Pranesh Chatterjee
email: pranesh.chatterjee@bwk.kuleuven.ac.be

Geert Degrande
email: geert.degrande@bwk.kuleuven.ac.be

Didier Clouteau
email: clouteau@mss.ecp.fr

Tahmeed Al-Hussaini

Maarten Arnst

Ramzi Othman
email: othman@mss.ecp.fr

Numerical modeling of ground borne vibrations from underground traffic

Keywords: soil structure interaction, finite element, boundary element, Floquet transform, Green-Floquet function

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I. Introduction

The passage of trains in tunnels generates ground-borne vibrations that propagate to adjacent structures and may produce vibrations and re-radiated noise. The amplitude of vibrations depends on several factors as the vehicle characteristics, the train speed, the track and wheel irregularities, the properties of the tunnel, the propagation of waves through the soil and the properties of the structures.

Two-dimensional finite element models with (local) absorbing boundary conditions are frequently used [START_REF] Chua | Groundborne vibrations due to trains in tunnels[END_REF], [START_REF] Chua | Building response due to subway train traffic[END_REF], [START_REF] Rücker | Einwirkung von U-Bahnerschütterungen auf Gebäude; Anregung, Ausbreitung und Abschirmung[END_REF], but necessitate important simplifications to translate the three-dimensional moving load into an equivalent line load and do not allow to incorporate three-dimensional structures. Moreover, two-dimensional models underestimate radiation damping in the soil. The computational results may give a qualitative indication of vibration isolation efficiency (insertion loss) in the design phase. Underlying simplifying assumptions, however, compromise their use as absolute or quantitative predictions.

Within the frame of the EC-Growth project CONVURT ("The control of vibration from underground railway traffic"), an efficient and modular numerical prediction tool is being developed to predict vibration and re-radiated noise in adjacent buildings from excitation due to metro trains in tunnels for both new-build and existing situations. The model will be validated by means of elaborate in situ experiments at a site on the RER B line of RATP at Cité Universitaire in Paris and at another site on the Bakerloo line of London Underground in Regent's Park, London.

The three-dimensional (3-D) dynamic tunnel-soil interaction problem is solved by a coupled substructure approach [START_REF] Aubry | A subdomain approach to dynamic soil-structure interaction[END_REF] exploiting the invariance or the periodicity of the tunnel in the longitudinal direction [START_REF] Hunt | Modelling of rail vehicles and track for calculation of ground vibration transmission into buildings[END_REF], [START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF], [START_REF] Clouteau | Periodic BEM and FEM-BEM coupling[END_REF]. The paper describes a 3-D numerical model based on a finite element formulation for the tunnel structure with a modal reduction technique coupled with a boundary element formulation for a horizontally layered soil medium. The soilstructure interaction problem for the whole tunnel-soil system is restricted within the analysis of a generic cell by using Floquet transform. The impedance of the soil along the interface between the tunnel and the soil is computed with a boundary element formulation, using the Green-Floquet functions defined on a generic cell with the same width as the tunnel segment. Use is made of recent progress to model moving sources at the surface of a layered halfspace [START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF], [START_REF] Aubry | Modelling of wave propagation due to fixed or mobile dynamic sources[END_REF]. The modal coordinates are calculated in the frequency-wavenumber domain. The dynamic representation theorem is used to compute the radiated wave field in the soil, while the inverse Floquet transformation is used to obtain the response in the frequency domain.

Elaborate in-situ experiments have been performed at the site of Cité Universitaire in Paris for the purpose of site characterization as well as validation of the numerical model. A spectral analysis of surface waves (SASW) test has been performed to obtain the thickness and the dynamic characteristics of the shallow layers of the soil. The vibrations at different points in the tunnel, in the free field and in a five-story reinforced concrete building have been measured during the passage of service trains, as well as a test train at variable speed. Furthermore, transfer functions between the track and the free field and the building have been recorded, using an impact hammer.

The main objective of the paper is to describe the development of the numerical prediction model and to comment on the experimental results obtained at the site of Cité Universitaire in Paris.

II. Numerical prediction model

In this section, the stepwise development of the numerical prediction model for a 3-D dynamic coupled linear tunnel-soil interaction problem is discussed. The problem geometry, the Floquet transformation using the periodicity of the problem in longitudinal direction, the basic equations of the generic cell and the solution of the generic problem by periodic coupled FEM-BEM approach are discussed in the following subsections.

A. Problem geometry

A long tunnel is considered as an unbounded open set Ω t , with given elastic properties and assumed to be embedded in a layered elastic halfspace Ω. The horizontally layered soil domain is denoted by Ω s , while Ω (= Ω t ∪Ω s ) represents the global domain.

The tunnel Ω t is assumed to be periodic in the longitudinal y-direction, defined by the unit vector e y . Hence, the generic cell in the tunnel may be defined by:

Ω = {x ∈ Ω | 0 < x • e y < L} (1)
The tunnel Ω t is constructed as an assemblage of an infinite number of elementary bounded cells Ωtn . The definitions of Ω t and Ωtn are given as

Ω t = ∪ +∞ n=-∞ Ωtn (2) Ωtn = x ∈ Ω / x -nLe y ∈ Ωt (3)
The elementary soil domain Ωs in the reference cell has a similar definition.

The interface between the tunnel Ω t and the soil Ω s is denoted by Σ. In the reference cell, the interface between the tunnel and the soil segments Ωt and Ωs is denoted by Σ. The total interface Σ can be decomposed as follows:

Σ = ∪ +∞ n=-∞ Σn (4)
The boundary ∂ Ω of the generic cell Ω can be decomposed as follows:

∂ Ω = Γσ ∪ Γ∞ ∪ Σ 0 ∪ Σ L ( 5
)
where Γσ is the part of the boundary where Neumann boundary conditions are specified. The left and the right boundaries of the reference cell are denoted by Σ 0 and Σ L and defined as follows:

Σ 0 = x ∈ δ Ω / x • e y = 0 Σ L = x ∈ δ Ω / x • e y = L (6)
Sommerfeld's radiation conditions should hold on Γ∞ .

B. Floquet transformations

The problem of dynamic soil-tunnel interaction can be considered as a three-dimensional problem having a periodicity with period L in one direction e y . The position vector x ∈ IR 3 of any point in the problem domain Ω is conveniently decomposed as:

x = x ′ + ye y = x ′ + ỹe y + nLe y = x + nLe y [START_REF] Clouteau | Periodic BEM and FEM-BEM coupling[END_REF] where x ′ denotes the position vector in the (x, z)-plane perpendicular to the longitudinal direction e y , while x is the position vector in the reference cell Ω.

The Floquet transform f (x, κ) of a general non-periodic function f (x) = f (x + nLe y ) defined on a threedimensional problem domain Ω which is periodic with period L in the direction e y is defined as:

f (x, κ) = +∞ n=-∞ f (x + nLe y ) exp (+inLκ) (8)
f (x, κ) is a continuous function of κ in the interval κ ∈ ] -π/L, +π/L[. It is a periodic function of the second kind with respect to x or ỹ as the following relation holds:

f (x ′ , L, κ) = exp (-iκL) f (x ′ , 0, κ) (9)
If the Floquet transform f (x, κ) is known on the reference cell Ω, the function f (x + nLe y ) can be reconstructed for any x = x + nLe y using the inverse Floquet transform:

f (x + nLe y ) = L 2π +π/L -π/L
f (x, κ) exp (-inLκ)dκ [START_REF] Pyl | Determination of the dynamic soil characteristics with the SASW method at the site of Cité universitaire in Paris[END_REF] Equations ( 8) and [START_REF] Pyl | Determination of the dynamic soil characteristics with the SASW method at the site of Cité universitaire in Paris[END_REF] define the Floquet transform pair.

C. Generic problem

It has been discussed in the previous subsection that, for a three-dimensional problem defined on a threedimensional domain with periodicity L along longitudinal direction e y , the analysis can be restricted to the generic domain Ω. The boundary of ∂ Ω of the generic domain Ω is decomposed as ∂ Ω = Γσ ∪ Σ 0 ∪ Σ L where Γσ is the restriction of Γ σ on the generic cell and Σ 0 and Σ L are additional boundaries on which periodic conditions are imposed. Specifying the equilibrium equations in Ωs and Ωt , the boundary conditions on Γσs and Γσt and the coupling equations on Σ, the following generic problem have to be solved.

For every κ ∈] -π/L, +π/L[ and for every frequency ω ∈ IR, the soil displacements ũs (x, κ, ω) in Ωs should satisfy the following equations:

div σ s (ũ s ) = -ρ s ω 2 ũs in Ωs (11) t s (ũ s ) = 0 on Γσs (12) ũs (x) = exp (-iκL)ũ s (x -Le y ) on ΣLs (13)
The tunnel displacements ũt (x, κ, ω) in Ωt should satisfy the following equations:

div σ t (ũ t ) + ρ t b = -ρ t ω 2 ũt in Ωt (14) t t (ũ t ) = f t on Γσt (15) ũt (x) = exp (-iκL)ũ t (x -Le y ) on ΣLt (16)
On the boundary Σ between the tunnel Ωt and the soil Ωs , displacement continuity and stress equilibrium conditions are imposed by the following equations:

ũs = ũt on Σ (17) t s (ũ s ) + t t (ũ t ) = 0 on Σ (18)
The global solution is recovered using the inverse Floquet transform.

D. Numerical solution of the generic problem

The numerical solution of the problem stated in the previous subsection will be built using the classical domain decomposition approach using FEM for the structure and BEM for the soil.

D.1 The subdomain approach

As the tunnel domain Ωt is bounded, the displacement field ũt (x, κ, ω) in the tunnel can be decomposed on a basis of functions Ψ n (x, κ)(n = 1, . . . , N ) that are periodic of the second kind:

ũt (x, κ, ω) = N n=1 Ψ n (x, κ)c n (κ, ω) = Ψ(x, κ)c(κ, ω) (19)
The soil displacements ũs (x, κ, ω) are subsequently written as the sum of an elastodynamic wave field ũ0 (x, κ, ω) that is equal to zero on the interface Σ and a scattered wave field ũsc (x, κ, ω):

ũs (x, κ, ω) = ũ0 (x, κ, ω) + ũsc (x, κ, ω) (20)
The displacement ũ0 (x, κ, ω) is written as the sum of the incident wave field ũi (x, κ, ω) and the locally diffracted wave field ũd0 (x, κ, ω), so that equation ( 20) may be rewritten as:

ũs (x, κ, ω) = ũi (x, κ, ω) + ũd0 (x, κ, ω) + ũsc (x, κ, ω) (21)
The sum of the locally diffracted wave field and the scattered wave field is called the diffracted wave field. The displacement field ũs (x, κ, ω) in the soil can be further decomposed into:

ũs (x, κ, ω) = ũi (x, κ, ω) + ũd0 (x, κ, ω) + N n=1 ũdn (x, κ, ω)c n (κ, ω) (22)
The following boundary conditions hold on the restriction Σ of the interface Σ on the reference cell:

ũdn (x, κ) = Ψ n (x, κ) on Σ ũi (x, κ, ω) + ũd0 (x, κ, ω) = 0 on Σ (23)
A standard Galerkin procedure is used to write the stress equilibrium on in a weak sense for any function Ψ n (x, κ) in the basis and the following linear system of equations is obtained:

K t (κ) -ω 2 M t (κ) + K s (κ, ω) c(κ, ω) = F t (κ, ω) + F s (κ, ω) (24)
where the matrices K t (κ) and M t (κ) are the stiffness and mass matrices of the tunnel, the matrix K s (κ, ω) is the impedance matrix of the soil, the vector F t (κ, ω) is the generalized force vector transmitted from the track to the tunnel and the vector F s (κ, ω) is the generalized force vector due to the incident wave field.

D.2 The tunnel impedance

The kinematics of the structure is described by the dynamic eigenmodes Φ n (x) of the unit cell, that are the solutions of the generalized eigenvalue problem:

K FEM t Φ n = ω 2 n M FEM t Φ n ( 25
)
where K FEM t and M FEM t are the finite element stiffness and mass matrix of the unit cell. The eigenmodes Φ n (x) are periodic functions of the first kind on the unit cell Ωt as the following kinematical conditions are imposed:

Φ n (x + Le y ) = Φ n (x) ∀x ∈ ΣLt & ∀x + Le y ∈ ΣRt (26)
The fields Ψ(x, κ) are subsequently constructed as the following periodic functions of the second kind:

Ψ n (x, κ) = exp (-iκỹ)Φ n (x) (27)
The matrices K t (κ) and M t (κ) in the governing equation ( 24) are the stiffness and mass matrices of the tunnel:

K tIJ = Ωt σ(Ψ * I ) : ǫ(Ψ J )dΩ = Ψ * T I K FEM t Ψ J (28) M tIJ = Ωt ρ t Ψ * I • Ψ J dΩ = Ψ * T I M FEM t Ψ J (29)
while F t is the generalized force vector transmitted from the track to the tunnel:

F tI = Σ Ψ * I • f dV = Ψ * T I f (30) D.

The soil impedance

The matrix K t (κ, ω) in equation ( 24) is the impedance matrix of the soil, defined as:

K sIJ = Σ Ψ * I • t s (ũ dJ )dΣ (31)
where the stresses t s (ũ dI ) are calculated with a periodic boundary element formulation with Green-Floquet functions defined on the periodic structure with period L along the tunnel. In equation (24), F s (κ, ω) is the generalized force vector due to an incident wave field:

F sI = - Σ Ψ * I • (t(ũ i ) + t(ũ d0 )) dΣ (32)
The stress vector t(ũ d0 ) is also computed using a periodic boundary element formulation. In the case of dynamic tunnel-soil interaction problem where the external loading is applied on the tunnel invert, however, the force vector F s (κ, ω) can generally be disregarded. If u G ij (y, x, ω) denotes the Green's tensor or the reponse in the direction e j in the receiver x of a horizontally layered halfspace due to a unit harmonic force in the direction e i in the source point y, then the Green-Floquet tensor ũGF ij (ỹ, x, κ, ω) is defined as:

ũGF ij (ỹ, x, κ, ω) = +∞ n=-∞ exp (+inκL)u G ij (y + nLe y , x, ω) (33)
The dynamic representation theorem now becomes:

c( ξ)ũ i ( ξ, κ, ω) = Σ∪ ΣLs ∪ ΣRs tj (ũ)(x, κ, ω)ũ GF ij ( ξ, x, κ, ω)dΣ - Σ∪ ΣLs∪ ΣRs tGF ij ( ξ, x, κ, ω)ũ j (x, κ, ω)dΣ (34)
After discretization of the unknown displacement and traction vectors along the boundary Σ between the tunnel and the soil, the following system of equations is obtained:

ŨGF (κ, ω) t(κ, ω) = TGF (κ, ω)ũ(κ, ω) (35)
This system of equations is used to solve for the unknown tractions t s (ũ dJ ) in equation ( 31), which allows to compute the soil impedance matrix K t (κ, ω).

D.4 Wave propagation in the soil

Once the modal coordinates c(κ, ω) are computed from the general equation ( 24), the displacements ũt (x, κ, ω) of the tunnel are obtained from equation ( 19). The tractions tt (Ψ n )(x, κ, ω) imposed by each mode Ψ n (x, κ) are first computed using a periodic boundary element formulation. The traction vector tt (x, κ, ω) on Σ is finally obtained as:

tt (x, κ, ω) = N n=1 tt (Ψ n )(x, κ, ω)c n (κ, ω) (36)
The incident wave field ũinc ( ξ, κ, ω) is obtained by application of the dynamic representation theorem in the unbounded soil domain corresponding to the reference cell:

ũinc i ( ξ, κ, ω) = Σ ũGF ij ( ξ, x, κ, ω) ttj (x, κ, ω) -tGF ij ( ξ, x, κ, ω)ũ tj (x, κ, ω)dΣ (37)
where u GF ij ( ξ, x, κ, ω) and t GF ij ( ξ, x, κ, ω) are the Green-Floquet tensors and the integration is performed on the tunnel-soil interface Σ in the reference cell.

The incident wave field in the soil is obtained after evaluation of the Floquet transformation.

III. Numerical example

The metro tunnel on the line RER B of RATP at Cité Universitaire in Paris is a masonry cut-and-cover tunnel with two tracks at a shallow depth of about 9.3 m below the free surface and a width of 11.9 m (figure 1). The slab thickness is 0.6 m at the top and 0.4 m at the bottom, while the wall thickness is 1.5 m. The masonry is assumed to have a Young's modulus E t = 14000 MPa, a Poisson's ratio ν t = 0.15, a density ρ t = 2400 kg/m 3 and a hysteresis damping ratio β t = 0.02.

The tunnel is considered to be infinitely long with a cross section that is invariant in the longitudinal e y direction. The length L of the reference cell is chosen as 0.3 m. The reference cell is modelled with 8-node isoparametric brick elements (figure 2). Preliminary calculations only consider the transfer functions between the tunnel invert and the free field; therefore, the track is not yet included in the finite element model of the reference cell.

The eigenmodes Φ n (x) of the reference cell of the tunnel are subjected to the periodicity condition (26). Figure 3 shows two modes of the reference cell that involve flexural motion in the transverse (x, z)-plane. The modes Ψ(x, κ) are subsequently constructed using the periodicity condition (27). As calculations will be made for excitation frequencies upto 80 Hz, 30 eigenmodes of the tunnel segment (upto 225 Hz) are used to describe the kinematics of the tunnel.

The tunnel is embedded in a soil consisting of a shallow layer of 1.6 m of fill material, a layer of Beauchamp sand with a thickness of approximately 3.2 m and a stiffer layer with a thickness of 7.8 m with marl and gravel on top of a halfspace consisting of chalk. A Spectral Analysis of Surface Waves (SASW) test has been performed to determine the thickness and the dynamic characteristics of the soil layers [START_REF] Pyl | Determination of the dynamic soil characteristics with the SASW method at the site of Cité universitaire in Paris[END_REF] and revealed the presence of a shallow layer with a thickness of d = 1.4 m and a shear wave velocity C s = 115 m/s on top of a layer with d = 3.0 m and C s = 220 m/s on a halfspace with C s = 315 m/s. A Poisson's ratio ν s = 0.4, a density ρ s = 1700 kg/m 3 and a material damping ratio β s = 0.05 is assumed in all layers.

A. Response in the wavenumber-frequency domain

The response of the tunnel-soil system due to a unit harmonic load the tunnel invert in the frequency range between 0.1 Hz and 80.1 Hz are computed. The frequency step is equal to 2 Hz. Two forces are applied on both edges of the generic cell, resulting in a uniform wavenumber content. Computations are made in the wavenumber-frequency domain. For low frequencies upto 15 Hz, wavenumbers are sampled upto κ max = 0.6 m -1 with a step ∆κ = 0.015 m -1 , which is largely sufficent as the solution is shown to rapidly decay with increasing κ. For higher frequencies between 15 and 80 Hz, a sampling in the slowness domain corresponding to p max = κ max /ω = 0.004 s/m and ∆p = 0.0001 s/m is preferred. Figure 4 shows the real parts of the diagonal elements K t (6, 6) and K t [START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF][START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF] of the tunnel impedance, corresponding to the flexible bending modes 6 and 8, as a function of the slowness p = κ/ω and the frequency ω. At low values of the slowness and the frequency, the inertial term dominates the real part of the tunnel impedance, while at high values of the slowness and the frequency, the stiffness term dominates the tunnel impedance.

B. Impedance of the soil

The impedance matrix K s (κ, ω) of the soil is computed according to equation (31), using a periodic boundary element formulation. Figure 5 shows the real and the imaginary part of the elements K s [START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF][START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF] and K s [START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF][START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF] of the soil impedance matrix, corresponding to modes 6 and 8, as a function of the slowness p = κ/ω and the frequency ω.

The (negative) imaginary part represents attenuation due to radiation and material damping in the soil. At zero slowness or infinite phase velocity in the longitudi- Frequency(Hz) Slowness(s/m) Ks [START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF][START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF][real] a. Re (K s [START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF][START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF]). nal y-direction, the absolute value of the imaginary part increases, resulting in higher attenuation. For a fixed frequency, the absolute value of the imaginary part decreases for increasing slowness or decreasing phase velocity; it finally becomes very small, reflecting the absence of radiation damping in the soil at high values of the slowness.

C. Response of the tunnel

The solution of the general system of equations ( 24) in the wavenumber-frequency domain finally results in the modal coordinates c(κ, ω). Figure 6 shows the modulus of the modal coordinates c 6 and c 8 as a function of the slowness p = κ/ω and the frequency ω. The response of the tunnel in the wavenumber-frequency domain is obtained through the synthesis of the contribution of all modes according to equation (19). The stresses along the tunnel-soil interface are evaluated and the dynamic representation theorem (34) is used to compute the incident wave field in the soil in the wavenumber-frequency domain.

D. Response in the free field

In the present example, the response is computed in the soil at points on the free surface and at the level of the tunnel base, at both sides of the tunnel upto a distance of 24 m from the central axis of the tunnel, as well as on a vertical line between the tunnel roof and the free surface.

Figure 7a shows the modulus of the transfer functions between the tunnel and 4 points at the free surface in the same cross section where the force is applied. A dominant peak around 14 Hz can clearly be observed. Figure 8 shows the real part of the displacements of the tunnel and the soil due to a harmonic loading on the tunnel invert at 14 Hz. It is clearly demonstrated that the soil above the tunnel and the tunnel roof move in phase. The peak at 14 Hz on figure 7a is due to resonance that can be explained with a single degree of freedom system where the tunnel acts as a spring and the soil above the tunnel represents the mass. This is confirmed in figure 7b where the displacements of the tunnel roof and the soil above the tunnel are shown to be equal.

IV. Experimental results

The results of the free field transfer function measurements between the track and the free field are presented here. Before going into the discussion about the results, the characteristics of the track and the experimental setup are briefly reviewed below.

A. The characteristics of the track

Two classical ballast tracks supported on mono-block concrete sleepers are running in the tunnel. The mass per unit length of the UIC60 rail is 60 kg/m and the vertical bending stiffness is 6.4 × 10 6 Nm 2 . The rail is supported on grooved rubber pads with a thickness of 9 mm which are resting on the sleepers. Mono-block concrete sleepers (VXP U61) are used. Each sleeper has a length of 2.27 m, a width of 0.29 m and a of 0.14 m. The sleeper distance 0.6 m. The mass of each sleeper is 200 kg and Young's modulus is 30 GPa. The thickness of the ballast layer is 0.4 m.

The direct rail receptance, measured with a weak static pre-load and a small dynamic force amplitude, shows a first resonance around 80-90 Hz, from which a ballast stiffness per unit area of 100 MN/m 3 and a loss factor of 0.80 can be estimated. The second resonance frequency around 600 Hz allows to derive a dynamic rail pad stiffness of about 375 MN/m and a loss factor of 0.50 (measured at 600 Hz and a small force amplitude, without a static pre-load). The pinned-pinned frequency of the track is about 1200 Hz, which is a common value for tracks with a UIC60 rail and a sleeper distance of 0.60 m. Measurements of direct sleeper receptance confirm the values of the natural frequencies of the track as determined by the direct rail receptance.

B. The experimental setup

During the transfer function measurements, the excitation on the left rail of track 1 is generated with an impact hammer with a mass of 5.3 kg and a soft head as to excite in the low frequency range below 400 Hz. A load cell is placed on the impact hammer and measures the impact force on the rail head. Figure 11 shows the time history and frequency content of the vertical impact force. The peak value of the force is about 10.75 kN, while the frequency content is limited to the range below 400 Hz. Figure 12 shows the position of accelerometers for measuring transfer functions between the track and the free field. The impact force and corresponding accelerations are measured in 16 channels for 14 impacts of the hammer.

C. Response in the free field

Figure 13 shows the time history of the free field vertical accelerations as a function of the distance perpendicular to the tunnel for a hammer impact on the left rail of track 1. The function values in this seismogram have been rescaled so that the arrival times and the dispersive nature of the wave propagation can better be appreciated. On the contrary, the attenuation of the waves with increasing distance to the track is no longer visible.

Figures 14 and15 show the time history and frequency content of the free field vertical acceleration on the top of the tunnel (0 m) and at 64 m from the tunnel. The maximum amplitude of the acceleration a z (0, 0, 0, t) in the near field (figure 14a) is about 0.017 m/s 2 . The peak value of the acceleration a z (64, 0, 0, t) (figure 15a) in the far field is 0.001 m/s 2 . These results demonstrate the decrease in the amplitude of the acceleration with increasing distance from the tunnel due to material and radiation damping in the soil. It is observed from figures 14b and 15b that the dominant frequency range is between 30 to 120 Hz. At

D. Validation of the numerical model

Figure 16 shows the comparison of the computed and measured vertical seismograms in the free field. The computation has been performed with the numerical model described earlier in this paper. It is seen from figure 16 that at distances 4 m, 8 m, 12 m, 16 m, 24 m and 32 m, the computed vertical seismograms match reasonably well with experimentally obtained seismograms, exhibiting a similar For points farther away from the tunnel, the waves travel for longer distances in the third layer and then refract to the surface.

V. Conclusion

The theoretical basis for the development of the numerical model for prediction of ground borne vibrations has been discussed in this paper. The model is based on a coupled periodic FE-BE method using a substructure approach, where the Floquet transformation is used to exploit the periodicity of the tunnel in longitudinal direction.

The experimental validation in this paper is limited to a preliminary comparison of the transfer functions between the tunnel and the free field and the computed response. A reasonable good correspondence of the arrival times and dispersive nature of the waves is observed.

Fig. 1 .

 1 Fig. 1. Cross section of the metro tunnel on the line RER B of RATP at Cité Universitaire.

Fig. 2 .Fig. 3 .

 23 Fig. 2. (a) Cross section of the model of the metro tunnel with the soil stratigraphy and (b) finite element mesh of the reference cell and the applied forces.

  Re (K t[START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF][START_REF] Elhabre | Modélisation de l'interaction sismique sol-fluideparois moulées suivant une approche périodique[END_REF]).

  Re (K t[START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF][START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF]).

Fig. 4 .

 4 Fig. 4. Real part of the diagonal elements K t (6, 6) and K t (8, 8) of the tunnel impedance matrix as a function of the slowness p = κ/ω and the frequency ω.

  Re (K s[START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF][START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF]).

  Im (K s[START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF][START_REF] De Barros | Moving Green's functions for a layered visco-elastic half-space[END_REF]).

Fig. 5 .

 5 Fig. 5. Real and imaginary parts of the diagonal elements K s (6, 6) and K s (8, 8) of the soil impedance matrix as a function of the slowness p = κ/ω and the frequency ω.

Fig. 6 .

 6 Fig. 6. Modulus of the modal coordinates c 6 and c 8 of the tunnel as a function of the slowness p = κ/ω and the frequency ω.

  0,0) above the tunnel top Point (-2,0,0) above the tunnel top a. Free surface.

  5,-0.15,-9.25) (0,-0.15,-2.3) Point at tunnel top (0,-0.15,-2.3) Point above tunnel top (0,0,0) Point at tunnel floor (-2.5,-0.15,-9.25) b. Tunnel and free surface.

Fig. 7 .

 7 Fig. 7. Modulus of the transfer functions for points (a) at the free surface and (b) at the tunnel floor, the tunnel roof and the free surface.

Fig. 8 .

 8 Fig. 8. Real part of the displacements of the tunnel and the soil due to a harmonic excitation on the tunnel invert at 14 Hz at (a) t = 0, (b) t = T /8, (c) t = 2T /8, and (d) t = 3T /8.

Figure 9

 9 Figure9shows the real part of the displacements of the tunnel and the soil due to a harmonic loading on the tunnel invert at 80 Hz. Waves are propagating in the tunnel in the longitudinal and the transverse direction and radiated into the soil. The projection of the displacement field on the free surface in figure10clearly reveals elliptical wave fronts, as the waves that propagate in the transverse direction have

Fig. 9 .

 9 Fig. 9. Real part of the displacements of the tunnel and the soil due to a harmonic excitation on the tunnel invert at 80 Hz at (a) t = 0, (b) t = T /8, (c) t = 2T /8, and (d) t = 3T /8.

Fig. 10 .

 10 Fig. 10. Real part of the displacements at t = 0 at 80 Hz at the free surface.

  Fig. 11. (a) Time history and (b) frequency content of impact force.

Fig. 12 .

 12 Fig. 12. Location of the measurement points.

Fig. 13 .

 13 Fig. 13. Time history of the free field vertical accelerations as a function of the distance to the tunnel.

Fig. 14 .

 14 Fig. 14. (a) Time history and (b) frequency content of the free field acceleration on the tunnel top (0 m).

  Fig. 15. (a) Time history and (b) frequency content of the free field acceleration at 64 m.

Fig. 16 .

 16 Fig. 16. Comparison of computed and experimentally obtained vertical seismograms.

Acknowledgements

The results presented in this paper have been obtained within the frame of the EC-Growth project G3RD-CT-2000-00381 CONVURT ("The control of vibration from underground railway traffic"). The financial support of the European Community is gratefully acknowledged. In situ vibration measurements have been performed in collaboration with Vibratec and RATP.


Gérard H E Duchamp

Christophe Tollu

G Érard

H E Duchamp

Sweedler's duals and Schützenberger's calculus

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

This paper is entirely devoted to questions of rationality which arose, seemingly independently, in automata theory (Schützenberger's calculus) and in the dualization of multiplication (Sweedler's duals). As in the classical (univariate) case, rationality shows itself twofold : on functions and on expressions. The rationality framework exposed here can be considered as the noncommutative analogue of linear recurrences [START_REF] Fliess | Matrices de Hankel[END_REF]. It is known, (see [START_REF] Connes | Noncommutative geometry[END_REF][START_REF] Lando | Lectures on generating functions[END_REF]), that it is equivalent to state that the coefficients of the Taylor expansion of a function satisfy a linear recurrence or that the generating function itself is rational (i.e. the quotient of two polynomials). This equivalence has a counterpart in the theory of rational expressions and this (nowadays classical) theory can be considered as "localized at zero" (i.e. analogous to the theory of rational functions without a pole at zero). The paper is organized as follows.

In section 2, one sets out the theory of representative functions which were introduced in [START_REF] Sweedler | Hopf algebras[END_REF] and are now standard in the theory of algebraic groups [START_REF] Hochschild | Basic theory of algebraic groups and Lie algebras[END_REF]. Most of the material of this section (preparatory to the subsequent ones) is not new and can be found in several domains (but we believe that, beyond the needs of exposition, the description of the link itself will be of some use to these domains). The scope of proposition (2.1) can be harmlessly extended to semigroups [START_REF] Abe | Hopf algebras[END_REF]. As representative functions on the free monoid are the core of automata theory [START_REF] Berstel | Rational series and their languages[END_REF][START_REF] Eilenberg | Automata, languages and machines[END_REF][START_REF] Schützenberger | On the definition of a family of automata[END_REF][START_REF] Schützenberger | On a theorem of R. Jungen[END_REF], the domain of their scalars can (and, in fact, had to) be considerably enlarged to include structures allowing matrix (with unit) computations. This is the domain of semirings [START_REF] Golan | Power Algebras over Semirings with Applications in Mathematics and Computer science[END_REF][START_REF] Golan | Semirings and Affine Equations over Them: Theory and Applications[END_REF], the scalars of automata theory. Section 4 prepares the link between representative functions and rational expressions by means of the notion of star (which is the "positive analogue" of the inverse [START_REF] Cayley | On certain results related to quaternions[END_REF][START_REF] Heyting | Die Theorie der linearen Gleichungen in einer Zahlenspezies mit nichtkommutativer Multiplikation[END_REF][START_REF] Lewin | Fields of fractions for group algebras of free groups[END_REF][START_REF] Passman | The algebraic structure of group rings[END_REF][START_REF] Richardson | Simultaneous linear equations over a division ring[END_REF]) and star closure (which is the analogue of rational closure in classical algebra1 ). The framework is then ready for a correct exposition of rational expressions which is the main concern of section 4. In section 5, we apply what has been constructed to the dualization of bialgebras, then recovering known results. In section 6, we show how to apply this "rational calculus" to solve the carrier problem in combinatorial physics.

Representative functions on a semigroup

The aim of this section is to discuss the dualization of bi-algebras and Hopf algebras. This problem, solved by Sweedler's duals, is the following. Let k be a field and (B, ., ∆, 1 B , ε) be a k-bialgebra ; we know that, if B is finitedimensional (resp. graded in finite dimensions), the dual (resp. graded dual) endowed with the transpose structure is a bialgebra and that, in case B is a Hopf algebra this statement carries over. Now the question can be asked. What is the good notion of retricted dual for the general (i.e. ungraded finite or infinite dimensional) case ? Analysing the dualization of the structural operations (., ∆, 1 B , ε) of B, one sees at once that only the dualization of the multiplication is problematic as, in the general case, the codomain of the transpose of . is larger than B * ⊗ B * . The first result follows (and somehow extends) [START_REF] Abe | Hopf algebras[END_REF]. To state it, we need the notion of (left and right) shifts of functions on a semigroup. Let k be a field, (S, .) a semigroup and

f ∈ k S . For each s ∈ S define f s : x → f (sx) (right shift of f ) and s f : x → f (xs) (left shift of f ), s f t : x → f (txs) (bi-shift of f ).
Then, we have. Proposition 2.1 (see also [START_REF] Abe | Hopf algebras[END_REF] section 2.2) Let k be a field, (S, .) a semigroup and f ∈ k S . The following are equivalent : i) The family (f s ) s∈S is of finite rank in k S ii) The family ( s f ) s∈S is of finite rank in k S iii) The family ( s f t ) s,t∈S is of finite rank in k S iv) There exists a double family (g i , h i ) 1≤i≤n of functions such that

∀x, y ∈ S f (xy) = n i=1 g i (x)h i (y) (1)
v) There exists λ ∈ k 1×n , γ ∈ k n×1 and µ : (S, .) → (k n×n , ×) a morphim of semigroups such that (∀s ∈ S)(f (s) = λµ(s)γ). Moreover, if S admits a neutral (i.e. is a monoid), its image by µ of (v) above can be chosen to be the unity matrix.

Proof -Omitted The elements of k S which fulfill the above conditions will be called representative functions on S and denoted R(k, S, .) [START_REF] Abe | Hopf algebras[END_REF][START_REF] Chari | A guide to quantum groups[END_REF][START_REF] Duchamp | Hopf Algebras in General and in Combinatorial Physics: a practical introduction[END_REF]

then f ∈ k S is representative iff f | T is so.
In particular, if S = T (1) , the monoid obtained by adjunction of a unity to T , one has that f

∈ k S is representative iff f | T is.
vi) The proof of proposition (2.1) can also be found in [START_REF] Hochschild | Basic theory of algebraic groups and Lie algebras[END_REF] where it does not use the structure of group.

Semirings

Throughout the text "monoid" stands for "semigroup with unit". iii) The semiring ([-∞, +∞[, max, +), called in the literature "(max,plus)-semiring". iv) In the semiring ([0, +∞[, +, ×), the laws are continuous at infinity and then can be completed. We obtain a semiring ([0, +∞], +, ×) which is suited for multiplicities arising in repeated additions of positive values during iterations.

Example 3.2

The following example is fundamental and will be used in the definition of CM-modules. Let (M, +) be a commutative monoid, then (End(M), +, •) (defined as though M were a group) is a semiring. The units are respectively, the constant mapping

M ∋ m → 0 M for + and Id M : m → m for •.
The structure of semiring defines a category larger than that of rings, the morphisms being defined similarly.

Let (k i , + i , × i ), i = 1, 2 be two semirings, a mapping φ : k 1 → k 2
is called a morphism of semirings iff it is a morphim for the two structures of monoids (additive and multiplicative), then compatible with the laws and units of k 1 and k 2 .

The definition of modules (here called CM-modules as they are constructed on Commutative Monoids as vector structure) follows also the classical pattern.

The structure of a (left) k-CM-module is given by the following data -a commutative monoid (M, +)

-a morphism (the scaling morphism) of semirings s : k → End(M).

The structure of (right) k-CM-module is defined by replacing End(M) by End op (M) the opposite semiring (constructed with the opposite multiplicative law). Bi-and multimodules are defined as in [START_REF] Bourbaki | Algebra, chapter III[END_REF] and follow the general philosophy of "structures with operators".

Example 3.3 Let X be a set, then k X , the set of all functions X → k is naturally endowed with a structure of kk bimodule defined as in the case when k is a ring. So is k (X) , the set of finitely supported functions of k S (actually a sub-k-k bimodule of k S ).

The free monoid generated by a set X (finite of infinite) is the set of words (i.e. finite sequences of elements of X comprising the empty one denoted by 1 X * ) endowed with the concatenation law.

4 Shift operators and rational closure Let (M, .) be a (commutative or not) monoid. For a function f : M → k and a ∈ M, we define the following shift operators [START_REF] Abe | Hopf algebras[END_REF][START_REF] Berstel | Rational series and their languages[END_REF][START_REF] Eilenberg | Automata, languages and machines[END_REF][START_REF] Jacob | Représentations et substitutions matricielles dans la théorie matricielle des semigroupes[END_REF] • f a : x → f (ax) (right shift)

• a f : x → f (xa) (left shift)
We also have to describe the analog, for CM-algebras, of full subalgebras and full subalgebraic closures (see [START_REF] Bourbaki | Théories spectrales[END_REF] Ch. 1.1.4) and this requires the notion of summability [START_REF] Berstel | Rational series and their languages[END_REF].

Definition 4.1 A family (f i ) i∈I of functions M → k is called summable iff, for each m ∈ M, (f i (m)) i∈I is finitely supported. Then, the mapping m → i∈I f i (m) is denoted i∈I f i and called the sum of (f i ) i∈I . As a consequence, it is easily checked that, if M is locally finite ([14] Vol. A VII.4) and f : M → k is without constant term (i.e. f (1 M ) = 0 k ), then the family (f n ) n∈N (convolutional powers) is summable and its sum n∈N f n (2)
will be denoted f * and called the star of f . Note 4.2 There is a lot of literature about the star problem (see [START_REF] Golan | Power Algebras over Semirings with Applications in Mathematics and Computer science[END_REF][START_REF] Golan | Semirings and Affine Equations over Them: Theory and Applications[END_REF]). For a general discussion of star-type solutions in a semiring, see [START_REF] Duchamp | Algebraic elimination of ǫ-transitions[END_REF]. Now, we are in the position of stating the Kleene-Schützenberger theorem.

Theorem 4.3 Let M = X * be a free monoid, k a semiring and f ∈ k M . The following are equivalent i) the family (f w ) w∈M belongs to a finitely generated shift-invariant left-submodule ii) the family ( w f ) w∈M belongs to a finitely generated shift-invariant right-submodule iii) there exist a row λ ∈ k 1×n , a column γ ∈ k n×1 and a representation (of monoids)

µ : M → (k n×n , ×) such that (∀w ∈ M)(f (w) = λµ(w)γ).
If X is finite, then (i-iii) above are also equivalent to: iv) f lies in the rational closure of X (i.e. the smallest subalgebra of k X closed under the star operation and containing X).

Remark 4.4 i) Rational elements in the sense of (iii) infinitely many (linearly) independent shifts. That is why finitely generated shift-invariant submodules are needed in the general case. As an example one may consider

S = (a * ) 2 ∈ N a .
One can check at once that a -k S = k.(a * )+S and then for all k, a -(k+1) S / ∈ span(a -s S) k s=0 . ii) One can remove the hypothesis of freeness of M if k is a field. Indeed, in this case, the submodule can be taken as generated by the shifts (right or left) of f and the representation is automatically compatible with the relations of M. iii) Here the star is used as the localization at one (i.e. with positive formulas) of the inverse function. Indeed, with coefficients in a ring, if we are at the neighbourhood of 1, the condition

(1 -x)(1 + y) = 1 (resp. (1 + y)(1 -x) = 1
) is equivalent to y = x + xy (resp. y = x + yx). These self-reproducing positive conditions are taken as the definition of "y is a star of x" in a semiring (see [START_REF] Duchamp | Algebraic elimination of ǫ-transitions[END_REF]). iv) The condition (iv) in theorem (4.3) is known under the name of Kleene-Schützenberger theorem as, when k is specialized to B, this is actually Kleene's theorem. In this sense, this theorem lies at the frontier of harmonic analysis (the set of representative functions is dense in the Fourier space of compact groups), spectral theory (the notion of full subalgebra closure comes from this theory [START_REF] Bourbaki | Théories spectrales[END_REF]) and theoretical computer science (the notion of star was developped as a computational model of iteration and the notion of a semiring was developped to cope with general scalars as diverse as the ones arising in stochastic automata theory and shortest path problems).

In the general case (X not necessarily finite), Kleene-Schützenberger's theorem has to be modified as follows.

Theorem 4.5 Let M = X * be a free monoid, k a semiring and f ∈ k M . The following are equivalent i) the family (f w ) w∈M belongs to a finitely generated shift-invariant left-submodule ii) the family ( w f ) w∈M belongs to a finitely generated shift-invariant right-submodule iii) there exist a row λ ∈ k 1×n , a column γ ∈ k n×1 , and a representation (of monoids) µ : M → (k n×n , ×) such that (∀w ∈ M)(f (w) = λµ(w)γ). iv) the function f lies in the rational closure of kX = { x∈X α(x)x} α∈k X (i.e. the smallest subalgebra of k X closed under the star operation which contains kX).

Remark 4.6 The rational closure of X is, in fact, the intersection of the set of elements characterized by (i-iii) (i.e. Sweedler's dual of k X ), and the algebra

F ⊂X F f inite
k F of the series whose support involves a finite alphabet.

Rational expressions

The construction of [START_REF] Champarnaud | Derivatives of rational expressions and related theorems[END_REF] was localized at zero, we extend it here to any localization i.e. for any mapping Λ : X → k. As the rational closure involves a unary law (the star) partially defined, the definition of universal formulas for this closure needs some caution. Indeed, we need to build in parallel a "character" (the constant term) const so that all proper expressions should have a star. One first defines, as in [START_REF] Champarnaud | Derivatives of rational expressions and related theorems[END_REF] the completely free expressions (or formulas) as the terms of the universal algebra defined on X ∪ {0 E } (0 E , which does not belong to X will serve as a null or void expression and will be mapped to the zero series). This algebra will be denoted E cf (X, k). More precisely

-If x ∈ X ∪ {0 E } then x ∈ E cf (X, k). -If E, E 1 , E 2 ∈ E cf (X, k), and λ ∈ k then E 1 + E 2 ∈ E cf (X, k), E 1 • E 2 ∈ E cf (X, k) λE ∈ E cf (X, k), Eλ ∈ E cf (X, k).
-

If E ∈ E cf (X, k) then E * ∈ E cf (X, k).
The partial function const : E cf (X, k) → k (constant term) is constructed as follows:

1. If x ∈ X const(x) = Λ(x) and const(0

E ) = 0 k . 2. If E, E i ∈ E cf (X, k), i = 1, 2 and λ ∈ k then const(E 1 + E 2 ) = const(E 1 ) + const(E 2 ), const(E 1 • E 2 ) = const(E 1 ) • const(E 2 ) const(λE) = λconst(E), const(Eλ) = const(E)λ. 3. If const(E) = 0 k then const(E * ) = 1 k .
The domain of const will be called rational expressions and denoted E Λ (X, k). For example 0 * E ∈ E Λ (X, k). Let now Θ : X → k A be a mapping such that, for every x ∈ X, the constant term of Θ(x), i.e. the coefficient of 1

A * in Θ(x), is equal to Λ(x) (in symbols, ∀x ∈ X), [1 A * ]Θ(x) = Λ(x)
). Following recursively (1-2-3) above, we can construct a polymorphism φ Θ : E Λ (X, k) → k A which is a morphism for the laws (2 internal and 2 external) and the star. Moreover δ 1 A * • φ Θ = const (i.e. const can be considered as a "constant term function" for the expressions). The image of φ Θ is exactly the rational closure of the set {Θ(x)} x∈X .

Dual laws and bialgebras

Let ∆ : A → A ⊗ A be any comultiplication (i.e. A is a k-coalgebra). It is known that its dual (A * , t ∆) is an algebra and if A is coassociative (resp. cocomutative, counital), A is associative (resp. commutative, unital) [START_REF] Abe | Hopf algebras[END_REF]. We would like here to enlarge the framework of [START_REF] Duchamp | Direct and dual laws for automata with multiplicities[END_REF]. If A is an algebra, let us call dual law on A * a law of the form t ∆ for some (not necessarily coassociative) comultiplication on A. In [START_REF] Duchamp | Direct and dual laws for automata with multiplicities[END_REF] were considered the dual laws on k X ≃ k X * in order to prove that the Hadamard and Infiltration products, which were known to preserve rationality, were essentially the only (along with an interpolation between the two) alphabetic (associative and unital) dual laws between series. The notion of dual law provides an implementation scheme for the automata so that the rationality preservation is naturally effective. Theorem 6.1 Let A be a k-algebra and ∆ : A → A ⊗ A be a comultiplication which is a morphism of algebras. Then

i) If k is a field, Sweedler's dual A • of A is closed under the dual law t ∆. ii) If k is a semiring and A = k X , A • is closed under the dual law t ∆. Note 6.2 i)
The theorem is no longer true if ∆ : A → A ⊗ A is arbitrary (i.e. not necessarily a morphism) as shows the following counterexample. With ∆ : Q

[x] → Q[x] ⊗ Q[x] such that ∆(x) = 1 n! (x n ⊗ x n ), one has t ∆( 1 1 -x , 1 1 -x ) = exp(x).
ii) In (i) above, the restriction on scalars (to be a field) can be extended to inductive limits of PIDs. iii) Comultiplications different from morphisms can preserve rationality. For example, let ∆ : k X → k X ⊗ k X be a morphism and ∆ 1 : k X → k X ⊗ k X be a linear mapping which coincides with ∆ except for a finite number of words of X ⋆ . It can be checked that ∆ 1 , although not a morphism, preserves rationality.

Let us now return to the case of a bialgebra (B, ., ∆, 1 B , ε). The following proposition says that, if a linear form on B is transformed by t m (m : B ⊗ B → B is just the multiplication mapping) into an element of B * ⊗ B * , it must be of the form exhibited in (iv) of Proposition 2.1. Let us state that precisely.

Proposition 6.3 Let (B, m, ∆, 1 B , ε) be a bialgebra and f ∈ B * . 1) The following are equivalent i) t m(f ) ∈ B * ⊗ B * (for the canonical embedding B * ⊗ B * ֒→ (B ⊗ B) * ) ii) f ∈ R(k, B, .)
iii) ker(f ) contains a finite-codimension one-sided ideal iv) ker(f ) contains a finite-codimension two-sided ideal v) There exist λ ∈ k 1×n , γ ∈ k n×1 and µ : (B, +, .) → (k n×n , +, ×) a morphim of kalgebras (associative with units) such that (∀x ∈ B)(f (x) = λµ(x)γ).

2) Moreover, let B 0 be the set of linear forms which are decomposable as 1 -(i) above. Then (B 0 , t ∆, t m, t ε, t 1 B ) is a bialgebra, and if B admits an antipode σ (i.e. is a Hopf algebra), one has t σ(B 0 ) ⊂ B 0 and (B 0 , t ∆, t m, t ε, t 1 B , t σ) is a Hopf algebra.

An application of rational expressions to Combinatorial Physics

In a joint work with J. Katriel, one of us gave the solution of the problem of matrix coefficients in the Fock space of carriers between two levels. We give here a brief review of these continued fractions-type formulas and provide a sketch of their proof.

In [START_REF] Katriel | Ordering relations for q-boson operators, continued fractions techniques, and the q-CBH enigma[END_REF] was considered, as a Fock space, a general vector space V over a field k with basis |e n n = 0, 1, • • •, equipped with its natural grading

V = ⊕ n∈Z V n with V n := ke n ; V -n-1 := {0} for n ≥ 0 (3)
We consider the words in f, g:

w(f, g) := f p 1 g q 1 f p 2 g q 2 • • • f pn g qn (5)
the degree (excess) π e (w) of which is n k=1 (q kp k ) (this is, for algebraists, the degree of the graded operator f p 1 g q 1 f p 2 g q 2 • • • f pn g qn ). This provides a representation µ of a twoletter free monoid {b -, b + } * on V by µ(b -) = f ; µ(b + ) = g, which is graded for the weight on {b -, b + } * . In order to keep the reading of a word from left to right, one performs the action on the right. Thus V becomes a {b -, b + } * right module by e 0 .b -= 0 ; e n+1 .b -= α n+1 e n ; e n .b + = β n+1 e n+1 , n ≥ 0 [START_REF] Cayley | On certain results related to quaternions[END_REF] one is interested by the matrix elements 2 .

e n .{b -, b + } i |e m = ω (i) n→m . (7)
Define W (i) n→m as follows

W (i) n→m = {w ∈ {b -, b + } i | (π e (w) = m -n) and (w = uv =⇒ π e (u) ≥ -n)} . (8)
The following proposition characterizes W (i)

n→m as an universal transporter between level n and level m.

Proposition 7.1 i) If all the weights α

n ; n ≥ 1, β n , n ≥ 0 are nonzero, W (i)
n→m is exactly the set of words of length i such that e n .w|e m = 0. ii) In all cases the latter is a subset of W

(i) n→m i. e. {w ∈ {b -, b + } i | e n .w|e m = 0} ⊂ W (i) n→m (9) Indeed, W (i)
n→m admits factorizations as "noncommutative continued fractions". In order to do this, we recall for the reader the definition of Dyck, positive Dyck and negative restricted (by depth) Dyck codes, respectively:

D = {w ∈ π -1 e (0)|w = uv and ε / ∈ {u, v} =⇒ π e (u) = 0} D + = {w ∈ D|w = uv and ε / ∈ {u, v} =⇒ π e (u) > 0} D (n) -= {w ∈ D -| min uv=w (π e (u)) = -n} . (10)
Now, identifying the subsets defined previously with their characteristic series in N b -, b + , one has the following factorizations (here, equalities between rational expressions of b±, D + and D

(n) -).

2 An interpretation of a similar coefficient in terms of paths, namely for the computation of +∞ -∞

x i p n p m w(x)dx

where (p n ) n∈N is a family of orthogonal polynomials for the weight w(x) can be found in [START_REF] Viennot | Une théorie combinatoire des polynômes orthogonaux[END_REF].

Proposition 7.2 Set W n→m = i≥0 W (i)
n→m . 1) One has the following factorizations for W n→n+k

(D + + D (n) -) * (b + D * + ) k = → (k-1) i=0 ((D (n+i) - ) * b + ) (D + + D (n+k) - ) * if k ≥ 0 (D + b -) k (D + + D (n+k) - ) * = (D + + D (n) -) * → (-k) i=1 ((b -(D (n-i) - ) * ) otherwise (11)
2) One has the following self-reproducing equations for the Dyck codes

D + = b + (D + ) * b -; D (n) -= b -(D (n-1) - ) * b + ; D (0) -= ∅ (12)
The statements of this proposition can be considered as a "noncommutative continued fraction" expansion of W n→m . As a result [START_REF] Katriel | Ordering relations for q-boson operators, continued fractions techniques, and the q-CBH enigma[END_REF], with µ(W (i) m→n )|n = ω (i) n→m |m and T n→n+k :=

i≥0 t i ω (i) n→n+k , (13)
using proposition (7.2) and the representations µ t (b±) = tb± (observe that µ 1 = µ), one can expand T n→n+k as a product of continued fractions. Let

F + n = 1 1 - t 2 α n+1 β n+1 1 - t 2 α n+2 β n+2 1 - t 2 α n+3 β n+3 1 -• • • = 1 1 -E + n F - n = 1 1 - t 2 α n β n 1 - t 2 α n-1 β n-1 1 - t 2 α n-2 β n-2 1 -• • • = 1 1 -E - n ( 14
)
and

F n = 1 1 -E + n -E - n . (15)
Then, if k ≥ 0, we obtain

T n→n+k = t k F n+k k-1 i=0 F - n+i = t k F n k i=1 F + n+i ( 16
)
and, if k ≤ 0

T n→n+k = t -k F n+k -k-1 i=0 F + n-i = t -k F n -k i=1 F - n-i (17)

Conclusion

The dualization problem solved by Sweedler's duals has striking relations with language theory (for this last point, see [START_REF] Berstel | Rational series and their languages[END_REF]). A true "engineer-like" calculus was developped in order to handle the rational closure mentionned above. This set of formulas is mainly based on a recursion to compute the "star of a matrix" (the formulas and a complete discussion can be found in [START_REF] Duchamp | Algebraic elimination of ǫ-transitions[END_REF] and are reminiscent of general formulas giving the inverse of a matrix decomposed in blocks). This calculus is powerful enough to be the main ingredient in investigating rationality properties within various domains (see [START_REF] Connes | Noncommutative geometry[END_REF][START_REF] Duchamp | Un critère de rationalité provenant de la géométrie noncommutative[END_REF] for noncommutative geometry, [START_REF] Cauchon | Séries de Malcev-Neumann sur le groupe libre et questions de rationalité[END_REF] for functions on the free group and [START_REF] Berstel | Rational series and their languages[END_REF][START_REF] Eilenberg | Automata, languages and machines[END_REF] for automata theory) and sufficiently expressive to give exact developments of some transfer coefficients in Combinatorial Physics [START_REF] Katriel | Ordering relations for q-boson operators, continued fractions techniques, and the q-CBH enigma[END_REF]. These rational expressions are generic in the sense that any Sweedler's dual can be described by them. In this way, as for Dirac's notation, we can hope to inherit the computational skill developped through fourty years of practice.

-a set k 3 -

 3 Semirings are the structures adapted to matrix (with unity) computation. A semiring (k, +, ×) consists of the following data two binary laws +, × on k such that -(k, +) and (k, ×) are monoids, the first being commutative, their neutrals will be denoted respectively 0 k and 1 k -× is left and right distributive over + -0 k is an annihilator i. e. (∀x ∈ k)(0 k .x = x.0 k = 0 k ) Example 3.1 i) Any ring. ii) The boolean semiring B = {0, 1} endowed with the laws x ⊕ y = x + yxy and x ⊗ y = xy.

  and scalar product defined by e n |e m = δ n,m . Let f, g be two linear operators on V of degrees -1, +1, respectively. Generically, they read for n ≥ 0; f |e 0 := 0; f |e k+1 = α k+1 |e k ; g|e k := β k+1 |e k+1

  .

	Remark 2.2 i) When k is only a PID, proposition (2.1) above still holds with the five equivalent conditions, orbits and ranks being computed in kS ( k is the fraction field of k).
	ii) If S is finite, R(k, S, .) = k S and if S is a group, one has
	R(k, S, .) = k S ⇐⇒ S is finite
	iii) If S is a semigroup, the equivalences above are false in general as shown by the
	following counterexample. Let G be a finite group and endow S = N × G with the law
	(n, g) * (m, h) = (0, gh). It can be easily checked that (S, * ) is a semigroup and that
	R(k, S, * ) = k S .
	iv) When W is a shift-invariant subspace of k S and f ∈ W ∩ R(k, S, .), the families
	(f s ) s∈S , ( s f ) s∈S , ( s f t ) s,t∈S are of course in W ∩ R(k, S, .) (and are of finite rank). Two
	useful examples of such "relative representative functional spaces" are with W = C(S)
	(continuous functions, S being a topological semigroup) and W = S * (linear forms, S
	being an algebra with its product as semigroup law).
	v) If T ⊂ S is a subsemigroup with finite set-theoretical complement (S\T is finite)

and "sous-algèbres pleines" in spectral theories[START_REF] Bourbaki | Théories spectrales[END_REF].

Acknowledgements

The authors wish to acknowledge support from Agence Nationale de la Recherche (Paris, France) under Program No. ANR-08-BLAN-0243-2 as well as support from "Projet interne au LIPN 2009" "Polyzêta functions".