Thiernithi Variddhisaï

Danilo P Mandic
email: d.mandic@ic.ac.uk

On an RLS-Like LMS Adaptive Filter

Keywords: Online predictor, least mean square, recursive least squares, quaternions, widely linear model, steepest descent, conjugate gradient, Markov conjugate

, which relates LMS with the Kalman filter. The proposed alternative to the RLS is favored when the matrix inversion lemma is not useful, such as the case of multivariate or multichannel data where the input is not a vector. Furthermore, all the derivations are conducted in the quaternion domain and are hence generalizable to complexand real-valued models. The resulting algorithm has a neat form and a similar complexity to the RLS. Through experiments, the method is shown to exhibit performance close to or even better than the RLS algorithm. Other aspects, such as the choice of descent directions and variable stepsizes, are also discussed to support the analysis.

I. INTRODUCTION

The least mean square (LMS) algorithm has long been a workhorse of adaptive filtering and has, since its conception in 1959 [START_REF] Widrow | Adaptive switching circuits[END_REF], found a number of successful applications. Based on the minimization of an instantaneous approximation to the mean square error, the LMS converges to Wiener-Hopf solution w op (i.e. the optimal solution), while its 'instantaneous' nature also makes it an algorithm of choice for non-stationary signal processing. Another popular way to reach the Wiener solution, w op , is by using the method of least squares, through the algorithm called the recursive least squares (RLS) [START_REF] Sayed | Fundamentals of Adaptive Filtering[END_REF]. Owing to no approximations in the derivation, the RLS is capable of converging in much fewer iterations than the LMS, and achieve better tracking performance. It is nevertheless not without drawbacks, such as poorer ability to track nonstationary processes or processes with outliers. Moreover, it is less numerically stable and more computationally expensive, both due to the use of matrix inversion lemma [START_REF] Sayed | Fundamentals of Adaptive Filtering[END_REF]. To date, there has been no clear consensus on whether to use the LMS or the RLS, and the answer depends on the applications at hand, while much research effort has been dedicated to improving both algorithms. For LMS, most of its variants have focused on varying the step size, in an attempt to accelerate the convergence and reduce steady-state misalignment [START_REF] Bismor | Review and comparison of variable step-size LMS algorithms[END_REF], while for RLS, the main focus has been on effective implementation and numerical robustness [START_REF] Diniz | Adaptive Filtering: Algorithms and Practical Implementation[END_REF].

Many methods were proposed in an attempt to create alternative online linear algorithms which inherit the virtues of both LMS and RLS, while mitigating some of their drawbacks. Albeit introduced under seemingly different concepts(e.g. [START_REF] Chang | Analysis of conjugate gradient algorithms for adaptive filtering[END_REF], [START_REF] Diene | Adaptive filtering algorithms designed using control Lyapunov functions[END_REF], [START_REF] Zhao | Conjugate gradient algorithm design with RLS normal equation[END_REF]), these RLS-like methods would benefit from being undertaken within one unifying framework. This paper intends to provide a rigorous and systematic derivation of a unifying algorithm in this class, based on such a unified approach, so that it can be more accessible and more easily utilized. The other important contribution here is that the framework was derived in the quaternion domain which offers twofold advantages; first, since quaternions are generalization of real and complex numbers so that the derived algorithms will also hold true for both these number systems; second, the adoption rate of quaternions is increasing in a variety of applications, such as modeling of 3-D rotational data (computer graphics [START_REF] Rabbani | Automatic reconstruction of industrial installations using point clouds and images[END_REF] and array signal processing [START_REF] Miron | Quaternion-MUSIC for vectorsensor array processing[END_REF], [START_REF] Tao | A novel combined beamformer based on hypercomplex processes[END_REF], [START_REF] Zhang | Quaternion-valued robust adaptive beamformer for electromagnetic vector-sensor arrays with worst-case constraint[END_REF]), color image processing [START_REF] Chen | Color image analysis by quaternion-type moments[END_REF], [START_REF] Xu | Vector sparse representation of color image using quaternion matrix analysis[END_REF] and source separation [START_REF] Chan | Complex and quaternionic principal component pursuit and its application to audio separation[END_REF]. Hence, deriving the proposed framework in the quaternion domain offers more rigor and intuition in terms of physical interpretation, algebraic generality and algorithmic elegance.

II. BACKGROUND

A. Quaternions and GHR Calculus

Since there is no shortage of excellent tutorials on basic quaternion algebra [START_REF] Kantor | Hypercomplex Numbers: An Elementary Introduction to Algebras[END_REF], [START_REF] Ward | Quaternions and Cayley Numbers: Algebra and Applications[END_REF], we start from the generalized HR (GHR) calculus, the recent concept which enables our proposed method to have an elegant form in the quaternion domain which is consistent with its real-and complex-valued counterparts. For detailed information, we refer to [START_REF] Xu | Enabling quaternion derivatives: The generalized HR calculus[END_REF], [START_REF] Xu | Optimization in quaternion dynamic systems: Gradient, Hessian, and learning algorithms[END_REF].

Consider a general case of functions f (q) : H M ×1 → H, where q = (q 1 , q 2 , ..., q M ) T ∈ H M ×1 . Then, the quaternion gradient and conjugate gradient are respectively given by [START_REF] Xu | Optimization in quaternion dynamic systems: Gradient, Hessian, and learning algorithms[END_REF] ∇ q f ∂f ∂q

T = ∂f ∂q 1 , ..., ∂f ∂q M T ∈ H M ×1 , ∇ q * f ∂f ∂q * T = ∂f ∂q * 1 , ..., ∂f ∂q * M T ∈ H M ×1 .
Similar to the complex domain, it was shown that the conjugate gradient ∇ q * f yields the steepest descent direction of the function f [START_REF] Mandic | A quaternion gradient operator and its applications[END_REF]. This makes the conjugate GHR derivatives a natural choice for the optimization of our proposed algorithm.

Within the GHR calculus, the conjugate derivative is defined based on a generalized orthogonal basis {1, i µ , j µ , k µ } where µ is a rotating quaternion. In our work, a simplified case where µ = 1 suffices and yields a neater form of the HR conjugate derivative [START_REF] Mandic | A quaternion gradient operator and its applications[END_REF]. The left and right derivative operators, which arise out of the non-commutative nature of quaternions, are respectively defined as follows

∂f ∂q * = 1 4 ∂f ∂q a + ∂f ∂q b i + ∂f ∂q c j + ∂f ∂q d k , ∂ r f ∂q * = 1 4 ∂f ∂q a + i ∂f ∂q b + j ∂f ∂q c + k ∂f ∂q d .
where q = q a + iq b + jq c + kq d , q a , q b , q c , q d ∈ R and i 2 = j 2 = k 2 = -1. Observe that when f is real-valued, both left and right derivatives are identical, which is the case with our analysis, and from now on, we will therefore implicitly refer to the left derivative.

The key benefits of the GHR calculus are the novel product rule and chain rule (not present in the HR calculus). These two rules are integral in the derivation of the proposed framework. For the product rule, if the function f, g : H → H are realdifferentiable, then so too is their product, that is,

∂(f g) ∂q * = f ∂g ∂q * + ∂f ∂q g * g (1)
where q g * = gq * g -1 is a quaternion rotation [START_REF] Xu | Enabling quaternion derivatives: The generalized HR calculus[END_REF]. For the chain rule, if g : S → H and f : T → H are real-differentiable at the respective interior points, q ∈ S ⊆ H and g(q) ∈ T ⊆ H, then the derivative of the composite function f (g(q)) is given by ∂f (g)

∂q * = ∂f ∂g ∂g ∂q * + ∂f ∂g i ∂g i ∂q * + ∂f ∂g j ∂g j ∂q * + ∂f ∂g k ∂g k ∂q * = ∂f ∂g * ∂g * ∂q * + ∂f ∂g i * ∂g i * ∂q * + ∂f ∂g j * ∂g j * ∂q * + ∂f ∂g k * ∂g k * ∂q * (2)

B. Adaptive Filters

For completeness, we next briefly summarize the LMS and RLS algorithms in their most general form -the widely linear quaternion model (for general non-circular data [START_REF] Took | Augmented second-order statistics of quaternion random signals[END_REF]).

Denoted by y n , x n ∈ H , n = 1, ..., N the desired (output) signal and input signal, respectively, the linear MSE estimator of y n , denoted by ŷn , can then be expressed as [START_REF] Papoulis | Probability, Random Variables and Stochastic Processes[END_REF]

ŷn = ŵ, q n ŵH q n ( 3
)
where ŵ is an estimate of the optimal solution w op , q n is defined as

q n = [x T n x iT n x jT n x kT n ] T (4)
and

x n = [x n , x n-1 , ..., x n-M +1 ]
T for a filter of order M . For a simple linear model (i.e. data is circular [START_REF] Took | Augmented second-order statistics of quaternion random signals[END_REF]),

q n = x n (5)
The goal of an online linear predictor is to minimize the mean square error (MSE)

J n ( ŵ) = E{ ξ n ( ŵ) 2 } (6a) ξ n ( ŵ) = y n -ŷn = y n -ŵH q n , (6b)
recursively, with the weight update expressed as [START_REF] Xu | Enabling quaternion derivatives: The generalized HR calculus[END_REF] w

n = w n-1 -α n ∇ ŵ * J n ( ŵ)| ŵ=wn-1 . (7)
In the case of the LMS algorithm, eq. (6a) becomes an instantaneous estimate, that is,

J n ( ŵ) ≈ ξ n ( ŵ) 2 , (8)
which as a result gives the following update [START_REF] Took | The quaternion LMS algorithm for adaptive filtering of hypercomplex processes[END_REF] w

n = w n-1 + α n q n e * n ( 9
)
where e n is a priori error, defined as

e n ξ n (w n-1 ) = y n -w H n-1 q n (10)
and α n ∈ R is an adaptive stepsize. For the normalized LMS, α n can be found via the minimum disturbance principle [START_REF] Sayed | Fundamentals of Adaptive Filtering[END_REF] as

α n = 1 q n 2 + ( 11
)
where > 0 is a regularizer which prevents numerical instability when q n 2 → 0.

On the other hand, the RLS filter iteratively solves the least square solution, ŵ, of the given set of training pairs y n , x n via the following deterministic cost function [START_REF] Jahanchahi | The widely linear quaternion recursive least squares filter[END_REF],

Φ n ( ŵ) = n k=1 λ n-k ξ k ( ŵ) 2 (12)
where 0 < λ < 1 is a forgetting factor used to suppress the effect of early data which may be no longer relevant to the current estimate. The core idea of the proposed method is to derive an LMS algorithm which approximates MSE in a manner similar to eq. ( 12). We did not describe the whole RLS algorithm owing to page limits.

III. AN RLS-INSPIRED LMS ALGORITHM

A. Derivation in Quaternions

Inspired by the RLS cost function in eq. ( 12), we reformulate the estimation of the MSE in eq. (6a) to

J n ( ŵ) ≈ 2Φ n ( ŵ) (13)
Here, the factor 2 is present to normalize the expressions involving GHR derivatives [START_REF] Xu | Optimization in quaternion dynamic systems: Gradient, Hessian, and learning algorithms[END_REF] (which actually can be absorbed into the stepsize α n ). Then, we substitute eqs. ( 3) and (6b) into eq. ( 13) to yield

J n ( ŵ) = ŵH R n ŵ -2R{ ŵH r n } + n k=1 λ n-k |y k | 2 (14)
where R{•} is a real-part operator with

R n = n k=1 λ n-k q n q H n = λR n-1 + q n q H n , (15)
r n = n k=1 λ n-k q n y * n = λr n-1 + q n y * n . ( 16
)
We now follow the optimization path of LMS with the MSE estimate as in eq. ( 12). Through eqs. ( 1) and ( 2), the a priori gradient g n|n-1 is defined as

g n|n-1 ∇ ŵ * J n ( ŵ)| ŵ=wn-1 = R n w n-1 -r n . (17)
Then, the expression of the quaternion weight update becomes

w n = w n-1 + d n α n ( 18
)
where d n is the descent direction, while the stepsize α n ∈ H, unlike eq. ( 7), is non-commutative and has to post-multiply d n so that it will not lead to Sylvester's equation which has no closed-form solution [START_REF] Shpakivskyi | Linear quaternionic equations and their systems[END_REF]. One convenient way to find α n is by using exact line search (equivalent to method of minimum disturbance in the NLMS case [START_REF] Morgan | On a class of computationally efficient, rapidly converging, generalized NLMS algorithms[END_REF]), where ∂J n (w n )/∂α n is set to zero. Upon substituting into eq. ( 7), we obtain

α n = - d n , g n|n-1 d n , h n|n (19)
where

h n|n = R n d n . (20)
Observe that d n , h n|n is real-valued. Now, to reuse past calculations efficiently, a posteriori gradient g n|n is defined as

g n|n ∇ ŵ * J n ( ŵ)| ŵ=wn = R n w n -r n , (21)
and after some manipulation, we arrive at the following dual recursion

g n|n-1 = λg n-1|n-1 -q n e * n ( 22
)
g n|n = g n|n-1 + h n|n α n . (23)

B. Choice of Descent Directions

Most works concerning the RLS-like LMS algorithm use the conjugate gradient (CG) direction because it is the fastest first-order solver of a quadratic-form equation; however, due to the time-varying statistics of R n , most proposed CG techniques fail to sustain the conjugacy property (see [START_REF] Chang | Analysis of conjugate gradient algorithms for adaptive filtering[END_REF] and reference therein). To overcome this issue, some methods were introduced to relax the conjugacy constraint, like control Lyapunov functions [START_REF] Diene | Adaptive filtering algorithms designed using control Lyapunov functions[END_REF] and a CG-like technique [START_REF] Zhao | Conjugate gradient algorithm design with RLS normal equation[END_REF]. All these methods can be considered under one unifying umbrella termed Markov conjugacy. Definition 1. A set of descent directions {d 1 , d 2 , ..., d n } is Markov conjugate if, at any iteration i,

d H i R i d i-1 = 0 for i = 1, 2, ..., n. (24)
In the follow-up paper, we show this idea can easily generalize into a more sophisticated technique. With only one degree of freedom in eq. ( 24), the Markov conjugate direction is expressed as

d n = -g n|n-1 + d n-1 β n . (25)
Pre-multiplying with (d n-1 ) H R n and substituting eq. ( 24) into eq. ( 25) yields

β n = d n-1 , v n d n-1 , h n|n-1 (26)
where

h n|n-1 = R n d n-1 (27)
v n = R n g n|n-1 . (28)
and after some manipulation, we arrive at the another dual recursion

h n|n-1 = λh n-1|n-1 + q n (q H n d n-1 ) (29)
h n|n = -v n + h n|n-1 β n . (30)
An interesting simplification is the steepest descent scheme where β n = 0, ∀n so that d n in eq. ( 25) becomes

d n = -g n|n-1 (31)
and it can be shown that α n will become real-valued, i.e.,

α n = 1 π n = g n|n-1 2 g n|n-1 , v n . ( 32
)
This is very beneficial in that all the techniques of variable stepsize proposed in the real domain [START_REF] Lee | Performance comparison of variable step size NLMS algorithm[END_REF] are directly applicable in quaternions, i.e.

α n = 1 π n + n (33)
where n ∈ R is an adaptive regularizer [START_REF] Mandic | A generalized normalized gradient descent algorithm[END_REF]. The unified routine of the RLS-like LMS algorithm, which we shall refer to as the m-NLMS i.e. the NLMS with memory, is summarized in Algorithm 1, where δ > 0 is used as a stopping criterion. Update q n by x n according to eqs. ( 4) and ( 5);

6 ŷn -(w n-1 ) H q n ; 7 e n = y n -ŷn ; 8 R n = λR n-1 + q n q H n ; 9 g n|n-1 = λg n-1|n-1 -q n e * n ; 10 v n = R n g n|n-1 ; 11 if Markov conjugate scheme then 12 h n|n-1 = λh n-1|n-1 + q n (q H n d n-1 ); 13 β n = dn-1,vn dn-1,h n|n-1 (β 1 = 0); 14 d n = -g n|n-1 + d n-1 β n ; 15 h n|n = -v n + h n|n-1 β n ; 16 α n = - dn,g n|n-1
dn,h n|n ;

IV. NUMERICAL EXPERIMENTS

A set of numerical case studies was performed to benchmark the validity of the m-NLMS algorithm against the original LMS and RLS algorithms.

For rigor, the simulations were conducted within a widely linear quaternion framework which has found its potential in MIMO channel identification [START_REF] Ding | Widely linear sphere decoder in MIMO systems by exploiting the conjugate symmetry of linearly modulated signals[END_REF]. Here, a widely linear quaternion moving average of order 3 -WLQMA(3) was used, where the input x n was drawn from 1000 samples of the same distribution N (0, 1) for each component of x n , with a moderate SNR of 40dB. Every experiment was conducted over 100 independent realizations and the m-NLMS used in the experiments, if not specified otherwise, was the steepest descent (SD) scheme. For a fair comparison, the stepsizes α n for the original LMS routines was calculated using the minimum disturbance principle as in eq. ( 11) in which the original LMS yields NLMS. For convenience, we shall omit the term 'quaternion', as our primary attention is paid to the algorithmic mechanism. The metric used to benchmark the algorithms was the normalized misalignment, η n , defined as In the first experiment, we tested how our new algorithm performed against the NLMS and RLS with different parameter settings but with a fixed n = 0.01 just to ensure numerical stability. As seen in Fig. 1, the m-NLMS converged faster and achieved lower steady-state misalignment than the standard NLMS. At λ = 0.99, the m-NLMS also converged faster than RLS while at λ = 0.95, the m-NLMS converged slightly slower but achieved better misalignment than RLS. Fig. 2 demonstrates the performance of the Markov-conjugate (MC) m-NLMS. At λ = 0.99, the MC scheme converged slightly faster than the SD scheme and achieved the same misalignment as the RLS while at λ = 0.95, it converged even faster than the RLS but achieved worse misalignment than the SD scheme. In other words, the m-NLMS, regardless of its descent schemes, behaves on par with the RLS. Fig. 3 shows the results of the second experiment which examined the variable-stepsize version of the m-NLMS. Three fixed-stepsize routines (RLS and m-NLMS) were benchmarked against three variable-stepsize ones (NLLS and m-NLMS with GSER scheme [START_REF] Lee | Performance comparison of variable step size NLMS algorithm[END_REF]). By inspection of m-NLMS at λ = 0.99 and 0.95 as well as comparison of NLMS between Fig. 3 and Fig. 1, it is clear that making n variable reduces misalignment but also convergence rate. A closer inspection of m-NLMS between λ = 0.99 and 0.95 reveals that the reduction in Fig. 4: Misalignment of RLS and m-NLMS algorithms with different value of with λ and variable stepsizes both with and w/o GNGD scheme, averaged over 100 trials, for the identification of a WLQMA(3) process, at an SNR of 40dB misalignment for the case λ = 0.95 is more significant than that for λ = 0.99. It is also feasible to make the variable stepsize more effective by tracking whether the learning process is in the transient or steady state. A technique like GNGD [START_REF] Mandic | A generalized normalized gradient descent algorithm[END_REF] could be incorporated to render the algorithm achieving even lower misalignment, without trading off convergence speed, as illustrated by a dashed line in Fig. 4. We did not provide further details on GNGD m-NLMS and experiments on realworld signals, like 3-D wind, which will be covered in the follow-up paper and supported with more analysis. Intuitively however, it would behave as consistently as in this paper.

η n w n -w op 2 w op 2 . (34) Time (Samples)

V. CONCLUSION

In an effort to explore the space of algorithms spanned by the LMS and RLS, we have built on the work in [START_REF] Mandic | On the intrinsic relationship between the least mean square and kalman filters [Lecture Notes[END_REF], based on the Kalman filter, to provide a unified and generalized approach to a class of RLS-inspired LMS adaptive filters. This has resulted in the proposed m-NLMS framework which has been derived in the quaternion domain so that the resulting algorithm, which has the complexity comparable to the RLS, can be straightforwardly simplified to complexand real-valued cases. Two options of descent direction have also been derived rigorously, under the unifying notion of Markov conjugacy proposed in this paper. Since the m-NLMS is technically a variant of LMS, it is more numerically stable than RLS, as no matrix inversion is involved in the update. The experiments have demonstrated that the m-NLMS has been able to achieve performance which is more or less identical to the RLS, regarding both convergence rate and misalignment. Sharing the virtues of both LMS and RLS, the m-NLMS has been shown to admit further improvement through a variable stepsize. The experiments in the system identification have supported the analysis.

Algorithm 1 : 4 n

 14 The m-NLMS Algorithm Input : x n , y n , M , λ, , δ and update scheme Output: ŷn and w op 1 Initialize w 0 = d 0 = g 0|0 = h 0|0 = 0 and R 0 = 0; 2 n = 0; 3 do = n + 1;

5

 5

17 else steepest descent scheme 18 d 1 2 ; 20 end 21 w 22 g

 17181202122 n = -g n|n-1 ;19 α n = 1πn+ n where π n = g n|n-1 ,vn g n|n-n = w n-1 + d n α n ; n|n = g n|n-1 + h n|n α n ; 23 while e n > δ or n = N ; 24 w op = w n .

Fig. 1 :TimeFig. 2 :

 12 Fig. 1: Misalignment of NLMS, RLS, and m-NLMS algorithms with different λ, averaged over 100 trials, for the identification of a WLQMA(3) process, at an SNR of 40dB

NLMSFig. 3 :

 3 Fig. 3: Misalignment of NLMS, RLS, and m-NLMS algorithms with different value of with λ and either fixed or variable stepsizes, averaged over 100 trials, for the identification of a WLQMA(3) process, at an SNR of 40dB

  w/o GNGD (solid) VS with GNGD (dashed) RLS, λ = 0.95 m-NLMS, λ = 0.99 m-NLMS, λ = 0.95 GNGD m-NLMS, λ = 0.95 GNGD m-NLMS, λ = 0.95 RLS, λ = 0.95 m-NLMS, λ = 0.95 m-NLMS, λ = 0.99


TRANSFORMATIONS OF STATE'S USE OF FORCE IN EUROPE

 Fabien Jobard Centre Marc Bloch, BerlinThis chapter deals with the transformations that have occurred in the course of the last 40 years in Europe in the state's use of force. This topic calls for two preliminary remarks.

The first concerns the narrow terms "use of force". In what follows, we will make extensive use of this expression, which is commonly used to describe the use of (lethal) force by public police officers. We will also use a classic conception of the state, i.e. a set of institutions aiming at a monopoly of the use of violent and coercive means [START_REF] Weber | [END_REF][START_REF] Whitman | The Comparative Study of Criminal Punishment[END_REF], including prison and all other means of imprisonment1 . Second, we will consider the approximate date of 1970 as the appropriate time-scale for assessing the "transformation" which the paper attempts to track, considering that the end of the 1960s and the begin of the 1970s signalled the start of an overall concern in Western democracies for the problem of crime control and of the diffusion of a given "culture of control" [START_REF] Garland | The culture of control : crime and social order in contemporary society[END_REF], characterised by a willingness among the public and the political elites to move away from the dominant "penal-welfare culture". A useful date in this respect (even though it is taken from outside of the European world) is 1972: that year, the US detention rate reached its lowest level in almost half a century; 40 years later, it has been multiplied by five (Hinds 2005, 48).

In what follows, we will try to determine whether state powers have been transformed, with regards to the use of coercion and force. We will first focus on some broad indicators of punitiveness to assess whether punitiveness has increased in Europe, and if it has, for what reasons. We will then try to figure out how state violence is governed in Europe.

Part 1. Governing through crime (or not)

Reading the academic literature, one might think a spectre is haunting Europe: the spectre of the so-called "penal state". Loic Wacquant, one of the leading scholar promoting this notion, defines it in 'Ordering Insecurity' (Wacquant 2008, 21) as "the likely contours of the future landscape of the police, justice, and prison in the European and Latin American countries that have embarked on the path of 'liberating' the economy and reconstructing the state blazed by the American leader". In the wake of 1930s German social scientist [START_REF] Rusche | Punishment and Social Structure[END_REF], this notion sketches an explanatory link between the transformation of social structures and the transformation of the state, pointing to a parallel increase in the US in detention rates and in the promotion of neo-liberal economic and social doctrines (see [START_REF] Harcourt | The illusion of free markets[END_REF]. Now, is this double assertion (increased punitiveness on the one hand and a robust correlation between social and economic policies on the other hand) congruent with contemporary European states? 1.1. Increased punitiveness: not all forms of punitiveness, not everywhere Changes have occurred in Europe, with respect to crime control strategies, but on much more limited scale than in the US, and not in all domains of public violence. Police rates (i.e. the ratio police personnel to the population) have been remarkably stable in the European Union, as shown in Hinds' work (2005): there were 2,93 public police officer for 1000 inhabitants at the beginning of the 1970s: there are 3,05 thirty years later. At the same time, the overall US police personnel increased 75% (from 1,97 to 3,4), although this rate has since then been affected by the cities fiscal crisis of the 2000s, and brought down to 2,9 in 2011 (CJS Sourcebook). Meanwhile, detention rates, which are universally taken as the indicator par excellence of punitiveness [START_REF] Hamilton | Reconceptualizing Penality Towards a Multidimensional Measure of Punitiveness[END_REF], have reached exceptionally high levels in Europe, jumping from 65 prisoners for 100.000 inhabitants to 87 in the EU member states [START_REF] Hinds | Crime control in Western countries, 1970 to 2000[END_REF], from 75 to 150 in England & Wales (1970-2010), from 100 to 150 in Ireland, from 50 to 115 in France, and from 75 to 85 in Germany (SPACE).

Two things should be taken into account, though. First, even though detention rates have risen up in Europe in the course of the last 30-40 years, they go nowhere near the US rate: in the US, detention have risen from 130/100.000 at the beginning of the 1970s to almost 750/100.000. Any comparison between Europe and the US is between two incomparable worlds. Second, the European situation is rather uneven. In "Nordic" countries, it seems that there has been no wave of punitiveness: detention rates remain at very low levels (between 65 and 75/100.000 [START_REF] Pratt | Contrats in Punishment. An Explanation of Anglophone Excess and Nordic Exceptionalism[END_REF]. Some countries experienced a decrease in the use of imprisonment as a penal tool, like Germany after the Unification (-15% during the first 2000 decade) or like Portugal, where detention rates fell from 150 in the mid-90s to 110 now, after an upward wave in the 1980s.

In contrast with the US, the situation in Europe is not just very uneven, it is also miles away in terms of police rates and more importantly detention rates. Jonathan Simon, in his very influential books Governing through Crime (2007) and From the New Deal to the Crime Deal (XXXX) has argued that contemporary democracies have experienced and even encouraged a shift in political cultures, placing crime issues at the very centre of the political game and making punitiveness the most efficient tool to win democratic competitions. Can this groundbreaking thesis be helpful to make sense of the heterogeneity of the European landscape? In other words, there is a reverse relationship between welfare and penal states, as Simon (XXXX) or [START_REF] Garland | The culture of control : crime and social order in contemporary society[END_REF] had already pointed out for common law countries. Leppi-Seppälä's work goes a step further, however, by showing that neither homicide rates, nor victimisation rates or rates of fear of crime are correlated with detention rates in democratic states. Social expenditures is actually the most correlated variable with detention rates. Levels of welfare states and levels of penal states are then strongly related to one another. And [START_REF] Downes | Welfare and punishment in comparative perspective[END_REF] have shown that the divide between welfare and penal states increased during the 1990s.

The states and the political game

In fact, the crucial role played by levels of welfare in explaining differences in punitiveness across the western world, and especially between Europe and the US, is reinforced by government's organisation and the rules of the political game.

Governments' organization and the power to punish

The broad divide between welfare and penal states overlaps with another one, between strong and weak states. David Downes provided 25 years ago a clear insight into the relation between state structures and punishment [START_REF] Downes | Contrasts in Tolerance[END_REF]. In his seminal work, he studied the difference in punitiveness in England and in the Netherlands. Both countries had been confronted with rising crime rates after the end of WW2. But while prison rates doubled in the UK from 1945 to 1975, they were cut by two during in the Netherlands. No difference in "crime" or "insecurity" explains the difference between the two states' answer to the crime issue.

The major difference, Downes argued, was anchored in how the power to punish is organised in the two countries. In England & Wales, there was at the time no involvement of the government in penal matters, no "penal capacity": state authorities only had limited influence over justice: decisions and sentences were pronounced by fully independent judges and magistrates. The rise in imprisonment in England and Wales was the uncoordinated result of judiciary decisions. On the contrary, there was a strong penal structure in the Netherlands, as well as a robust coordination between the Ministry of Justice, the prosecutors, and the "professionals"especially criminologists, who helped give birth to the so-called "Utrecht school of criminology", which would become very influential at the European level. The implementation of a "call-up system" i.e. the strict prohibition of any new prison sentence in a context of prison overcrowding was possible in the Netherlands, but not in England and Wales where government was not supposed to interfere with judges' decisions.

Dutch detention rates were at an all-time low at the beginning of the 1980s (around 25/100.000), even in comparison with Nordic countries today. However, since David Downes published his book, Dutch rates have jumped up to 100/100.000 in the mid-2000s.

But if anything, this recent increase reinforces the state organisation's thesis. Dutch public opinion has converged with that of Anglophone countries in making crime victims and safety two central themes of political discourse since the end of the 1980s [START_REF] Boutellier | The convergence of social policy and criminal justice[END_REF]. As a consequence, the same state structures that adopted an abolitionist path in the 1960s and 1970s produced the policies expected by the shift in public opinion: compensation funds for victims were created in 1975, prison capacity has dramatically increased (from 4.000 in 1986 to 18.000 prison cells in 2000), task penalties for adults were implemented (there were 15.000 a year already in 2000), reactions to criminal behaviour were internalised by the criminal justice system (instead of being externalised to social and welfare agencies). Finally, there has been an overall growth of state capacities, with the Ministry of Justice's budget tripling between the mid-80s and the beginning of the 2000s.

At the same time, convergent efforts were made by Labour government in the UK and Democrat government in the US to reassert the state's power to punish: both governments strengthened judges' punitiveness and helped local governments pass "zero tolerance" policies (such as the Morgan Act and the US Crime Act in the US, or the Crime and Disorder Act, the Crime Reduction Partnership and Home Office national objectives in the UK - [START_REF] Gilling | Community safety and social policy[END_REF][START_REF] Garland | The limits of the sovereign state. Strategies of crime control in contemporary society[END_REF]. As a "punitive turn" was hitting European public opinions, states with a strong criminal justice system and states with strong judges' and local authorities' autonomy alike implemented punitive policies: the former by moving from a welfare to a punitive culture, the latter by regulating judges' powers with mandatory sentencing systems and by allocating more resources to the most repressive local governments.

The rules of the political game

The existence of a state-monitored criminal justice system (which we summed up with the notion of "strong state") or the existence of a myriad of independent judges and local authorities are not the only cleavage that overlaps with the welfare vs. penal distinction.

Voting systems and the rules of the political game play just as important a role as state structures. In short, we should add Lijphart's name to Esping-Andersen's in the discussion.

Lappi-Seppälä, 2014.

Leppi-Seppälä has shown that there is a stronger variable than social expenditures in predicting detention rates in Anglophone, Southern, Christian-conservative, and Nordic countries, namely Lijphart's executive-parties index. Lijphart distinguishes between "consensus" and "majoritarian" democracies, which display very different institutional arrangements: consensus democracies have a larger number of political parties, a proportional electoral system, and minority or broad-based coalition governments, with the consequence that political decision making is based on negotiation seeking processes.

Scandinavian countries and Germany are today's best examples of such systems.

Majoritarian democracies like the UK and the US are characterized by antagonistic political systems. These political antagonisms are reflected in media culture, because majoritarian, adversarial democracies encourage crisis discourses that are supported by hegemonic media coverage encouraging repressive policies and one-sided answers to crime [START_REF] Green | When Children Kill Children. Penal Populism and Political Culture[END_REF][START_REF] Cavadino | Penal systems: A comparative approach[END_REF].

This cleavage in political cultures has been strongly supported by the influential book of Nicola Lacey, The Prisoner's Dilemma (2008), which opposes northern Europe's coordinated systems to adversarial, liberal political systems: in the latter, criminal policy is formed by a government accountable to an electorate within a two-party system which encourages the politicisation of criminal justice, creating a volatile policy-making environment, particularly where both parties have embraced crime as a major policy platform [START_REF] Hall | [END_REF]. [START_REF] Lacey | The rule of law and political economy of criminalisation. an agenda for research[END_REF] insists that the great division between these two political systems not only produces different detention rates, but moreover reveals opposite cultures. According to her, adversarial democracies support legislation based on character, or status-based perceptions of the criminal (such as the "bad character" disposition in Labour legislation), which singularizes the political path taken in criminal justice matters by these political systems. One of the best examples here is the devolution process in Scotland, combining an alignment of the local Labour with the English Labour, and a massive shift of Scottish towards English criminal justice outcomes (McAra 2011, 97).

In conclusion, a more or less massive "punitive turn", as exemplified by detention rates and repressive legislations, has occurred in Western democratic systems during the last 3 decades. Still, this turn, which reinforced the coercive capacities of the states, is strongly connected to a greater division between welfare states and coordinated economic systems anchored in consensus political systems opposed to liberal and free market economic systems echoing adversarial political systems. The power to punish is then rooted in a political economy of punishment along with the rules organising the political competition.

"Cultures of control" [START_REF] Garland | The culture of control : crime and social order in contemporary society[END_REF], which have universally developed since the beginning of the 1980s, were either strongly, either minimally supported by the state in the US and in Europe. The "from the new deal to the crime deal" thesis (Simon XXXX) met the resistance of state's structures, wherever welfare social systems and consensus political regimes were strong enough.

Still, this overall description of the transformation (or stability) of the state's coercive capacities has been so far too much oriented by a single variable, the detention rates. A full analysis of this transformation calls for a more multi-centred approach based on a multiplicity of variables. As Zedner (2002, 356) has put it: "Popular punitivism may be a key feature of contemporary penal politics, but it is not the whole story".

Part 2. Sources of coercion

Detention rates give a first insight into the state's coercive powers across Western Europe and North America. But they only account for part of these powers. In the US, but only there, the state's monopoly of violence still coincides with the power to administer capital punishment. In some penal systems, prisons are still viewed as places of rehabilitation, and in other systems, they are only places for incapacitation and confinement. Finally, police powers are not the same in these different systems, and do not necessarily coincide with imprisonment capacities.

To give a better view of the state's coercive powers, we shall now examine how states are limited by the law (2.1) and by the society (2.2), and how police organisations form coercive archipelagos within coercive institutions (2.3).

The state in Europe: The crucial role of the law

Punitiveness has increased in Western democracies, with few (but notable) exceptions.

Detention rates are in many countries at an all-time hight and penal legislation is more and more at the centre of public debate, with strong implications for every part of the criminal justice system, starting with police forces. But these numbers have risen in the context of a growing concern for due process of law and human or fundamental rights. At least in Europe, the rise in penality is highly contradictory with the concern for human rights, and this makes a large difference with the US situation.

Human rights and coercion in Europe

What is punitiveness? If one follows the leading (Anglophone) authors, punitiveness is embodied in the fact that crime control infringes on the due process of law. From this point of view, punitiveness in Europe has been limited by the growing concern for fundamental rights, at least since the adoption of the European Convention on Human Rights (ECHR) in 1954 by the European Council. In countries where the protection of human rights was already a cornerstone of the state, like in Germany with its 1949 Fundamental Law, the adoption of the ECHR introduced no substantial change. But in countries where governments traditionally enjoyed large legislative discretion like in the UK and France, resistance against the individual ability to seize the European Court in Strasbourg (first introduced in 1981 in France) or against the adoption of the Human Rights Act in England (1998) were strong (and they still are: Home minister Theresa May is thus in favour of replacing it with a British Bill on Human Rights, even at the cost of withdrawing from the ECHR). Such resistances demonstrate the subversive power of the ECHR [START_REF] Feldman | The Impact of Human Rights on the UK Legislative Process[END_REF].

EU member states have imported the substance of fundamental rights from the Council by considering the ECtHR as a kind of Supreme Court, and adopted in 2000 the Charter of fundamental rights (Nice Treaty), which considers criminal law as a mean to protect fundamental rights rather than as a mean to widen the penal net, and the Union as a space of freedom, security, and justice (Lisbon Treaty).

A cornerstone of criminal justice policies in Europe is the absolute prohibition of death penalty, torture and ill treatment (Kamninski, [START_REF] Kaminski | Mutations dans le champ des peines et de leur exécution[END_REF][START_REF] Snacken | Resisting punitiveness in Europe?[END_REF]. Death penalty is one of the few indisputable requirements that must be met to enter the EU. A few European states (among which France) have already been sentenced by the ECtHR for "torture", with decisive consequences for the control of prison and police cells.

Moreover, the European Council's view of detention (of any kind) is based on defiance.

Not only should imprisonment be used in last resort (it has even been defined in 2013 as an "ultima ratio" by German Supreme Court jurisprudence), framed within the iron law of due process (legality, legitimacy, proportionality). But even when detainees are in prison, rather than being objects of shame and degradation, they are considered to be objects of rehabilitation or at least of protection: as the European Prison Rules stated in 2006: "All forms of detention shall be managed in a way that facilitates the reintegration into free society of persons who have been deprived of their liberty". In a variety of domains (work, education, family visits, syringe exchange, health care, high security units, prison overcrowding, suicide…), the Council's provisions have paved the way for a European regime of punitiveness, which of course pressurizes the state (for a French example, see [START_REF] Cliquennois | A new risk management for prisoners in France: The emergence of a death-avoidance approach[END_REF].

National states are indeed the targets of the ECtHR and of the European Council's Committee for the Prevention of Torture: "The CPT is regarded as a regional prisons inspectorate of considerable sophistication, and no other region has a human rights court that gives such comprehensive and binding judgements on matters affecting prisoners' human rights. This European commitment to protecting human rights of prisoners has been described as one of the major mechanisms explaining the growing divide in harshness between penal policies in Europe and the USA (Snacken et al. 2014, 437)." Contrary to the US, where prison is used as a tool of incapacitation and discipline, where "punishment is an act of sovereign might, a performative action which exemplifies what absolute power is all about (Garland 1996, 460)", European system might turn to a "post-disciplinary" institution [START_REF] Chantraine | Rights in prison[END_REF], in which all public efforts are devoted to minimizing the coercive dimension of imprisonment.

Human rights and the protection of the body and soul of individuals

The sacralisation of human rights in Europe has also paved the way for a reinforcement of state punitiveness. In a 1979 decision, the ECtHR gave birth to the "theory of positive obligations", which states that a violation of the Convention committed by a private person can be indirectly attributed to the state when the state has made it possible or probable.

One of the most famous examples of this theory is to be seen in the field of child protection. In the "X & Y vs. Netherlands" decision of 1987, applicants complained because they were unable to have criminal proceedings instituted against the person who had raped a young girl with learning disabilities that made her unable to lodge a complaint herself. The Court asserted the protection of private life (art. 8 ECHR) as an obligation for the state. In A v. UK 1993, the Court ruled that Article 1 of the ECHR requires states "to take measures designed to ensure that individuals within their jurisdiction are not subjected to treatment proscribed by Article 3 (torture, inhuman and degrading tretament) ,including such treatment administered by private individuals". The court went on to find that "children and other vulnerable individuals are entitled to protection in the form of effective deterrence against such serious breaches of personal integrity". In the 1987 case already, the Court had issued that in some cases "effective deterrence can be achieved only by criminal law provision [START_REF] Kinkelly | Protecting Children's Rights under the ECHR: The Role of Positive Obligations[END_REF][START_REF] Mowbray | The Creativity of the European Court of Human Rights[END_REF]."

The ECtHR's efforts to provide "effective" human rights is often seen as placing direct obligations on state power (Banach-Gutierrez & Harding 2012). The notion of "effective remedy" is one among many notions that force the state to downsize its discretionary powers and the scope of its tools of coercion. In the case of Yase v. Turkey 1998, the ECtHR stated that the applicants could not establish that the applicant had been attacked and his uncle killed by state security forces, but that Turkey should be condemned on the basis of Art. 2 ECHR (right to life) because state authorities had not conducted an effective investigation into the circumstances of the complaint. To quote another example, Osman v UK overruled the UK court's decision in Hill v West Yorkshire that public bodies could not be held to be negligent. In short, all aspects of the state's power to kill, to detain, to investigate, to question, etc. are now under the scrutiny of the ECtHR. What the theory of positive obligations adds to this, is the obligation for the state to enlarge the scope of criminal law to prevent any crime committed by an individual against another individualwhich gives in return more power to the state, as it is now seen both as a potential threat to the enjoyment of fundamental rights and as the best tool to enforce them. As a consequence, all crimes that jeopardize the rights promoted in the ECHR, specifically crimes against individual integrity like drug crimes, violent crimes, sexual crimes or crimes against youths, minorities, women etc. are criminal fields where the state has been more and more requested in the last 30 years by supranational authorities and courts.

At the beginning of the 1980s, Anthony Bottoms coined the notion of "bifurcation policy" to differentiate between penalties for serious and minor offences [START_REF] Bottoms | Neglected features of contemporary penal systems[END_REF]. After a wave of de-criminalization during the 1970s (de facto de-criminalization of cannabis use in different European countries, de lege de-criminalization of abortion, of "bad behaviour" in public places -for instance "grober Unfug" in Germany, of rubber cheques in France, of men having sex with male minors, of shoplifting up to a certain amount, etc.), some crimes were promptly considered as serious offences calling for serious sanctions, in particular crimes committed in the domestic sphere: violence or harassment against children, violence or harassment against a spouse or partner (including marital rape), and other forms of domestic abuses. One of the most immediate impacts of this is the invasive power of state authorities, in particular prosecution and the police (see above), in the sphere of intimacy. Until the 1980s, criminal policies dealt with the protection of individual property. Now, the state has abandoned this purpose for the benefit of insurance companies, but is on the front line to protect individual bodies and minds. The evolution of motives for imprisonment in France in the last 35 years illustrates this transformation of the targets of state power. Half of the 20.000 prison sentences pronounced in France had to deal with property crime (theft, handling stolen goods, fraud, breach of trust, etc.) in 1980, but by 2013 this proportion had fallen down to a quarter of the 22.000 prison sentences. This evolution is not as spectacular in other countries, like in Germany where property crimes and traffic offenses have always been severely sanctioned. Property crimes represented two thirds of the offences for which people were sent to prison in 1970 in Germany; they are now less than half of the offences for which people are sent to prison.

In the same period, violent offences and homicides have doubled, and so have sex offences (Drenkhahn 2013, 74).

More generally, the sacralisation of the body, as exemplified by the "bifurcation policy" and the development of transnational positive obligations, has led throughout Europe to increased security against offences considered as the most severe forms of invasion of privacy and intimacy, such as sexual offences and assault against children. As a consequence, one sees the resurgence of long-term imprisonment, like in Germany where lifelong prisoners doubled from 1977 to 2008 (Dünkel et al., 2010, 23 - Violent, drug and sex offenders now face in Europe more stringent legislation, enhanced prosecution, longer sentences and restrictions of parole. One major explanation for the rise of prisoners rates in European countries is the growing acceptance by politicians and courts alike that penal severity means crime deterrence [START_REF] Snacken | Prisons and punishment in Europe[END_REF].

In sum, human rights and the protection of individual lives have had two kinds of consequences on the evolution of state coercive powers in Europe. On the one hand, to the British government's delight, supra-national courts and laws constrain the discretion and intensity of state powers. On the other hand, the sacralisation of the body and of the notion of intimacy in France contributed to an increased criminalization of violence-related offences, specifically against women and children, and to increased punitiveness by the state.

Meanwhile, what could be a restored power of the state to protect, even under supranational scrutiny, appears to be faceing a severe crisis. Two issues challenge state coercive power: their ability to deal with a growing demand, and the increased use of contract and diversion.

The penal state between stifling and diversion

Punitiveness is growing, and the state finds itself at the foreground under transnational pressure and scrutiny. But this growth of state coercive powers inhabits a twilight aspect.

The state no longer uses punitiveness as an act of sovereign will. Its acts and decisions, far from being one-sided decisions taken in magnificent isolation, are now mere contracts where the state sometimes only plays a notarial role (2.2.1.). In addition, state authorities are under pressure from victims and, above all, victims' representatives, for getting hard on crime and insecurity issues. This leads to an overloaded process where the state is reduced to a mere mechanism vainly aiming at reducing the uninterrupted flow of complaints, claims, and procedures (2.2.2.).

State authority in the shadow of contractualisation, diversion and victims' claims.

A common view on the growing coercive capacity of public authorities is that the state punishes more when economic conditions worsen. Although this may be true, there is a broader evolution that can also be related to a neo-liberal framing, and that is the growing tendency of contractualisation in criminal justice matters. 'Contractualisation' encompasses all forms of bargaining between the offender, and the victim (whereas the former is, in the classical criminal justice doctrine, the sole target of a one-sided decisions taken once and enforced by the state).

One of the best examples here are the 'youth offenders contracts' introduced by the Youth Justice and Criminal Evidence Act in England & Wales (1999). When a youth offender pleads guilty, he is sent to a panel to face a decision. Neither judges nor civil servants of any kind are members of this panel, but trained community volunteers. The aim of this form of youth justice is to deal with overloaded penal systems, but also to consider the young offender like a responsible citizen and to elaborate a decision (not a mere sentence) in accordance with his sense of responsibility and the damage he could have done to the local community.

On the other hand, victims also play a greater role in the production of penality, and contribute to weaken state sovereignty in this policy area. Victims are at the core of Simon's argument: "We are crime victims. We are loved ones of crime victims. Above all, we are those who live in fear that we or those we care for will be victimized by crime" (Simon 2007, 109). The figure of the crime victim has ascended to become the idealized political subject of legislation. This is an ambivalent evolution, because it first legitimizes law and order stances and government's authority. The public in general has become an "allegorical victim" (Kaminski el al., 2007), in the name of which governments reinforce their penal agenda. French president Nicolas Sarkozy, who won the 2002 presidential race practically on the basis of a law and order campaign which helped him to gain the extremeright voters back [START_REF] Roché | Criminal Justice Policy in France: Illusions of Severity[END_REF], created a "Ministry for victims' rights" as soon as he became head of state. Compassion is now seen as a central aspect of criminal hearings, and criminal process law was strongly reoriented to allow victims to interfere in judicial decisions, especially when the judge pronounces a moderate sentence.

Herein lies the ambivalence of the rise of victim's role, with respect to the state. First, it actually gives more power to government than to the state. As a matter of fact, victims are used by politicians to win electoral races, but they do not contribute to reinforce state structures: judges' authority has been lowered, while "communities" have been reinforced (either directly through programs like "community safety programs" in Englands.

Gillings 2001, Tonry 2010, or indirectly through media coverage of criminal cases, which places the criminal justice system under a new kind of pressuresee Lappi-Seppälä 2014 for a comparative perspective). Moreover, the victim's rhetoric is one of the consequences of the dramatic rise of crime rates since the beginning of the 1960s and the parallel reduction of clearance rates by the police forces, a twin evolution which has increasingly jeopardized "the myth of sovereign state control", that is "the myth that the sovereign state is capable of providing security, law and order, and crime control within its territorial boundaries" (Garland 1996, 448). Populist posturing by governments do not necessarily imply an empowered state. To some extent, the growing role of victims in the criminal justice system could lead to what the "abolitionists" of the 1970s (Mathiesen, Houlsman, etc.s. Brodeur) had imagined: a substitution of civil litigation (i.e. of contract between individuals) over criminal law (i.e. unilateral power of the state) through victim/offender mediation, reparation boards, reparation sentencing, healing circles, involvement of local communities in the sentence and, more to the point, in the punishment, which is becoming a form of community service and reparation (Kaminsky et al. 2007)… As Garland has recently noted, populist states are in a way "weak" states, because populist policies are enforced by elected politicians who won electoral races on the basis and in an environment of scepticism, if not overt defiance, towards the state [START_REF] Garland | Penalty and the penal State[END_REF]. One of the most eloquent aspect of this duplicity is the withdrawal of the Ministry for victims' rights in the French government, after the scandal caused by its support of young victims in a huge network of sex offenders, most of them sentenced to long-term prison, which turned out to be fake. A common interpretation of the scandal was that it had been caused by the compassionate turn of local criminal justice, which had worked under heavy pressure of the public, the press and politicians in the "century's paedophilia case".

The state and the demand for security.

The failed myth of the sovereign state, according to Garland, is based on the state's growing inability to deal with an unstoppable demand for security in response (or simply in parallel) to the enormous growth of delinquency and petty crime from the beginning of the 1960s to the beginning of the 2000s. Recorded crimes have been multiplied by 10 during this period in France and in Netherlands [START_REF] Boutellier | The convergence of social policy and criminal justice[END_REF], 365, Robert 2013), and police clear-up rates have symmetrically fallen down: in reasonable proportions in Germany (from 55% to 45% of all recorded crimes from 1965 to 1988, according to [START_REF] Entorf | Socioeconomic and demographic factors of crime in Germany: Evidence from panel data of the German states[END_REF] or in huge amounts in France (down to 25% at the beginning of the 2000s). At the other end of the penal complex, overloaded criminal justice systems are less and less able to prosecute: and when they are, they are less and less willing or able to sentence offenders, as shown in the graphs below. with it at a time of budget restrictions has two consequences. The first is the increased complexity of the system and the disorientation of the public with regard to penal justice, between diversion, negotiation, plea-bargaining, disposal, conditional disposal, penal order, accelerated proceedings, etc. [START_REF] Mouhanna | l'interaction entre justice et monde pénitentiaire en matière d'enfermement[END_REF]). The most blatant example in this respect is the French example, as displayed in the following picture (Jehle 2008, 14). Public disorientation has reinforced the general populist claim for a more punitive justice, which means a simplified justice system more and more relying on prosecutorial and police powers. Allegedly lenient judges are blamed for their supposed soft hand on petty crime, as elected politicians try to shift the balance of power within the criminal justice system to the prosecution, as well as to the police [START_REF] Jehle | Coping with overloaded criminal justice systems: the rise of prosecutorial power across Europe[END_REF]and 2008[START_REF] Luna | Prosecutors as judges[END_REF][START_REF] Choe | Discretion at the Pre-Trial Stage: A Comparative Study[END_REF]. This is the second aspect of the long-term decline of the state's capacity to deal with rising crime rates: the reinforcement of discretion in the criminal justice system (embodied in the power of the prosecutor to prosecute, most of the times exclusively relying on the sole information given by the police).

At the time when elected politicians theatrically call for simplified and harsher criminal justice, overloaded justice systems deal with more and more petty crime (Aubusson de Cavarlay 2013), and penal laws with more and more obscure proceedings and methods of decisions, which contribute to an overall feeling of loss of sovereignty by the state over crime issues. In the meantime, since politicians want to give prosecution more powers visà-vis allegedly compassionate judges, and since they are the entry gate to overloaded systems, prosecutors (and with them the police) are more and more "replacing the judge in all but the most serious (criminal) cases" (Jehle & Wade 2006, 115), reinforcing the state's discretional power in criminal matters. These are the three cornerstones of the undermining of state authorities dealing with crime: a decline in public faith in state ability to protect them and their property, a rise of diversion and other ways to bypass executed prison sentences, a rise of discretional powers carried by prosecutors and police officers. As such, the reduced ability of the state to deal with high crime rates does not necessarily coincide with a vanishing of sovereignty, no matter how high the esteem in which the public holds the state. This reinforced role of prosecution and of police powers has gained so much importance over the last three decades that it needs a more detailed report now.

A semi-archipelago of sovereignty: The insularity of the police and prosecution

Coercive power of the state embodied in the criminal justice system has undergone many transformations in the course of the last decades, as we have seen. These transformations prove difficult to cast on a scale going from a "lenient" to a "repressive" gradient, or from a soft to a hard line. The new and decisive concern for bodily integrity is a clear incentive to adopt new protection measures for the benefit of individuals, more and more embodied in criminal law provisions. But at the same time, this worry for bodily integrity limits state power in that it also applies to the protection of individuals in the hands of state authorities, as exemplified in the transformation of prisons in Europe. The same contradictory transformation can be observed at the level of police authorities in Europe: their power to coerce has been significantly harnessed, and at the same time the central place of punitiveness in the political game has led to an increase of discretional powers in the hands of the police, at least in some major European countries.

As far as police powers are concerned, two major factors have had an influence on the level of discretion and brutality that police agents in Europe are allowed to exercise. The first does not need to be developed, because it is strongly related to the growing influence of judicial and supra-national control over state powers very similar to the one observed at the level of the criminal justice system, especially prison organizations. Resounding sentences given by the ECtHR against major countries like France or the UK in cases of police brutality (France in the first place) or unfair police investigative powers (the UK in the first place) have led to significant changes in the way police skills are trained and performed, as well as in criminal proceedings [START_REF] Jobard | Counting violence committed by the police : raw facts and narratives[END_REF].

The second aspect is more specific to the police, and is a direct consequence of the riot wave that hit the UK in the first half of the 1980s, and France in 1981 and again from the beginning of the 1990s onwards. Police discretion and police powers, exemplified in stops & search powers, suddenly gained greater attention from the press and the public (Waddington et al. 2009). Police undoubtedly became a source of prestige for government and candidates. France is the most blatant example, where in the last two decades being a Home minister has become a necessary step for any candidate to the presidency. As such, the game of give-and-take played by police and politicians is congruent with the "governing through crime model". But slipping to the centre of political attention, police scandals have become at the same time a great source of concern for elected politicians, and as a consequence the game appears more complex and risky for politicians who know that they cannot avoid being held accountable in case of a deadly encounter with the police. Politicians cannot leave the police out of control any more, as was much more the case before the 1980s riots or some police scandals like the Guildford Four and the Maguire Seven in England. Police-government relationships are not as one-sided as some declarations make it soud, like the motto of "being the first cop of the nation" adopted by the most recent French ministers of Interior. They are constrained by the following paradox, often unseen by observers of crime politics: the more an elected politician builds his/her political capital on the police, the more he/she has to make police authorities comply with the rule of law despite his/her claim to free the police from its hindrances in the fight against crime, due to the fact that police have primarily to deal with contingency and as such are much exposed to trouble that might develop into scandals [START_REF] Jobard | Proposition on the theory of policing[END_REF][START_REF] Manning | Democratic policing in a changing world[END_REF].

Beyond the growing willingness in countries like England and France (contrary to Germany or Italys. Nelken 2005) to make police a full piece of the political game, the inability of state authorities, and the police in the first place, to deal with petty crime and daily disorders have indeed led to increased powers given to the police, like anti-social behaviour orders introduced in the UK or the ability for the justice to sanction the mere gathering of small groups of people in housing estates entrance halls in France. This last example is quite significative: the bill introduced in 2002 aiming to restore hope and faith in politics within deprived housing estates facing the daily gatherings of hostile youths in their halls was almost impossible to enforce, because the criminal justice system is reluctant to restore older penal concerns like vagrancy or to consider collective penal responsibility, let alone grant police discretion in such matters. The case of the much controversial and debated English ASBOs is more complex and far-reaching than French attempts, but as a matter of fact, and whatever their degree of implementation, they more or less restore the former (and far more discretional) "bind over" power enjoyed by the police for more than a millennium, before it was little by little narrowed under the influence of the ECtHR (Hashman & Harrup v. UK), and before the High Court magistrates and the Law Commission set in 1994 [START_REF] Ramsay | Vulnerability, sovereignty, and police power in the ASBO[END_REF]). Despite their controversial dimension, highly related to the importance given to them by PM Tony Blair, ASBOs are definititely not a breach in the history of the state's ability to discipline and punish. The main change introduced by ASBOs in England as far as state sovereignty is concerned is really a shift in justificative rhetoric, with the claim for the victims rights replacing the sheer affirmation of state authority.

ASBOs otherwise exemplify the attempt by state authorities to identify and sanction conducts that are not criminal, but socially objectionable in social areas exposed to multiple deprivation where police are reluctant or even unable to deal with sources of disorders and annoyance. These administrative penalties, often dealt or bargained with the author of the trouble (like the British "youth offenders contracts" defined in the 1999 Youth Justice and Criminal Evidence Act), attempt to restore the link between a troublemaker and his community [START_REF] Crawford | Contractual governance' of deviant behaviour[END_REF][START_REF] Donoghue | Antisocial behaviour orders (ASBOs) in Britain: contextualizing risk and reflexive modernization[END_REF], and give indeed a large interpretative discretion power to authorities in identifying those who are "likely to cause harassment, alarm or distress" (in the words of the ASBOs' definition). These continental forms of US "quality of life" definitely strengthen the multi-secular power over people that the police define as "police property" [START_REF] Lee | Some structural aspects of police deviance in relation to minority groups[END_REF]) and contribute to a "criminalization of social policies" in which the disadvantaged individual (typically an unemployed migrant youth leaving in social housing) is made responsible for his present life and summoned to sign up an asymmetrical contract with the local community (in fact, the police or the administrative power); if he breaks it, he will face a judge or even a sentence across-theboard (Bonelli 2006[START_REF] Tonry | The costly consequences of populist posturing: ASBOs, victims 'rebalancing' and diminution in support for civil liberties[END_REF]).

As we have seen, one of the striking aspects of this recent reinforcement of police discretion and power lies in the fact that all new powers conceded to police authorities are given in the name of local community support or victim's rights. Allocations or subsidies supposedly given to local communities were actually captured by local police forces in the name of their multi-secular competence in fighting disorders, despite their blatant failure in their confrontation with mass delinquency since the beginning of the 1960s. Management reforms undertaken since the mid-1990s under the label of "new public management" contributed to reinforce the centralization of the police and government's control over them, through the rhetoric of "performance management", despite promises made to local communities involvement into police matters (de Maillard & Savage 2012, Brogden & Ellison 2013). Added to this, the growing concern for the protection of the individual's body and life throughout Europe, as already shown in many examples, also contributed to reinforce police coercive powers through the criminalization of traffic offenses and, above all, the penalization of unintended homicides and injuries and the penalization of drivers under the influence of substances. In France for instance, controls of drivers for drugs and alcohol use by French police and gendarmerie are today 35 times more frequent than 10 years ago (Obradovic 2014).

All these transformations contribute both to a widening and to a strengthening of the penal net: the penal net is widening through incentives to deal with petty offences and to ameliorate crime statistics through better clearance rates (as shown by ASBOs in the UK or the rise of recorded use of cannabis in France, which have been multiplied by 6 since the beginning of the 1990s), and it is getting stronger through the insistence put on the criminalization of repeat offending, so that the "usual suspects" (mainly the young deprived migrants, as well as males sentenced for domestic abuse and for violence or outrage against police officers) see their existence now marked by a long-term relationship with the criminal justice system. In this domain also the growing influence of victims plays a role, because a sentenced individual can be free from any penal measure, but still has to pay reparation to the victim and then is under the sword of Damocles of reintegrating the criminal justice circuit. To conclude, the shift of powers within the criminal justice system to the advantage of the prosecution has made the prosecutors more dependent on police officers in their daily routine, so that they are more reluctant to interfere in police mattersas much in the way the police bring them cases, as in the clearance of eventual police brutality cases, contributing to a tension between the requirements expressed by the ECtHR on the one hand and the daily distribution of power in the criminal justice systems on the other hand. And the crucial symbolic role played by the police in the political races have strengthened the ability of police unions to bargain with elected politicians and with prosecutors, which does not help either to enforce the international provision on the use of force and on fairness in judicial investigations.

Police and prosecution authorities, which personify the coercive power of the state, have benefited from the overloading of criminal justice systems in Europe and reinforced their autonomy. But their autonomy really consists in a kind of semi-archipelago of sovereignty.

First, although prosecutors enjoy greater power, their decisions are more and more clinched in automatic or semi-automatic sanctions or diversions imposed by legislators to deal with mass crime and show a hard stance against crime. Second, although the weight of police authorities both in the criminal justice system and in the political competition has surely strengthened the power of police authorities and unions towards magistrates and government members, it also place these authorities under public and specifically media scrutiny, bearing on their discretional power and discretional use of force. As such, prosecutorial and police authorities are at best described as semi-archipelagos of sovereignty in the geography of a broader state transformation. Also, one should keep in mind that while these contradictory tensions exist in European countries where the state has been under populist pressures, like in France and the UK, they are weaker in countries like Germany or Italy where penality is no part of the political game.

CONCLUSION

The internal coercive powers of the state have undergone a muddled transformation in the last decades, which has been far from homogeneous. To take the most common indicator, detention rates have risen up in Great Britain, France and the Netherland, slowly decreased in Germany, dramatically decreased in Finland, and are less than twice as high in Nordic countries as they are in England & Wales. But these discrepancies seem to be related to a trade-off in state's organization. A bipartisan, majoritarian political system together with low amounts of social expenditures are highly predictive of the power to incarcerate. On top of it, two common features influence the state's coercive powers in Europe. The first is the sacralisation of individual integrity and dignity as a cornerstone of penal rules and practices. The concern for individuals has contradictory consequences for state violence. It legitimizes the criminalization of both verbal and physical violence and gives greater power to the victims, as exemplified in the preoccupation for domestic abuse and sexual offenses. As such, it reinforces the penal capacity of the state, but under the pressure of the victim and the victim's stakeholders. The second consequence is a strengthening of control and oversight over authorities with the power to use force and coercion, in the first place prison authorities, and this in a great contrast to what is going on in the US, where the rising influence of victims has no counterpart. The second main source of transformation of the coercive power of the state is the normality of high crime numbers in overloaded criminal justice systems. This contributes to insularising the courts and the police, and to a widening as well as a strengthening of the penal net. But even in this domain, organisational insularity does not necessarily lead to a rise in the discretional use of force.

The greater role symbolically played by the police in the political game forces the governments to control them and to avoid any kind of blatant cases of police brutality or racism. Instead, the rise of prosecutorial and police power manifests itself in the reinforced capacity of criminal justice systems to have a close hand on the existence of precarious beings exposed or willing to resort to violence, substance abuse, disorder or simply contemporary forms of vagrancy and idleness. More than ever, they are the subjects of today's penal state.

  Tapio[START_REF] Lappi-Seppälä | Imprisonment and penal demands. Exploring the dimensions and drivers of systematic and attitudinal punitiveness[END_REF] has recently shown that Esping-Andersen's famous typology of liberal, chistian-conservative and social-democratic states turns out to have strong explanative power with respect to the manufacturing of western penal systems. Liberal states display high levels of punitiveness, as illustrated by their detention rates: Australia stands at 115, England, Wales and Scotland at 150, New Zealand at 190 (New Zealand), and the US at 750. Christian-conservative states display lower levels of punitiveness, with Switzerland standing at 75, Germany at 85, and France and Austria at 115. Socialdemocratic states have the lowest rates, with Iceland at 50 and Denmark and Norway at 75.

  Figure 1.

Figure 2 .

 2 Figure 2.

  they also doubled in France in the same period) or in Belgium and the Netherlands in the aftermath of the Dutroux scandal, but also in other countries like England (see Criminal Justice Act 2003).

  Figure 3

Figure 6

 6 Figure 6

Since it is dealt in an other paper of the book, my paper will not address the issue of the specific use of force against foreigners (power to arrest, to confine and to deport).

Violence & Homicide Unintended violence & death Property crimes Sex offences

Violence & Homicide Unintended violence & death Property crime Sex offences