Alena Soloshenko
email: alena.soloshenko@etu.unistra.fr

Language, Cognition and Culture in Eco-Narratives

Keywords: ecolinguistics, cognitive semantics, conceptual metaphor, discourse, narrative

This paper represents the on-going project in ecolinguistics titled "ECOWord." The main objective of the project is to bring about suitable answers to the questions of human-nature relationships reflected in eco-related texts.

Introduction

The emergence of various ecological approaches in linguistics that developed since publication of Haugen's "The Ecology of Language" [START_REF] Haugen | The Ecology of Language[END_REF] generated several significant ways to study this interdisciplinary field. Because the language of ecology and specifically "the environment of language" open an infinite world of possibilities [START_REF] Steffensen | Ecolinguistics: the state of the art and future horizons[END_REF][START_REF] Cameron | Metaphor analysis: research practice in applied linguistics, social sciences and the humanities[END_REF], it is important to define how we approach language-environment constructs primarily from the linguistic point of view. Such an approach can be described nowadays more as juxtaposed and less as opposing. The on-going dynamics demonstrate that language can be both a tool in analysis of eco-related issues and be the object of ecological research. On the one hand, language conceptualizes as a "living organism" that influences the way we perceive the world and interact with it. On the other hand, language is shaped by the environment, in which it is functioning.

Eco-narrative is understood as a form of environmental writing, a text that passes ecological tenets through rhetorical, linguistic and cognitive strategies.

Discourses on literature and the environment, the dynamics of environmental perception and the interaction between humans and the world around them as a solution to the many problems of such relationships has been the main focus of ecolinguistics and ecocriticism. From this point of view, the current ecological crisis created environmental thinking and it is important to be fully aware of what kind of knowledge we have on this issue (do we need more), what power generates critical thinking, what are the desires and intentions of humans.

The ecological frameworks in linguistics use different terms for researching the relationships between the language of humans when describing the environment, such as "language ecology", "the ecology of language" or "ecolinguistics" but it is generally accepted that the worldview on the interaction between humans and the rest of life on earth has a dual nature: anthropocentric (nature as machine) and biocentric (nature as mother) [START_REF] Verhagen | Worldviews and metaphors in the human-nature relationship: an ecolinguistic exploration through the ages[END_REF][START_REF] Bruner | Acts of Meaning: Four Lectures on Mind and Culture (The Jerusalem-Harvard Lectures)[END_REF][START_REF] Cameron | Metaphor analysis: research practice in applied linguistics, social sciences and the humanities[END_REF][START_REF] Eco | Six Walks in the Fictional Woods[END_REF].

The biocentric worldview is a "set of beliefs, assumptions, biases or ideologies that place the biosphere at the centre of a person's life, thought and feeling" [START_REF] Verhagen | Worldviews and metaphors in the human-nature relationship: an ecolinguistic exploration through the ages[END_REF][START_REF] Hutchins | Cognitive ecology[END_REF]. The role of humans in the anthropocentric worldview as dominant over nature and seeing nature as no more than "an instrument for human ends" [START_REF] Verhagen | Worldviews and metaphors in the human-nature relationship: an ecolinguistic exploration through the ages[END_REF][START_REF] Eco | Six Walks in the Fictional Woods[END_REF] evokes the series of questions such as "What is the impact of humans on nature sustainability of the environment?" and "What can we do about it?" Discussions related to cognitive assessment of the global environment and the impact of humans on nature and vice versa have brought an interesting insight to the topic of language and cognition. In the words of Hutchins (2010), it is now apparent that "cognitive science is shifting from viewing cognition as a logical process to seeing it as a biological phenomenon" [START_REF] Hutchins | Cognitive ecology[END_REF]705]. Although the ways and outcomes of the human-environment interaction are one of the most important topics in cognitive ecology and ecocriticism, understanding of the issue of language and psychology in this sense is lacking in depth. Further research in this application should meet this challenge. Summarising all the above, it seems obvious that the key solutions on the topic should be found in an interdisciplinary approach. Only linguistic or strictly cultural approaches are not sufficient to research such complex constructs of human-nature relationships. Thus, I propose to focus on three main areas, each highlighting a particular sector of an interdisciplinary framework involving linguistics, cognitive psychology and cultural studies: (i) biosphere as expressed in eco-narratives of different cultures; (ii) metaphorical density (i.e., a number of coded conceptual metaphors per document) and diversity in conceptualization of ecology related issues in environmental writing; and (iii) psychological evaluation of eco-narratives in society in different cultures.

To attain these goals, the following questions should be prioritized: (i) How does the language system transport ecological ideas in eco-literature? (ii) How does language shape the environmental cognition of people in different cultures? and (iii) How images (e.g., visual perception of travel experience imagined or experienced) and words (e.g., metaphors, emphatic descriptive constructions) embody thoughts on human-nature interaction?

Both anthropocentric and biocentric worldviews are significant in this kind of analysis.

However, to fully reveal the issues behind a "partnership model" [START_REF] Verhagen | Worldviews and metaphors in the human-nature relationship: an ecolinguistic exploration through the ages[END_REF][START_REF] Hutchins | Cognitive ecology[END_REF] between humans and nature in the age of global environmental crises, the anthropocentric view is considered as the more important with its various gradations.

Methodological Approach and Project Management

According to [START_REF] Howarth | The Ecocriticism Reader: Landmarks in Literary Ecology[END_REF], nature is known though images and words [START_REF] Howarth | The Ecocriticism Reader: Landmarks in Literary Ecology[END_REF] and thus validity is found through data (i.e., "in words") involving semiotic perspective of text (discourse analysis) as well as extra linguistic data, such as intentions or attitudes embedded in texts (textual attitude analysis) and thought construction tools as image-schema or conceptual metaphor (conceptual cross-domain mapping). Thus, the source material of such complex study should reflect its diversity. Consequently, the corpus of environmental writings is divided into three sections: (i) literature on nature, ecosystems;

(ii) exploratory and adventure literature; and (iii) literature on climate change, population and global sustainability.

Apart from eco-narratives covering topics on nature and environment, specific attention is paid to travel writings. Starting from travel literature of ancient Greece (Pausanias, 2nd century CE) or the Song Dynasty of medieval China, this genre continued to tell the stories of the unknown via maritime diaries of writers like Richard Hakluyt or well-known explorers such as Captain James Cook. Both fictional literature in the form of essays and poems and guides of travel writers in the forms of diaries served as sources of information about unknown lands, inevitably including notes on nature and the environment. In spite of quite an impressive historical background, the study of travel literature of various kinds as an independent study started to gain power only in the mid-1990s [START_REF] Mansfield | Researching Literary Tourism[END_REF]. Nowadays such literature is transformed into independent studies on environment and travelling involving, for example, nature literature and exploratory literature.

The methodological framework is two-fold: the grounded theory principles of qualitative data analysis [START_REF] Glaser | Theoretical Sensitivity: Advances In Methodology of Grounded Theory[END_REF] are employed in the cognitive linguistic part of the research using computer-assisted qualitative data analysis software (CAQDAS). A software-based methodology for studying concepts, conceptual metaphors in discourse [START_REF] Kimmel | Optimizing the analysis of metaphor in discourse[END_REF][22][23] was already applied successfully to trace discourse dynamics [START_REF] Cameron | Metaphor analysis: research practice in applied linguistics, social sciences and the humanities[END_REF]. The source material is analysed to disclose and interpret linguistic, cognitive and discursive functions of eco-related texts. It is possible to research all functions using CAQDAS because the structure and capacity of CAQDAS are designed for profound qualitative analyses like this project.

Psychological evaluation of eco-narratives in society of different cultures

Ecocriticism from the psychological perspective is a way of finding specifics of human psychology in the age of environmental crises. Unique folk psychology is created and expressed through narrative, when narrative construction of reality is strongly connected with social intelligence [START_REF] Bruner | Acts of Meaning: Four Lectures on Mind and Culture (The Jerusalem-Harvard Lectures)[END_REF]. Environmental issues are relevant for the stylistic categories of fiction and non-fiction narrative because even fictionalized accounts of imaginary or historical events is a part of human interaction with the world.

In the case of fiction narrative, ecological, social or other principles are "mapped" in the narrative and are informed by such to see how society might look under such principles.

More recently the emphasis has shifted towards psychological meaning construction [START_REF] László | Historical Tales and National Identity: An Introduction to Narrative Social Psychology[END_REF], which has a direct relation to the expression of the way people organize their relationships with the world. Narrative organization and psychological organization are current topics to disclose the relationships of a speaker's perspective and experience of text. Narrative composition here refers to psychological relationships (emotions, attitudes) between characters in the narrative. Originating from Peirce's concept of unlimited semiosis (1931), Eco [START_REF] Eco | Six Walks in the Fictional Woods[END_REF] developed an interesting insight on the role of a reader in the process of making meaning from texts, emphasising in his Theory of the Model Reader (1998) that a reader may actualize various meanings and contents to decode the narrative. For a reader to acquire awareness on various eco-related topics "encoded" in text, one needs to actualise the meaning-content of text. Here the notions developed by Eco -the empirical reader and the model reader -find their application to answer the question on the extent of general awareness and/or encyclopaedic knowledge on the subject.

Concluding remarks

What I expect to achieve is a profound analysis of eco-thematic texts in a variety of genres. The ecorelated information embedded in the written language is something that can be used for better understanding of the way people conceptualize ecological reality and store it for future retrieval. This seems especially useful for educational purposes. Furthermore, conceptual coding of eco-related

information of the eastern and western writers will show contrast between cultural backgrounds on the topic.


F Chapoton

Free pre-Lie algebras are free as Lie algebras

We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.

Introduction

A pre-Lie algebra is a vector space V endowed with a bilinear map :

V ⊗ V → V such that (x y) z -x (y z) = (x z) y -x (z y), (1)
for all x, y, z ∈ V . This kind of algebra has been used for a long time in various areas, see [START_REF] Burde | Left-symmetric algebras, or pre-Lie algebras in geometry and physics[END_REF] for a survey. In geometry, there is a pre-Lie product on the space of vector fields of any variety endowed with an affine structure. In algebra, this kind of product appears for instance in relation with deformation theory, operads and vertex algebras.

In a previous article [START_REF] Chapoton | Pre-Lie algebras and the rooted trees operad[END_REF], the free pre-Lie algebras have been completely described in terms of rooted trees. The language of operads was a convenient setting for this.

In any pre-Lie algebra, the bracket [x, y] = x y -y x satisfies the Jacobi identity and therefore defines a Lie algebra on the same vector space.

The main goal of this article is to prove that free pre-Lie algebras are free Lie algebras for the Lie bracket [ , ]. This result has been obtained before with different methods by Foissy [START_REF] Foissy | Finite-dimensional comodules over the Hopf algebra of rooted trees[END_REF]. We give here a new proof using a spectral sequence and obtain a natural filtration on the space of generators.

The article is written using the language of S-modules. We will briefly recall the main features of this theory. This is the natural category for operads and provides a clean way to deal with free algebras.

The main result is the fact that the S-module for pre-Lie algebras is isomorphic (as a Lie algebra in the category of S-modules) to the composition of the S-module for Lie algebras with a S-module X. This implies the statement above about free pre-Lie algebras. We also give a description of the S-module X and a relation between X and the anticyclic structure on the PreLie operad.

Thanks to Jérome Germoni and Muriel Livernet for their interest in this work and useful suggestions.

Notations and S-modules

Let K be a fixed ground field of characteristic zero.

We will work in the Abelian category of S-modules over K or sometimes with complexes of S-modules over K. Recall that a S-module P can either be seen as a functor P from the category of finite sets and bijections to the category of vector spaces over K or as a sequence (P (n)) n≥0 of K-modules over the symmetric groups S n . We will use freely one or the other of these equivalent definitions.

The category of S-modules is symmetric monoidal for the following tensor product:

(F ⊗ G)(I) = ⊔ I=J⊔K F (J) × G(K), (2)
where I is a finite set.

There is another monoidal structure on S-modules which is non-symmetric and defined by

(F • G)(I) = ⊔ ≃ F (I/≃) × J∈I/≃ G(J), (3)
where I is a finite set and ≃ runs over the set of equivalence relations on I. An operad Q is essentially a monoid in the monoidal category of S-modules with • as tensor product, i.e. the data of a map from Q • Q to Q which is associative.

We will consider S-modules endowed with various kinds of algebraic structures. One can define associative algebras, commutative algebras, exterior algebras, Lie algebras, Hopf algebras and pre-Lie algebras in the category of Smodules by using the symmetric monoidal product ⊗ and the usual diagrammatic definitions. This works in particular for algebras over any operad Q. This kind of object is sometimes called a twisted algebra or a left-module over the corresponding operad. We will simply call them algebras.

If Q is an operad and P is an S-module, a structure of Q-algebra on P is given by a map from Q • P to P .

If Q is an operad and P is an S-module, the S-module

Q • P is the free Q-algebra on P . The map from Q • Q • P to Q • P is deduced from the map from Q • Q to Q defining the operad Q.
For instance, let us consider the following S-modules. Let λ be a partition of an integer n. Then λ is associated with an irreducible representation of the symmetric group S n , that will be denoted S λ . This can be seen as a S-module concentrated in degree n. As a special case, S 1 is the trivial module over S 1 . Let S be the direct sum of all S-modules S n corresponding to trivial modules over the symmetric groups.

Let T be the S-module defined as the direct sum of all the regular representations T n over the symmetric groups.

Let Λ be the direct sum of all S-modules Λ n corresponding to alternating modules over the symmetric groups.

There are morphisms of S-modules S ⊗ S → S, T ⊗ T → T and Λ ⊗ Λ → Λ, which define associative algebras in the category of S-modules. These algebras are essentially versions of the polynomial algebra, the tensor algebra and the exterior algebra.

For example, if I is a finite set, T (I) is the vector space spanned by total orders on I. One can concatenate total orders on two sets I and J to get a total order on I ⊔ J. This defines the map from T ⊗ T to T .

Let P be an S-module. The generating series f P associated with P is defined by

f P = n≥0 dim(P ({1, . . . , n})) x n n! . (4)
There is a more refined object, which is a symmetric function Z P recording the action of the symmetric groups:

Z P = n≥0 χ(P ({1, . . . , n})), ( 5
)
where χ is the image of the character of the S n -module P ({1, . . . , n}) in the ring of symmetric functions. This symmetric function is a complete invariant, in the sense that two S-modules are isomorphic if and only if they share the same symmetric function.

For more information on S-modules and operads, see for instance [START_REF] Livernet | From left modules to algebras over an operad: application to combinatorial Hopf algebras[END_REF][START_REF]Operads: Proceedings of Renaissance Conferences[END_REF].

Relation with usual algebra

Each S-module P defines a functor from vector spaces to vector spaces as follows:

P (V ) = ⊕ n≥0 P (n) ⊗ Sn V ⊗n . (6)
The tensor product ⊗ of S-modules has the following property:

(F ⊗ G)(V ) ≃ F (V ) ⊗ G(V ), ( 7
)
where V is a vector space. The tensor product • of S-modules has the following property:

(F • G)(V ) ≃ F (G(V )), ( 8
)
where V is a vector space. If the S-module P is an operad and V is a vector space, then P (V ) is the free P -algebra on V .

Then, if P is a S-module with such an algebraic structure, i.e. a Q-algebra for some operad Q, the vector space P (V ) is endowed with the corresponding structure of Q-algebra in the category of vector spaces.

For instance, the classical algebra structures on S(V ), T (V ) and Λ(V ) (respectively the polynomial algebra, the tensor algebra and the exterior algebra) come from the morphisms of S-modules S ⊗ S → S, T ⊗ T → T and Λ ⊗ Λ → Λ that were introduced above.

The pre-Lie S-module W

A rooted tree on a set I is a connected and simply-connected graph with vertex set I together with a distinguished element of I called the root. Note that a rooted tree can be canonically decomposed into its roots and a set of rooted trees obtained when the root is removed. Let W be the S-module which maps a finite set I to the vector space spanned by the set of rooted trees on I.

It has been shown in [START_REF] Chapoton | Pre-Lie algebras and the rooted trees operad[END_REF] that W is the operad describing pre-Lie algebras and in particular that W (V ) is the free pre-Lie algebra on a vector space V . There is a natural morphism : W ⊗ W → W corresponding to the PreLie product. This means that W is a pre-Lie algebra in the category of S-modules.

Let us recall briefly the definition of the map .

Given a rooted tree S on a finite set I and a rooted tree T on a finite set J, one can define S T (which is a sum of rooted trees on the set I ⊔ J) as follows: consider the disjoint union of S and T , then add an edge between the root of T and a vertex of S and sum with respect to the chosen vertex of S. The root is taken to be that of S.

As there exists a morphism from the Lie operad to the PreLie operad given by

[x, y] = x y -y x, (9)
there is a morphism [ , ] : W ⊗ W → W which makes W into a Lie algebra in the category of S-modules.

The universal enveloping algebra of a Lie algebra is a well-defined associative algebra in the category of S-modules, which has most of the classical properties of the usual construction for vector spaces. Let U(W ) be the universal enveloping algebra of W .

In the sequel, we will call W -module a module over the Lie algebra W or equivalently a right module over U(W ). This notion is defined in the obvious way in the symmetric monoidal category of S-modules.

There are two distinct W -module structures on the S-module W . The first one is given by the adjoint action of the Lie algebra W on itself. It will be denoted by W ad . The other one is given by the PreLie product and will be denoted by W pL . The fact that the map is a right action results from the PreLie axiom (1).

Let us now recall results from [CL01, Th. 3.3].

There is an isomorphism ψ of W -modules between W pL and the free U(W )-module on the S-module S 1 . By the description of W in terms of rooted trees and the decomposition of a rooted tree into its root and its set of subtrees, the module W pL is isomorphic as an S-module to S 1 ⊗ S • W . The S-module S • W is spanned by forests of rooted trees. The isomorphism ψ can be written as Id ⊗φ between S 1 ⊗ S • W and S 1 ⊗ U(W ). This defines an isomorphism φ of W -modules from S • W to U(W ), where the action ⊳ of W on S • W is deduced from the product and is given by

(t 1 t 2 . . . t k ) ⊳ t = t 1 t 2 . . . t k t + k i=1 t 1 t 2 . . . t i-1 (t i t)t i+1 . . . t k , (10)
where the t's stand for some rooted trees. We will from now on identify by the mean of φ the W -module U(W ) with the W -module S • W with this action ⊳. One can see from the explicit shape of the action ⊳ that the W -module U(W ) has a decreasing filtration by the number of connected components of the forest.

The associated graded module is given by the action

(t 1 t 2 . . . t k ) t = k i=1 t 1 t 2 . . . t i-1 (t i t)t i+1 . . . t k , (11)
where we have slightly abused notation by using the symbol for the action. This is easily seen to be the natural W -module structure on S • W pL obtained by extending the W -module W pL by derivation. This is also the symmetric algebra on the W -module W pL .

One can see that there is only one other term in the filtered action (10), which is of degree 1 with respect to the graduation by the number of connected components and is just a product:

t 1 t 2 . . . t k ⊗ t → t 1 t 2 . . . t k t. ( 12
)
3 Two spectral sequences

Let us consider the usual reduced complex computing the homology of the Lie algebra W with coefficients in the W -module U(W ). This is the tensor product of the exterior algebra on W with the module

U(W ) ≃ S • W . As an S-module, this complex is (S • W ) ⊗ (Λ ⊗ W ).
The differential ∂ is the usual Chevalley-Eilenberg map, which uses both the bracket or the action as a contraction:

∂(x 1 x 2 . . . x k ⊗y 1 ∧y 2 ∧• • •∧y ℓ ) = ℓ j=1 ±(x 1 x 2 . . . x k )⊳y j ⊗y 1 ∧• • •∧ y j ∧• • •∧y ℓ + 1≤i<j≤ℓ ±x 1 x 2 . . . x k ⊗ [y i , y j ] ∧ y 1 ∧ • • • ∧ y i ∧ • • • ∧ y j ∧ • • • ∧ y ℓ , (13)
where the signs are given by the Koszul sign rule.

But U(W ) is a free W -module by definition, hence the homology is concentrated in homological degree 0 and is given by the S-module S 0 .

Let us now use the filtration on U(W ) to define two spectral sequences computing the same homology. In fact, we will first define a bicomplex and then consider its two associated spectral sequences.

We have to introduce a triple grading on the complex (S • W ) ⊗ (Λ ⊗ W ). Let us denote by n the internal degree of the S-module (seen as a collection of modules over S n ), by p the degree with respect to the graduation of the symmetric algebra and by q the homological degree with respect to the graduation in the exterior algebra. As W has no component with n = 0, one has p ≥ 0, q ≥ 0 et p + q ≤ n.

Let r be n-p-q. We will use the triple grading by (n, p, r). The differential ∂ is of degree 0 with respect to the first grading. Hence one can consider separately each part of fixed first degree n.

The differential ∂ on (S •W )⊗(Λ⊗W ) decomposes into two pieces according to the decomposition of the action ⊳ into the action coming from (11) plus another term coming from (12).

The first part is defined as follows:

∂ pL (x 1 x 2 . . . x k ⊗y 1 ∧y 2 ∧• • •∧y ℓ ) = ℓ j=1 ±(x 1 x 2 . . . x k ) y j ⊗y 1 ∧• • •∧ y j ∧• • •∧y ℓ + 1≤i<j≤ℓ ±x 1 x 2 . . . x k ⊗ [y i , y j ] ∧ y 1 ∧ • • • ∧ y i ∧ • • • ∧ y j ∧ • • • ∧ y ℓ . ( 14
)
One can recognize the differential ∂ pL of degree (0, 0, 1) computing the homology of the Lie algebra W with coefficients in the graded W -module (S • W pL , ).

The remaining terms of the differential are

∂ K (x 1 x 2 . . . x k ⊗y 1 ∧y 2 ∧• • •∧y ℓ ) = ℓ j=1 ±x 1 x 2 . . . x k y j ⊗y 1 ∧• • •∧ y j ∧• • •∧y ℓ . ( 15
)
The map ∂ K has degree (0, 1, 0) and does no longer use the bracket of the Lie algebra W . This is nothing but the differential in the Koszul complex relating the exterior algebra Λ• W on W and the symmetric algebra

S • W on W [Pri70].
The two differentials ∂ pL and ∂ K are of degree (0, 0, 1) and (0, 1, 0) and their sum is also a differential. Hence they define a bicomplex and one can consider the two spectral sequences associated with this bicomplex.

Proposition 3.1 The spectral sequence beginning with ∂ K degenerates at first step.

Proof. As it is known that the exterior and symmetric algebras are Koszul dual of each other and Koszul, the homology of ∂ K is concentrated in homological degree 0 and is given by S 0 . Before studying the other spectral sequence, one needs the two following results. Let us first recall a classical lemma: Lemma 3.2 Let A be a Hopf algebra and let N be a right A-module. Then N ⊗ A is isomorphic as a right A-module to the free right A-module generated by N .

Proof. The argument uses the antipode of A to define an isomorphism, see for instance [Jan03, §3.6 and §3.7].

We will also need the following property of some W -modules. Proposition 3.3 Let λ be a non-empty partition. Then S λ • W pL is a projective U(W )-module.

Proof. Recall that W pL is isomorphic to the free U(W )-module S 1 ⊗ U(W ).

Hence, for each integer k ≥ 1, the W -module

T k • W pL is isomorphic to the module T k ⊗ (T k • U(W ))
, where W acts on the right factor only. Here, we have used the property of the S-module T that T

• (A ⊗ B) ≃ (T • A) ⊗ (T • B).
As U(W ) is a Hopf algebra, one can apply Lemma 3.2, which implies that

T k • U(W ) is a free U(W )-module. It follows that T k • W pL is also a free U(W )- module.
Now, for each partition λ, S λ is usually defined as a direct factor of T |λ| , where |λ| is the size of λ.

So S λ • W is a direct factor of T |λ| • W . As T |λ| • W is a free U(W )-module, S λ • W is a projective U(W )-module.
Remark: it may be that all the U(W )-modules S λ • W are in fact free.

Let us now go back to the bicomplex. To illustrate the computation of the horizontal spectral sequence starting with ∂ pL , we will draw the component of fixed first degree n of the bicomplex in the first quadrant, with the p grading increasing from bottom to top and the r grading from left to right.

We have the following description of the homology with respect to ∂ pL .

The bottom line (p = 0) of the bicomplex is the complex S 0 ⊗ (Λ • W ) computing the homology of the Lie algebra W with coefficients in the trivial module. This is what we would like to compute.

The other lines (p > 0) of the bicomplex are the complexes (S p • W pL ) ⊗ (Λ • W ) computing the homology of the Lie algebra W with coefficients in the modules (S p • W pL ). As we know that these modules are projective by Prop. 3.3, the homology is concentrated in degree q = 0 i.e. r = n -p.

So the first step of the spectral sequence looks as follows: * 0 * 0 0 * 0 0 0 * 0 0 0 0 * * * * * * * ,

where a * means a possibly non-zero homology group.

As the spectral sequence converges to S 0 because the homology of the total complex is S 0 , one deduces from the shape above that the homology of the bottom line is concentrated in degree q = 1 i.e. r = n -1. So in fact, the first step looks like that:

* 0 * 0 0 * 0 0 0 * 0 0 0 0 * 0 0 0 0 * 0. ( 17
)
Let us denote by H(n, p, r) these homology groups. We will denote by X the S-module corresponding to the collection of S n -modules H(n, 0, n -1) for n ≥ 1. Note that one does not include the degree 0 component in this definition.

Proposition 3.4 Let n ≥ 1. The dimension of H(n, 0, n -1) is (n -1) n-1 . The dimension of H(n, p, n -p) is n p (p -1)(n -1) n-p-1 for 1 ≤ p ≤ n.
There is a filtration on the unique non-vanishing homology group H(n, 0, n -1) of the bottom line whose graded pieces are isomorphic to the homology groups H(n, p, n -p) of the other lines.

Proof. Using once again the fact that the spectral sequence converges to S 0 , one can see that the component of first degree n of the horizontal spectral sequence cannot degenerate before the nth step and that the successive pages of the spectral sequence provide the expected filtration on H(n, 0, n -1).

Let f W be the generating series associated with W :

f W = n≥1 n n-1 x n n! . ( 18
)
The function -f W (-x) is usually called the Lambert W function [CGH + 96].

Let us consider now the graded S-module S • W . We introduce a variable s to encode the p grading. The associated generating series is

f S•W = e sfW . ( 19
)
For the graded exterior S-module Λ•W , we introduce a variable -t to encode the q grading. The minus sign in front of t is convenient here ; specialization at t = 1 gives the Euler characteristic. The associated generating series is

f Λ•W = e -tfW .
(20)

The associated generating series for the bicomplex is then given by

f (S•W )⊗(Λ⊗W ) = e sfW e -tfW = e (s-t)fW = 1+(s-t) n≥1 (n+s-t) n-1 x n n! . (21)
Here the Taylor expansion is a classical result, see for example [CGH + 96, Formula (2.36)].

Taking the horizontal Euler characteristic is given by substituting t = 1:

e sfW e -fW = e (s-1)fW = 1 + (s -1)

n≥1 (n + s -1) n-1 x n n! . ( 22
)
As we known by ( 17) where the horizontal homology of the bicomplex is concentrated, computing the Euler characteristic is enough to get the dimension of the homology.

Let us first compute the constant term of (22) with respect to s. One gets

1 - n≥1 (n -1) n-1 x n n! . (23)
Hence the dimension of H(n, 0, n -1) is (n -1) n-1 as expected.

One can also easily compute the coefficient of s p for p > 0 and get the expected formula for the dimension of H(n, p, n -p).

Remark: the filtration on H(n, 0, n -1) with quotients H(n, p, n -p) for 1 ≤ p ≤ n gives a nice interpretation of the classical identity

(n -1) n-1 = n p=1 n p (p -1)(n -1) n-p-1 , (24)
which can be found for instance in [CDG00, Prop. 2] and goes back to Cayley [START_REF] Cayley | On the Analytical Forms Called Trees[END_REF].

4 Description of X One can get not just the dimensions of H(n, 0, n -1) but a description of the action of the symmetric groups on X.

In [Cha07, Prop. 7.2], the symmetric function Z Λ•W was computed, with a parameter -t accounting for the cohomological grading. Putting t = 1 in the formula there and removing the constant term, one finds the symmetric function for X.

Let us denote by p λ the power-sum symmetric functions. If λ is a partition, let λ k be the number of parts of size k in λ and f k (λ) be the number of fixed points of a permutation of type λ. Let z λ be the product over k of k λ k (λ k )!, a classical constant associated to a partition.

1+(-t) λ,|λ|≥1 (λ 1 -t) λ1-1 k≥2 (f k (λ) -t k ) λ k -kλ k (f k (λ) -t k ) λ k -1 p λ z λ , (25)
and the symmetric function Z X has the following expression

λ,|λ|≥1 (λ 1 -1) λ1-1 k≥2 (f k (λ) -1) λ k -kλ k (f k (λ) -1) λ k -1 p λ z λ , (26)
where the sums are over the set of non-empty partitions λ.

Recall that it was proved in [START_REF] Chapoton | On some anticyclic operads[END_REF] the the PreLie operad is an anticyclic operad. This implies in particular that there is an action of the symmetric group S n+1 on the space PreLie(n). Let us denote by W the corresponding S-module and let us compute the symmetric function Z W describing this action of the symmetric group S n+1 on the space PreLie(n).

From [Cha05, Eq. ( 50)], this symmetric function is characterized by the relation 1

+ Z W = p 1 (1 + Z W + 1/Z W ). ( 27
)
Proposition 4.2 The symmetric function Z W is given by

λ,|λ|≥1,λ1 =1 (λ 1 -1) λ1-2 k≥2 (f k (λ) -1) λ k -kλ k (f k (λ) -1) λ k -1 p λ z λ . (28)
Proof. One has (see [START_REF] Chapoton | Hyperarbres, arbres enracinés et partitions pointées[END_REF])

  p 1 exp   - k≥1 p k /k     • Z W = p 1 . (29)
Let us introduce new variables:

y ℓ = p ℓ • Z W , (30)
for ℓ ≥ 1. Then the inverse map is given by

p ℓ = y ℓ exp - k y kℓ /k . (31)
Let λ be a partition with longest part at most r. To compute the coefficient of p λ in the symmetric function 1 + Z W , it is enough to compute the residue

(1 + Z W ) r i=1 dp i p λi+1 i , (32)
which is equal by formula (27) to

p 1 (1 + Z W + 1/Z W ) r i=1 dp i p λi+1 i . (33)
One can assume without restriction that all variables y j and p j vanish if j > r.

We will change the variables to get instead a residue in the variables y. One has to use the formula

r i=1 dp i = exp - i k y ik /k r i=1 (1 -y i )dy i . (34)
We therefore have to compute the residue

y 1 exp - k y k /k (1 + y 1 + 1/y 1 ) exp i λ i k y ik /k r i=1
(1 -y i ) y λi+1 i dy i .

(35) Gathering the exponentials and reversing the order of summation, one finds

exp k (f k (λ) -1)y k /k (1 + y 1 + y 2 1 ) r i=1 (1 -y i ) y λi+1 i dy i . (36)
This integral decomposes as a product of residues in each variable y i . Let us discuss first the integral with respect to y 1 : exp ((λ 1 -1)y 1 ) (1 -y 3 1 ) y λ1+1 1 dy 1 .

(37) This is the sum of two terms:

exp ((λ 1 -1)y 1 ) 1 y λ1+1

1 dy 1 = (λ 1 -1) λ1 λ 1 ! , (38)
and exp ((λ 1 -1)y 1 ) (-y 3 1 ) y λ1+1

1 dy 1 . = - (λ 1 -1) λ1-3 (λ 1 -3)! . ( 39
)
Note that some care must be taken in the second term when λ 1 ≤ 2. The resulting sum is zero if λ 1 = 1 and

(λ 1 -1) λ1-2 λ 1 ! if λ 1 = 1. ( 40
)
Let us then discuss the integral with respect to

y k for k ≥ 2. exp ((f k (λ) -1)y k /k) (1 -y k ) y λ k +1 k dy k . ( 41
)
This is also the sum of two terms

(f k (λ) -1) λ k /k λ k /λ k ! -(f k (λ) -1) λ k -1 /k λ k -1 /(λ k -1)!. ( 42
)
This can be rewritten as

(f k (λ) -1) λ k -kλ k (f k (λ) -1) λ k -1 k λ k λ k ! . ( 43
)
Gathering all terms and removing the contribution of the empty partition gives the result.

From this, one deduces the following relation.

Theorem 4.3 One has Z X -p 1 = (p 1 ∂ p1 -Id)Z W , (44)
i.e. the action of S n on X(n) is obtained from the action of S n on W (n) by taking the inner tensor product with the reflection module of dimension n -1 of S n .

Proof. This is a computation done using Prop. 4.1 and 4.2. Indeed, the operator (p 1 ∂ p1 -Id) acts by multiplication of p λ by λ 1 -1. There is one subtle point to check, though. In formula (28) for Z W , the summation is over all partitions of size at least 2 with λ 1 = 1, whereas in Formula (26) for Z X -p 1 , the summation is over all partitions of size at least 2 without further condition. Let λ be any partition of size at least 2 with λ 1 = 1. Let k be the size of the next-to-smallest part of λ. Then f k (λ) = kλ k + 1 and hence the expression

(f k (λ) -1) λ k -kλ k (f k (λ) -1) λ k -1 (45)
vanishes. It follows that all such partitions do not contribute to Z X -p 1 and that one can deduce the expected equation. That the operator (p 1 ∂ p1 -Id) corresponds to the inner tensor product by the reflection module is a classical fact in the theory of symmetric functions.

Freeness from homology concentration

We will use the knowledge of the homology of the bottom line of the bicomplex to show that the free pre-Lie algebras are free as Lie algebras.

General setting

Let P be an operad and assume that P (1) = K1 and let P + be the S-module such that P = K1 ⊕ P + . Let A be a P -algebra in the category of S-modules.

The structure of P -algebra on A is given by a morphism µ :

P • A → A.
Let us define for each k ≥ 0 a subspace A ≥k of A. Let A ≥0 be A. By induction, let A ≥k be the image by µ of P + • A ≥k-1 . This is a decreasing filtration of A by subspaces. By construction, this filtration is in fact a filtration of P -algebra.

We will furthermore assume that this filtration is separating, which is true for instance if A has some auxiliary grading concentrated in positive degrees.

Let us define H 0 (A) to be the degree 0 component A ≥0 /A ≥1 of the associated graded P -algebra grA.

Let us choose a section of H 0 (A) in A. Let Free P (H 0 (A)) be the free Palgebra on H 0 (A). Then there exists a unique morphism θ of P -algebra from the free P -algebra Free P (H 0 (A)) to A extending the chosen section.

Proposition 5.1 The morphism θ is surjective.

Proof. Because the filtration is assumed to be separating, it is enough to prove that the associated morphism of graded algebras is surjective. This is done by induction on the degree associated with the filtration. This is true in degree 0, because the map θ comes from a section.

Let now [x] be a class in grA k with k ≥ 1. Pick a representative x ∈ A ≥k of the class [x]. Then by hypothesis, x can be written as a sum a µ(y a , z a ),

where y a ∈ P + and z a ∈ A ≥k-1 .

Then it follows from the fact that we used a filtration of P -algebra that the class [x] itself can be written as a µ(y a , [z a ]),

where y a ∈ P + and z a ∈ A ≥k-1 . Each class [z a ] in grA k-1 belongs to the image of θ by induction. Therefore [x] belongs to the image of θ.

To show that the morphism θ is an isomorphism, an argument of equality of dimension (in some appropriate sense) between A and Free P (H 0 (A)) is therefore sufficient.

Application

Let us apply the general setting above to the case that we are studying. The operad P is the operad Lie. The P -algebra A is the S-module W . As W has no component in degree 0, one can apply the previous construction. The space X is exactly the homology group H 0 (A).

Theorem 5.2 The S-module W is isomorphic as a Lie algebra in the category of S-modules to the free Lie algebra Lie •X on the S-module X.

Corollary 5.3 For any vector space V , the free pre-Lie algebra on V is isomorphic as a Lie algebra to the free Lie algebra on X(V ).

Proof. As follows from section 5.1, there is a map θ from Lie •X to PreLie which is surjective. To prove the theorem, it suffices to compare the dimensions.

Let f X be the generating series

n≥1 (n -1) n-1 x n n! , (48)
which is associated with the S-module X of indecomposable elements of PreLie as a Lie algebra. We have to check that the generating series of the free Lie algebra Lie •X on X is equal to f W . As the series f Lie is -log(1 -x), this amounts to the equality

-log(1 -f X ) ? = f W , (49)
and Representation Theory held at the AMS Meeting in Hartford, CT, March 4-5, 1995, and from the Conference on Operads and Homotopy Algebra held in Luminy, May 29-June 2, 1995.

[Pri70] S. B. Priddy. Koszul resolutions. Trans. Amer. Math. Soc., 152:39-60, 1970.

Proposition 4. 1

 1 The symmetric function Z Λ•W has the following expression

which can be rewritten as

The constant term in x is 1 on both sides. Therefore it is enough to compare the derivatives with respect to x:

where the right equality is by comparison of the Taylor expansions ( 18) and ( 48). But we have by definition f W = xe fW , hence

This proves the expected equality and hence the Theorem.


Aldo Napoli

A HYBRID APPROACH FOR THE ANALYSIS OF ABNORMAL SHIP BEHAVIORS

Keywords: Maritime Domain Awareness, Ontology, Data Mining, Geovisual Analytics

.

We present in this paper a hybrid approach for analysing dangerous behaviours of ships based on AIS data. This approach is based on supervised and unsupervised analyses and it was developed in the frame of PhDs. Our approach is based on three complementary methodologies: (1) data mining for knowledge and pattern extraction for behaviour modelling, (2) ontological modelling of behaviour for an unsupervised detection and (3) geovisual analysis of large volume of data for a supervised detection of threats at sea. Three prototypes were developed to test our approach.

INTRODUCTION

The maritime surveillance centres (MRCC) integrate monitoring systems in order to have in real time a large vision of the traffic of ships. Data acquisition operates heterogeneous data like AIS data, meteorological data, bathymetry data, but all the data are not integrated in a unique interface. Moreover, the surveillance systems are in general not equipped with analysis support or decision support allowing to detect dangerous behaviours. Since 2007, in the frame of PhD and R&D projects, our research centre focus its work on the design and the development of maritime surveillance systems and specially on automatic or supervised detection of abnormal behaviours of ships. We can identify two kinds of behaviours:

-Known behaviours that we can model with expert knowledge [START_REF] Roy | Anomaly detection in the maritime domain[END_REF][START_REF] Van Laere | Evaluation of a workshop to capture knowledge from subject matter experts in maritime surveillance[END_REF];

-Unknown behaviours or behaviours interpretable with difficulties, to be characterized with automatic methods like data mining [START_REF] Pallotta | Vessel pattern knowledge discovery from AIS data: a framework for anomaly detection and route prediction[END_REF].

Then, the modeled behaviours can be integrated into a real-time system of detection of abnormal or dangerous behaviours.

The hybrid approach is designed on both unsupervised and supervised analysis methods. We present in the following sections the components of the approach:

-An unsupervised method for the analysis of the abnormal behaviours, based on data mining; -An ontological framework for the modelling of behaviours; this framework is used into a real-time detection system of abnormal behaviours, relying on case-based reasoning; -A geovisual analytics frameworks allowing supporting the operator in his task of behaviour analysis; -A maritime surveillance system integrating all the components of the methodology.

A DATA MINING SUPPORT SYSTEM

The unknown behaviours can be founded with data mining methods that can detect characteristic patterns. The patterns define a ship behaviour that can be modelled for further analysis. In our research (PhD of Idiri), we developed a prototype of AIS data analysis system based on data mining methods [START_REF] Idiri | Méthodologie d'extraction de connaissances spatiotemporelles par fouille de données pour l'analyse de comportements à risques -Application à la surveillance maritime[END_REF][START_REF] Idiri | The automatic identification system of maritime accident risk using rule-based reasoning[END_REF]. This system named ShipMINE (Fig. 1) is inspired by the system MoveMine [START_REF] Li | MoveMine: Mining Moving Object Databases[END_REF] that integrates algorithms for the analysis of different kind of trajectories (animals, cyclones, etc.).

Different algorithms are integrated into ShipMINE for the detection of:

-Zones with a lot of accidents (algorithm DBSCAN), -Abnormal trajectories into a group of trajectories (algorithm TROAD), (Fig. 2), -Usual trajectories of ships (algorithm TRACLUSS), -Parallel trajectories like trajectories of parallel fishing that is forbidden (algorithm CONVOY).

The patterns detected with ShipMINE are interpreted by experts of the domain. The useful patterns are then used for the modelling of abnormal or dangerous behaviours.

AN ONTOLOGICAL FRAMEWORK

Ontologies and the semantic enrichment of maritime trajectories are necessary in order to characterize the sequences of the behaviours. In the frame of the PhD of Vandecasteele [START_REF] Vandecasteele | Modélisation ontologique des connaissances expertes pour l'analyse de comportements à risque -Application à la surveillance maritime[END_REF][START_REF] Vandecasteele | A semisupervised learning frameworkbas ed on spatio-temporal semantic events for maritime anomaly detection and behavior analysis[END_REF], we based our research on the extension of the ontological framework developed by Yan and his colleagues [START_REF] Yan | Semantic Trajectories: Mobility Data Computation and Annotation[END_REF]. We decided to consider not only the spatio-temporal positions of the trajectory but also semantic trajectory units (e.g. begin, stop, moves, end). These semantic units can be enriched with different types of knowledge (e.g. spatio-temporal, geographic, domain) to provide end-users with high-level semantic descriptions of trajectories and a better understanding of the situation (step (a) in Fig. 3). This step allows performing further analyses of trajectories identifying potential alerts related to abnormal movement (step (b) in Fig. 3). Then, the proposed framework allows interpreting vessels' activities and behaviours (step (c) in Fig. 3), using a case-based reasoning approach to compare previous behaviours defined by operators with the current facts. Then the semantics behaviours are integrated through a specific user interface that provides a better understanding of the situation (step (d) in Fig. 3). This approach was integrated into the prototype OntoMAP (Fig. 4).

A GEOVISUAL ANALYTICS SUPPORT SYSTEM

The community of geovisual analytics develops a lot of visualizing methods for geographical information [START_REF] Kraak | Beyond geovisualization[END_REF][START_REF] Andrienko | Visual analytics of movement: an overview of methods, tools and procedures[END_REF][START_REF] Willems | Visualization of vessel traffic[END_REF][START_REF] Riveiro | Visual analytics for maritime anomaly detection[END_REF]. During the PhD of Vatin [START_REF] Vatin | Formalisation d'un environnement d'aide à l'analyse géovisuelle -Application à la sécurité et sûreté de la maritimisation de l'énergie[END_REF][START_REF] Vatin | High-Level Taxonomy of Geovisual Analytics Tasks for Maritime Surveillance[END_REF][START_REF] Vatin | Geovisual Analytics Environment for Supporting the Resilience of Maritime Surveillance System[END_REF][START_REF] Vatin | Using Ontologies for Proposing Adequate Geovisual Analytics Solutions in the Analysis of Trajectories[END_REF], we developed a methodological approach of ship trajectory analysis based on geovisual analytics (Fig. 5).

Our system proposes to an operator, information and visualizing method the most adapted to the analysis of risky situations. Our approach is based on (1) the modelling of the normal or abnormal situations to be analyzed, (2) the modelling of the information and the visualizing methods to be used and (3) the modelling of the profile of the operator and his capacity to handle the information and the visualizing methods.

Because the operator of the MRCC is not accustomed to handling these methods, our system proposes to him, based on the analyses of his capacity and of the situation to be detected (pirate attacks, etc.), the visualizing methods the most adapted to the context. We developed a system allowing a real-time analysis as well as delayed analysis of ship traffic. It can be adapted to every situation and every user profile. This prototype is integrated into our maritime surveillance system FishEYE.

A MARITIME SURVEILLANCE SYSTEM

In order to monitor data in real time and to test the researches introduced in previous sections, we developed a prototype of maritime surveillance system named FishEYE (Fig. 6). This system integrates heterogeneous data (ship localizations, meteorological and oceanic data, bathymetry, etc.) and different functions (analysis filters, drift models, etc.). Prototypes presented in the previous sections have been integrated as modular functions of the system. The design and the development of FishEYE are still in progress and improved continually.

Fig. 6: The prototype FishEYE

CONCLUSION

The proposed approach allows analyzing identified and unknown behaviours thanks to supervised and unsupervised analysis methods. Unknown behaviours are detected with unsupervised methods (data mining). Similarly to known behaviours, the detected behaviours can be interpreted with a system based on supervised methods (geovisual analytics). Then, the behaviours can be formalized within an ontological framework.

Different prototypes were developed, integrating the methods chosen (data mining, geovisual analytics, case based reasoning). Finally the maritime surveillance system FishEYE was developed to integrate all methods into a unique interface, easier to be handled by an operator.

Next step of our research will focus on a geocollaborative support system allowing supporting operators to solve a problem like a huge accident or oil spill. This framework will be integrated into FishEYE.

Fig. 1 :Fig. 2 :

 12 Fig. 1: The prototype ShipMINE

Fig. 3 :Fig. 4 :

 34 Fig. 3: Conceptual architecture of the system. Raw data are extracted, enriched and then analyzed to identify potential abnormal behavior (in [11])

Fig. 5 :

 5 Fig.5: The Geovisual analytic interface in[START_REF] Vatin | Formalisation d'un environnement d'aide à l'analyse géovisuelle -Application à la sécurité et sûreté de la maritimisation de l'énergie[END_REF]


EXPLOITING THE POTENTIAL OF THE FUTURE "MARITIME BIG DATA"

Bernard GARNIER*, Aldo NAPOLI** *BlueSolutions Consulting SAS, Sophia Antipolis, France **MINES ParisTech, PSL Research University, CRC, Sophia Antipolis, France ABSTRACT Today, most of the operational "abnormal behaviour" detection algorithms primarily operate kinematic rules on the vessel tracks provided by the AIS [START_REF] Pallotta | Vessel pattern knowledge discovery from AIS data: a framework for anomaly detection and route prediction[END_REF], [START_REF] Vandecasteele | From Movement Data to Objects Behavior Using Semantic Trajectory and Semantic Events[END_REF], [START_REF] Idiri | The automatic identification system of maritime accident risk using rule-based reasoning[END_REF]. To be more effective, they must be associated to additional "context data", however insuring this data access is today a challenge, due to the disparity of registers and the fragmentation of actors… For example, detecting suspicious "associations of ships" requires already a complex data mining to detect indirect common ownership through the myriad of cascaded legal entities used for formal ownership and registration; as another example, maritime security actors know the importance of environmental factors (sea state, fog, clouds, moon etc) when assessing the risk of illegal passages or piracy attacks at night -but gathering the right local weather forecast data in association with abnormal behaviour detection algorithms for maritime radar systems is also a challenge.

While many more data and metadata should be browsed from all the existing maritime reporting systems and data repositories to cross-check systematically the declared versus actual behaviour of commercial, fishing and leisure ships, we shall start developing a new data access thinking to benefit from the progressive deployment of the EU-wide CISE which is approaching its pre-operational validation milestone and should be largely mature by 2020, materializing a break-through in terms of data access for the maritime security communities. In parallel, another key enabler is the capacity to collect the "local picture" gathered by genuinely cooperating shipping (sightings and nav radar) with the "big picture" (VMS, AIS, S-AIS, LRIT, satellite imaging…). The VDES will provide a very effective data uplink as an alternative to broadband maritime SatComs at the same 2020 horizon. Other planned technological gapfillers deserve to be integrated in future data processing strategies: new space projects aim at solving the data synchronicity challenge by co-locating SAR, S-AIS and VDES payloads for specialized maritime surveillance constellations; EDRS allows downloading LEO maritime surveillance data streams in near real-time from anywhere on Earth; smart and fast embarked data processing will downsize the "rising tide" of data, extracting and tagging straight away the mere fraction of data requiring prompt human attention… This paper will aim at delivering a sort of "wake-up call" to integrate this new data access paradigm in the current research on maritime knowledge discovery associated to the detection of safety and security threats: 2020 is tomorrow, we shall think, develop and test our toolbox at the whole scale of this "Big Data". Index Terms-Maritime surveillance, early detection, heterogeneous correlation, CISE, weak signals analysis

CURRENT STATE OF PLAY

While maritime surveillance has been radically transformed by the introduction of the Automatic Identification System (AIS) in 2002 through the IMO SOLAS Agreementsuddenly populating the screen of the vessel traffic management systems (VTMS) well beyond the range of the coastal radars, the operational maritime surveillance capabilities have not much evolved since. Voluntary reporting systems (mainly AIS, VMS for fishery vessels and LRIT in distant sea lanes) remain the essential source of vessel monitoring, leaving in the shade the smaller boats… and the deliberately cheating ones. Furthermore, noncooperative ship detection provided by maritime radars (on board ships and on the coast) is now automatically fused with AIS, no more supported by additional VHF voice contact and binoculars to confirm the vessel identity and its planned route.

The recent development of commercial satellite payloads designed to collect AIS signals from ships well beyond coastal VHF horizon (S-AIS) is a welcomed "plus" to overcome the deficient cover of coastal AIS receivers in some regions, but remains overall a moderate contribution to the VTMS operation as S-AIS largely recoups LRIT.

International cooperation has become routine, but again most if not all maritime traffic data exchange agreements relate to the data of these voluntary reporting systems; as they are often faulty and sometimes cheated, this means building the common operational maritime traffic picture on sand! Space observation systems have been promoted as a new way to ascertain the maritime traffic data. Indeed, medium and high resolution synthetic aperture radar (SAR) on low Earth orbit (LEO) satellites provide a theoretical capability of ship detection anywhere in high seas, but its use as a daily maritime traffic monitoring tool remains problematic (no persistence, limited refresh, significant cost, latency of several hours). It is technically impossible to acquire synchronously SAR radar and S-AIS, rendering the correlation of these data tedious and often uncertain. As a consequence, its operational use is limited to specific fishing grounds and trafficking areas (where and what to look are known ahead of data collection and processing).

With these limitations, the blue borders remain largely porous to all sorts of trafficking (drugs, arms, migrants…) while endangered fishing species are still poached at high scale (IUU fishing remains as high as 25 to 30% of all catches).

In the field of Defence, Intelligence Agencies are actively seeking for illegal arms trade, intrusions in territorial waters and security threats of all sorts; however cross-border information exchanges (e.g. under the auspices of NATO) are most often limited to share a "list of usual suspects" designating about 2000 vessels of specific interest to be jointly monitored. In parallel, anti-drug operations are also most of the time driven by human intelligence (HUMINT), leaving 90% of the traffic undetected. In short, the challenge relate to the large number of "unknown unknown" threats.

To overcome the incapacity of the current state-of-play to anticipate the creativity and flexibility of criminal schemes, there is a clear need for changing the maritime surveillance paradigm: monitoring ship tracks on big screens is not enough!

LOOKING FOR WEAK SIGNALS

Time is long gone where seas appeared as the ultimate area of freedom, where the Master was invested of every power… after God. All maritime activities are today explicitly regulated, even in High Seas, as a result of international treaties and agreements (IMO…), regional agreements (Baltic…), national and local regulations. The respective authority of Flag States, Port States, Coastal States etc. is internationally agreed and results into massive data collection from all operators: every ship voyage generates dozens of massive files on the ship itself, the voyage, the cargo and the people on board.

Each of these files are directed to a particular Maritime Authority which screens the documents, clears the corresponding ship operation and triggers possible controls.

Criminal gangs are thus used to provide "clean" documents to avoid controls, e.g. cargo or fish catch declarations that will look "business as usual" for the custom officer or fishery inspector respectively. In the same time, these data are not today available in parallel to the serious crime investigators that could detect inconsistencies or possible correlations with their own investigations.

This "fragmentation" of the State controls of maritime activities is inherited from the absence of an holistic vision of the maritime economy common to almost all States: a comprehensive survey undertaken by DG Mare confirmed a split of the maritime authority prerogatives between more than 10 different administrations all across the 21 EU maritime nations, with many different Ministries involved (Transport, Energy, Environment, Agriculture, Interior, Economy, Defence…). Furthermore, this organizational mix differs significantly form a country to the next, with hardly a cross-border match between mandates and legal prerogatives, hence making inter-administration cooperation furthermore complex. This results into what is usually called "data silos", each of them only exploited under a single angle.

In essence, maritime surveillance operations are not distinct from any business, and the general approach of "Strategic Early Warning Systems" theorized from 1975 ref [START_REF] Ansoff | Managing Strategic Surprise by Response to Weak Signals[END_REF][START_REF] Gilad | Early Warning: Using Competitive Intelligence to Anticipate Market Shifts, Control Risk, and Create Powerful Strategies[END_REF] to provide on-time strategic reaction capabilities is perfectly applicable. The central element is that disruptions do not emerge without warning, however these warnings remain most often undetected as they don't come from the expected channels of business information. These warning signs are described as "weak signals" [START_REF] Ansoff | Managing Strategic Surprise by Response to Weak Signals[END_REF], a concept aimed at early detection of those signals which could lead to strategic surprises --events which have the potential to jeopardise an organization's strategy. Brison and Wybo [START_REF] Brizon | The life cycle of weak signals related to safety[END_REF] represent the life cycle of weak signals as four successive steps associated with barriers that the weak signal has to overcome. These four steps are: Detection, Interpretation, Transmission and Priority setting. The extraction of such weak signals from the massive data and meta-data collected on Internet by the GAFA turns to be potentially extremely profitablecurrently turning as the 21th century gold rush… Everyone has experienced already how effectively e-advertising can be targeted by processing the heterogeneous navigation data and metadata of your internet browser.

There are already demonstrative contributions of Open Source data mining and weak signals analysis in the maritime security domain, such as the ConTraffic webservice of the JRC able to alert Custom Authorities on particular containers associated with "abnormal" voyage histories (https://contraffic.jrc.ec.europa.eu/).

The largest potential relate to the application of weak signals detection over widely heterogeneous data possibly correlated. Ref [START_REF] Zuech | Intrusion detection and Big Heterogeneous Data: a Survey[END_REF] proposes an interesting application of this approach to the detection of cyber-intrusions, which is not dissimilar to our own preoccupation. Proper "Features Detection" comes as a cornerstone of efficient heterogeneous correlation. (http://journalofbigdata.springeropen.com/articles/10.1186/s 40537-015-0013-4 ).. As another inspiring example, heterogeneous correlation will soon offer a totally secure substitute to traditional passwords: a very clear signal has been given by Google at its annual I/O conference 2016, announcing the availability of a "Authentication APS developer kit" by the end of 2016 based upon "behavioural biometrics" (Abacus project). This component of Chrome will allow elaborating a "trust score" to enable the user login, achieving its authentication through a comprehensive pattern of behavioural features currently captured by the smartphone own sensors: how you type, where you are, how fast you move, your voice intonations etc. -none of them "mediated" by an explicit identification request (https://www.newscientist.com/article/2091203-googleplans-to-replace-smartphone-passwords-with-trust-scores/ ). The transposition of these advanced IT concepts to cross-correlate the data respectively collected by Port Authorities, Customs, VTMS, fishery control agencies, Defence, Law enforcement agencies etc. seems straightforward: this shall allow directing the operator's focus on few "abnormal/suspicious" seafarers trying to hide in the global maritime traffic while clearing without any further investigation most of the tracked vessels.

THE REVOLUTION TO COME: EASING DATA ACCESS

The key point to enable weak signal analysis is to access as many data as possible without any prior selection: searching the "unknown unknown" means that the data owner has no clues to pre-select what is worth being shared with its partnering maritime authorities -so it is a totally distinct approach from current "need to know" (and even the emerging "dare to share") the data he has already identified as suspicious.

The second characteristic of this activity is to process all sorts of data collected in the global framework of "maritime management", including the associated metadata. This is totally distinct from building a "common maritime picture" by fusing all the ship tracks collected across the community of maritime administrations: instead of aggregating every possible data of the same nature (ship tracks), the purpose is to browse with pre-determined search strategies (e.g. to build a confidence index) all possible layers of data (e.g. ship owner, previous ports of call, container numbers, crew list, average speed, mix of cargo, berthing records, occurrence of encountering with ship Y etc etc)

The conjunction of the total dematerialization of all the shipping documents (e-maritime, single national window…) and of the go-ahead for the Common Information Sharing Environment (CISE) is creating the framework of a massive "maritime Big Data" which is the pre-requisite for deploying advanced data mining tools underlying the weak signal analysis approach. This requires however building effectively a CISE with all the features of the original vision of DG Mare in terms of seamless access to all relevant national/sectoral data repositories.

Fig. 1 is a familiar conceptual view of CISE (excerpt from the Deloitte report ref [START_REF]Study on the current surveillance IT landscape and the resulting options for the Common Information Sharing Environment for Surveillance in the Maritime Domain[END_REF]) showing the various "User Community Layers" expected to rally the common exchange environment. On this graphic, developing heterogeneous correlations would come as achieving a seamless permeability between any layers, to conduct the correlation by picking data of all 7 colours and detect anomalies that no layer would ever suspect. There is currently a risk that a number of User Communities (UCs) still consider the horizontal permeability as the principal scope for CISE, materializing into a collection of "common sectoral operational picture" built from the data considered by each data owner as "interesting to share" within the same UC for improving cross-border cooperation. At a time where the Pre-Operational Validation project CISE-2020 is launching the procurement of critical IT software bricks of the future CISE, opening this discussion seems critical.

THE VDES OPPORTUNITY

The AIS is currently under revision at international level (IALA, IMO) to incorporate the capability of broadband data transfer (Very high frequency Data Exchange System, VDES) while improving as well the capture of AIS signals by satellites (S-AIS).

Planned to enter into service by 2020, as for CISE, the VDES will provide all reporting vessels with a capability to contribute to the global maritime surveillance picture. It shall be seen having the potential of a Copernican revolution: moving from the era of unilateral reporting toward VTMS operators with little if no feedback, ship masters will be able to exchange all sorts of maritime safety and security notices with the neighbouring ships (via VHF) and with the whole community (via S-AIS) at no cost (compared to the SatCom broadband links currently needed). EU Maritime Authorities should monitor more closely and possibly influence the current phase of standard VDES messaging definition to secure the effective contribution of every cooperative ship to report its local environment as a contribution to the grand picture currently compiled by National Authorities, Regional Commissions, EMSA and NATO. Early reporting of the sighting by a cargo or a ferry of "strange" ships around her (fishing vessels out of fishing grounds, old cargo much too low over the water, towed pateras or RHIBs, unusual routes, apparent rendez-vous at sea, low flying plane etc) could help directing patrols well before incidents might trigger alerts. VDES has the potential to turn every cooperative ship as an "in-situ" maritime surveillance sensor, able to transmit messages, pictures, radar screenshots, open comments etc.

RECOMMENDATIONS

Maritime authorities will not transform their data sharing and data access policies overnight. In the same time, major enablers of a new way to think the maritime surveillance data analysis are at stake now depending on short term decisions at EU level (CISE) or UN/IMO (VDES) to freeze the technical requirements of these new systems.

Our R&D community has the duty to launch now very imaginative and convincing "vertical" data mining experiments (wrt Fig. 1) to demonstrate the power of weak signal analysis, the way to build unprecedented "trust scores" and the consequence of this transformation of the notion of "abnormal behaviour detection" on the requirements (including at some stage underlying legal agreements) of both CISE and VDES. Without offering attractive use-cases, we might miss this challenging 2020 milestone, with the risk of not meeting again before long such opportunity.

Fig 1 :

 1 Fig 1: "CISE Landscape", from ref [8] p.189


Pablo Arrighi
email: pablo.arrighi@univ-amu.fr.

Gilles Dowek
email: gilles.dowek@inria.fr.

What is the Planck constant the magnitude of?

The Planck constant is the minimal area of one bit.

Introduction

The constant c is not just a constant occurring in some formulae, it is also the magnitude of something: of the speed of light. So it is easy to answer the question: what is the constant c the magnitude of? Answering the same question for the Planck constant seems to be more difficult. Two signs point at this difficulty. The constant c does not have a name. It is neither called "the Rømer constant", nor "the Einstein constant". To name it, we just say what it is the magnitude of: "the speed of light". But we do not seem to know what the Planck constant is the magnitude of, so we call it "the Planck constant". Moreover, there is only one constant c, while at least two versions of the Planck constant are in competition:

and h = 2π , each being more or less convenient, depending on the formula we consider.

We suggest here that Bekenstein's postulate, according to which the amount of information stored in a region of space is bounded, would allow us to give an answer to this question: the Planck constant is the minimal area of one bit.

Units

The Planck constant is expressed in kg m 2 s -1 . But since Special Relativity, we have been knowing that seconds and kilograms are just other units for distance, like inches and ångströms. Indeed, a duration t can just as well be expressed in meters, as the distance ct travelled by light during the duration t. In this respect, c = 2.998 10 8 m s -1 is no more a fundamental constant than the constant 0.0254 m in -1 which transforms inches in meters.

When distances and durations are expressed in meters, a gravitational field E no longer needs to be expressed in m s -2 , it may be substituted by e = E/c 2 which is expressed in m -1 . Then the gravitational field created by a point mass M at a distance d is

e = E c 2 = GM/c 2 d 2 = m d 2
where m = GM/c 2 is the mass expressed in meters. Again, notice that here G, or more precisely G/c 2 = 7.426 10 -28 m kg -1 , is no more a fundamental constant than the constant that permits to transform inches in meters. The Planck constant is expressed in m 2 kg s -1 . But as seconds and kilograms are just other units for distance, it can as well be expressed in m 2 . Its valuea in m 2 is

a P = G c 2 1 c = 2.612 10 -70 m 2
more often called "the Planck area". Bekenstein's postulate

Bekenstein's postulate states that the amount of information that can be stored in a sphere of radius R is bounded, and that the bound is

I max = 1 4 ln(2) c 3 G 4πR 2 = 1 4 ln(2)a P 4πR 2
This postulate, and in particular the fact that this amount of information grows like R 2 , and not R 3 , is still controversial, but if we assume it, then we can deduce that the minimal area to store one bit is 4 ln(2)a P . Thus, the Planck constant expressed in m 2 , a P , modulo the factor 4 ln(2), is just the minimal area of one bit. This answers the question of what the Planck constant is the magnitude of. So, the meaningful version of the Planck constant does not seem to be , h, or a P , but rather b = 4 ln(2)a P = 7.243 10 -70 m 2 , which is the minimal area of one bit. In the usual units, the meaningful value does not seem to be or h, but 4 ln(2) = 2.924 10 -34 m 2 kg s -1 .

Defining the constant b as the minimal area of one bit makes Bekenstein's formula

I max = 4πR 2 b
very natural: the maximal amount of information that can be stored in a sphere is its area divided by the minimal area of one bit. Just like the speed of light stops to be a fundamental constant when we express durations in meters, the constant b stops to be a fundamental constant if we use it to define a natural distance unit. This unit is not the Planck length l p = √ a p = 1.616 10 -35 m, but l = √ b = 2 ln(2)l p = 2.691 10 -35 m. Using this unit to measure the radius of a sphere, the maximal information that can be stored in a sphere is just its area.


Laurent Molez

Yves Berthaud

Denis Beaupre

Benoit Bissonnette

D Beaupré

Structural concrete behavior: coupled effect of drying and loading

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

INTRODUCTION

The field of concrete repairs has made considerable great strides for the few years and the tendency does not display any sign of deceleration. If search and knowledge are passably advanced in the field of new construction, however that of repair remains relatively misunderstood.

A very large amount of reinforced concrete structures world-wide are in a state of advanced deterioration. The defects are due to various causes, like errors in construction, accidental overloading, foundation failure, corrosion of reinforcement, fire attack or chemically hostile environment. Repairs should fulfil four basic requirements: arresting the deterioration of the structure, restoring the structural integrity, providing an aesthetically acceptable finish and, the last but not the least, being durable [START_REF] Saucier | La durabilité de l'adhérence des réparations en béton[END_REF]) [START_REF] Emmons | Factor affecting the durability of concrete repair: the contractor's viewpoint[END_REF] [START_REF] Bissonnette | Le fluage en traction: un aspect important de la problématique des réparations minces en béton[END_REF] . However, at the same time, it must also be conceded that many repaired structures are severely damaged after only a few years after being repaired [START_REF] Vaysburd | Concrete problems, repair solutions and confusions -an attempt to establish the facts[END_REF]. The cracking of the repair material causes the most serious deterioration processes leading to repair failures, since the aggressive agents can then penetrate through these cracks.

RESEARCH SIGNICANCE

The purpose of this investigation is to study the overall behaviour of repaired elements. We want to understand which are the significant parameters (hydration, shrinkage, creep, bond, etc.), and their relative importance, leading to the degradation of concrete repairs.

We want to answer two questions. How to control the cracking of repairs? Is it possible to restore the structural integrity of repaired elements?

Facing to this problem, an important experimental study was undertaken. The generated data will feed a numerical analysis model that will permit to study various parameters and configurations.

THE CAUSE FOR CONCERN

A concrete repair can be defined as a thin concrete layer ( 50 to 150 mm) cast on an existing structure to replace the part close to the surface which is deteriorated. These two concrete layers have different mechanical and physical characteristics. The old one is quasi-stable with time. The new one change with time (hydration, drying shrinkage, creep). Some stresses are built-up as the sum of chemically, hygrally, thermally and mechanically induced strains increases (figure 1). The shrinkage and thermal strains are restrained by the bond with the old layer.

Tensile creep can partly relaxe these induced strains [START_REF] Bissonnette | Le fluage en traction: un aspect important de la problématique des réparations minces en béton[END_REF]).

EXPERIMENTAL WORK 4.1 Specimens

Previous studies carried out at Laval University [START_REF] Pigeon | Steel fiber reinforced shotcrete repair of concrete structures affected by alkali-silica reaction: laboratory assessement[END_REF] have shown that, with respect to crack development, it is important to perform the experiments on samples that are as close as possible to a real structure dimension-wise. The scale effects are very important and this is why the study has been done on large samples. All the selected materials (see section 4.3) were used to repair 4-meter long laboratory made reinforced concrete beams. The size and reinforcement details of the beams are given in figures 2 and 3. Thirty (five for each material) 200 mm × 300 mm × 4000 mm beams were cast and moistcured during 3 days. Ready mix concrete was used in the beams. The water to cement ratio was 0.40.

The slump was 100 mm and the air content was 5%. The average compression strength at 28 days was 42 MP a. The shrinkage magnitude was about 800 μm after 300 days. The beams were manufactured in august 1996 and exposed to outside conditions during 30 months and to laboratory conditions for 6 months before testing in order to simulate real aging.

Repair procedure

The same surface preparation technique was used before repairing all the beams. The bottom portion (tension strength zone) of the beam was chipped off, with a 7.5 kg jack-drill, to a depth of 75 mm on a length of 2 meters. Details can be seen in figure 3. The chipped surface was sandblasted to remove rust on reinforce- ment, carbonated concrete and loose particules. Before repairing, the surface was continuously humidified for 24 hours. Then, beams were repaired with the different materials (see section 4.3). The beams were wet-cured by covering them with wet burlap for 3 days and then allowed to dry during throughout the experiments.

Materials

Six different concretes were used to repair the damaged reinforced concrete beams. The choice of materials was carried out in order to obtain, on the one hand, similar mechanical properties and, on the other hand, a broad spectrum of amplitude of shrinkage and creep. Then, the water to cement ratio was fixed at 0.40. In order to obtain various values of shrinkage and creep, we adjusted the cement paste content of concretes [START_REF] Neville | Properties of concrete[END_REF]. The interface quality was "adjusted" by the setting technic. Previous study held at Laval University showed that ordinary cast-in-place concrete gives a poor quality interface, self-leveling concrete gives a good quality interface and shotcrete gives a very good quality interface [START_REF] Lacombe | Rheology and bonding characteristics of selfleveling concrete as a repair concrete[END_REF]. Composition of the repair materials can be seen in table 1. Two cast-in-place ordinary concrete were used (OC and SRA-OC). The second one has quite the same composition but with shrinkage reducing agent added. Two self-leveling concrete (SLC and LCC-SLC) were used. The first one can be considered as a "ordinary" self-leveling concrete. In the second one -low cement content self-leveling concrete -a part of cement was replaced by fly ash. The two last repair materials were shotcrete. The first one was a wet process shotcrete and the second one was a dry process shotcrete.

Materials characteristics

The mechanical and hygrometric properties of the materials were characterized. Concrete cylinders of 100 mm × 200 mm were taken from ordinary and self-leveling concrete to measure compressive strength, splitting tensile strength and modulus of elasticity. For shotcrete, cores of 75 mm diameter were taken from 500 mm × 500 mm × 75 mm moulds. Drying shrinkage was monitored on 100 mm × 100 mm × 350 mm prisms (cast or sawn). Tensile creep behaviour was obtained from 75 mm × 75 mm × 400 mm. Tensile creep experimental procedure can be seen in [START_REF] Bissonnette | Le fluage en traction: un aspect important de la problématique des réparations minces en béton[END_REF].

Figure 4 give compressive strength versus time. It can be seen that we achieved to have six "similar" ma- 4.5 Beams submitted to drying only Repaired beams were monitored for 3 to 6 months in the laboratory. Two of the five beams repaired with each material were exposed to drying. After 3 days of curing, they were allowed to dry during throughout the experiments. For each beam, the crack propagation pattern was recorded. The surface cracks were monitored. A crack microscope with a magnification of 20X was used. 5 μm-cracks can be detected and cracks larger than 25 μm can easily be measured. The behavior of the repairs will be thus quantified by a series of crack parameters including the spacing, the opening and the length of cracks. Cracks were generally perpendicular to the principal axis of the beam. They usually appeared where stirrups were located. The crack opening was measured in five points across the beam width. The average crack opening is calculated by adding each width measurement for each crack on the average repair length and then dividing by the number of cracks.

Figure 9 gives the evolution of crack width for beams submitted to drying only. It has to be noted that beams repaired with ordinary concrete (OC) never crack even after 6 months of drying. The same behaviour is observed for ordinary concrete with shrinkage reducing agent (SRA-OC) after 28 days of drying. For all the other mix, first cracks appear after 5 or 7 days of drying. After 3 months of drying, the average crack width is between 20 μm for DSC and 45 μm for WSC. For WSC, after two months of drying, crack width seems to be stabilized. On the contrary, for SLC, LCC-SLC and DSC crack width continues to increase. The stabilization can be explained for WSC by the fibers content in this mix (see table 1). Fibers prevent large opening cracks by developing multi-cracked pattern. It can be noted that the mixtures that present the larger cracks are those who exhibit the more important drying shrinkage (see figure 6). SLC show the more important shrinkage values, but not the more important crack width. It can be explained by the high value of the creep to shrinkage ratio (see figure 8). The number of cracks after 6 months of drying is noted between () in figure 9.

4.6 Beams submitted to drying and to long term loading After 10 days of drying, two other beams, of each repair material, were subjected to drying and mechanical loading. They were tested as simply supported beams under four-point loading with a total loaded span of 3700 mm. They were loaded up to 33% of the theoretical ultimate load (equivalent to a service load in a high chemical aggressive environment). The last beam for each repair material, was loaded up to 66% of the calculated ultimate load (equivalent to a service load in a non-aggressive environment) after 45 days of drying.

Test apparatus

The test apparatus can be seen in figure 10. Two beams were placed back to back and loaded under four-point loading as can be seen in figure 10. The load is applied at each end through springs. Those springs can partially compensate the effects of creep. With this type of equipment, a long-term load can be applied without monopolizing a closed loop testing system. It can be used in laboratory or outside in order to expose the repaired elements to severe environmental conditions. Each spring system was calibrated on a testing machine. Thus, the applied load could be known by measuring the deformation of springs. The load can be adjusted to 0.15 kN and is relatively constant in time. It had to be adjusted only twice during the 5 months of testing.

Beams loaded to 33% of P ult

Upon loading, the average crack opening was about 50 μm, and after 4 months of drying, it increased to about 85 μm for OC and DSC and 100 and 125 μm for LCC-SLC and SLC respectively. It should be noted that almost no crack is lately created during the loading. The cracks previously created during the 10 days of drying open more at loading. The crack width is quite similar for WSC and OC. On the other hand, WSC presents more cracks than OC. It can be explained by the larger shrinkage of WSC and the fibers content in WSC mix. SLC and LCC-SLC exhibit larger cracks than OC and WSC. This difference that appears just at loading, and up to a certain point, is due to mechanical properties mismatch. Finally, we could notice that during the four months of drying after loading, the crack width in- crease is quite similar for all mixtures and is equal to about 50 μm.

Beams loaded to 66% of P ult

Upon loading, the average crack opening was about 100 μm to 125 μm. Firstly, we must note that WSC and SLC present the least open cracks. However, they display the greatest number of cracks. It can be explain by the fibers content, again, for WSC. The important creep to shrinkage ratio of SLC may explain these facts. Secondly, OC present again less cracks than other mixtures. Finally, the crack width increase is equal to 25 μm for all mixtures. This last test was conducted 45 days after setting, and drying shrinkage kinetic is slower.

Flexural testing

Previous tested beams (drying only series and 66% loaded series) were tested to rupture. The beams were loaded in four-point bending as previously with an hydraulic testing machine. Deflection was recorded at midspan using two displacement transducers. The development and location of cracks was continuously observed until failure occurred. The load-deflection curves for control beams and repaired beams is presented in figure 13. We extracted, from corresponding curves for the 8 repaired beams and 2 control beams, the load at first crack P cr , the load at yield of reinforcement P y , and the ultimate load P u . Results are summarized in table 2. In this table P control u is defined by the average ultimate load for the two control beams. All repaired beams achieved the theoretical ultimate flexural capacity of 65 kN, presumably because failure was controlled by properties of parent concrete. Ductile flexural failures occurred, no interfacial bond breakdown was observed. The larger decrease (about 5 to 7%) of ultimate load was observed for OC beams.

These tests confirm that it is possible to restore the flexural behaviour of damage beams. No interfacial bond breakdown was observed as opposed to what literature usually says [START_REF] Emberson | Significance of property mismatch in the patch repair of structural concrete. part 1,2,3[END_REF].

TOWARDS A NUMERICAL ANALYSIS TOOL

Experimental results presented previously are often difficult to interpret. It is difficult to make distinction between the different effects of each parameter. A numerical analysis tools may help to understand the real behaviour of repaired beams and define global parameters to respect for good design of repaired elements.

We try to write a model that take into account the different physical phenomena. Phenomenological models are used to not complicate the numerical tool.

Mechanical damage

In the present analysis, we use an isotropic scalar damage model [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture de béton de structure[END_REF]. In this model, the mechanical effect of progressive micro-cracking due to external loads is described by a single internal variable which degrades the Young's modulus of the material. The constitutive relations are:

σ ij = (1 -d)Λ ijkl ε kl (1)
where σ ij and ε kl are the components of the stress and strain tensors respectively (i, j, k, l ∈ [1, 3]), are the initial stiffness moduli, and d is the damage variable. The material is initially isotropic, with E and ν the initial Young's modulus and Poisson's ratio respectively. Damage is a function of the positive strains which means that it is mainly due to micro cracks opening in tension mode. In order to avoid ill-posedness due to strain softening, the mechanical model has to be enriched with an internal length (Pijaudier-Cabot and Bažant 1987).

Non linear water diffusion equation

Moisture movements in concrete are of two types: a water flux in liquid form and a water flux in va-por form. The water consumption due to hydration is neglected. It is useful to write these movements as a function of the relative humidity H [START_REF] Bažant | Non-linear water diffusion in non-saturated concrete[END_REF]. The relationship between H and w (the water content) at constant temperature T and at a fixed degree of hydration is given by the desorption or adsorption isotherms. The slope of isotherm is almost constant over the range of humidities 0.15 ≤ H ≤ 0.95 [START_REF] Alvaredo | Drying shrinkage and crack formation[END_REF]. Then moisture movement can be written by a phenomenological diffusion equation:

∂H ∂t = D(H) • gradH (2)
The diffusion coefficient depend on H. When H is important, a reduction of the relative humidity causes a strong reduction of D. When H ≤ 0.70 to 0.80 D(H) is almost constant. We choose the following expression of D(H) [START_REF] Bažant | Non-linear water diffusion in non-saturated concrete[END_REF]:

D(H) = D 0 ⎛ ⎝ α 0 + 1 -α 0 1 + 1-H 1-Hc n ⎞ ⎠ (3)
where α 0 , n, and H c are adjustable parameters specific to material. This parameters can be determined from the "weight loss -time" relation.

Boundary conditions of the convective type are imposed at the surface Γ in contact with air at the relative humidity H a :

q = β(H Γ -H a ) ( 4
)
where q is the moisture flux normal to the exposed surface, β is the coefficient of surface hygral transfer and H Γ is the relative humidity at the surface.

Shrinkage

Drying shrinkage can be explain by three theoretical mechanism: capillary depression, Gibbs-Bangham theory (surface tension) and disjoining pressure theory. A study was undertaken to take into account all these phenomena [START_REF] Lassabatere | Couplage hydromécanique en milieu poreux non saturé avec changement de phase : application au retrait de dessiccation[END_REF]). This study shows that drying shrinkage can be well evaluated by a simple proportional equation:

Δε sh = α sh ΔH (5)
where ΔH is the variation of relative humidity at the studied point and α sh is the of hydrous dilatation coefficient, by analogy with the thermal dilatation coefficient.

First results

Nonlinear finite element simulations of repaired beams was carried out. We used CASTEM 2000, the CEA finite element code. First results shown a good accuracy for prediction of humidity profile and shrinkage. The crack pattern seemed to be in good agreement with experimental observations. A calculated crack pattern can be seen in figure 14.

CONCLUSIONS

• An important experimental study on real structure was undertaken. This study permitted the creation of a bank of data on concrete repairs.

• The effects of drying shrinkage do not to be neglected by repairs designers.

• Tensile creep can be a positive parameter. Tensile creep can partly relaxe induced strains. Creep to shrinkage ratio seems to be a good indicator for the choice of repair materials.

• Short term flexural test showed that it possible to restore flexural capacity of damage beams. Ductile flexural failures occurred in all studied repaired elements. No interfacial bond breakdown was observed.

• A numerical tool was proposed. This approach was based on phenomenological models including damage model, non linear diffusion theory and a shrinkage model. The finite element results were shown to be in good agreement with shrinkage behaviour and crack pattern.

• Much work remains to be done. On the experimental side, interfacial data have to be extracted. On the analytical side the finite element will benefit from an extension to account for creep, partially debonded interfaces and hydration effects.

Figure 1 :

 1 Figure1: Schematical illustration of the stress buildup in repairs[START_REF] Saucier | The challenge of numerical modeling of strains and stresses in concrete repairs[END_REF]).

FigureFigure 3 :

 3 Figure 2: Test specimens.

Figure 5 :Figure 8 :

 58 Figure 5: Tensile strength evolution.

Figure 9 :

 9 Figure 9: Crack width evolution -Drying only.

FigureFigure 11 :

 11 Figure 10: Loading setup

Figure 12 :

 12 Figure 12: Crack width evolution -Loaded at 66%.

Figure 13 :

 13 Figure 13: Flexural testing (drying only series).

Figure 14 :

 14 Figure 14: Damage due to restrained shrinkage in repaired beam

Table 1 :

 1 Concrete mixture characteristics * with the air content/without the air content

		OC	SRA-OC WSC *	SLC	LCC-SLC DSC *
	Cement T10 (kg)	400	400		412	235	400
	Cement T10 SF (kg)			475
	Silica fume (kg)				37	20	40
	Fly ash (kg)					214
	Water (kg)	162	158	190	180	188	176
	Aggregate 10-2.5 (kg)	957	957	495	769	766	210
	Sand (kg)	767	767	850	677	672	1450
	Fiber (kg)			18			1
	Air entraining agent (ml)	80	80	119	powder	powder	powder
	SP(ml)	1800	1200	1675	powder	powder
	Colloidal agent				powder	powder
	Shrinkage reducing agent (ml)		8000
	Air (%)	8	8	14	12	8	6
	E/L	0.40	0.40	0.40	0.40	0.40	0.40
	S/(G+S) Cement paste content (%) * *	0.440 36/31	0.44 36/31	0.630 48/39.5	0.459 45/37	0.458 45/40	0.869 38/34

* before projection *

Table 2 :

 2 Load deflection characteristics from shortterm static tests

	Repaired beams P cr (kN)	P y	P u	P u /P control u
	Control 1	24	67.6 70.4	0.99
	Control 2	-	68	71.6	1.01
	OC-0	18	62	67.2	0.95
	OC-66	-	62	66	0.93
	WSC-0	20.4	68.8 72.8	1.02
	WSC-66	-	68.8	72	1.01
	SLC-0	21.6	65.8	70	0.99
	SLC-66	-	68	72	1.01
	LCC-SLC-0	24	65.6 71.6	1.01
	LCC-SLC-66	-	64	67.6	0.95