F Lemarchand
email: francois.lemarchand@onera.fr

P Beauchêne

B Laine

F Chinesta

A multi-scale method to predict residual stress appearance in the process of on-line consolidation of thermoplastic composites

Keywords: Numerical simulation, Composites welding, Themomechanical model, Model reduction, Meshless methods, Multi-scale modeling, Homogenization

On-line consolidation process is a promising non-autoclave technique to design large size parts in aeronautical industry. Residual stress growth during the process is locking significantly the process validation. In this paper, we present a multi-scale modeling based on the constrained natural element method (C-NEM) and on the proper orthogonal decomposition method which allows to evaluate residual stress growth. The method allows to take into account the thermomechanical properties evolution of the material during the process, which is a dominating factor in residual stress development.

Introduction

Processing of thermoplastic composites involves residual stress development. Residual stresses can lead to distortion of finished components, matrix cracking, and inter-ply delamination. Tensile stresses may even reach to a significant fraction of the tensile strength. Therefore, to ensure reliable and satisfactory performance, residual stresses within a composite part should be known before the part is put into use. The tape placement process is one of the few techniques that have the potential to continuously process thermoplastic composites in large scale industrial production. In the process, an incoming composite tape is bonded to the previously laid and consolidated laminate under heat and pressure locally applied to the interface (figure 1). By laying additional layers in different directions, a part with desired thickness and properties can be fabricated. Since heat is mainly supplied to the region where the tape and the laminate meet, the process is highly non-isothermal and, residual stresses may become critical. The process poses a very challenging problem of determining residual stresses for the following reason the tape placement is a continuous process. As such, when a new layer is being placed, the previously laid and consolidated layers are again subjected to heating. If the temperature of these layers exceeds the glass transition temperature, annealing will ensue and the residual stresses developed during previous tape placement could relax. Most of the previous studies on the prediction of residual stresses developed during thermoplastic composite processing concentrated on press molding [START_REF] Chapman | Prediction of process-induced stresses in thermoplastic composites[END_REF][START_REF] Jeronimidis | Residual stresses in carbon fibre-thermoplastic matrix laminates[END_REF][START_REF] Manson | Prediction of residual stress in continuous fiber semicrystalline in thermoplastic composites : a kinetic viscoelastic approach[END_REF][START_REF] Wang | Thermoviscoelastic analysis of residual stresses and warpage in composite laminates[END_REF][START_REF] Unger | The effect of cooling rate and annealing on residual stress development in graphite fibre reinforced PEEK laminates[END_REF][START_REF] Lee | Optimal cool down in nonlinear thermoviscoelasticity with application to graphite/PEEK (APC-2) laminates[END_REF][START_REF] Li | A plane-strain finite element model for process-induced stresses in a graphite/PEEK composite[END_REF][START_REF] Wang | Thermoelastic behavior of PEEK thermoplastic composite during cooling from forming temperatures[END_REF][START_REF] Domb | The effect of cooling rate on free-edge stress development in semicrystalline thermoplastic laminates[END_REF][START_REF] Ersoy | Measurement of residual stresses in layered composites by compliance method[END_REF][START_REF] Ersoy | Prediction and measurement of residual stresses in layered composites[END_REF]. On the other hand, [START_REF] Nejhad | Prediction of process-induced stresses for in-situ thermoplastic filament winding of cylinders[END_REF] proposed a model to predict process induced stresses in filament winding. They modeled filament winding as a transient process. Since this led to excessive computational time, residual stresses could not be obtained within a reasonable time. Moreover, Residual stress analysis using a thermoviscoelastic finite element model was carried out by [START_REF] Sonmez | Analysis of process-induced residual stresses in tape placement[END_REF]. [START_REF] Sonmez | Analysis of process-induced residual stresses in tape placement[END_REF]

Figure 1. The thermoplastic composite tape placement process [from

Description of the occurring phenomena

During the welding, the changes of state of the matrix lead to the coexistence of three components (fiber, amorphous matrix and crystalline matrix) with different thermomechanical properties in a zone with high thermal gradients (see figure 2).

Figure 2. Schematic diagram of the on-line consolidation process and a modeling domain diagram representative of the phenomena occurring during the process

Unacceptable high levels of stress may arise because of two factors. First of all, thermal expansion behavior of fiber-reinforced thermoplastic composites is highly anisotropic. This is due to the large difference in the thermal expansion coefficients of the matrix and fiber materials. In a typical carbon-reinforced unidirectional thermoplastic composite, a temperature increase induces considerable expansion of the laminate in the transverse direction but almost none along the fiber direction. Besides, fibers, being themselves anisotropic, contribute to this effect. The other factor is the temperature gradients induced during the cooling process. This is because of the fact that, when different regions of the composite experience different temperature histories and stiffness properties of the material vary with the temperature, deformation behavior will be incompatible. Although elastic properties of fibers are almost unaffected by temperature, influence of temperature on the properties of the polymer matrix is drastic. Moreover, high processing temperatures typical for high performance thermoplastic composites exacerbate both the effect of temperature gradients and the effect of anisotropic thermal expansion behavior. Crystallization and melting phenomena occurring in the composite during process due to welding cause important evolution of the material properties. Indeed material thermomechanical properties depend on temperature and rate of crystallinity. These evolutions must be taken into account in the calculation, by using a multi-scale modeling.

Thermomechanical modeling

Introduction

We consider that the viscous relaxation times are greater than the characteristic process time. Therefore, a thermoelastic behavior can be considered for the simulations. Moreover, a plane strain 2D analysis will be performed. The process simulation at the macroscopic scale required homogenized thermal parameters at each point and time. On the other hand, a microscopic modeling will be required to define the homogenized thermomechanical parameters (to be employed in the macroscopic analysis) as well as to computed the microscopic thermomechanical fields evolution allowing to check the criteria of default appearance. Thus, both scale analysis must be performed simultaneously, the macroscopic one in the whole macroscopic domain, and the microscopic one in some representative domains, from which the solutions could be interpolated everywhere in the whole domain. Thus, the proposed strategy is not a purely homogenization technique, but a multi-scale analysis where the microscopic analysis also serves to computed the homogenized (and then macroscopic) thermomechanical parameters. This modeling results coarser that a fully microscopic analysis as the one performed in [START_REF] Ladeveze | A micro-macro and parallel computational strategy for highly heterogeneous structures[END_REF][START_REF] Ladeveze | On a multiscale computational strategy with time and space homogenization for structural mechanics[END_REF][START_REF] Farhat | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF] by using high performance computing (domain decomposition running in parallel computing platforms). On the other hand, it differs also of standard multi-scale homogenization techniques in which the microscopic domain only serves to compute homogenized parameters but not for defining the whole time evolution of microscopic fields. Moreover it is well known that these last techniques have serious difficulties to represent real boundary conditions. In the technique here proposed the microscopic domains are strongly connected with the macroscopic mesh, in fact they result by zooming some finite elements of the macroscopic mesh. Thus, the scales connection is very simple. Of course in the case of an eventual incompatibility between the micro and macro meshes, the use of the Arlequin approach (Ben [START_REF] Dhia | Problèmes mécaniques multi Echelles : la méthode Arlequin[END_REF] or any superposition method (S-FEM) [START_REF] Takano | Three-scale finite element analysis of heterogeneous media by asymptotic homogenization and mesh superposition methods[END_REF]) could be envisaged. It can be noticed that the particular choice of the microscopic domains allows for (i) the imposition of real boundary conditions on their boundaries ; (ii) compute the whole time evolution of the microscopic thermomechanical fields ; and (iii) compute the homogenized thermomechanical parameters related to the elements allocating the microscopic domains, from which these parameters can be interpolated everywhere in the macroscopic domain. The microscopic zones must be representative, and then they must contain at least the tape which will be laid and the previous one (figure 3). Firstly, we define the locations of the microscopic domains which will allow us to describe the microscopic state of the composite during the process. A mesh is assigned to each microscopic domain. The large nodal density differences between the microscopic and macroscopic are accurately accounted thanks to the meshless character of the natural element approximation considered for the interpolation of the different fields [START_REF] Sukumar | The natural elements method in solid mechanics[END_REF][START_REF] Yvonnet | The constrained natural element method (C-NEM) for treating thermal models involving moving interfaces[END_REF][START_REF] Yvonnet | A new extension of the natural element method for non-convex and discontinuous domains : the constrained natural element method (C-NEM)[END_REF]. Microscopic domains are coupled to the macroscopic one through their boundaries. As described later, the natural element interpolation is based on the use of Voronoi cells instead of Delaunay triangles (used in the finite element framework) for defining the fields approximation. It has been proved in some of our former works [START_REF] Martinez | A meshless simulation of injection processes involving short fibers composites[END_REF] that the quality of the Delaunay triangulation (dual of the Voronoi diagram) does not affect the quality of the computed solution in the NEM framework, even when one proceeds in very distorted meshes.

General algorithm

The global solution procedure is presented on figure 4. After the welding of two composite layers we assume known the initial temperature (which results from the numerical modelling of the welding operation) as well as the boundary conditions related to the particular industrial process considered. We suppose that we want to describe the state of the composite in the time interval {t 0 , t f }. The different steps of the calculation are presented below. We realize a partition of the time interval {t 0 , t f } in m time intervals by means of a time step Δt Macro

t f = t 0 + mΔt Macro [1] 3.2.1. Macroscopic thermal calculation
We consider a regular mesh composed of bilinear quadrilateral elements. We perform the thermal calculation in the macroscopic time intervals {t k , t k + Δt Macro } , k ∈ {0, . . . , m -1} by taking into account the crystallization kinetics as well as the homogenized thermomechanical parameters. This thermal calculation allows us to compute the macroscopic thermal field that is then transfered to the microscopic calculation.

Microscopic calculation

We consider a time step Δt micro , in general much lower than Δt Macro , i.e. Δt micro << Δt Macro . We perform p microscopic calculations (time steps) in a time 5. On the boundary of each microscopic cell considered for the microscopic calculation we prescribe the temperature from the temperatures known at the four nodes defining that cell at t n = nΔt Macro and t n+1 = (n + 1) Δt Macro , assuming a linear evolution in both space -on the cell boundary -and time. Thus, at time t = t n + iΔt micro , for a point x located on the segment defined by nodes k and l we can write

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T i k = T n k + iΔt micro Δt Macro T n+1 k -T n k T i l = T n l + iΔt micro Δt Macro T n+1 l -T n l T i (x) = T i k + s L kl T i l -T i k [2]
where s is the distance between the considered point and the node k and L kl is the distance between both nodes. We assume that a macroscopic time step Δt Macro corresponds with p microscopic time steps, i.e. Δt Macro = pΔt micro . In this manner the solution of the thermal problem offers the field T (X, x,

t = t n + iΔt micro ) , i ∈ [1, . . . , p],
where X denotes the macroscopic position related to the microscopic domain (center of gravity of the microscopic cell) an x the point in the microscopic domain. Moreover we can evaluate the microscopic stress fields in any point of the microscopic domains and for any time, in order to evaluate the failure criteria.

Homogenization

In the considered microscopic time intervals, thermomechanical properties practically don't evolve. Therefore we can realize the homogenization of the thermomechanical properties in the time interval {t k , t k + Δt Macro } , k ∈ {0, . . . , m -1} by means of the defined representative volume elements. The homogenized thermomechanical properties are then used in the macroscopic scale at next time increment in order to evaluate the macroscopic fields. The principle of the homogenization method and the transfer of the microscopic information towards the macroscopic scale is presented later.

Macroscopic stress calculation

The macroscopic stresses are calculated by means of the thermomechanical homogenized properties. This calculation will be described later.

Thermal modeling

In this section, we present the thermal modeling used for the simulation.

Energy balance

The equation to be solved, the transient 2D heat transfer equation including the heat generated by crystallisation, may be written as

div (k (T ) GradT) = ρ (χ) C p (T ) ∂T ∂t + m m H c ∂χ ∂t [3]
where ρ is the material density (kg.m -3 ), C p is the material specific heat (KJ.g -1 .K -1 ), k is the material thermal conductivity (W.m -1 .K -1 ), χ the relative degree of crystallinity, m m the matrix mass fraction, and H c the crystallisation enthalpy (KJ.g -1 ).

Crystallisation kinetics

An Avrami crystallisation kinetics law has been used to describe time crystallisation evolution

χ (T, t) = χ ∞ (1 -exp (-K (T ) t c n )) [4]
with

K (T ) = K 0 exp - E a RT [5]
where E a is the activation energy (KJ.mol -1 ), n the Avrami coefficient and t c the time elapsed from the beginning of the polymer crystallisation.

Matrix formulation

The thermal equation [3] can be written as

M ∂T ∂t + KT = F [6] with M = Ω ρC p N T NdΩ [7] K = Ω B T kBdΩ + S ϕ hN T NdS [8] F = Ω N T ρ (χ) m m H c ∂χ ∂t dΩ + S ϕ N T hT amb dS [9]
N is the vector containing shape function, B contains the shape function derivatives, h is the thermal transfer coefficient (W.m -2 .K), and T amb is the ambient temperature (K).

Model reduction

Computing times can be significantly alleviated by using model reduction, like the proper orthogonal decomposition. The aim of the model reduction is to define a transformation matrix to get few basis functions to describe the spatially distributed state, with few degree of freedom. We present in this section a method which has been used to reduce computing times based on the proper orthogonal decomposition.

The proper orthogonal decomposition

We assume that the evolution of a certain field T (x, t) is known. In practical applications, this field is defined at the nodes of a spatial mesh x i and for some times

t m = mΔt, with i ∈ [1, . . . , N]. We introduce the notation T (x i , t m ) = T m (x i ) ≡ T m i (t m
). T m defines the vector containing the nodal degrees of freedom (temperatures) at time t m . The main idea of the proper orthogonal decomposition is to obtain the most typical or characteristic structure φ (x) among these T m (x) , ∀m. The maximization of

α = m=M m=1 i=N i=1 φ (x i ) T m (x i ) 2 i=N i=1 (φ (x i )) 2 [10] leads to m=M m=1 ⎡ ⎣ i=N i=1 φ (x i ) T m (x i ) ⎛ ⎝ j=N j=1 φ (x j ) T m (x j ) ⎞ ⎠ ⎤ ⎦ = α i=N i=1 φ (x i ) φ (x i ) ; ∀ φ [11]
which can be rewritten in the form

i=N i=1 ⎧ ⎨ ⎩ j=N j=1 m=M m=1 T m (x i ) T m (x j ) φ (x j ) φ (x i ) ⎫ ⎬ ⎭ = α i=N i=1 φ (x i ) φ (x i ) ; ∀ φ [12]
Defining the vector φ such that its i-component is φ (x i ), [12] results in the eigenvalue problem [13], whose eigenvectors related to the highest eigenvalues define the characteristic structure of T m (x),

ψ T cφ = α φ T φ; ∀φ ⇒ cφ = αφ [13]
where the two points correlation matrix is given by

c ij = m=M m=1 T m (x i ) T m (x j ) ⇔ c = m=M m=1 T m (T m ) T [14]
which is symmetric and positive definite. If we define the matrix Q containing the discrete field history

Q = ⎛ ⎜ ⎜ ⎜ ⎝ T 1 1 T 2 1 . . . T M 1 T 1 2 T 2 2 . . . T M 2 . . . . . . . . . . . . T 1 N T 2 N . . . T M N ⎞ ⎟ ⎟ ⎟ ⎠ [15]
then it is easy to verify that the matrix c in [13] results

c = QQ T [16]

A posteriori reduced modelling

We solve the eigenvalue problem defined by [13] selecting the eigenfunctions φ k associated with the eigenvalues belonging to the interval defined by the highest eigenvalue and that value divided by a large enough value. In practice n is much lower than N . Now, we could try to use these n eigenfunctions φ k for approximating the solution T m (x) , ∀m. For this purpose we need to define the matrix

B = [φ 1 . . . φ n ] B = ⎛ ⎜ ⎜ ⎜ ⎝ φ 1 (x 1 ) φ 2 (x 1 ) . . . φ n (x 1 ) φ 1 (x 2 ) φ 2 (x 2 ) . . . φ n (x 2 ) . . . . . . . . . . . . φ 1 (x N ) φ 2 (x N ) . . . φ n (x N ) ⎞ ⎟ ⎟ ⎟ ⎠ [17]
Now, if we consider the linear system of equations resulting from an implicit or semiimplicit time discretization of the thermal model defined by [6]

G m T m+1 = H m [18]
expressing

T m+1 = i=n i=1 ζ m+1 i φ i = Bζ m+1 [19]
[18] results

G m T m+1 = H m ⇒ G m Bζ m+1 = H m [20]
and by multiplying both terms by B T we obtain

B T G m Bζ m+1 = B T H m [21]
where the size of B T G m B is n × n, instead N × N . When n << N as is the case in numerous physical models, the solution of [21] is preferred because its reduced size.

Enriching the approximation basis

Obviously, accurate simulations require an error evaluation as well as the possibility of adapting the approximation basis by introducing new functions able to describe the solution features. Ryckelynck proposed in [START_REF] Ryckelynck | A priori hyper-reduction method : an adaptive approach[END_REF] an adaptive procedure, able to construct or enrich the reduced approximation basis. For this purpose, he proposed to enrich the reduced approximation basis by adding some Krylov subspaces generated by the equation residual. Despite the fact that this enrichment tends to increases the number of approximation function, when it is combined with a POD decomposition that continuously reduces this number, the size of the problems is quickly stabilized. This strategy, which was successfully used in [START_REF] Ammar | On the reduction of kinetic theory models related to finitely extensible dumbbells[END_REF] as well as in [START_REF] Ryckelynck | On the 'a priori' model reduction : overview and recent developments[END_REF], is summarized in the present section. We assume the solution accurately described in the interval ] 0, t s ] by using the reduced basis B. Now, the solution evolution is performed in the time interval ] t s , t s+1 ], with t s+1 = t s + SΔt solving [21] making use of the reduced approximation basis defined by matrix B

ζ m+1 = B T G m B -1 B T H m [22]
When time t s+1 is reached, a control step is performed in order to evaluate the accuracy of the solution computed using the reduced basis. The control step is performed at t s+1 by computing the residual

R = G t s+1 -Δt Bζ t s+1 -H t s+1 -Δt [23]
If the norm of the residual is small enough R < (being a small enough parameter) the computed solution can be assumed as good, and the time integration goes on in ] t s+1 , t s+2 ], with t s+2 = t s+1 + SΔt, using [22] without changing the reduced approximation basis.

On the contrary, if R > Bζ M , the approximation must be improved. For this purpose, we propose to enrich the reduced approximation basis by introducing the just computed residual (or some Krylov's subspaces generated by the residual, as proposed in [START_REF] Martinez | A meshless simulation of injection processes involving short fibers composites[END_REF]))

B ← [BR]

[24] and now, the evolution is recomputed in the ] t s , t s+1 ] using [22] with the just updated reduced basis B. When the convergence criterion is satisfied (i.e. R < ) we apply a POD on the entire past history ] 0, t s+1 ] for extracting the reduced basis able to represent all the past evolution of the unknown field, which results in an updated B that can be considered as the optimal reduced basis to describe the evolution of the unknown field in ] 0, t s+1 ]. Using the just updated reduced basis B the time evolution given by [22] is performed in ] t s+1 , t s+2 ] and the solution accuracy is checked at t = t s+2 .

The enrichment tends to increase the size of the reduced basis, but the POD reduces its size. The combination of both procedures allows to stabilize the size of the model as we noticed in some of some former works [START_REF] Ammar | On the reduction of kinetic theory models related to finitely extensible dumbbells[END_REF].

Microscopic description : dealing with highly heterogenous nodal distributions

In the present method, we define in the macroscopic mesh some microscopic domains which will allow us to determine the microscopic thermomechanical fields evolution as well as the homogenized thermomechanical properties during the simulation. In order to use the information coming from the macroscopic scale we must ensure the transition between the macroscopic and the microscopic scales. A microscopic domain is localized in a macroscopic element. The size of the representative volume element must be smaller then the macroscopic element. So the microscopic mesh nodal density evolves suddenly when ones moves from a microscopic domain to the transition one (which corresponds to the macroscopic scale). The nodal density inside the microscopic domain must be enough to represent accurately the microstructure, whereas in the transition domain it must be enough to represent the structure response. Thus, usually, the nodal distribution becomes highly heterogenous leading to very distorted meshes inadmissible if one proceeds in the finite element framework. For this reason, the use of a meshless discretization technique seems appropriate, because it is much less sensible to the aforementioned mesh distortions (see figure 6). In the next paragraphs, we briefly touch upon the basis of the natural element method that will be used for discretizing the variational formulation given by [3].

The natural element method is an appealing choice among the different meshless methods, because essential boundary conditions can be imposed directly, without detriment to the other properties (linear consistency, smoothness,...). Thus the finite element and natural element interpolations coupling becomes direct. It was originally proposed by Traversoni [START_REF] Traversoni | Three-scale finite element analysis of heterogeneous media by asymptotic homogenization and mesh superposition methods[END_REF], Sambridge et al. [START_REF] Sambridge | Geophysical parameterization and interpolation of irregular data using natural neighbors[END_REF] and was widely investigated in elastostatic problems by Sukumar [START_REF] Sukumar | The natural elements method in solid mechanics[END_REF]. Moreover, it was successfully applied in fluid dynamic simulations involving polymer injection and metal forming simulations using updated Lagrangian formulations [START_REF] Martinez | A meshless simulation of injection processes involving short fibers composites[END_REF][START_REF] Martinez | Updated lagrangian free surface flow simulations with natural neighbour Galerkin methods[END_REF]Alfaro et al., 2006), in problems involving cracks [START_REF] Yvonnet | A new extension of the natural element method for non-convex and discontinuous domains : the constrained natural element method (C-NEM)[END_REF] and in phase change problems involving moving discontinuities [START_REF] Yvonnet | The constrained natural element method (C-NEM) for treating thermal models involving moving interfaces[END_REF].

The constrained natural element interpolation revisited

We briefly touch upon the foundation of Sibson's natural neighbor coordinates (shape functions) that are used in the natural element method. For a more in-depth discussion on the Sibson interpolant and its application for solving second-order partial differential equations, the interested reader can refer to Braun and Sambridge [START_REF] Sambridge | Geophysical parameterization and interpolation of irregular data using natural neighbors[END_REF] and Sukumar et al. [START_REF] Sukumar | The natural elements method in solid mechanics[END_REF]. The NEM in- For the sake of simplicity we only consider the 2D case, the 3D case being a direct extension. Let S = {n 1 , n 2 , . . . , n N } be a set of nodes in 2 . The Voronoi diagram is the subdivision of 2 into regions T i (Voronoi cells) defined by

T i = x ∈ 2 : d (x, x i ) < d (x, x j ) , ∀j = i , ∀i [25]
where x i are the coordinates of node n i and d (., .) denotes the Euclidean distance. The Sibson coordinates of x with respect to a natural neighbor n i (see figure 7) is defined as the ratio of the overlap area (volume in 3D) of their Voronoi cells to the total area (volume in 3D) of the Voronoi cell associated with the point x

ψ i (x) = Area (af ghe) Area (abcde) [26]
If the point x coincides with the node n i , i.e. x = x i then ψ i (x i ) = 1, and all the other shape functions vanish, i.e. ψ i (x j ) = δ ij (δ ij being the Kronecker's delta). The properties of positivity, interpolation and partition of unity are then verified [START_REF] Li | A plane-strain finite element model for process-induced stresses in a graphite/PEEK composite[END_REF]) where n is the number of neighbor nodes of point x.

⎧ ⎨ ⎩ 0 ≤ ψ i (x) ≤ 1 ψ i (x j ) = δ ij i=n i=1 ψ i (x) = 1 [27]
The natural neighbor interpolation satisfies the local coordinate property [START_REF] Sibson | A vector identity for the Dirichlet tesselations[END_REF], namely

x = i=n i=1 ψ i (x) x i [28]
which combined with [27], implies that the natural neighbor interpolant spans the space of linear polynomials (linear completeness).

Natural neighbor shape functions are C ∞ at any point except at the nodes, where they are only C 0 , and on the boundaries of the Delaunay circles (spheres in 3D) where they are C 1 , because of the discontinuity in the neighbor nodes across these boundaries.

Another important property of this interpolation is its ability to reproduce linear functions on the boundary of convex domains. The proof can be found in Sukumar et al. [START_REF] Sukumar | The natural elements method in solid mechanics[END_REF], that we illustrate in figure 7(right) due to the fact that the Voronoi cells areas associated to points on the boundary become infinite, the contribution of internal points vanishes in the limit when the point approaches the convex boundary, and the shape functions associated with nodes n 1 and n 2 become linear on the segment (n 1n 2 ). This is not the case when non convex boundaries are considered where appropriate treatments are required [START_REF] Cueto | Imposing essential boundary conditions in the natural element method by means of density-scaled Alpha-shapes[END_REF][START_REF] Yvonnet | A new extension of the natural element method for non-convex and discontinuous domains : the constrained natural element method (C-NEM)[END_REF]. This drawback will be considered later. Consider an interpolation scheme for a vectorvalued function U h (x) : Ω ⊂ 2 → 2 , in the form

U h (x) = i=n i=1 ψ i (x) U i [29]
where U i are the vectors defining the nodal degrees of freedom at the n natural neighbors of point x and ψ i (x) are the Sibson coordinates defined in [26] associated with each node n i . It can be noticed that [29] defines a local interpolation scheme which will be used to define both the trial and the test functions considered in the discretization of different variational formulations.

A recent development in the NEM, the constrained natural element method (C-NEM), was proposed in [START_REF] Yvonnet | The constrained natural element method (C-NEM) for treating thermal models involving moving interfaces[END_REF][START_REF] Yvonnet | A new extension of the natural element method for non-convex and discontinuous domains : the constrained natural element method (C-NEM)[END_REF] in order to circumvent the problems induced by non convex domains. In this approach, a visibility criterion is introduced to restrict the natural neighbors (influent nodes). For this purpose the constrained Voronoi diagram is constructed, from which the shape functions can be easily computed. In this manner, linear interpolation is recovered along the boundary of non-convex domains, making possible the introduction of essential boundary conditions as well as the treatment of fixed or moving discontinuities.

In the C-NEM framework, the interpolation can be expressed by

U h (x) = i=V i=1 ψ C i (x) U i [30]
where V is the number of natural neighbors visible from point x and ψ C i is the constrained natural neighbor shape function, which is actually the Sibson interpolant computed using the constrained Voronoi diagram [START_REF] Yvonnet | A new extension of the natural element method for non-convex and discontinuous domains : the constrained natural element method (C-NEM)[END_REF].

Transferring microscopic information towards the macroscopic scale

In this section we present the homogenization strategy. One of the contributions of the present work is the calculation of the thermomechanical homogenized properties by the application of the model reduction method presented in section 4.3. This method allows to use real boundary conditions to calculate the homogenized properties. The thermomechanical properties are then extended to the transition domain (where the macroscopic analysis is performed) by using an interpolation method. The transition medium has a particular role in the multi-scale model, because it ensures the connection between all the microscopic patterns.

Homogenization strategy

Let's Ω be the domain of interest, i.e. the whole macroscopic domain. We define in this macroscopic domain the elements Z i=1...N for which we will place a microscopic pattern (i.e. a representative volume element) in order to determine the microscopic state and the thermomechanical homogenized properties (see figure 8 left). An independent mesh Γ i=1...N is built for each macroscopic elements Z i=1...N with the constrained natural element method (C-NEM) (see figure 8 right). A microscopic model (Ω i ) i=1...N is built in each mesh Γ i=1...N . Each Ω i is a microscopic pattern. These patterns are either surrounded by a transition medium Ξ i (see figure 8 right). The transfer between macroscopic scale and microscopic scale is realized by the four nodes of the macroscopic element Z i (see figure 8 right) and the transition domain Ξ i=1...N which surrounds the representative volume element (Ω i ) i=1...N . The thermomechanical properties are assumed to be perfectly defined at the microscopic scale inside each one of the microscopic patterns (Ω i ). In the transition domain we assume that the spatial variations of the averaged (or macroscopic) thermomechanical properties is smooth enough. In the next section we present a new method to realize the

Thermal conductivity homogenization

We assume known the temperature field T (X, x, t = t n + iΔt micro ), i ∈ [1, . . . , p], where X denotes the macroscopic position related to the microscopic domain (center of gravity of the microscopic cell) an x the point in the microscopic domain. We construct the matrix Q as presented in [15]. From the application of the model reduction method presented in section 4.3 we select the two most representative eigenfunctions φ 1 and φ 2 (related to the highest eigenvalues). Now, in a fist approximation the temperature evolution at the microscopic level can be approximated as

T (x, t) ≈ β 1 φ 1 (x) + β 2 φ 2 (x) [31]
where the β 1 and β 2 coefficients can be computed by minimizing the integral

E (β 1 , β 2 ) = t n+1 t n Ω i [T (x, t) -β 1 φ 1 (x) -β 2 φ 2 (x)] 2 dΩdt [32]
Now, with the beta coefficients just computed we can define the first and second temperature contributions Φ 1 and Φ 2 respectively, with

Φ 1 = β 1 φ 1 and Φ 2 = β 2 φ 2 .
We define the corresponding local gradients

g 1 (x) = GradΦ 1 (x) g 2 (x) = GradΦ 2 (x) [33]
as well as the average ones

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ G 1 (X) = 1 |Ω i | Ω i g 1 (x) G 2 (X) = 1 |Ω i | Ω i g 2 (x)
[34]

The scale transfer tensor M (X, x) is then defined by

g (x) = M (X, x) G (X) [35]
and applying this expression to both gradient contributions, the four components of tensor M (X, x) at each point x ∈ Ω i can be computed

g 1 (x) = M (X, x) G 1 (X) g 2 (x) = M (X, x) G 2 (X) [36]
Where (X) serves to locate Ω i in Ω. Finally, the homogenized conductivity tensor corresponding to the microscopic domain Ω i results

k i (X) = 1 |Ω i | Ω i k (x) M (X, x) dΩ [37]
The thermal conductivity at a point M in the transition domain Ξ i of the microscopic pattern Γ i is

k |M ≈ k i [38]
The simplest manner for defining the behavior at any point in the macroscopic domain Ω consists of using C-NEM interpolation constructed on the basis of the Voronoi diagram related to the cloud of points consisting of the center of gravity of the macroscopic elements Z i . Let's ψ i be the shape function related to the center of gravity of Z i which is perfectly defined in the whole domain Ω. Thus, the macroscopic law results

k |M = i=V i=1 ψ i (M ) k i [39]
Therefore the thermal conductivity is perfectly defined at each point M of the macroscopic domain Ω.

Remark 1 : if the macroscopic thermal gradient G 2 is too small, we construct an artificial G 2 in order to calculate the scale transfer tensor. Indeed [36] leads to

M (X, x) = G -1 (X) × g (x) = = ⎡ ⎢ ⎢ ⎣ G 1X (X) G 1Y (X) 0 0 0 0 G 1X (X) G 1Y (X) G 2X (X) G 2Y (X) 0 0 0 0 G 2X (X) G 2Y (X) ⎤ ⎥ ⎥ ⎦ -1 ⎡ ⎢ ⎢ ⎣ g 1x (x) g 1y (x) g 2x (x) g 2y (x) ⎤ ⎥ ⎥ ⎦ [40]
The tensor G (X) must be invertible, i.e. det [G (X)]

= 0 i.e. G 1X (X) •G 2Y (X) = G 1Y (X) •G 2X (X).
For this purpose, we define the microscopic thermal gradient g 2 (x) on each point x of the RVE such as g 2 (x) ⊥g 1 (x). Thus we can calculate the transfer scale tensor M (X, x) and the homogenized conductivity with [37].

Remark 2 :

We can apply the model reduction directly on the microscopic thermal gradients. For this purpose, we construct the matrix of the microscopic

thermal gradients Q = ⎛ ⎜ ⎜ ⎜ ⎝ GradT 1 1 Grad T 2 1 . . . Grad T M 1 Grad T 1 2 Grad T 2 2 . . . Grad T M 2 . . . . . . . . . . . . Grad T 1 N Grad T 2 N . . . Grad T M N ⎞ ⎟ ⎟ ⎟ ⎠ and build the matrix c = Q.Q T .
The application of the model reduction gives the eigenvectors φ 1 and φ 2 and therefore the homogenized conductivity with [37].

Remark 3 : The same procedure can be used to compute the homogenized constitutive law. Obviously, in the linear case one could perform a standard homogenization by solving three linear boundary problems. However, the non-linear case could be addressed within the framework just described.

Stress calculation

The residual stress calculation requires the calculation of the thermal stress field and the crystallization induced stress field within the composite during the process. These stresses are induced during the cooling phase.

Thermal induced stress calculation

The macroscopic thermal stress at a point M of the macroscopic domain is defined by the relationship

σ th |M ≡ C |M th |M [41]
with C |M the macroscopic constitutive behavior at the point M and th |M the macroscopic thermal strain at a point M . The macroscopic thermal strain at a point M of the macroscopic domain is calculated by the relationship

th |M = α th |M T |M -T ref [42]
with α th |M is the thermal expansion at the point M , T |M the temperature at the point M and T ref the reference temperature for which there is no thermal expansion. The thermal expansion α th |M at the point M is calculated by the relationship

α th |M = i=V i=1 ψ i (M ) α th i [43]
The average thermal expansion on each microscopic domain Ω i is calculated by evaluating the average thermal strain on the microscopic domain. The average thermal strain on the microscopic domain Ω i can be expressed as

th i = 1 |Ω i | Ω i th |P dΩ [44]
with the microscopic thermal strain Moreover the average thermal strain can be expressed by

th i = α th i ΔT i [46]
And the average temperature variation on the microscopic domain Ω i can be expressed by

ΔT i = 1 |Ω i | Ω i ΔT dΩ [47]
Therefore

α th i . ΔT i = 1 |Ω i | Ω i α th (P ) (T (P ) -T ref (P )) [48]
Homogenized thermal expansion α th i and the thermal stress field can thus be calculated.

The thermal stress field induces nodal forces in the macroscopic mesh Ω which can be explained by

F th = Ω B T σ th dΩ [49]
with B is the matrix which contains the shape function derivatives.

Crystallization induced stress calculation

The volume change ΔV c|M at a point M of the macroscopic domain Ω can be expressed by

ΔV c|M = V 0|M ρ 0 |M ρ |M -1 [50]
By considering an isotropic material, the volume change at the point M is

ΔV c|M V 0|M = 1 + Δl c|M l 0|M 3 -1 [51]
and therefore the crystallization induced strain at the point M is

crist |M = Δl c|M l 0|M 1 = ΔV c|M V 0|M + 1 1/3 -1 1 [52]
with 1 is the identity tensor. Finally, the crystallization induced strain at the point M is

crist |M = ρ 0 |M ρ |M 1/3 -1 1 [53]
At the microscopic level, only the matrix density varies. As the density of amorphous and crystalline phases is known, the actual density of a point P in the matrix is

ρ |P = χρ c + 1 -χ |P ρ a [54]
where ρ a is the density of the amorphous phase and ρ c the density of the crystalline phase. The average material density of a microscopic domain Ω i is calculated by

ρ i = 1 |Ω i | Ω i ρ |P dP [55]
and the macroscopic material density at each point M of the macroscopic domain Ω is calculated by

ρ |M = i=V i=1 ψ i (M ) ρ i [56]
The macroscopic induced crystallization stress at each point M of the macroscopic domain Ω is then defined by

σ crist |M ≡ C |M crist |M [57]
The induced crystallization stress field induces nodal forces in the macroscopic mesh Ω which can be explained by

F crist = Ω B T σ crist dΩ [58]
with B is the matrix which contains the shape functions derivatives.

Macroscopic stress calculation

In order to predict the residual stress field in the composite at the end of the process, we have to evaluate the macroscopic stress field in the structure. The global stress is calculated by the equation

σ |M = C |M |M -crist |M -th |M [59]
where |M is the macroscopic global strain, crist |M is the macroscopic crystallization induced strain and th |M is the macroscopic thermal induced strain.

[59] can be rewritten

σ |M = C |M |M -σ crist |M -σ th |M [60]
where σ crist

|M

is the macroscopic crystallization induced stress ([57]) and σ th |M is the macroscopic thermal induced stress ([41]). The macroscopic global strain |M is calculated by the relationship

|M = B |M U |M [61]
with U |M is the global displacement field at the point M , obtained by solving the equation

KU = F [62] with K = Ω B T CBdΩ [63] F = Ω B T σ th dΩ + Ω B T σ crist dΩ + Ω N T f V dΩ + S Ω N T f S dS Ω [64]

Numerical examples

In this section we present the results of the simulation of a cooling of two prepreg layers laid one on the other. We consider a graphite fiber reinforced composite similar to APC2 [START_REF] Cogswell | Thermoplastic aromatic polymer composites[END_REF] (see table 1). The geometrical and boundary conditions of the simulation are given on figure 9. The results are presented for a time of 10 -1 s. The temperature field is presented on figure 10. The degree of crystallinity field is presented on figure 11. The thermal stress field is presented on figure 12. According to 41, the thermal stress components are negative. The crystallization stress field is presented on figure 13. We note that the values of the crystallization stress field are low. Residual stress field is then mainly induced by thermal stress field. Finally the global stress field [59] (σ Y Y component) is presented on figure 14. We see that in agreement with theory the top prepreg layer is in compression (σ Y Y < 0).

Matrix Thermal Properties

Values

Conductivity k (W.m -1 .˚C

Conclusion

In this paper, we have proposed a coupled two-scales modelling for thermomechanical problems involving microstructure, which allows to evaluate residual stress growth during the on-line consolidation process. The method allows to take into account the thermomechanical properties evolution of the material during the process, which is a dominating factor in residual stress development. We use a meshless constrained natural element interpolation for taking into account the large and very localized variations in the nodal density. This technique allows considering simultaneously both scales, the one related to the microstructure description and the one related to its evolution (the macroscopic one). For this purpose the domain of interest is divided in some microscopic zones representing the microstructure evolution in the whole macroscopic domain, which do not cover the entire domain, and a transition domain. The thermomechanical properties are defined only in the microscopic zones at the microscopic scale. The thermomechanical properties are homogenized with the real boundary conditions by addressing proper orthogonal decomposition. The homogenized laws accurately defined at the microscopic level are then extended, using an appropriate technique, to that transition domain. In contrast to the vast majority of homogenization techniques, our approach allows an accurate description of the boundary Comput. Methods Appl. Mech. Engrg., vol. 195, n˚48-49, p. 6661-6675, 2006.

Figure 3 .

 3 Figure 3. Definition of the modeled zone

Figure 4 .Figure 5 .

 45 Figure 4. Schematic diagram of the multi-scale calculation algorithm

Figure 6 .

 6 Figure 6. Mesh of a microscopic cell and of the intermediate mesh realized with the method of the constrained natural elements (C-NEM)

Figure 7 .

 7 Figure 7. Construction of the Sibson shape functions

Figure 8 .

 8 Figure 8. Domain definition

  th |P = α th (P ) ΔT |P = α th (P ) (T (P ) -T ref (P ))[45]

Figure 9 .Figure 10 .

 910 Figure 9. Definition of the simulation example

Figure 11 .Figure 12 .

 1112 Figure 11. Degree of crystallinity field (%)

Figure 13 .Figure 14 .

 1314 Figure 13. Crystallization stress field (MPa)

  -1 ) 0.3 Expansion coefficient α (˚C -1 )50.e-6 Graphite Fibers Thermal Properties Values Transverse Conductivity k t (W.m -1 .˚C -1 ) 3.0 Longitudinal Conductivity k l (W.m -1 .˚C -1 ) 16.0 Transverse Expansion coefficient α t (˚C -1 ) 10.e-6 Longitudinal Expansion coefficient α l (˚C -1 ) -1.e-6 Thermal properties and mechanical properties of the material used for simulation

	Matrix Mechanical Properties	Values
	Young's modulus E (GPa)	3.6
	Shear modulus G (GPa)	1.3
	Poisson's ratio ν	0.4
	Graphite Fibers Mechanical Properties Values
	Transverse Elastic modulus E t (GPa)	15.0
	Longitudinal Elastic modulus E f (GPa)	230.0
	Shear modulus G f (GPa)	20.0
	Torsion modulus G t (GPa)	5.0
	Poisson's ratio ν t	0.4
	Poisson's ratio ν tf	0.013
	Poisson's ratio ν f t	0.25
	Tableau 1.


Diana Lupan
email: diana.lupan@cti.pub.ro

Stefan Bobocescu-Kesikis
email: stefan.bobocescu-kesikis@cti.pub.ro

Mihai Dascalu
email: mihai.dascalu@cti.pub.ro

Stefan Trausan-Matu
email: stefan.trausan@cs.pub.ro

Philippe Dessus
email: philippe.dessus@upmf-grenoble.fr

Predicting Readers' Emotional States Induced by News Articles through Latent Semantic Analysis

Keywords: automatic evaluation of news articles, emotional state, Latent Semantic Analysis

With the increasing spread of the social web, identifying emotions in texts has proved to have various applications in fields like opinion mining or market analysis. Emotion recognition from written statements does not only reveal information about the person who wrote them, but can also be used in predicting how the emotional state of the readers can be affected. We propose a novel automatic method for analyzing texts that predicts how reading a news article can influence in turn the emotional state of the reader. This method integrates several word-count approaches and natural language processing techniques, such as Latent Semantic Analysis. Moreover, our implemented system contains a module designed to personalize the provided feedback according to the reader's current emotional state. A preliminary validation has been performed and results are promising.

Introduction

Emotions play an important role in our lives as they can be identified in various contexts (e.g., arguing with someone, reading an article, receiving or giving bad news). Usually, emotions can be easily identified when analyzing body language or voice features (tone, rhythm, frequency). On the contrary, a particular case is when none of the previous cues are valid, as the persons involved are not present face-to-face or not even able to communicate verbally, but only through written communication. We will focus on this situation and analyze how a news article can influence the emotional state of the person reading it.

The system we built, Emo2 (Emotions Monitor) [START_REF] Lupan | Analyzing emotional states induced by news articles with Latent Semantic Analysis[END_REF], was implemented in two phases. Firstly, we enforced a context independent approach that predicted the emotional state by evaluating the content and the title of a news article using different natural language processing techniques. Afterwards, we added a second stage of analysis (results personalization), based on the current emotional state of the user who is reading the news articles. In comparison to the first release of the system [START_REF] Lupan | Analyzing emotional states induced by news articles with Latent Semantic Analysis[END_REF], the number of considered articles increased dramatically and a multitude of fine-tunes were performed in order to increase the system's precision. Also, another validation was performed in order to verify the validity of the results.

The remainder of the paper presents similar work, the system's architecture and details about the actual evaluation steps. The last two sections are focused on the presentation of results and conclusions.

Related Work

A similar approach -UA-ZBSA" [START_REF] Kozareva | UA-ZBSA: A Headline Emotion Classification through Web Information[END_REF] -tries to classify a set of news headlines into six types of emotions: "anger", "disgust", "fear", "joy", "sadness", "surprise" using the idea that "words which tend to co-occur across many documents with a given emotion are highly probable to express that emotion". From a different point of view, UPAR7 [START_REF] Chaumartin | UPAR7: A knowledge-based system for headline sentiment tagging[END_REF] focused on analyzing news articles based on the idea that in a news title all the words can express emotions; therefore, the goal is to identify the main topic within the title.

Furthermore, research was conducted to determine the emotional state of the person who wrote an article. For example, a study was made on blog articles and revealed that angry authors tend to use negative words while joyful ones used more positive words [START_REF] Gill | The Language of Emotion in Short Blog Texts[END_REF]. Other experimental scenarios used blogs where the authors themselves tag an article with their emotional states as baselines in evaluating the system's performance [START_REF] Leshed | Understanding How Bloggers Feel: Recognizing Affect in Blog Posts[END_REF].

A more recent approach is SenticNet [START_REF] Cambria | Sentic computing for social media analysis, representation, and retrieval[END_REF][START_REF] Cambria | Semantic multi-dimensional scaling for open-domain sentiment analysis[END_REF] whose aim is to make conceptual and affective information expressed in natural language more easily accessible even to machines, by enforcing a multi-dimensional hierarchy consisting of a multitude of emotional states with different strengths.

Overview of the Architecture

The system's architecture is presented in Fig. 1. The "Current emotional state" module was introduced in the second phase of development and covers the feedback personalization, while all the others tend to focus on identifying the potential emotional state induced by the article, as expressed from the author's point of view.

The system's processing can be divided in 2 steps. Firstly, we perform an objective analysis in the "Core evaluation" module; then, the obtained results are personalized in the "Estimate induced emotional state" module, using input from "Current emotional state" that provides the estimation of the user's state. As it can be observed from Figure 1, the "Core evaluation" module has multiple inputs consisting of the enriched Affective Norms for English Words (ANEW) database [START_REF] Bradley | Affective norms for English words (ANEW): Stimuli, instruction manual and affective ratings[END_REF], a pre-trained Latent Semantic Analysis (LSA) [START_REF] Landauer | A solution to Plato's problem: the Latent Semantic Analysis theory of acquisition, induction and representation of knowledge[END_REF] semantic vector space, a corpus of articles that express a dominant emotional state (also called "pure") and the actual article to be evaluated. The ANEW database (containing around 1,000 terms) has a special influence on our system because it gives the scales on which the final results are expressed upon. Furthermore, it contains for each concept 3 dimensions that entangle its potentially induced emotional state (Happy/Unhappy, Calm/ Excited and Controlled/In-Control). The first (affective) and second (arousal) dimensions are considered to form the core emotional state, while the third (dominance) is less strongly related [START_REF] Mehrabian | An approach to environmental psychology[END_REF]. The Happy/Unhappy axis expresses the presence or absence of a feeling of content, joyfulness. The Calm/Excited dimension shows if a person reading the article is interested in reading more about that news, or might be interested in similar topics. The last dimension shows if the reader feels that he/she is controlled by the information described in the article, or if he/she controls the situation.

The news articles from our experiments were either directly extracted from RSS feeds or crawled from CNN.com using Apache Nutch (http://nutch.apache.org/) with 2 enforced filters. Firstly, a limit of 1 MB was set because many pages had to be ignored due to a large amount of irrelevant content (pictures, links, metadata) and we only extracted the tags corresponding to the title and content. Secondly, a regex was used to limit the crawled domain to general news, extracted from the international version of CNN available at http://edition.cnn.com. In the end, a corpus of around 15,000 articles was built.

Moreover, the objective analysis was further split into two steps: 1/ semantic similarity between the contents of articles and the documents used as a baseline (expressing a pure emotional state) by means of LSA and 2/ mining key concepts. The "Current emotional state" module uses a questionnaire and a set of 5 similar articles that are evaluated at startup by a reader in order to determine his potential emotional state when reading a news.

Context-Independent Evaluation

The context-independent approach contains two dimensions: 1/ an analysis of an article in comparison with the baseline (documents that express a pure emotional state), therefore a semantic approach, and 2/ a key concepts mining process that takes into account the values from the enriched ANEW database, in other words a lexical approach. The documents used as baseline (10 documents per each state) were chosen from general interest news so they could express various emotions. Also, they imply a pure and powerful emotional state. Below is an example of a text expressing a pure state of happiness:

"I visited Egypt last year what an amazing place the food the people seeing the pyramids the mosques the museums it was a magical trip the people there are so friendly it was a tour that I did with other people from all over the world we also did a cruise on the Nile river which was amazing I have been a lot of places but Egypt was one of my favorites what an amazing sight to be in the hotel pool and to see the pyramids in the background".

Semantic similarity between documents is computed using Latent Semantic Analysis (LSA) [START_REF] Landauer | A solution to Plato's problem: the Latent Semantic Analysis theory of acquisition, induction and representation of knowledge[END_REF] through the cosine measure between the vector representations of concepts or documents within the semantic space. In order to best fit the specificity of the analysis, the LSA vector space was trained on a large corpus of news articles using the Reuters newswire from 1987, available online at http://kdd.ics.uci.edu/databases/reuters21578/reuters21578.html. Moreover, LSA has proven to have a similar behavior to human evaluators, so we considered using it to enrich the initial set of concepts from the ANEW database that were experimentally determined through questionnaires. As drawback, LSA is based on a "bag of words" approach [START_REF] Foltz | Latent semantic analysis for text-based research[END_REF], meaning that the actual word order or the grammatical structure of a sentence is not taken into account.

In order to address the second approach, focused on key concepts mining, the ANEW database was enriched in 2 steps. Firstly, synonyms for each concept are determined using synsets from WordNet [START_REF] Miller | WordNet: A lexical database[END_REF], and only the ones stored in the ANEW database are considered to be relevant (if their emotional value is known). Nevertheless, WordNet has a major limitation as it focuses on strict relations between common words having the same corresponding part of speech, so it cannot be used alone as a method to enrich data based on the similarity between concepts. By referring solely to the synonymy relationships from WordNet, the values for a new concept are computed using a weighted mean of the values of its synonyms. Secondly, LSA is also used to determine the closest

! 4!
concepts from within the enriched database in order to express the valences for a new concept through co-occurrence induced relationships. The actual number of closest concepts from within the pre-trained LSA space considered for expressing the valence of a new word was determined experimentally and, after multiple iterations, we opted to choose the three most dominant and similar concepts.

The overall valences of each sentence are determined as the normalized sum of valences of contained words. Afterwards, the individual results for each article are computed using Equation 1that gives a greater value to the concepts from the title, rather to the ones from the contents of the article. Since we are dealing with rather short articles whose titles are most often suggestive, we considered that the title is designed to attract the potential users' attention, so it conveys more valuable information than the actual content of the article.

!"#$% !"#!!"#!$%&' = ! * !"#$%&'$(% + 1 -! !"#$%#$&'()% (1) ContentValue and TitleValue contain key-concepts valences, with all concepts having the same importance. p was determined experimentally by using increments of 0.05 within the range [0.55; 0.9] and 0.6 proved to be the best alternative for weighting the message in terms of the title, while considering the overall mood induced by the content.

The final results for the context-independent approach are obtained by using Equation 2 that augments the importance of the values obtained through key concepts mining, as the semantic approach is limited by the bag of word approach from LSA:

!"#$%&'()%* = ! * !"#$% !"#!!"#!$%&' + 1 -! !"#$%! !"#$%&'()*"+#! (2)
The value of p was determined experimentally using values from [0.55; 0.9] and was set to 0.65.

Results Personalization

This second phase is used to adapt and personalize the previous results from the context independent evaluation based on the estimation of the current emotional state of the person reading the article. For example, a person who is unhappy tends to be more affected by sad news than someone who is already happy. Also, another factor that proved to be relevant is geo-localization, as people directly involved in the presented situation will be more influenced by it (for example if an accident took place nearby the user's current location). From a more general perspective, the effects of reading a news article are also inter-linked with the reader's psychological traits and further experiments will try to take into consideration more dimensions that impact the estimation of the reader's emotional state.

Taking into account the previous observations, a user must evaluate 5 news articles (see Figure 2) chosen to be as similar as possible in terms of LSA similarity to the news displayed in the end. In other words, for visualizing the main news that will be displayed to the user and their corresponding valence estimations, s/he must beforehand evaluate the top five most similar articles from the corpus.

Additionally, in order to fine-tune even further the displayed results, the user must introduce the continent where he/she lives in. The concepts from the news are analyzed with a geolocalization API (GeoNames -http://www.geonames.org/) that gives access to over 8 million toponyms (various kinds of geographical concepts: countries, cities, continents etc.) and to functions to retrieve the connections between them. Afterwards, our system determines whether the article refers to a location or event from the continent introduced by the user. In case of a positive result, the values for an article are modified so the ones greater than the median value are augmented, while the others are decreased.

!"#$%&'" = !"#$%"&' 1 ± !".!"!!"#"$%&'!!"!"!!"#$%&'!!"!!"#! ! !!!"#$%#&#$ !".!"!!""!!"#"$%&'!!"#$!!"#$%&' (3) ! 5!

User Interface and Results

An output sample for a given news article is displayed in Figure 3. The 3 dimensions are each represented as a progress bar positioned between the two images of the corresponding pure emotional states.

Figure 3. Graphical interface providing feedback on a specific article

We have also opted for presenting the results as a radar graph that introduces the valences for each dimension. Only the dominant subcomponent for each axis is presented on a [0; 4] scale (for example, in Figure 4 we have an article with sad = 3, excited = 1 and in-control = 2).

The results were validated by performing a comparison with the values obtained from a survey conducted on 10 participants from different backgrounds (educational science and computer science). The survey simulated the interaction of a user with the application: firstly, each person was asked to assess 5 news articles that were automatically evaluated to be the closest ones from the corpus, to the ones that were later on presented to the user; afterwards, the user provided details about his/her geographical location and was asked to manually annotate 10 news articles on the 3 dimensions.

A preliminary drawback of our method was highlighted from the evaluation of the latest article (the 10 th ) whose semantic similarities with all the documents from our corpus were low (less than 0.1). In this context, this induced noise within our initial assessment, as correlations were rather low (less than 0.2). After eliminating this article from the evaluation based on objective reasons, as it covered different topics from the ones contained within our "pure" emotional documents, the results dramatically improved. Nevertheless this can be considered a positive aspect, as this exception within the evaluation corpus clearly showed means of improvement of the analysis and a clear indicator when the evaluation is prone to noise (the semantic similarity is below a minimum threshold).

While considering only 9 news articles for the context independent evaluation, correlations were computed between the automatic valences determined by our system and the mean valences of all evaluators. The reason why these computations were performed resides in the fact that multiple perspectives from different users converge to the overall induced emotional state. The Pearson correlations are as follows: r affective =.44, r arousal =.42 and r dominance =.47.

For the second phase that presumes the personalization of results, the initial 5 news articles were used to adapt the results for the latter 9 articles. In this case we have opted to perform a correlation for each user between his/her personalized scores and the manual annotations. In order to better grasp the results, the mean and standard deviation for each dimension were determined: µ(r affective )=.49, σ(r affective )=.28, µ(r arousal )=.16, σ(r arousal )=.17, and µ(r dominance )=.36, σ(r dominance )=.31. There are some situations where the second approach has significantly inaccurate results compared to the context-independent approach. A possible explanation is the subjectivity of the evaluators participating at the survey or the misunderstanding of the interpretation of the arousal and dominance dimensions that are harder to grasp in terms of affect estimations. Moreover, by considering the underlying computations, another plausible explanation takes into account that the news from the initial questionnaire were not similar enough to the ones from the results, so the algorithm was misguided and, instead of adjusting the values, it modifies them incorrectly.

Conclusions

We have implemented an automatic system that predicts how a person is affected by reading a news article and preliminary validation results, despite the intrinsic subjectivity of the task at hand, are encouraging. We firstly focused on the modules that compute an objective, context-independent value for a news article and then we developed a personalization module that adapts the results according to the emotional state of the person when reading the article. Several natural language techniques, with emphasis on Latent Semantic Analysis, were integrated within our approach. Moreover, based on the conducted survey we have identified exceptions that needed to be addressed and means of improving the overall results.

As future research, we intend to conduct more surveys in order to have more feedback and also to add a trust management module to make sure the validation of the system's results is done adequately. For example, if a user rates two similar news very differently (on purpose), the results will not be relevant as the system's logic is based on how similar news are evaluated. Also, not only the results might not be relevant, but also less accurate than without the personalization module. This can create the false impression that the system is faulty and our aim is to limit as much as possible this side effect. Also, we consider that increasing the corpus of "pure" emotional documents can lead to more accurate results, as they are considered a baseline in our analysis.

Mihai DASCALU was head of promotion 2009, University Politehnica of Bucharest, and currently holds 2 master degrees -one in Internet systems engineering, UPB, and one in knowledge extraction, University of Nantes. He's in the last year of his double PhD in computer science (UPB) and educational sciences (University Grenoble Alpes) and has experience in national and international projects (FP7 LTfLL, FP7 ERRIC and CNCSIS K-TEAMS). He has more than 40 published papers, including top computer education conferences (ITS, CSCL) and other renowned international conferences (ICALT, EC-TEL, ICWL, ISPDC, AIMSA). Complementary to his competencies in natural language processing, technology-enhanced learning and discourse analysis, Mihai holds a multitude of professional certifications (e.g. PMP, RMP-PMI, CEH, CISSP). Philippe DESSUS, PhD, is a full professor in educational sciences at the Teacher Education Institute (IUFM) of Univ. Grenoble Alpes. He is head of the Laboratory of Educational Sciences in the same university. He was a former post-doc researcher in educational sciences at Liège University (Belgium), as well as an invited researcher at TECFA, Geneva University (Switzerland) and at LIG-MeTAH, Univ. Grenoble Alpes. Philippe Dessus was involved in the LTfLL EC project (7th PCRD-STREP) and in several French research projects. His research interests are to design and implement experiments involving ICTs in educational contexts, related to two main themes: computer-aided reading/writing environments and instructional design. Most of these experiments have used Latent Semantic Analysis as a tool for delivering automated assessments to the learners. He has published over 40 international articles in conferences, proceedings, invited talks or reports on that theme.

Figure 1 .

 1 Figure 1. Emo2's Architecture

Figure 2 .

 2 Figure 2. Graphical interface for the initial assessment of articles later used in the personalization step

Stefan

  Trausan-Matu, PhD (http://www.racai.ro/~trausan) is a full professor at the Computer Science Department of the "Politehnica" University of Bucharest, and principal researcher at the Institute of Artificial Intelligence of the Romanian Academy. He is lecturing Algorithms Analysis and Design, Human-Computer Interaction, Natural Language Processing, Adaptive and Collaborative Systems. He was a Fulbright post-doc at Drexel University, Philadelphia, USA, was invited professor and lectured in USA, Netherlands, France, San Marino, Germany, Puerto Rico, etc. His current research interests are: Discourse Analysis, Intertextuality, Creativity Fostering, Computer-Supported Collaborative Learning, Human-Computer Interaction, and Philosophy. Prof Trausan-Matu has authored or edited 17 books, 25 book chapters and more than 200 peer-reviewed papers.


Amador Martin-Pizarro

Martin Ziegler
email: ziegler@uni-freiburg.de

Martin Ziegler Equational Theories

Martin-Pizarro Amador

Martin And

Ziegler

EQUATIONAL THEORIES OF FIELDS

Keywords: 1991 Mathematics Subject Classification. 03C45, 12H05. Key words and phrases. Model Theory, Algebraically closed fields, Separably closed fields, Equationality

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Introduction

Consider a sufficiently saturated model of a complete theory T . A formula ϕ(x; y) is an equation (for a given partition of the free variables into x and y) if the family of finite intersections of instances ϕ(x, a) has the descending chain condition (DCC). The theory T is equational if every formula ψ(x; y) is equivalent modulo T to a Boolean combination of equations ϕ(x; y).

Quantifier elimination implies directly that the theory of algebraically closed fields of some fixed characteristic is equational. Separably closed fields of positive characteristic have quantifier elimination after adding to the ring language the so-called λ-functions [START_REF] Delon | Idéaux et types sur les corps séparablement clos[END_REF]. The imperfection degree of a separably closed field K of positive characteristic p encodes the linear dimension of K over K p . If the imperfection degree is finite, restricting the λ-functions to a fixed p-basis yields again equationality.

It is unknown whether equationality holds for the theory of a separably closed field K of infinite imperfection degree, that is, when K has infinite linear dimension over the definable subfield K p .

Another important (expansion of a) theory of fields having infinite linear dimension over a definable subfield is the theory of an algebraically closed field with a predicate for a distinguished algebraically closed proper subfield. Any two such pairs are elementarily equivalent if and only if they have the same characteristic. They are exactly the models of the theory of Poizat's belles paires [START_REF] Poizat | Paires de structures stables[END_REF] of algebraically closed fields.

Deciding whether a particular theory is equational is generally not obvious. So far, the only known natural example of a stable non-equational theory is the free non-abelian finitely generated group [START_REF] Sela | Free and Hyperbolic groups are not equational[END_REF]. In this short paper, we will prove that the theory of belles paires of algebraically closed fields of some fixed characteristic is equational. In Section 5 we provide an alternative proof in characteristic 0, by showing that definable sets are Boolean combination of certain definable sets, which are Kolchin-closed in the corresponding expansion DCF 0 . A similar approach appeared already in [START_REF] Günaydın | Topological study of pairs of algebraically closed fields[END_REF] using different methods.

Equations and Indiscernible sequences

Most of the results in this section come from [START_REF] Pillay | Closed sets and chain conditions in stable theories[END_REF][START_REF] Junker | A note on equational theories[END_REF][START_REF] Junker | The indiscernible topology: A mock Zariski topology[END_REF]. We refer the avid reader to [START_REF] O'hara | An introduction to equations and equational Theories[END_REF] for a gentle introduction to equationality.

We work inside a sufficiently saturated model of a complete theory T . A formula ϕ(x; y), with respect to a given partition of the free variables into x and y, is an equation if the family of finite intersections of instances ϕ(x, a) has the descending chain condition (DCC). The theory T is equational if every formula ψ(x; y) is a Boolean combination of equations ϕ(x; y). Typical examples of equational theories are the theory of an equivalence relation with infinite many infinite classes, completions of the theory of R-modules and algebraically closed fields. If ϕ(x; y) is an equation, then so is ϕ -1 (y; x) = ϕ(x; y). Also is ϕ(f (x); y) an equation, whenever f is a ∅-definable map. Finite conjunctions and disjunctions of equations are again equations.

Equationality is preserved under unnaming parameters and bi-interpretability [START_REF] Junker | A note on equational theories[END_REF]. It is unknown whether equationality holds if every formula ϕ(x; y), with x a single variable, is a boolean combination of equations.

Equationality implies stability [START_REF] Pillay | Closed sets and chain conditions in stable theories[END_REF]. Observe that, if ϕ is an equation, the ϕdefinition of a type p over a set of parameters A is particularly simple: The intersection ϕ(x;a)∈p ϕ(x, a) is a formula ψ(x) over A contained in p. If suffices to set

d p ϕ(y) = ∀x (ψ(x) → ϕ(x; y)) .
Instances of equations coincide with Srour-closed sets. A definable set X is Srourclosed if the family of finite intersections of conjugates of X under automorphisms of the ambient model has the DCC. Furthermore, a formula ϕ(x; y) is an equation if and only if every instance ϕ(x, a) is Sour-closed. Srour-closed definable sets are exactly the indiscernibly closed definable sets [START_REF] Junker | The indiscernible topology: A mock Zariski topology[END_REF]Theorem 3.16]. A definable set is indiscernibly closed if, whenever (a i ) i≤ω is an indiscernible sequence such that a i lies in X for i < ω, then so does a ω .

By extending the indiscernible sequence so that it becomes a Morley sequence over an initial segment, we conclude the following: Remark 2.1. In a stable complete theory T , a definable set ϕ(x, a) is indiscernibly closed if, for every elementary substructure M and every Morley sequence

(b i ) i≤ω over M such that a | ⌣ M b i with b i |= ϕ(x, a) for i < ω,
then b ω realises ϕ(x, a) as well.

Types and Independence

In this section, we will provide a presentation of the theory T P of proper pairs of algebraically closed fields in some fixed characteristic. Most of the results mentioned here appear in [START_REF] Poizat | Paires de structures stables[END_REF][START_REF] Ben-Yaacov | Lovely pairs of models[END_REF].

Work inside a sufficiently saturated model (K, E) of T P in the language L P = L rings ∪ {P }, where E = P (K) is the proper subfield. We will use the index P in order to refer to the expansion T P .

A subfield

A of K is tame if A is algebraically independent from E over E A A | ⌣ EA E,
where E A is E ∩ A. Tameness was called P -independence in [START_REF] Ben-Yaacov | Lovely pairs of models[END_REF], but in order to avoid a possible confusion, we have decided to use a different terminology.

The L P -type of a tame subfield of K is uniquely determined by its L P -quantifierfree type.

Recall that two subfields L 1 and L 2 of K are linearly disjoint over a common subfield F , denoted by

L 1 | ld ⌣ F L 2 ,
if, whenever the elements a 1 , . . . , a n of L 1 are linearly independent over F , then they remain so over L 2 . Linear disjointness implies algebraic independence and agrees with the latter whenever the base field F is algebraically closed. Let us note that linear disjointness is a transitive relation:

If F ⊂ D 2 ⊂ L 2 is a subfield, denote by D 2 • L 1 the field generated by D 2 and L 1 . Then L 1 | ld ⌣ F L 2 if and only if L 1 | ld ⌣ F D 2 and D 2 • L 1 | ld ⌣ D2 L 2 .
Expand the language L P as follows:

L D = L rings ∪ {R n (x 1 , . . . , x n ), λ i n (x; x 1 , . . . , x n )} 1≤i≤n∈N
, where we define the relation R n as

K |= R n (a 1 , . . . , a n ) ⇐⇒ a 1 , . . . , a n are E-linearly independent,
and the λ-functions take values in E and are defined by the equation

a = n i=1 λ i n (a; a 1 . . . , a n ) a i , if K |= R n (a 1 , . . . , a n ) ∧ ¬R n+1 (a, a 1 , .
. . , a n ), and are 0 otherwise. Clearly, a field A is closed under the λ-functions if and only if it is linearly disjoint from E over E A . Thus, a fraction field K of an L D -substructure is automatically tame and the theory T P has quantifier elimination [START_REF] Delon | Élimination des quantificateurs dans les paires de corps algébriquement clos[END_REF] in the language L D . The theory T P is ω-stable of Morley rank ω. Since every subfield of E is automatically tame, the induced structure on E agrees with the field structure, so E has Morley rank 1. If A is a tame subfield, then the L P -definable closure coincides with the inseparable closure of A and its L P -algebraic closure is the field algebraic closure acl(A) of A. This implies that E aclP (A) = acl(E A ). Proof. By the previous remark, it suffices to show that A and E • B are linearly disjoint over E A • M . Furthermore, we may assume that A is a tame algebraically closed subfield of K, that is

A | ld ⌣ EA E.
Thus, let a 1 , . . . , a n in A and z 1 , . . . , z n in E • B, not all zero, such that

n i=1 a i • z i = 0.
We may clearly assume that all the z i 's lie in the subring generated by E and B, so

z i = m j=1 ζ i j b j ,
for some ζ i j 's in E and b 1 , . . . , b m in B, which we may assume to be linearly independent over E.

A straight-forward heir argument implies the existence of some ξ i j 's in E, not all zero, and c 1 , . . . , c m in M linearly independent over E such that

n i=1 a i m j=1 ξ i j c j = 0.
Since A is linearly disjoint from E over E A , we may assume that the ξ i j 's lie in E A . Since the ξ i j 's are not all zero, we have that, for some

1 ≤ i ≤ n, m j=1 ξ i j c j = 0,
as desired.

In order to isolate certain formulae which will determine the type of a tuple in the theory of belles paires of ACF 0 , we require certain basic notions from linear algebra (cf. [4, RÃľsultats d'AlgÃĺbre]).

Let V be a vector subspace of E n with basis {v 1 , . . . , v k }. Observe that

V = {v ∈ E n | v ∧ (v 1 ∧ • • • ∧ v k ) = 0 in k+1 E n }. Thus, the vector v 1 ∧ • • • ∧ v k depends
only on V , up to scalar multiplication, and determines V completely. The PlÃijcker coordinates Pk(V ) of V are the homogeneous coordinates of this vector with respect to the canonical basis of

k E n . Let Gr k (E n ), the k th -Grassmannian of E n , denote the collection of PlÃijcker co- ordinates of all k-dimensional subspaces of E n . Clearly Gr k (E n ) is contained in P r-1 (E), for r = n k . The k th -Grassmannian is Zariski-closed. Indeed, given an element ζ of k E n , there is a smallest vector subspace V ζ of E n such that ζ belongs to k V ζ . The
vector space V ζ is the collection of inner products e ζ, for e in k-1 (E n ) * . Recall that the inner product is a bilinear map

: k-1 (E n ) * × k (E n ) → E. A non-trivial element ζ of k E n determines a k-dimensional subspace of E n if and only if ζ ∧ (e ζ) = 0,
for every e in k-1 (E n ) * . Letting e run over a fixed basis of k-1 (E n ) * , we see that the k th -Grassmannian is the zero-set of a finite collection of homogeneous polynomials.

Fix some enumeration (M i (x)) i=1,2,... of all possible monomials in s variables. Given a tuple a of length s, denote

Ann n (a) = (λ 1 , . . . , λ n ) ∈ E n n i=1 λ i • M i (a) = 0 .
Notation. Given tuples x and y of length n, we will denote by x • y the scalar multiplication for all e from a a fixed basis of k-1 (E n ) * , and

n i=1 x i • y i . Therefore, Ann n (a) = {λ ∈ E n | λ • (M 1 (a), . . . , M n (a)) = 0}.
j≤m p j (ζ) = 0.
In particular, the tuple ζ is not trivial, so we conclude that the above is a tame formula.

By compactness, we conclude the following:

Corollary 3.5. In the theory T P of proper pairs of algebraically closed fields, every formula is a Boolean combination of tame formulae.

Let us conclude with an observation regarding projections of certain varieties.

Remark 3.6. Though the theory of algebraically closed fields has elimination of quantifiers, the projection of a Zariski-closed set need not be again closed. For example, the closed set

V = {(x, z) ∈ K × K | x • z = 1} projects onto the open set {x ∈ K | x = 0}.
An algebraic variety Z is complete if, for all varieties X, the projection X × Z → X is a Zariski-closed map. Projective varieties are complete.

Equationality

In order to show that the stable theory T P of proper pairs of algebraically closed fields is equational, it suffices to show by Corollary 3.5 that tame formulae, according to some fixed partition of the variables, are equations, or equivalently, that every instance of such formulae is indiscernibly closed.

We beging with a special case as an auxiliary result. Remember that x and a are finite tuples of variables and parameters respectively. for some polynomials p 1 , . . . , p m with integer coefficients and homogeneous in ζ.

Express each of the monomials in a appearing in the above equation as a linear combination of a basis of K over E. We see that there are polynomials q 1 , . . . , q s with coefficients in E, homogeneous in ζ, such that

ϕ(x, a) ∧ x ∈ P = ∃ ζ ∈ P r ¬ζ . = 0 ∧ j≤s q j (x, ζ) . = 0 .
Working inside the algebraically closed subfield E, the expression inside the brackets is a projective variety, which is hence complete. By Remark 3.6, its projection is again a Zariski-closed variety, as desired. Suppose now that the formula ϕ(x, a) has the form

ϕ(x, a) = ∃ ζ ∈ P r ¬ζ . = 0 ∧ j≤m p j (x, a, ζ) . = 0 ,
for polynomials p 1 , . . . , p n with integer coefficients and homogeneous in ζ. By appropriately writing each monomial on a in terms of the basis c, we have that

ϕ(x, a) = ∃ ζ ∈ P r ¬ζ . = 0 ∧ j≤s q j (x, ζ) • c . = 0 ,
where the tuples of polynomials qj have now coefficients in E(M ) and are homogeneous in ζ. Hence, linearly disjointness implies that

|= ∃ ζ ∈ P r ¬ζ . = 0 ∧ j≤s q j (b i , ζ) . = 0 for i < ω.

Rewrite this as

|= ∃ ζ ∈ P r ¬ζ . = 0 ∧ j≤t M(m, b i ) • s j (e, ζ) . = 0 for i < ω
for given tuples e in E and m in M , a certain enumeration M(X, Y ) of all monomials appearing in the the polynomials qj 's and tuples of polynomials s j (U, Z)'s with integer coefficients and homogeneous in ζ.

Observe that the sequence (M(m, b i )) i≤ω is indiscernible over ∅. Thus, it suffices to show that

ϕ 1 (x, y; e) = ∃ ζ ∈ P r ¬ζ . = 0 ∧ j≤t M(x, y) • s j (e, ζ) . = 0
is indiscernibly closed. Lemma 4.1 yields the desired result, since the formula ϕ 1 (x, y; u) ∧ u ∈ P is an equation.

An alternative proof in Characteristic 0

In this section, we will provide an alternative proof to the equationality of the theory T p of belles paires of algebraically closed fields in characteristic 0, by means of differential algebra, based on an idea of Günaydın [START_REF] Günaydın | Topological study of pairs of algebraically closed fields[END_REF].

A differential field consists of a field K together with a distinguished additive morphism δ satisfying Leibniz rule δ(xy) = xδ(y) + yδ(x). Analogously to Zariski-closed sets for pure field, one defines Kolchin-closed sets in differential fields as zero sets of systems of differential-polynomials equations, that is, polynomial equations on the different iterates of the variables under the derivation. For a tuple x = (x 1 , . . . , x n ) in K, denote by δ(x) the tuple (δ(x 1 ), . . . , δ(x n )). Ritt-Raudenbush's theorem implies that the Kolchin topology in differential fields of characteristic zero is Noetherian. Noetherianity holds in arbitrary characteristic and is easily proved: Lemma 5.1. In any differential field (K, δ), an algebraic differential equation p(x, δx, . . . ; y, δy, . . .) . = 0 is an equation in the model-theoretic sense.

Proof. Write p(x, δx, . . . ; a, δa, . . .) as q(M 1 (x), . . . , M m (x)), where q(u) is a linear polynomial with coefficients in K and the M i are differential monomials in x. The result now follows from the trivial fact that linear equations q(u) . = 0 are Srour closed in the model-theoretic sense and differential monomial are 0-definable in (K, δ).

Perfect fields of positive characteristic cannot have non-trivial derivations. In characteristic zero though, any field K which is not algebraic over the prime field has a non-trivial derivation δ. Furthermore, if K is algebraically closed, the set of constants E = {a ∈ K | δ(a) = 0} is a proper algebraically closed subfield, so (K, E) is a model of T P . We work from now on in a saturated algebraically closed differential field (K, δ), with non-zero δ. For example in a saturated model of DCF 0 , the elementary theory of differential closed fields of characteristic zero.

In order to show that the theory of proper extensions of algebraically closed fields in characteristic 0, it suffices to show, by Proposition 3.4, that every instance of a tame formula determines a Kolchin-closed set in (K, δ). We first need a couple of auxiliary lemmata to describe the differential ideal associated to a system of polynomial equations. Lemma 5.2. Let v be a vector in K n . Then the K-vector space generated by v, δ(v), . . . , δ n-1 (v) has a basis in E n .

Proof. Let V denote the span of v, δ(v), . . . , δ n-1 (v) and k be minimal such that v can be written as v = a 1 e 1 + • • • + a k e k for a i ∈ K and e i ∈ E n . Clearly the e i are linearly independent and k ≤ n. We claim that e 1 , . . . , e k is a basis of V . Since

δ j (v) = δ(a 1 )e 1 + • • • + δ(a k )e k ,
the vector space V is contained in the span of the e i . If the dimension of V is smaller than k, then v, δ(v), . . . , δ k-1 (v) are K-linearly dependent, that is the rows of the Wronskian

     a 1 a 2 . . . a k δ(a 1 ) δ(a 2 ) . . . δ(a k ) . . . . . . δ k-1 (a 1 ) δ k-1 (a 2 ) . . . δ k-1 (a k )     
are linearly dependent over K. It follows that a 1 , . . . , a k are linearly dependent over E. So there are ξ i ∈ E, not all zero, such that

ξ 1 a 1 + • • • + ξ k a k = 0. Then the vector space k i=1 b i e i k i=1 ξ i b i = 0
contains v, has a basis from E n and its dimension is strictly smaller than k. This contradicts the choice of the e i .

Corollary 5.3. Let V ⊂ K n be the smallest subspace of K n containing the vectors v 0 , . . . , v m-1 and closed under δ. Then V is generated by {δ j (v i )} i<m j<n and has a basis from E n .

Every polynomial over K [X] is a K-linear combination of monomials. Furthermore, homogeneous polynomials of a given degree are K-linear combinations of the corresponding monomials of the same degree. Thus, in order to apply the previous result, equip the polynomial ring K [X] in n variables with a derivation D obtained by differentiating the coefficients of a polynomial in K. Elements in E [X] have Dderivation 0. We say that an ideal

I of K [X] is differential if it is closed under D.
Clearly, any ideal generated by polynomials from E [X] is differential. Given a (finite) collection of polynomials {h i } i∈I of K [X], the ideal generated by {D j (h i )} i∈I j∈N is finitely generated and a differential ideal. An easy application of Corollary 5.3 provides an upper bound for the set of generators of the latter. Analogously, we conclude the corresponding result for homogeneous ideals.

Corollary 5.5. Given homogeneous polynomials h 0 , . . . , h m-1 in K [X] of a fixed degree d, there exists an integer k in N (bounded only in terms of d and the length of X) such that the ideal generated by {D j (h i )} i<m j<k is a differential and homogeneous ideal.

We now have all the ingredients in order to show that tame formulae are equations.

Lemma 3 . 1 .

 31 Given two subfields A and B of K containing an L P -elementary substructure M of K such that A | ⌣ P M B, then the fields E • A and E • B are linearly disjoint over E • M .

Lemma 3 . 2 .Proposition 3 . 4 .

 3234 Two tuples a and b have the same type in T P if and only if for every n in N, we have that ldim E Ann n (a) = ldim E Ann n (b) and the type tp(Pk(Ann n (a))) equals tp(Pk(Ann n (a))) (in the pure field language). Proof. We need only prove the right-to-left implication. Since Pk(Ann i (a)) is determined by Pk(Ann n (a)), for i ≤ n, we obtain an automorphism of E mapping Pk(Ann n (a)) to Pk(Ann n (b) for all n. This automorphism maps Ann n (a) to Ann n (b) for all n and hence extends to an isomorphism of the rings E[a] and E[b]. It clearly extends to a field isomorphism of the tame subfields E(a) and E(b), mapping a to b and E to itself, so a and b have the same T P -type, as required. Definition 3.3. Let x be a tuple of variables. A formula ϕ(x) in the language L P is tame if there are polynomials p 1 , . . . , p m in Z[X, Z], homogeneous in the variables Z, such that ϕ(x) = ∃ ζ ∈ P r ¬ζ . Two tuples a and b have the same type in T P if and only if they satisfy the same tame formulae. Proof. Let p 1 (Z), . . . , p m (Z) be homogeneous polynomials over Z. By Lemma 3.2, it suffices to show that « Ann n (a) has a k-dimensional subspace V such that j≤m p j (Pk(V )) = 0 » is expressible by a tame formula. Indeed, it suffices to guarantee that there is an element ζ in Gr k (E n ) such that (e ζ) • (M 1 (a), . . . , M n (a)) = 0

Lemma 4 . 1 .

 41 Let ϕ(x, a) be an instance of a tame formulae. The set ϕ(x, a) ∧ x ∈ P is Zariski-closed in E. In particular, the formula ϕ(x; y) ∧ x ∈ P is an equation. Proof. Suppose that the formula ϕ(x, a) has the form ϕ(x, a) = ∃ ζ ∈ P r ¬ζ . = 0 ∧ j≤m p j (x, a, ζ) . = 0 .

Proposition 4 . 2 .

 42 Tame formulae are equations.Proof. We need only show that every instance ϕ(x, a) of a tame formula is indiscernibly closed. By Remark 2.1, it suffices to consider a Morley sequence (b i ) i≤ω over an elementary substructure M of (K, E) witha P | ⌣ M b i with |= ϕ(b i , a) for i < ω.In particular, the sequence (b i ) i<ω is Morley over M ∪ {a}. By Lemma 3.1, the fields E(M, a) and E(M, b i ) are linearly disjoint over E(M ) for every i < ω. A basis c of E(M, a) over E(M ) remains thus linearly independent over E(M, b i ).

Corollary 3 . 5 andTheorem 4 . 3 .

 3543 Proposition 4.2 yield the desired result: The theory of proper pairs of algebraically closed fields of a fixed characteristic is equational.

Corollary 5 . 4 .

 54 An ideal of K [X] is differential if and only if it can be generated by elements from E [X].

Proposition 5 . 6 .Corollary 5 . 7 .

 5657 Every tame formula in (K, E) with parameters describes a Kolchinclosed set in (K, δ).Proof. Suppose that the tame formula ϕ(x) is of the formϕ(x) = ∃ ζ ∈ P r ¬ζ . = 0 ∧ i<m p i (x, ζ) . = 0 ,for polynomials p 0 (X, Z), . . . , p m-1 (X, Z) over K homogeneous in Z of some fixed degree d. Let k be as in Corollary 5.5. For a tuple b in K of length |x|, writeD j (p i (b, Z)) = p i,j (b, . . . , δ j (b), Z),for polynomials p i,j (X 0 , . . . , X k , Z) over K, homogeneous in Z. By Corollary 5.5, the ideal I b (Z) generated by {p i,j (b, . . . , δ j (b), Z)} i<m j<k has a generating set consisting of homogenous polynomials g 0 (Z), . . . , g s-1 (Z) with coefficients in E [Z]. Now, since ζ ranges over the constant field, the tuple b realises ϕ(x) if and only if(K, E) |= ∃ ζ ∈ P r ¬ζ . = 0 ∧ I b (ζ) . = 0 ,which is equivalent to(K, E) |= ∃ ζ ∈ P r ¬ζ .Since the field E is an elementary substructure of K, this equivalent toK |= ∃ ζ ¬ζ . = 0 ∧ i<s g i (ζ) . = 0 ,which is again equivalent toK |= ∃ ζ ¬ζ . = 0 ∧ I b (ζ) . = 0 .Since I b is homogeneous, the Zariski-closed set it determines is complete, hence its projection is given by a finite number of equations X(b, . . . , δ k-1 (b)). Thus, the formula ϕ(b) holds if and only if(K, δ) |= X(b, . . . , δ k-1 (b)),which clearly describe a Kolchin-closed set, as desired.By Corollary 3.5, we conclude the following: The theory of proper pairs of algebraically closed fields in characteristic 0 is equational.

Research partially supported by the program MTM2014-59178-P. Additionally, the first author conducted research with support of the program ANR-13-BS01-0006 Valcomo.


Fran De

Fran De Aquino

Asteroid Redirect

Keywords: Asteroid Redirect, Gravitational Spacecraft, Gravitational Mass, Gravity

Asteroids are a great threat to mankind. Here we will show that it is possible to redirect them from their trajectories by means of a strong gravitational repulsion, produced by the gravitational interaction between the asteroid and a Gravitational Spacecraft positioned close to the asteroid.

Previously, I have published a paper where it is shown a new type of quantum device for controlling gravity, called Quantum Controller of Gravity [START_REF] De Aquino | Quantum Controller of Gravity[END_REF], which is basically a spherical capacitor connected to a specific voltage source. This device acts controlling at subatomic level, the gravitational mass of a thin spherical shell at the outer plate of the spherical capacitor. This thin shell works as a Gravity Controller, in such way that if the gravity acceleration at the inner border of the Gravity Controller is g (See Fig. 1) then the gravity acceleration outside the Gravity Controller becomes g g χ = ′

(assuming that the Gravity Controller is sufficiently far from other bodies in such way that the intensity of their gravitational fields are negligible in the region); In addition, if 0 < χ and then the expression above can be rewritten as follows

odd is n ( ) ( ) ( ) 1 2 0 2 r m G r m G g g S i n S g n n χ χ χ + ≅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - = = ′
This means that if a Gravitational Spacecraft with (odd) Gravity Controllers is positioned close to an asteroid, then the asteroid will be repelled from it with a gravity acceleration , then the gravity acceleration, g ′ , acting on the asteroid due to the Gravitational Spacecraft, will be (repulsive in respect to the spacecraft).

/ 6 . 0 s m g + ≅ ′
The idea of generation of a repulsive gravitational force field using Gravity Controllers is not new. In a previous paper we have showed a similar method [START_REF] De Aquino | Repulsive Gravitational Force Field[END_REF].

* In this case, there is also a contribution due to the spherical capacitors, but it can be inconsiderate if the capacitors are very thin (thick << 1mm); n << 100;

Fig. 1 -

 1 Fig.1 -Schematic diagram of a Gravity Controller

2

 2

Fig. 2 -

 2 Fig.2-Asteroid Redirect. For example, if 29 = n , 3 -= χ

  n concentric spherical capacitors, each one with one Gravity Controller) g

Available at: https://hal.archives-ouvertes.fr/hal-01128520