Rifat Sipahi

Silviu-Iulian Niculescu
email: niculescu@hds.utc.fr

Fatihcan M Atay
email: atay@member.ams.org

Effects of Driving Aggressiveness to Stability of Traffic Flow with Drivers Utilizing Short-Term Memory

Keywords: driver aggressiveness, short-term memory, distributed delay, traffic, stability

The main objective in this paper is to understand the stability features of traffic flow from the perspective of driver aggressiveness. We study this over a fundamental carfollowing traffic model, which is further enhanced by taking into account the short-term memory of human drivers. Physically, memory indicates that human drivers continuously receive stimuli from a set of continuous points distributed over the history. Specifically, it is of practical interest, when aggressive drivers are present, to reveal stability features of traffic flow in the parametric domain defining the physics of drivers' memory. Various traffic scenarios where vehicles are arranged in linear and ring configurations are presented and some interesting physical interpretations are discussed.

I. INTRODUCTION AND PROBLEM STATEMENT

Traffic flow is one of the main research directions in Physics, Engineering and Mathematics since the 1930s. The main reason for this is that irregular traffic flow has direct impacts on human life (injuries, hospitalization), economy (time losses) and the environment (emission issues), [START_REF] Helbing | Traffic and Related Self-Driven Many-Particle Systems[END_REF]. There exists numerous mathematical models in the literature, [START_REF] Bando | Analysis of Optimal Velocity Model with Explicit Delay[END_REF], [START_REF] Chandler | Traffic Dynamics: Analysis of Stability in Car Following[END_REF], [START_REF] Davis | Modifications of the Optimal Velocity Traffic Model to Include Delay Due to Driver Reaction Time[END_REF], [START_REF] Gazis | Car Following Theory of Steady State Traffic Flow[END_REF], [START_REF] Gazis | Non-Linear Follow the Leader Models of Traffic Flow[END_REF], [START_REF] Rothery | Transportation Research Board (Trb) Special Report 165[END_REF], [START_REF] Treiber | Memory Effects in Microscopic Traffic Models and Wide Scattering in Flow-Density Data[END_REF]- [START_REF] Treiber | Congested Traffic States in Empirical Observations and Microscopic Simulations[END_REF], which describe various phenomena arising from traffic flow dynamics. Clearly, one of the most important research directions is along the lines of studying the stability of traffic flow, [START_REF] Bando | Analysis of Optimal Velocity Model with Explicit Delay[END_REF], [START_REF] Davis | Modifications of the Optimal Velocity Traffic Model to Include Delay Due to Driver Reaction Time[END_REF], [START_REF] Edie | Following and Steady-State Theory for Non-Congested Traffic[END_REF], [START_REF] Gazis | Car Following Theory of Steady State Traffic Flow[END_REF], [START_REF] Gazis | Non-Linear Follow the Leader Models of Traffic Flow[END_REF], [START_REF] Orosz | Hopf bifurcation calculations in delayed systems with translational symmetry[END_REF], [START_REF] Orosz | Global Bifurcation Investigation of an Optimal Velocity Traffic Model with Driver Reaction Time[END_REF], [START_REF] Treiber | Delays, Inaccuracies and Anticipation in Microscopic Traffic Models[END_REF], [START_REF] Treiber | Congested Traffic States in Empirical Observations and Microscopic Simulations[END_REF], since this research will help understand how to avoid traffic accidents, reduce traffic jams and lower emissions.

Systems in which humans actively participate in evolution of dynamics are richer and more complicated as they can be seen as human-in-the-loop dynamics [START_REF] Murray | Control in an Information Rich World: Report of the Panel on Future Directions in Control, Dynamics, and Systems[END_REF]. Furthermore, humans exhibit delayed reactions as they need some time to become conscious, make decision and perform an action. From this perspective, we can state that the presence of human drivers makes traffic flow dynamics inherently time delayed. This is a critical observation as it is known that time delays may invite poor performance and even instability in Inspired by authors earlier work [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF], a more realistic mathematical model of traffic flow is deployed by assuming all human drivers utilize their short-term memory. Next, the main contribution of this paper is presented over this new model: Considering presence of relatively aggressive drivers, we elaborate on the stability robustness of traffic flow in the parametric domain defining the features of shortterm memory. This is practically important as it is aimed to understand risk levels (instability) in traffic when, for various reasons, some humans drive aggressively. See also [START_REF] Sipahi | Effects of Short-Term Memory of Drivers on Stability Interpretations of Traffic Flow Dynamics[END_REF] for a study in which the effect of number of vehicles (where vehicles are arranged around a ring) to stability is studied assuming identical driver aggressiveness.

The paper is organized as follows: Notations conclude this section. Mathematical modeling of traffic flow and delayed reactions of human drivers are summarized In Section II. In Section III, the relevance of the proposed approach and main objectives are detailed. Inspired by [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF], the framework of the stability analysis is explained in Section IV. Practical scenarios are considered in Section V from stability point-of-view. Assuming all drivers utilize their memory for their control actions, we specifically study how stability is affected with respect to driver aggressiveness. Two standard configurations, ring and linear, are chosen to demonstrate the results. Section VI presents the conclusions and future research directions.

Notation. The notations are standard. We use R for the set of real numbers, where an additional + orsign as a subscript indicates the positive and negative real numbers, respectively. Similarly, C + , and C -denote complex numbers whose real parts are positive and negative, respectively. The imaginary axis in the complex plane is denoted by jR, where j = √ -1. We use s ∈ C for the Laplace variable, whose values on the imaginary axis are denoted by s = jω where ω ∈ R. ith eigenvalue of a matrix A is represented by λ i (A).

II. MATHEMATICAL MODELING

A. Discrete delay modeling

Time delay (known to appear in a traffic flow model first time in [START_REF] Chandler | Traffic Dynamics: Analysis of Stability in Car Following[END_REF]), essentially originates from the time needed for human drivers to become conscious of the changes in the environment, make decisions, and perform control actions, see [START_REF] Bando | Analysis of Optimal Velocity Model with Explicit Delay[END_REF], [START_REF] Davis | Modifications of the Optimal Velocity Traffic Model to Include Delay Due to Driver Reaction Time[END_REF], [START_REF] Gazis | Car Following Theory of Steady State Traffic Flow[END_REF], [START_REF] Gazis | Non-Linear Follow the Leader Models of Traffic Flow[END_REF], [START_REF] Sipahi | Analytical Stability Study of a Deterministic Car Following Model under Multiple Delay Interactions, at Invited Session Traffic Dynamics under Presence of Time Delays[END_REF], [START_REF] Treiber | Delays, Inaccuracies and Anticipation in Microscopic Traffic Models[END_REF] for time-delayed mathematical models and [START_REF] Green | How Long Does It Take to Stop?" Methodological Analysis of Driver Perception-Brake Times[END_REF] for quantitative level of such delays. Although the above argument is straightforward, modeling of time delays representing the behavior of human drivers is a challenge. The open literature responds to this problem by suggesting discrete delay models [START_REF] Bando | Analysis of Optimal Velocity Model with Explicit Delay[END_REF], [START_REF] Davis | Modifications of the Optimal Velocity Traffic Model to Include Delay Due to Driver Reaction Time[END_REF], [START_REF] Sipahi | Analytical Stability Study of a Deterministic Car Following Model under Multiple Delay Interactions, at Invited Session Traffic Dynamics under Presence of Time Delays[END_REF], [START_REF] Sipahi | Some Remarks on the Characterization of Delay Interactions in Deterministic Car Following Models[END_REF], [START_REF] Treiber | Memory Effects in Microscopic Traffic Models and Wide Scattering in Flow-Density Data[END_REF], which may give some insight on the characteristics of traffic flow dynamics. Discrete delays indicate that an action of a human driver at time t is based on what is experienced at time t -τ , τ ≥ 0. Following this idea, numerous studies appear in the literature [START_REF] Bando | Analysis of Optimal Velocity Model with Explicit Delay[END_REF], [START_REF] Davis | Modifications of the Optimal Velocity Traffic Model to Include Delay Due to Driver Reaction Time[END_REF], [START_REF] Orosz | Hopf bifurcation calculations in delayed systems with translational symmetry[END_REF], [START_REF] Orosz | Global Bifurcation Investigation of an Optimal Velocity Traffic Model with Driver Reaction Time[END_REF], [START_REF] Sipahi | Analytical Stability Study of a Deterministic Car Following Model under Multiple Delay Interactions, at Invited Session Traffic Dynamics under Presence of Time Delays[END_REF], [START_REF] Sipahi | Some Remarks on the Characterization of Delay Interactions in Deterministic Car Following Models[END_REF], [START_REF] Treiber | Memory Effects in Microscopic Traffic Models and Wide Scattering in Flow-Density Data[END_REF]- [START_REF] Treiber | Congested Traffic States in Empirical Observations and Microscopic Simulations[END_REF].

Discrete-delay models, on the other hand, exhibit some drawbacks, mainly due to the assumption that they are fixed (rather than representing a distribution) and that they disregard possible "memory" effects inherently present in the dynamics. From the application point-of-view, in traffic flow, one needs to incorporate memory effects arising from the physical capacities of the human drivers. Mathematically speaking, such effects can be modeled by using certain distribution functions (such as uniform and gamma distributions) which, in parallel, help represent the heterogeneity of the distribution of non-identical drivers. For these reasons, distributed nature of the time delay will yield a better representation of reality. Such an argument has also some potential to open new research directions in understanding human behavior [START_REF] Murray | Control in an Information Rich World: Report of the Panel on Future Directions in Control, Dynamics, and Systems[END_REF].

Motivated by the previous arguments, short-term memory effects will be taken into account in mathematical modeling of traffic flow. In this section, a summary of authors' preliminary work is presented in order to guide the reader, see [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF] for details.

B. Short-term memory -Distributed delay modeling

Short-term memory can be modeled by a uniform distribution f (τ ), Figure 1.

f (τ ) = 1/δ if h < τ < h + δ 0 otherwise (1)
where δ > 0 and h ≥ 0. Such a model is quite satisfactory for representing an average of the information available in the short-term memory as well as for introducing some heterogeneity among the drivers. See [START_REF] Atay | Delayed-feedback control of oscillations in non-linear planar systems[END_REF], [START_REF] Atay | Distributed delays facilitate amplitude death of coupled oscillators[END_REF], [START_REF] Cushing | Integrodifferential Equations and Delay Models in Population Dynamics[END_REF], [START_REF] Haldane | A contribution to the theory of price fluctuations[END_REF] for theory and applications with different forms of distributed functions.

The uniform distribution can be defined by a combination of any of the two physical parameters, dead-time (h, before which information in the memory can be utilized for a control action), memory horizon (τ 2 = h + δ, representing the oldest possible information available in the memory) and memory window (δ, indicator of the size of the memory). Note that, (1) converges to discrete delay case as δ → 0 (where f (τ ) becomes a dirac function).

C. Traffic flow modeling

We consider a single-lane continuous-time microscopic car following model, in which a chain of vehicles travel at a constant velocity, the so-called quasi steady-state, without changing lanes, [START_REF] Chandler | Traffic Dynamics: Analysis of Stability in Car Following[END_REF], [START_REF] Helbing | Traffic and Related Self-Driven Many-Particle Systems[END_REF], [START_REF] Rothery | Transportation Research Board (Trb) Special Report 165[END_REF],

vi (t) = κ i (v i-1 (t -τ ) -v i (t -τ )), (2)
where v i is the velocity of the ith vehicle, and τ ≥ 0 is the discrete time delay. The parameter κ i > 0 denotes the aggressiveness (also known as sensitivity) of the i th driver against the velocity difference between his vehicle and the one in front. Equation ( 2) is also similar to consensus problems [START_REF] Fang | Information Consensus of Asynchronous Discrete-Time Multi-Agent Systems[END_REF], [START_REF] Lee | Agreement with Non-Uniform Information Delays[END_REF], [START_REF] Olfati-Saber | Consensus Problems in Networks of Agents with Switching Topology and Time-Delays[END_REF], [START_REF] Ren | A Survey of Consensus Problems in Multi-agent Coordination[END_REF], in which drivers penalize the relative velocity between two vehicles in order to achieve the agreement of

v i (t) → v i-1 (t).
Notice that the model given above carries discrete delay, τ , only. We further improve this model by combining memory effects from [START_REF] Atay | Delayed-feedback control of oscillations in non-linear planar systems[END_REF]. Hence, the generalized dynamics considered in this work becomes

vi (t) = κ i ∞ 0 f (τ )(v i-1 (t -τ ) -v i (t -τ )) dτ, (3)
i = 1, . . . , n,, where n is the number of vehicles. We assume that the delay kernel f (τ ) is a measurable function of exponential order. For instance, the choice of a Dirac distribution for f (τ ) gives the discrete delay model ( 2) above.

Eq. ( 3) defines a traffic dynamics in which drivers perform their decisions based on what they continuously observed during a memory window, and the information in the memory is retained and used in the decision-making process. Clearly, the decisions of the drivers are limited by the capacity of this short-term memory, i.e. the size of the memory window δ.

The system (3) can be expressed in vector form as

v(t) = ∞ 0 Jv(t -τ )f (τ ) dτ (4)
where v = (v 1 , . . . , v n ) and J ∈ R n×n is a configuration matrix weighted by the driver sensitivities κ i . We assume that J is diagonalizable, that is, its eigenvectors form a basis for R n .

In the circular configuration, we identify v 0 = v n in (3), and the matrix J takes the form

J =      -κ 1 0 • • • 0 κ 1 κ 2 -κ 2 0 • • • 0 . . . . . . . . . . . . . . . 0 • • • 0 κ n -κ n      (5)
In the linear configuration, the vehicle in front (denoted with the index i = 1) travels at a constant velocity, i.e. v1 = 0. Hence, the linear configuration can be derived from the circular one by setting κ 1 = 0, and the resulting matrix is denoted by J .

D. Spatial configuration of traffic flow.

We consider two spatial configurations which are widely utilized in the literature, [START_REF] Orosz | Hopf bifurcation calculations in delayed systems with translational symmetry[END_REF], [START_REF] Orosz | Global Bifurcation Investigation of an Optimal Velocity Traffic Model with Driver Reaction Time[END_REF], [START_REF] Sipahi | Analytical Stability Study of a Deterministic Car Following Model under Multiple Delay Interactions, at Invited Session Traffic Dynamics under Presence of Time Delays[END_REF]: (a) vehicles traveling around a ring, and (b) vehicles arranged along a linear path. It will be interesting to obtain the stability of the traffic flow with respect to these two different configurations, and give some reasoning why one is more favorable than the other from the stability point-of-view.

III. OBJECTIVE, PRACTICAL RELEVANCE, APPROACH

The main objective in this paper is to elaborate on how the stability of ( 4) is affected if aggressive drivers populate the traffic flow. In other words, it is of interest to see how stability of traffic flow defined in physical parameters of the short-term memory, (δ, h), vary as drivers become more aggressive. Recall that κ i denotes driver aggressiveness and it will be used as one of the main parametric domain in presenting how stability is affected. We will particularly focus on understanding how memory window size of drivers, δ, varies as κ i increases for particular drivers.

The above study will be investigated both over ring and linear configuration of vehicles, as explained earlier. The scenarios considered are practically of interest, as the results will help interpret how stability is strongly/weakly robust for these two configurations. Arising results can be generalized for various dynamical systems that are also in the form of (4); see for instance consensus problems in [START_REF] Fang | Information Consensus of Asynchronous Discrete-Time Multi-Agent Systems[END_REF], [START_REF] Lee | Agreement with Non-Uniform Information Delays[END_REF], [START_REF] Olfati-Saber | Consensus Problems in Networks of Agents with Switching Topology and Time-Delays[END_REF], [START_REF] Ren | A Survey of Consensus Problems in Multi-agent Coordination[END_REF].

The approach for the stability analysis of the dynamics (4) comes from an interesting idea of rewriting the corresponding characteristic equation of (4) in a relatively different form, which decouples some parameters of interest for stability. This coupling is a representation of an interconnection scheme that enables the analytical results. We further discuss in the following section regarding how interconnection scheme idea works effectively.

IV. STABILITY ANALYSIS

Although undesirable, the presence of time-delays in the process of decision making and performing a control-action by human drivers is neither avoidable nor negligible. Therefore, one needs to understand the stability features of traffic dynamics (4) in presence of delays.

A. Characteristic equation

To perform the stability analysis, the characteristic equation (4) in Laplace domain is obtained first

χ(s) = det[sI -JF (s)] = 0, (6)
where F (s) is the Laplace transform of the distribution function f (τ ),

F (s) = e -sh (1 -e -sδ ) sδ . (7)
Note that when δ → 0, discrete-delay model ( 2) with τ = h is recovered.

One can rewrite (6) as

χ(s) = n i=1 (s -λ i (J) e -sh (1 -e -sδ ) sδ ) = 0 (8)
where λ i (J) is the ith eigenvalue of J. The roots corresponding to the ith factor of χ, i.e., the solutions s of this equation will determine the stability/instability of traffic dynamics (4).

Notice that the configuration matrix J has a simple zero eigenvalue and all its remaining eigenvalues are complex conjugates with negative real parts, [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF]. This simple zero creates an s = 0 pole of the dynamics which corresponds to a rigid body motion of the entire configuration of the vehicles. Furthermore, s = 0 has no influence on the stability/instability of the dynamics, thus we will assume that s = 0 in the remaining of the paper, see details in [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF].

B. Approach for stability analysis, Interconnection scheme

Stability analysis of the dynamics (4) is not trivial due to inherent challenges in the corresponding characteristic equation ( 8), thus we propose a different way to approach the problem. The need for this approach naturally arises within the following complications listed:

(a) Eq. ( 8) possess two independent time delay parameters, (δ, h), which complicate the analysis. Thus one needs to find the regions of stability in the (h, δ) parameter plane.

(b) In [START_REF] Fang | Information Consensus of Asynchronous Discrete-Time Multi-Agent Systems[END_REF], one of the time delays, δ not only appears in an exponential term, but also in the denominator of the characteristic equation. This is an additional complication for stability analysis.

(c) A reasonable size of a traffic scenario would contain at least ten vehicles, n = 10, however, the corresponding characteristic equation ( 8) becomes extremely complicated for n > 2. To extract stability features from this equation is quite cumbersome and computationally inefficient. To overcome this difficulty, one may study the spectra of ( 8) with respect to each eigenvalue λ i (J), separately, obtaining the stability/instability features. The superposition of the spectra corresponding to individual eigenvalues will depict the complete stability characteristics. Hence, we propose to study the spectra of the following characteristic equation for each counter value i = 2...n.

χ i (s) = s -λ i (J)e -sh 1 -e -sδ sδ = 0, (9)
Clearly, studying ( 9) is much simpler, since χ i (s) is a 2 nd order dynamics (if expanded by s). Hence, the dimension of the problem is reduced from analyzing an n th order dynamics to repeatedly (i = 2...n) analyzing one of a 2 nd order. Clearly, such a reduction is quite beneficial when studying large scale traffic flow.

(d) Although, the idea of studying ( 9) reduces the dimensional difficulty of the analysis, this equation now carries complex coefficients since λ i (J) ∈ C -, [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF]. This prevents one to benefit from many stability analysis techniques available in the literature [START_REF] Gu | Stability of Time-Delay Systems[END_REF], [START_REF] Niculescu | Delay Effects on Stability: A Robust Control Approach[END_REF], [START_REF] Stepan | Retarded Dynamical Systems: Stability and Characteristic Function[END_REF].

Since s = 0, one can now rearrange [START_REF] Gazis | Car Following Theory of Steady State Traffic Flow[END_REF] as

H i (s) • ∆(s) = 1, (10)
where

H i (s) = λ i (J)e -sh s , ∆(s) = 1 -e -sδ sδ (11)
The advantage of the form (10) lies mainly on the separation of the parameter δ and the eigenvalues of J. This new form is an interconnection scheme which represents the characteristic equation ( 9) in two separate blocks. By doing so, parameters of interest (δ and h for instance) are decoupled, and consequently an effective technique can be proposed for the stability analysis.

Remark 1 (Continuity of roots):

The continuity of the roots of the interconnection scheme [START_REF] Gazis | Non-Linear Follow the Leader Models of Traffic Flow[END_REF] holds with respect to parameters defining the short-term memory (δ and h), [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF], therefore, the stability may be lost only when the interconnection scheme [START_REF] Gazis | Non-Linear Follow the Leader Models of Traffic Flow[END_REF] attains s = jω.

The method for stability analysis can be summarized as follows, [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF]. (i) Check the stability of the delay-free dynamics by setting h = δ = 0 in [START_REF] Fang | Information Consensus of Asynchronous Discrete-Time Multi-Agent Systems[END_REF]. (ii) Next, determine all s = jω roots of the interconnection scheme in the domain of delays (h, δ). (iii) Check the sensitivity of s = jω with respect to h and δ, in order to determine if s = jω roots transit to stable C -or to unstable C + . (iv) Label those (h, δ) regions for which there exist no roots in C + as stable.

C. Detection of s = jω roots of [START_REF] Gazis | Non-Linear Follow the Leader Models of Traffic Flow[END_REF] The outline of the framework for detecting the imaginary roots of ( 10) is presented as follows. If s = jω is a root of the characteristic function χ(s), then the following should be satisfied,

H i (jω) • ∆(jω) = 1. ( 12
)
Although there are several ways to analyze the interconnection scheme, we prefer the following as it also describes a practical geometric perspective. The above interconnection can be studied for its magnitude and argument in complex domain. Without loss of generality, we assume ω > 0, and notice that |e -jhω | = 1, ∀(h, ω) ∈ R 2 , thus the algorithm is formed by the following steps:

Step 1: Define first an intermediary variable z = δω/2. Notice that there does not exist an explicit function which maps from ω to δ. Hence, the additional parameter z is used to achieve this mapping. Clearly, to obtain the correspondence δ ←→ ω, a sweeping on the parameter z is quite practical (see below), and such an idea resembles frequency sweeping techniques presented in [START_REF] Niculescu | Delay Effects on Stability: A Robust Control Approach[END_REF].

Step 2: From the magnitude condition, one can easily find the relation

δ = 2z 2 |λ i (J)||sin(z)| , ( 13
)
from which δ can be obtained by sweeping z. See [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF] where we prove it is sufficient that 0 ≤ z ≤ π/2 for extracting the stability boundary of the dynamics in (h, δ) ∈ R + × R + parameter space.

Step 3: By definition of z and given δ, ω can be solved as

ω = |λ i (J)||sin(z)| z . ( 14
)
Step 4: Next, by deploying the argument condition on ( 12), h values are obtained from

h = - 1 ω ∠ δs 2 λ i (J)(1 -e -sδ ) s=jω ∓ 2π . ( 15
)
Clearly, solution set (ω, h, δ) contains all possible imaginary roots of [START_REF] Gazis | Non-Linear Follow the Leader Models of Traffic Flow[END_REF], and one has to establish the guidelines for extracting those that form the boundary of the stability regions in (h, δ) domain. The sensitivity study on the imaginary root s = jω will enable this, see Subsection D below.

D. Imaginary root transitions in h domain for fixed δ

Given λ i (J) and δ, the tendency of the imaginary roots s = jω across the imaginary axis as the corresponding h in (15) increases, can be found by the help of the following sensitivity function

S i (h, ω) = ds dh = - ∂χ i (s) ∂h ∂χ i (s) ∂s -1 . (16)
For analyzing the tendency across the imaginary axis, one needs to study the real part of S i (h, jω). If the real part is positive (or negative), it indicates the root s = jω crosses the imaginary axis along the direction from stable C -to unstable C + (or vice versa) of the complex plane. In other words, if (S i (h, jω)) > 0 (or < 0), number of roots on right half complex plane increases (or decreases) as h in (15) increases infinitesimally. Readers are directed to [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF] for the details as well as some invariance properties of the sensitivity function.

Remark 2 (Geometric approach): The analytical framework explained above ultimately enables the complete stability/instability characterization of the traffic flow dynamics in (h, δ) domain. Such a geometric approach can also be supported by an analytical framework which detects the stability boundaries completely and analytically. The characterization and its proof are suppressed here to maintain the flow of the paper, see [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF] for details.

V. ILLUSTRATIVE EXAMPLES

In the following, we present two different example scenarios for the traffic dynamics in (4). For both scenarios, linear and chain configuration of vehicles are considered. It is important to note that all results and interpretations mentioned this section are based on the particular numerical example and mathematical model considered.

First scenario takes ten drivers (number of vehicles), n = 10, and presents the effects of driver aggressiveness, κ i to the memory window size, δ of human drivers. All drivers are assumed to be identical and their aggressiveness is the range of κ i = κ ∈ [1.5, 2.5], which is realistic according to literature, [START_REF] Bando | Analysis of Optimal Velocity Model with Explicit Delay[END_REF].

The second scenario alters the first one by assuming there exists one, two or three relatively more aggressive drivers among the ten drivers. It is assumed that aggressiveness, κ i of these drivers is 20% larger compared with any κ chosen for the remaining drivers. Assuming these aggressive drivers are present in traffic, the change in maximum allowable memory window size of drivers is studied with respect to the first scenario (where there are no aggressive drivers). Arising results are interesting as they allow us to interpret stability robustness in the two different vehicle configurations considered for the traffic flow.

In the case studies, the number of vehicles n will be kept fixed for comparison reasons. Readers interested with how n affects the stability are directed to a recent work of the authors in [START_REF] Sipahi | Effects of Short-Term Memory of Drivers on Stability Interpretations of Traffic Flow Dynamics[END_REF].

A. Stability regions with respect to driver aggressiveness, κ i As summarized above, in this scenario we consider n = 10 number of identical drivers (vehicles), κ i = κ = 2.0, in both ring and linear configurations. Before we proceed to the main result of this subsection, the stability picture below will be helpful for the reader, Figure 2. This figure represents the stability regions (shaded) of traffic flow in the parametric domain defining the memory of the drivers given κ i = κ = 2.0. Clearly, the linear configuration of the vehicles allows larger stability regions in this parametric domain, R 1 ∪ R 2 , whereas in ring configuration the stability region shrinks to R 2 .

It is proven in [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF] that for the traffic flow model considered, the maximum memory window size on the stability boundary, max(δ) = δ, such as in the above figure, occurs when dead-time is zero, h = 0, independent of ring or linear configuration of vehicles. Moreover, the stability boundaries shown in this figure are monotonically decreasing, [START_REF] Sipahi | Stability of Traffic Flow with Distributed Delays Modeling the Memory Effects of the Drivers[END_REF]. Starting from this, we will use δ as a measure of stability, and that linear configuration of vehicles for the selected range of κ allows larger memory size for the drivers without stability of the traffic flow is lost. However, as drivers become more aggressive, it can be seen in this figure that maximum allowable memory size rapidly decreases for both ring and linear configurations.

B. Effects of Presence of Aggressive Drivers to Stability

Let us now alter the above scenario. Instead of assuming all drivers are identical, some drivers are assumed to be 20% more aggressive, i.e., their κ i is 20% larger, while remaining drivers will still be assumed identical. In order to represent how maximum memory window size is affected under presence of these relatively more aggressive drivers, arising results are compared with the previous scenario where all drivers are at same aggressiveness level, see Figure 4 and Figure 5 for ring and linear configurations, respectively. The results in the above figures are very interesting. In ring configuration: as the number of relatively more aggressive drivers increases, the percentage of change (decrease but its sign is kept as positive) in maximum memory window size increases. This is intuitively an expected results, as more aggressive drivers may disfavor stability measure. However, the linear configuration of vehicles is quite interesting: The decrease in memory window size is independent of the number of aggressive vehicles (decrease but shown with positive sign in figure).

Finally, we increase the number of aggressive drivers for the ring configuration, by taking four, five or six relatively more aggressive drivers. The change of maximum memory window size (decreasing but sign is kept positive) with respect to the case where all drivers are at same level of aggressiveness is depicted in Figure 6. For this particular numerical example, when compared with Figure 4, we observe that increasing number of drivers disfavor stability measure even further.

VI. CONCLUSIONS

Stability of a traffic flow dynamics is studied under various scenarios in which some drivers are relatively more aggressive and all drivers utilize their memories for their control actions. In particular, short-term memory modeling is developed and effects of presence of more aggressive drivers to stability is presented in the parametric domain defining the physics of short-memory. Both linear and ring configuration of vehicles is considered and interpretations on the arising results are presented. In ring configuration, as expected, it is found that stability measure narrows down in presence of more aggressive drivers. On the other hand, we observe that stability measure does not narrow down depending on the number of aggressive drivers. To the best of our knowledge, these results are new in the context of studying driver aggressiveness to the stability of traffic flow in which drivers utilize their memory.

Fig. 1 .

 1 Fig. 1. Uniform distribution model for memory effects

Fig. 2 .Fig. 3 .

 23 Fig. 2. Comparison of stability regions for linear and ring configurations with n = 10 vehicles and driver sensitivity κ i = κ = 2.0

Fig. 4 .Fig. 5 .

 45 Fig. 4. Percentage of decrease (shown with positive sign) in maximum memory window size with respect to identical driver case, if 1, 2 or 3 drivers among 10 are 20% more aggressive. Ring configuration.

Fig. 6 .

 6 Fig. 6. Percentage of decrease (shown with positive sign) in maximum memory window size with respect to identical driver case, if 4, 5 or 6 drivers among 10 are 20% more aggressive. Ring configuration.

Acknowledgements: This research started in November, 2005. Part of the research of Dr. Sipahi is supported by Chateaubriand Post-Doctoral Bourse of the French Government in the academic year 2005-2006 at HeuDiaSyC (UMR CNRS 6599). Dr. Sipahi is with the Northeastern University, Boston starting in 2006-2007 academic year. Rifat Sipahi would like to acknowledge the financial support by Centre National Recherche de Scientifique (CNRS) for his visit to Max Planck Institute, Leipzig, where he had fruitful discussions with Fatihcan M. Atay on distributed delays and memory effects.

Some parts of this research is supported by Chateaubriand Scholarship 2005-2006 during RIFAT SIPAHI's post-doctoral stay at HeuDiaSyC (UMR CNRS 6599),


Women at the Meeting An immense wasteland. Deafening music. A colossal scene. Blinding projectors. A crowd of onlookers mingles at the meeting point of militants. Dozens of Land Cruiser VX, Hilux, gleaming new GXs, seek to come closer to the center of action. Drivers honk their horns and young women shout, waving the poster of a candidate. Some motion wildly from the back of pick-ups. All of them wearing light purple veils [melehfa], the winning color of the candidate who will soon become president. Some of them wear caps with photos of the candidate, others, the most coquettish, have swapped the official headgear for a purple scarf that they tie on their head in the manner of a pirate, or more likely a RnB star. Even the security guards, overwhelmed, try to impose their authority, contrite in their purple ties... "a woman's idea" comments a friend later on. The place is invaded by young girls, young women, and older ladies. Activists. And those pretending to be. Some laugh. Others sing. They discuss. In a space, in an instant ... Nouakchott blushes. 2 Since the country's independence in 1960, women have had an undeniable presence in Mauritanian political life. Their engagement and will to act is acknowledged by everyone, as is their determination to have a place in debates. However, their weak insertion into the spheres of decision-making is also well known in spite of a multiplication of discourses supporting women and related laws put into place in the 1990s concerning the "promotion of women to electoral mandates and election functions" 3 (2006).

Thus, two distinct realities characterize the equation between woman and politics in Mauritania: on one hand, there is the strength of women's commitment, the ability of women to have a voice; and on the other hand, the limited place and minor role they are actually granted within institutions of power. If, during the last fifty years, female citizens are gradually but undeniably given the opportunity to leave their tents to go to the well -and this is proven by the statistics of the evolution of their representation -I believe that the fact of gradually imposing women in politics urges us to look at the opposite issue: that of imposing politics on women. On which women? To serve what objectives? This passage of women from the backstage to the stage qualifies less as an argument of gender equality "in progress" -one of the trappings of democracy -than strategies of a patronage power seeking to maintain itself by manipulating at all costs the individualistic tendencies of female political and economic actors, also in search of an opportunity for success. This article is based upon research that was undertaken as a part of a post-doctoral project (ANR PRANO) examining the trajectories of political and economic elite in Mauritania from independence until 2008. Ethnographic data collection was conducted by the author, and included in-depth observation as well semi-structured interviews with over 100 participants (male and female former ministers and members of parliament, political party members, and women and men active in civil society). The results of this research are presented here contextualized and updated by additional observations and interviews conducted in Mauritania in 2013-2014.

WHEN WOMEN SPEAK OUT

Women in the Campaign

The female activists sleep-in after debating and moving around the city late into the night on the lookout for electoral trends ... While male activists get up early, or at least earlier, after a long night meeting in various public spaces of Nouakchott. They gather around breakfast to hear, share, and discuss the news of the previous evening and predict the nominations, improvise political analyses. In late afternoon, after managing the household affairs (the driver will pick up the kids at school, snacks for "political" visitors need to be provided...), some female activists put on their most beautiful purple veils/boubous to start the door-to-door campaign to convince those who are still hesitant. They give promises, count, and toil to persuade voters while going from house to house... They participate in private meetings, keep track of this or that press conference... Who rallied who… They pass on the news to relatives and friends... The telephone network is saturated, just like the paved streets of Nouakchott. Banners, posters, garlands, tents, music, cars painted with the colors of a candidate...The city celebrates the campaign. Elections are also an opportunity for entertainment... for men and for women. 4 This female Moorish "militancy" is not new. In tribal battles and other internal disputes, women encourage men in combat and participate in fighting through an exchange of poetry, including lullabies [t'mari] in which they boast of the courage of their men and the cowardice of their opponents. Since the birth of the Islamic Republic of Mauritania, this female dynamic has not disappeared.

In 1958, two political parties come into conflict over the future of independent Mauritania. The colonial territories of sub-Saharan Africa, governed by the 'Framework Law' since 1957, prepare for the autonomization process. They constitute a Franco-African community in pursuit of the adoption of a constitution. This constitution is to be ratified through a referendum by the "people of France." Mauritanians are divided into two camps (those supporting and those opposing the constitution), and enter into a tense campaign leading up to the consultation planned for September 28, 1958. The members of the PRM (Mauritanian Republican Party) advocate the acceptance of the constitution in contrast to the Nahda 5 movement that opposes it. In the campaign supporting the constitution, the PRM, led by Mokhtar Ould Daddah, tries to mobilize its countrymen but are met by a pocket of resistance: Adrar, a regional stronghold of the Nahda. The emir of Adrar and many local figures enter the campaigning process, among them, a female trader and an activist of the PRM : "Mokeltoum traveled between Atar and Chinguetti to campaign for Mokhtar [Ould Daddah], she made a campaign as women do today, she met people, she went to their homes. (...) She explained that they should vote yes, "She was very supportive of Mokhtar (...) .She proved to be a popular woman there, and had strong ability to convince (...) "," She was a trader and a politician (...) She was truly a great lady" (Lesourd, 2010, 2014).

Since independence in 1960, the women's movement emerges but without much social impact.

The UFM (Union of Mauritanian Women) is born in 1964 under the influence of the French wife of the President of the Republic (Marie-Thérèse Daddah) and becomes the MNF (National Women's Movement) in 1966. This movement, promoted by the First Lady in 1961, is constituted by "unemployed" young mothers and offers courses in childcare, nutrition, and civic education of illiterate women.

"She began her social work early, first with the few female inhabitants of the capital, then with those of Ksar [old urban center], many of whom belonged to all social strata of the country. Then gradually this educational activity was extended to the entire country, especially since 1964, with the development of the Mauritanian People's Party" (Ould Daddah, 2003:177). This group then joins the PRM (the government party) and all women's groups are integrated into a formal movement called the UFM (Union of Women of Mauritania). These associations flourish quickly because they are composed mainly of the wives of wealthy men and political leaders. However, the margins of maneuver for the UMF movement are reduced as the single party cannot disagree with religious conservatives who are not in favor of the women's demands.

Consequently, the group of women fails to reach the bâdiyya [countryside 6 ]. The discourse of these female activists does not echo outside the capital and does not even reach the urban Haalpular and Soninke women -since one of the main demands of the movement is to fight against polygamy [START_REF]Baydhâniyatrefuse polygamy, which is not the case of other Mauritanian women belonging to so-called «Negro-Mauritanian» communities, namely the Haalpulars, the Soninkés, and the Wolofs[END_REF] . With respect to his wife's disputed political engagement, President Daddah writes:

"Some of the assumed responsibilities of Mariem [or Marie-Thérèse, his wife] involved particularly sensitive areas for us: the evolution of women and ideology, domains that remain close enough when they are not intermingled. Hence the reluctance -if not hostility -that she met with. So much so that in addressing these issues, she started with two specific handicaps: her origin and being a woman. But in our country, one hundred percent Muslim and conservative in its quasi-totality, a woman should not take an active interest in an area reserved for men: politics (...). In this context, what a "bombshell" it was when Mariem began to speak of the necessary evolution of women, promotion of women through labor, and generalized education for girls." (Ould Daddah, 2003: 178) Despite this partial failure -that the former president and his wife attribute to the difficulty of promoting women in a country like Mauritania -other women's movements start to emerge in parallel and achieve great popular success. They constitute successive movements of female contestations that oppose the central powers. In 1961, pro-Moroccan members of the Nahda Party, banned in the aftermath of independence, go underground and demand the organization of new elections. During this period, the new Islamic Republic of Mauritania has to stand up to the territorial claims of Morocco. Many opponents of Ould Daddah enter into exile in the Sharifian kingdom of Morocco. Against this backdrop of unrest and distrust, women invest in politics and express their political preferences, disapproval, and anger through chanting poetry and lullabies [t'mari] in which they address social issues. This is a genuine feminine political expression.

Through their poems, supposedly destined to put children to sleep, women rise as genuine political agents who address pertinent political issues. The production of lullabies, which, as their name suggests, were intended to rock babies to slumber, use small children, not as a subject of songs, but as a pretext for claims and political demands. A soothing balm, lullabies are a female outlet 8 . At this tumultuous time, the baydhâniyyat mock the oppression that targets opponents. They also mourn the difficult exile, denunciation, military surveillance, and lack of freedom. "(...) Ever since, harmless and seen as frivolous or irrelevant texts, lullabies reach the rank of political dignity to the point of being banned by the political power" (Z. Ould Ahmed Salem, 1995: 780). It was by the same process later on in the 1970s that women express their opinions and participate in the new struggle. As genuine political actors, it is through songs that they convey kaddihin ideas [START_REF]Clandestine Maoist protest movement, driven by students and trade unions, which constitute the Party of Kaddihin Mauritanians[END_REF] . Some of the revolutionary demands are indeed granted, including the revision of economic agreements between France and Mauritania, the nationalization of the mining company "MIFERMA" (which becomes SNIM), the creation of a national currency (ouguiya), the creation of a central bank, and the end of a one-party system:

«My son of mine, this excellent boy He promotes dialogue here He boasts national unity In a national front Here is the program That he strongly advocates He calls for freedom of thought And freedom of expression He also has a main claim That of freedom of association My son of mine, this valorous one He claims the nationalization of iron He also wants to quickly End our misery He also demands insistently The lowering of the price of tea He also wants bluntly That the prisoners be released He also demands that without delay

The price of sugar be decreased And he also wants uncompromisingly Support for the people of Sahara 10 And he even wants that by the force of arms Our economy be liberated.» 11 (Cited by Z. Ould Ahmed Salem, 1995: 784) This "rhymed guerrilla" (Z. Ould Ahmed Salem, 1995: 785) is accompanied by real covert actions that lead some women to prison. In this regard, Meïnatou recalls: "I hid leaflets under my veil, here on my stomach [...] I went from house to house: information and the leaflets had to circulate. Leaflets and newspapers had to move across Mauritania. And also reach the prisoners [...] Even when I traveled to Ayoune, I carried this with me [...]. We were many women to support kaddihin in Tidjikja [...] We were well organized but we had no means and we were watched very closely by the police [...] There were even kaddihin moms whose husbands were in the secret information service! Moms were muzzled, women were covered with gasoline; they wanted to scare them. Some suffered so much [...] Kaddihin was a personal choice; family had nothing to do with it, not like now."

Almost thirty years later, just a few days before the first round of presidential elections on March 18, 2007, the same Meïnatou supports Ould Maouloud and votes for him: "It's true, he is a relative, we are from the same region. But I voted for him because this is the old politics, a genuine party for equality that gathers all Mauritanians. It's just like kaddihin [...] He is a former member of the party; an old underground. He wants equality between men and women, equality between blacks and whites [...] This is the kaddihin, the same thing."

During this interview, another woman, Aisha, speaks; she was a high school student in the 1970s and then supported then the MND, the party of Maouloud, but more than thirty years later, the front of her house features two huge posters of Zein Ould Zeidane: "He is a young candidate. He has ideas. Otherwise, if he was not there, I would have voted Maouloud, but Zein has more chance, Inshallah." It is in these terms that Mauritanian women can express themselves in politics. "Of course women are in politics!" They participate, advocate, get interested, invest... Their opinion is respected, heard, considered. Their opinions have sometimes put them in danger and great humiliation. Mauritanian women have a way of doing politics in the feminine. Speech channels of their own. The female role is primarily to "say" to "circulate" to "hold" to "phone" to "visit homes" to "circulate leaflets ..." "With this campaign, I am at home, in the market. There are constantly people who walk by, departures to interior regions that constantly bring drivers here. This is the HQ. Can you imagine the pain it takes to prepare everything, to always be available. And the door-to-door in the evening. Me, I am tired with all this, I hope we will win."

But rare are women who go on from the campaign to take on positions of political power…

WOMEN ON THE POLITICAL STAGE

Women start to enter the corridors of state power towards the end of the 1980s, a period during which the first feminist discourses begin to flourish. Departments to which they have access, sparingly, are confined to specific areas: health, status of women. The order established in 2006 by the military government of transition provides a numerical increase in women's participation in local and national decision-making positions. They can access the position of counselor, but rarely that of mayor. Some women, mostly Moorish, have higher-level positions, but these remain tailored to fit "appropriate female roles".

Small Political Steps to Promote Women

By the late 1960s, some female citizens can access the highest ranks of the Mauritanian successive governments mobilize 14 women, which is not much, and in positions essentially thought to be "female-appropriate" (Status of Women, Health, Marital Status, Civil Service, and Youth). 14 Even in the PRDS (the party of former President Taya), a few months before the coup of 2005, the political advancement of women is without results: if we cannot obtain any figures on male and female adherents, or even an idea of a gender--based distribution, we have some evidence to assess the access of women to key positions in the organization of the party. The Secretary General (Boulah Ould Mageyya) is a man; the congress which elects the executive office members has seven women out of 67 members (10 percent female representation); and the same executive office has three women out of 13 members. In the five cabinets [START_REF]These cabinets surround the Secretary General and constitute the "staff" of the PRDS; through cabinets, they managed specific areas: relations with civil society, training of cadres and activists, follow-up of the popular structures of the party... 16 Department, district[END_REF] that the party runs, no woman is ever chair and in the 15 services that the cabinet counts, only three women have positions of responsibility (consultants or leaders of a mission).

At the national level representation of the party, of the 53 sections it counts (one section = a moughataa 16 ), only one of them is headed by a woman, who is no other than the President of the General Women's Committee of the PRDS, also a member of the Executive board and a Senator. We find in this structure, which should be a pioneer in the political integration of women, the same inequalities and deficiencies as in the political world in general and in the public service at large. Women are present in female structures (National Women's Party Committee) but senior management positions (Secretary General, Director of Staff) are not allocated to them; they remain limited in the best case, to the functions of project manager, counselor or intermediary decision-making positions. Finally, at the national level, one can speak of a virtual absence of female representation. Thus, the presence of women in the state apparatus and the official organs of power remain very symbolic despite 20 years of a pro-women discourse ...While women engage in politics, politics do not engage with women ...

The Quota Maneuver

The first transitional government (set up on August 9, 2005) The female presence is similar to that in the previous governments: three women and a majority of secretaries of state occupying the usual female positions. But the CMJD decides that those who will come after must go further. Accordingly, on August 22, an order "on the organic law relative to the promotion of women's access to electoral mandates and elective functions" is born.

Municipal Mandates:

Article 2: For the municipal elections, women are entitled to a minimum quota of 20 percent seats in municipal councils. Article 3: To promote the election of women as indicated in Article 2 above and in case no woman is nominated, the candidates in municipal elections lists must be drawn so as to place women in the eligible seats, according to the number of councilors provided.

These lists should include at least: 2 candidates for boards of 9 and 11 councilors 3 candidates for boards of 15 and 17 advisers 4 candidates for boards of 19, 21 and more councilors

Parliamentary Mandates: Article 4: In parliamentary elections, women are entitled to a minimum quota place (...) defined as indicated below: 1 in constituencies with two seats the candidate lists will include one candidate of each sex; 2 in electoral districts with three seats, the candidate lists will include at least one female candidate in the first or second position on the list; 3 in constituencies with more than three seats, each candidate list is made alternately from one candidate of each sex (...)

Elections to the Senate:

In the constituencies of Nouakchott, the candidate lists will include at least one female candidate in the first position on the list (...) Article 6: The political parties or groups, which, by appropriate measures, will elect women in a proportion greater than those established above, may obtain a financial incentive. (...) 17 Quickly, rumors, fantasies, and ignorance of specific texts capture a very targeted law and turn it into a political generalization. Shortcuts and ignorance spread like a flash in a pan.

Everyone talks about the "new law that requires a quota of 20 percent women in politics." But the 20 percent concerns only mayors and councilors. For parliamentary seats, the accounting is different. Thus, the very elaborate order is too hastily renamed "quota" and erroneous comments spread quickly: "One MP in four will be a woman", "The Senate will count 20 women", "Women will now have more power." The citizens, male or female, informed or not of the totality of the measures, choose their camp. The public opinion is divided between those who are "for the quota" and those who are "against the quota." Naha Mint Mouknass, president of the party and former Advisor to the President of the Republic expresses her displeasure in the press: "This is a big joke [...]. Why not a quota for the disabled or youth under 34? I am against hollow and immature slogans. I am for merit; not for filling the assembly and city halls with quotaized women." 18 Numerous women who invest in politics speak to me along the same lines: "Women are already in politics, we are already here; the only thing is that we do not fit into the boxes of international statistical agencies that have decided that with so many women in parades, Mauritania would be a real democracy [...]" They decided that women should do politics but we are already doing it! [...] We cannot say that all is well over here ... We must appoint professionals, elect competent people but all women are not ready [...] we impose in a bad way [...] and too much at once [...]" [START_REF]NGO coordinators[END_REF] Lack of training is always invoked by opponents of the quota, whether male or female: «One cannot engage in politics if you cannot read or write. Here, there are women here in big parties and in a good position, who never even went to school, they can count, read a little, but that's not enough! We should not demystify responsibilities! […]We cannot continue with everything and anything!» 20 .

The second argument, which is integral to the first one, is the lack of women in administration and decision-making positions (private or public companies), the lack of women who are able to take over and ensure the places offered to them: "Who will be women in politics? illiterate traders? Women who sleep all day? There is no stock of women who are trained and competent in Mauritania, who can take the places that Ely [either CMJD and the order] offer them [...]. Who can really be able to do that? [...] So it will be anyone [...] and we will say that it is the fault of women?" [START_REF][END_REF] Those who are favor of the quota see it first and foremost part of a logical continuity: "In our country, women do a lot of politics, now they will have jobs," and a proof of change to come: "Bringing women in politics is a great democratic progress, it enriches the discussion, brings it closer to people and their concerns. I am happy for women! And I support them!" 22 Some women claim that they deserve this power because "Mauritanian women are different! strong women! they know politics well!." As for the "pro-quota" camp, talk is not about lack of training for girls, the collapse of the Mauritanian school system, poor access of women to decision-making positions, or their difficulty in accessing jobs in the administration… [START_REF]Women are rare when it comes to access to positions of responsibility in the public administration[END_REF] In the aftermath of the October 2006-March 2007 elections, it is clear from the outset that, despite the media hype built around this positive discrimination, women are not truly well represented within institutions of power. True that the municipal councils are full of women (sometimes up to more than 30 percent, because only municipalities are subject to the notorious minima. The question is, once elected, what type of responsibilities will these women have? As for the office of mayor, it remains in the hands of men: only three women (5 percent) win this seat (in 53 moughataa). Three women from 1222 competing lists … And it is in urban areas, specifically in Nouakchott, that these positions are provided since the newly elected women have to manage the municipalities of El Mina, Sebkha, and Tevragh-Zeina. The first two are areas mostly inhabited by Harâtîn 24 and black Mauritanians. The two women who manage these areas were known-just like the party they represented (including APP [START_REF]APP is the party of Messaoud Ould Boulkheir, constituted of harâtîn and the Nasserists[END_REF] ) -to embody the religious ideas and/or identity of these poor and in big trouble districts. In contrast, it should be noted that the third city hall was won "by accident": Ould Hamza, first on the winning municipal RFD list, is elected to head the urban community, which places, according to the agreement signed by the CFCD, 26 the reformer Yaye Ndaw Koulibali at the head of the municipality of Tevragh-Zeina [START_REF]The mayors of each municipality must vote among themselves for the President of the Urban Community of Nouakchott. O/Hamza being elected in Tevragh Zeina then to the presidency of the CUN, the seat goes back to Mrs Koulibali[END_REF] .

Thanks to affirmative action, nine women access the new senate which counts 56 members; a 16 percent female representation. During the last mandate of President Taya, only three women are senators. Again, this phenomenon limited to the urban capital inflates the figures and the actual scope of the 2006-29 Order because the law stipulates that senatorial lists should be led by a woman in the electoral districts of Nouakchott, but the rule does not apply to the rest of the country. Consequently, the Senate is feminized due to Nouakchott (eight women senators out of nine) as the first names on the lists were de facto elected … In this case, the authority of the chiefs of the bâdiyya is not threatened... One does not talk politics under a tent. 17 women out of 95 seats (17.8 percent) enter the National Assembly [START_REF]Three deputies under Taya out of 75 deputies[END_REF] . Four of them are elected in Nouakchott (hence almost a quarter) and the other three in the national lists 29 , which relativizes the amplitude of the phenomenon touted by the press and politicians. Again, the urban effect is clear. But even in Nouakchott, on a total of 41 lists in competition, only four of them are headed by a woman! This suggests that the majority of MPs are elected indirectly... involuntarily... The women are there, in conformity with the law, on a list ... their presence meets a very specific logic.

Towards a More Feminized Government?

Pursuant to the notorious order of more women," it is upon a careful gleaning of the successive government formations from Sidi Ould Cheikh Abdallahi to Ould Abdel Aziz, that the conventional arithmetic mode of recruitment established by predecessors perdures. Women are not more numerous in the government than before the "quotas"; they represent nine percent of ministers between 2008 and 2009, and they are assigned the same positions for 40 years (social affairs, women's status, new technologies, health ...). Likewise, the various ministerial formations of the Aziz system count 11 percent female ministers. The positions provided for women remain within the "feminine" scope. However, Coumba Ba was appointed by Ould Abdel Aziz as Minister Delegate to the Prime Minister in charge of African Affairs in 2010. The case of Coumba Ba is also an opportunity to highlight -since independence, and in the image of the political class in general-that Negro-Mauritanian women are poorly represented in the government. Although the first female minister of the Islamic Republic of Mauritania was Haalpular, the trend becomes very quickly to focus on, among women, the appointment of baydhâniyyat 30 .

In addition, it is worthy of note that the "tanks" of recruitment for these three particular social categories -often referred to by more acerbic Mauritanians under the terms "maids" -offer numerous similarities for twenty years. Take for example the case of women who access the ministerial function, either recruited by cooptation within the National Import and Export Corporation (SONIMEX) -this is the case, for example, of Fatimetou Mint Khattri, Minister of Sidi Ould Cheikh Abdallahi and Mintata Mint Hedeid under Ould Taya -or having a close parental relationship with a politician (usually it is "daughter of"). Hence in the Ould Daddah era, Khadijettou Aicha Mint Ahmed, appointed

CONCLUSION: THE TENT CLOSER TO THE WELL

Female Political Showcases

Much ado about nothing, numerous speeches, promises, and yet women remain more political "animators" than actors in their own right. If the evolution of their representation is obvious -we move from a total absence of women ministers in the 1960s, to a single nomination in the 1970s, and then a rate of 11 percent of women ministers at the beginning of the year 2014women gently access the public scene but are more confined to second-level activities: they animate tents, go door-to-door, and create the ambiance in meetings. Indeed, this female parapolitical investment helps to attract onlookers and spectators, and transform electoral campaigns into genuine festive appointments: "We must emphasize the link with voters, proximity. [...] The tents, and all the trimmings, help the visibility of VIPs. For the youth, it is an occasion to dredge, and for us, it is useless [START_REF] Interview | political leader of an opposition party[END_REF] .

The more women there are in the tents, the more visitors increase in numbers. It is an opportunity to meet and exchange a few words. It is a seduction operation (Lesourd , 2010b). In a sense, women participate in some sort of political marketing. A screen that they embody in the service of power -for their own interests?

Elected or appointed, women function as a smokescreen thrown to the donors, who are proud of the good results, as can be read on many official reports in recent years: "A regional high-level meeting for the promotion of women's political participation for the achievement of gender equality and women's empowerment, the third of the eight Millennium Development Goals (MDG 3) was organized in collaboration with the Ministry of Social Affairs, Childhood and Family and the United Nations Programme for Development (UNDP) [...]. It led to the adoption of a consensual agenda with strategic and operational policy proposals that will contribute to improving the political participation of women. It must remembered that Mauritania has achieved substantial progress in the field of gender in recent years. It has adopted a series of regulations that target positive discrimination to improve women's representation in elected office. This has fostered a representation of women in Parliament and 19 percent of the seats in municipal councils 33 ».

Access to positions of power represent tangible evidence of democratization in the same way as that of the appointment or the victory of a token Hartâni or Negro-Mauritanian candidate (A. Antil & C. Lesourd, 2012). Negro-Mauritanians, Harâtîn, women nominated because the power is required to do so. Another example is the fact that the nomination of a woman for the presidential office is perceived as an opening, a novelty: evidence of social change and political mutation, as it is implied in this excerpt from the Moroccan press: "Mrs. Mint Moulaye Idriss, 57 years old, holds a doctorate in financial engineering. She is currently the Chair of the Board of Directors of the Mauritanian Agency of Information (MAI official). She is the second woman to be a candidate in a presidential election in Mauritania after Aicha Mint Jedeine who postulated in 2003 against former President Maaouiya Ould Taya (1984-2005). In Mauritania, an Islamic country where Sharia is in effect, women occupy only 20 percent of elected positions in elected institutions 34 ".

Finally, if the "quota" can certainly propel more women in social positions -which can lead to certain privileges -such as titles like senator, congressman, councilor, women fit more readily with secondary responsibilities than true positions of power. A municipal councilor -even if he/she takes advantage of his/her symbolic status and other benefits tied to his/her function -is not a mayor. The 26 women parliamentarians may debate in their respective chambers on behalf of a particular region or village -and, likewise, take advantage of this prestigious position -in the bâdiyya, men hold the reins of local power in the eyes of all, and, therefore, retain their place in the system of power. If Nouakchott seems to offer more visibility and opportunities to Mauritanian women -and that women proudly perceive themselves as invested with a potential for power and opportunity -, it seems that the sensibilities and prerogatives of the chiefs of the interior of the country are consciously managed by the intricacies of the Order. Outside the capital, the men remain at the well and the women under the tent.

Making the Best of All Opportunities

Under the presidency of Colonel Taya (1984-2005), the necessary democratization of the 1990s and the use of clientelism by reactivating/reinventing identity affiliations, including tribal and regional [START_REF] Marchesin | [END_REF] ones, launches an era during which the kleptocracy of public funding supporting (and even encouraging) legal, and especially illegal, enrichment participates in building and retaining the politico-commercial clique of the leader. The man is deemed sober and unengaging with splendor, but it is in this court context, connected to all the possible captures of annuities (starting of course with that of the state) that the mode of sultanic government sarcastically reviewed by Abdel Wedoud Ould Cheikh (2003: 153) unfolds:

"The lightning ascensions brutal fall in the highest ranks of the members of the "court", the miraculous returns of the disgraced, the enormous weight of financial predation and the more or less hidden ramifications of the "Sultan's cash register" are reminiscent of some Babylonian lottery à la Borges, of a sort of a universal and compulsory game "legitimately" peppered for the benefit of the sultan and his satraps, where "negative items" (removal, expropriation, imprisonment, torture ... ) would mix under the sardonic eye of the game masters, to risky profits."

Money and the promise of access to resources somehow takeover from the one-party system in that they become the preferred modes by the power to tighten the grip on men, to empower in every sense of the term, and to influence action.

It is at this time, in this mode of specific governing, that the first major political operations advancing women take place: a context of the revitalization of a necessary clientelism to maintain power. In this card game, the female element is not reduced to the single argument of "gender": women belong to a family, embody a tribe, reflect a region... To promote a woman is in this way to provide an opportunity to a group (Lesourd C., 2007b). Conversely, to nominate a woman, or make her a candidate for a party (within a quota system, for example) is a guarantee of the support of her clients. The "quotas" in the end reinforce this trend. Thus, in the legislative elections of November 2006, Ahmed Ould Daddah, leader of the Rally of Democratic Forces (RFD), chooses Lematt Mint Mogueya over a longtime activist to occupy the fourth position on his list. The membership of this tradeswoman to a recent opposition party that suddenly becomes of the leaders in the race for power: and lasts until the overthrow of President Ould Taya by the military (August 2005), Lematt features clearly in the ranks of the presidential party... For the health of his business in the transition to democracy provided by the Military Council for Justice and Democracy (CMJD), businesswomen, like their male counterparts, do not spare any effort to reckon with the powers to come.

As an activist, or if possible a candidate, the race to opportunism invites them to choose, without a transition, an official camp. With the Order of 2006, the parties are hence obligated to include more women and propel them to the front. They preferably need to recruit women with financial and material resources that they can use for the campaign, but also women who are known and influential, capable of attracting the public opinion. Available for this system as they mostly are, the next thing to do is to position themselves in the game of mutual manipulation. For a woman, support is a quest: to obtain a position, means, and prestige. For the power, to be supported is crucial to its continuation. In this way, men and women are in the same boat. Each one ensuring their position and managing their opportunities. This individualism does not encourage women in positions of power to embrace wider causes, including the very cause of women...

Notes

1 The title is reminiscent of the imaginary -based Baydhân [Moor] that "serious" conversations -including politics -are related to the well, masculine space by excellence, while most trivial discussions are related to the tent, the female space. A different version of this paper was published in French in L'Année du Maghreb, 2007. I would like to warmly thank Fatima Sadiqi and Kelley Sams (CNE) for the translation.

2 Fieldwork notes, Nouakchott, March 2007.

3 Ordinance 2006-029, soon renamed by everyone "female quota law". 4 Fieldwork note, Nouakchott, March 2007.

5 Nahda 6 Which is not the city, the countryside, the bush

  People's Party (MPP). The strong presence of President Ould Daddah's wife was perhaps behind this. But the first woman minister is appointed only in 1975: Aïssata Kane, a Haalpular Brakna woman, then takes the reins of the Ministry of Social Affairs. After the coup of July 10, 1978, successive Praetorian governments do not include women until 1986. "All citizens (male and female) became players in Mauritania's national future, especially women, given the impact they have on children as those primarily responsible for their education and guardians of social values. For this, and as prescribed by our Holy Religion Islam, women must assimilate and take on the demands of the new productive society and get rid of all social barriers that have hindered the progress of our people now free and egalitarian. Mauritanian women, just like Mauritanian men, have the right to education, work, responsibility [...] This is the political will of the national leadership 12 That same year Khadijetou Mint Ahmed, a former director at the central bank, is appointed Minister of Mines and Industry. She remains in this function until 1991. In 1991, Mariem Mint Ahmed Aicha becomes Secretary of State for the Status of Women and holds this post until the Cabinet reshuffle on January 5, 1995. She is at that time the only woman in the government. Another woman then takes her position: Seniya Mint Sidi Haiba. Between 1986 and 1995, during which eight ministerial teams hold office, these three women are the sole representatives of the Mauritanian women despite the president's call in Nema. After five cabinet reshuffles, three years later, on November 17, 1998 , two women appear on the list of nominations: the first one is Secretary of State in charge of Civil Status (Mrs. Khadijetou Mint Boubou), and the second one is appointed to the Secretariat of the Status of Women (Mrs Mintata Mint Hedeïd). On August 4, 2000, Fatimetou Mint Mohamed Saleck is assigned a new Secretariat, in charge of the Development of New Technologies, affiliated to Prime Minister's office. This newcomer joins her two colleagues and thus increases the number of female Secretaries of State. Between 2000 and 2005, seven new women hold various governmental positions: Ba Diyyé -Head of the Department of Health and Social Issues -, Serka Mint Bilal OuldYamar-Minister of Public Service and Employment, Fatimetou Saleck Mint Ahmed -Minister of New Technologies -, Naha Mint Mouknass 13 -Advisor to the President -,Aïchatou Mint Mahamham and Zeinabou Naha Mint Mint Ahmed -Secretaries of State for the Status of Women -and Elbetrigha Mint Kaber who succeeds Ms. Mint Ahmed Saleck -Secretary of State at the Department of New Technologies. Thus, in 45 years (including 15 years of promoting women)

  put in place by Colonel Ely Ould Mohamed Vall and the CMJD includes three women: a Minister of Youth and Sports (Mehia Mint Ahmed) and two secretaries of state, as always for the Status of Women and New Technologies, (respectively Noubghouha Mint Ettelamid and Meyena Mohamed Sow Deyna).

  Minister of Mines is the daughter of an executive of the Mauritanian People's Party (MPP); then, later on under Taya, we see the presence of Ba Diye, daughter of Ba Mamadou Samboly (Minister under Daddah); Fatimetou Mint Mohamed Saleck, 31 daughter of Moustava Saleck Ould Ahmed; Naha Mint Mouknass, daughter of a former minister and founder of a political party under Daddah. During the transition period, the CMJD appoints Sow Meyana Mohamed Deyna, daughter of a former member/mayor/minister. Finally, under Mohamed Sidi Ould Cheikh Abdallahi, Noubgouha Mint Mohamed Vall, daughter of a famous businessman and accomplice in the aborted coup attempt March 16,1981, Selma Mint Teguedi, sister of Limam Ould Teguedi Secretary General of the National Assembly, then Minister under Ould Taya, Sidi Ould Cheikh Addallahi).