Natalia Lyubova

David Filliat

Serena Ivaldi

Improving object learning through manipulation and robot self-identification

Keywords: developmental robotics, incremental learning, robot self-identification, interactive object exploration

We present a developmental approach that allows a humanoid robot to continuously and incrementally learn entities through interaction with a human partner in a first stage before categorizing these entities into objects, humans or robot parts and using this knowledge to improve objects models by manipulation in a second stage. This approach does not require prior knowledge about the appearance of the robot, the human or the objects. The proposed perceptual system segments the visual space into proto-objects, analyses their appearance, and associates them with physical entities. Entities are then classified based on the mutual information with proprioception and on motion statistics. The ability to discriminate between the robot's parts and a manipulated object then allows to update the object model with newly observed object views during manipulation. We evaluate our system on an iCub robot, showing the independence of the self-identification method on the robot's hands appearances by wearing different colored gloves. The interactive object learning using self-identification shows an improvement in the objects recognition accuracy with respect to learning through observation only.

I. INTRODUCTION

Future service robots will need the ability to work in different human environments that cannot be predicted in advance. Serving humans will require a capability to detect many different objects and to learn about them. Ideally, robots should be able to learn about objects without constant or dedicated supervision, but rather like children do, during interaction with adults and by manipulating objects [START_REF] Perone | Learning how actions function: The role of outcomes in infants' representation of events[END_REF].

Objects appearances can be learned through observation. However, more complete objects representations are required, when a robot needs to exploit objects for accomplishing tasks. This information can be essentially retrieved through active object exploration [START_REF] Oakes | Manual object exploration and learning about object features in human infants[END_REF]. Manipulation provides an opportunity to gather an object appearance from different viewing angles and scales by turning the object and approaching to a camera. However, during manipulations, objects are often partly covered by a robot's or human hand, and thus the ability to distinguish between features that belong to the robot, the human, and the manipulated object, is crucial. This paper focuses on this issue: we propose an approach to enhance learning through object manipulation and categorization of visible entities into robot's parts, human parts, and manipulable objects. The interplay of the implemented modules is shown in Fig. 1. Self-identification has been used in various applications, as it endows the robot with a better control of its body [START_REF] Natale | Linking action to perception in a humanoid robot: A developmental approach to grasping[END_REF], it facilitates interaction with humans and objects. The ability to distinguish between several individuals or sources of motion also gives the robot an opportunity to understand the dynamic of its environment and to interact with several persons [START_REF] Michel | Motion-based robotic selfrecognition[END_REF].

Among the variety of robot self-recognition methods, most algorithms are based on local approaches or prior knowledge. Some strategies impose restrictions on the change of motors configurations during self-recognition. Others exploit a predefined appearance of the robot's body or a predefined pattern of the robot's motion that simplifies the selfidentification [START_REF] Natale | Linking action to perception in a humanoid robot: A developmental approach to grasping[END_REF]. Since these techniques are not independent of the appearance of the robot's body and behavior, they cannot be easily generalized over new end-effectors and they cannot recognize robot's parts extended by grasped tools, that would be useful to increase the robot capabilities.

Following a developmental robotics approach, we take inspiration from the sensorimotor developmental stage in humans. Observations show, that at the beginning of life, infants learn about own body through simple repetitive movements, and then spend a lot of time by exploring surrounding objects through interaction [START_REF] Piaget | Play, dreams and imitation in childhood[END_REF]. These exploratory actions become effective, once toddlers learn to control and recognize their own body [START_REF] Asada | Cognitive developmental robotics: A survey[END_REF]. Our preliminary experiments investigating this issue with the iCub robot are presented in [START_REF] Ivaldi | Perception and human interaction for developmental learning of objects and affordances[END_REF].

In this paper we propose a self-identification, categorization, and learning method which is able to differentiate and to memorize appearances of objects, humans, and robot's parts. The algorithm builds upon our previous learning approach [START_REF] Lyubova | Developmental approach for interactive object discovery[END_REF] and introduces new elements integrating the robot's actions into the learning process and improving the final learning performance. Our algorithm does not require prior knowledge about the robot or objects appearances, robot's body model (kinematics or dynamics), nor the functional description of its joints, and is thus easily adaptable to different robots.

The paper is organized as follows: Section 2 gives a brief overview of the related work on robot self-discovery and its applications; the proposed approach is detailed in Section 3; the performed experiments and their evaluation are reported in Section 4; the last Section is devoted to conclusions and future work.

II. RELATED WORK Self-identification has been performed using several approaches. It can be achieved based on a known robot's appearance, or a predefined pattern of the robot's motion [START_REF] Natale | Linking action to perception in a humanoid robot: A developmental approach to grasping[END_REF]. The identification of a robot's hand can be also based on temporal contingency, for example, by learning the time delay between the initiation of the robot's movements and the emergence of its parts in the visual field, as proposed in [START_REF] Michel | Motion-based robotic selfrecognition[END_REF]. However, methods based on time delay are often limited to one active motion source at a time.

The identification of robot's parts without prior knowledge can be based on correlation between the proprioceptive and sensory information. This information can be analyzed during head-arm movements, as performed in [START_REF] Saegusa | Action learning based on developmental body perception[END_REF]. The authors analyse the speed of visual motion and of the robot's joints to recognize the robots arms and learn its appearance.

A system discovering robot's hands during natural interaction with a human is presented in [START_REF] Kemp | What can i control?: The development of visual categories for a robots body and the world that it influences[END_REF]. Mutual information is used to identify which salient region of the visual space can be influenced by the robot's actions: the algorithm analyzes the visual input and proprioceptive sensing. Since it is designed to detect humans and robot's parts, it focuses on regions that are close to the sensor and move fast.

Assuming knowledge of the robot's body, several studies exploit the robot's actions for object exploration. The decomposition of scene into objects by means of interactive actions is proposed in [START_REF] Van Hoof | Maximally informative interaction learning for scene exploration[END_REF]. In [START_REF] Katz | Interactive perception of articulated objects[END_REF], perception and interaction are integrated for autonomous acquisition of kinematic structures of rigid articulated objects. The interactive learning of objects features and object-specific grasping knowledge is performed in [START_REF] Kraft | Development of object and grasping knowledge by robot exploration[END_REF]. Robots actions are also used to improve object recognition in ambiguous situations. Having several similar objects, interaction can be used to turn one object into a representative perspective that allows to recognize it [START_REF] Browatzki | Active object recognition on a humanoid robot[END_REF].

In our approach, we do not focus on the selection of a particular action to act on objects, or use of actions for object segmentation; we rather attempt to learn objects appearances in between actions and during manipulations, while the objects are grasped. As a consequence, the discrimination between manipulated objects, the robot's and human parts is fundamental.

III. PROPOSED METHOD

Our approach detects proto-objects as salient regions of the visual space, incrementally encodes their appearance, and as-sociates them with physical entities. The learning algorithm is based on our previous work on object learning through observation [START_REF] Lyubova | Developmental approach for interactive object discovery[END_REF], but it has been improved with Bayesian filtering in order to enhance temporal coherency of object recognition and enhanced with a capability of categorization and interactive learning. Entities are classified into robot's hands, human hands, and manipulable objects. The pose of each object entity, its dimensions, and its localization in the robot's space are estimated in order to plan the robot's actions. Finally, the object learning is improved through manipulation using the outcome of categorization.

As input data, we use color and depth images from a RGB-D sensor (Kinect) and robot's motors states. The complete experimental setup will be described in section IV.

A. Segmentation of the visual space

The visual attention in our approach is based on motion; we therefore begin proto-object detection by estimating moving regions by image differencing. Among all moving regions, we ignore whose located far from the robot according to the constraints of the reachable area. In remaining regions, GFT-points are extracted and grouped into clusters of coherently moving points. Each cluster is considered a proto-object and tracked in time. The contours of protoobjects are refined based on the depth variation of the visual field. The processing steps are detailed in [START_REF] Lyubova | Developmental approach for interactive object discovery[END_REF] and summarized in Fig. 2.

B. Robot actions

Before interaction, we localize proto-objects in the operational space of the robot, estimate their orientations and dimensions. By retrieving the depth information from the RGB-D sensor and processing it as a point cloud, we compute each proto-object's 3D position relative to the sensor before transforming it to the operational space. The protoobject's axes orientations are obtained from eigenvectors and eigenvalues of the covariance matrix of the proto-object's points giving three orthogonal reference directions for which we compute the proto-object's dimensions.

Since this study is aimed at learning objects appearances, the robot should perform actions that help to explore different object perspectives. Thus, we use both simple actions, like reach, push, take, and more complex manipulations, T akeLif tF all and T akeObserve, that are aimed at revealing new object perspectives. Both manipulations are composed from a sequence of action primitives. T akeLif tF all includes reaching an object, taking it, lifting, and releasing that turns the object into a random perspective, when it falls on the table. T akeObserve consists of reaching an object, taking it, turning, approaching to the camera, and returning to the table; during this manipulation, the robot perceives several object perspectives and its visual details.

C. Object model learning

The proto-objects appearances are learned incrementally based on the algorithm presented in [START_REF] Lyubova | Developmental approach for interactive object discovery[END_REF]. Our system acquires all information iteratively by analyzing low-level image features and synthesizing them into higher-level representations. As low-level features, we extract SURF points [START_REF] Bay | Speeded-up robust features (surf)[END_REF] and colors of superpixels [START_REF] Micusik | Semantic segmentation of street scenes by superpixel co-occurrence and 3d geometry[END_REF] that correspond to nearly homogeneous image regions segmented with some regularity. Mid-features are constructed as pairs and triples of low-level features nearest in the visual space. All extracted features are quantized into vocabularies of visual words. Since an entity appearance can vary between its perspectives, we learn its model as a set of views E i = {v j }, each view being encoded by the occurrence frequencies of its mid-features v j = {m k }.

In [START_REF] Lyubova | Developmental approach for interactive object discovery[END_REF], views are recognized through a voting method based on TF-IDF (Term-Frequency -Inverse-Document Frequency [START_REF] Sivic | Video google: Text retrieval approach to object matching in videos[END_REF]) of mid-features and a maximum likelihood approach:

L(v j ) = m k ∈vj tf (m k )idf (m k ), (1)
where tf (m k ) is the frequency of the mid-feature m k , and idf (m k ) is the inverse view frequency for this mid-feature. Since several objects can have similar views, we introduce a Bayesian filter that improves temporal consistency of recognition between consecutive images therefore reducing potential confusion between objects. The probability of recognizing a view is estimated recursively based on its likelihood, its probability computed in the previous image, and its tracking:

p t (v j ) = ηL(v j ) l p(v j |v l )p t-1 (v l ), ( 2
)
where η is the normalization term; L(v j ) is the current likelihood of the view v j ; p t-1 (v l ) is the probability of the view v l computed in the previous image; p(v j |v l ) is the probability that the view v j appears, when the view v l was recognized in the previous images. This probability is fixed to 0.8 when v j = v l , and otherwise 0.2/N v with N v being the total number of views.

The recognized view is then associated with a physical entity. If the entity tracking from previous image was successful, the view is associated to the same entity. When tracking fails, the current entity is recognized through a maximum likelihood approach similar to the view recognition but based on the occurrence frequency of views among entities:

L(E i ) = tf (v j )idf (v j ), (3)
where tf (v j ) is the frequency of the view v j , and idf (v j ) is the inverse entity frequency for the view v j .

Since our experiments are based on object manipulation, it is important to recognize connected physical entities moving together, while the robot or the human interacts with an object. For this purpose, we use a double-check recognition. In the first stage, the most probable view is identified. In the second stage, features that don't belong to the most probable view (see Fig. 3) participate in the voting method again to identify a second possible view. Thus, each moving region of the visual space is recognized either as a single entity or two connected entities. Since objects are partly covered by hands during manipulations, the double-check recognition allows to prevent erroneous updates of objects models with hand features. The information about connected physical entities is also used by the categorization module described in the Section III.D and during interactive object learning presented in the Section III.E.

D. Categorization

The categorization procedure is aimed at identifying the nature of physical entities detected in the visual space, while the robot learns objects through interaction with a human partner. First, the parts of the robot's body are discriminated among all entities, and then, the rest of single entities are distinguished either as a human part or a manipulable object category. As a result, each entity is associated with one of the following categories (see Fig. 4): a robot c r , a human c h , an object c o , an object grasped by the robot c o+r , an object grasped by the human c o+h , or unknown c u category that will be identified later, when more statistics is gathered.

1) Robot self-identification: The robot's body identification is based on mutual information (MI) between visual data and proprioception. As proprioceptive information, we analyze the robot's arm and torso motors states. We acquire states of the following arm joints (see Fig. 4): shoulder (pitch, roll, and yaw), elbow, wrist (pronosupination, pitch, and yaw) and torso joints (pitch, roll, and yaw). Finger joints are not considered, since their movements do not produce a significant visual displacement of the hand.

The visual space is quantized regularly by applying a grid (12x10) producing 120 visual clusters. The position of each physical entity is quantized to the closest visual cluster. Each time a new image is acquired from the visual sensor, we acquire the robot's arms and torso joints values. The joints values are incrementally quantized into a vocabulary of arm-torso configurations, where each entry is encoded as a vector of joints values. During quantization, if the minimal L2 distance between the current vector of joints values and each vocabulary entry exceeds a threshold, a new configuration is stored in the vocabulary; otherwise, the current vector of joints values is recognized as the closest arm-torso configuration from the vocabulary. In our experiments, we obtain in average 37 arm-torso configurations.

As in [START_REF] Kemp | What can i control?: The development of visual categories for a robots body and the world that it influences[END_REF], MI is used to evaluate the occurrence dependency between the robot's arm-torso configuration A c and the physical entity localization L Ei :

M I(L Ei ; A c ) = H(L Ei ) -Hc(L Ei |A c ), (4)
where H(L Ei ) is the marginal entropy, and Hc(L Ei |A c ) is the conditional entropy computed in the following way:

H(L Ei ) = - l p(l Ei )log(p(l Ei )), (5)
Hc(L Ei |A c ) = - ac p(a c ) l E i p(l Ei |a c )log(p(l Ei |a c )), (6)
where p(l Ei ) is the probability of the entity localization l Ei , p(a c ) is the probability of the arm-torso configuration a c , and p(l Ei |a c ) is the probability of the entity localization l Ei given the arm-torso configuration a c .

Since we change the appearance of the robot's hands during experiments, M I(L Ei ; A c ) is estimated for each robot's arm and for each physical entity. Thereby, the robot category c r can be associated with several entities that correspond to different appearances of the hand (for example, with and without wearing gloves); while views of each entity describe the hand appearance in different postures (see Fig. 5). The threshold identifying the robot category is selected empirically by analyzing the MI distribution for robot's and non-robot's parts on a small labelled database. If MI is higher than th r = 40%, the physical entity is identified as a robot category c r ; otherwise, its category is identified according to the algorithm of the following section.

2) Discrimination of human and object categories: The discrimination between human parts and manipulable objects is based on statistics on entities motion: human parts often move by themselves while objects are static most of time, and they are rather displaced by the robot or the human.

Since our vision module is able to detect and to categorize connected entities moving together, we identify objects during manipulations based on the statistics of their simultaneous motion with entities categorized as robot's parts. During the experiment, we count the number of times each entity E i moves alone as a non-robot category, and the number of times the same entity moves connected to a robot's entity and estimate the associated occurrence frequencies:

• f s = N c E i =cr Nc E i
is the occurrence frequency of a nonrobot's entity moving alone,

• f c = Nc E i ,c E i2 =cr Nc E i ,c E i2
is the occurrence frequency of an entity moving together with a connected entity E i2 categorized as a robot's part. Since objects usually do not move alone, the frequency f s should be low and f c should be high for the object category. Therefore, a non-robot's entity is identified as:

• the object category c o , if f c > th o .c. and f s < th o .s.;
• the human category c h , otherwise. Gathering these statistics require the identification of the robot hand category c r , therefore all entities are temporarily associated with the unknown category c u before c r is identified. Once the robot's body is identified, all single entities are categorized as c o , c h , or c r . In the case of connected entities, the category of each individual entity is retrieved from the categorization statistics and the connected entity is categorized as an object grasped by the robot category c r+o or an object grasped by the human category c h+o .

E. Object model update during interaction

The outcome of the categorization module is used to improve object learning during manipulation. The interaction with an object starts when the robot detects an object entity in a reachable distance. In case of a successful grasp, the model of the grasped entity E g is updated during manipulation. This is a kind of self-supervision, where the object is supposed to be the same during manipulation.

The perceptual system continuously detects entities in the visual space and categorizes them. In the case of detecting connected entities with one entity identified as a robot category, the categories of both connected views are verified. We link each connected view with a set of physical entities {E i } that have this view in their models. The category c Ei of each entity is retrieved from the categorization statistics, and each connected view is identified as:

• a robot's view, if at least one linked entity is identified as the robot category (∃i, c Ei = c r ); • a non-robot's view, if none of linked entities is identified as the robot category (∀i, c Ei = c r ). If during manipulation, a proto-object is identified as a robot's view connected to a non-robot's view,the manipulated entity model is updated with the non-robot's view. If a protoobject identified as a robot's view contains a large amount of features that do not correspond to this entity, a new view is stored with these features. If this newly created view is identified again later, it will be added to the manipulated entity model. Therefore, interactive learning allows to update the object model with both newly created and recognized non-robot's views.

IV. EXPERIMENTS

The proposed approach is evaluated on an iCub robot interacting with a human partner, as demonstrated in Fig. 6a, and manipulating objects, as shown in Fig. 6b. Objects used in the experiments are shown in Fig. 7. We design experiments for two purposes: first, to evaluate the categorization algorithm and then to analyse the accuracy of objects learning through manipulation and to compare it with the results of learning through observation.

A. Camera calibration

In our experiments, the visual input is acquired from an RGB-D sensor mounted above the robot (see Fig. 6b). This sensor is chosen due the precision of depth data compared to stereo vision. Since in our scenario, the robot performs actions in its operational space, the visual sensor is calibrated with respect to the robot, like described in [START_REF] Ivaldi | Perception and human interaction for developmental learning of objects and affordances[END_REF]. In this procedure, a calibration pattern is placed on the table and the robot moves its hand to the origin of the pattern in order to acquire its position in the operational space H pat→rob . The OpenCV library is used to estimate the sensor position relative to the pattern H sen→pat , and the transformation matrix from the target to the robot's space is computed:

H sen→rob = H pat→rob × H sen→pat .

B. Evaluation of categorization

In this experiment, a human manipulates objects and produces simple hand movements in the visual field of the robot. The robot performs simple actions, like reach, take, push, and manipulations with and without objects, as described in the Section III.B. The self-identification method is evaluated based on the robot's hands positions estimated by the forward kinematics model.

During evaluation, the categorization module was able to identify the robot's hand within first 10 seconds of its motion in the visual field. The average self-recognition rate was about 98.2%. Our self-identification method is also evaluated with changing the robot's hand appearance by wearing colored gloves (see Fig. 5). The system has shown to be independent on the robot's hand appearance and to recognize 98.1% of the robot's hands in the blue gloves and 98.0% of the robot's hands in the pink gloves. The slightly lower self-recognition accuracy in the case of changing the hand appearance can be explained by a large sizes of the gloves that reduce visibility of hand motion.

The system's ability to identify an object category is evaluated in an interactive scenario, while the robot is asked to interact with entities detected at a reachable distance. As shown in Fig. 8, each object has been successfully identified during within 5-10 seconds of interaction with it. Human parts have been correctly identified in 89% of images.

C. Evaluation of object learning

We evaluate the accuracy of objects learning through interaction and compare it with the results of learning through observation. During observation, a human demonstrates objects to the robot (about 700 images per object). Then, during manipulation, the robot performs T akeLif tF all action (about 800 images per object) in order to improve its knowledge about objects appearances.

Since our experiments are based on interaction with objects, it is difficult to evaluate the system using existing image databases. Thus, we created a database of 50 images for each object shown from different perspectives. This database is processed after each experiment in order to estimate the object recognition rate based on the number of times an object is identified as its most frequently associated entity.

Learning through manipulation improved the recognition rate for several objects compared to the results of learning through observation (see Fig. 9). This improvement slightly depends on the robot's hand appearance; the best results have been achieved with the robot's hand appearance the most different from all objects appearances, i.e. without wearing gloves. Gloves produce a larger occlusion of object features, making it less visible and leading to less updates of object models and smaller learning improvement.

Using only observation, several objects whose appearance significantly varies between perspectives are associated with multiple physical entities. It occurs when the human partner takes an object out of the visual field while demonstrating different perspectives, making it impossible to track the object and therefore to associate all its views with a single entity. For these objects (O 2 , O 4 , O 5 , O 8 , O 9 ), learning during manipulation has been especially useful as several entities created during observation have been merged into a single entity during interactive learning, thus leading to better object recognition. Moreover, the system was able to memorize new views while manipulating objects O 1 , O 6 , O 8 , thus improving the informativeness of objects models.

V. CONCLUSION AND FUTURE WORK

The proposed developmental approach allows a robot to explore its close environment in purely unsupervised way, to identify its body and to categorize other visible physical entities as human parts or manipulable objects. Based on these categories, it is possible to learn objects through observation and to improve their visual models through manipulation.

Important aspects of our model are its capacity to extract new information about an object during and in between manipulations and its adaptability to the modification of the robot's appearance. The system works online and gathers all information in an incremental manner.

Future work will include the use of weak supervision by integrating the audio information in our system. We plan to take advantage of naming objects, like in infant directed speech, in order to learn objects names and to improve object recognition in more complex interactive scenarios.

1Fig. 1 .

 1 Fig. 1. The main modules of the proposed system.

Fig. 2 .

 2 Fig.2. The segmentation of the visual space into proto-objects p 0 , p 1 , p 2 . See[START_REF] Lyubova | Developmental approach for interactive object discovery[END_REF] for a complete description.

Fig. 3 .

 3 Fig. 3. Recognition of connected views: the mid-features (in this case, pairs of superpixels) found in the most probable view are shown by the green color, the mid-features found in the connected view are red, and the rest of extracted mid-features are blue.

Fig. 4 .

 4 Fig. 4. The categorization algorithm: single entities are categories as cr, c h or co based on mutual information between the visual data and proprioception and statistics on entities motion; connected entities are categorized as c o+r or c o+h based on the entities categorization statistics throughout the whole experiment

Fig. 5 .

 5 Fig. 5. The representation models of three entities that correspond to different appearances of the robot's hands.

Fig. 6 .

 6 Fig. 6. The context of the experiments: a) learning through observation; b) learning through manipulation.

Fig. 7 .

 7 Fig. 7. Objects used in the experiments.

Fig. 8 .

 8 Fig. 8. Identification of five objects based on their probability p(c E = co) of being an object category co; each object is marked as an unknown category cu, when it appears, and as co, when it is identified.

Fig. 9 .

 9 Fig. 9. Object recognition rate (with initial robot's hands appearances): the results after objects observation are shown by the blue color, and the improvement after manipulation is shown by the yellow color.

www.macsi.isir.upmc.fr

ACKNOWLEDGMENT

Our work is performed as a part of MACSi 1 (Modele pour l'apprentissage du comportement sensorimoteur d'iCub) project supported by the French ANR program (ANR 2010 BLAN 0216 01).


Preface: Special Section on Foundations of Coordination Languages and Software Architectures (Selected Papers from FOCLASA '10) This section contains extended versions of selected papers from the 9 th International Workshop on the Foundations of Coordination Languages and Software Architectures (FOCLASA'10).

Computer systems combining concurrent, distributed, mobile and heterogenous components are omnipresent and managing the complexity of their interaction raises challenging problems for system designers and integrators. Coordination languages and software architectures are recognised as fundamental approaches to tackle these problems, improving software productivity, enhancing maintainability, advocating modularity, promoting reusability, and leading to systems that more tractable and more amenable to analysis, validation and verification.

The objective of the FOCLASA workshops is to bring together researchers and practitioners of the aforementioned fields, in order to share and identify common problems, and to devise general solutions in the contexts of coordination languages and software architectures. FOCLASA'10 the was the 9 th edition in this series of workshops and was held in Paris (France) on the 4 th of September, 2010.

From the eight research paper that were presented at FOCLASA'10, an initial selection of papers was made by the Program Committee, and their authors were invited to submit an extended version to this special section. These extended papers went through a rigorous peer review process; the revised versions of two papers were finally accepted and are included in this special section. We believe that the papers presented here provide key insights into different aspects (quality of service, compositionality, fault-tolerance) of coordination and interaction in concurrent, mobile and distributed systems.

The first article in this special section, "A Compositional Model to Reason about end-to-end QoS in Stochastic Reo Connectors", by Young-Joo Moon et al., presents a compositional semantics for the channel-based coordination language Reo that enables the analysis of quality of service (QoS) properties of service compositions. The authors also propose Stochastic Reo Automata as an extension of Reo automata, in order to compositionally derive a QoS-aware semantics for Reo. A translation of Stochastic Reo Automata to Continuous-Time Markov Chains (CTMCs) is presented in order to allow the use of third-party CTMC verification tools to carry out an end-to-end performance analysis of service compositions. Finally, this article discusses to what extent Interactive Markov Chains (IMCs) can serve as an alternative semantic model for Stochastic Reo.

The second article, "Formal Development of Wireless Sensor-Actor Networks", by Maryam Kamali et al., presents a model of a distributed recovery algorithm in Event-B that addresses the Wireless Sensor and Actor Network (WSAN) partitioning problem caused by actor node failures. They prove, via refinement, that this distributed algorithm is correct and that it terminates in a finite number of steps. Proofs are carried out using the RODIN platform, an integrated development framework for Event-B. This article also proposes a generalisation of the formal development strategy, which can be reused in the context of a wider class of networks by using the notions of refinement patterns and pattern-driven formal development.

Many people have contributed to this special section, without whose effort this special section would not have been possible. Besides the authors of the papers, we would like to thank the members of the Program Committee of the workshop: Farhad Arbab (CWI, The Netherlands), Luis Barbosa (University of Minho, Portugal), Antonio Brogi (University of Pisa, Italy), Carlos Canal (University of Málaga, Spain), Vittorio Cortellessa (University of L'Aquila, Italy), Gregor Goessler (Inria, France), Ludovic Henrio (Inria, France), Jean-Marie Jacquet (University of Namur, Belgium), Alexander Knapp (University of Augsburg, Germany), Ronaldo Menezes (Florida Institute of Technology, USA), Ernesto Pimentel (University of Málaga, Spain), Pascal Poizat (University of Evry, France), Jaco van de Pol (University of Twente, The Netherlands), Antonio Ravara (Technical University of Lisbon, Portugal), Marjan Sirjani (Reykjavik University, Iceland), Carolyn Talcott (SRI International, USA), Emilio Tuosto (University of Leicester, UK), Mirko Viroli (University of Bologna, Italy), Danny Weyns (Katholieke Universiteit Leuven, Belgium), Erik de Vink (Eindhoven University of Technology, The Netherlands). Our thanks go also to the anonymous referees who kindly agreed to help us with the selection and reviewing of the papers in this special section and carried out an excellent job during this lengthy and laborious process.


François G Schmitt

Explicit predictability and dispersion scaling exponents in fully developed turbulence

Keywords: turbulence, intermittency, multifractal, scaling exponents PACS: 47.27.-i, 47.53.+n, 47.27.Eq

We apply a simple method to provide explicit expressions for different scaling exponents in intermittent fully developed turbulence, that before were only given through a Legendre transform. This includes predictability exponents for infinitesimal and non infinitesimal perturbations, Lagrangian velocity exponents, and dispersion exponents. We obtain also new results concerning inverse statistics corresponding to exit-time moments.

Introduction

One of the main feature of homogeneous and isotropic fully developed turbulence, corresponding to very large Reynolds numbers, is the fact that statistical properties at small scales are scaling and universal [1], and possess intermittent fluctuations [2]. This is classically characterized using structure functions scaling exponents, describing the probability density of velocity fluctuations at all scales belonging to the inertial range [START_REF] Frisch | Turbulence; the legacy of A. N. Kolmogorov[END_REF]. This concerns the Eulerian velocity and its statistics, but the characteristic features of fully developed turbulence is not only intermittency and multiple scales, it is also the loss of predictability, chaotic behaviour, dispersion and mixing properties, and a large number of degrees of freedom. In the last ten years, many of these properties have been revisited taking into account intermittency to consider how different scaling laws related to these issues are modified (for a review of several results in this framework see [START_REF] Boffetta | [END_REF]). The scaling exponents obtained have often been expressed as a Legendre transform of some function of velocity exponents, and numerical applications need a numerical Legendre transform involving the dimension function of velocity fluctuations. Here we present a simple method to simplify Legendre transform expressions and to replace them with parametric functions that provide explicitely the scaling exponents when the Eulerian velocity scaling exponent structure functions are known. In the following the general framework is presented in section 2 and applied to different applications from predictability to ǫ-entropy studies in section 3. Let us note that the results presented here correspond to the ideal situation of homogeneous and isotropic turbulence; for realistic flows, this property cannot be expected to hold perfectly since the influence of boundary conditions may introduce statistical inhomogeneity.

The multifractal framework and Legendre transform

We recall first here the multifractal framework for velocity intermittency in turbulence, classically characterized using statistical moments of velocity increments. We recall the Legendre transform introduced to relate scaling moment function to the codimension function of velocity singularities. The latter is subsequently systematically used to simplify the Legendre transform of some general expression of the codimension function.

Singularities, moment functions and Legendre transform

In fully developed turbulence, intermittency is classically characterized by ζ(q), the scaling exponent of spatial structure functions. Denoting ∆V ℓ = |V (x + ℓ) -V (x)| the increments of the velocity field at a spatial scale ℓ, their fluctuations are characterized, in the inertial range, using the scale invariant moment function ζ(q) (see [START_REF] Frisch | Turbulence; the legacy of A. N. Kolmogorov[END_REF]5] for reviews):

< ∆V q ℓ >∼ ℓ ζ(q) ( 1
)
where q is the order of moment. Kolmogorov's initial proposal [1], for a nonintermittent constant dissipation, leads to ζ(q) = q/3. For intermittent turbulence, ζ(q) is a cumulant generating function, and is nonlinear and concave; only the third moment has no intermittency correction:

ζ(3) = 1 (see below).
On the other hand, taking into account intermittency for the Eulerian velocity, one can write locally:

∆V ℓ ∼ ℓ h (2)
where h is a singularity of the velocity fluctuations. There is a continuous range of possible h values, and they are characterized through their codimension function c(h) defined as [START_REF] Frisch | Turbulence; the legacy of A. N. Kolmogorov[END_REF]5]:

p(∆V ℓ ) ∼ ℓ c(h) (3)
where p(∆V ℓ ) is the probability density of velocity fluctuations at scale ℓ.

Here the codimension is used for convenience, instead of the more classical dimension f (h) = d -c(h), where d is the dimension of the space (d = 1 or d = 3 in most studies), and p(∆V ℓ ) is the probability density of velocity fluctuations. Let us note that, while h can be both negative or positive (but it is most of the time positive), the codimension function c(h) is positive and decreasing. This continuous range of h values justified the use of the term "multifractal" for such process [START_REF] Parisi | Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics[END_REF].

The moments write:

< ∆V q ℓ > = ∆V q ℓ p(∆V ℓ ) (4) ∼ ℓ qh+c(h) p(h) ∼ ℓ ζ E (q) (5)
using a saddle point argument [START_REF] Parisi | Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics[END_REF], this gives the classical Legendre transform, between ζ E (q) and the codimension function:

ζ(q) = min h {qh + c(h)} (6)
This can also be written in the following way, emphasizing the one-to-one relation between orders of moment q and singularities h:

   qh + c(h) = ζ(q) q = -c ′ (h) (7)
The last relation can be replaced by h = ζ ′ (q). Equations ( 7) are used below.

Estimates of structure function scaling exponents up to moments of order 8

The scaling moment function ζ(q) is proportional to the cumulant generating function of the variable log |∆V ℓ |; it is nonlinear and concave. There are two "fixed points" for this curve, corresponding to a piori exact values:

ζ(0) = 0 and ζ(3) = 1.
The latter relation is a consequence of the Kolmogorov 4/5 law [START_REF] Kolmogorov | [END_REF] for the third order longitudinal structure function (without absolute values):

< ∆V 3 ℓ >= - 4 5 ǫℓ (8)
The other moments do not have well-accepted theoretical values; the analytical curves depend on the stochastic model chosen for the statistics of velocity fluctuations. In agreement with Kolmogorov's initial hypothesis concerning the universality of velocity fluctuations in the inertial range, experimental estimates are rather stable for small and medium orders of moments and do not depend on the Reynolds number: the exponents ζ E (q) have been estimated experimentally for more than twenty years, for different types of flows (see [8][9][START_REF] Schmitt | Nonlinear Proc. Geophys[END_REF][START_REF] Arnéodo | [END_REF][12][13]), and are now considered as rather stable and almost universal until moments of order 7 or 8 [START_REF] Schmitt | Nonlinear Proc. Geophys[END_REF][START_REF] Arnéodo | [END_REF]. Let us note that the moment of order 7 seems to be close to 2, for any reasons that did not receive up to now any theoretical founding.

Since these estimates are rather stable up to moments of order 8, we chose in the following for simplicity for ζ E (q) average values of the estimates published in Ref. [9,[START_REF] Arnéodo | [END_REF][12][13]: see Table 1. These are estimates obtained from many experiments with R λ going from 35 to 5 000. The experimental average which is used here is quite close to all these experimental estimates.

3 Applications to different predictability and dispersion turbulence functions

Predictability of small perturbations in the intermediate dissipation range of turbulence

Let us first consider the predictability property of small perturbations in turbulence. It is well-known that the viscous cutoff, corresponding to the scale for which the Reynolds number is of order 1, is different from the Kolmogorov scale η when intermittency is taken into account: more intense singularities have a smaller local cutoff scale and weak fluctuations have larger cutoff scales. This range of scales is the intermediate dissipation range, where there is a mixture of inertial and dissipation effects. Using Eq. ( 2) and the Reynolds number Re = V L/ν, this writes [14]:

η h = LRe -1/(1+h) (9)
where L is a large scale and η h is the cutoff scale associated to singularity h. This spatial scale is associated to a temporal cutoff scale τ h through the velocity v h = (η h ) h : τ h = η h /v h = (η h ) 1-h giving the singularity dependent cutoff time scale:

τ h = T Re h-1 h+1 ( 10
)
where T is the large time scale associated to L. This temporal cutoff scale is the smallest scale of the system: below these scales viscosity dominates.

In fully developed turbulence, the positive maximum Lyapunov exponent λ, characterizing the exponential growth rate of an infinitesimal disturbance, is proportional to the inverse of the smallest characteristic time of the system. Since there is a range of time scales associated to the continuous range of h values through Eq.( 10), the Lyapunov exponent may be given by [15,16]:

λ ∼ 1 τ h dp(h) ∼ Re 1-h h+1 Re -c(h) h+1 dh (11)
giving finally:

λ ∼ Re α (12)
with the exponent α given by a saddle-point argument since the Reynolds number is large [15,16]:

α = max h 1 -h -c(h) 1 + h (13)
This was solved numerically by Crisanti et al. [15] to provide α ≃ 0.46 through a numerical Legendre transform using experimental estimates of the function c(h). This result is slightly smaller than the one provided by Ruelle [17] for nonintermittent turbulence:

α = 1/2.
Let us go further and show here how to obtain this exponent, considering, for more generality, the moments of order q > 0 of the inverse times:

< 1 τ q h >∼ Re α(q) (14)
where, using as done above a saddle-point argument, the exponents α(q) are given by:

α(q) = max h q(1 -h) -c(h) 1 + h (15)
In general, the singularity h 0 for which this condition is met verifies G ′ (h 0 ) = 0 where

G(h) = q(1 -h) -c(h) 1 + h (16)
Estimating the derivative G ′ (h 0 ) = 0 and using Eq. ( 7) to introduce the moment q 0 associated to the singularity h 0 leads to:

ζ(q 0 ) = 2q -q 0 ( 17
)
This provides a unique value of q 0 and hence of h 0 , for a given value of q. This equation, together with another use of the Legendre transform, gives the function α(q) = G(h 0 ):

α(q) = G(h 0 ) = q(1 -h 0 ) -(ζ(q 0 ) -q 0 h 0 ) 1 + h 0 = q 0 -q
This gives finally the following parametric relation between α(q) and the moment function ζ(q):

   ζ(q 0 ) = 2q -q 0 α(q) = q 0 -q (18)
The nonintermittent curve obtained for ζ(q) = q/3 is α(q) = q/2. The explicit parametric relation (18) can be used to represent the curve α(q) for experimental estimates of ζ(q): this is shown in Fig. 1 using the average ζ(q) curve. One can see from Fig. 1 that up to moments of order 2.5 the linear non-intermittent curve is a very good approximation. The value of α(1) = α is smaller than 1/2 (close to 0.48) but the intermittent correction is very small. For q = 1 we have α = α(1) = q 0 -1, where q 0 is given by ζ(q 0 ) = 2 -q 0 . For q = 2 one obtains q 0 = 3 since ζ(3) = 1, and hence α(2) = 1 is a fixed point, non affected by intermittency, recovering a result already given in Ref. [15]. One can also obtain the following estimates:

α(3) ≃ 1.56, α(4) ≃ 2.2, α(4.5) ≃ 2.5 (coming from ζ(7) ≃ 2)
, and α(5) ≃ 2.85.

Predictability of noninfinitesimal perturbations in turbulence

The same approach can be used to express scaling exponents characterizing time scale statistics associated to noninfinitesimal perturbations. As proposed in Ref. [18] and developed in Ref. [START_REF] Boffetta | [END_REF], and using the notations of the latter reference, we consider a perturbation of size δ of the velocity field, for δ belonging to the inertial range (hence the term noninfinitesimal). The time scale associated to the eddy of typical velocity δ can also be considered as the decorrelation time associated to the perturbation. In this framework, the inverse time is 1/τ ∼ δ/ℓ and since using the singularity h, one has ℓ ∼ δ 1/h , we obtain locally the relation between the time scale and the perturbation δ:

1 τ ∼ δ h-1 h (19)
As before, this can be used to consider the moments of order q > 0 of the inverse times, but here we will consider their scaling property as a power-law function of the velocity perturbation δ. Using the same type of argument as above, the moments of the inverse time are expressed as an integral over all singularities h an we consider all functions as power-law functions of δ so that using a saddle-point argument, one has a scaling law:

< 1 τ q >∼ δ -β(q) ( 20
)
where the exponents β(q) are given by the following relation (using the fact that the probability density writes p(h) ∼ ℓ c(h) ∼ δ c(h)/h ):

-β(q) = max h q(h -1) + c(h) h (21)
where the negative sign has been introduced for convenience to have positive final numerical values for β(q) (see below). As done above for infinitesimal perturbations, this can be expressed explicitely introducing the singularity h 0 that maximises the expression into brackets. With the same procedure as above: involving a differenciation and the introduction of the moment of order q 0 associated to h 0 , we obtain the final result as a parametric relation between β(q) and ζ(q):

   ζ(q 0 ) = q β(q) = q 0 -q = ζ (-1) (q) -q (22)
The exponent β(q) is simply linked to the reciprocal of ζ, denoted ζ (-1) . The nonintermittent curve obtained for ζ(q) = q/3 is β(q) = 2q. Using Eq.( 22) we can see that ζ(3) = 1 gives β(1) = 2, which is a fixed point for β(q), non affected by intermittency corrections [18]. Figure 2 shows the resulting function compared to the straight line of equation 2q. It is visible that β(q) grows very fast. Indeed, the relation ζ( 7) ≃ 2 gives the value β(2) ≃ 5. We can see that for q < 1, β(q) < 2q and for q > 1, β(q) > 2q. The result β(1) = 2 was checked using shell-model simulations [START_REF] Boffetta | [END_REF].

Lagrangian velocity structure functions' in turbulence

As an analogy with Kolmogorov's dimensional analysis in the Eulerian framework, Landau [START_REF] Landau | Fluid Mechanics[END_REF] proposed in 1944 a 1/2 law for the temporal increments of the Lagrangian velocity ∆V τ = |V (t + τ ) -V (t)|. This was later generalized by Novikov, with a Lagrangian intermittency framework for the velocity [START_REF] Novikov | [END_REF]:

< ∆V q τ >∼ τ ζ L (q) (23)
As for the Eulerian case, for a constant dissipation one obtains the "mean field" expression, neglecting intermittency: ζ L (q) = q/2. In this framework, the third order moment for the Eulerian velocity is analogous to the second order moment for the Lagrangian velocity: in case of intermittency ζ L (q) is nonlinear and concave, and the non-intermittent function is valid only for

q = 2: ζ L (2) = 1.
Recently some authors have proposed an hypothesis helping to relate Eulerian and Lagrangian structure functions [21][22][23][24], following an earlier proposal by Borgas [25]. They consider the velocity advecting Lagrangian trajectories as a superposition of different velocity contributions coming from different eddies having different characteristic times. After a time τ the fastest eddies, of scale smaller than ℓ, are decorrelated so that at leading order, they assume [21][22][23]:

∆V τ ∼ ∆U ℓ (24)
writing ∆U ℓ ∼ ℓ/τ and using ∆U ℓ ∼ ℓ h , gives the time and space local correspondence τ ∼ ℓ 1-h . Introducing this relation inside the integral of Eq.( 7), one obtains for the moments of the Eulerian velocity increments:

< ∆U q ℓ >∼ τ qh+c(h) 1-h dp(h) (25)
Using a saddle point argument and using Eqs.( 24) and ( 23), this gives the Lagrangian structure function as a Legendre transform [21]:

ζ L (q) = min h qh + c(h) 1 -h (26)
The authors that explored this relation did not go further to express ζ L (q) in the general case [21][22][23].

In fact, as we showed elsewhere for a Lagrangian turbulence study [26], using an approach analogous to the ones above, one can obtain an explicit relation between Eulerian and Lagrangian structure functions' scaling exponents. We introduce the singularity h 0 that maximises the expression into brackets, and with the same procedure involving a differenciation and the introduction of the moment of order q 0 associated to h 0 we obtain the final result (see also [26]):

   ζ L (q) = ζ E (q 0 ) q = q 0 -ζ E (q 0 ) (27)
This relation provides ζ L (q) when the fonction ζ E (q) is known, the second lign giving the link between q and q 0 . We can check that for q 0 = 3, the second line gives q = 2 and the first one,

ζ L (2) = ζ E (3) = 1.
We can also consider the case q 0 = 7 and using ζ E (7) ≃ 2, the approximate result ζ L (5) ≃ 1.

Let us note that an analogous expression is provided in [25] for singularities and (with our notations) codimension functions c(h) in his Appendix A (Equation (A 4)). Borgas used for this other types of arguments, including a Reynolds number scaling expression for Eulerian and Lagragian statistics; however, he did not go to the moment framework to provide an equivalent expression for structure functions scaling exponents. Let us also note that some comparisons are performed in [21] between their shell model Lagrangian values and Eulerian shell model estimates, transformed through a numerical Legendre transform using the analytical expression of a fit of c(h). The same type of comparison is performed in [23] between their DNS estimates and a numerical Legendre transform of Eulerian values. The approach proposed here is more direct since we test an explicit parametric relation corresponding to different hypothesis.

The Euler-Lagrange relation given here is non-linear, so that one cannot expect to obtain the same type of statistical models in Eulerian and Lagrangian frameworks. For example, as explained by Chevillard [24] using Borgas' relation, in this framework, the statistics cannot be lognormal for both Eulerian and Lagrangian velocity fields. Indeed, for ζ E (q) = aq -bq 2 with a = (2 + µ)/6 and b = µ/18, one obtains ζ L (q) = (a -1 + (1 -a) 2 + 4qb -2bq)/2b. For more details see our paper [26] where different proposals for the Lagrangian structure functions scaling exponents are tested.

In Figure 3, the Lagrangian prediciton is compared to Lagrangian estimates published in recent papers [21,27,22]. These papers report different types of such estimates: Ref. [27] gives the values of experimental estimates for four flows going from R λ of 510 to 1 000, and Direct Numerical Simulations (DNS) estimates for R λ =75 and 140. Ref. [21] reports shell model estimates and Ref.

[23] DNS corresponding to R λ =280. In order not to have a too heavy figure, we plotted in Figure 3 four different types of Lagrangian values: an average value of the 4 experiments shown in Ref. [27], the DNS simulation corresponding to R λ =140 of the same publication, the shell model estimates of Ref. [21] and the DNS estimate of Ref. [23]. The average Eulerian curve is transformed and compared to these four Lagragian estimates. Several comments can be proposed from Figure 3. First, let us note that these experimental and numerical estimations for Eulerian and Lagrangian scaling exponents come from different types of flows with different Reynolds numbers. For Eulerian estimates, the exponents have been carefully compared and are now considered as rather stables up to moments of order 8, but for Lagrangian estimates, this work is still to do; measurements are only available for a few years, and the scaling ranges are still small, so that the range of variability for these exponents is expected to narrow in the future. Indeed, as for the Eulerian case, one can expect that for high Reynolds numbers these exponents are rather stable and universal, up to a given order of moment that may be smaller than the Eulerian case, for which this critical order of moments is about 7-8.

With these comments in mind, we can provide some preliminary conclusions from the observation of Figure 3: (i) up to moment of order 3, the prediction are close to all Lagrangian data; (ii) for larger moments, there is too much scatter in the estimates to conclude about the validity of the transformation given by Eq. (27). A possible explanation of such scatter is the fact that scaling can be strongly affected by trapping in vortex filaments [23]. In this case, after a filtering of high frequencies, the data may be closer to the multifractal prediction.

Relative turbulent dispersion

We next consider the influence of intermittency on relative dispersion: this deals with the statistics of particle pair dispersion, advected by an intermittent Eulerian velocity field. This topic is close to the Lagrangian case. let us consider R(t), the distance between a pair of particles advected by an intermittent turbulence. Following Ref. [START_REF] Novikov | [END_REF]28,29], one can introduce ∆V τ (R), the velocity difference along the Lagrangian trajectories. This can be written:

dR dt = ∆V R (28)
where ∆V R is the increment of the Lagrangian velocity is evaluated for a distance R associated to a time scale τ = R/∆V R . Then using the same hypothesis as for the Lagrangian case (Eqs. ( 23) and ( 24)), we obtain the time evolution of the moments of the pair distance:

< R q >∼< (τ ∆V τ ) q >∼ τ H(q) ( 29
)
where H(q) is the scaling exponent characterizing the intermittent pair dispersion. Using Eq.( 26) the exponents H(q) can be given as a Legendre transform of the Eulerian velocity, as provided by Boffetta et al. [28,29]:

H(q) = min h q + c(h) 1 -h (30)
In fact Eq.( 29) shows also directly that the dispersion exponent is simply related to the Lagrangian velocity exponent:

H(q) = q + ζ L (q) (31)
Equations ( 27) and ( 31) then give:

   H(q) = q 0 q = q 0 -ζ E (q 0 ) (32)
This can also be written H(q -ζ(q)) = q. This relation could also have been obtained from Eq.( 30) using the same approach as above. With q 0 = 3 we obtain q = 2 and H(2) = 3, corresponding to Richardson's law < R 2 >∼ τ 3 [30], which is thus a fixed point for which there is no intermittency correction, as already noticed [START_REF] Novikov | [END_REF]28,29]. In Fig. 4, we represent the transform (32) compared to the non-intermittent line of equation 3q/2, compared also to the shell model estimates published in Ref. [29]. This shows that the nonlinearity of H(q) does not seem very strong. Furthermore, H is concave, and the values give by Eq. ( 32) are quite close to the ones reported in Ref. [29].

Exit-time moments and ǫ-entropy

We finally consider exit-time moments [31], also called inverse structure functions [32]. We consider a time series of a turbulent quantity X(t) having intermittent fluctuations characterized by singularities h such as ∆X τ ∼ τ h with codimension c(h): p(h) ∼ τ c(h) . The moments are assumed to scale as < ∆X q τ >∼ τ ζ(q) with Eq.( 9). One then studies the dynamics of such signal by considering exit times (also called distance structure functions or inverse turbulence functions): considering a threshold value δ = ∆X τ , let us denote τ (δ) the first time needed to obtain an increment δ = ∆X τ . A new time series (τ i ) can be obtained this way, whose statistics will scale as the threshold value δ:

< τ q >∼ δ χ(q) (33

)
where χ(q) can be obtained, as before, as a Legendre transform involving c(h).

Writing τ ∼ δ 1/h and inserting this into the integral of the moment estimation gives [32,31]:

χ(q) = min h q + c(h) h ( 34
)
Here as before, we estimate this function explicitely. A derivative of G(h) = (q + c(h))/h leads to χ(q) = G(h 0 ) with ζ(q 0 ) = -q. We then obtain

   χ(q) = -q 0 ζ(q 0 ) = -q (35)
The result then writes simply:

χ(-ζ(q 0 )) = -q 0 ( 36
)
which is exactly a result given in Ref. [33] based on an exact result for a special case (a multifractal Cantor set), and experimentally verified using shell model turbulence simulations, but with no formal proof of the formulae in the general case. A similar expression obtained in the very different context of Laplacian random walk is given in Ref. [34].

We must note here, as already noticed in [33], that Eq.( 36) involves for either the original series or for the return time series, negative moments. Let us consider first positive moments of return times: q > 0 in Eq. (33). We need here the hypothesis of the existence of negative moments of the velocity structure functions, but it may not be the case, since this needs in practice extremely precise measurements. It is argued in [35] that negative moments for q < -1 do not exist since the pdf of velocity increments does not vanish at 0. The derivative writes G ′ (h 0 ) = -(q + ζ(q 0 ))/h 2 0 . For q > 0, if negative moments q 0 such that ζ(q 0 ) = -q < 0 are not accessible in the experimental time series, then the derivative stays strictly negative, G(h) does not reach a relative minimum, and the minimal value selected by Eq.( 34) will be obtained for the maximal value of h, which is a "minimal" singularity of the velocity fluctuations, and hence denoted here h min : χ(q) = G(h min ). Since this value of h is the same for all q's, the result is a linear behaviour:

χ(q) = q + c(h min ) h min (37)
The singularity h min is associated to small moments and the smallest fluctuations detectable in experimental time series; it is not clear whether this maximum singularity reaches a fixed value for Eulerian turbulence, and what is this value. This maximal singularity also depends on the precision of the experimental measurements which are analyzed. To check this, we consider numerical and experimental analysis of such inverse statistics for q > 0. We plot in Fig. 5 the data for χ(q) with q > 0 reported in Refs. [32,36]. The latter are experimental estimates, while the former are obtained from shell model calculations. Both are rather linear (the published experimental estimates contain only the first three moments), and seem to confirm a linear behaviour such as Eq. (37). Recently inverse statistics have been tested using wind-tunnel experimental data, confirming a linear behaviour for positive (relative) exponents [35].

Let us now focus on negative moments of the time statistics: q > 0 in Eq.( 33). This corresponds to positive moments of inverse times. We have ζ(q 0 ) = -q and χ(q) = G(h 0 ) = -q 0 . The result will be simply written introducing p = -q and q 0 (p) = -χ(q):

< 1 τ p >∼ δ -q 0 (p) (38)
with:

ζ(q 0 ) = p (39)
The exponent obtained with the inverse statistics is in fact the inverse (in the sense of reciprocal) of the structure function:

q 0 (p) = ζ (-1) (p) (40)
For Eulerian velocity (with Taylor's hypothesis to interprete time statistics as being close to spatial statistics), a time series measured at a fixed location will be characterized by ζ = ζ E . Thus q = 1 selects q 0 = 3, non affected by intermittency corrections. The non-intermittent curve is q 0 = 3q, and taking into account intermittency we have also the approximate value q 0 (2) ≃ 7. The corresponding curves for Eulerian turbulence and for passive scalar turbulence (data from Ref. [37] corresponding to an average of several experimental estimates) are shown in Fig. 6: they are increasing fast, especially for the passive scalar case. Let us note also that the experimental results published in [35] do not confirm Eq.( 40), except for p = 1.

Exit-time statistics have also been used in Ref. [38,39,[START_REF] Boffetta | [END_REF] to characterize the ǫ-entropy of turbulent signals. The concept of ǫ-entropy has been introduced to quantify the predictability and degree of complexity of continuously varying signals such as turbulent signals. The continuous signal is transformed into a symbolic time series using a grid in the phase space, of resolution ǫ.

The classical Shannon entropy h of the resulting symbolic time series is then studied, as a function of the grid size ǫ (see [40,41]). Abel et al. [38,39] have proposed to study the ǫ-entropy of turbulent signals h(ǫ) using the exit-times t(ǫ), the time for the signal to undergo a fluctuation of size ǫ. They obtained the following expresion

h(ǫ) = h Ω (ǫ) < 1 τ > (41)
where h Ω (ǫ) is the exit-time ǫ-entropy, being bounded and having only a logarithmic variation with ǫ. The leading ǫ behaviour of h(ǫ) is then given by < 1 τ >, where the average is done here by considering a succession of exit time events, and is thus related to Eq. ( 38). This gives using Eq.( 38) and ǫ = δ:

h(ǫ) ∼ ǫ -β (42)
with β = q 0 (1) as given by Ref. [38,39]. But we can write also using Eq.( 40):

β = ζ (-1) (1) (43)
Hence for a turbulent time series recorded at a fixed location, but using Taylor's hypothesis, corresponding to Eulerian turbulence, β = 3 [38,39,[START_REF] Boffetta | [END_REF]. On the other hand, for Eulerian temperature turbulence (still with Taylor's hypothesis), one will have a larger value since ζ θ (3) ≃ 0.8; using the values published in Ref. [37], we have ζ θ (4) ≃ 1 and hence we obtain for passive scalar Eulerian turbulence β θ ≃ 4. For times series of Lagrangian velocity and passive scalar turbulence, both characterized by ζ(2) = 1, we have β = 2. This recovers the result obtained in Ref. [40], which have been obtained using a dimensional argument, thus corresponding implicitely to Lagrangian velocity. Let us also note that Eq.( 43) recovers the results of Kolmogorov [42]: for Brownian processes with scaling power spectra of the form E(ω) ∼ ω -(1+2H) , he gave β = 1/H. Equation (43) gives the same result, since for such process, ζ(q) = qH, hence

β = ζ (-1) (1) = 1/H.

Conclusion

We have introduced a simple procedure helping to express explicitely scaling exponents expressed as Legendre transforms in multifractal turbulence. We have applied this idea to several problems involving intermittent corrections to predictability and dispersion studies. Even if the basic idea is simple, this produced several interesting new results: several functions are obtained as parametric transform of the Eulerian velocity structure function. We have confirmed and generalized some previously known results and provided some predictions.

More precisely, in this paper we have expressed the scaling moment function of infinitesimal perturbations, that characterize the inverse times statistics in the intermediate dissipation range, as function of the Reynolds number. For finite size perturbations, we have also provided the scaling moment function β(q) of inverse times, as function of the size δ of perturbations, showing that this exponent is very simply related to the reciprocal of the velocity structure functions exponent ζ(q). We veryfied the known result β(1) = 2 and obtain approximate value β(2) ≃ 5. We have also considered Lagrangian velocity structure functions ζ L (q) and showed how to relate them explicitely to the Eulerian curve ζ E (q). We compared the predicted Lagrangian curve to recent experimental values. We noticed the approximate value ζ L (5) ≃ 1.

We also considered the scaling exponent H(q) for pair dispersion which is very simply related to the Lagrangian scaling exponents. The obtained curve was compared to recent numerical estimates. We finally considered exit-time moments, and confirmed in the general case a result obtained recently on experimental grounds and analytically for a special case. We showed that, for exit-time positive moments, the scale invariant function χ(q) may often be linear, due to finite precision of experimental measurements. This is in agreement with several published experimental results. We then proposed to focus on inverse statistics considering negative moments, hence positive moments of inverse times. In this framework we showed that the resulting exponent q 0 (q) is the reciprocal of ζ(q). Using some recent results linking exist time statistics to ǫ-entropy expression, we obtain a new general result for the ǫ-entropy exponent of multi-affine signals β = ζ (-1) (1) confirming β = 3 for Eulerian turbulence, and giving as prediction β = 2 for Lagrangian turbulence (velocity and passive scalar), β ≃ 2 for financial time series, and the approximate value β ≃ 4 for passive scalar Eulerian turbulence.

Table 1 Some recent experimental estimations for ζ E (q) and the average value used here.

Ref. [9]: Re from 6,000 to 47,000; Ref. [START_REF] Arnéodo | [END_REF]: various experiments, R λ going from 35 to 5,000; Ref. [12]: turbulent wake, R λ =230; Ref. [13]: various experiments, R λ going from 340 to 800. [27]; DNS from [27], with R λ =140; shell model results from [21]; and DNS from [23]. The agreement is good for low orders of moment; for larger moments experimental and numerical estimates of scaling exponents have a larger scatter, and it is difficult to conclude concerning the validity of Eq. ( 27). . This is also compared to the shell-model estimates published in [29]. The agreement is quite good. The curve H is only slightly nonlinear and concave. Fig. 5. Representation of data for χ(q) for q > 0 reported in Refs. [32,36]. These correspond to numerical estimates from a shell model [32] and experimental estimates with R λ = 582 for velocity (longitudinal structure functions) and temperature turbulence [36]. These values are compatible with a linear behaviour (linear fits are displayed as dotted lines), as predicted by Eq.( 37), although experimental estimates are provided here for only three values.

Fig. 1 .Fig. 2 .

 12 Fig. 1. The curve α(q) (open dots: experimental values; continuous line: lognormal fit) compared to the non-intermittent curve of equation q/2 (dotted line). It is visible that α(1) = α < 1/2, but the intermittency correction is very small. The value α(2) = 1 is a fixed point not affected by intermittency. One obtains numerically α(3) ≃ 1.56 and α(4) ≃ 2.2.

Fig. 3 .

 3 Fig.3. The curve ζ L (q) (open dots: experimental values; continuous line: lognormal fit) compared to the non-intermittent curve of equation q/2 (dotted line), and to four Lagrangian recent estimates: average of 4 experiments published in[27]; DNS from[27], with R λ =140; shell model results from[21]; and DNS from[23]. The agreement is good for low orders of moment; for larger moments experimental and numerical estimates of scaling exponents have a larger scatter, and it is difficult to conclude concerning the validity of Eq. (27).

Fig. 4 .

 4 Fig.4. The curve H(q) (open dots: experimental values; continuous line: lognormal fit) compared to the non-intermittent curve of equation 3q/2 (dotted line). This is also compared to the shell-model estimates published in[29]. The agreement is quite good. The curve H is only slightly nonlinear and concave.

Fig. 6 .

 6 Fig. 6. The curve q 0 (p) = ζ (-1) (p) representing the scaling exponent function for the moments of order p > 0 of the inverse exit times, for the velocity (open dots: experimental values; continuous line: lognormal fit) compared to the non-intermittent curve of equation 3q (dotted line); also shown: the same curve for passive scalar turbulence (continuous line and closed dots, data from Ref. [37]). The passive scalar curve is very close to the velocity one for weak moments; it increases very quickly. We also represent the ǫ-entropy exponent β = ζ (-1) (1), to emphasize the clear different values for velocity and passive scalar Eulerian turbulence

1.78 1.75 1.79 1.782 1.78

2.01 1.97 2.01 2.007 2.00

2.22 2.10 2.22 2.217 2.19

Acknowledgements

Useful suggestions by the referees are acknowledged.