Eric Martin
email: emartin@cse.unsw.edu.au

Jean Sallantin

A dialectic approach to problem-solving

We analyze the dynamics of problem-solving in a framework which captures two key features of that activity. The first feature is that problem-solving is a social game where a number of problem-solvers interact, rely on other agents to tackle parts of a problem, and regularly communicate the outcomes of their investigations. The second feature is that problem-solving requires a careful control over the set of hypotheses that might be needed at various stages of the investigation for the problem to be solved; more particularly, that any incorrect hypothesis be eventually refuted in the face of some evidence: all agents can expect such evidence to be brought to their knowledge whenever it holds. Our presentation uses a very general form of logic programs, viewed as sets of rules that can be activated and fire, depending on what a problem-solver is willing to explore, what a problem-solver is willing to hypothesize, and what a problem-solver knows about the problem to be solved in the form of data or background knowledge. Our framework supports two fundamental aspects of problem-solving. The first aspect is that no matter how the work is being distributed amongst agents, exactly the same knowledge is guaranteed to be discovered eventually. The second aspect is that any group of agents (with at one end, one agent being in charge of all rules and at another end, one agent being in charge of one and only one rule) might need to sometimes put forward some hypotheses to allow for the discovery of a particular piece of knowledge in finite time.

Introduction

The last century has seen the advent of a number of frameworks that place rationality at the heart of the process of scientific discovery; still none of those frameworks has endowed epistemology with a definitive mathematical foundation. The seminal research of Herbert Simon on the logical theorist and Mc-Carthy's research on commonsense reasoning are two prominent examples of general attempts at interfacing the thinking of rational agents and the dynamics of scientific discovery, before more specific approaches, tackling more specific problems, have appeared. Departing more or less from Boole's framework, a plethora of logics, mainly developed in the AI community, have grounded various rational approaches to problem solving; some of them have been validated by the implementation of tools, successfully applied to the resolution of a broad class of problems. Formal Learning Theory, PAC learning and Query learning, developed around the prominent work of Gold [START_REF] Gold | Language identification in the limit[END_REF], Valiant [8] and Angluin [START_REF] Angluin | Learning from different teachers[END_REF], offer theoretical concepts, rooted in recursion theory, statistics and complexity theory, to describe the process of data generalization. Induction has been studied from different angles, in particular by Pierce and Suppes, before the Inductive Logic Programming community suggested a more practical approach. Numerous investigations on automatic or semi-automatic scientific discovery have taken place [START_REF] Langley | Rediscovering chemistry with the BACON system[END_REF][START_REF] Soldatova | An ontology for a robot scientist[END_REF][START_REF] Chavalarias | Bottom-up scientific field detection for dynamical and hierarchical science mapping-methodology and case study[END_REF][START_REF] Afshar | Aristotle's square revisited to frame discovery science[END_REF]. Finally, let us mention the more recent work on the relationships between scientific discovery and game theory; but these pointers do not by far exhaust the whole body of work on the relationship between rationality and scientific discovery.

Our approach is based on an extension of Parametric logic [START_REF] Martin | Deduction, induction and beyond in parametric logic[END_REF], a new framework that unifies logic and formal learning theory, developed along three dimensions, two of which are illustrated in this paper.

-The first dimension is formal. 2 The logical framework

An illustrative example

Imagine the following game. Countably many copies of every card in a deck of 52 cards are available to a game master. The game master chooses a particular ω-sequence of cards. For instance, she might choose the sequence consisting of nothing but the ace of spades. Or she might choose the sequence where the queen of hearts alternates with the four of spades, starting with the latter. A number of players, who do not know which sequence has been chosen by the game master, can make requests and ask her to reveal the nth card in the sequence, for some natural number n. The players aim at eventually discovering which sequence of cards has been chosen by the game master, or to discover some of its properties. The game illustrates the process of scientific discovery, with the game master playing the role of Nature, and the players the role of the scientists. A feature of the game is that unless the game master has explicitly ruled out a large number of possible sequences, the players usually cannot, at any point in time, know whether their guesses are correct: they might at best be able to converge in the limit to correct guesses. We have not precisely defined what a "guess" is. There has to be a language where some properties of a sequence of cards can be described, and the expressive power of the language is crucial in circumscribing what the players can or cannot achieve. Let us refer to such a description as a theory, in analogy to the work of scientists whose aim is to discover theories that correctly describe or predict some aspects of the field of study. In this paper, we will let logic programs play the role of theories.

Logical background

N denotes the set of natural numbers and Ord the class of ordinals. We consider a finite vocabulary V consisting of a constant 0, a unary function symbol s, the observational predicate symbols, namely, the unary predicate symbols hearts spades diamonds clubs ace two . . . ten jack queen king and a number of other predicate symbols. For all nonzero n ∈ N, we denote by n the term obtained from 0 by n successive applications of s; n will refer to the nth card. We denote by Prd(V) the set of predicate symbols in V. Given n ∈ N, we denote by Prd(V, n) the set of members of Prd(V) of arity n. We fix a countably infinite set of (first-order) variables and a repetition-free enumeration (v i ) i∈N of this set. We need a notation for the set of all possible sequences of cards.

Definition 1. We call possible game any set T of closed atoms such that:

for all n ∈ N, T contains one and only one member of hearts(n), spades(n), diamonds(n), clubs(n); -for all n ∈ N, T contains one and only one member of ace(n), two(n), . .

. , ten(n), jack(n), queen(n), king(n); -T contains no other atom.

We consider a notion of logical consequence that is best expressed on the basis of a forcing relation , based on both principles that follow.

-The intended interpretations are Herbrand structures: every individual has a unique name (a numeral); this is because intended interpretations are ωsequences of cards-n being the name of the nth card in the sequence.

-Disjunction and existential quantification are constructive: an agent will derive a disjunction iff she has previously derived one of the disjuncts, and she will derive an existential sentence iff she has previously derived one of the closed instances of the sentence's matrix.

We denote by L ωω (V) the set set sentences, that is, closed first-order formulas over V. Given two sets S and T of sentences, we write S T iff S forces all members of T .

Logic programs and occurrence markers

A formal logic program provides, for every n ∈ N and ℘ ∈ Prd(V, n), two rules: one whose head is ℘(v 1 , . . . , v n ), and one whose head is ¬℘(v 1 , . . . , v n ). This is at no loss of generality since the left hand side of the rules can contain equality and the intended interpretations are Herbrand. So to define a formal logic program, we only need the left hand side of both rules associated with a predicate symbol and its negation. It is convenient, and fully general as well, to assume that all variables that occur free on the left hand side of a rule also occur on the right hand side of the rule.

Definition 2. A logic program (over V) is defined as a family of pairs of formulas over V indexed by Prd(V), say

((ϕ + ℘ , ϕ - ℘ )) ℘∈Prd(V) , such that for all n ∈ N and ℘ ∈ Prd(V, n), fv(ϕ + ℘ ) ∪ fv(ϕ - ℘ ) is included in { v 1 , . . . , v n }. An important particular kind of logic program is the folllowing. Definition 3. Let a logic program P = ((ϕ + ℘ , ϕ - ℘ )) ℘∈Prd(V) be given. We say that P is symmetric iff for all ℘ ∈ Prd(V), ϕ - ℘ = ∼ϕ + ℘ .
To distinguish between agents, we need the key notion of occurrence marker, which intuitively is a function that selects some occurrences of literals in some formulas. Let ψ be a nullary predicate symbol or the negation of a nullary predicate symbol. An agent could select an occurrence o of ψ in a formula ϕ because she wants to (provisionally) assume that ψ is either true or false, at least in the particular context of ψ occurring in ϕ at occurrence o. We will see that social agents will make use of the opportunity of assuming that ψ is false, whereas empirical agents will make use of the opportunity of assuming that ψ is true. Actually, ψ does not have to be nullary for these ideas to be developed (we will need more generality anyway), so the definitions that follow deal with arbitrary literals, not only literals built from a nullary predicate symbol. The underlying idea is the same, though it was more easily explained under the assumption that ψ is nullary.

We want to be able to select occurrences of literals in the left hand sides of the rules of a logic program. This justifies the definition that follows.

Definition 4. Let a logic program P = ((ϕ +

℘ , ϕ - ℘ )) ℘∈Prd(V) be given. An occurrence marker for P is a sequence of the form

((O + ℘ , O - ℘ )) ℘∈Prd(V)
where for all members ℘ of Prd(V), O + ℘ and O - ℘ are sets of occurrences of literals in ϕ + ℘ and ϕ - ℘ , respectively.

What we need is to be able to replace some occurrences of literals in some formulas by some other formulas. Given a formula ϕ and a partial function ρ from the set of occurrences of literals in ϕ to L ωω (V), we denote by ϕ[ρ] the result of applying ρ to ϕ. For instance, if ρ is the function that maps the first occurrence of p in ϕ = p ∧ (q ∨ p) to r ∧ s, then ϕ[ρ] = (r ∧ s) ∧ (q ∨ p).

Independent and social agents

Let a logic program P and an occurrence marker Ω for P be given. Suppose that V contains n predicate symbols for some nonzero n ∈ N, so there are 2n rules in P, n positive rules and n negative rules, say R 0 , . . . R 2n-1 . Imagine that for all m < 2n, R m is 'under the responsability' of some agent A m (a single agent might be responsible for many rules in P, possibly all of them). Let m < 2n be given. Some occurrences of literals in R m might be marked by Ω. Intuitively, these are the occurrences of literals that A m 'does not bother to' or 'is not able to' directly deal with: a marked occurrence of literal in R m is assumed by A m to be false unless A m is told otherwise (expectedly by another agent, but possibly by himself. . . ), for instance because those literals are not under A m 's responsibility-they are instances of rules whose right hand side are under the responsibility of other agents. The definitions that follow formalize these ideas.

Definition 5. Let a formula ϕ, a set O of occurrences of literals in ϕ, and a set E of literals be given. Let ρ be the function from

O into L ωω (V) such that for all o ∈ O, n ∈ N, ℘ ∈ Prd(V, n) and terms t 1 , . . . , t n , 3 ρ(o) = { 1≤i≤n t i = t i | ℘(t 1 , . . . , t n ) ∈ E } if ℘(t 1 , . . . , t n ) ∈ o, { 1≤i≤n t i = t i | ¬℘(t 1 , . . . , t n ) ∈ E } if ¬℘(t 1 , . . . , t n ) ∈ o.
We let O E ϕ denote ϕ[ρ]. Definition 6. Let a logic program P = ((ϕ + ℘ , ϕ - ℘ )) ℘∈Prd(V) , a possible game T , and an occurrence marker Ω = ((O + ℘ , O - ℘ )) ℘∈Prd(V) for P be given. We inductively define a family ([ P, T, Ω ] α ) α∈Ord of sets of closed literals as follows. For all ordinals α, [ P, T, Ω ] α is the ⊆-minimal set of literals that contains T and such that for all n ∈ N, ℘ ∈ Prd(V, n) and closed terms t 1 , . . . , t n ,

-℘(t1, . . . , tn) ∈ [ P, T, Ω ]α iff [ P, T, Ω ]α O + ℘ S β<α [ P, T, Ω ] β ϕ + ℘ [t1/v1, . . . , tn/vn]; -¬℘(t1, . . . , tn) ∈ [ P, T, Ω ]α iff [ P, T, Ω ]α O - ℘ S β<α [ P, T, Ω ] β ϕ - ℘ [t1/v1, . . . , tn/vn].
We set

[ P, T, Ω ] = α∈Ord [ P, T, Ω ] α .
The independent agent does everything by herself; she does not rely on anyone. If we assume that she works 'nonstop' then her behavior is captured by the empty occurrence marker.

Definition 7. Let a logic program P = ((ϕ +

℘ , ϕ - ℘ )) ℘∈Prd(V) and a possible game T be given. Let Ω = ((O + ℘ , O - ℘ )) ℘∈Prd(V) be the occurrence marker for P such that for all ℘ ∈ Prd(V), O + ℘ and O - ℘ are empty. We write [ P, T ] for [ P, T, Ω ]. The next result shows that social agents, irrespective of how their responsibility has been defined, discover the same information, no less, not more, as the independent agent.

Theoretical and empirical agents

In the previous section, we have allowed agents to interact transfinitely many times: in [ P, T, Ω ] α , we allow α to be an infinite ordinal. In this section, we tackle the following issue: is it possible to derive all derivable information in finite time, irrespective of how social agents share their work, or of how single agents organize their work? Obviously, this requires a way of 'working' different to what the concepts that have been defined so far accept. In this section, we will allow agents to make hypotheses. If an agent can assume that some literals in the bodies of some rules are true, she might be able to speed up the derivations she can perform. Such hypotheses should abide stringent conditions. We suggest that a hypothesis should eventually either be confirmed, that is, proved correct, or refuted, that is, proved wrong. Let us first precisely define what 'making a hypothesis' means. A pleasant feature of this notion is that it is again based on the notion of occurrence marker. This time, we use occurrence markers to select some occurrences of literals on the left hand side of some rules to make them the targets of some hypotheses. Definition 8. Let a formula ϕ, a set O of occurrences of literals in ϕ, and a set E of literals be given. Let ρ be the function from O into the set of formulas such that for all o ∈ O, n ∈ N, ℘ ∈ Prd(V, n) and terms t 1 , . . . ,

t n , ρ(o) is equal to { ℘(t 1 , . . . , t n ), n i=1 t i = t i | ℘(t 1 , . . . , t n ) ∈ E } if ℘(t 1 , . . . , t n ) ∈ o, and to { ¬℘(t 1 , . . . , t n ), n i=1 t i = t i | ¬℘(t 1 , . . . , t n ) ∈ E } if ¬℘(t 1 , . . . , t n ) ∈ o. We let O E ϕ denote ϕ[ρ].
An agent willing to assume that the literals in E are true provided that they occur on the left hand side of the rules of a logic program P, as selected by the occurrence marker Ω for P, essentially decides to work on the basis of the logic program P + Ω E introduced in the definition that follows.

Definition 9. Let a logic program P and an occurrence marker Ω for P be given. Write P = ((ϕ

+ ℘ , ϕ - ℘ )) ℘∈Prd(V) and Ω = ((O + ℘ , O - ℘ )) ℘∈Prd(V)
. Given a set E of literals, the sequence ((

O + ℘ E ϕ + ℘ , O - ℘ E ϕ - ℘ )) ℘∈Prd(V) is denoted P + Ω E.
Our aim is to show that making hypotheses can pay off. Definition 10. A logic program P = ((ϕ + ℘ , ϕ - ℘ )) ℘∈Prd(V) is acceptable iff the following holds. Let V * be V without the observational predicate symbols.

-For all possible games T , [ P, T ] is a complete set of literals.

-The restriction of P to V * is symmetric.

-For all ℘ ∈ Prd(V),

• if ℘ is observational then both ϕ + ℘ and ϕ - ℘ are equal to ∅, • either ℘ is nullary or no quantifier occurs in ϕ + ℘ , and • all quantified formulas that occur in ϕ + ℘ have one quantifier only. Here is an example of part of an acceptable logic program. The proposition that follows shows that it is possible to enrich V into a vocabulary V , transform P into a logic program P over V , and make some assumptions such that all possible games T , all members of [ P, T ] can be derived after a finite number of steps. Moreover, P is such that it is safe to make any set of assumptions; indeed, any set of assumptions that is inconsistent with P and a possible game will proved inconsistent after finitely many inferences. Proposition 2. Let V * be V without the observational predicate symbols. For all acceptable logic programs P, there exists a finite set E of nullary predicate symbols that do not belong to V and there exists a logic program P over V ∪ E whose restriction to V * ∪ E is symmetric such that for all possible games T , there exists an occurrence marker Ω for P with the following properties. for all possible games T and for all occurrence markers Ω and Ω for P , if

[ P + Ω E, T ] = [ P , T ] then n∈N [ P + Ω E, T, Ω ] n is inconsistent.
The transformation of P to P amounts to replacing some complex formulas in the bodies of some rules of P by some new nullary predicate symbols, themselves defined thanks to a new pair of rules-a form of predicate invention-that can play the role of hypotheses and enjoy a refutation property. With the previous example of acceptable logic program, E could consist of two nullary predicate symbols, say p and q, and P could be defined as

∀v1 ((hearts(v1) ∨ diamonds(v1)) → red(v1)) ∀v1 ((spades(v1) ∨ clubs(v1)) → black(v1)) ∀v0 (red(v0) ↔ black(s(v0))) → p p → alternatedColors ∀v0 red(v0) → q (q ∨ ∃v0 (queen(v0) ∧ clubs(v0))) → allRedsOrAQofC
An agent would then have four options, depending on whether she would assume p or q in the bodies of the 4th and 6th rules, respectively. For any possible game T , one of these options would be appropriate and allow the agent to discover whether T is a sequence of cards where black and red alternate, or whether T is a sequence consisting of nothing but red cards, unless it contains a queen of clubs. Any wrong set of hypotheses would be guaranteed to be eventually refuted in the limit on the basis of a finite subset of T .

Conclusion

We have presented a framework where fundamental questions about the nature of scientific discovery can be formulated and studied. The basic working hypothesis is that a purely logical approach to scientific discovery and problem solving is possible, in a way that can shed light on the nature of those activities. We believe that our approach can address a whole range of questions related to the nature of scientific discovery or problem solving, always within the boundaries of a pure logical setting. For instance, Angluin proposes a binary categorization of agents, with learners and teachers, and she proves robustness results about their interaction; how does this categorization translate into our setting? Starting from a fixed language, we have to a certain extent accounted for predicate invention in the last proposition, allowing agents to make a rational use of hypotheses expressed in an extension of the original language, but how does predicate invention relate to postulates? Surely, logic is not an iron collar, but it can potentially strive far beyond the territories where it has been confined to.

Proposition 1 .

 1 For all logic programs P, possible games T and occurrence markers Ω for P, [ P, T, Ω ] = [ P, T ].

  ∀v1 ((hearts(v1) ∨ diamonds(v1)) → red(v1)) ∀v1 ((spades(v1) ∨ clubs(v1)) → black(v1)) ∀v0 (red(v0) ↔ black(s(v0))) → alternatedColors (∀v0 red(v0) ∨ ∃v0 (queen(v0) ∧ clubs(v0))) → allRedsOrAQofC

-

  [ P, T ] and the restrictions of [ P , T ] and [ P + Ω E, T ] to V are equal; -for all occurrence markers Ω for P , [ P +Ω E, T ] = S n∈N [ P +Ω E, T, Ω ]n;

In case n = 0, the replacing expression is W˘V ∅ ¯if ℘ ∈ E, andW ∅ if ℘ / ∈ E.Note thatW˘V ∅ ¯is logically equivalent to V ∅.


Marc Louis

Maurice François
email: francois@lmt.ens-cachan.fr

Marc Franc

Behavior of cracked materials

Keywords: Rough fracture, concrete, damage, quasi-brittle material

Due to their microstructure, quasi brittle materials present rough cracks. Under sliding of the crack lips, this roughness involves in one hand induced opening and in the other hand some apparent plasticity which is due to the interlocking of the crack lips combined with Coulomb's friction. The proposed model is written under the irreversible thermodynamics framework. Micromechanics uses the Del Piero and Owen's structured deformation theory. Opening of the crack depends upon the crack shape and the relative sliding of the crack lips. The thermodynamic force associated to the sliding has the mechanical meaning of the force acting in order to make the crack slide. Yield surface is defined as a limitation of this force with respect to the Coulomb's friction and the Barenblatt cohesion. The crack orientation is defined as the one for which the criterion is reached for the lowest stress level. A decreasing cohesion, respect to sliding is supposed. Tension and compression reference cases are envisaged.

Introduction

This work is in continuity with [START_REF] ¸ois | Structured deformation of damaged continua with cohesive-frictional sliding rough fractures[END_REF], in which the problem of sliding rough fracture is treated is the case of compressive and shearing testing, relevant to the behavior of civil engineering structure response under earthquake. The goal was to extend this approach to cases of tension and compression, predict the crack's orientation and the global response of the specimen.

Micromechanics

In case of periodic fracture, a micro-mechanical structured deformation analysis [START_REF] Del | Structured deformations of continua[END_REF][START_REF] ¸ois | Structured deformation of damaged continua with cohesive-frictional sliding rough fractures[END_REF] [START_REF] Del | Structured deformations of continua[END_REF]. The sliding of the crack is defined by a variable s and the opening respect to the function θ(s).

Considering the orthonormal basis

[e 1 , e 2 ]
, where e 2 defines the normal to the (average) crack plane and e 1 the sliding direction.

ε = ε e + ε s
(1)

ε s = θ(s) (e 1 ⊗ e 1 ) + sθ(s) 1 2 (e 1 ⊗ e 2 + e 2 ⊗ e 1 ) (2)
The micro-mechanical point of view is sketched in the Fig. 1. The sliding s induces the opening of the crack, respect to the opening function θ(s). This one depends upon the crack's shape aω (a is a physical length). In case of sinusoidal crack shape, naming ξ the (metric) abscissa of the crack along e 2 , A and p the mathematical amplitude and periodicity, we have (a vanishes in θ):

ω ξ a = -A cos 2π ξ ap , θ(s) = A sin πs p (3)

Thermodynamics: state

The free energy ρΨ is related to the elastic energy stored by the sound material (with C the Hook stiffness tensor). The derivative of it respect to state variables lead to the thermodynamic forces: the stress σ and the force S (Eq. 5), that has the clear meaning of the projection of the stress σ that have the will make the crack slide.

2ρΨ(ε, s) = (ε -ε s ) : C : (ε -ε s ) (4) σ = C : ε, S = θ ′ (s)σ 11 + σ 12 (5)
Elasticity domain for a crack The yield surface f (σ, S) is defined (in the thermodynamic forces space) as a limitation of the force S respect to two phenomena: one cohesive and the other frictional. The fist, is a Barenblatt adhesive term B 0 0 that represents cohesion, i.e. when crack does not exist; this cohesion can be seen as the cement paste role in a concrete. The second member introduces the term ϕ 0 that can be seen as a Coulomb's friction (at the macro scale level).

B -B B ϕ Bθ' ϕ σ 11 σ 12 -ϕ-θ' ϕ-θ' B>0 0 B=0 σ 12 σ 11 0 -ϕ-θ' ϕ-θ' s≠0 s=0 s=0 s≠0
f (σ, S) = |S| -B 0 + ϕσ 11 (6)
The elasticity domain is then defined as f < 0 and the dissipative evolution (siding of the fracture) occurs when f = 0. Using Eq. ( 5), the combination of them, f 0, define the yield function (Eq. 7) and the admissibility condition (Eq. 8), in the e i coordinates:

|θ ′ (s)σ 11 + σ 12 | B 0 -ϕσ 11 (7) σ 11 B 0 ϕ (8)
The elasticity domain is represented onto Fig. 2. Non admissible domain is hatched. It can be seen as the case when tension becomes large enough in order to separate crack's faces. When B 0 = 0, the yield surface tends to a Mohr-Coulomb's one, inclined respect to the slope θ ′ (s).

Due to mathematical properties of the crack shape (Eq. 3), the opening function θ(s) is even and the maximum M = max(θ ′ (s)) is obtained for s = kp, then for s = 0. Then the derivative at s = 0 belongs to sub-differential defined as θ ′ (0) ∈ [-M, M]. The corresponding yield surface is non convex (Fig. 2); it can be seen as the collection of all the possible yield surface for any s.

Thermodynamics: dissipation

For shake of simplicity we retain an associated model: the potential of dissipation is the elasticity domain. The flow rule is deduced from the normality rule (in which λ 0) i.e. ṡ = df /dS λ = sign(S) λ. Then comes that S and ṡ have the same sign, proving that the thermal dissipation Eq. 9 is positive under the admissibility condition (Eq. 8).

Ḋ = S ṡ = (B 0 -ϕσ 11 )| ṡ| 0 (9)

Yield stress in the tension and compression case

The stress is σ = σ n ⊗ n; with n = cos(α) e 1 + sin(α) e 2 and (with no restriction) 0 α π/2. In the specimen's orthonormal basis [ n, m], the axial strain writes (from 1, 2):

ε nn = σ E + θ(s) cos 2 (α) + s sin(α) cos(α) (10)
Following in some way [START_REF] Francfort | Une approche variationnelle de la mécanique du défaut[END_REF], we consider that all the cracks are possible in the material; the one that will open first will be the one whose orientation α leads to the highest value of the yield function. The condition of yielding f = 0 (Eq. 7) writes:

|σ||θ ′ (s) cos 2 (α) + sin(α) cos(α)| = B 0 -ϕσ cos 2 (α) (11)
The positiveness of the second member represents the admissibility condition (Eq. 8); it is always true in compression but represents a limit in tension. It can be shown that (Eq. 9) yields ṡ 0, in tension and ṡ 0 in compression. Then, the crack initializes while considering θ ′ (0 + ) = M in tension and θ ′ (0 -) = -M in compression. The crack angle and the stress value is obtained in the two cases while minimizing the (absolute) stress level and we obtain (with superscript t for tension and c for compression:

σ t = 2B 0 ϕ + M + 1 + (ϕ + M) 2 tan(2α t ) = 1 ϕ + M (12)
σ c = 2B 0 ϕ + M -1 + (ϕ + M) 2 tan(2α c ) = -1 ϕ + M (13)
The ratio -σ c /σ t appears to depend only upon the value of ϕ + M. Considering a Coulomb's friction ϕ = 0.5 and M = 1 (approximatively relevant of measured roughness [START_REF] Schmittbuhl | Development of roughness in crack propagation[END_REF]), this ratio is 10.9, typical for concrete [START_REF] Kupfer | Behavior of concrete under biaxial stresses[END_REF]. The angle of cracks are respectively α t =17 and α c =73 degrees for tension and compression (Fig. 4) and these value are reasonable respect to experiments.

Evolution

The post-peak response can then be obtained from (Eq. 11), as angles α are set. The Barenblatt constant B is supposed to depend upon the sliding s, in order to represent the damage of the material. Using a linear dependancy, we suppose that no more cohesion remains after an half step:

B(s) = B 0 1 - 2 max t |s| p + (14)
The obtained stress to strain curve (for p = 5 10 -3 , E = 70 GPa, B 0 = 6 GPa, Fig. 3) exhibits softening in both tension and compression. The zero stress state is obtained for a larger strain in tension than in compression. These two facts lead to consider that the proposed model is only relevant to yield stress but not yet to describe anelastic behavior.

Conclusion

This approach seems to be able to predict yield stress and crack angle in tension and compression. An extension in case of bi-tension or compression will be done. The anelastic behavior needs to be enhanced in order to be predictive. Softening behavior will have to be considered with respect to some material scale in order to avoid localization problems (mesh dependancy); this micro-scale being already present in the model allow to think that it is possible.

Figure 1 :

 1 Figure 1: sketch of the micromechanics

Figure 2 :

 2 Figure 2: Yield surface (active crack), for any s and for s = 0 (light gray)

Figure 3 :

 3 Figure 3: Tension to compression curve Figure 4: Crack kinematics


Martin Quinson -Nancy
email: martin.quinson@loria.fr

SimGrid: a Generic Framework for Large-Scale Distributed Experiments

Introduction

Distributed computing has become mainstream and arises today in a wide variety of domains for a wide range of applications. These include parallel scientific simulations running on clusters, large-scale "grid" applications running on platforms aggregating high-end resources across institutions, scientific applications running on "desktop grids" that harness the idle cycles of individually owned computers, multi-tier business applications based on Web services, and peer-to-peer applications for content dissemination and sharing. The number of such applications in production is rapidly growing. Alongside these development activities, distributed computing is an extremely active and broad research area covering distributed algorithms and protocols, distributed resource and content management, and resource and application scheduling. Many challenges remain for improving distributed applications (better performance, better resilience to faults, better scalability, etc.) and new challenges constantly arise due to the joint evolution of technology and of applications.

A key issue in distributed computing is to scientifically assess the quality of several solutions with respect to a particular metric (task throughput achieved by a scheduling heuristic, probability of availability of a service, response time of lookup queries, etc). In some (rare) cases this assessment can be addressed solely via theoretical analysis. Unfortunately, in most cases assessment can at best be obtained for very stringent and ultimately unrealistic assumptions regarding the underlying platform and/or the application. Therefore, most research results in these areas are obtained via empirical evaluation through experiments.

The use of real-world platforms proves time-and laborintensive while not allowing reproducibility. Emulators, although attractive, often prove too cumbersome and are rarely used by researchers in our area. Packet-level simulators from the networking community prove poorly adapted to our purpose. Consequently, the majority of published results are obtained in simulation.

Simulation has been used extensively in several areas of computer science for decades, e.g., for microprocessor design and network protocol design. In these areas, several widely used and acknowledged simulation frameworks are available. By comparison, the use of sound simulation frameworks for distributed applications on distributed computing platforms is not as developed, certainly without any standard simulation tool (although network simulation is a key component of distributed application simulation). An inordinate number of in-house, shortlived and highly specialized simulation tools have been developed to fulfill the need of the community. Expectedly, these tools are difficult to use by others in other contexts than the ones for which they were intended. Moreover, tool proliferation hinders the comparison of (often unreproducible) published results.

In this talk we describe a comprehensive simulation framework, SimGrid, for the simulation of distributed applications on distributed platforms. Our goal is to describe the salient capabilities of SimGrid and explain how they allow users to perform simulations for a wide range of applications and platforms. Some of the key features of SimGrid are: (1) A scalable and extensible simulation engine that implements several validated simulation models, and that makes it possible to simulate arbitrary network topologies, dynamic compute and network resource availabilities, as well as resource failures; (2) High-level user interfaces for distributed computing researchers to quickly prototype simulations either in C or in Java; (3) APIs for distributed computing developers to develop distributed applications that can seamlessly run in "simulation mode" or in "real-world mode."

This paper is organized as follows. Section 2 provides an overview of SimGrid, while Section 3 provides some insights on quality of SimGrid-based simulations, in term of speed, scalability and accuracy. Finally, Section 4 concludes with perspectives on future work.

The SimGrid Framework

The SimGrid project was initiated in 1999 to allow the study of scheduling algorithms for heterogeneous platforms. Ten years after, it has evolved to a generic and modular framework allowing the study of almost any distributed applications.

SimGrid offers four user interfaces which can be sorted in two categories: SimDag and MSG are intended to the researchers wanting to study their algorithms while GRAS and MSG are intended to developpers wanting to improve their applications. The simulation kernel is modular, and several models are pluggable. It supports dynamic re-source availabilities and failures.

The SimDag API is designed for the investigation of scheduling heuristics for applications as task graphs. It allows to create tasks, add dependencies between tasks, retrieve information about the platform, schedule tasks for execution on resources, and compute the DAG execution time. The MSG interface allow the study of CSP applications. It is the most widely used API, perfectly usable for scheduling algorithms, P2P or desktop grids. Java bindings of this interface are provided allowing user reluctant to program in C to still use SimGrid. The GRAS system allows the development of distributed applications (e.g., a resource information service) within the simulator, but that are then seamlessly deployable on real-world platforms without code modification. To enable this, the same API is implemented twice (once for the simulated world, once for the real world), and user code simply has to be linked against the appropriate library. SMPI enables the direct simulation of unmodified MPI applications by intercepting MPI primitives.

SURF, the simulation engine, was designed with two main goals in mind. First, it must be highly modular to allow the implementation (and comparison) of several resource models. The default one implements a MaxMin sharing strategy for both the network and the CPU. Vegas and Reno models implement the corresponding TCP algorithms thanks to analytical models. A model for the execution of parallel tasks is also provided. New models can be plugged seamlessly by researchers. In addition, as it constitutes the basis of the whole SimGrid framework, SURF is carefully optimized so as not to hinder simulation speed.

XBT is a generic toolbox. SimIX is an internal module factorizing efforts to implement the abstraction of multiple concurrent processes on top of SURF.

Experimental Evaluation

Simulator Scalability and Speed The classical metric for a simulator's scalability is the number of simulated nodes/processes that are allowed to start. In our case, this is only a matter of available memory since nodes are not mapped into threads by default, but into UNIX98 contexts. This removes any limitation on the amount of threads from the kernel and has allowed us to run simulation with up to 2,000,000 simulated processors on a computer with 16Gb of memory.

We feel however that computation time is more representative of usability in practice. The network simulation speed is mesured by the speed to simulate a given amount of flows of a given size. Experiments show that SimGrid clearly outperform the GTNetS packet-level simulator in any scenario. The running time of GTNetS is linear in both the number of flows and their sizes while the running time of SimGrid is only linear in the number of flows. To measure the overall simulation speed, we compute the amount of task per second handled by a classical master/slaves scenario on a on a 2Ghz 32-bit Laptop with 1Gb of memory. Experiments show that the simulator delivers more than 10,000 tasks per second. This timings do not depend on the amount of processes in the scenario. The Java bindings deliver about 4,000 tasks per second, which is due to the cost of JNI.

Simulation Accuracy We mesure the simulation accuracy by comparing the timing computed by our simulator to the ones provided by the GTNetS packet-level simulator in several scenarios. For the sake of discussion it is assumed that GTNetS results are 100% correct and any discrepancy between the SimGrid results and the GTNetS results is considered SimGrid "error.". Overall, experiments presented show that: (1) Since SimGrid does not account for TCP's slow-start mechanism, its accuracy is poor for very short messages; (2) When a flow's bandwidth is limited by the flow's RTT, the SimGrid error is very low (below 1%); (3) When a flow's bandwidth is limited by the actual link bandwidth, there seems to be a fixed error of roughly 5%; (4) When network congestion rises, SimGrid leads to optimistic results (i.e., flows receive more bandwidths than they do with GTNetS). Amongst other causes, this comes from the fact that packet dropping by overloaded routers is not modeled in SimGrid. The error seems to be heavy-tailed: even in an highly contented network, we observe that for 50% of the flows the error is below 10%, while for 90% of them, the error remains below 50%. Some outliers however exhibit error up to 100%.

Conclusion

We presented the SimGrid simulation framework whose goal is to provide a generic evaluation tool for large-scale distributed computing. Its main components are: two APIs for researchers who study algorithm and need to prototype simulations quickly, and two for developers who can develop applications in the comfort of the simulated world before deploying them seamlessly in the real world. SimGrid employs a modular simulation kernel that supports the addition and use of new resource models without changes in the user code. We used this feature ourselves to implement several simulation models and even to integrate the GTNetS packet-level simulator.

SimGrid is freely available (LGPL license) from its Web page1 and comes with several example programs and tutorials. SimGrid uses an extensive regression testing suite ensuring a reasonable level of software quality. It is ported to Linux, Windows, Mac OS X and AIX.

SimGrid is a very active project, both in terms of research and development, and we envision many directions for future work. We plan to further extend its scalability. A first solution of course would be to implement an efficient but simplistic network model that returns constant communication times. Another direction currently under investigation is the ability to dispatch simulated nodes over several physical machines to go beyond the memory limit of a single computer.

http://simgrid.gforge.inria.fr/


Julien Chenal

Lie Algebras

The object of this note is to define the generalized flag geometry of a graded Lie algebra which corresponds to the generalized projective geometry in the case of 3-gradings. Then we construct a structure of manifold on this generalized flag geometry. This result generalizes a result known for 3-graded Lie algebras to the more general case of (2k + 1)-graded Lie algebras.

Résumé

Géométries de drapeaux généralisées et variétés associées aux algèbres de Lie graduées en dimension quelconque.

L'objet de cette note est de définir la géométrie de drapeaux généralisée d'une algèbre de Lie graduée, qui correspond à la géométrie projective généralisée dans le cas des 3-graduations, puis de construire une structure de variété différentielle sur cette géométrie. Ce résultat généralise au cas des (2k + 1)-graduations un résultat déjà connu pour les 3-graduations.

Version française abrégée

Dans une première partie, nous considérons le groupe projectif élémentaire G := P E(g) ⊂ Aut(g) d'une algèbre de Lie graduée g = g k ⊕ • • • ⊕ g 0 ⊕ • • • ⊕ g -k . C'est le groupe engendré par les deux groupes U + := e ad(x) , x ∈ k i=1 g i et U -:= e ad(x) , x ∈ k i=1 g -i (cf. [START_REF] Loos | Elementary Groups and Stability for Jordan Pairs[END_REF], [START_REF] Bertram | Projective Completions of Jordan Pairs. Part I: The generalized projective Geometry of a Lie Algebra[END_REF]) ; puis on considère trois espaces homogènes : X + := P E(g)/P -, X -:= P E(g)/P + , où pour σ ∈ {+, -}, P σ := U σ H avec H le sousgroupe des éléments du groupe projectif élémentaire qui préservent la graduation, et M := P E(g)/H.

Email address: chenal@iecn.u-nancy.fr (Chenal, Julien).

Alors M s'injecte dans X + × X -et plus précisément, M est isomorphe à l'orbite du point de base (P -, P + ) dans X + × X -sous l'action du groupe projectif élémentaire. Enfin, sur X + × X -, on définit une relation de transversalité : le couple (x, y) où x ∈ X + et y ∈ X -, est dit transverse si (x, y) ∈ M . Alors (X + , X -), avec la relation de transversalité forme ce que nous appellons la géométrie de drapeaux généralisée associée à la graduation de g.

Dans une deuxième partie, nous utilisons les filtrations de g, i.e les drapeaux de sous-espaces de g de la forme 0

= n k+1 ⊂ n k ⊂ n k-1 ⊂ • • • ⊂ n 0 ⊂ n -1 ⊂ • • • ⊂ n -k+1 ⊂ n -k = g avec [n i , n j ] ⊂ n i+j ,
pour obtenir une réalisation géométrique des espaces X + et X -. On obtient X + et X -comme les orbites sous l'action du groupe projectif élémentaire des filtrations associées à la graduation de g :

X + ∼ = G.n -et X -∼ = G.n + où n + := (g k ⊂ g k ⊕ g k-1 ⊂ • • • ⊂ g k ⊕ • • • ⊕ g 0 ⊂ • • • ⊂ g k ⊕ • • • ⊕ g -k+1 ) et n -:= (g -k ⊂ g -k ⊕ g -k+1 ⊂ • • • ⊂ g -k ⊕ • • • ⊕ g 0 ⊂ • • • ⊂ g -k ⊕ • • • ⊕ g k-1 ).
Enfin, dans une dernière partie, nous utilisons la notion de calcul différentiel sur un corps topologique non discret (cf. [START_REF] Bertram | Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings[END_REF]) pour construire une structure de variété différentielle sur X + . Cette construction généralise celle de [START_REF] Bertram | Projective Completions of Jordan Pairs. Part II: Manifold Structure and symetric Spaces[END_REF] pour les algèbres de Lie 3-graduées. Pour cela, on suppose que n + 1 et n - 1 sont des algèbres de Lie topologiques et nous définissons les opérateurs de Bergman généralisés, qui généralisent la notion d'opérateur de Bergman dans une paire de Jordan (cf. [START_REF] Loos | Jordan Pairs[END_REF]) :

pour x ∈ n + 1 , w ∈ n - 1 , et 1 ≤ j ≤ k, on pose : B + (x, w) j = pr n + j • (e -ad(x) e -ad(w) ) • ι n + j et B -(w, x) j = pr n - j
• (e ad(w) e ad(x) ) • ι n - j où pr n σ j est la projection de g sur n σ j et ι n σ j l'inclusion de n σ j dans g, pour σ ∈ {+, -}. Alors, nous montrons que sous des hypothèses (H1) et (H2) formulées ci-dessous, il existe sur X + une structure de variété différentielle (Theorem 3.2).

Definition of generalized flag geometries

In the sequel, σ ∈ {+, -} and we assume that K is a commutative ring in which the integers are invertible.

Z-graded Lie algebra (over K) is a Lie algebra over K of the form g = n∈Z g n , with [g i , g j ] ⊂ g i+j .

A (2k + 1)-graded Lie algebra is a Z-graded Lie algebra g = n∈Z g n such that g j = 0 if |j| > k.

Let g be a (2k + 1)-graded Lie algebra. The map D : g → g defined by DX = iX if X ∈ g i is a derivation of g called the characteristic derivation of the grading. If D = ad(E), E ∈ g, D is said an inner derivation and E is called an Euler operator. Finite dimensional real simple graded Lie algebras have been classified in [START_REF] Kaneyuki | Graded Lie Algebras and Generalized Jordan Triple Systems[END_REF].

For x ∈ V + := k i=1 g i , or x ∈ V -:= k i=1
g -i , we define the operator

e ad(x) := 2k j=0 1 j! ad(x) j .
By our assumption on K, e ad(x) is well-defined and is an automorphism of g. We define the groups generated by these operators, and the group generated by U + and U -:

Definition 1.1
The group G := P E(g, D) :=< U + , U ->⊂ Aut(g) is called the projective elementary group of (g, D) (cf. [START_REF] Loos | Elementary Groups and Stability for Jordan Pairs[END_REF] and [START_REF] Bertram | Projective Completions of Jordan Pairs. Part I: The generalized projective Geometry of a Lie Algebra[END_REF]).

We define also the automorphism group of (g, D) by

Aut(g, D) := {g ∈ Aut(g), g • D = D • g}
and the subgroups

H := G(D) ∩ Aut(g, D) and P σ := HU σ = U σ H.
The elements of H commute with the derivation D, so they preserve the grading, and normalize U σ so P + and P -are subgroups of G. Finally, we define homogeneous spaces:

X + := G/P -, X -:= G/P + and M := G/H
We note (o + , o -) the base point (P -, P + ) of X + × X -.

Definition 1.2 On X + × X -, we define the relation of transversality: for x ∈ X + and y ∈ X -, x and y are said transversal if and only if (x, y) are conjugate under G to the base point. If x and y are transversal, we will write x⊤y. We define also

(X + ×X -) ⊤ := {(x, y) ∈ X + ×X -, x⊤y} and for x ∈ X σ , x ⊤ := {y ∈ X -σ , x⊤y}.
The data (X + , X -, ⊤) is called a generalized flag geometry of k-graded type.

Then the relation of transversality can be defined using the space M thanks to the following:

Lemma 1.3 i) If x = g • o + ∈ X + and y = g ′ • o -∈ X -, then
x and y are transversal if and only if there exists z ∈ V + such that x = g ′ e ad(z) • o + . ii) We have

P + ∩ P -= H. Hence V + ⊂ X + via x → e ad(x) • o + and (o -) ⊤ ∼ = V + .
iii) The space M is isomorphic to (X + × X -) ⊤ , i.e H is the stabilizer of the base point (o + , o -).

Geometric realization and filtrations of a Lie algebra

In this section, we will give a geometric realization of the homogeneous spaces X + and X -together with the transversality relation. Let g be a Lie algebra over K. A (2k + 1)-filtration of g is a flag of subspaces

0 = n k+1 ⊂ n k ⊂ n k-1 ⊂ • • • ⊂ n 0 ⊂ n -1 ⊂ • • • ⊂ n -k+1 ⊂ n -k = g with [n i , n j ] ⊂ n i+j .
Such a flag will be denoted by n = (n k ⊂ • • • ⊂ n -k+1 ), and the space of (2k + 1)-filtrations of g will be denoted by F . If n ∈ F, then ad(X) with X ∈ n 1 is nilpotent and hence the automorphism e ad(X) is well-defined. We denote by

U (n) := e ad(n1) = {e ad(X) , X ∈ n 1 } ⊂ Aut(g) the corresponding group. Since [n i , n j ] ⊂ n i+j , the group U (n) preserves the filtration n. To any (2k + 1)-grading, g = g k ⊕ • • • ⊕ g 0 ⊕ • • • ⊕ g -k of g,
with characteristic derivation D, we may associate two (2k + 1)-filtrations of g, called the associated plus-and minus-filtration, defined by

n + (D) := (g k ⊂ g k ⊕ g k-1 ⊂ • • • ⊂ g k ⊕ • • • ⊕ g 0 ⊂ • • • ⊂ g k ⊕ • • • ⊕ g -k+1 ) and n -(D) := (g -k ⊂ g -k ⊕ g -k+1 ⊂ • • • ⊂ g -k ⊕ • • • ⊕ g 0 ⊂ • • • ⊂ g -k ⊕ • • • ⊕ g k-1 ).
A (2k + 1)-filtration n of g is said inner if n = n + (ad(E)), E ∈ g. The space of inner (2k + 1)-filtrations of g will be denoted by F. Definition 2.1 Let (m, n) be two (2k + 1)-filtrations of g. We say that m and n are transversal if, for -k + 1 ≤ i ≤ k, we have g = n i ⊕ m -i+1 . If m and n are transversal, we will write m⊤n.

By construction, the filtrations associated to a derivation D, n + (D) and n -(D) are transversal. Conversely, we will prove that any pair of transversal filtrations arises in this way. In this case, X σ ∼ = G • n -σ , and the two notions of transversality (Definitions 1.2 and 2.1) are compatible: Theorem 2.2 There is an embedding X + × X -֒→ F × F which is G-equivariant and compatible with transversality. More precisely, i) the orbit of the filtration n σ under the action of G is isomorphic to X -σ i.e P σ is the stabilizer of the filtration n σ :

P σ = {g ∈ G, g.n σ = n σ }. Hence H = {g ∈ G, g.(n + , n -) = (n + , n -)}.
ii) Two inner filtrations are transversal if and only if they come from a grading of g. iii) Let n ∈ F. Then e ad(n1) acts simply transitively on n ⊤ := {m ∈ F, m⊤n}. iv) We have an injection n + 1 → X + , x → e ad(x) • o + . Then the map

n + 1 × H × n - 1 → Ω + , (v, h, w) → e ad(v) he ad(w) is a bijection, where Ω + := {g ∈ G, g.o + ∈ n + 1 }. Proof.
Here, we will only prove ii) which is the key property to prove the other points. First, we prove that for E ∈ g and n = n + (ad(E)), we have e ad(n1

) E = E + n 1 .
Clearly, e ad(n1) E ⊂ E + n 1 . In order to prove the other inclusion, let X ∈ n 1 . We will prove by induction that for all n ∈ {3, . . . , k +1}, there exists

Y n ∈ n 1 such that e ad(Y n ) E = E + X + R n where R n ∈ n n . If n = k + 1, then n k+1 = 0 so R k+1 = 0 and e ad(Y k+1 ) E = E + X. Since X ∈ n 1 , we have X = k i=1 X i with X i ∈ g i . Let Y 3 := k i=1 - 1 i X i .
Then e ad(Y

3 ) E = E + X + R 3 , where R 3 := k j=2 1 j! ad(Y 3 ) j E ∈ n 3 .
Now, let n ∈ {3, . . . , k}. We assume that there exists

Y n ∈ n 1 such that e ad(Y n ) E = E + X + R n with R n ∈ n n . Since R n ∈ n n , we have R n = k i=n R n i with R n i ∈ g i . Let Y n+1 := Y n + 1 n R n n ∈ n 1 .
Then we have e ad(Y n+1

) E = E + [Y n , E] + 1 n [R n n , E] + k j=2 1 j! ad(Y n + 1 n R n n ) j E. So e ad(Y n+1 ) E = e ad(Y n ) E -R n n + R n . Since the first term in R n is Y n , 1 n R n n , E = -[Y n , R n n ] ∈ n n+1 , we have R n ∈ n n+1 . Hence e ad(Y n+1 ) E = E + X + k i=n+1 R n i + R n = E + X + R n+1 with R n+1 := k i=n+1 R n i + R n ∈ n n+1 . Finally, there exists Y := Y k+1 ∈ n 1 such that e ad(Y ) E = E + X. So E + n 1 ⊂
e ad(n1) E and then e ad(n1) E = E + n 1 . Now let us prove ii). We have already remarked that n + (ad(E)) and n -(ad(E)) are transversal. In order to prove the converse, let (m, n) be two transversal inner filtrations. Since n is inner, there exists

E ′ ∈ g such that n = n + (ad(E ′ )). Let g = g k ⊕ • • • ⊕ g 0 ⊕ • • • ⊕ g -k
be the (2k + 1)-grading of g associated to ad(E ′ ). As m⊤n, we have g = n 1 ⊕ m 0 , so there exists Z ′ ∈ n 1 such that E ′ -Z ′ ∈ m 0 . But there exists Z ∈ n 1 such that e ad(Z) E ′ = E ′ -Z ′ since e ad(n1) E ′ = E ′ + n 1 . Then let Let g be a (2k + 1)-graded Lie algebra and n + and n -the plus-and minus-filtration of g. For x ∈ n + 1 and g ∈ Aut(g), we define the denominators and co-denominators of g, for 1 ≤ j ≤ k by:

E := e ad(Z) E ′ = E ′ -Z ′ ∈ m 0 .
d g (x) j = pr n + j • (e -ad(x) g -1 ) • ι n + j ∈ End(n + j ) and c g (x) j = pr n - j • (ge ad(x) ) • ι n - j ∈ End(n - j )
where pr n σ j is the projection of g onto n σ j and ι n σ j is the inclusion of n σ j into g. Then, for all 1 ≤ j ≤ k, n + 1 → End(n + j ), x → d g (x) j and n + 1 → End(n - j ), x → c g (x) j are polynomial maps.

Definition 3.1 If x ∈ n + 1 and w ∈ n - 1 , let B + (x, w) j := d e ad(w) (x) j and B -(w, x) j := c e ad(w) (x) j , for 1 ≤ j ≤ k. These operators are called the generalized Bergman operators. These operators can be seen as a generalization of the Bergman operator in Jordan Pairs (cf. [START_REF] Loos | Jordan Pairs[END_REF] and [START_REF] Bertram | Projective Completions of Jordan Pairs. Part I: The generalized projective Geometry of a Lie Algebra[END_REF]). Now, we assume that K is a non discrete topological field and n + 1 and n - 1 are topological Lie algebras over K. Then under the following conditions on (n + 1 , n - 1 ), we can build on X + a structure of manifold modeled on n + 1 (cf. [START_REF] Bertram | Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings[END_REF] for the definition of differential calculus over a general field and [START_REF] Bertram | Projective Completions of Jordan Pairs. Part II: Manifold Structure and symetric Spaces[END_REF] for the case of a 3-graded Lie algebra). It should be remarked that we do not consider a topology on g 0 .

(H1) First, we assume that the maps when i + j = 0 and k + l + m = 0 are continuous.

g i × g j → g i+j

3 .

 3 Let us prove that n = n + (ad(E)) and m = n -(ad(E)). Indeed, since Z ∈ n 1 , we have n+ (ad(E)) = n + (ad(e ad(Z) E ′ )) = e ad(Z) n + (ad(E ′ )) = e ad(Z) n = n. So n = n + (ad(E)).the other hand, by construction E ∈ m 0 , so the filtration m is ad(E)-stable. Moreover, n = n + (ad(E)) is also ad(E)-stable. So g = n -k+1 ⊕ m k and this decomposition is ad(E)-stable. As n -k+1 = Ker(ad(E) -kId) ⊕ • • • ⊕ Ker(ad(E) + (k -1)Id), we have m k = Ker(ad(E) + kId) i.e m k = n -(ad(E)) k .With the same argument, it follows that m -i+1 = n -(ad(E)) -i+1 . So m = n -(ad(E)) and finally, n = n + (ad(E)) and m = n -(ad(E)).✷ Structure of manifold on X +

  (x, y) → [x, y] and g k × g l × g m → g k+l+m (x, y, z) → [[x, y], z]

(H2) In addition, we assume that the set denoted by (n + × n -) × and defined by

ii) For all g ∈ G and for all x ∈ n + 1 , the action of d g (x) on n + 1 is continuous. iii) There exists on X + a structure of a smooth manifold modeled on n + 1 , uniquely defined by the fact that A = (g(n + 1 ), ϕ g ), g ∈ G is an atlas of X + , where ϕ g :

The proof of this result will be given elsewhere. It relies on Theorem 2.2. Let us add the remark that there is reasonable topology neither on G nor on H. So our proof cannot use the classical arguments establishing a manifold structure on the quotient of a Lie group by some (closed) subgroup. Indeed, if g is real or complex, finite dimensional, and semisimple, it is enough to show that P + is closed to have a strucure of manifold on X + . But, in Theorem 3.2, there is no assumption on the dimension of g and K is not only R or C. Finally, concerning the condition (H2), if K is R or C and if n + 1 and n - 1 are Banach spaces (of finite dimension or not), then under the condition (H1), the generalized Bergman operators B + (x, w) 1 and B -(w, x) 1 to the Banach algebra L(n + 1 ) or L(n - 1 ) of the continuous linear operators on n + 1 or n - 1 and inversion in these spaces is continuous (cf. [START_REF] Upmeier | Symmetric Banach Manifolds and Jordan C*-algebras[END_REF]), so (H2) is automatically true.

am grateful to the referee for his helpful remarks.