Inverse Scattering for Soft Fault Diagnosis in Electric Transmission Lines

Qinghua Zhang, Michel Sorine and Mehdi Admane Abstract-Today's advanced reflectometry methods provide an efficient solution for the diagnosis of electric transmission line hard faults (open and short circuits), but they are much less efficient for soft faults, in particular, for faults resulting in spatially smooth variations of characteristic impedance. This paper attempts to fill an important gap for the application of the inverse scattering transform to reflectometry-based soft fault diagnosis: it clarifies the relationship between the reflection coefficient measured with reflectometry instruments and the mathematical object of the same name defined in the inverse scattering theory, by reconciling finite length transmission lines with the inverse scattering transform defined on the infinite interval. The feasibility of this approach is then demonstrated by numerical simulation of lossless transmission lines affected by soft faults, and by the solution of the inverse scattering problem effectively retrieving smoothly varying characteristic impedance profiles from reflection coefficients.

Index Terms-inverse scattering, fault diagnosis, transmission line, telegrapher's equations, Zakharov-Shabat equations.

I. INTRODUCTION

The fast development of electronic devices in modern engineering systems and in consumer products comes with an increasing number of electric wires in these equipments, and also inevitably with more and more failures related to electric connections. This fact has motivated research projects on methods for the diagnosis of faults in electric transmission lines. In this context, the technology of reflectometry has been extensively studied by different research groups [START_REF] Smith | Analysis of spread spectrum time domain reflectometry for wire fault location[END_REF], [START_REF] Auzanneau | A simple and accurate model for wire diagnosis using reflectometry[END_REF]. It consists in injecting electric signals from one end or from both ends of a line and in analyzing the reflected electric waves. At least for laboratory experimentations, currently this technology is able to detect and to locate hard faults (open circuit or short circuit) up to an accuracy of about 10 cm. For soft faults, however, no satisfactory result has been reported, to our knowledge.

If the term "hard fault" refers to (almost) open or short circuits, any other kind of conduction faults may be qualified "soft". In [START_REF] Griffiths | The invisible fray: A critical analysis of the use of reflectometry for fray location[END_REF] the difficulty for detecting soft faults has been investigated, mainly assuming weak (but abrupt) characteristic impedance changes caused by soft faults (such faults are called "frays" in [START_REF] Griffiths | The invisible fray: A critical analysis of the use of reflectometry for fray location[END_REF]). The reported results show that wave reflections caused by the discontinuities at the borders of fray segments are comparable to, or even smaller than, the effects of noises and disturbances (cable movements, vibrations, etc.). For this reason, it seems impossible to detect and to locate such faults with reflectometry-based methods in real application environments.

In this paper, the considered soft faults correspond to spatially smooth and arbitrary variations of characteristic impedance. Though there is no impedance discontinuity when such a fault occurs, the (smooth) impedance inhomogeneity can cause wave reflections. In addition to the difficulty related to the weakness of such reflections, it is conceptually difficult to deal with such faults, because they cannot be located in terms of impedance discontinuities. Moreover, no particular form of the characteristic impedance variations is assumed a priori. It is clear that any reflectometry-based method aiming at detecting and locating impedance discontinuities is not suitable for this problem.

The purpose of this paper is not to handle the problem reported in [START_REF] Griffiths | The invisible fray: A critical analysis of the use of reflectometry for fray location[END_REF], namely weak reflections of soft faults drowned out by noises and disturbances. Instead, the particular case of "smooth soft faults" are investigated, under the assumption that noises and disturbances are limited by well controlled experimental conditions. Though this assumption is not realistic in many applications, the considered non-trivial problem merits investigations, while waiting for a satisfactory solution to noise and disturbance attenuation. Indeed, by making experiments at the ends of a transmission line, it is obviously difficult to inspect spatially smooth and arbitrary variations of characteristic impedance.

If computing reflection coefficients from specified transmission line characteristics is called a direct problem, its inverse problem corresponds to the purpose of this paper, namely retrieving distributed characteristics of a transmission line from reflection coefficients. This problem is related to the inverse scattering theory. Some early investigation on this topic was reported in [START_REF] Moses | Properties of dielectrics from reflection coefficients in one dimension[END_REF]. It then attracted the attention of more researchers in 1970's and 1980's. For instance, time domain approaches were studied with continuous and discrete transmission line models respectively in [START_REF] Gopinath | Inversion of the telegraph equation and the synthesis of nonuniform lines[END_REF] and [START_REF] Bruckstein | Inverse scattering for discrete transmission-line models[END_REF]. The frequency domain inverse scattering approach is adopted in this paper for its direct connection with the measurements of frequency domain network analyzers. By transforming the telegrapher's equations to Schrödinger equation and to Zakharov-Shabat equations, Jaulent's pioneer work [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF] has founded the theoretic basis for the inverse scattering problem of transmission lines in frequency domain. More recently, multiconductor lines have been studied in [START_REF] Knockaert | Recovering lossy multiconductor transmission line parameters from impedance or scattering representations[END_REF], but uniform transmission lines were assumed.

Despite these theoretic results, mostly dated a quarter of a century ago, to our knowledge, no successful experiment on their application to real transmission line fault diagnosis has been reported. The studies presented in this paper constitute an important step towards practical applications. Following Jaulent's frequency domain approach [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF], the first contribution of this paper is to clarify the relationship between the reflection coefficient measured in practice and the mathematical object of the same name in the inverse scattering theory, by reconciling finite length transmission lines with the inverse scattering transform defined on the infinite interval. The feasibility of this approach is then illustrated by numerical simulations of lossless transmission lines with smooth variations of characteristic impedances, and by the solution of the inverse scattering problem, which effectively retrieves the characteristic variations of the simulated transmission lines.

Only lossless transmission lines are considered in this paper. In practice, the lossless assumption is reasonable for high quality transmission lines of moderate length. The more general lossy case, shortly discussed in Section IV, is currently under study. Some preliminary results will be reported in [START_REF] Tang | Inverse scattering for lossy electric transmission line soft fault diagnosis[END_REF].

This paper is organized as follows. Section II is about the formulation of the inverse scattering problem for transmission lines, with, in particular, Section II-D clarifying the relationship between the reflection coefficient measured with reflectometry instruments and the mathematical object of the same name defined in the inverse scattering theory. Simulation results are presented in Section III. Concluding remarks are made in Section IV.

II. TRANSMISSION LINE AND INVERSE SCATTERING

This section is devoted to the transformations of the telegrapher's equations leading to the Zakharov-Shabat equations and the related inverse scattering problem. The basic result has been established by Jaulent [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF], but it was not clear how to reconcile finite length transmission lines with the inverse scattering transform defined on the infinite interval. The main purpose of this section is to fill this gap.

A. The telegrapher's equations and the reflection coefficient

In this subsection the telegrapher's equations for lossless transmission lines will be recalled, so will the reflection coefficient as defined in electric engineering.

Consider a lossless transmission line connected to an alternating voltage source of frequency k at the left end and to a load at the right end, as illustrated in Figure 1 (k will be treated as a wavenumber in the inverse scattering problem). Let L(z) and C(z) be respectively the distributed inductance and capacitance along the longitudinal axis z of the transmission line, then, following [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF], the voltage V (k, z) and the current I(k, z) at any point z along the longitudinal axis of the line are governed by the telegrapher's equations

d dz V (k, z) -ikL(z)I(k, z) = 0 (1a) d dz I(k, z) -ikC(z)V (k, z) = 0 ( 1b
)
where i is the imaginary unit.

In Figure 1, if the source connected to the left end of the transmission line is an instrument for reflection coefficient measurement, typically a network analyzer, then the measured reflection coefficient is

r e (k) = Z(k, z S ) -Z S Z(k, z S ) + Z S (2)
where

Z(k, z S ) = V (k, z S ) I(k, z S ) (3)
is the input impedance of the transmission line, and Z S is the internal impedance of the source.

In textbooks on transmission lines, reflection coefficients are often defined at the load end or at the source end of uniform transmission lines, such as in [10, page 17] and in [11, page 193]. These reflection coefficients characterize more the adaptation of the load or the source than the transmission line itself. The reflection coefficient r e (k) expressed in (2) characterizes better the transmission line. Its definition is related to the S-parameters of transmission lines [12, chapter 13].

The quantity r e (k), as a function of k, will be referred to as engineering reflection coefficient in this paper, as it corresponds to a measurement frequently used in engineering practice.

B. From telegrapher's equations to Zakharov-Shabat equations

To connect the telegrapher's equations to the scattering theory, the first step is to replace the space coordinate z by the wave propagation time through the Liouville transformation

x(z) = z 0 L(s)C(s)ds. (4)
As 1/ L(s)C(s) is the wave propagation velocity at the point s, x(z) is clearly the wave propagation time from the position 0 to the position z.

After this coordinate change from z to x, by abuse of notation, L(z(x)) will be simply written as L(x), and similarly for C(x), V (k, x), I(k, x). Then the telegrapher's equations (1) become

dV (k, x) dx = ikZ 0 (x)I(k, x) (5a) dI(k, x) dx = ikZ -1 0 (x)V (k, x) (5b)
with

Z 0 (x) = L(x) C(x) (6)
being the characteristic impedance of the lossless transmission line at the point x. Accordingly, the circuit of Figure 1 is redrawn, by simply replacing z, z S and z L respectively by x, x S = x(z S ) and x L = x(z L ), as shown in Figure 2.

In the new coordinate system, define the reflected and incident waves1

ν 1 (k, x) = 1 2 Z -1 2 0 (x)V (k, x) -Z 1 2 0 (x)I(k, x) (7a) ν 2 (k, x) = 1 2 Z -1 2 0 (x)V (k, x) + Z 1 2 0 (x)I(k, x) (7b)
Some direct computations from ( 5) and ( 7) then lead to

dν 1 (k, x) dx + ikν 1 (k, x) = q(x)ν 2 (k, x) (8a) dν 2 (k, x) dx -ikν 2 (k, x) = q(x)ν 1 (k, x) (8b) with q(x) = - 1 4 d dx ln L(x) C(x) = - 1 2Z 0 (x) d dx Z 0 (x). ( 9
)
These computations have implicitly used the following assumption.

Assumption 1: For x ranging within the transmission line, the functions L(x) and C(x) are such that the characteristic impedance Z 0 (x) as defined in ( 6) is differentiable.

Equations ( 8), known as Zakharov-Shabat equations, constitute the main mathematical object connecting the electric transmission line to the scattering theory. The related inverse scattering transform computes the so-called potential function q(x) from the reflection coefficient (in a sense to be recalled in this paper). It is clear that the characteristic impedance along the x-axis as defined in equation ( 6) can be determined from this potential function q(x) through the ratio L(x)/C(x). Hence the inverse scattering transform provides a powerful mathematical tool for the monitoring of characteristic impedance variations in electric transmission lines.

C. Jost solutions of the Zakharov-Shabat equations and the theoretic reflection coefficient

In this subsection recalling some concepts of the scattering the pair ν 1 (k, x) and ν 2 (k, x) denotes a solution of the Zakharov-Shabat equations [START_REF] Knockaert | Recovering lossy multiconductor transmission line parameters from impedance or scattering representations[END_REF], and for the moment let us forget their relationship with V (k, x) and I(k, x) through [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF].

In [START_REF] Lamb | Elements of Soliton Theory[END_REF]- [START_REF] Sobey | Scattering analysis and synthesis of wave trains[END_REF], inverse scattering problems are studied for systems modeled by equations ( 8) with x ranging over the infinite interval (-∞, +∞). It is assumed that the potential function q(x) involved in (8) decays sufficiently fast when x → ±∞, so that the solutions of the Zakharov-Shabat equations [START_REF] Knockaert | Recovering lossy multiconductor transmission line parameters from impedance or scattering representations[END_REF] have their limiting behaviors expressed in terms of exp(±ikx).

Consider a particular solution of (8) (not necessarily given by ( 7)) with its limiting behaviors satisfying

lim x→+∞ ν 1 (k, x) = 0 (10a) lim x→-∞ ν 2 (k, x) exp(-ikx) = 1 (10b)
This solution is known as a Jost solution. Its left limiting behavior defines a reflection coefficient r th (k):

r th (k) = lim x→-∞ ν 1 (k, x) ν 2 (k, x) exp(2ikx) (11)
This coefficient r th (k) is sometimes referred to as the left reflection coefficient, because it represents the ratio between the incident wave ν 2 (k, x) coming from -∞ and the reflected wave ν 1 (k, x). Similarly, the right reflection coefficient can also be defined.

Through the inverse scattering transform, the potential function q(x) can be retrieved from r th (k) [START_REF] Lamb | Elements of Soliton Theory[END_REF]- [START_REF] Sobey | Scattering analysis and synthesis of wave trains[END_REF]. This quantity r th (k), defined with the limiting behaviors of a Jost solution of the Zakharov-Shabat equations, will be referred to as the theoretic reflection coefficient in this paper.

D. The relationship between the two reflection coefficient definitions

The purpose of this subsection is to clarify the relationship between the definitions of the engineering reflection coefficient (2) and the theoretic reflection coefficient [START_REF] Weisshaar | Transmission lines[END_REF].

The theoretic reflection coefficient r th (k) has been defined with a solution of the Zakharov-Shabat equations for x ranging from -∞ to +∞. To relate this definition to the finite length transmission line circuit illustrated in Figure 2, the circuit will be replaced by an equivalent circuit as specified in the following proposition.

Proposition 1: Build a new circuit, as shown in Figure 3, by modifying the circuit of Figure 2 as follows:

• insert a uniform transmission line of length a with characteristic impedance Z 0 (x) = Z S between the source and the left end; • insert a uniform transmission line of length b with characteristic impedance Z 0 (x) = Z L between the right end and the load; • add a phase shift -ka to the source voltage. Then, for any positive values of a and b, the new circuit of Figure 3 and the one of Figure 2 As this proposition holds for any (arbitrarily large) positive values of a and b, the quantities V (k, x), I(k, x) and Z 0 (x) of the extended circuit are virtually defined for all x ∈ R, so are the related ν 1 (k, x), ν 2 (k, x) and q(x).

To ensure well defined potential function q(x) as formulated in (9) at the two connection points x S and x L in the extended Fig. 3. The same transmission line as in Figure 2, but now the source is connected through a uniform line of length a with characteristic impedance Z 0 (x) = Z S , the load is connected through a uniform line of length b with characteristic impedance Z 0 (x) = Z L , and the source phase is shifted by -ka.

circuit of Figure 3, the characteristic impedance Z 0 (x) should be smooth at these two points, implying the following assumption.

Assumption 2: The transmission line in Figure 2 is locally adapted at its left and right ends such that

lim x→x + S dZ 0 (x) dx = lim x→x - L dZ 0 (x) dx = 0
and

Z S = Z 0 (x S ) lim x→x + S Z 0 (x) (12)
Z L = Z 0 (x L ) lim x→x - L Z 0 (x). ( 13
)
Proposition 2: When a and b both tend to +∞, for the circuit of Figure 3, the pair ν 1 (k, x) and ν 2 (k, x) formed by [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF] constitutes a solution of the Zakharov-Shabat equations ( 8) for all x ∈ R. The theoretic reflection coefficient r th (k) defined in [START_REF] Weisshaar | Transmission lines[END_REF] with the limiting behaviors of ν 1 (k, x) and ν 2 (k, x) is related to the engineering reflection coefficient r e (k) defined in (2) through the equality

r th (k) = r e (k) exp(2ikx S ) (14)
Proof of Proposition 2. This proof will consist of the following steps: (i) The pair ν 1 (k, x) and ν 2 (k, x) is a solution of the Zakharov-Shabat equations for all x ∈ R.

(ii) The pair ν 1 (k, x) and ν 2 (k, x) is a Jost solution, up to a coefficient common to ν 1 (k, x) and ν 2 (k, x). (iii) The limit in the definition (11) of the theoretic reflection coefficient is reached at x = x S . (iv) r th (k) = r e (k) exp(2ikx S ).

All the transformations and equations ( 4)-( 9) are valid for the circuit of Figure 3 with arbitrary a and b. It means that the Zakharov-Shabat equations ( 8) are derived from the telegrapher's equations with ν 1 (k, x) and ν 2 (k, x) defined in [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF] for all x ∈ [x S -a, x L + b]. It is then clear that the pair ν 1 (k, x) and ν 2 (k, x) is a solution of the Zakharov-Shabat equations (8) for x ∈ [x S -a, x L + b]. When a and b tend to +∞, the solution is valid for x ∈ R.

It will be shown in the following that the pair ν 1 (k, x) and ν 2 (k, x) satisfies the limiting conditions [START_REF] Dworsky | Modern transmission line theory and applications[END_REF], up to a factor common to ν 1 (k, x) and ν 2 (k, x).

Let us first examine the value of ν 1 (k, x) for any x ∈ [x L , x L +b]. As the load Z L is matched to the extended uniform line, the equalities

V (k, x) = Z 0 (x)I(k, x) = Z L I(k, x) hold for any x ∈ [x L , x L + b]. Therefore, following (7a), ν 1 (k, x) = 1 2 Z -1 2 0 (x)Z 0 (x)I(k, x) -Z 1 2 0 (x)I(k, x) = 0 for all x ∈ [x L , +∞) when b → +∞. It is then clear that the first limiting condition (10a) is satisfied. Now consider the case of x ∈ [x S -a, x S ].
As the characteristic impedance Z 0 (x) = Z S is constant over the uniform line extension, the potential function q(x) = 0, according to [START_REF] Tang | Inverse scattering for lossy electric transmission line soft fault diagnosis[END_REF]. Hence the Zakharov-Shabat equations (8) become

dν 1 (k, x) dx + ikν 1 (k, x) = 0 dν 2 (k, x) dx -ikν 2 (k, x) = 0
for all x ∈ (-∞, x S ], when a → +∞. These are two decoupled first order differential equations, whose solutions are the reflected and the incident waves

ν 1 (k, x) = c 1 (k) exp(-ikx) (15) ν 2 (k, x) = c 2 (k) exp(ikx) (16)
with some coefficients c 1 (k) and c 2 (k) independent of x.

As equation ( 16) holds for all x ∈ (-∞, x S ], the second limiting condition (10b) is satisfied up to the coefficient c 2 (k). The limiting condition [START_REF] Dworsky | Modern transmission line theory and applications[END_REF] would be fully satisfied if ν 1 (k, x) and ν 2 (k, x) were both divided by c 2 (k). This normalization is not necessary in view of the definition of the theoretic reflection coefficient as formulated in [START_REF] Weisshaar | Transmission lines[END_REF], as ν 1 (k, x) and ν 2 (k, x) share the same factor. It is thus confirmed that the limiting behaviors of ν 1 (k, x) and ν 2 (k, x) are in agreement with the definition of the theoretic reflection coefficient.

Because ( 15) and ( 16) both hold for all x ≤ x S , the limiting value in the definition [START_REF] Weisshaar | Transmission lines[END_REF] of the theoretic reflection coefficient is reached at x = x S . Hence this definition can be replaced by

r th (k) = ν 1 (k, x S ) ν 2 (k, x S ) exp(2ikx S ) (17)
It then follows from [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF] that

r th (k) = Z -1 2 0 (x S )V (k, x S ) -Z 1 2 0 (x S )I(k, x S ) Z -1 2 0 (x S )V (k, x S ) + Z 1 2 0 (x S )I(k, x S ) exp(2ikx S ) = V (k, x S ) -Z 0 (x S )I(k, x S ) V (k, x S ) + Z 0 (x S )I(k, x S ) exp(2ikx S ) = V (k, x S ) -Z S I(k, x S ) V (k, x S ) + Z S I(k, x S ) exp(2ikx S )
where the last equality is due to the fact that Z 0 (x S ) = Z S following [START_REF] Orfanidis | Electromagnetic Waves and Antennas[END_REF]. Rewrite (3) as Z(k, x S ) = V (k, x S )/I(k, x S ), then

r th (k) = Z(k, x S ) -Z S Z(k, x S ) + Z S exp(2ikx S ).
Compare this result with (2) by reminding that Z(k, x S ) and Z(k, z S ) denote the same quantity under different coordinate systems, the relationship between the two reflection coefficients expressed in ( 14) is then proved.

This result indicates that the engineering reflection coefficient r e (k) and the theoretic reflection coefficient r th (k) are identical up to a phase shift of 2kx S . It appears that the definition of r th (k) depends on the choice of the origin of the x-axis. By choosing the origin such that x S = 0, the two reflection coefficients coincide exactly.

E. Inverse scattering for characteristic impedance monitoring

To retrieve the potential function q(x) (and thus the characteristic impedance Z 0 (x)) from the reflection coefficient only, it is assumed that the Zakharov-Shabat equations ( 8) have no bound state (square integrable solution for x ∈ R), following [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF].

The inverse scattering transform consists of the following steps for computing the potential function q(x) and L(x)/C(x) = Z 2 0 (x), from the reflection coefficient r th (k) as defined in [START_REF] Weisshaar | Transmission lines[END_REF]. See [START_REF] Lamb | Elements of Soliton Theory[END_REF], [START_REF] Eckhaus | The inverse scattering transformation and the theory of solitons: an introduction[END_REF] for more details.

1) Let the Fourier transform of the reflection coefficient r th (k) be

ρ(x) = 1 2π +∞ -∞ r th (k) exp(-ikx)dk (18)
2) Solve the integral equations (known as Gel'fand-Levitan-Marchenko equations) for its unknown kernels A 1 (x, y) and A 2 (x, y):

A 1 (x, y) + x -y A 2 (x, s)ρ(y + s)ds = 0 A 2 (x, y) + ρ(x + y) + x -y A 1 (x, s)ρ(y + s)ds = 0
3) Compute the potential function q(x) through

q(x) = 2A 2 (x, x) 4
) By inverting equation ( 9), compute

L(x) C(x) = L(x S ) C(x S ) exp -4 x x S q(s)ds
In practice, the integral equations have to be solved numerically, by discretizing the kernels A 1 (x, y) and A 2 (x, y) over a grid in the x-y plane. By choosing the origin of the x-axis such that x S > 0, the potential q(x) = 0 for x < 0. It is then sufficient to compute the kernels A 1 (x, y) and A 2 (x, y) in the region x ≥ |y|. The numerical inverse scattering algorithm used for producing the simulation results presented in the next section is the one of [START_REF] Xiao | An efficient algorithm for solving Zakharov-Shabat inverse scattering problem[END_REF], which is an improved variant of the algorithm of [START_REF] Frangos | A numerical solution to the Zakharov-Shabat inverse scattering problem[END_REF]. Both algorithms produce similar results in our simulations, but the one of [START_REF] Xiao | An efficient algorithm for solving Zakharov-Shabat inverse scattering problem[END_REF] is much faster.

III. SIMULATION STUDY

In this section, results of simulation will be presented to confirm the validity of the approach presented in this paper. The first step is to implement a numerical simulator generating the reflection coefficient from the specified profiles L(z) and C(z) of a transmission line.

A. Transmission line simulator

For the circuit of Figure 2, let

Z(k, x) = V (k, x) I(k, x) (19)
be the apparent impedance at any point x ∈ [x S , x L ]. It is then straightforward to check that the telegrapher's equations [START_REF] Gopinath | Inversion of the telegraph equation and the synthesis of nonuniform lines[END_REF] imply that Z(k, x) satisfies the Riccati equation

d dx Z(k, x) = ikZ 0 (x) -ikZ -1 0 (x)Z 2 (k, x) (20)
where Z 0 (x) is the characteristic impedance as defined in [START_REF] Bruckstein | Inverse scattering for discrete transmission-line models[END_REF]. For a given value of k, by initializing Z(k, x) with the load impedance at the right end, namely 20) can be solved for x < x L . In particular, the value Z(k, x S ) at the left end of the transmission line is obtained. The reflection coefficient r e (k) is then computed with (2), and r th (k) with [START_REF] Eckhaus | The inverse scattering transformation and the theory of solitons: an introduction[END_REF].

Z(k, x L ) = Z L equation (
It then appears that the main computation of the simulator consists of the numerical solution of the Riccati equation (20) for the value of the apparent impedance Z(k, x S ). This computation is repeated with different values of k to cover a sufficiently large spectrum.

B. Simulation results

For the first example, a smoothly increasing L(z) profile is simulated. The capacitance is kept to a constant value C = 0.1nF/m, and the corresponding ratio L/C is depicted in Figure 4, in both z and x coordinates, where the units for L/C are µH/nF. The simulated reflection coefficient r th (k) (modulus and phase) is shown in Figure 5. The L/C profile computed through the inverse scattering transform is compared to the original simulated profile in Figure 6.

Remark that the inverse scattering transform computes the ratio L/C as a function of x. In practice it would be more useful to inspect the ratio as a function of z, the true spatial coordinate of the transmission line. Like all reflectometry methods, the information obtained by observing incident and reflected waves is related to the wave propagation time x. Without knowing the wave propagation velocity (varying along the line in the considered case), it is impossible to convert the result from x to z. However, for moderate variations of the ratio L/C (under the soft fault assumption), the two profiles in x and in z are similar, as shown in Figure 4 for the considered example. It is thus practically reasonable to assume a constant wave propagation velocity to convert the computed L/C ratio to the z-coordinate. Fig. 6. The smoothly increasing L/C profile computed by inverse scattering and compared to the simulated profile. The two curves may not be distinguishable if printed in black and white. For the second example, a hump-shaped L/C profile is simulated, as illustrated in Figure 7. The simulated reflection coefficient is plotted in Figure 8 assumption of perfect model and noise-free measurement. To evaluate the robustness of the method to random uncertainties, noises can be added in the simulation study. The second simulation example is then repeated by adding a Gaussian noise to each simulated value of the reflection coefficient. The standard deviation of the added noise is equal to about 6% of the maximum absolute value of the simulated r th (k). The noise-corrupted reflection coefficient (modulus and phase) is also illustrated in Figure 8 (dashed lines). The hump-shaped L/C profile is again computed by inverse scattering, but now from the noise-corrupted reflection coefficient. The result is illustrated in Figure 10. Though the reflection coefficient has been significantly corrupted, the inverse scattering transform is still capable of retrieving reasonably the L/C profile. This example is mainly to investigate the numerical stability of the inverse scattering method, not to simulate the disturbances in real application environments, like those studied in [START_REF] Griffiths | The invisible fray: A critical analysis of the use of reflectometry for fray location[END_REF]. Though in theory the computation of the Fourier transform of the reflection coefficient requires the values of r th (k) for k ranging from 0 to +∞ (in [START_REF]Analytical and numerical solution to the two-potential Zakharov-Shabat inverse scattering problem[END_REF] it ranges from -∞ to +∞, but r th (-k) = r * th (k), since ρ(x) is real), in the above simulation examples the values of r th (k) are truncated to 20MHz. The frequency bandwidth after truncation is related to the details of the characteristic impedance to be reconstructed: measurements of larger bandwidth capture more details.

IV. CONCLUSION

Though the theoretic basis for the inverse scattering problem of electric transmission lines has been founded about a quarter of a century ago, there has been a gap between such theoretic results and their practical applications. By reconciling finite length transmission lines with the inverse scattering transform defined on the infinite interval, this paper constitutes an attempt to fill this gap. Moreover, the simulation studies reported in this paper confirm the feasibility of this approach.

The success of practical applications of this method will depend on the accuracy of the reflection coefficients used for solving the inverse problem. Though simulations have shown that the inverse scattering algorithm behaves reasonably when the reflection coefficients are slightly disturbed, in practice, various noises and disturbances may hide the effects of small faults. Some efficient noise and disturbance attenuation methods remain to be developed, so that methods for soft fault diagnosis, like the one presented in this paper, can be widely applied.

The lossless assumption made in this paper is only reasonable for high quality transmission lines of moderate length. For lossy transmission lines, as shown by Jaulent in [START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF], after the transformations adopted in this paper, an extra step transforms the telegrapher's equations for lossy transmission lines to a pair of Zakharov-Shabat equations similar to equations ( 8), but with two distinct potential functions in the two equations. Similar Gel'fand-Levitan-Marchenko integral equations can be established in this case [START_REF] Eckhaus | The inverse scattering transformation and the theory of solitons: an introduction[END_REF]. Though the numerical algorithms solving these equations, like the one of [START_REF]Analytical and numerical solution to the two-potential Zakharov-Shabat inverse scattering problem[END_REF], remain similar to those of the single-potential Zakharov-Shabat equations, their numerical stability in their applications to lossy transmission lines is more delicate. Some preliminary results of our studies on lossy transmission lines, extending the results presented in this paper, will be reported in [START_REF] Tang | Inverse scattering for lossy electric transmission line soft fault diagnosis[END_REF]. APPENDIX: PROOF OF PROPOSITION 1.

In Figure 3, because the extended uniform line at the right side of x L is connected to the matched load Z L , the total impedance of the extended line plus the matched load is equal to Z L . Therefore, viewed from the left side of x L , the extended line and the matched load behave as the simple load Z L of Figure 2.

When the telegrapher's equations ( 5) is applied to the circuit of Figure 3

I(k, x S ) = 1 Z(k, z S ) + Z S V S (k) (22)
On the other hand, for the circuit of Figure 2, the transmission line and the load Z L can be viewed from the source as an equivalent load of impedance Z(k, z S ). With such an equivalent circuit, it is easy to compute the values of V (k, x S ) and I(k, x S ), which happen to coincide with those of equations ( 21) and ( 22). It means that V (k, x S ) and I(k, x S ) have the same values in Figure 2 and Figure 3. Consequently, for any x ∈ [x S , x L ], V (k, x) and I(k, x) have the same values in these two circuits.

Fig. 1 .

 1 Fig. 1.A transmission line connected to a voltage source and a load, illustrated in z-coordinate. The source internal impedance Z S = Z 0 (z S ) and the load impedance Z L = Z 0 (z L ) are locally matched to the characteristic impedance of the line.

Fig. 2 .

 2 Fig. 2. The circuit of Figure 1 re-illustrated in x-coordinate.

  are equivalent, in the sense that they have the same values of V (k, x) and I(k, x) for anyx ∈ [x S , x L ].Proof of Proposition 1. See the Appendix at the end of this paper.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Simulated smoothly increasing L/C profile in zand x-coordinates.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Simulated hump-shaped L/C profile in zand x-coordinates.

Fig. 9 .

 9 Fig.9. The hump-shaped L/C profile computed by inverse scattering and compared to the simulated profile. The two curves may not be distinguishable if printed in black and white.

  computed by inverse scattering.

Fig. 10 .

 10 Fig. 10. The hump-shaped L/C profile computed by inverse scattering from noise-corrupted reflection coefficient.

2 Z 2 V

 22 for x ∈ [x S -a, x S ] where Z 0 (x) = Z S , the solution writesV (k, x) = α(k) exp(-ikx) + β(k) exp(ikx) I(k, x) = 1 Z S [-α(k) exp(-ikx) + β(k) exp(ikx)]in which α(k) and β(k) are determined by the boundary conditionsV S (k) exp(-ika) -V (k, x S -a) = Z S I(k, x S -a) V (k, x S ) = Z(k, z S )I(k, x S )with Z(k, z S ) being the apparent impedance at the point x S of Figure3, which is also equal to the input impedance of the circuit of Figure2. It then yieldsα(k) = 1 (k, z S ) -Z S Z(k, z S ) + Z S V S (k) exp(ikx S ) β(k) = 1 S (k) exp(-ikx S )andV (k, x S ) = Z(k, z S ) Z(k, z S ) + Z S V S (k)(21)

The pair ν 1 (k, x)-ν

(k, x) defined here is slightly different from the corresponding notation Y in[START_REF] Jaulent | The inverse scattering problem for LCRG transmission lines[END_REF], to be better in agreement with the definition of scattering parameters in engineering practice. Accordingly, a negative sign is also added to the potential function q(x) defined in[START_REF] Tang | Inverse scattering for lossy electric transmission line soft fault diagnosis[END_REF].

This work has been supported by the ANR SEEDS project (Smart Embedded Electronic Diagnosis Systems).


Housem Boulmezaoud

Johannes Kellendonk

Comparing different versions of tiling cohomology

We establish direct isomorphisms between different versions of tiling cohomology. The first version is the direct limit of the cohomologies of the approximants in the Anderson-Putnam-Gähler system, the second is the recently introduced PV-cohomology of Savinien and Bellissard and the third is pattern equivariant cohomology. For the last two versions one can define weak cohomology groups. We show that the isomorphisms extend to the weak versions. This leads to an alternative formulation of the pattern equivariant mixed quotient group which describes deformations of the tiling modulo topological conjugacy.

1 Introduction.

There exist various isomorphic pictures of what one now calls tiling cohomology. Originally it was defined as the groupoid cohomology of the tiling groupoid [START_REF] Kellendonk | The local structure of tilings and their integer groups of coinvariants[END_REF] or as the Čech cohomology of the tiling space (the hull) [START_REF] Anderson | Topological invariants for substitution tilings and their associated C * -algebras[END_REF]. If the tiling space can be seen as an inverse limit of finite simplicial (or cellular) complexes. the cohomology of the tiling space is a direct limit of simplicial (or cellular) cohomology groups [START_REF] Anderson | Topological invariants for substitution tilings and their associated C * -algebras[END_REF], [START_REF] Gähler | Talk given at the conference[END_REF][START_REF] Sadun | Tilings Spaces are Inverse Limits[END_REF]. An intuitively simpler version was later proposed using the concept of pattern equivariant exterior forms [START_REF] Kellendonk | Pattern-equivariant Functions and Cohomology[END_REF][START_REF] Kellendonk | The Ruelle-Sullivan map for actions of R n[END_REF] and then modified using pattern equivariant simplicial (or cellular) cochains [START_REF] Sadun | Pattern-Equivariant Cohomology with Integer Coefficients[END_REF]. Finally, a recent version making full use of the simplicial structure defined by the tiling goes under the name of PV-cohomology [START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF]. Each version has his own advantages and disadvantages. Computationally the groupoid cohomology approach was most successful for cut & and project patterns [START_REF] Forrest | Topological Invariants for Projection Method Patterns[END_REF] and the direct limit approach for substitution tilings [START_REF] Anderson | Topological invariants for substitution tilings and their associated C * -algebras[END_REF]. The newer versions led conceptionally to new insight, especially the pattern equivariant de Rham cohomology. The latter comes about as it naturally offers different choices of cohomology groups based on the distinction between algebraic modules and their topological closure. The original tiling cohomology is isomorphic to strongly pattern equivariant cohomology, the algebraic version. The topologically closed version is the so-called weakly pattern equivariant cohomology, and this one is isomorphic to the tangential cohomology of the hull (seen as a lamination) [START_REF] Kellendonk | The Ruelle-Sullivan map for actions of R n[END_REF]. It has been shown that the theory of deformations of tilings is related to the mixed group of strongly pattern equivariant cochains modulo weakly pattern equivariant coboundaries [START_REF] Kellendonk | Pattern Equivariant Functions, Deformations and Equivalence of Tiling Spaces[END_REF]. It is therefore of quite some interest to determine the latter. However, apart from substitution tilings [START_REF] Clark | When Shape Matters: Deformations of Tiling spaces[END_REF], we have at present no general method to offer to compute this mixed group.

The present article sheds new light on the different versions of pattern equivariant cohomology in that it provides analogues of the weak and the mixed version in the framework of PV-cohomology. To obtain these we provide explicit descriptions of the isomorphisms between the various cohomology groups. We establish in particular an explicit formula for the isomorphism between pattern equivariant de Rham cohomology and PVcohomology and show that it carries the right continuity properties for it to extend from the algebraic to the topological closed versions of the cohomology groups.

Preliminaries.

In this section we provide some known background material on tilings. In particular we recall the definitions of the following versions of tiling cohomology:

1. as direct limit of the simplicial cohomologies of the Anderson-Putnam-Gähler complexes [START_REF] Anderson | Topological invariants for substitution tilings and their associated C * -algebras[END_REF][START_REF] Gähler | Talk given at the conference[END_REF][START_REF] Sadun | Tilings Spaces are Inverse Limits[END_REF] (we refer to that simply as DL-cohomology), 2. as PV cohomology [START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF],

3. as PE de Rham cohomology [START_REF] Kellendonk | Pattern-equivariant Functions and Cohomology[END_REF][START_REF] Kellendonk | The Ruelle-Sullivan map for actions of R n[END_REF],

4. as PE simplicial cohomology [START_REF] Sadun | Pattern-Equivariant Cohomology with Integer Coefficients[END_REF].

We also recall or introduce weak versions of PV and PE cohomology. PE stands for pattern equivariant, s-PE for strongly and w-PE for weakly pattern equivariant.

Tilings and tilings spaces

A tiling of the euclidean space R d is a covering of the space by closed topological disks (its tiles) which overlap at most on their boundaries. We consider here simplicial tilings of (translational) finite local complexity. This means that the tiles are simplices of R d which touch face to face and that there are only finitely many simplices up to translation; we present more details in Section 2.1.1. In the context of tilings of finite local complexity the restriction to simplicial tilings is topologically not a restriction, since there are ways to associate to any tiling of finite local complexity a simplicial tiling which carries the same topological information.

The tiling space or hull of a tiling P is

Ω P := {t x (P)|x ∈ R d } D ,
the completion of the translation orbit of P w.r.t the metric given by

D(P 1 , P 2 ) = inf r>0 { 1 r + 1 |∃x, y ∈ B 1 r (0) : B r [t x (P 1 )] = B r [t y (P 2 )]}
where P 1 and P 2 are tilings of R d . Here t x denotes the translation action t x (Q) = Q -x where Q is a point or a geometric object and we use the notation B r [P] = B r [P ′ ] to indicate that the tiling P and P ′ coincide1 on the r-ball around 0. Tiling spaces can be obtained as inverse limits of simpler spaces. Such a construction was given for substitution tilings by [START_REF] Anderson | Topological invariants for substitution tilings and their associated C * -algebras[END_REF] and then later generalized to all tilings of finite local complexity in [START_REF] Gähler | Talk given at the conference[END_REF][START_REF] Sadun | Tilings Spaces are Inverse Limits[END_REF] and a somewhat similar construction given in [START_REF] Bellissard | Spaces of tilings, finite telescopic approximations and gap-labeling[END_REF]. We present the details of the construction using k-neighborhoods.

Let P be a tiling of R d and t be the tile (or the patch) of P. The first corona of t is the set of its nearest neighbors, the second corona of t is the set of the second nearest neighbors, and so on out to the k th corona. The kneighborhood of a tile is the collection of its jth coronae for j ≤ k. Consider two tiles t 1 , t 2 in P, we say that t 1 is equivalent to t 2 , if t 1 is translationally congruent to t 2 , the equivalence class is called prototile. The condition of finite local complexity implies that there are finitely many prototiles. A kcollared tile is a tile labelled with its k-neighborhood. The labelling of the tile does not change the set covered by the tile, which we call its support, but it is only a decoration of it. A k-collared prototile is the translational congruence class of the k-collared tile. Again, finite local complexity implies that the number of k-collared prototiles is finite. Let f 1 and f 2 to be faces of two k-collared prototiles t1 and t2 . We say f 1 is equivalent to f 2 if t1 has a representative t 1 and t2 has a representative t 2 in P such that the edges corresponding to f 1 and f 2 coincide. The disjoint union of the supports of the k-collared prototiles quotiened out by this equivalence relation on the faces is the set Γ k . Equipped with the quotient topology we consider it as a topological space. It inherits the structure of a complex from P and is therefore called an Anderson-Putnam-Gähler complex.

Let ρ k+1 k : Γ k+1 -→ Γ k the map which associates to a point of a k + 1-collared prototile the same point in the k-collared prototile obtained by forgetting about the (k + 1) th corona in its label. The inverse system (Γ k , ρ k k-1 ) k∈N is called the Anderson-Putnam-Gähler system. Its importance lies in the fundamental result that its inverse limit is homeomorphic to the tiling space

Ω P ∼ = lim ←- k Γ k .
A point of Γ k tells us how to place a patch corresponding to a k-collared tile around the origin of R d . By definition of the inverse limit the surjections ρ k k-1 induce a continuous surjection

ρ k : Ω P -→ Γ k (2.1)
which can be described as follows. Consider the tile of ω ∈ Ω P together with its k-neighborhood which lies on the origin 0 R d . Taking congruence classes w.r.t. translations, the point in the tile above corresponds to a point in a k-collared prototile. ρ k (ω) is this point viewed as a point in Γ k . By the definition of the equivalence for Γ k , this is well defined also in the cases in which a boundary point of a tile lies on the origin. Related to the above is the surjection ρk :

R d -→ Γ k ρk (x) = ρ k (t x (P)). (2.2)

Simplicial tilings and ∆-complexes

For their definition of PV-cohomology Savinien and Bellissard consider tilings whose tiles are finite ∆-complexes which are compatible in the sense that the intersection of two tiles are sub-∆-complexes. Intuitively the reader should simply have in mind a tiling whose tiles are simplices which meet face to face and such that simplices can be pattern equivariantly oriented. We call such tilings simplicial tilings. In more technical tems, a ∆-complex structure (see [START_REF] Hatcher | Algebraic Topology[END_REF]) on a topological space X is a collection of continuous maps σ : ∆ n → X, where

∆ n = {(x 1 , x 2 , • • • , x n+1 ) ∈ R n+1 | n+1 i=1 x i = 1, x i ≥ 0}
is the standard n-simplex in euclidean space. It is supposed that these maps are injective on the interior of ∆ n . They are referred to as the characteristic maps and their images are the so-called cells (of dimension n) of X. It is also supposed that each point of X lies in the interior of exactly one cell. ∆ n has n + 1 faces of dimension n -1, we write ∂ i ∆ n for the face described by coordinates (x 1 , x 2 , • • • , x n+1 ) for which x i = 0 and call it the ith face. We write ∂ i σ for the restriction of σ to ∂ i ∆ n . It is also a characteristic map, but for an n -1 cell. When cells are next to each other, touching along lower dimensional faces, or when characteristic functions are not injective on the boundary but identify different faces, then the corresponding characteristic maps of the lower dimensional faces are identified. The vertices of ∆ n are ordered, the ith vertex being the one having coordinate x i = 1, and this order passes to the faces. It is assumed that the identification of characteristic maps preserves the ordering of the vertices. They are further compatibility conditions among the characteristic maps for which we refer to [START_REF] Hatcher | Algebraic Topology[END_REF][START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF]. We will use the phrasing "the simplex σ of X" to refer to the characteristic map or to the cell it defines in X.

A simplicial complex is special kind of ∆-complex. In [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF] this notion is used for ∆-complexes discribing subsets of euclidean spaces whose cells are actually simplices, i.e. convex hulls of n + 1 vertices (in general position) if n is their dimension. Lemma 2.1. Any tiling of finite local complexity is mutually locally derivable with a tilings whose tiles are the top-dimensional simplices of a simplicial complex. Moreover, the simplicial structure is pattern equivariant in the following sense. If t 1 and t 2 are tiles which differ by a translation, i.e. t 2 = t x (t 1 ) for some x, then the characteristic functions σ 1 and σ 2 of t 1 and t 2 , respectively, differ by the same translation,

σ 2 = t x • σ 1 .
Proof. It is well known that any tiling of finite local complexity is mutually locally derivable with a tiling whose tiles are polyhedra and match face to face. One may then subdivide the polyhedra into simplices in a mutually locally derivable way to obtain a tiling whose tiles are simplices and match face to face. The issue is now to order the vertices of the simplices in a pattern equivariant way. This can be done as follows: By the finite local complexity the exists a unit vector ν in R d (the ambient space) such that none of the edges of the tiles is perpendicular to ν. Then the order on the vertices defined by v 1 ≤ v 2 provided ν, v 1 ≤ ν, v 2 is total and translation invariant in the sense that the pair (v 1 -x, v 2 -x) (if it occurs in the tiling) has the same order than (v 1 , v 2 ). We may therefore define the characteristic maps to be translation invariant in the same sense.

We call a tiling with this property a simplicial tiling. We denote by P n the n-skeleton of the simplicial complex it defines, that is, P n is the union of all faces of dimension smaller or equal to n. In particular

P d = R d .
If X carries in addition the structure of a differentiable manifold and we want to integrate n-forms over certain n-dimensional submanifolds it is useful to use charts defined by the cells. For that purpose we shall not only require the characteristic maps to be diffeomorphisms, but even that they extend to diffeomorphisms from an open neighborhood of ∆ n to an open neighborhood of the corresponding cell; in the terminologie of [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF] this means that the simplicial complex defines a smooth triangulation of X. This is clearly possible for the simplicial complex defined by a tiling in R d .

The tiling defines a ∆-complex for the topological spaces X = Γ k . This is obvious from the construction of Γ k . In fact, if σ is a characteristic map of a k-collared tile then σ := ρk •σ is a characteristic map for the corresponding k-collared prototile in Γ k . By Lemma 2.1 this is independent of the choice of representative for the prototile. Note that σ is not necessarily injective. We denote by S n k the set of characteristic maps of the n-simplices on Γ k . An n-chain on X is a formal linear combination of oriented simplices from X, and we denote the set of n-chains by C n (X). It is thus the free Z-module generated by elements σ . An n-cochain with coefficients in an abelian group A is an element of the Z-module C n (X, A) = Hom Z (C n (X), A). Let a ∈ A. We denote by a σ the morphism defined by

a σ ( τ ) = a if σ = τ 0 else .

The ∆-transversal of a tiling

The canonical transversal or discrete hull of a tiling is defined with the help of punctures for the tiles of the tiling [START_REF] Kellendonk | Non commutative geometry of tilings and Gap Labelling[END_REF]. In [START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF] punctures for the faces of tiles are introduced as well so that one gets a complex defined by transversals of different dimension.

A puncture of a face of a tile is a chosen point in the face, for instance its barycenter. The pair given by the face and its puncture is called a punctured face. All punctures are supposed to be chosen in such way that two faces of tiles in the tiling which agree up to translation have their punctures at the same relative position. Then the puncturing passes to the equivalence classes so that the simplices of Γ k are also punctured. We denote by P n,punc the set of punctures of all n-faces of P. Definition 2.2. [START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF] The n-dimensional ∆-transversal is

Ξ n = {P ′ ∈ Ω P |0 ∈ P ′ n,punc }.
The disjoint union Ξ = ˙ d n=0 Ξ n is called the ∆-transversal of the tiling P. In [START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF] the notation Ξ ∆ for the ∆-transversal was used instead which distinguishes it from the usual notation for the canonical transversal, but we see no danger of confusion here. The sets Ξ n are totally disconnected and compact subsets of Ω P . Their topology is generated by the following clopen subsets, referred to as acceptance zones for faces of Γ k .

Definition 2.3. The acceptance zone of an

n-simplex σ of Γ k is the subset Ξ(σ) of Ξ n given by Ξ(σ) = ρ -1 k (p σ
) where p σ is the puncture of σ. The surjections ρ k are defined in (2.1).

Remark 2.4. Recall that a d-simplex σ in Γ k corresponds to (the congruence class of ) a tile t σ in P labelled with its k-neighborhood. In a similar fashion a n-simplex σ in Γ k corresponds to an n-dimensional face of a tile in P together with its k-neighborhood. Then Ξ(σ) can be seen as the subset of tilings which contain a patch corresponding to a k-collared face defined by σ whose puncture lies on 0 ∈ R d .

We denote the restriction of D to Ξ by D 0 . The family of clopen balls {B D 0 (P, ε)} P∈Ξ , of radius ε around P, is a base of the topology of the space (Ξ, D 0 ). We denote

B n D 0 (P, ε) = B D 0 (P, ε) ∩ Ξ n .
Lemma 2.5. The family of acceptance zones {Ξ(σ)} σ∈Γ k ,0≤k≤d is a base for the topology of the metric space (Ξ, D 0 ).

Proof. Let σ be an n-simplex of Γ k and T a tiling in Ξ n (σ). Recall that D 0 (T , T ′ ) ≤ 1 R+1 means that T and T ′ agree on the ball B R (0) of R d of radius R around the origin. Choose R > 0 large enough such that B R (0) covers the k-neighborhood of the face of T at the origin. Then D 0 (T , T ′ ) ≤ 1 R+1 implies that T ′ contains as well that face with its k-neighborhood and hence belongs to the acceptance zone Ξ(σ). We thus have proved ∃R > 0:

T ∈ B n D 0 (T , 1 R+1 ) ⊂ Ξ(σ).
Conversely, let T be a tiling of Ξ n and R > 0. Choose k large enough so that the k-neighborhood of the face f at the origin of T covers B R (0). Let σ f ∈ Γ k be the n-simplex corresponding to this k-collared face. Then T ′ ∈ Ξ(σ f ) implies that T and T ′ agree on B R (0) and hence D 0 (T , T ′ ) ≤ 1 R+1 . We thus have proved that there exists an acceptance zone

Ξ(σ f ) such that T ∈ Ξ(σ f ) ⊂ B n D 0 (T , 1 R+1 ).
We denote by C lc (Ξ n , A) the Z-module of A-valued locally constant functions over the n-dimensional ∆-transversal. It constitutes the degree n part of the complex defining PV cohomology which we will consider below.

It follows from Lemma 2.5 that any element of C lc (Ξ n , A) is a finite sum of elements a Ξ(σ) where a ∈ A and

a Ξ(σ) (ξ) = a if ξ ∈ Ξ(σ) 0 else .

Cohomology as a direct limit (DL-cohomology)

As mentioned in the introduction, one way to define the cohomology of a tiling P is as Čech cohomology Ȟ(Ω P , A) of Ω P . Since Ω P is an inverse limit of spaces (Γ k ) k∈N its Čech cohomology can be expressed as a direct limit of groups

Ȟ(Ω P , A) = Ȟ(lim ←- l Γ l , A) = lim -→ l Ȟ(Γ l , A).
The main advantage is that Ȟ(Γ k , A), the Čech cohomology of Γ k , is simple to compute. As Γ k is a finite ∆-complex its Čech cohomology is isomorphic to its simplicial cohomology. Furthermore, the maps ρ k k-1 preserve the ∆complex structure and so the maps they induce on simplicial cohomology can easily be handled. In its original version this was the method to calculate the tiling cohomology for substitution tilings in [START_REF] Anderson | Topological invariants for substitution tilings and their associated C * -algebras[END_REF] (although they used cellular cohomology).

2.3 Savinien and Bellissard's approach to tiling cohomology

2.3.1 PV-cohomology.
Recall that S n 0 denotes the (finite) set of characteristic maps of the nsimplices in Γ 0 . Let σ ∈ S n 0 and choose a characteristic map σ of an nsimplex of P such that σ = ρ0 • σ. We define the vector

x σ,i = p ∂ i σ -p σ ∈ R d
which is the difference vector pointing from the puncture of the cell defined by σ to the puncture of its ith face. This is independent of the choice of σ by Lemma 2.1. We denote by t σ,i : Ξ

(σ) → Ξ(∂ i σ) the restriction of t x σ,i to Ξ(σ), t x σ,i (ξ) = ξ + p σ -p ∂ i σ .
The following definition is essentially that of [START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF]. Definition 2.6. Let σ ∈ S n 0 . Define the linear map θ σ,i :

C lc (Ξ n-1 , A) → C lc (Ξ n , A) by θ σ,i (f ) = ı • t * σ,i (f Ξ(∂ i σ) ) where t * σ,i : C lc (Ξ(∂ i σ), A) → C lc (Ξ(σ), A)
is the pull back of t σ,i and ı : C lc (Ξ(σ), A) ֒→ C lc (Ξ n , A) the inclusion. The strong PV-cohomology of P is the cohomology of the complex (C lc (Ξ * , A), d P V ) where

d P V : C lc (Ξ n , A) -→ C lc (Ξ n+1 , A) is given by d P V = σ∈S n+1 0 n+1 i=0 (-1) i θ σ,i .

We denote the strong PV-cohomology with coefficients in A by

H * P V (Γ 0 , C lc (Ξ, A)). H * P V (Γ 0 , C lc (Ξ, Z)
) is isomorphic to the Čech cohomology of the hull [START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF].

Relation with pattern groupoid cohomology

PV-cohomology can effectively seen as the continuous cocycle cohomology (after [START_REF] Renault | A Groupoid approach to C * -algebras[END_REF]) of the pattern groupoid GP, which is the variant of the discrete tiling groupoid defined in [START_REF] Forrest | Topological Invariants for Projection Method Patterns[END_REF]. More precisely, the PV-complex is a reduction of the complex of continuous cocycle cohomology which has the advantage of being trivial in degrees larger than the dimension of the tiling. This is similar to using a finite resolution of length d instead of the standard resolution for the cohomology of the group Z d . Let us explain this. By definition, GP is the reduction to Ξ (0) of the transformation groupoid Ω P ⋊R d . Its elements are pairs (ω, v) ∈ Ω P ×R d which satisfy ω, t v (ω) ∈ Ξ (0) . The continuous cocycle cohomology of GP is the cohomology of the complex of continuous functions

C(GP (n) , Z) δ -→ C(GP (n+1) , Z) • • • where GP (0) = Ξ 0) and GP (n) is the subset of elements (ω, v 1 , • • • , v n ) ∈ Ω P × (R d ) n which satisfy ω ∈ Ξ (0) and ω -j i=1 v i ∈ Ξ (0)
for all j. Its topology is the subspace topology of the product topology. The differential is given by δf (ω, v) = f (t v (ω)) -f (ω) in degree 0, and in degree n > 0 by

δf (ω, v 1 , • • • , v n+1 ) = f (t v 1 (ω), v 2 , • • • , v n+1 ) + (-1) n+1 f (ω, v 1 , • • • , v n ) + n i=1 (-1) i f (ω, v 1 , • • • , v i + v i+1 , • • • , v n+1 )
We define the map α : Ξ

(n) → GP (n) by α(ω) = (ω, σ(x 1 ) -σ(x 0 ), • • • , σ(x n ) -σ(x n-1 ))
where σ(x i ) is the ith vertex of the n-simplex of ω whose puncture is on 0. Clearly α is injective. It is continuous as the σ(x i ) -σ(x i-1 ) are locally defined. A straightforward computation shows that the pullback of α satisfies α * • δ = d P V • α * . It hence yields a surjective chain map. It thus induces a surjective map from the continuous cocycle cohomology of the pattern groupoid to PV-cohomology. As we know that the two cohomologies are isomorphic, α * has to be an isomorphism at least in the case that the cohomology is finitely generated.

Weak PV-cohomology

We now consider the situation in which the abelian group A carries a metric δ w.r.t. which it is complete. Then C(Ξ n , A) is a complete module w.r.t. the metric δ(f, g) := sup ξ∈Ξ n δ(f (ξ), g(ξ)).

Lemma 2.7. The P V -differential extends to a continuous map

d P V : C(Ξ n , A) → C(Ξ n+1 , A).
Proof. First observe that t σ,i is a partial translation. θ σ,i is essentially its pull-back and since domain and range of t σ,i are clopen θ σ,i (f ) is continuous for continuous f . Since δ is translation invariant the pull back of a partial translation is a partial isometry. Hence θ σ,i is continuous. Since d P V is a finite sum of θ σ,i the result follows.

Definition 2.8. The weak P V -cohomology is the cohomology of the so-called weak P V -complex (C(Ξ * , A), d P V ). It is denoted by

H * P V (Γ 0 , C(Ξ, A)).
While for discrete groups, like Z, the strong and the weak PV-complex coincide they differ vastly in the case that A is a continuous group, like R. Although the inclusion C lc (Ξ n , A) ⊂ C(Ξ n , A) is a chain map, there is no evident relation between the two cohomologies in the continuous case.

PE de Rham cohomology

In this section, we recall the definition of the pattern equivariant de Rham cohomology of a tiling. Definition 2.9. Let P be a tiling of finite local complexity. A function

f on R d is pattern equivariant with range R > 0 if ∀x, y ∈ R d , B R [t x (P)] = B R [t y (P)] ⇒ f (x) = f (y).
It is called strongly pattern equivariant (s-PE) if it is pattern equivariant with some finite range R. 2We denote by A n b (R d ) the space of bounded (smooth) n-forms over R d . Such forms can be seen as smooth functions from R d into Λ n R d * , the exterior algebra of the dual of R d (equipped with some norm). We may therefore consider A n s-P (R d ) the space of s-PE n-forms over R d . Together with the standard exterior derivative (A * s-P (R d ), d) is a subcomplex of the usual de Rham complex of differential forms on R d .

The cohomology of the differential complex (A * s-P (R d ), d) is called the strongly pattern equivariant cohomology and denoted by H * s-P (R d ). As ordinary de Rham cohomology it is a cohomology with real coefficients. H * s-P (R d ) is isomorphic to the Čech cohomology of the tiling space with real coefficients [START_REF] Kellendonk | The Ruelle-Sullivan map for actions of R n[END_REF].

Strongly pattern equivariant functions are in some sense algebraic, namely the size R of the patch around a point x which has to be inspected to obtain the value of the function is fixed and finite. This naturally calls for looking at functions which are obtained as limits of strongly pattern equivariant functions. These are called weakly pattern-equivariant. More precisely, the space of smooth weakly pattern equivariant n-forms

A * w-P (R d ) is the closure of A * s-P (R d ) in A * b (R d )
with respect to the Fréchet topology given by the family of semi norms s k ,

s k (f ) = sup |κ|≤k ||D κ f || ∞ where κ = (κ 1 , . . . , κ n ) ∈ N n , |κ| = n i=1 κ i , D κ f = ∂ κ 1 ∂x κ 1 1 • • • ∂ κn ∂x κn n f and
||.|| ∞ is the norm of the uniform convergence. The weakly pattern equivariant cohomology H * w-P (R d , R) is the cohomology of the differential complex

(A * w-P (R d ), d). H * w-P (R d , R
) is isomorphic to the tangential cohomology of the tiling space [START_REF] Kellendonk | The Ruelle-Sullivan map for actions of R n[END_REF]. We shall provide below an example which confirms our expectation that the weak pattern equivariant cohomology of aperiodic tilings is always infinitely generated.

In the last section, we will be interested in deformations of tilings. These deformations are parameterized3 (in [START_REF] Kellendonk | Pattern Equivariant Functions, Deformations and Equivalence of Tiling Spaces[END_REF]) by a group called the pattern equivariant mixed group. It is defined by the quotient

Z 1 s-P (R d )/B 1 w-P (R d ) ∩ Z 1 s-P (R d )
of closed strongly PE 1 forms modulo those 1 forms which are weakly exact,

Z 1 s-P (R d ) = A 1 s-P (R d ) ∩ ker d and B 1 w-P (R d ) = d(A 0 w-P (R d )).

PE simplicial cohomology

Since P is a simplicial tiling it defines a simplicial complex in R d . The concept of pattern equivariance can equally well be defined for cochains (with values in any abelian group A) over this complex. In fact, a cochain is s-PE if there is some k > 0 such that its value on σ depends only on the kneighborhood of the simplex σ of P. We denote the degree n s-PE cochains with coefficients in A by C n s-P (P, A). Clearly the coboundary operator maps s-PE cochains to s-PE cochains and so we may consider what is called the s-PE simplicial cohomology, which is the simplicial cohomology of the complex (C * s-P (P, A), d S ). We denote it by H * s-P (P, A). It has been shown in [START_REF] Sadun | Pattern-Equivariant Cohomology with Integer Coefficients[END_REF] that s-PE simplicial cohomology is isomorphic to the Čech cohomology of the tiling space.

Just as one can define weak PE de Rham cohomology by using a completion w.r.t. a natural family of semi-norms, one can define weak PE simplicial cohomology by means of completion in the (metric) topology of uniform convergence. More precisely we define the w-PE cochains C w-P (P, A) (with coefficients in A) as the closure of C s-P (P, A) in the set of bounded cochains C b (P, A) w.r.t. to the metric δ(c, c ′ ) = sup σ δ(c( σ ), c ′ ( σ )). Clearly the simplicial coboundary-boundary operator is continuous in the topology and defines the w-PE complex (C * w-P (P, A), d S ) whose cohomology we call the w-PE simplicial cohomology and denote by H * w-P (P, A).

PE simplicial cohomology and PV cohomology

We start with comparing the strong versions of PE simplicial cohomology and PV cohomology. This material is essentially known though not written in this form. At the end of the section we compare the weak versions. This is new.

PV cohomology as DL cohomology

We need to discuss in some detail the construction of the isomorphism between DL cohomology and PV cohomology. It essentially establishes that PV-cochains yield a model for the direct limit of the simplicial cochains of the APG complexes. We denote in this section the pull back of ρ k+1 k : Γ k+1 → Γ k by π k k+1 . Recall that the direct limit of the directed system C n (Γ k , A)

π k k+1 -→ C n (Γ k+1 , A) is a module (denoted lim -→ C n (Γ k , A)) together with module maps π k such that the diagram C n (Γ k , A) π k k+1 =ρ * k+1 k / / π k & & N N N N N N N N N N N C n (Γ k+1 , A) π k+1 w w o o o o o o o o o o o lim -→ l C n (Γ l , A)
commutes. It has moreover the universal property that, whenever another module M with module maps π ′ k :

C n (Γ k , A) → M yields a commuting diagram C n (Γ k , A) π ′ k ) ) π k k+1 / / C n (Γ k+1 , A) π ′ k+1 t t M
then the π ′ i factor uniquely through the direct limit [START_REF] Lang | Algebra(Revisited Third Edition[END_REF]. The latter means that there exists a unique homomorphism h : lim

-→ C n (Γ k , A) -→ M satisfying h • π k = π ′ k for all k ∈ N. Define the module maps η k : C n (Γ k , A) -→ C lc (Ξ n ∆ , A) by η k (a σ ) = a Ξ(σ) .
These maps induce a new commuting diagram

C n (Γ k , A) η k ( ( π kj / / C n (Γ j , A) η j v v C lc (Ξ n , A)
By the universal property of the direct limit there exists a unique homomorphism

η : lim -→ l C n (Γ l , A) -→ C lc (Ξ n , A) such that η • π k = η k k ∈ N.
Lemma 3.1. The maps η k are one-to-one.

Proof. Let ξ in Ξ(σ 0 ) for some σ 0 ∈ S n k . Hence ξ contains a patch at the origin which is translationally congruent to the face with its k-neighborhood encoded by σ 0 . Thus

a Ξ(σ) (ξ) = a if σ = σ 0 , 0 if σ = σ 0 . Let γ ∈ C n (Γ k , A).
Then the above shows that η k (γ)(ξ) = γ( σ 0 ). Letting ξ vary we see that η k (γ) = 0 implies γ = 0.

Lemma 3.2. For all γ ∈ C(Ξ n , A) there is k ∈ N such that γ ∈ im(η k ).
Proof. By Lemma 2.5 every element of C(Ξ n , A) is a finite sum of a Ξ(σ) , a ∈ A and σ ∈ Γ k for some k. Since a Ξ(σ) ∈ im(η k ) and im(η k ) ⊂ im(η k+1 ) the statement follows. Proof. Recall the definition of the simplicial coboundary operator d S on Γ k : Given σ ∈ S n k we have

d S (a σ ) = τ ∈S n+1 k n+1 i=0 (-1) i δ σ ∂ i τ a τ .
Hence

η k • d S (a σ )(ξ) = n+1 i=0 (-1) i δ σ ∂ i σ k ξ
where σ k ξ is the k-collared simplex of ξ on 0. On the other hand

d P V • η k (a σ ) = τ ∈S n+1 0 n+1 i=0 θ τ,i (a Ξ(σ) ) = τ ∈S n+1 0 n+1 i=0 ı • t * τ,i (a Ξ(σ) Ξ(∂ i τ ) ). Now ı • t * τ,i (a Ξ(σ) Ξ(∂ i τ ) )(ξ) = a Ξ(σ) (t τ,i (ξ)) if ξ ∈ Ξ(τ ) 0 else = 1 if σ k ξ-x τ,i = σ and σ 0 ξ = τ 0 else and since σ k ξ-x τ,i = σ and σ 0 ξ = τ can happen iff ∂ i σ k ξ = σ the claim follows.
Recall that the cohomology of lim

-→ C(Γ k , A) is the direct limit lim -→ H(Γ k , A) together with module maps H(π k ) : H(Γ k , A) → lim -→ H(Γ k , A) such that H(π k+1 ) = H(π k ) • H(π k k+1 ).

Theorem 3.4 ([18]). η is a module isomorphism. It induces an isomorphism in cohomology.

Proof. As the η k intertwine the differentials they induce homomorphisms in cohomology H(η k ) :

H(Γ k , A) → H P V (Γ 0 , C lc (Ξ, A)) such that H(η k+1 ) = H(η k )•H(π k k+1
H P V (Γ 0 , C lc (Ξ, A)) commute. Hence H(η) • H(π k )([ω]) = H(η k )([ω]) = [η k (ω)] = [η • π k (ω)]
where we have denote by [•] cohomology classes.

Injectivity: Let x ∈ lim -→ H(Γ l , A) such that H(η)(x) = 0. There exists k ∈ N and a cochain in Γ k such that x = H(π k )[ω]. Thus 0 = [η k (ω)]. Let γ ∈ C lc (Ξ, A) satisfy η k (ω) = d P V γ. By Lemma 3.2, there is l ∈ N and α ∈ C(Γ l , A), such that γ = η l (α). By Lemma 3.3 d P V η l (α) = η l (d S α).
We may suppose that l ≥ k. Then the above implies that η l (ω) = η l (d S α) and hence, by Lemma 3.

1 ω = d S α. Surjectivity: Let [γ] ∈ H(Γ 0 , C lc (Ξ, A)). Then γ = η(ω) for some ω ∈ lim -→ C(Γ l , A). There exists k ∈ N and a cochain ω ∈ Γ k such that ω = π k [ω]. Hence [γ] = [η • π k (ω)] = H(η) • H(π k )([ω]).

PE simplicial as DL cohomology

Recall the definition (2.2) of ρk : R d → Γ k . In particular, to each simplex σ of P corresponds a simplex ρk • σ ofΓ k . Pulling these back over cochains we obtain module maps ρ *

k : C * (Γ k , A) → C * s-P (P, A) ρ * k (c)( σ ) = c( ρk • σ ).
They induce a new commuting diagram

C n (Γ k , A) ρ * k ) ) π kj / / C n (Γ j , A) ρ * j v v
C n s-P (P, A)

By the universal property of the direct limit there exists a unique homomorphism ρ * : lim

-→ l C n (Γ l , A) -→ C n s-P (P, A) such that for all k ∈ N, ρ * • π k = ρ * k .
As Sadun explains in [START_REF] Sadun | Pattern-Equivariant Cohomology with Integer Coefficients[END_REF], every s-PE cochain on P can be viewed as the pull back of a cochain on Γ k provided k is sufficiently large and vice versa. In other words, the analogue of Lemma 3.1 and Lemma 3.2 are true and ρ * is an isomorphism. It is trivial that the ρ * k intertwine the differentials as they are defined in essentially the same way. Hence the same proof as that for Thm. 3.4 yields that ρ * induces an isomorphism between DL cohomology and s-PE simplicial cohomology.

PE simplicial and PV-cohomology

The last two sections can be summarized by saying that the strong versions of PE simplicial and PV-cohomology are abstractly the same thing: in both cases the cochains arise as models for the direct limit of the cochains of APG-complexes. In fact, by the universal property the module morphism φ : C s-P (P, A) → C lc (Ξ, A):

φ = η • (ρ * ) -1
is an isomorphism which, by Lemma 3.3 and its analogon intertwines the differentials. It hence induces an isomorphism between the strong versions of PE simplicial and PV cohomology. By construction φ(ρ * k (a σ )) = η k (a σ ) = a Ξ(σ) and hence φ(c) is on the dense orbit through P given by

φ(c)(P -p) = c( σ p ) (3.1)
where σ p is the simplex of P whose puncture is on p.

Lemma 3.5. φ is an isometry and thus extends to an isomorphism, which we also denote by φ, between C w-P (P, A) and C(Ξ, A).

Proof. That φ is an isometry is direct. The continuous extension of a bijective isometry is bijective as well.

Corollary 3.6. φ induces a isomorphism between the weakly pattern equivariant cohomology H w-P (P, A) and the weak PV-cohomology

H P V (Γ 0 , C(Ξ, A)).
Proof. By continuity φ intertwines the differentials. Hence φ is a bijective chain map.

PE de Rham versus PE simplicial cohomology

A PE de Rahm theorem

The de Rham theorem for smooth manifolds states that, given a smooth triangulation of the manifold and hence a simplicial decomposition, then the de Rham cohomology of the manifold and the simplicial cohomology of the complex are isomorphic. Moreover, the isomorphism has a very simple form, it is induced from a chain map which associates to a k-form the k-cochain whose evaluation on the generator associated with a k-simplex is given by integrating the form over the simplex. A very careful exposition of the proof is given in [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF] for the case of finite simplicial complexes in euclidean spaces. We shall adapt this proof to obtain a PE version of the de Rham theorem the role of the manifold being played by R d with its simplicial decomposition defined by the tiling P. This then will lead to an explicit isomorphism between the s-PE de Rham cohomology of R d and the s-PE simplicial cohomology of P with values in R. We furthermore show that the maps involved are continuous so that the isomorphism extends to an explicit isomorphism between the weak versions.

Recall that P defines a smooth triangulation on R d but even a smooth triangulation. This allows us to integrate a k-form ω ∈ A k b (R d ) over a ksimplex σ. In fact we write σ ω to mean ± the integral of ω over the set imσ where the sign is + provided the orientation of σ is the same than that induced on im(σ) by the (chosen) orientation of R d , andotherwise. We equip C b (P, R) with the sup-norm: c = sup σ |c( σ )|. This makes it a complete real vector space on which the simplicial differential acts continuously.

Definition 4.1. Let J l : A l b (R d ) → C l b (P, R) be given by J l (ω)( σ ) = σ ω
for any l-simplex σ.

Our aim is to prove the following theorem Theorem 4.2. J l induces an isomorphism between H l s-P (R d ) and H l s-P (P, R) and also between H l w-P (R d ) and H l w-P (P, R).

We should say that the first part could also be derived from the de Rham type theorems for transversally locally constant tangential forms [START_REF] Kellendonk | The Ruelle-Sullivan map for actions of R n[END_REF] or for branched manifolds [START_REF] Sadun | Pattern-Equivariant Cohomology with Integer Coefficients[END_REF], but these proofs use the Čech-de Rham double complex and hence yield a priori isomorphisms with Čech cohomology. We provide here a proof avoiding Čech cohomology and double complexes at all. So it is more elementary but also longer. But it has the advantage of extending directly to the topological closures (weak versions). Lemma 4.3. J l has the following properties:

1. J l+1 • d = d S • J l . 2. J l is continuous. 3. If ω is s-PE then J l (ω) is s-PE. 4. If ω is w-PE then J l (ω) is w-PE. Proof. The first statement is Stokes theorem. Continuity follows from J l (ω) ≤ sup σ | σ ω| ≤ ω ∞ sup σ vol(σ).
The third statement is evident and implies the last one by continuity of J l .

The first step in showing a de Rham theorem is the construction of right inverse maps for J l . We proceed as in [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF].

Lemma 4.4. There are module maps α l : C l b (P, R) → A l b (R d ) which satisfy the following properties:

1. J l • α l = id. 2. α l+1 • d S = d • α l . 3. α l is continuous. 4. If c is s-PE then α l (c) is s-PE. 5. If c is w-PE then α l (c) is w-PE.
Proof. We follow the construction of α l given in [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF]. It is based on a partition of unity {g v } v∈P (0) which is subordinate to the covering given by the stars of the vertices v: The star St(v) of v is simply the interior of the 1-neighborhood of v, i.e. the union over all open simplices touching v. Let F v be the compact set points of St(v) which have at least distance 1 n+1 from the boundary of St(v) and G v the closed set of points which have at most distance 1 n+2 from the complement of St(v). As is well known there exists a smooth positive function ǧv which vanishes on G v and equals 1 on F v . Clearly we can require that the choice of ǧv be made so that if v and v ′ have the same 1-neighborhood up to translation then ǧv and ǧv ′ coincide up to translation by v -v ′ . Now g v = ǧv / v ′ ∈P (0) ǧv ′ and α l is given as follows.

Let σ be a l-simplex and v i the ith vertex, i.e. the image under σ of x i ∈ ∆ l . Then

α l (1 σ ) = ω σ := l! l i=0 (-1) i g v i dg v 0 ∧ • • • dg v i • • • dg v l .
A proof of the first two properties can be found in [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF]. The above formula shows that

α l (1 σ ) = t x * α l (1 σ ′ ) (t x * denotes the pull back of t x ) if σ = t x (σ ′ ) and their 2-neighborhoods coincide. α l (c) is therefore s-PE provided c is a s-PE cochain.
To show continuity let c = σ∈P (l) r σ 1 σ , r σ ∈ R. Then

s k (α l (c)) ≤ |κ|≤k D κ σ∈P (l) r σ ω σ ∞ .
Since the support of ω σ is contained in St(v) and the number of lsimplices in a star is uniformly bounded, let's say by N l , we have

sup x |D κ σ∈P (l) r σ ω σ (x)| ≤ N l sup σ∈P (l) |r σ | sup x |D κ ω σ (x)| and hence s k (α l (c)) ≤ N l c max σ∈P (l) s k (ω σ ).
Hence α l is continuous from which the last statement follows by continuous extension.

Corollary 4.5. J l induces a surjective homomorphism between H l s-P (R d ) and H l s-P (P, R) and also between H l w-P (R d ) and H l w-P (P, R).

The proof injectivity of these maps is essentially based on Poincaré's lemma. In contrast to the usual case we need however good control on the contracting homotopies involved. Let U be an open star-shaped subset of R d and u ∈ U a center, i.e. for all x ∈ U we have

∀0 ≤ t ≤ 1 : tx + (1 -t)u ∈ U . Then we denote by h U,u k-1 : A k b (U ) → A k-1 b (U ), k > 0, the following module map: h U,u k-1 (f dx i 1 ∧• • • dx i k )(x) = I u k-1 (f )(x) k j=1 (-1) j (x i j -u i j )dx i 1 ∧• • • dx i j • • • dx i k
where

I u k-1 (f )(x) = 1 0 t k-1 f (tx + (1 -t)u)dt.
Lemma 4.6. Let U be bounded and as above. h U,u k have the following properties

1. h U,u k • d = d • h U,u k-1 for k ≥ 1, 2. h U -x,u-x k = t x * • h U,u k , 3. h U,u k is continuous.
Proof. h U,u k-1 is a standard homotopy for U and the first statement (Poincaré lemma for U ) proven for instance in [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF]. The second statement is a direct consequence of the definition. As for the third, continuity of h U,u k-1 follows from continuity of

I u k-1 since the form ω(x) = k j=1 (-1) j (x i j -u i j )dx i 1 ∧ • • • dx i j • • • dx i k does not
depend on f and has bounded seminorms given that U is bounded. Now

D κ I u k-1 (f ) ∞ = 1 0 t k-1+|κ| (D κ f )(tx + (1 -t)u)dt ∞ ≤ D κ f ∞ shows that I u k-1 is indeed continuous.
Let σ be an k-simplex. We denote by [σ] ǫ the ǫ-neighborhood of im(σ) and by [∂σ] ǫ the ǫ-neighborhood of its boundary ∂im(σ). We denote by Z r b (U ) the closed bounded r-forms on

U . For r = k, r, k ≥ 1 let D r ǫ (σ) = {(ω, τ ) ∈ Z r b ([σ] ǫ ) ⊕ A r-1 b ([∂σ] ǫ ) : ω [∂σ]ǫ = dτ } and for r = k ≥ 1 let D k ǫ (σ) = {(ω, τ ) ∈ Z k b ([σ] ǫ ) ⊕ A k-1 b ([∂σ] ǫ ) : ω [∂σ]ǫ = dτ, σ ω = ∂σ τ }.
Lemma 4.7. There exists an ǫ > 0 and a module map β r (σ) :

D r ǫ (σ) → A r-1 b ([σ] ǫ ) satisfying 1. τ ′ = β r (σ)(ω, τ ) coincides with τ on [∂σ] ǫ , 2. dτ ′ = ω, 3. If t x (σ) is a simplex then β r (t x (σ)) = t x * • β r (σ), 4. β r (σ) is continuous.
Proof. The proof is by induction. For that we need to consider a second module map. For

r = k, r ≥ 0, k ≥ 1 let E r ǫ (σ) = Z r b ([∂σ] ǫ and for r = k -1 ≥ 0 let E k ǫ (σ) = {ω ∈ Z k b ([∂σ] ǫ : ∂σ ω = 0}.
We aim to show that there exists a module map γ r (σ) :

E r ǫ (σ) → Z r b ([σ] ǫ ) which satisfies 1. γ r (σ)(ω) coincides with ω on [∂σ] ǫ 2 , 2. If t x (σ) is a simplex then γ r (t x (σ)) = t x * • γ r (σ), 3. γ r (σ) is continuous. γ 0 (σ) Let ω ∈ E 0 ǫ (σ)
. Since dω = 0, ω is constant on any connected subset of ∂σ. If k = 1 ∂σ is connected and hence ω constant. If k = 1 the extra condition σ ω = 0 implies that ω is constant. Thus for all choices of k we can define γ 0 (σ)(ω) to be the constant extension of ω to σ. The properties stated are obviously satisfied.

β r (σ) We assume the existence of γ r-1 (σ) satisfying the above properties. This is certainly correct for r = 1 and then will follow inductively in view of the next step. Let (ω, τ ) ∈ D r ǫ (σ). Let b σ be the barycenter of σ and h σ = h [σ]ǫ,bσ . Define

β r (σ)(ω, τ ) := h σ (ω) -γ r-1 (σ)(h σ (ω) [∂σ]ǫ -τ ).
The first two properties are verified in [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF]. The other two follow immediately from the properties of γ r-1 (σ) and Lemma 4.6.

γ r (σ) We assume the existence of β r (σ) satisfying the properties stated in the lemma. This follows inductively in view of the preceding step.

Let ω ∈ E r ǫ (σ). Let v be the first vertex of σ (w.r.t. the ordering of the vertices) and σ ′ the corresponding k -1-simplex of the boundary of σ, i.e. the face which does not contain v. Then [∂σ\σ ′ ] ǫ is star-shaped with center v. We let

µ ′ σ = h [∂σ\σ ′ ]ǫ,v (ω [∂σ\σ ′ ]ǫ] ) and µ ′′ σ = β r (σ)(ω [σ ′ ]ǫ , µ 0 [∂σ ′ ]ǫ
). µ ′ σ and µ ′′ σ coincide on their common domain and hence define an r -1-form µ σ on [∂σ] ǫ . It follows from Lemmata 2.1 and 4.6 and the induction hypothesis for β r that µ t x (σ) = t x * µ σ provided t x (σ) is a simplex of the tiling. Now consider a family of smooth functions b σ : [σ] ǫ → R, defined for all k-simplices, which is s-PE in the sense that b t x (σ) = t x * b σ . We suppose that each b σ is identically 1 on [∂σ] ǫ 2 and identically 0 on the complement of

[∂σ] ǫ . Then γ r (σ)(ω) := d(b σ µ σ ) satisfies γ r (σ)(ω) = ω on [∂σ] ǫ 2 ,
as is shown in [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF], and γ r (t x (σ)) = t x * • γ r (σ). Continuity of γ r (σ) follows from the continuity of h [∂σ\σ ′ ]ǫ,v , β r (σ) and d, and smoothness of b σ . Let D r ǫ (P (k) ) be defined as D r ǫ (σ) except replacing the simplex by the k-skeleton. If r = k we demand the integral equation σ ω = ∂σ τ to hold for all k-simplices. We then define β r : D r ǫ (P (k) ) → A r-1 b ([P k) ] ǫ ) by glueing the β r (σ)(ω [σ]ǫ , τ [∂s]ǫ ) defined for the k-simplices σ of P (k) together which is possible as they agree with (ω, τ ) along the ǫ-neighborhood of P (k-1) .

Corollary 4.8.

1. If (ω, τ ) ∈ D r ǫ (P (k) ) is s-PE then β r (ω, τ ) is s-PE. 2. If (ω, τ ) ∈ D r ǫ (P (k) ) is w-PE then β r (ω, τ ) is w-PE. Lemma 4.9. Let ω ∈ Z l b (R d ), l ≥ 1, and J l (ω) = d S c. Then there exists τ ∈ A l-1 b (R d ) such that ω = dτ . Moreover, 1.
If ω and c are s-PE then τ can be chosen s-PE.

2. If ω and c are w-PE then τ can be chosen w-PE.

Proof. We follow again [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF] constructing τ by a composition of module maps

τ = βn • • • • • β0 (ω).
β0 (ω) is an l-1-form on the ǫ-neighborhood of P (0) . We may therefore define it through its restrictions to the connected components of [P (0) ] ǫ , namely the restriction to [v]ǫ is given by

h [v]ǫ (ω) -δ l1 (h [v]ǫ (ω)(v) -c(v)). If k > 0 then βk (ω) = β l (ω, βk-1 (ω)) -δ l-1 k α l-1 (J l-1 (β l (ω, βk-1 (ω)) -c).
This construction of τ follows exactly the lines of the proof of [START_REF] Singer | Lecture Notes on Elementary Topology and Geometry Undergraduate texts in mathematics[END_REF] and so the proof given there for the first statement applies also in our context.

= φ • J : A w-P (R d ) → C(Ξ, R) an isomorphism H(ψ) between H w-P (R d ) and H P V (Γ 0 , C(Ξ, R)) which restricts to an isomorphism between H s-P (R d ) and H P V (Γ 0 , C lc (Ξ, R)). Let ω ∈ A n w-P (R d ). An explicit formula for ψ(ω) ∈ C(Ξ n , R) is given by ψ(ω)(P -p) = σp ω,
where σ p is the simplex of P whose puncture is on p ∈ P n,punc .

Proof. Combine Cor. 3.6 with Thm. 4.2 and Eq. 3.1.

5 Some applications

Weak cohomology of cut & project tilings with dimension and codimension equal to one

We saw that the weak PE de Rham cohomology, the weak PE simplicial cohomology with real values and the weak PV cohomology with real values are all isomorphic. We refer to this cohomology simply as the weak tiling cohomology. The following calculation shows that this cohomology is infinitely generated even for the simplest aperiodic tilings of finite local complexity. Our calculation is based on an old result, namely that the transversally continuous tangential cohomology of a Kronecker foliation on a torus is infinitely generated in degree 1 [START_REF] Mostow | Continuous cohomology of spaces with two topologies[END_REF]. We expect that weak tiling cohomology is always infinitely generated for aperiodic tilings of finite local complexity. As explained in [START_REF] Savinien | A spectral sequence for the K-theory of tiling spaces, Ergodic Theory Dynam[END_REF] for one dimensional tilings PV-cohomology is just the cohomology of the group Z with coefficients in C(Ξ 0 , Z) where the action is induced by shifting the tiling by a tile (this is more precisely the first return map into the 0-dimensional ∆-transversal Ξ 0 of the R-action by translation of the tilings). The same argument applies to weak PV-cohomology upon replacing C(Ξ 0 , Z) by C(Ξ 0 , R).

For cut & project tilings with dimension and codimension equal to one the dynamical system (Ξ 0 , Z) can be described as follows [START_REF] Forrest | Topological Invariants for Projection Method Patterns[END_REF]: The cut & project tiling is determined by two data, θ and K. θ is an irrational number defining the slope of a line E ⊂ R 2 w.r.t. a choice of orthonormal base. K is a subset of the orthocomplement of E which is supposed to be compact and the closure of its interior. These data define a one dimensional tiling of E ∼ = R. Indeed the tiles are intervals whose end points are given by the ortho-projection onto E of all points in Z 2 ∩ (K + E) (Z 2 is the lattice generated by the base and it is supposed that no point of that intersection lies on the boundary of K + E). Since θ is irrational the tiling is aperiodic.

On the 0-dimensional ∆-transversal of this tiling Ξ 0 the group Z acts via the first return map of the R-action by translation of the tilings. This dynamical system (Ξ 0 , Z) factors onto the dynamical system (S 1 = R/Z, Z) given by the rotation around θ. The factor map µ : Ξ 0 → S 1 is almost one-to-one. The points where it is not one-to-one, the preimages of so-called cut points, form a union of orbits under the Z-action. On these orbits µ is two-to-one and we denote such points by a = (a + , a -).

Theorem 5.1. The first weak tiling cohomology of a cut & project tiling with dimension and codimension equal to one is infinitely generated.

Proof. By the last theorem and the above remark the first weak tiling cohomology is isomorphic to H 1 (Z, C(Ξ 0 , R)), the cohomology of the group Z with values in C(Ξ 0 , R). The pull back of the factor map µ : Ξ 0 → S 1 gives rise to a Z-equivariant short exact sequence of Z-modules

0 → C(S 1 , R) µ * -→ C(Ξ 0 , R) → Q → 0
where Q = C(Ξ 0 , R)/C(S 1 , R) naturally inherits a Z-action. We claim that H 0 (Z, Q) = 0 so that the long exact sequence in group cohomology contains the short exact sequence

0 → H 1 (Z, C(S 1 , R)) → H 1 (Z, C(Ξ 0 , R)) → H 1 (Z, Q) → 0.
We prove the claim: For a pre-image of a cut point a = (a + , a -) ∈ Ξ 0 let τ a : C(Ξ 0 , R) → R be given by τ a (f ) = f (a + ) -f (a -). This is a module homomorphism which vanishes on functions of µ * (C(S 1 , R)) and hence extends to a homomorphism τ a : Q → R. H 0 (Z, Q) is the group of invariant elements. Assuming that H 0 (Z, Q) is non-trivial suppose that f ∈ C(Ξ 0 , R) projects onto a non-trivial invariant element in Q. Then there exists an a such that τ a (f ) = τ a (f • ϕ) = 0. Choose 0 < 4ǫ < |τ a (f )|. There exists g ∈ C lc (Ξ 0 , R) such that f -g ∞ < ǫ. This implies that |τ a (f

• ϕ n ) -τ a (g • ϕ n )| < 2ǫ, for all n ∈ Z. Hence |τ a (f ) -τ a (g • ϕ n )| < 2ǫ < 1
2 |τ a (f )|, for all n ∈ Z, and therefore |τ a (g • ϕ n )| > 1 2 |τ a (f )|, for all n ∈ Z. It follows that g has infinitely many jumps. But by compactness of Ξ 0 the elements of C lc (Ξ 0 , R) have only finitely many jumps. This being a contradiction we must have H 0 (Z, Q) = 0.

Hence H 1 (Z, C(Ξ 0 , R)) is an extension of H 1 (Z, Q) by H 1 (Z, C(S 1 , R)). Now by the results of Mostow [START_REF] Mostow | Continuous cohomology of spaces with two topologies[END_REF][page 98f], H 1 (Z, C(S 1 , R)) is infinitely generated, as θ is irrational.

The mixed group in PV-cohomology

There are two approaches to the deformation theory of tilings. The first is based on the description of tiling cohomology as the direct limit lim → H((Γ l , R d ) and asymptotic negligible cocycles [START_REF] Clark | When Shape Matters: Deformations of Tiling spaces[END_REF], and the second on pattern equivariant 1-forms [START_REF] Kellendonk | Pattern Equivariant Functions, Deformations and Equivalence of Tiling Spaces[END_REF]. In the context of the second approach a deformation of a Delone set P into a Delone set P ′ is given by a bi-Lipschitz smooth function ϕ : R d -→ R d which has a strongly pattern equivariant differential and satisfies P ′ = ϕ(P). Moreover, if two such functions differ on P by a constant vector (which amounts to a global translation) then they define the same deformation. The deformations of P are therefore parameterized by the elements of Z The following corollary yields a description of the parameter space of deformations modulo topological conjugacy in terms of a mixed PVcohomology group. We denote by Z * P V (Γ 0 , -) and B * P V (Γ 0 , -) PV coycles and PV coboundaries, respectively.

Lemma 3 . 3 .

 33 The η k intertwine the differentials, i.e. η k • d S = d P V • η k .

  1 s-P (R d , R d )/N 1 s-P (R d , R d ) where N 1 s-P (R d , R d ) = {dϕ ∈ B 1 s-P (R d , R d )|ϕ is constant on P}.Deformations of tilings can be seen as deformations of their set of punctures. Denote by B ||.||∞ (ω, ε) the open ε-ball around an element ω ∈ Z 1 s-P (R d , R d ) w.r.t. the uniform norm. For sufficiently small ǫ the elements of Z 1 s-P (R d , R d ) ∩ B ||.||∞ (did, ε) define invertible deformations (in sense of [11]) of P. If two such elements differ by an element of B 1 s-P (R d , R d ) then their images on P are mutually locally derivable. If their difference is in B 1w-P (R d , R d ) then their images on P define pointed topological conjugate dynamical systems. The elements near the class of did in the mixed quotient groupZ 1 s-P (R d , R d )/B 1 w-P (R d , R d ) ∩ Z 1 s-P (R d , R d )parameterize therefore the set of small deformations modulo bounded deformations which are in the same pointed conjugacy class.

Corollary 5 . 2 .

 52 The pattern equivariant mixed quotient groupZ * s-P (R d , R d )/B * w-P (R d , R d ) ∩ Z * s-P (R d , R d ) is isomorphic to Z * P V (Γ 0 , C lc (Ξ, R d ))/B * P V (Γ 0 , C(Ξ, R d )) ∩ Z * P V (Γ 0 , C lc (Ξ, R d )). Proof. H * (ψ) : H * w-P (R d ) → H * P V (Γ o , C(Ξ, R)) is injective and hence its restriction to Z * s-P (R d )/B * w-P (R d ) ∩ Z * s-P (R d ) ⊂ H * w-P (R d ) is injective as well. H * (ψ) : H * s-P (R d ) → H * P V (Γ o , C lc (Ξ, R)) is surjective and ψ(B * w,P (R d )) ⊂ B * P V (Γ 0 , C(Ξ, R)). Itfollows that the restriction of H * (ψ) to Z * s-P (R d )/B * w-P (R d ) ∩ Z * s-P (R d ) is surjective.

e.g. their sets of boundary points of their tiles coincides on the r-ball around 0

This definition does not only apply to tilings but to any kind of pattern of R d , but it captures what we want only in the case of finite local complexity.

Strictly speaking the parameterization is in terms of this group with coefficients in R d .


W Miladi
email: walidmiladi@hotmail.com

M Racila
email: mracila@yahoo.com

Mathematical model of fluid flow in an osteon Influence of cardiac system

Keywords: Human cortical bone, fluid-structure interaction, Brinkmann equation

Introduction

Numerical simulations of the behavior of the osteonal structure are more and more acute and an important parameter is the pressure of the bony fluid. Haversian and Volkmann canals contain blood vessels that transport oxygen and nutrients necessary for the cellular activity. The pressure in these vessels must be taken into account. While it is possible to estimate the value of this pressure, there is no information on the effect of the vessel wall that may have in the transmission of pressure. This point is important for the applications that can be done with the SiNuPrOs model [START_REF] Predoi-Racila | Human cortical bone: the SiNuPrOs model. Description and macroscopic results[END_REF][START_REF] Racila | SiNuPrOs: human cortical bone model with a fluid-structure interaction[END_REF] since the pressure variations are very low in such a structure (because its small size). Moreover, taking into account the mesh of the wall in the simulations increases the size of the problem and the cost of its solving.

Methods

We consider a femoral bone for which one knows the parameters of its architecture. The SiNuPrOs model allows computing firstly the macroscopic homogenized elastic properties and then the elastic properties of an osteon and of surrounding interstitial system. This study is based on the localization aspect of the SiNuPrOs model. Knowing a macroscopic loading on this bone, it's possible to compute, by a FEM, all the mechanical fields at the macroscopic scale and to determine these fields at the scale of one osteon.

The osteon {O} is considered as a cylinder embedded in a parallelepiped box of interstitial system {IS} and is crossed by the haversian channel {H} where a cylindrical hole represents a blood vessel (Figure 1). The flow is modeled by a Brinkmann equation in {H}, a conservation equation with the Darcy's law in {O} U {IS} :

G v ) v pId ( div          0 )] g P ( [ div       
where p, v,  ,  ,  and G are respectively the pressure, the velocity, the permeability, the viscosity, the porosity and the gravitational force. Some conditions of coupling are added. On the interface between {H} and in {O} U {IS} one considers an equality of pressure and a correlation depending of the porosity between the velocities. On the interface between the vessel wall and {H}, one imposes equality between the normal component of the stresses and the fluid pressure and equalities between the fluid velocity and the velocity of the external boundary of the wall. A periodic function is applied on the fluid inside H in order to simulate the blood pressure for which the minimal and maximal values are respectively 9.69 and 9.71Kpa (Figure 2).

Results and Discussion

On a mathematical point of view, it is possible to prove that the two previous problems have unique solutions in velocity and pressure. In order to evaluate the effect of the wall, we study the evolution of the pressure in several points located inside the osteon but not in the immediate vicinity of the haversian channel in the framework of a comparative analysis between the cases with and without wall effect. The considered parameters are the fluid viscosity and the elastic properties of the wall. The fluid viscosity varies between 1 and 4 mPa.s, the Young's modulus between 0.1 and 10 MPa and the presented results (Figure 3) are those corresponding to the maximal value of the blood pressure.

The only difference that we can observe in each point between the two cases only deals with the intensities of the considered physical quantities. There in no major differences between the various points.

We find that the wall vessel absorbs a part of the blood pressure and for a Young modulus higher than 0.5 MPa, the absorption is almost constant for each viscosity. One can see on Figure 3 the similarity of the curves. It should be noted that the presence of the wall induces a decreasing of the pressure which can be important (until 0.8 kPa). This effect cannot be neglected since the mean value of the order of magnitude of the pressure in the osteonal structure is 1-10 kPa. It seems that this influence is constant for every Young's modulus that is higher than 1 MPa. In fact, it is not exactly constant and since the local variations in this osteonal structure are low (about 10-80 Pa), this variations can have some importance. These variations for a Young's modulus of the vessel wall varying between 1 et 10 MPa are summarized on Table 1.

Viscosity (mPa.s) 1 2 3 4 Pressure variation (Pa) 10 39 82 100 Table 1. Variation of the pressure for a Young's modulus of the vessel wall varying between 1 et 10 MPa as function of the fluid viscosity.

The last part of this study concerns the effect of the pulsatile blood flow. This pulsatile aspect is conserved in the osteonal structure but there is a smoothing of the values and a decreasing of the maximal value due to the low permeabilities of the osteonal and interstitial system structures when the mineralization degree is important.

Conclusions

This study shows that good quantification of pressure in the osteonal structure requires the taking into account of blood vessels in the channels of Havers and Volkmann and the introduction of an amplitude pressure to find the effect of pulsatile blood flow.

There is no need to mesh the wall, which would be very expensive and it is only necessary to change the value of the applied pressure.

The multi scale analysis developed by SiNuPrOs allows various investigations using datasets consistent with each other.

Figure 1 .

 1 Figure 1. Osteonal structure Two cases are studied: in the first one only these domains and in the second, a thin layer is introduced inside the haversian channel to model the wall of the vessel.Some conditions of coupling are added. On the interface between {H} and in {O} U {IS} one considers an equality of pressure and a correlation depending of the porosity between the velocities. On the interface between the vessel wall and {H}, one imposes equality between the normal component of the stresses and the fluid pressure and equalities between the fluid velocity and the velocity of the external boundary of the wall. A periodic function is applied on the fluid inside H in order to simulate the blood pressure for which the minimal and maximal values are respectively 9.69 and 9.71Kpa (Figure2).

Figure 2 .

 2 Figure 2. Evolution of the blood pressure during a cardiac cycle

Figure 3 .

 3 Figure 3. Decreasing of the pressure induced by the vessel wall (stationnary case)

Acknowledgments

Grant PN-II-RU-RP-2008-4-7 of the Romanian MEC We thank J.M. Crolet for his remarks.