Femke Van Wageningen-Kessels

Ludovic Leclercq

Winnie Daamen

Serge P Hoogendoorn

The Lagrangian coordinate system and what it means for two-dimensional crowd flow models

Keywords: Crowd flow, Model, Simulation, Lagrangian coordinates

A continuum crowd flow model is solved using the Lagrangian coordinate system. The system has proven to give computational advantages over the traditional Eulerian coordinate system for (one-dimensional) road traffic flow.

Our extension of the model and simulation method to (two-dimensional) crowd flow paves the way to explore the advantages for crowd flow simulation. This paper provides a first exploration and shows that a model and simulation method for two-dimensional crowd flow can be developed using Lagrangian numerical techniques and that it leads to accurate simulation results.

Introduction

Crowd flow models are used to describe, understand and predict collective behaviour of crowds. Roughly, three types of models exist [START_REF] Bellomo | Modeling crowd dynamics from a complex system viewpoint[END_REF][START_REF] Duives | State-of-the-art crowd motion simulation models[END_REF]: microscopic models in which the movements of individual pedestrians are described and traced, mesoscopic models in which position and velocity of individual pedestrians is described using probability distributions and macroscopic (or continuum) models in which variables are aggregated and which are the focus of this study.

Continuum models describe the dynamics of crowds as a continuum flow, in terms of average speed, velocity and density [START_REF] Hughes | A continuum theory for the flow of pedestrians[END_REF]. Recent research shows that continuum models can reproduce self-organisation and certain important dynamic phenomena such as lane formation and diagonal striping [START_REF] Jiang | Numerical simulation of a continuum model for bi-directional pedestrian flow[END_REF][START_REF] Hoogendoorn | Continuum modelling of pedestrian flows: From microscopic principles to selforganised macroscopic phenomena[END_REF][START_REF] Hoogendoorn | Continuum theory for pedestrian traffic flow: Local route choice modelling and its applications[END_REF], which is in contrast to previous claims. Furthermore, simulations based on continuum models have the potential to significantly reduce computation time whilst keeping high accuracy. This makes them useful for a larger range of applications, including real time state estimation and prediction for crowd management and optimisation of control strategies, under normal operating conditions and for evacuations. For this, the solutions to the model equations need to be calculated using efficient (i.e. both fast and accurate) computational methods. The Lagrangian coordinate system has been applied to traffic flow models [START_REF] Aw | Derivation of continuum traffic flow models from microscopic follow-the-leader models[END_REF][START_REF] Leclercq | The Lagrangian coordinates and what it means for first order traffic flow models[END_REF]. [START_REF] Van Wageningen-Kessels | Lagrangian formulation of a multi-class kinematic wave model[END_REF], Yuan et al. 1 VAN WAGENINGEN-KESSELS, Femke, LECLERCQ, Ludovic, DAAMEN, Winnie, HOOGENDOORN, Serge P., 2016, The Lagrangian coordinate system and what it means for two-dimensional crowd flow models, Physica A: Statistical Mechanics and its Applications, 443, ELSEVIER, pp. 272-285, DOI: 10.1016/j.physa.2015.09.048 [2012] show that the system has many advantages over the traditional Eulerian coordinate system, including more accurate reproduction of shock waves and more efficient state estimation based on trajectory data. It is expected that the advantages also hold in the case of two-dimensional crowd flow models resulting in more efficient simulation results than with methods based on the Eulerian coordinate system such as proposed by [START_REF] Dogbé | On the numerical solutions of second order macroscopic models of pedestrian flows[END_REF], [START_REF] Huang | Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm[END_REF], [START_REF] Jiang | Numerical simulation of a continuum model for bi-directional pedestrian flow[END_REF], [START_REF] Hänseler | A macroscopic loading model for time-varying pedestrian flows in public walking areas[END_REF]. In this contribution we focus on showing how crowd simulations can be done using the Lagrangian coordinate system. We also give some insight into the advantages of Lagrangian crowd simulations. Future studies will have to provide a definitive answer to the question under which circumstances the Lagrangian coordinate system is superior to the Eulerian coordinate system for crowd simulation.

Our main contribution is the proposal of an extension of Lagrangian simulation methods previously applied for one-dimensional traffic flow models to two-dimensional crowd flow models. We introduce the reformulation of the two-dimensional continuum crowd flow model into Lagrangian coordinates (Section 2), develop a numerical simulation method (Section 3) and show that it leads to meaningful simulation results (Section 4). The contribution is summarised and open questions and future research directions are discussed in Section 5.

Crowd flow model and its Lagrangian formulation

In a traditional, Eulerian (fixed) coordinate system the density at location ⃗

x and time t is calculated:

ρ = ρ(⃗ x, t) (1)
In contrast, the main idea of the Lagrangian (or moving) coordinate system is that the location ⃗ x of the n-th particle (i.e. a vehicle or a pedestrian) is calculated at time t:

⃗ x = ⃗ x(n, t) (2)
In vehicular traffic the n-th vehicle usually follows the (n-1)-th vehicle. Such a simple ordering of pedestrians is usually not possible and limits the model formulation. We

show how the Lagrangian model can be formulated nevertheless, but without a closedform conservation equation.

Conservation model

Our approach includes a physical explanation of the conservation of particles. In

Eulerian coordinates, the one-dimensional conservation equation is well-known:

∂ρ ∂t + ∂q ∂x = 0 (3)
with ρ the density (vehicles per meter), and q the flow (vehicles per second unit). The

Lagrangian equivalent of the one-dimensional conservation equation ( 3) is as follows:

∂(1 ρ) ∂t + ∂v ∂n = 0 (4)
with v = q ρ the vehicle speed (meter per second). To derive the Lagrangian equivalent (4) of the one-dimensional conservation (3) equation in Eulerian coordinates, a Moskowitz function can be defined [START_REF] Newell | A simplified theory of kinematic waves in highway traffic (part I-III)[END_REF]. Therefore, a numbering of the vehicles n = 1, . . . , N is used and their position as a function of vehicle number x(n)

is defined [START_REF] Aw | Derivation of continuum traffic flow models from microscopic follow-the-leader models[END_REF][START_REF] Leclercq | The Lagrangian coordinates and what it means for first order traffic flow models[END_REF][START_REF] Van Wageningen-Kessels | Lagrangian formulation of a multi-class kinematic wave model[END_REF].

However, for crowd flow, such a numbering of the pedestrians is not possible and pedestrians can not be numbered consistently. This prevents a continuous mapping of the pedestrians with respect to space and time. Therefore, an alternative approach is proposed.

Two-dimensional conservation model

The two-dimensional extension of the Eulerian conservation equation ( 3) is wellknown:

∂ρ ∂t + ∂q x ∂x + ∂q y ∂y = 0 (5)
with ρ the density (pedestrians per square meter), and q x and q y the flow in x-and y-direction, respectively (pedestrians times meter per second).

Two-dimensional Lagrangian conservation model

We use the Lagrangian approach to solve the conservation equation in two dimensions

(5). Therefore, fixed portions of particles (i.e. groups of vehicles or pedestrians) are defined and their position and speed is traced over time, see Figure 1. Note that the portion does not necessarily include an integer number of particles but instead the number of vehicles or pedestrians can take any positive value. In one-dimensional road traffic flow, the portion of vehicles occupies a certain length of road L(t) at time t. In two-dimensional crowd flow, the portion of pedestrians occupies a certain surface A(t) at time t. The evolution of the size of such portions (i.e. the evolution of L(t)

or A(t) over time) and their location only depends on the movement of the particles located at the boundary. Therefore, we have to assume that the boundary particles remain at the boundary and that there is no switching of positions, i.e. vehicles or pedestrians do not overtake each other. This assumption is true if the speed profile is smooth enough and the evolution is followed over a small time period. The inflow into the portion and the outflow out of the portion are by definition zero. Therefore, we can apply the conservation principle and we find for one-dimensional flow:

L(t 1 ) = L(t 0 ) + t1 t0 v first (t) -v last (t) dt (6)
with v first and v last the speed of the first and the last vehicle in the portion, respectively. The same principles can be applied to find the area occupied by a portion of the pedestrian in two-dimensional crowd flows. Therefore, we first consider the movement of one pedestrian j, which is at location ⃗ x j (t 0 ) at time t = t 0 and moves with velocity ⃗ v j (⃗ x, t). At time t = t 1 > t 0 its location will be:

⃗ x j (t 1 ) = ⃗ x j (t 0 ) + t1 τ =t0 ⃗ v j (τ ) dτ (7)
Secondly, we consider a region Ω i with N i (t) = ∫ ⃗ x∈Ωi(t) ρ(⃗ x, t) d⃗ x pedestrians in it. The area of this region equals:

A i (t) = ⃗ x∈Ωi(t) 1 d⃗ x = ∮ ⃗ x∈Γi(t) ⃗ x d⃗ x (8)
with Γ the boundary of the region. In the Lagrangian coordinate system, the region moves with the pedestrians, the pedestrians on the boundary, are assumed to stay on the boundary, the flux over the boundary is zero, and thus the number of pedestrians does not change (N (t 1 ) = N (t 0 )). However, the region and its boundary move if the speed of the pedestrians on the boundary is nonzero (Γ i (t 1 ) ≠ Γ i (t 0 )). Using both ( 7) and ( 8), we can compute the area A(t 1 ) at time t = t 1 > t 0 , see also Figure 1(b):

A i (t 1 ) = ∮ ⃗ x∈Γi(t1) ⃗ x d⃗ x new area = ∮ ⃗ x∈Γi(t0) ⃗ x old area + t1 τ =t v(⃗ x, τ ) dτ d⃗ x change in area = A i (t 0 ) + ∮ ⃗ x∈Γi(t0) t1 τ =t0 v(⃗ x, τ ) dτ d⃗ x (9)
The above concept of change in area occupied by a group of pedestrians according to the velocity of the pedestrians at the boundary, forms the basis of the numerical computation method, as outlined in Section 3.

q max = ρ crit v free ρ crit ρ jam (

Velocity: speed and route choice

To complete the model, we also need to describe the velocity ⃗ v(⃗ x, t). We propose a simple velocity model, which may be applied either in Lagrangian or in Eulerian coordinates. The velocity is calculated using a simplified version of the approach proposed by [START_REF] Hoogendoorn | Continuum modelling of pedestrian flows: From microscopic principles to selforganised macroscopic phenomena[END_REF][START_REF] Hoogendoorn | Continuum theory for pedestrian traffic flow: Local route choice modelling and its applications[END_REF]:

⃗ v(⃗ x, t) = V (ρ(⃗ x, t)) ⃗ e(⃗ x, t) = V (ρ(⃗ x, t)) ⃗ e stat (⃗ x, t) + ⃗ e dyn (ρ(⃗ x, t)) ⃗ e stat (⃗ x, t) + ⃗ e dyn (ρ(⃗ x, t)) (10)
with V (ρ) the fundamental relation and ⃗ e the route choice (i.e. walking direction).

The route choice ⃗ e is composed of the static route choice component ⃗ e stat and the dynamic (local, state dependent) route choice component ⃗ e dyn .

We are well aware that other, possibly more realistic, speed and route choice models exist. However, all models include key variables such as speed, density and density gradient, just like our simple model. This allows us to develop a numerical method, which could also be applied to more refined models. We emphasise that the simple version of the velocity model will suffice to illustrate the numerical method, which is the focus of our current study.

Speed

In our simplified approach, the speed only depends on density, following a bi-linear fundamental relation, see also Figure 2:

V (ρ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ v free if ρ ∈ [0, ρ crit ] ρ crit v free ρ jam -ρ crit ρ jam ρ -1 if ρ ∈ (ρ crit , ρ jam ] (11)
with v free free flow speed, ρ crit critical density and ρ jam jam density.

Route choice: static and dynamic

The static route choice component ⃗ e stat depends on the layout of the facility and possibly other factors. It can, for example reflect the shortest route to the destination.

In the simulations (Section 4) we apply three different static route choices, which are described in more detail there.

For the dynamic route choice we apply a simple model reflecting that pedestrians avoid high density regions: with β dyn ≥ 0 the weight of the dynamic route choice. The dynamic route choice component steers pedestrians in the direction with the steepest negative density gradient., i.e. away from other pedestrians.

⃗ e dyn = -β dyn ∇ρ (12) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵ ▵

Simulation approach

The main idea of the simulation approach using the Lagrangian coordinate system is that a moving grid is applied. In particular, Lagrangian cells correspond to a fixed portion of the crowd and the grid moves with the pedestrians in that portion. In the following we discuss how the grid is constructed, how the velocity of the pedestrians and the vertices of the grid is computed and how the grid is moved each time step ∆t. Furthermore, we discuss when and how the grid is reconstructed (remeshing).

The grid

We assume triangular grid cells for simplicity, but the methods can be extended to other polygon-shaped grid cells. The grid is staggered, meaning that some variables are calculated at the cell centres, others at the vertices, see Figure 3. We do not consider any variables calculated at the edges. The cell centres are defined as the centroid of the cells. Variables calculated at the cell centres are indicated with a subscripted triangle (▵). Variables calculated at the vertex are indicated with a subscripted circle (○). The main variable to be calculated at the cells centres is density (ρ ▵ ). The main variable to be calculated at the vertices is velocity (⃗ v ○ ). This is consistent with the cell definition: calculating the velocity at the vertices allows the cells to be moved, calculating the density at the interior is natural as each cell contains a fixed number of pedestrians.

Velocity of grid cell vertices

At the start of each time step, the location of the grid cells and their vertices and the number of pedestrians within each cell is known from the previous time step. Recall that the number of pedestrians in the portion can take any positive value and is not integer. From the location of the vertices and the number of pedestrians in a cell, we approximate the velocity of the grid vertices, as outlined below. We first discuss the method for an inner vertex, later the adaptations for a boundary vertex are discussed.

Density in grid cells

The average density in the grid cell is calculated:

ρ ▵ = N ▵ A ▵ (13)
and serves as an approximation of the density at the grid cell centre. To simplify notation, we leave out the time index in this section wherever possible. N ▵ is the number of pedestrians in the grid cell (remains constant over time). A ▵ is the area of the grid cell (may change each time step). In general, the area is computed as in ( 8), but because the grid cell is triangular we can use:

A △ = 1 2 (x j1 -x j3 )(y j2 -y j1 ) -(x j1 -x j2 )(y j3 -y j1 ) (14)
with the locations of the vertices {j 1 , j 2 , j 3 } ∈ J of the triangle (see also Figure 4(a)):

⃗ x ○,j = ⎛ ⎝ x j y j ⎞ ⎠ (15)

Speed at vertices

To obtain the speeds from the fundamental relation, the densities at the vertices are needed. The density at the vertex is an interpolation of the densities at the centroids of the surrounding cells, see Figure 4. The centroids (cell centres) ⃗ x ▵,i are calculated as the centre of mass of the cells:

⃗ x ▵,i = 1 3 j∈Ji ⃗ x ○,j (16)
with j ∈ J i the indices of the vertices of cell i. The density at a vertex ρ ○ (x ○ , y ○ ) is calculated as a linear interpolation of the densities in the surrounding cells ρ ▵ (x ▵ , y ▵ ).

This results in:

ρ ○ (x ○ , y ○ ) = c 1 x ○ + c 2 y ○ + c 3 . ( 17
)
with parameters c 1 , c 2 and c 3 determined by the linear interpolation function, which uses a truncated Taylor series [D 'Errico, 2012]. Subsequently, the speed of the vertices are approximated using the fundamental relation v ○ = V (ρ ○ ), see equation (11). In this simple algorithm we do not take into account anisotropy (the fact that pedestrians react differently to pedestrians in front than to those to the side and to the back).

The method could possibly be improved by including anisotropy using weights for each of the surrounding cells.

Route choice at vertices

For the dynamic route choice component we need the density gradient at the vertices.

These can be obtained directly from the parameters of the linear interpolation (17), see also Figure 4:

∇ρ ○ = ⎛ ⎝ c 1 c 2 ⎞ ⎠ (18)
We use ⃗ e dyn,○ = -β dyn ∇ρ ○ (see equation ( 12)) to calculate the dynamic route choice component.

The static route choice component ⃗ e stat is supposed to be given at any location and time.

1

2 3 A ▵ , ρ ▵ , ⃗ x ▵ (

Velocity of vertices

We can now approximate the velocity of the vertex (see equation ( 10)):

⃗ v ○ = v ○ ⃗ e stat,○ -β dyn ∇ρ ○ ⃗ e stat,○ -β dyn ∇ρ ○ (19)

Boundary conditions

At the boundaries, vertices are only partly surrounded by cells and density and density gradient can not be computed straightforwardly as in ( 17) and ( 18). A simple approach to deal with boundaries is introduced. Alternative, more accurate numerical boundary conditions may be developed in future research. In the current approach, ghost cells are introduced, see Figure 5. They have their centroids opposite of the centroids of the surrounding cells of the boundary vertex. Furthermore, since we are only dealing with free boundaries in the test cases in the next section, the ghost cells have zero densities. To calculate the density gradient, the ghost cells are included in the linear regression model as in ( 17) and the density gradient is as in (18). However, when calculating density, ghost cells are only included in the regression model ( 17) if they are located in the walking direction according to the static route choice.

static route choice

V 0 C 1 C 2 C 3 G 1 G 2 G 3

Moving of grid cells

Once the velocity of the vertices is known, they can be moved. Therefore, we assume that the velocity remains as computed in (19) during the time step. The vertices are moved as follows:

⃗ x new ○ = ⃗ x old ○ + ∆t⃗ v ○ (20)
This is illustrated in Figure 6. We assume that the velocity of the edges is a linear interpolation of the velocities of the neighbouring vertices. Therefore, edges remain straight and the cell remains triangular.

Remeshing the grid

The vertices of an individual grid cell, will usually have different velocities. Therefore, the grid cell will change its shape. If this deforms the grid cells 'too much' and the grid becomes too irregular, it is be necessary to remesh. Furthermore, densities may become too large (i.e. above jam density) and remeshing becomes necessary for this reason. Finally, if vertices move in opposite directions or if they have large speed differences, the cell may 'flip' (see Figure 7).

In the remeshing process a new mesh is formed and the pedestrians in the old grid cells, are assigned to new grid cells in such a way that the density at each location ⃗

x remains (approximately) the same and the total number of pedestrians in the computational domain does not change (conservation of pedestrians). Too achieve a computationally efficient remeshing process, only vertices and cells that satisfy certain remeshing criteria are remeshed, while others are left unchanged.

The objective of the remeshing is twofold: the density in each cell is to be kept below (or at) jam density and the grid is to be kept regular with the shape of the cells 'close enough' to an equilateral triangle. To achieve the first objective, grid cells that have densities close to jam density are enlarged, see Figure 8(a). To achieve the second objective vertices are moved if the are 'too close' to other vertices, see Figure 8(b).

The combination of both the cell enlargement and the moving of vertices, prevents the cells from flipping. The remeshing procedures are detailed below.

Enlarging grid cells

A grid cell is enlarged if its density becomes too large:

ρ > α ρ ρ jam (21)
with α ρ ∈ ρ crit ρ jam , 1 a parameter. Too enlarge the grid cells, all vertices of the cell are moved outward. They follow the line from the centroid (⃗ x c ) of the grid cell through the original position of the vertex (⃗ x i ), see Figure 8(a). The new position is:

⃗ x * i = (1 + β ρ )⃗ x i -β ρ ⃗ x c ( 22
)
with β ρ > 0 a parameter. Currently, an appropriate value for β ρ is determined by trial and error for the simulations. A more rigorous approach, possibly taking into account the actual traffic state can be developed in the future.

Moving vertices to keep grid regular

Furthermore, a vertex ⃗ x i is moved if its position is far away from the centroid of the surrounding polygon ⃗ x c , compared to its distance to any of its neighbouring vertices ⃗ x j :

⃗ x i -⃗ x c > α c max j ⃗ x i -⃗ x j ( 23
)
with α c > 0 a parameter. The vertex is then moved towards the centroid of its surrounding polygon, see Figure 8(b). Its new position is:

⃗ x * i = (1 -β c )⃗ x i + β c ⃗ x c (24)
with β c ∈ (0, 1) a parameter. Just like with β ρ , the value of β c is currently determined by trial and error for the simulations, leaving room for more efficient approaches.

A 1 A * 1 A 2 A * 2 A 3 A * 3 A 4 A * 4 A 5 A * 5 A 6 A * 6 A 7 A * 7
Figure 9: Example of recalculating densities after remeshing. The old mesh consists of (lightly coloured) triangles Ai. One triangle of the new mesh is drawn with thick black lines. It overlaps with 7 triangles of the old mesh. The intersections of the old and new triangles are brightly coloured and are indicated with A * i .

Recalculation the densities

After the new grid has been determined, the density in each of the reshaped grid cells is computed. Figure 9 shows an example. For the remeshing, density in the j-th triangle of the new mesh ρ remeshed j is calculated as follows:

ρ remeshed j = N remeshed j A remeshed j = ∑ i A * i Ai N i A remeshed j (25)
with A i the area of the i-th triangle of the old mesh, N i the number of pedestrians in that triangle and A * i the area of the intersection between the i-th triangle of the old mesh and the j-th triangle in the new mesh.

Simulation and results

We perform simulations with the newly developed method to show that crowds can indeed be simulated using Lagrangian techniques and that this approach gives good solutions. We consider three test cases. In the first and second case we show that the numerical solution using the newly developed Lagrangian simulation method approximates the analytical solution well. Furthermore, the second test case shows the influence of the weight of the dynamic route choice component. The third test case shows a more realistic setting where a group of pedestrians rounds a corner.

Test set up

In all three test cases we use remeshing settings and a bi-linear fundamental diagram with parameters as in Table 1 and initial state and static route choices as in Figure 10. The other settings are different for each of the cases, as Table 1 shows, and they are described in more detail below.

Case 1 ('straight')

Test case 1 is used to show the numerical accuracy, that is: the ability of the Lagrangian simulation method to approximate the exact solution of the model equations.

In this simple test case, initially, there is a high density area, which could resemble a jam in a wide corridor. At the front of the jam, pedestrians start walking and the jam dissolves. We ignore the influence of the dynamic route choice (i.e. we set its weight to zero: β dyn = 0), which enables us to calculate the analytical solution using kinematic wave theory and compare it with the simulation result. Furthermore, we compare the results for different time step sizes to get more insight into the accuracy.

Case 2 ('circular')

Test case 2 is also used to show the numerical accuracy, now with taking the dynamic route choice term into account. It also shows the influence of the weight of the dynamic route choice on the outcome of the model. In this test case, initially, there is a circular high density area. The static route choice represents an attractor at the centre of this area: that is where the pedestrians would want to go. The static route choice component is normalised such that its length always equals one: ⃗ e stat = 1.

However, the pedestrians will also walk away from the centre because it is too crowded, which is represented by the dynamic route choice term. For different weights of the dynamic route choice term, the stationary state in which the crowd does not move anymore is computed analytically and compared to the simulation results.

Case 3 ('corner')

Test case 3 is used to show that the numerical method also works for (slightly) more realistic cases in which a corner is involved. The static route choice is chosen according to equidistant trajectories around the corner. And again, the static route choice component is normalised such that its length always equals one: ⃗ e stat = 1. The analytical solution is not computed, but we will discuss the plausibility of the results.

Initialization

For the initialisation of the test cases, a mesh is created manually, depending on the initial positioning of the pedestrian. The initial meshes are shown in Figure 11. A certain number of pedestrians is assigned to each cell according to the initial density.

The values are shown in Table 1. pedestrians start walking. Figure 13 also includes a comparison with the analytical solution. This comparison shows that there is some delay in when the pedestrians start to walk, resulting in the front of the congestion wave travelling backward slower than expected. Comparison over different time step sizes, shows that this delay decreases for small time step sizes and that more accurate results can be obtained using smaller time steps, which indicates that the method is consistent.

Test results

Test

The results for Case 2 (Figure 14) show that the numerical method is well able to approximate the stationary solution. After 15 seconds the solution barely changes anymore and this is considered to be the numerical steady state. The results on the left show that the numerical steady state (blue lines) and analytical steady state makes the pedestrians spread out over a much larger area (in this example with radius of about 43 meter). Therefore, in most applications, it will be important to calibrate this parameter well.

The results for Case 3 (Figure 15) show that the method is also capable of simulations with slightly more complex layout. The pedestrians first walk to the right, while spreading out a bit more than in the initial solution. They round the corner in a way similar to observations with low densities. The results show that the numerical method allows for changes of the shape of the pedestrian group. Furthermore, the numerical scheme reproduces what we expect from the behavioural hypothesis in the model.

Summary and future work

A numerical simulation method for crowd flow based on the Lagrangian coordinate system is introduced. The method uses a triangular mesh that moves according to rules dictated by the traffic flow model. In order to prevent too high densities and too much deformation of the mesh, a remeshing procedure is introduced. Test results

show that this leads to plausible simulation results and that high accuracy can be obtained.

The Lagrangian simulation method can be applied to solve continuum crowd flow models. It can be applied to a simple model as presented here, but also to more advanced models. Further research is necessary to quantify the efficiency in terms of accuracy and computational speed, and to compare the efficiency with that of alternative methods such as those based on Eulerian grids fixed in space [START_REF] Hänseler | A macroscopic loading model for time-varying pedestrian flows in public walking areas[END_REF]. Certain aspects can be studied further, such as the approach to free boundaries and the remeshing procedure. This can include a sensitivity analysis of the remeshing parameters. Furthermore, future research will focus on alternative settings of the numerical method, including meshes consisting of generic polygons and alternative remeshing procedures. Future extensions of the method include the introduction of closed boundary conditions (including walls and obstacles), inflow and outflow boundary conditions and multiple groups that have different walking or route choice characteristics or destinations.

Figure 1 :

 1 Figure 1: Following of the evolution of a fixed portion of particles (vehicles or pedestrians) over time. Gray lines are trajectories.

Figure 2 :

 2 Figure 2: Two representations of the bi-linear fundamental diagram.

Figure 3 :

 3 Figure 3: A staggered triangular grid. Density ρ▵ is calculated at the cell centres, velocity ⃗ v○ is calculated at the vertices.

  a) Vertex numbering to compute area, density and centre of a cell. ∇ρ ○ , ρ ○ Cell numbering to compute density and density gradient at vertex.

Figure 4 :

 4 Figure 4: Examples of vertex and cell numbering. This is used to compute the area, density and centre of a cell (a) and to compute the density and density gradient at a vertex (b).

Figure 5 :

 5 Figure 5: Example of boundary with ghost cells. C1, C2 and C3 are the centroids of the cells neighbouring boundary vertex V0. The centroids of their respective ghost cells are located at G1, G2 and G3. Only ghost centroids G2 and G3 are in the static route choice walking direction and they are taken into account when computing the density at boundary vertex V0. All ghost cells are taken into account when calculating the density gradient.

Figure 6 :

 6 Figure 6: Example of movement of the grid during one time step. The old grid is shown in thin blue lines. The new grid in thick black lines. One grid cell is highlighted and the figure shows how it moves and deforms, following the velocities of its corners (red arrows).

Figure 7 :

 7 Figure 7: Cells 'flip' if the velocities of their vertices (red arrows) are very different. Blue cell with thin lines: old, gray cell with thick lines: new.

Figure 8 :

 8 Figure 8: Illustrations of the remeshing approaches. Thin black line indicates the old mesh, thick blue ones the new mesh, arrows indicate the movement of the vertex or vertices. In Figure 8(a), the dot indicates the centroid of the cell that is to be enlarged, in Figure 8(b), the dot is the centroid of the polygon consisting of all triangles shown in the figure.

  Case 3: pedestrian are initially positioned in ellipse, will walk to right and then turn right around a corner.

Figure 10 :

 10 Figure 10: Initial state and static route choice. Gray area indicates original position of pedestrians. Thick blue arrows indicate static route choice walking direction.

Figure 11 :

 11 Figure 11: Initial meshes.

Figure 12 :Figure 13 :

 1213 Figure 12: Case 1, densities at different times, with time step size ∆t = 1 s.

  Weight dynamic route choice β dyn = 5. Weight dynamic route choice β dyn = 20. Weight dynamic route choice β dyn = 80.

Figure 14 :

 14 Figure 14: Densities for Case 2, for various weights of the dynamic route choice. Left: average density at certain radii. Shown every second: light colors for first time steps, gradually getting darker. Black thick line: density in steady state according to analytical solution. Blue: density after 15 seconds. The plot shows that the density barely changes anymore and this is considered the numerically computed steady state. Right: space-density plots. Only density after 15 seconds (numerical steady state) is shown. Black circle indicates radius of group in steady state according to analytical solution.

(

  black lines) are almost identical. Furthermore, the test results show that the solution depends heavily on the weight of the dynamic route choice. This can be seen best on the right hand side, where the radius of the circle depends on the weight. If it has a low value (β dyn = 5, Figure14(a)), the pedestrians stay closely together (in this example in an area with radius of about 17 meter), while a high value (β dyn = 80, Figure14(c))

Table 1 :

 1 Parameters of test cases

	Fundamental diagram parameters
	free flow speed				v free	1.3	m/s
	critical density				ρ crit	1.35	peds/m 2
	jam density				ρ jam	5.4	peds/m 2
	Remeshing settings
	threshold enlarge cell			α ρ	1.0
	factor enlarge cell				β ρ	0.1
	threshold move vertex towards centroid	α c	0.33
	factor move vertex towards centroid		β c	1.0
	Case dependent settings: route choice
				Case 1	Case 2	Case 3
	static	⃗ e stat		1 0	-1 x 2 +y 2	x y	see Figure 10(c)
	dynamic	β dyn		0	[5, 10, 20, 40, 80]	20	m 2 /ped
	Case dependent settings: initial condition
				Case 1	Case 2	Case 3
	density	ρ(x, 0) 0.75ρ jam = 4.05	0.4ρ crit = 0.54	0.6ρ crit = 0.81	peds/m 2
	position			L x = 80	R = 25	R x =-30 R y =35 C x =10 C y =15	m
	nr of peds N tot		3637	1058	380
	Case dependent settings: mesh and numerical settings
				Case 1	Case 2	Case 3
	initial cell area A	2.1	[10.9, 14.3]	[7.3, 9.6]	m 2
	peds per cell	N	8.4	[5.9, 7.7]	[5.9, 7.8]
	nr of cells		I	434	150		54
	nr of vertices	P	240	91		37
	time step size	∆t [0.2, 0.5, 1, 2]	0.1		0.5

Acknowledgements

This research is performed as part of the The Netherlands Organisation for Scientific Research (NWO) Aspasia grant of Daamen (Daamen, Van Wageningen-Kessels, grant number 016.125.440) and as part of the NWO-VICI project 'Travel Behavior And Traffic Operations In Case Of Exceptional Events' (Hoogendoorn, grant number 453-08-006).


David Marquez-Carreras

Carles Rovira
email: carles.rovira@ub.edu

Samy Tindel
email: tindel@iecn.u-nancy.fr

David Márquez-Carreras

Asymptotic behavior of the magnetization for the perceptron model

Keywords: spin glasses, perceptron model, cavity method

Introduction

One of the basic problems in neural computation theory is the supervised learning for pattern recognition by a simple perceptron (we refer to [START_REF] Hertz | Introduction to the Theory of Neural Computation[END_REF] and references therein for a detailed account on that theory, from a physical and numerical point of view). This problem can be summarized in the following way: for two given integers N, M ≥ 1 and i ∈ {1, . . . , N }, the input of the perceptron is η = (η 1 , . . . , η N ) with η i ∈ {-1, 1}, and its output is of the form

O k = u i≤N w i,k η i , k ≤ M,
for a collection of real weights {w i,k ; i ≤ N, k ≤ M }, and where u : R → R should be, for numerical purposes, a step function. Note however that, in many case, u is taken as a regularization of the sign function, whose derivatives are allowed to grow at exponential speed with N . This is assumed for mathematical convenience, and also because it reproduces better the actual behavior of the brain synapses, from which this kind of model is inspired. The supervised learning procedure consists then in providing p inputs η 1 , . . . , η p , jointly with p desired output patterns ζ 1 , . . . , ζ p (called the training set), and, if possible, try to find the weights w i,k matching the inputs and the outputs, i.e. such that

u i≤N w i,k η l i -ζ l k = 0, k ≤ M, l ≤ p. (1)
Before computing those weights, a natural and classical question is to get some information about the theoretical capacity of our model, that is the maximal number of patterns for which the problem (1) can be solved. Without any a priori knowledge about w i,k and η l , the following assumptions can be done:

1. The inputs η are uniformly distributed in Σ N = {-1, 1} N .

2. The hyperplanes defined by the weights w i,k are also taken as "uniformly" distributed random planes, which is equivalent to model w i,k as an i.i.d. family of standard Gaussian random variables, normalized by N -1/2 in order to get a term of order 1. That is, one chooses

w i,k = g i,k N 1/2 ,
where {g i,k ; i ≥ 1, k ≥ 1} is a family of independent standard Gaussian random variables.

However, it is a hard task to solve the system [START_REF] Bardina | The p-spins interaction model with external field[END_REF]. Thus, in that context, one step further in the analysis is to replace it by the study of an associated Gibbs measure, a usual trick known as the finite temperature approximation.

In our case, if one wishes to know on how many bits M the output can be coded, a natural measure to consider is the one whose density with respect to the uniform measure µ N on Σ N is given, for a typical configuration σ = (σ 1 , . . . , σ N ) ∈ Σ N , by Z -1 N,M exp(-H N,M (σ)), with

-H N,M (σ) = k≤M u N -1/2 i≤N g i,k σ i , Z N,M = σ∈Σ N exp (-H N,M (σ)) .
(2) Notice that, in order to get the formula [START_REF] Hertz | Introduction to the Theory of Neural Computation[END_REF], one has to remove all the constant weights ζ l k , and to include the usual inverse of the temperature parameter β into the function u, which are also two standard tricks.

It is now easily seen that a phase diagram for the model given by ( 2), known as the perceptron model, would give some essential information about the capacity of the original model. This phase diagram is far from being complete from a mathematical point of view. However, the deep study initiated in [4, Chapter 3] (see also [START_REF] Shcherbina | Rigorous solution of the Gardner problem[END_REF] in a slightly different context) leads to a rigorous replica symmetric solution when M is a (small enough) proportion of N , i.e. M = αN with α small enough. This replica symmetric solution indicates that the system does not behave too chaotically when N → ∞, and hence that the capacity of the perceptron is not reached yet.

Another way of looking at the regularity of the system when N → ∞ is to investigate the asymptotic behavior of the magnetization of an arbitrary family of spins σ 1 , . . . , σ k for k ≥ 1, that is the limit of the laws of the random variables σ 1 , . . . , σ k , where • designates the average with respect to the measure defined by (2). In fact, in most engineering applications, the classical way to illustrate the self averaging phenomenon for this kind of model is to show that those quantities converge to some independent and identically distributed centered random variables that can be clearly identified, by analogy with the fact that the magnetization vanishes for the Ising model at high temperature. Therefore, our aim in this paper will be to give an answer to that natural problem, which can be summarized in the following theorem: Then, there exists a constant r ≥ 0, that will be defined at Section 2, such that, for any fixed m ≥ 1,

i≤m E σ i -tanh r 1/2 z i 2 ≤ κ m N 1/2 ,
where κ m is a positive constant, and (z 1 , . . . , z m ) is a family of independent standard Gaussian random variables (which still depend, however, of the randomness contained in g).

Let us give now some hints about the way one can get that result: of course, our proof will be inspired by the methods introduced in [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF], and particularly by the smart path method, which is an efficient way of interpolating two Gaussian vectors, once the limiting local fields corresponding to (2) can be guessed. In our case, if the natural path is quite easy to figure out (see (8) for a heuristic justification), the computations along that path are long and sometimes involved. In particular, the path (18), which seems to be the obvious one at first sight, lead us to a series of technical problems. Like in [START_REF] Bardina | The p-spins interaction model with external field[END_REF], we have chosen to give some details of our calculations for sake of readability, hoping that the final aspect of our paper will not be too scary.

Notice also that the optimal expected rate of convergence in our theorem would be in N -1 . We stopped at N -1/2 and did not check the computations allowing to get this optimal rate, but we believe that it could be reached along the same lines we have developped in this paper. In any case, let us recall again that our result answers a natural question for the perceptron model, and corresponds to one of the basic facts that practitioners are looking for, as mentioned in [START_REF] Hertz | Introduction to the Theory of Neural Computation[END_REF]. Observe also that our way to derive Theorem 1.1 is certainly not the only one, and we hope that, in the future, some easier proofs can be elaborated. And it is worth pointing out that the methods contained in [START_REF] Shcherbina | Rigorous solution of the Gardner problem[END_REF] could potentially yield one of these simplifications. However, the latter paper is based on some convexity assumptions on the function u, and some spherical hypothesis on the spin model, which makes it hard to adapt to our situation.

Our paper is divided as follow: in the next section, we will introduce the basic notations concerning the perceptron model, and recall some results taken from [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF]. Section 3 is devoted to state an important intermediate result (Proposition 3.1) on our way to the proof of Theorem 1.1, to the definition of the Gaussian path we will use in the sequel, and to the first computations along this path. Some new Gibbs measures appear naturally in those computations. We will introduce them at Section 4. The main step towards Theorem 1.1 will then be achieved with an asymptotic expansion of a general term involving the measures introduced at Section 4. This will be done at Section 5. Then, we will apply those results in order to prove Proposition 3.1 at section 6. Theorem 1.1 will be proved at Section 7. Eventually, let us mention that, in the remainder of the paper, κ, c, c 1 , . . . will stand for some positive constants, that can change from line to line.

Preliminary results

As mentioned in the introduction, the state space of our model will be Σ N = {-1, 1} N for a given N ∈ N * , and setting µ N for the uniform measure on Σ N , we will consider the random measure whose density with respect to µ N is given by Z -1 N,M exp(-H N,M (σ)), where Z N,M has been introduced at relation (2), and where we recall that H N,M (σ) is defined by

-H N,M (σ) = k≤M u N -1/2 i≤N g i,k σ i .
In the above formula, M stands for a positive integer such that M = αN for a given α > 0, u will be a continuous function defined on R satisfying conditions (H1) and (H2) given in the introduction, and {g i,k ; i ≥ 1, k ≥ 1} is a family of independent standard Gaussian random variables. Note that u and α will depend on N , though this dependence will be kept implicit for sake of readability.

We will first give an account on the results obtained in [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF] concerning the perceptron model. Those results are based on the smart path method, that is a way to single out continuously the last spin σ N from the other ones in the definition of H N,M . This smart path can be described as follows: for

t ∈ [0, 1], σ ∈ Σ N , set S k,t (σ) = 1 N 1/2 i≤N -1 g i,k σ i + t N 1/2 g N,k σ N , (3)
and consider the Hamiltonian

-H N,M,t (σ) = k≤M u (S k,t (σ)) + σ N (r(1 -t)) 1/2 Y, (4)
where r is a positive coefficient whose exact value will be given later on, and Y is a standard Gaussian random variable independent of all the other randomness in H N,M . For a given function f : Σ n N → R, set now f t for the Gibbs mean of f with respect to the Hamiltonian H N,M,t , and

ν t (f ) = E [ f t ] .
Then, on one hand, some simplifications in the computation of ν t (f ) can be done when t = 0: Lemma 2.1 Let f -be a function defined on Σ n N -1 , and I be a subset of {1, . . . , n} of cardinality |I|. Then

ν 0 f - l∈I σ l N = E (tanh(r 1/2 Y )) |I| ν 0 (f -).
Notice that, in the previous result, σ 1 , ..., σ n are understood as n independent configurations under G N , usually called replicas of σ.

On the other hand, the derivative of ν t (f ) with respect to the parameter t can be computed explicitely.

Proposition 2.2 Suppose that u is twice differentiable. Write w = u ′′ + (u ′ ) 2 . Then, for 0 < t < 1 and f : Σ n N → R, we have ν ′ t (f ) = 3 j=1 K j ,
where

K 1 = α 2 l≤n ν t w(S l M,t )f - α 2 nν t w(S n+1 M,t )f , K 2 is defined by K 2 = α 1≤l<l ′ ≤n ν t σ l N σ l ′ N u ′ (S l M,t )u ′ (S l ′ M,t )f -αn l≤n ν t σ l N σ n+1 N u ′ (S l M,t )u ′ (S n+1 M,t )f +α n(n + 1) 2 ν t σ n+1 N σ n+2 N u ′ (S n+1 M,t )u ′ (S n+2 M,t )f , and
K 3 = -r 1≤l<l ′ ≤n ν t σ l N σ l ′ N f -nν t σ l N σ n+1 N f + n(n + 1) 2 ν t σ n+1 N σ n+2 N f .
The drawback of Proposition 2.2 is that some terms involving u ′ and u ′′ appear in the fomula, while one would like to have only a mild control on those quantities. However, using some integration by parts arguments, some asymptotic results for the perceptron model are obtained in [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF]: first of all, define, for two replicas σ l , σ l ′ of σ, the quantity

R l,l ′ = 1 N i≤N σ l i σ l ′ i .
This quantity, called the overlap of σ l and σ l ′ , will be, as usual in the spin glass theory, a central object in the study of the model. It is then essential to obtain some sharp results on the asymptotic behavior of R 1,2 , that we will describe now: let q and r (notice that r has already been introduced in ( 4)) be the solution to the system

q = E tanh 2 (r 1/2 Z) , r = αE Λ 2 (q 1/2 Z, (1 -q) 1/2 ) , (5)
where Λ is defined on

R 2 by Λ(x, y) := E [ξ exp(u(x + ξy))] yE [exp(u(x + ξy))] = E [u ′ (x + ξy) exp(u(x + ξy))] E [exp(u(x + ξy))] ,
and where the second equality is obtained by a Gaussian integration by parts. In the preceding formulae, Z and ξ stand for two independent standard Gaussian random variables. Observe that, under hypothesis (H1), the system (5) admits a unique solution. Then the following inequalities hold true (notice that these inequalities are shown in [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF] under sligthly weaker assumptions):

Proposition 2.3 Under hypothesis (H1) and (H2), we have, for any k ≥ 0,

ν (R 1,2 -q) 2k ≤ Lk N k .
Furthermore, we have

|ν (R 1,2 -q)| ≤ L N . (6)
Notice that the computations leading to this exponential self-averaging property for the overlap also yield the replica symmetric solution for the model. We refer to [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF] for this last formula. Let us end this section by recalling an elementary integration by parts formula we will use throughout the paper: for p ≥ 1, let (g 1 , . . . , g p ) be a Gaussian vector in R p , and F : R p → R be a C 1 function having at most exponential growth together with its gradient. Then

E[g 1 F (g 1 , . . . , g m )] = l≤m E[g 1 g l ]E[∂ x l F (g 1 , . . . , g m )]. ( 7
)
3 Definition of the path A first step in order to prove Theorem 1.1 is to achieve an asymptotic decomposition of σ N separating, on one hand, the randomness contained in {g N,k ; k ≤ M }, and on the other hand, some terms involving

{g i,k ; i ≤ N -1, k ≤ M }.

First of all let us introduce a new Hamiltonian

-H N -1,M (σ) = k≤M u N -1/2 i≤N -1 g i,k σ i = k≤M u (S k,0 ) ,
where the quantities S k,t have been defined at (3), and let • -denote the average with respect to the Gibbs measure induced by H N -1,M .

Then, the following heuristic steps can be performed: we have

-H N,M (σ) ≃ -H N -1,M (σ) + N -1/2 k≤M u ′ (S k,0 )g N,k σ N ,
and hence

σ N ≃ sinh 1 N 1/2 k≤M g N,k u ′ (S k,0 ) - cosh 1 N 1/2 k≤M g N,k u ′ (S k,0 ) - . (8)
We will show in the sequel that this crude estimate holds true in the limit N → ∞, and in fact, using a direct analogy with the self averaging properties of the SK model, we will prove the following result:

Proposition 3.1 Under hypothesis (H1) and (H2), there exists a strictly positive constant κ such that

E   σ N -tanh 1 N 1/2 k≤M g N,k u ′ (S k,0 ) - 2   ≤ κ N 1/2 . (9)
Most of the remainder of the paper will be devoted to prove (9), and let us begin here with some simple considerations, that is, the definition of the Gaussian path we will use: for t ∈ [0, 1], let us consider the function

ϕ(t) = E ( σ N t -tanh Y t ) 2 ,
where

Y t = t N 1/2 k≤M g N,k u ′ (S k,0 ) -+ [r(1 -t)] 1/2 Y.
Notice then that ϕ(1) is the left hand side of (9), and on the other hand, using that σ N 0 = tanh(r 1/2 Y ), it is easy to check that ϕ(0) = 0. By means of the elementary equation

ϕ(1) -ϕ(0) = 1 0 ϕ ′ (t)dt,
we can see that (9) will be achieved as soon as we can show that

|ϕ ′ (t)| ≤ κ N 1/2 ,
for any t ∈ (0, 1). Hence, our first task will be to compute the derivative of ϕ. In fact, we will start by decomposing ϕ into three terms:

ϕ(t) = ϕ 1 (t) + ϕ 2 (t) + ϕ 3 (t), (10)
with

ϕ 1 (t) = E σ N 2 t , ϕ 2 (t) = E tanh 2 (Y t ) , ϕ 3 (t) = -2E [ σ N t tanh Y t ] .
Then, we will compute the derivatives of ϕ 1 , ϕ 2 and ϕ 3 separately.

Using two replicas of σ, we get ϕ

1 (t) = ν t (σ 1 N σ 2 N ) .
Then, as a direct consequence of Proposition 2.2, we get the following lemma. Lemma 3.2 For any t ∈ [0, 1], we have

ϕ ′ 1 (t) = k≤6 A 1,k (t), with A 1,1 (t) = α ν t u ′′ + (u ′ ) 2 (S 1 M,t ) σ 1 N σ 2 N , A 1,2 (t) = -α ν t u ′′ + (u ′ ) 2 (S 3 M,t ) σ 1 N σ 2 N , A 1,3 (t) = α ν t u ′ (S 1 M,t ) u ′ (S 2 M,t ) , A 1,4 (t) = -4α ν t u ′ (S 1 M,t ) u ′ (S 3 M,t ) σ 2 N σ 3 N , A 1,5 (t) = 3α ν t u ′ (S 3 M,t ) u ′ (S 4 M,t ) σ 1 N σ 2 N σ 3 N σ 4 N , A 1,6 (t) = -r 1 -4ν t σ 1 N σ 2 N + 3ν t σ 1 N σ 2 N σ 3 N σ 4 N .
Differentiating the function ϕ 2 (t) and using the integration by parts formula (7), we also obtain the following result.

Lemma 3.3 For any t ∈ [0, 1], we have ϕ ′ 2 (t) = A 2,1 (t) + A 2,2 (t), with A 2,1 (t) = αE 1 -tanh 2 (Y t ) 1 -3 tanh 2 (Y t ) u ′ (S M,0 ) 2 -, A 2,2 (t) = -rE 1 -tanh 2 (Y t ) 1 -3 tanh 2 (Y t ) .
Now, differentiating ϕ 3 (t) and using again the integrate by parts formula (7) we get the next lemma. Lemma 3.4 For any t ∈ [0, 1], we have

ϕ ′ 3 (t) = -2 i≤4 A 3,1,1,i (t) - i≤5 A 3,1,2,i (t) + A 3,2,1 (t) -A 3,2,2 (t) , with A 3,1,1,1 (t) = α 2 E σ N u (2) (S M,t ) + (u ′ (S M,t )) 2 t tanh Y t , A 3,1,1,2 (t) = - α 2 E σ 2 N u ′ S 1 M,t u ′ S 2 M,t t tanh Y t , A 3,1,1,3 (t) = α 2 E u ′ (S M,t ) t u ′ (S M,0 ) -1 -tanh 2 (Y t ) , A 3,1,1,4 (t) = - r 2 E 1 -tanh 2 (Y t ) ,
the terms A 3,1,2,i (t) being defined by

A 3,1,2,1 (t) = α 2 E σ 1 N u (2) S 2 M,t + u ′ S 2 M,t 2 t tanh Y t , A 3,1,2,2 (t) = α 2 E σ 1 N u ′ S 1 M,t u ′ S 2 M,t t tanh Y t , A 3,1,2,3 (t) = -αE σ 1 N σ 2 N σ 3 N u ′ S 1 M,t u ′ S 2 M,t t tanh Y t , A 3,1,2,4 (t) = α 2 E σ 1 N σ 2 N u ′ S 2 M,t t u ′ (S M,0 ) -1 -tanh 2 Y t , A 3,1,2,5 (t) = -rE [ σ N t tanh Y t ] + rE σ 1 N σ 2 N σ 3 N t tanh Y t - r 2 E σ 1 N σ 2 N t 1 -tanh 2 Y t .
The terms A 3,2,1 (t) and A 3,2,2 (t) are respectively obtained by

A 3,2,1 (t) = α 2 E u ′ (S M,t ) t u ′ (S M,0 ) -1 -tanh 2 (Y t ) - α 2 E σ 1 N σ 2 N u ′ S 1 M,t t u ′ (S M,0 ) -1 -tanh 2 (Y t ) -αE σ N t u ′ (S M,0 ) 2 -tanh Y t 1 -tanh 2 (Y t ) , and
A 3,2,2 (t) = r 2 E 1 -tanh 2 (Y t ) - r 2 E σ 1 N σ 2 N t 1 -tanh 2 (Y t ) -rE σ N t tanh Y t 1 -tanh 2 (Y t ) .

Introducing a new measure

All the preceding considerations show that, in order to prove relation (9), we should be able to expand and control any term of the form

f (σ 1 , ..., σ n 1 )U 1 S 1 M,t , ..., S n 1 M,t t U 2 S 1 M,0 , ..., S n 2 M,0 -, (11)
for

n 1 , n 2 ∈ N, U 1 : R n 1 → R, U 2 : R n 2 → R and t ∈ (0, 1].
In this section, we will begin to develop some new tools that should allow us to handle those terms. More precisely, we will introduce some new Gibbs measures in the following way: first of all, for t ∈ [0, 1], σ ∈ Σ N , set

-H * N,M,t (σ) = k≤M u(S k,t (σ)) + σ N b * t ,
where we recall that Y is a standard Gaussian random variable independent of the remaining randomness contained in H * N,M,t , and

b t = (r(1 -t)) 1/2 Y, b * t = b t 1 (0,1] (t). (12)
Write then G * t for the Gibbs measure related to this Hamiltonian, and notice that

H N -1,M (σ) = - k≤M u(S k,0 (σ)) = H * N,M,0 (σ).
For a fixed (t 1 , ..., t n ) ∈ [0, 1] n , let us consider now a new Gibbs measure G * t 1 ,...,tn on Σ n N , defined by its average on functions f : Σ N → R:

f (t 1 ,...,tn) := σ 1 ,...,σ n f exp l≤n k≤M -1 u(S l k,t l ) + σ l N b * t l σ 1 ,...,σ n exp l≤n k≤M -1 u(S l k,t l ) + σ l N b * t l
, where f = f (σ 1 , ..., σ n ). Now, the terms of the form (11) can be expressed using a Gibbs measure of the form G * : let us consider the notation n := n 1 + n 2 , as well as

U 1 := U 1 S 1 M,t , ..., S n 1 M,t , U 2 := U 2 S 1 M,0 , ..., S n 2 M,0 and f = f (σ 1 , ..., σ n 1 ), U 2 := U 2 S n 1 +1 M,0 , ..., S n M,0 . Then, for t ∈ [0, 1], f U 1 t U 2 - = σ 1 ,...,σ n 1 f U 1 exp l≤n 1 k≤M u(S l k,t ) + σ l N (r(1 -t)) 1/2 Y σ 1 ,...,σ n 1 exp l≤n 1 k≤M u(S l k,t ) + σ l N (r(1 -t)) 1/2 Y × σ 1 ,...,σ n 2 U 2 exp l≤n 2 k≤M u(S l k,0 ) σ 1 ,...,σ n 2 exp l≤n 2 k≤M u(S l k,0 )
, and using the relation

-H * N,M,t (σ) = -H * N,M -1,t (σ l ) + u(S M,t ), we get, for t ∈ [0, 1], f U 1 t U 2 - = σ 1 ,...,σ n f U 1 U 2 exp -l≤n 1 H * N,M,t (σ l ) -n 1 <l≤n H * N,M,0 (σ l ) σ 1 ,...,σ n exp -l≤n 1 H * N,M,t (σ l ) -n 1 <l≤n H * N,M,0 (σ l ) = f U 1 U 2 exp l≤n 1 u(S l M,t ) + n 1 <l≤n u(S l M,0 ) (t
where in (t, ..., t, 0, ..., 0) we have n 1 times t and n 2 times 0. On the other hand, the measures G * t 1 ,...,tn share some of the good properties of the original Gibbs measure, starting from the one concerning widely spread sequences of Gaussian random variables, a crucial notion that we recall briefly here (see however [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF] for a detailed account on that topic). Definition 4.1 For n ≥ 1, a centered Gaussian family (X 1 , . . . , X n ) is said to be widely spread if for each l ≤ n we can write

X l = Xl + k =l c l,k X k ,
where the c l,k are some real coefficients, Xl is independent of the random variables X k for k = l, and

E ( Xl ) 2 ≥ κ 1 ,
for a fixed positive constant κ 1 .

Then, as in Lemma 3.3.3 of [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF] we can prove that the measures G * t 1 ,...,t n satisfy: Lemma 4.2 Assume u satisfies (H1) and (H2), fix (t 1 , ..., t n ) ∈ [0, 1] n , and set, for any t ∈ [0, 1] and σ ∈ Σ N ,

g t (σ) = 1 N 1/2 i≤N -1 g i σ i + t N 1/2 g N σ N ,
where {g i ; i ≤ N } is a sequence of independent standard Gaussian random variables, also independent from the randomness contained in G * t 1 ,...,tn . Then, there exists some constants κ, c 1 > 0 such that

G * t 1 ,...,tn (σ 1 , ..., σ n ) ∈ Σ n N ; g t 1 (σ 1 ), ..., g t n (σ n ) is not widely spread ≤ κ n exp (-c 1 N ) .
Let us recall that the constant q ∈ [0, 1] has been introduced in (5) and define, for l ≤ n, the quantity

θ l = q 1/2 W + (1 -q) 1/2 ξ l , ( 14
)
where W and {ξ l ; l ≥ 1} are independent standard Gaussian random variables. For a fixed t ∈ [0, 1], consider also the variables

g l v,t = v 1/2 S l M,t + (1 -v) 1/2 θ l , (15)
for any v ∈ [0, 1]. In the sequel, we will denote by E ξ the partial expectation with respect to the randomness contained in {ξ l ; l ≥ 1}. With Lemma 4.2 in mind, one can get the following proposition Proposition 4.3 Assume u satisfies (H1) and (H2), and fix (t 1 , ..., t n ) ∈ [0, 1] n . Then we have 1. The overlap R 1,2 self-averages into q:

E (R 1,2 -q) 2 (t 1 ,t 2 ) ≤ κ N . ( 16
)
2. Consider some integers n ≥ 1, s 1 , . . . , s n and r 1 , . . . , r n ∈ {0, 1} such that, setting s = j≤n s j and r = j≤n r j , we have s, r ≤ 6. Set

V (x 1 , . . . , x n ) = exp l≤n u(x l ) , U (x 1 , . . . , x n ) = ∂ s s 1 ,...,sn V (x 1 , . . . , x n ) P (x 1 , . . . , x n ) = j≤n (u ′ (x l )) r j .
Then, for f :

Σ n N → R such that |f | ≤ 1, we have, for any v ∈ [0, 1], f E ξ U g 1 v,t 1 , . . . , g n v,tn P S 1 M -1,t 1 , . . . , S n M -1,tn (t 1 ,...,tn) E ξ V g 1 v,t 1 , . . . , g n v,tn (t 1 ,...,tn) ≤ κ e 2nD |f | + e -c 2 N , ( 17
)
for some positve constants κ, c 2 .

Proof: This proof is a long and easy elaboration of the computations performed at [4, Sections 3.3 and 3.4], that we won't include here. However, notice that we have to assume here that the derivatives of u are (midely) bounded up to order 12. This is due to the presence of both U (g 1 v,t 1 , . . . , g n v,tn ) and P (S 1 M -1,t 1 , . . . , S n M -1,tn ) in the left hand side of (17). In order to treat those terms, we will separate them using Schwarz's inequality, that will bring out up to 12 derivatives of u.

An asymptotic expansion

In this section, we will develop the tools that we will need in order to get an expansion of the terms of the form (11), which is a key step in the proof of Theorem 1.1. In fact, once the averages f (t 1 ,...,tn) are defined, we will try, for a fixed (t 1 , . . . , t n ), to find a natural path that will allow us to expand any term of the form (11), asymptotically in N .

Recall first that S k,t is defined by (3), g v,t by (15), θ l by ( 14), b t and b * t by (12). Observe that, using the notation introdued at Section 4, we have

Y t = t N 1/2 k≤M g N,k u ′ (S k,0 ) exp u(S M,0 ) (0) exp u(S M,0 ) (0) + b t .
We can turn now to the definition of the path we will use in order to get our expansion: for v ∈ [0, 1], set

Ȳv = t N 1/2 k≤M -1 g N,k E ξ u ′ (S k,0 ) exp u(g v,0 ) (0) E ξ exp u(g v,0 ) (0) + b t . (18)
In particular it is easy to check that

Ȳ0 = t N 1/2 k≤M -1 g N,k u ′ (S k,0 ) (0) + b t , Ȳ1 = Y t - t N 1/2 g N,M u ′ (S M,0 ) exp u(S M,0 ) (0) exp u(S M,0 ) (0) . (19)
Using this auxiliary path, we will get the announced expansion as follows:

Proposition 5.1 Let u be a function satisfying (H1) and (H2). Consider an integer m, nonnegative integers s 1 , . . . , s m with s = l≤m s l ≤ 2. Set also

V (x 1 , . . . , x m ) = exp l≤m u(x l ), U (x 1 , . . . , x m ) = ∂ s V ∂x s 1 1 • • • ∂x sm m .
Let f be a function on Σ m N such that |f | ≤ 1 and Ψ a bounded function with bounded derivatives of any order. Then, if we define, for

t l ∈ {0, t}, l ≤ m, Q = E f U (S 1 M,t 1 , . . . , S m M,tm ) (t 1 ,...,tm) V (S 1 M,t 1 , . . . , S m M,tm ) (t 1 ,...,tm) Ψ(Y t ) ,
we have

Q = E f (t 1 ,...,tm) Ψ( Ȳ0 ) E U (θ 1 , . . . , θ m ) E ξ V (θ 1 , . . . , θ m ) + O N -1/2 .
In particular, we have

E f (t 1 ,...,tm) Ψ(Y t ) = E f (t 1 ,...,tm) Ψ( Ȳ0 ) + O N -1/2 .
The proof of this proposition will we based on a technical lemma that we will present now. First of all, let us label some basic but useful results that we will use further on: for l, l ′ ≥ 1, t ∈ [0, 1], we have

E (S l M,t ) 2 = 1 + t -1 N , (20)
E S l M,t S l ′ M,t ′ := ∆ t,t ′ l,l ′ = R l,l ′ + tt ′ -1 N σ l N σ l ′ N . (21)
Lemma 5.2 Let u be a function satisfying (H1) and (H2). Consider integers n, s 1 , . . . , s n with s = l≤n s l ≤ 6. Consider also the functions Û and V defined by

Û (x 1 , . . . , x n ) = ∂ s V ∂x s 1 1 • • • ∂x sn n , V (x 1 , . . . , x n ) = exp l≤n u(x l ).
Let f be a function on Σ n N such that |f | ≤ 1 and Φ a bounded function with bounded derivatives of any order. Then, for t l ∈ {0, t}, l ≤ n, v ∈ [0, 1], we have

Ξ := E    E ξ f P Û (g 1 v,t 1 , . . . , g n v,tn ) ∂ v g 1 v,t 1 (t 1 ,...,tn) E ξ V (g 1 v,t 1 , . . . , g n v,tn ) (t 1 ,...,tn) Φ( Ȳv )    = O N -1/2 , (22) where
, P = 1 or

P = u ′ (S i M -1,t l )u ′ (S j M -1,t l ′ ), with l, l ′ , i, j ∈ {1, . . . , n}, i = j.
Proof of Lemma 5.2: In the sequel, we will write Û and V instead of Û (g 1 v,t 1 , . . . , g n v,t n ) and V (g 1 v,t 1 , . . . , g n v,tn ), respectively. We will also denote u ′ (x)e u(x) by U 1 (x). By means of ( 14) and (15) we can write

Ξ = Ξ 1 -Ξ 2 -Ξ 3 , (23)
where

Ξ 1 = 1 2v 1/2 E E ξ f P Û S 1 M,t 1 (t 1 ,...,tn) E ξ V (t 1 ,...,tn) Φ( Ȳv ) , Ξ 2 = 1 2(1 -v) 1/2 E E ξ f P Û q 1/2 W (t 1 ,...,tn) E ξ V (t 1 ,...,tn) Φ( Ȳv ) , Ξ 3 = 1 2(1 -v) 1/2 E E ξ f P Û (1 -q) 1/2 ξ 1 (t 1 ,...,tn) E ξ V (t 1 ,...,tn)
Φ( Ȳv ) .

We will now give a sketch of our computations: according to (20) and ( 21), an integration by parts argument with respect to S 1 M,t 1 or W or ξ 1 , and a little algebra bring out some terms of the form

Ξ1 ≡ t 1 -1 2N E E ξ f P ∂ x 1 Û (t 1 ,...,tn) E ξ V (t 1 ,...,tn) Φ( Ȳv ) ,
which can be handled quite easily: by Schwarz's inequality, we have E ξ V V * (t 1 ,...,tn,t 1 ,...,tn) , where f * = f (σ n+1 , . . . , σ n+p ), Û * = Û (g n+1 v,t 1 , . . . , g 2n v,tn ), and P * , V * are defined in a similar manner. Hence, using inequality (17), we get Ξ1 = O N -1 .

E E ξ f P ∂ x 1 Û (t 1 ,...,tn) E ξ V (t 1 ,...,tn) Φ( Ȳv ) ≤ Φ ∞ E 1/2 E ξ f f * P P * ∂ x 1 Û ∂ x 1 Û * (t
The second kind of term we have to deal with is of the type

Ξ2 = 1 2 n l=2 E E ξ f P ∂ x l Û ∆ t 1 ,t l 1,l -q (t 1 ,...,tn) E ξ V (t 1 ,...,tn) Φ( Ȳv ) ,
and here again, (17) and Schwarz's inequality allow us to bound Ξ2 by sum of terms of the type

κ E 1/2 |f 2 | ∆ t 1 ,t 2 1,2 -q 2 (t 1 ,...,tn) + e -cN ,
for some integer n and some positive constants κ, c. Then, recalling definition (21), using the fact that |f | ≤ 1, and invoking inequality (16), we get Ξ2 = O(N -1/2 ), which ends the proof.

Let us go back now to the main point of this section.

Proof of Proposition 5.1: Define the function

τ (v) = E E ξ f U (g 1 v,t 1 , . . . , g m v,tm ) (t 1 ,...,tm) E ξ V (g 1 v,t 1 , . . . , g m v,tm ) (t 1 ,...,tm)
Ψ( Ȳv ) .

We have

τ (1) = E f U (S
Ψ( Ȳ1 ) , τ (0) = E f (t 1 ,...,tm) Ψ( Ȳ0 ) E U (θ 1 , . . . , θ m ) E ξ V (θ 1 , . . . , θ m ) .
Then, the proof of this proposition simply consists in checking that

|Q -τ (0)| ≤ κ N 1/2 .
Notice first that (Ψ(Y t ) -Ψ Ȳ1 ) .

|Q -τ (1)| = E f U (S
Thus, using Schwarz's inequality, the bound (17), the fact that Ψ has bounded derivatives, and the relation ( 19) between Ȳ1 and Y t , we get

|Q -τ (1)| ≤ κ N 1/2 .
So, the proof of our proposition will easily follow from the inequality:

1 0 τ ′ (v) ≤ C N 1/2 , (24)
that we will show now. In the sequel of the proof, we will write, for notational convenience, U = U (g 1 v,t 1 , . . . , g m v,tm ) and V = V (g 1 v,t 1 , . . . , g m v,tm ), and recall that we have also set U 1 (x) = u ′ (x)e u(x) . Then, we can write

τ ′ (v) = ζ 1 (v) -ζ 2 (v) + t N 1/2 k≤M -1 ζ k 3 (v) -ζ k 4 (v) ,
where

ζ 1 (v) = l≤m E E ξ f ∂ v g l v,t l ∂ x l U (t 1 ,...,tm) E ξ V (t 1 ,...,tm) Ψ( Ȳv ) , ζ 2 (v) = l≤m E E ξ f ∂ v g m+1 v,t l U U 1 (g m+1 v,t l ) (t 1 ,...,tm,t l ) E ξ V e u(g m+1 v,t l ) (t 1 ,...,tm,t l ) Ψ( Ȳv ) , and
ζ k 3 (v) = E g N,k E ξ f ∂ v g m+1 v,0 u ′ (S m+1 k,0 ) U U 1 (g m+1 v,0 ) (t 1 ,...,tm,0) E ξ V e u(g m+1 v,0 ) (t 1 ,...,t m, 0) Ψ ′ ( Ȳv ) , ζ k 4 (v) = E g N,k E ξ f ∂ v g m+2 v,0 u ′ (S m+1 k,0 )e u(g m+1 v,0 ) U U 1 (g m+2 v,0 ) (t 1 ,...,tm,0,0) E ξ V e u(g m+1 v,0 )+u(g m+2 v,0 ) (t 1 ,...,tm,0,0) Ψ ′ ( Ȳv ) .
The first two terms are of the type (22) in Lemma 5.2, and hence applying this lemma we get that

ζ 1 (v) -ζ 2 (v) = O N -1/2 .
To study the last two terms we have to integrate by parts again. For instance, by symmetry and integrating by parts with respect to g N,M -1 , we obtain that ζ k 3 (v) can be expressed as a function of terms of type (22). Then, applying Lemma 5.2 we can easily finish the proof of (24).

6 Proof of Proposition 3.1

In this section, we will see that the proof of Proposition 3.1 can easily be deduced from some nice cancellations, that we will obtain by an extensive use of Proposition 5.1.

Let us first introduce some new notation that we will use throughout this section:

V k (x 1 , . . . , x k ) = exp k i=1 u(x i ) , U i,j k (x 1 , . . . , x k ) = ∂ 2 x i ,x j V k (x 1 , . . . , x k ).
(25) Recall also that definition (5) implies

E U 1,2 2 (θ 1 , θ 2 ) E ξ (V 2 (θ 1 , θ 2 )) = E   u ′ (θ 1 )e u(θ 1 ) E ξ (e u(θ 1 ) ) 2   = r α .
Using the same arguments, we can also get that for k = 3, 4,

E U 1,2 k (θ 1 , . . . , θ k ) E ξ (V k (θ 1 , . . . , θ k )) = E u ′ (θ 1 )u ′ (θ 2 ) e u(θ 1 )+u(θ 2 ) E ξ (e u(θ 1 )+u(θ 2 ) ) = r α .
On the other hand, we will set (recall that w

= u ′′ + (u ′ ) 2 ) E U 1,1 1 (θ) E ξ (V 1 (θ)) = E w(θ 1 )e u(θ 1 )
E ξ (e u(θ 1 ) ) := δ, and observe that, for k = 2, 3,

E U 1,1 k (θ 1 , . . . , θ k ) E ξ (V k (θ 1 , . . . , θ k )) = E   w(θ 1 )e k i=1 u(θ i ) E ξ e k i=1 u(θ i )   = δ.
Using the notation introduced in Section 3 and applying Proposition 5.1, we are now able to bound all the terms appearing in the derivative of ϕ.

Upper bound on ϕ ′ 1 : We will show now how the terms appearing in Lemma 3.2 will give raise to some nice cancellations, by a direct application of (13) and Proposition 5.1. Indeed, we have for instance

A 1,1 (t) = α E σ 1 N σ 2 N U 1,1 2 (S 1 M,t , S 2 M,t ) (t,t) V 2 (S 1 M,t , S 2 M,t ) (t,t) = α E U 1,1 2 (θ 1 , θ 2 ) E ξ (V 2 (θ 1 , θ 2 )) E σ 1 N σ 2 N (t,t) + O N -1/2 = α δE σ 1 N σ 2 N (t,t) + O N -1/2 .
Along the same lines, it can also be shown that A 1,2 (t) yields a contribution of -α δE σ 1 N σ 2 N (t,t) + O N -1/2 , which can be simplified with the one of A 1,1 (t). It is easily checked that the other terms defining ϕ ′ 1 (t) cancel out in the same way, which gives ϕ

′ 1 (t) = O N -1/2 . Upper bound on ϕ ′ 2 : Let Ψ 1 (y) = 1 -tanh 2 (y) 1 -3 tanh 2 (y) . Ap- plying again Proposition 5.1, we get A 2,1 (t) = αE U 1,2 2 (θ 1 , θ 2 ) E ξ (V 2 (θ 1 , θ 2 )) E Ψ 1 Ȳ0 + O N -1/2 = rE Ψ 1 Ȳ0 + O N -1/2 , A 2,2 (t) = -rE Ψ 1 Ȳ0 + O N -1/2 . Hence, ϕ ′ 2 (t) = O N -1/2 . Upper bound on ϕ ′ 3 : Set Ψ 2 (y) = tanh 2 (y)
Here again,we will expand the terms appearing in Lemma 3.4, thanks to Porposition 5.1, and look for some cancellations. An example of computation is

A 3,1,1,1 (t) = α 2 E σ N U 1,1 1 (S M,t ) (t) V 1 (S M,t ) (t) Ψ 2 (Y t ) = α 2 E U 1,1 1 (θ) E ξ (V 1 (θ)) E σ N (t) Ψ 2 Ȳ0 + O N -1/2 = α 2 δE σ N (t) Ψ 2 Ȳ0 + O N -1/2 ,
and following these steps, we obtain

4 i=1 A 3,1,1,i (t) = α 2 δE σ N (t) Ψ 2 Ȳ0 - r 2 E σ N (t) Ψ 2 Ȳ0 + O N -1/2 .
(26) The same kind of expansions also yield (28)

Then, from (26), ( 27) and (28) we get that ϕ ′ 3 (t) = O N -1/2 , which ends the proof of Proposition 3.1.

7 Proof of Theorem 1.1 Once Proposition 3.1 is proved, our main theorem will be shown along some more or less classical lines. However, we still have to prove here a technical lemma: M }. Furthermore, following again the steps of [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF]Lemma 2.4.14], it can be shown that E σ Ntanh r 1/2 z 2 ≤ κ N 1/2 . (29) Indeed, using Proposition 3.1, the Lipschitz property of the function x → tanh(x) and some easy algebraic manipulations, we obtain for some positive constant κ,

E σ N -tanh r 1/2 z 2 ≤ κE k≤M 1 N 1/2 - r 1/2 s g N,k u ′ (S k,0 ) - 2 + O N -1/2 = κE 1 N 1/2 - r 1/2 s 2 k≤M u ′ (S k,0 ) 2 -+ O N -1/2 ≤ κ r E s 2 N -r 2 + O N -1/2 .
Then notice that, using again the symmetry among the S k,t , we can write

E s 2 N -r 2 = E s 2 N -r α u ′ (S 1 M,0 )u ′ (S 2 M,0 ) --r = T 1 + T 2 + T 3 + r 2 ,
where

T 1 := α M -1 N E u ′ (S 1 M,0 )u ′ (S 2 M,0 ) -u ′ (S 1 M -1,0 )u ′ (S 2 M -1,0 ) -, T 2 := α N E u ′ (S 1 M,0 )u ′ (S 2 M,0 ) -u ′ (S 1 M,0 )u ′ (S 2 M,0 ) -, T 3 := -2αrE u ′ (S 1 M,0 )u ′ (S 2 M,0 ) -.
Clearly, T 2 = O N -1/2 . On the other hand, using Proposition 5.1 we have

T 3 = -2αrE U 1,2 2 (θ 1 , θ 2 ) E ξ (V 2 (θ 1 , θ 2 )) = -2r 2 .
So, in order to finish the proof of (29) it is enough to check that T 1 = r 2 + O N -1/2 . Althougth this result cannot be obtained as a direct consequence of our Proposition 5.1, it can be proved following exactly the same ideas and steps as in the proofs of this Proposition and Lemma 5.2. For the sake of simplicity we will not repeat here such proof.

Step 2: Along the same lines as in [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF]Theorem 2.4.12], the proof can be done by induction over m on the system of N -1 spins. Our hypothesis of induction yields that

i≤m E σ i --tanh r 1/2 -z i 2 ≤ κ N 1/2 , ( 30
)
where {z i ; i ≤ m} is a collection of independent standard Gaussian random variables depending only on the variables {g i,j } i≤N -1 , and r -is the solution to (5) when α is replaced by α -= M N -1 . Notice that it is easily seen that

|r -r -| ≤ κ N 1/2 . ( 31
)
Then the random variable z m+1 = z defined in the first step of the proof, is independent of the random variables {z i , i ≤ m}. The desired conclusion is then obtained by (30), (31) and Lemma 7.1.

Theorem 1 . 1

 11 Assume u satisfies the assumptions (H1) There exist a constant D > 0 and a large enough number L such that |u(x)| ≤ D for all x and Le LD ≤ 1. (H2) There exists a positive constant L * and a small enough constant c 3 such that, for any l ≤ 12, |u (l) | ≤ L * e L * D e c 3 N .

5 i=1A 3

 53 ,1,2,i (t) = α 2 δE σ N (t) Ψ 2 (Y 0 ) -r 2 E σ N (t) Ψ 2 (Y 0 ) + O N -1/2 ,(27)which cancels out with the contribution of 4 i=1 A 3,1,1,i (t). Finally, a similar kind of argument shows thatA 3,2,1 (t) -A 3,2,2 (t) = O N -1/2 .

  1 ,...,tn,t 1 ,...,tn)

  1 M,t 1 , . . . , S m M,tm ) (t 1 ,...,tm) V (S 1 M,t 1 , . . . , S m M,tm ) (t 1 ,...,tm)

 [START_REF] Bardina | The p-spins interaction model with external field[END_REF]Partially supported by DGES grant BFM2000-0607. 2 Partially done while the author was visiting the Université Paris 13 with a CIRIT grant. Partially supported by DGES grant BFM2000-0607 . 3 Partially written with a support of a fellowship grant at the Centre de Recerca Matemàtica (Bellaterra, Barcelona). The author would like to thank this institution for its warm hospitality.

Lemma 7.1 Let • -be the Gibbs average introduced at Proposition 3.1. Then, there exists a constant κ > 0 such that

Proof: For the proof of this lemma, we will need to consider again a slight variation of the Gibbs measures defined up to now: recall that b t has been introduced by equation ( 12). Then, for n ≥ 1, a test function

Notice that f • (t,...,t) = f t , but in general, f • (0) and f -do not coincide, because of the presence of the term b t in the first one. However, observe also that if

For t ∈ [0, 1], set now

Obviously, ψ(0) = 0. Then in order to finish our proof it suffices to show that ψ ′ 1 (t) and ψ ′ 2 (t) are of order N -1/2 . Indeed, ψ ′ 1 (t) can be computed through Proposition 2.2. Then, along the same lines as in [4, Chapter 3], one can show that ψ ′ 1 (t) is of order N -1/2 for any t ∈ [0, 1]. The term ψ ′ 2 (t) can be treated by the same kind of argument: in fact, it can be computed like in Proposition 2.2, except that we will differentiate only with respect to the first replica. We obtain

with

where

and

.

Then, ψ ′ 2 (t) can be bounded by κN -1/2 using again the same kind of arguments as in [START_REF] Talagrand | Spin Glasses: a Challenge for Mathematicians[END_REF]Chapter 3].

Let us turn now to the main aim of this section.

Proof of Theorem 1.1: This proof is now a slight variation of [4, Section 2.4], and we sketch it here for sake of completeness.

Step 1:

Since α > 0, r is bounded from below by a constant r 0 > 0 (see relation (5)). Set Then it can be shown, as in [4, Section 2.4], that z is a standard Gaussian random variable independent of the random variables {g i,k ; i ≤ N -1, k ≤


Thierno M Sow

G Cardano

Riemann and Mobius dust the Kakeya Problem

Keywords:

published or not. The documents may come

Riemann and Mobius dust the Kakeya Problem

THE KAKEYA NEEDLE PROBLEM

One hundred years ago, Soichi Kakeya asked whether there is a minimum area of a region D in the plane in which a needle of unit length can be turned through 360 degrees.

WHAT IS NEW?

All sciences lead to Riemann and the most elegant proof of the Riemann Hypothesis can be expressed as follows:

Theorem 1. There are infinitely many nontrivial zeros on the critical line and all these zeros have real part 1

/2.
Proof.

∞ k=1 1 2 + 1 p s k -1 = ζ(s), ( 1
)
where s ∈ C, and p ∈ P.

Now, the question is how to connect the Riemann hypothesis theorem to the Kakeya needle problem? We start the sketch of the proof by observing that we can make the following assumption. Let us consider n as the area of the given figure. We may also assume p is contained in the unit figure such that the tip of the needle corresponds to the smallest prime factor of n. It follows:

Theorem 2. The minimum area of a region D in the plane in which a needle of unit length can be turned through 360 degrees is zero.

Proof.

∞ k=1 p s k + 1 2 (p s k -1) = ζ(s) = ∞ n=1 n 2 -p 4 p 2 + n 2 -q 4 q 2 , ( 2
)
where n ≡ 0 (mod p) , n ≡ 0 (mod q) and p k denotes the k n-th prime.

It is specially true for all cases for which the Kakeya set corresponds to a prime-dimensional unit. Furthermore, for the general case, to obtain the precise value of the RHS of the Riemann zeta function we assume, a mere glance (the previous article) will show that, under suitable conditions

p s + 1 2 (p s -1) = 1 2 + 1 p s -1 . ( 3
)
3 WHAT IS ALREADY?

THE HORUS EYE THEOREM

We assume: by the properties of the light-sensitive cells, every image or object is inverted on our retina. This input-output relation corresponds to a quantum informations system which has deep connections with the field of mathematics of what has come to be known as geometric measure theory. According to some specialists in Optics like D. L. Anderson: "the size of the image projected on the retina is determined by the size of the object and the distance of the object from the perceiver...two objects of the same height will not project the same size image on the retina if one of them is further away than the other." Which means that the smallest Kakeya set depends more on the distance and remains zero, at least the set corresponds to the smallest length of a quantum light-sensitive cell. In times past, the Egyptians knew that the truly structure of the Horus eye corresponds to the set 1 n s .

THE KAKEYA-LISTING-MOBIUS THEOREM

Theorem 5. The minimum area of a region D in any n-dimensional space in which a needle of unit length can be turned through 360 degrees is zero. Proof.

µ (n) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 i fn = 1 (-1) k if n is the product of k n-th distinct primes ζ (s)
otherwise.

(

) 4
The results confirm the proof by the Riemann Hypothesis theorem since for all cases in which n is prime µ

(n) = ζ (s) = 0.
Ultimately, we can observe if a needle were to slip along the length of the "fancy clock" below, it would turn through 360 degrees having just traversed a short part-betwixt of the entire length of the ring without crossing an edge. Therefore, it is needless to wind up the Kakeya Clock. Namely, we improved successfully the results to any n-dimensional space and we doubled the number of 360 degrees tour inside, which makes the proof more elegant.

In troth, the old paths have joined the new.

MISCELLANEOUS

THE HOKUSAI THEOREM

In this section, would we introduce some masterpieces sank into oblivion which are truly connected to the Kakeya needle problem and the Riemann Hypothesis. First of all, let us recall the fact that the Japan has always had great mathematicians as well as great artists like the Haiku master Matsuo Basho (1644 -94), Harunobu (1725 -70) and his painting "Two Under an Umbrella", Ando Hiroshige (1797 -1858), and the famous master Hokusai (1760 -1849), among others. Indeed, two weeks ago, we were writing an article on, one of our favourite Japanese authors, Yoshikichi Furui when it has been brought to our attention that the Kakeya problem fulfilled perfectly the secret meanings of another Hokusai masterpiece "The Barrel-maker of Fujimihara" (c.1830).

At first, for many art historians, it was thought that the tale of this scene is about the life of a slender whacked man slanting the Nemesis's wheel of fortuna in opposite to the lyrical and charming view of the landscape behind him. Maybe! But, for the mathematicians who have a deep knowledge of the Art, let us say, there are wheels within wheels. Indeed, Mathematics play a central role in the Japanese Arts and Culture as well as in the Architecture. One can see an example of beautifully balanced asymetry in the plan of the Asukadera temple or the plan of Fujiwarakyo (694 -710), an Imperial City. Likewise, in Geometry the Japanese art of paper folding is well-known as Origami and the Japanese taste for Mathematics and the prime numbers can be seen in the seventeen-syllable Haiku or the thirty-one-syllable humorous form called Kyoka. An excellent example of their sophistication is the following thirty-one-syllable love poem by one Princess Kagami at the court of Emperor Tenji (r.662 -72):

Like the hidden stream trickling beneath the trees down the mountainside so does my love increase -more than yours, my Lord.

Ian Hideo Levy, Ten Thousand Leaves, Princeton.1981.vol.2.92.

To investigate further one can find out at the Library of Congress the History of Japanese Art by P. E. Mason (1993) and revised by D. Dinwiddie (2003).

Talking about Mathematics in the art of the master Hokusai, we was intrigued and aroused by the similar canvas and ceilings, like a keymaster in Cryptography, we both used. Indeed, in the painting below "The Well-Guided and the Arch of the Primes", we have tried to represent the modular frequency distribution of the primes (the wheels) along to the Riemann zeta function (the lighthouse) and the light comes out when the sieve revealed that the perspective of the wave inside the wheel is exactly the same than the master Hokusai used to depict the Mount Fuji and the character inside the disk of the barrel. Now, what does it means mathematically?

We do know that the Mount Fuji represents the immortality (infinity) and as a child Hokusai was fascinated by the mirrors, so, if Hokushinsai means the sparkle of star's room then it follows that the Hokusai problem can be expressed as follows: the infinite volume of any smooth surface whithout any thickness is paintable with a finite amount (coats) of paint because ζ (s) → 0 when n → +∞. Exactly what Hokusai did in his whole life, painting "infinitely" many times the Mount Fuji with finitely many numbers of coats of paint. Likewise, in Mathematical modeling, there exists deep connections between the Gabriel's horn paradox and the Riemann zeta function (more, see M. Lynch, and M. Henrikson -1965-).

For instance one can see an hint about the Euler product in the volume of the Gabriel's horn

V = π ˆp 1 1 n 2 dn = π 1 - 1 p .
the Euler seal [START_REF] Sow | [END_REF] So, Hokusai knew that the character and the Mount Fuji can be turned through 360 degrees inside the barrel as the Kakeya needle will do if the area of the Listing-Mobius ring equals to zero.

WHAT IS NEXT?

In the next article we will study the score of the music of the primes before to release the Next Prime Theorem and we will see the consequences of the Riemann Hypothesis theorem on the General Theory of Relativity and how Bernhard Riemann changes definitely the interplay between the new Quantum Number Theory and Modern Physics.


Thierno M Sow
email: rsat@one-zero.eu

RSA-T. The Oval Pylon

Keywords:

The present communication is a series of articles called SO PRIME. The goal of which is to provide a clear answer whether factoring is in polynomial time O (log(n)), in the light of a complete statement on the Riemann Hypothesis and it's applications beyond the RSA cryptography and the twin prime theorem. So, the question which remains is : where this Bernhard Riemann astonishing and great idea springs from?

INTRODUCTION

Small keys open large doors. We learn that from the celebrated formula of the master Albert Einstein E = mc 2 .

(1)

As the consequence of the article "Stealth Elliptic Curves and the Quantum Fields" we introduce a new factorization technique for the RSA set on the form

Q = P.N -1 -1 . ( 2
)
According to Gauss: "the problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic...Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated".

Furthermore, we propose a new and unconditional primality test protocol for any given integer. The goal of this article is to provide a clear answer whether factoring lies in polynomial time. So far, we assume, even if the Riemann Hypothesis is the answer, the consequence on the P versus NP problem remains, as a real challenge to design powerful discrete algorithms for the next generation of the computational intelligence. In the next article, we will release an innovative security protocol which fix the vulnerability of the RSA cryptosystem.

THE RIEMANN COMPLEX NUMBER

It's about the triumph of the order over the chaos. Indeed, the Riemann Hypothesis is the oval pylon of Mathematics which has shaped the number theory and provided a powerful understanding of the Sciences beyond the quantum fields.

Theorem 1.

For every odd number n there exists a complex number s such that

± 1 2 ± ˆ1 2 +it -1 2 +it i 2s -1 n 2 -1 ds = ζ(s), (3)
where

s = n 2 + 1 4 .
We can observe the perfect holomorphic function with a convergent series represented by the green line over the x-axis and the red polar which describes an elliptic curve around the [START_REF] Riemann | On the Number of Primes Less Than a Given Magnitude[END_REF] 2 coordinate. Regarding the parametric plot represented in purple, we can observe a chaos quantum particles movement surrounded between [-1, 1] as n → +∞.

THE PRIME NUMBER THEOREM

DEFINITION

We shall write (p) for the Riemann counting function for the number of primes less than p. For instance, the equivalent notation introduced by Gauss is π(N ).

THE RIEMANN COUNTING FUNCTION

In his celebrated article "On the number of Primes less than a given quantity", Bernhard Riemann recall the importance of his "investigation into the accumulation of the prime numbers; a topic which perhaps seems not wholly unworthy of such a communication, given the interest which Gauss and Dirichlet have themselves shown in it over a lengthy period". Follow up to Euler

p∈P 1 1 - 1 p s = n=1 1 n s . ( 4
)
Theorem 2. For every prime p there exists a complex number s on the form

s = p 2 + 1 4 = 1 2 + it, (5)
and

(p) = p 2 -1 4? ± β , ( 6
)
where β denotes the key remainder of the Riemann zeta score.

In fact, since the number 2 is not odd, it appears that for some special cases the even prime is not embedded in the Riemann counting function. So far, we recall the uncertainty relation in Quantum Physics which has the same similarity principle. For instance which is precisely the number of primes less than a given quantity. We should note that p must be absolutely prime. A smart algorithm will be released to determine how far β 1 when (p) → +∞.

VISUALIZING THE RIEMANN ZETA FUNCTION

It's very important to understand all the aspects of the Riemann zeta function, first. For instance, the Riemann theorem introduces a new virtualization concept in Mathematics which is called the high quality virtual (HQV). Indeed, if you integrate 1/x 2 in the complex plane, you will obtain an equivalent Riemann zeta function as well as if you reduce the expressions of the Riemann integral such that, with s = 2 and for any function

f (n) ˆ+∞ -∞ log(n) + 2 s log √ n -2 -1 f (n)dn = ζ(s) ˆ+∞ -∞ f (n)dn. ( 7
)
That is what we call the high quality virtual of the Riemann Hypothesis. We will observe insight into the next article how this property can be useful to build a powerful cryptographic protocol. Although, a conference key point is more suitable and appropriate to complete the statement on the HQV. So far, let us stay focus on the "real" Riemann zeta function.

Theorem 3. For every odd number n there exists finitely many complex numbers s such that

s = n 2 + α 4 4α 2 , ( 8
)
where α denotes every factor of n. This is the key which will be used to compute the Real Riemann zeta function.

THE 3D PLOT

According to the number of stairs, it appears clearly that there are 25 primes less than 100. Finally, the critical line of the Riemann zeta function is able to design both strips which separate primes from odd numbers.

THE PARAMETRIC PLOT

So, we have

x(t) = cos((t 2 + t 4 )/(4t 2 ))/2 ( 9 ) 11)

y(t) = t sin((t 2 + t 4 )/(4t 2 ))/2 (10) and x(t) = t cos((t 2 + t 4 )/(4t 2 ))/2 (
y(t) = sin((t 2 + t 4 )/(4t 2 ))/2 (12)
Side note: you need an additional line code to integrate the Riemann zeta function as illustrated in blue. You can also refine the interval from {t, -4π, 4π} to {t, -2π, 2π}.

THE POLAR

If we integrate the parametric formulas above in the polar of the complex plane, we can observe the output of the following figures which is the perfect illustration of an egg timer, an apple -pauca sed matura-, a butterfly and, don't ask me why, but above all, charming heart and love symbols.

THE SPECTRUM

In this plot, the red line denotes the real part and the green one the imaginary part of the Riemann zeta function which describes an egg timer around the pole when the density become higher.

THE RSA VULNERABILITY

Theorem 4. For every odd number on the form n = pq, where p and q are primes, we have

2n log(π) 2 log(π α ) + 2 log(π) log(π α ) ≡ 5 (mod 10), ( 13
)
where α ∈ P denotes the primes lower than n.

Proof. p and q are the only primes which generate solvable integers in ≡ 5 (mod 10).

So, we can consider the relation above as a suitable and strong primality test which is able to generate p and q instantaneously and provide a strong certificate whether a given integer is prime, since the relation fails only if n is prime. Although, that is an explicit proof which testify the very serious vulnerability of the RSA cryptographic technique.

THE COMMON FACTORS OF ABC

α a x c z + b y a x = β, ( 16
)
where β is a solvable integer if and only if α a, b, and c have a common factor. This completes the Andrew Beal theorem.

We can observe in the graph above the illustration of the deep connections between a, b and c. Finally, the question is now open to investigate how this relation is helpful to map the connections between different actors inside a given network and how it can be used by the counter-terrorism task forces. In modern Physics, does it implies the multi-verse theory? We don't know.

THE GENERALIZATION OF THE RH

To interpret random virtual zeta functions, we can observe the phenomenon described as the Pareidolia in the visual arts, for instance the paintings of the Mexican Octavio Ocampo and well earlier defined by Leonardo da Vinci as "an infinite number of things which you can then reduce into separate and well conceived forms." Then, ceteris paribus, to construct a cornerstone on the Riemann edifice facade, we need to recall the Ray Solomonoff general theory of inductive inference, or the principle of what has come to be known as the Occam's Razor, to distinguish between the virtual zeta functions either by separating unusual assumptions or cutting apart some similarities. So far, it turns out that the Euler product seems to be the most important breakthrough in the generalization of the Riemann Hypothesis. Although, let us try to build the final stone of the beautiful and remarkable Euler edifice.

THE EULER THEOREM

Theorem 6. For every prime p there are two complex numbers s = 1 2 and s = χ + such that

p∈P 1 s 1 - 1 p s = ζ(s). ( 17
)
The comparative odd and even sequences can be formulated as follows:

5 2 1 + e iπn = ± 5 2 (-1) n+1 , 5 2 (1 + (-1) n ) odd=0 . ( 18
)
With a discrete algorithm we can generate the following partition:

n, p 2 3 5 7 χ 6 3 2 2 3 3
0, 828427125 0, 232050808 0, 618033989 0, 2152504368 Nevertheless, even if the Euler product is a strong statement, it appears not very helpful as well for the prime number theorem as formulated by Bernhard Riemann himself.

THE CONVERGENT SERIES

In fact, we need a strong mathematical formula which is able to generate instantaneously the distribution of the zeros of the Riemann zeta function without any complex computational theory. So, "to hold them all", we need to define a new value δ = 34 ≈ π 5 9 ≈ π 3 + 3 which is called the Riemann ring. We assume that n δ is the roof for all set on the form 1 n . Finally, the Physicists have found that δ has the properties of a "magic number". Theorem 8. For every prime there exists a complex number s =

p 2 + 1 4 such that p∈P 1 s 1 - 1 p δ = ζ(s), ( 19
)
where

δ = π 5 9 .
From the previous theorem, we can now derive in a certain extend the Riemann Hypothesis with the perfect convergent series as follows:

Theorem 9. For every prime there exists a complex number s = 1 2 such that

p∈P 1 s 4 - 1 p δ = ζ(s), ( 20
)
where δ = π 5 9 .

THE PARAMETRIC SURFACES

With the following formulas we can generate the parametric surfaces of the Riemann Hypothesis.

x(u, v) = u • cos(5(1 + (-1) v ))/2 (21) y(u, v) = v • sin(5(1 + (-1) u ))/2 (22) z(u, v) = c • sin(5(1 + (-1) u ))/2 cos(5(1 + (-1) v ))/2 (23)
From Theorem 8 we have

x(u, v) = u • cos(1/((1/4(v 2 + 1))(1 -1/v 34 ))) (24) y(u, v) = v • sin(1/((1/4(u 2 + 1))(1 -1/u 34 ))) (25) z(u, v) = c • sin(1/((1/4(u 2 + 1))(1 -1/u 34 ))) . . . cos(1/((1/4(v 2 + 1))(1 -1/v 34 ))) (26)

THE RIEMANN THEOREM

At this point, we are now able to bring together the different materials which will be used to design"The oval pylon theorem" for the generalization of the Riemann Hypothesis. Moreover, we can observe some figures below which represent the Riemann throne and the central force particles (ODE) which is very similar to the Rutherford-Bohr Model.

Theorem 10. ∀n and prime p there are two complex numbers s = p 2 + 1 4 , π 5 9 and s = ±2 such that

∞ k=1 1 s 1 - 1 p s k = ζ(s) = s - ∞ n=1 1 s n -π 5 9 ± 1
.

where δ = π 5 9 = s (27) Proof. lim p→∞ 1 2 -2p 1 4 (-1-p 2 ) = 1 2 . ( 28
)

THE RIEMANN INTEGRAL

Follow up the Theorem 1 from the section 2, we have:

Theorem 11. For every n > 1 there exists a complex number s = n 2 + 1 4 such that the integral relation holds true if

ˆ1 2 +it -1 2 +it 2s n 2 -1 ds = ζ(s), ( 29
)
which can be refined and generalized such that for any function f (n) with n = 0 and for any complex number s = 0

ˆ+∞ -∞ n s + 1 2 (n s -1) f (n)dn = ζ(s) ˆ+∞ -∞ f (n)dn. ( 30
)
Proof.

lim n→±∞ n s + 1 2 (n s -1) = 1 2 . ( 31
)
Now we can validate this solution efficiently with a discrete algorithm which gives a fuller arithmetical overview of this torus which represents the threshold of an hexadecagon dome shape.

In the other hand we can observe how the magnetic quantum fields of the charged particles describe a perfect zeta figure as well as a truly mysterious fetus symbol.

THE FREQUENCY DISTRIBUTION OF PRIMES

The most important breakthrough for the Riemann Hypothesis lies in the frequency distribution of primes along to the brim of the Riemann zeta function. We can observe thoroughly that the zeros associated to the prime numbers follow a bolt of lightning path as primes describe an harmonic and symmetric complex phenomenon around the pole. We shown that the Theorem 11 (30) is a key point for the generalization of the Riemann theorem. Then, it follows:

Theorem 12. For every prime p and for any function ζ(s) there exists > 0 such that

ζ(s) ≡s ( mod prime) • n n + 1 = p + . ( 32
)
This completes efficiently the Riemann theorem and confirms all the previous results as we can observe it in the graph below.

THE UNIVERSAL PRIMES SPOT

Now we can write the unique formula which is able to generate all odd primes.

Theorem 13. For every odd prime they are α and β ∈ N 0 such that

∆ α 0 ± 1 ± ∆ β 1 = prime, ( 33
)
where ∆ 0 and ∆ 1 are not necessarily distinct primes.

BEYOND THE RIEMANN THEOREM

It turns out in number theory that there exists a popular myth in the existence of "one formula to rule them all ", in particular, we have no opinion to formulate about this issue. Fortunately, the Riemann theorem implies major conjectures in the related

THE FERMAT SUMS ON TWO SQUARES

From

p i = k , ( 35
)
where k denotes the Riemann constant 1 2 . Proof.

∞ k=1 P k pi>2 with pi-pj ≥2 and i =j p i = 0. ( 36
)
For instance, with a discrete algorithm we can obtain the k , dividing the 12265 th twin prime pair 1600789 by the sum of 528 consecutive twin primes (Λ•ctp) between {3, 14447}. Here we have the first perfect sequences of what we call the Riemann twin primes or the "Royal Primes".

13 3 + • • • + 11 4•ctp 139 3 + • • • + 59 12•ctp 1600789 3 + • • • + 14447 528•ctp • • • = 0. ( 37
)
Although, we can observe the shape of the twin primes charged particles orbits around the magnetic fields.

THE ROYAL PRIMES

The Royal primes or the Riemann twin primes are at a very interesting point which can be arranged into a perfect group. The Royal primes are specific case of the general case from the Theorem 16 in which the twin primes appear with a precise value will contribute significantly to the frequency distribution of primes. Each set of concentric royal twin primes corresponds to the following partition:

Although, there are some specific conditions for which the Royal primes hold true. Indeed, Λ • ctp ≡ 0 (mod 2), p = p 2 (χ + )

3 2 + p 2 k
, where ? p = . Finally, we assume: the numerator p is always the second twin of the given prime pair such that p-2 is prime which proves clearly that the smallest gap between consecutive primes corresponds to 2 such that

lim inf n→∞ (p n+1 -p n ) = 2. ( 38
)
where p n denotes the n-th prime.

For the other cases for which the numerator p is an isolated prime we can observe an high density of primes in the same interval which satisfy the Theorem 16. For instance, we listed 27 isolated primes for which the relation holds true between {3, 33619} which corresponds to 1.5312564 × 10 7 . Here we get a glimpse of some partitions:

FACTORING AND THE PvNP

Let n be odd and let s be a complex number on the form 1 2 + it such that

p z + q z 2 z = 1 2 + it, ( 42
)
where p and q are the prime factors of n.

Theorem 19. For every odd number on the form n = pq, where p and qare primes, there exists a complex number s such that ˆn prime ( -) f (α)dα = ζ(s) ˆn prime f (α)dα, (43) with = in 2 (2α p ) -2 + 2n -1 .α p 2 (44) and = in 2 (2α q ) -2 + 2n -1 .α q 2 , (45

)
where α ∈ P denotes the primes lower than n, i = ±1.

Proof. p and q have the same coordinate on the critical line such that

= i n 2 + α 4 4α 2 = = s. ( 46
)
From the Riemann theorem and the relation above we can build another powerful primality test protocol since every odd prime has a single and unique complex number on the critical line.

  Let us reformulate the RSA set on the form n = pq by the light of the Diophantine equation such that p + m p = n (14) and q + m q = n (15) such that a = p, q , b = m and c = n. Theorem 5. For every Diophantine equation a x + b y = c z there exists α and β positive integers such that

  From the Riemann theorem and the twin prime theorem we can reformulate a strong Goldbach theorem such that Theorem 18. Every even integer G can be expressed as the sum of two primes if and only ifG = Ω ≡0 ( mod 4) p pi>2 with pi-pj ≥2and i =j

  the Riemann theorem, we can formulate a new definition of a prime number.Theorem 14. An odd prime is an odd number which has a unique complex number on the critical line. Every odd prime p can be expressed as the sum of two squares if and only if, there exists for every odd prime p a unique complex number s on the critical line on the form

	Follow up to this square relation p 2 +	1 2 +p √ log π log ( π)	= (p + 1) 2 , it follows:
	Theorem 15. s =	p 2 + 1 2 2 .	(34)
	5.5.2 THE TWIN PRIME THEOREM
	Theorem 16. There are infinitely many odd twin primes P satisfying
	i→∞ lim inf	P
	pi>2 with pi-pj ≥2
	and i =j

THE ALGORITHM

The present project is called The Alice and Barack Lust EXchange (TABLEX). Any new technique isn't necessary to design the suitable recursive algorithm that runs in O (log(n)).

Finally, if the factorization is NP Complete (coNP) does it implies that

"Wir mussen wissen. Wir werden wissen".

Riemann Hypothesis Theorem The Cornerstone

The zeta function has infinitely many nontrivial zeros and they all have real part 1 2 .

where s ∈ C and p ∈ P.

We will see in the next articles the Next Prime theorem and how Riemann changes definitely the interplay between Mathematics and Modern Physics.