≻i ≻ (x ≺ x ≺ j-1 ) = (x ≻i ≻ x) ≺ x ≺ j-1 = x ≻i+1 ≺ x ≺ j-1 .
Expanding the product e ′ (-x)e(x) we get

x ≺n + • • • + (-1) j (x ≻i ≻ x ≺ j + x ≻i ≺ x ≺ j ) + • • • + (-1) n x ≺n
in degree n > 0. This element is 0 as a consequence of the preceding formula. The second formula is an immediate consequence of the first one.

Proposition 2.3 In the dendriform context, the inverse of the exponential series E(x) := e(x) -1 for composition is the series

L(x) := x ≺ (1 -x + x 2 -x 3 + • • • + (-1) n x n + • • • ) , called the logarithm without denominators. Proof. Let us write L(x) = x + ϕ 2 (x)+ • • • + ϕ n (x)+ • • • for the inverse of E(x). Since E(x) = x ≺ (1 + E(x))
, by replacing x by L(x) in this equality we get

x = L(x) ≺ (1 + x).
Hence we have ϕ 2 (x) = -x ≺ x and ϕ n+1 (x) + ϕ n (x) ≺ x = 0. By induction we suppose that ϕ n (x) = (-1) n-1 x ≺ x n-1 . We compute:

ϕ n+1 (x) = -ϕ n (x) ≺ x, = -(-1) n-1 (x ≺ x n-1 ) ≺ x, = (-1) n x ≺ (x n-1 ≺ x + x ≻ x n-1 ), = (-1) n x ≺ (x n ).

Remark

This proposition gives a quick proof of the fact that exp and log are inverse to each other. This proof is similar to the proof which uses the integral definition of the logarithm: log(1 + x) = dx 1+x .

Zinbiel algebras in characteristic p

Let us suppose that K is a characteristic p field and let us work in the Zinbiel framework. Since x n = n! x ≺n the logarithm becomes a polynomial which can be written

log(1 + x) = L(x) := p-1 ∑ i=1 (-1) i-1 x i i + (-1) p-1 (p -1)! x ≺p .
First, the element x ≺p is a divided power, that is (A, γ(x) := x ≺p ) is a divided power algebra, cf. for instance [START_REF] Dokas | Zinbiel algebras and commutative algebras with divided powers[END_REF]. Second, the first part of the logarithm is the so-called "one and a half logarithm" introduced by Maxim Kontsevich in [START_REF] Kontsevich | The 1 1 2 -logarithm. Appendix to:"On poly(ana)logs. I[END_REF]. In other words we have the following result.

Proposition 2.6

In a characteristic p divided power algebra (resp. Zinbiel algebra) the one and a half logarithm plus (-1) p-1 (p -1)! γ(x) is an invertible series whose inverse is the exponential series.

BCH-type formula for the exponential series without denominators

In a Zinbiel algebra we have the functional equation e(x)e(y) = e(x + y). But, in a dendriform algebra, x ≻ y = y ≺ x (even for x = y), hence there are nontrivial dendriform polynomials H n (x, y) in x and y such that e(x)e(y) = e(x + y

+ • • • + H n (x, y) + • • • ).
For instance we obviously have H 2 (x, y) = x ≻ yy ≺ x, which is the (left) pre-Lie product {x, y}. Our aim is to compute H n (x, y), in fact to show that it is a brace product. The aim is to prove that R(x, y) = 0. We use the following two relations: e(z) = 1 + z ≺ e(z) and e ′ (-x)e(x) = 1 (Lemma 2.2). On one hand we have Since R(0, 0) = 0 we see from this equation that the degree 1 part of R(x, y) is also 0, and, by induction, the degree n part of R(x, y) is 0 for any n. So R(x, y) = 0 and we are done.

The second formula follows from the fact that the involution τ :

Dend(V ) → Dend(V ) which sends ≺ to ≻, ≻ to ≺ and v 1 • • • v n to v n • • • v 1 is an isomorphism of dendriform algebras.
The Corollary is an immediate consequence.

Functional equation in one variable

In the associative case, if x = y, then the functional equation of the classical exponential series is the same as in the commutative case. However in the dendriform case, if x = y, then the functional equation is different from the Zinbiel case because x ≺ x = x ≻ x. As an immediate corollary of the main result the term H n (x, x) in the formula

e(x)e(x) = e 2x + ∑ n≥2 H n (x, x)
is as follows. Let us denote by lc i (resp. rc i ) the left comb (resp. right comb) with i leaves.

Corollary 3.4

In the dendriform context we have:

H n (x, x) = ∑ i+ j=n+1 i≥1, j≥1 (-1) j rc i ∨ lc j ∈ K[PBT n+1 ] f or n ≥ 2.
Proof. It follows from Corollary 3.2 and from the following equalities in the free dendriform algebra on one generator: rc i (x, . . . , x) = x ≺i-1 , lc i (x, . . . , x) = x ≻i-1 .

The multilinear case

For the classical exponential series in the associative case, the Baker-Campbell-Hausdorff formula takes the form It comes

exp(x) exp(y) = exp(x + y + • • • + BCH m (x, y) + • • • ) .
h(x 1 , . . . , x n ) = ∑ (-1) ℓ x i 1 ≻ x i 2 ≻ • • • (x i k ≻ x n ≺ x j ℓ ) ≺ • • • ≺ x j 2 ≺ x j 1 ,
where {i 1 • • • i k | j 1 • • • j ℓ } is a (k, l)-shuffle of {1, . . . , n -1}. It is clear that for n = 2 we get h(x 1 , x 2 ) = x 1 ≻ x 2x 2 ≺ x 1 = {x 1 , x 2 }. Then, by induction we get the expected result.

  e(y) = e x + e(x) ≻ y ≺ e ′ (-x) , which can be written e(x)e(y) = e(x + y + ∑ n≥1 {x, . . . , x n ; y}) , where {-, • • • , -; -} is the brace product.

Lemma 2 . 2

 22 In the dendriform context we have: e ′ (-x)e(x) = 1 = e(x)e ′ (-x).

Theorem 3 . 1 Corollary 3 . 2 1 (- 1 ) 1 ;

 3132111 In a dendriform algebra the following equalities hold: e(x)e(y) = e x + e(x) ≻ y ≺ e ′ (-x) , e ′ (x)e ′ (y) = e ′ e ′ (-y) ≻ x ≺ e ′ (y) + y , for e(x) := 1 + ∑ n≥1 x ≺n and e ′ (x) := 1 + ∑ n≥1 x ≻n . The dendriform polynomial H n is the brace product:H n (x, y) = ∑ i+ j=n-j (x ≺i ) ≻ y ≺ (x ≻ j ) = {x, . . . , x n-y} .Proof. [of Theorem 3.1] We prove the first relation. Let us define R(x, y) := e(x)e(y)e x + e(x) ≻ y ≺ e ′ (-x) .

  e x + e(x) ≻ y ≺ e ′ (-x) = 1 + x + e(x) ≻ y ≺ e ′ (-x) ≺ (e(x)e(y) -R(x, y)) = 1 + x ≺ (e(x)e(y)) + (e(x) ≻ y) ≺ (e ′ (-x)e(x)e(y)) -Φ(x, y), where Φ(x, y) := (x + e(x) ≻ y ≺ e ′ (-x)) ≺ R(x, y). Then we have e x + e(x) ≻ y ≺ e ′ (-x) = 1 + x ≺ (e(x)e(y)) + e(x) ≻ y ≺ e(y) -Φ(x, y) = 1 + x ≺ (e(x)e(y)) + e(x) ≻ E(y) -Φ(x, y), by Lemma 2.2. On the other hand we have: e(x)e(y) = e(x)(1 + E(y)) = e(x) + e(x) ≺ E(y) + e(x) ≻ E(y).Hence we computeR(x, y) = e(x) + e(x) ≺ E(y)) -1x ≺ (e(x)e(y)) + Φ(x, y) = e(x) + e(x) ≺ E(y) -1x ≺ (e(x)(1 + E(y)) + Φ(x, y) = e(x) + e(x) ≺ E(y) -1x ≺ e(x)x ≺ (e(x)E(y)) + Φ(x, y) = e(x) ≺ E(y)x ≺ (e(x)E(y)) + Φ(x, y) = e(x) ≺ E(y) -(x ≺ e(x)) ≺ E(y) + Φ(x, y) = 1 ≺ E(y) + Φ(x, y) = Φ(x, y) .So we have proved that R(x, y) satisfies the functional equation: R(x, y) = (x + e(x) ≻ y ≺ e ′ (-x)) ≺ R(x, y) .

Proposition 3 . 6

 36 One of the way to compute the Lie polynomial BCH m (x, y) is to consider the multilinear versionexp(x 1 ) • • • exp(x n ) = exp(x 1 + • • • + x n + • • • + BCH m (x 1 , . . . , x n ) + • • • ) .It is a result of Dynkin that BCH m (x, y) can be computed out of the multilinear part of BCH n (x 1 , . . . ,x n ) denoted eul(x 1 , . . . , x n ) ∈ Q[S n ].It is known, cf. for instance[START_REF] Loday | Série de Hausdorff, idempotents eulériens et algèbres de Hopf[END_REF], that eul(x 1 , . . . , x n ) is the Eulerian idempotent.Let us multilinearize similarly the functional equation of the series e(x) in the dendriform case:e(x 1 ) • • • e(x n ) = e(x 1 + • • • + x n + • • • + H m (x 1 , . . . , x n ) + • • •) .and denote by h(x 1 , . . . , x n ) the multilinear part of H n (x 1 , . . . , x n ). The dendriform polynomial h(x 1 , . . . ,x n ) is the iterated pre-Lie product: h(x 1 , . . . , x n ) = {x 1 , {x 2 , {• • • {x n-1 , x n }•}}},where {x, y} := x ≻ yy ≺ x. Proof. In order to compute e(x 1 ) • • • e(x n ) we can apply the functional equation of Theorem 3.1 iteratively. It comes immediately that the polynomial h(x 1 , . . . , x n ) is the multilinear part of e(x 1 ) ≻ e(x 2 ) ≻ • • • (e(x n-1 ) ≻ x n ≺ e ′ (-x n-1 )) • • • ≺ e ′ (-x 2 ) ≺ e ′ (-x 1 ).

  Considering free algebras on one generator x we get the polynomials K[x] for Zinb,Com and Ass, and we get n K[PBT n+1 ] for Dend:

		alg	/ / Brace-alg
	Com-alg	/ / Ass-alg	/ / Lie-alg