Francesca Guerriero

Luigi Di

Puglia Pugliese

Nathalie Mitton
email: nathalie.mitton@inria.fr

Luigi Di Puglia Pugliese

Efficient wireless sensor deployment at minimum cost

Keywords: sensor deployment, routing, wireless, cost efficiency

Introduction

These last years have witnessed the development of several wireless sensors deployment strategies, to be applied in many different fields. A large part of them focus on environmental monitoring and surveillance operations [START_REF] Farsi1 | Deployment Techniques in Wireless Sensor Networks, Coverage and Connectivity: A Survey[END_REF]. When deploying such a sensor network, the following main challenges arise. How many sensors need to be deployed to ensure the full area coverage at a minimum cost ? Where to deploy them to keep not only the proper monitoring of the area, but also to ensure the existence of a wireless multi-hop path between any pair of nodes in the network ? Then, once sensors deployed, how good is the quality of the communicating network yet constructed beyond the plain connectivity?

Covering problems have been investigated for several years by different research communities: robotics, networking and optimization groups. Most of robotics researchers mainly couple coverage with path planning and friction efforts; networking researchers mainly consider energy costs and connectivity while optimization researchers have generally neglected the connectivity issues. To the best of our knowledge, only very few approaches [START_REF] Rebai | Khoukhi Sensor Deployment Optimization Methods to Achieve Both Coverage and Connectivity in Wireless Sensor Networks Computers and Operation Research[END_REF][START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF][START_REF] Elloumi | Rovedakis Optimization of wireless sensor networks deployment with coverage and connectivity constraints[END_REF] from the literature consider jointly connectivity and quality of monitoring coverage. These approaches propose mixed integer linear programs to formulate the problem of deploying sensors ensuring coverage and connectivity. Efficient solutions are obtained, achieving very interesting and performing results, but they do not take into account the deployment cost. In addition, the solutions obtained by using the existing approaches are considered satisfactory if they achieve connectivity, whereas none of them evaluates the quality of the communication network obtained.

In this paper, we assume to have a limited number of sensors of different typologies, each characterized by a sensing range, a communication range and an operational cost. We add communication constraints in order to force sensors to be connected (permanently or intermittently) and form a communication network, allowing them to dynamically self-adapt to any environment aspect. We also consider the energy consumption and the travelled distance.

The main contributions of our work are reported in what follows:

-We address a generalized sensors deployment problem with coverage and connectivity constraints, considering different typologies of sensors. -We formulate the problem as a mixed integer program and we provide a deep evaluation of the solutions obtained, analyzing the efficiency and the effectiveness with respect to the deployment cost. -We assess the obtained sensor deployment, in terms of quality of communication topology.

The remainder of this paper is organized as follows. Section 2 summarizes the state of the art. In Section 3 we formally describe the addressed problem along with the mathematical formulation. Section 4 provides the numerical results collected on meaningful instances. Section 5 concludes the paper.

State of the art

The coverage and monitoring of a point or an area of interest have been studied by different research communities. i.e., networking, robotics, and optimization.

The robotics community refers to the techniques set up to cover an area as "formation control". They generally do not consider connectivity issues [START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF][START_REF] Wang | Moore Formation control in mobile actuator/sensor networks SPIE[END_REF].

In the optimization field, some works focus on multi-objective mathematical models, to determine the best placement of mobile nodes for different tasks; we cite, for example, the papers of Guerriero et al. [START_REF] Guerriero | Modelling and Solving Optimal Placement problems in Wireless Sensor Networks[END_REF] and Lambrou et al. [START_REF] Lambrou | Collaborative area monitoring using wireless sensor networks with stationary and mobile nodes[END_REF]. Guerriero et al. [START_REF] Guerriero | Modelling and Solving Optimal Placement problems in Wireless Sensor Networks[END_REF] focus on the optimal sensor displacement to allow the network to achieve high performance in terms of energy consumption and travelled distance.

In Lambrou et al. [START_REF] Lambrou | Collaborative area monitoring using wireless sensor networks with stationary and mobile nodes[END_REF], the authors provide a collaborative architecture between mobile and static sensors. The coverage objective described in [START_REF] Lambrou | Collaborative area monitoring using wireless sensor networks with stationary and mobile nodes[END_REF] is the same as the one taken into account in the present work. However, they do not consider the communication constraints. Yu et al. [17] applied Genetic Particule Swarm Optimization (GA-PSO) techniques to solve the optimal sensor deployment problem. However, the authors do not pursue the same goal as in this paper and do not address communication aspects.

Several challenging issues arise when quality of coverage and connectivity issues are considered simultaneously [START_REF] Khoufi | Survey of Deployment Algorithms in Wireless Sensor Networks: Coverage and Connectivity Issues and Challenges[END_REF]. From the networking perspective, self-deployment distributed and localized protocols are generally proposed to address these specific aspects. These protocols broadly aim at a single objective which is either cover a point of interest/target [START_REF] Kaur | Kaur An Approach of Mobile Wireless Sensor Network for Target Coverage and Network Connectivity with Minimum[END_REF] or a whole area of interest [START_REF] Al-Karaki | Gawanmeh The Optimal Deployment, Coverage, and Connectivity Problems in Wireless Sensor Networks[END_REF], that might be mobile [10]. Such approaches used different tools such as Virtual Forces [START_REF] Reynaud | Lassous Improving the performance of challenged networks with controlled mobility EAI International Conference on Ad Hoc Networks[END_REF], Swarm Optimization [START_REF] Laturkar | Coverage Optimization Techniques in WSN using PSO: A Survey International Journal of Computer Applications[END_REF] or graph theory tools [START_REF] Erdelj | Simplot-Ryl Covering Points of Interest with Mobile Sensors[END_REF][START_REF] Tolba | Topology control by controlling mobility for coverage in wireless sensor networks[END_REF].

Very few works, addressing jointly communication connectivity and quality of coverage, have been published by researchers operating in the optimization field. The main goal considered in [START_REF] Rebai | Khoukhi Sensor Deployment Optimization Methods to Achieve Both Coverage and Connectivity in Wireless Sensor Networks Computers and Operation Research[END_REF][START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF][START_REF] Elloumi | Rovedakis Optimization of wireless sensor networks deployment with coverage and connectivity constraints[END_REF] is very similar to the one taken into account in the present paper, since the authors aim at covering a sensing area, by deploying the minimum number of wireless sensors, while maintaining the connectivity between the deployed sensors. In [START_REF] Rebai | Khoukhi Sensor Deployment Optimization Methods to Achieve Both Coverage and Connectivity in Wireless Sensor Networks Computers and Operation Research[END_REF][START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF], the authors address the problem by reducing it to a two-dimensional critical coverage problem (an NP-Complete problem), and developing an integer linear programming formulation to obtain an optimal solution. In [START_REF] Elloumi | Rovedakis Optimization of wireless sensor networks deployment with coverage and connectivity constraints[END_REF] standard mixed integer linear programs are used to represent several variants of the problem. Although these works achieve very interesting results, they do not consider the cost of the deployment.

Most of the existing approaches that consider the cost of a wireless sensor network only focus on the deployment phase (cost of a sensor, number of sensors and sometimes distance to drop it), but, to the best of our knowledge, none of them take into account the operational cost (energy sensors will consume to send and forward data) and quality (quality of links).

Problem definition

Given a field, the objective is to deploy a set of sensors at minimum cost, such that full coverage and full connectivity are ensured.

We assume the availability of K different typologies of sensors. Each sensor k ∈ K is characterized by a sensing range r k , a communication range R k and a cost c k . In addition, we assume a limited number of sensors for each typology, i.e., n k . We assume a point is covered if it is within a sensing range of at least one sensor.

The full connectivity is guaranteed if each sensor is connected to a sink node o with either a 1-hop or multi-hop communication. In the latter case, the other sensors play the role of relay.

We formulate the problem over a complete graph G(N, A, C) where N is the set of possible location points in the field where the sensors could be deployed, A is the set of arcs connecting each point i ∈ N to the other points j ∈ N \ {i}, C is the set of locations to be covered (so that are in sensing range of at least one sensor).The Euclidean distance d uv is associated with each pair of positions u, v ∈ N ∪ C.

Mathematical formulation

To formulate the problem, we introduce the following decision variables:

x ij , ∀(i, j) ∈ A.

Continuous variables taking value greater than zero if a communication between position

i and j is active, zero otherwise. w ij , ∀(i, j) ∈ A. Binary variables taking value equal to one if a bilateral communication between i and j can be activated, i.e.,

d ij ≤ min{R k , R k },
where k and k are the typologies of sensors located in position i and j, respectively, zero otherwise.

y k i , ∀i ∈ N, k ∈ K.
Binary variables that assume value equal to one if a sensor of typology k is deployed in position i, zero otherwise. z i h , ∀h ∈ C, i ∈ N . Binary variables taking value equal to one if point h is covered by a sensor deployed in position i, zero otherwise.

The problem can be formulated as follows:

min i∈N k∈K c k y k i (1)
s.t. i∈N y k i ≤ n k , ∀k ∈ K, (2)
k∈K

y k i ≤ 1, ∀i ∈ N, (3)
w ij ≤ k∈K y k i R k d ij , ∀(i, j) ∈ A, (4)
w ij ≤ k∈K y k j R k d ij , ∀(i, j) ∈ A, (5)
x ij ≤ M w ij , ∀(i, j) ∈ A, (6)
(i,j)∈A x ij - (j,i)∈A x ji = k∈K j∈N y k j if i = o -k∈K y k i otherwise ∀i ∈ N, (7)
z i h ≤ k∈K y k i r k d ih , ∀h ∈ C, i ∈ N, (8)
i∈N z i h ≥ 1, ∀h ∈ C, (9)
x ij ∈ R |A| + , w ij ∈ {0, 1}, ∀(i, j) ∈ A, y k i ∈ {0, 1}, ∀i ∈ N, k ∈ K, z i h ∈ {0, 1}, ∀i ∈ N, h ∈ C. (10)
Equation ( 1) minimizes the cost of deployed sensors. Equation ( 2) allows the deployment of at most n k sensors for each typology k ∈ K. Equation ( 3) imposes that at most one sensor is deployed for each position i ∈ N . Equations ( 4) and ( 5) define the variables w ij . In particular, w ij is imposed to be zero in the case either the distance between the sensors deployed in positions i and j is greater than the minimum communication range between both sensors or no sensor is deployed in either position i or j. Equations ( 6) ensure that each arc (i, j) is active if and only if it is able to connect sensors deployed in positions i and j, where M is a big positive number. If w ij = 1 means that the sensors deployed at positions i and j can communicate. Since the value of x ij is not upper bounded, then it can assume any positive value ≤ M . Equations [START_REF] Khoufi | Survey of Deployment Algorithms in Wireless Sensor Networks: Coverage and Connectivity Issues and Challenges[END_REF] represent flow constraints aiming at defining the communication between the sink node o and all the deployed sensors. Equations ( 8) define the coverage of position h ∈ C. In particular, if the sensing range of a sensor of typology k located in position i is less than the distance d hi between h ∈ C and i ∈ N , then variable z i h is forced to assume the value zero. Equations [START_REF] Laturkar | Coverage Optimization Techniques in WSN using PSO: A Survey International Journal of Computer Applications[END_REF] impose that each point h ∈ C is covered by at least one sensor. Equation (10) defines the domain of the decision variables. We observe that equations [START_REF] Khoufi | Survey of Deployment Algorithms in Wireless Sensor Networks: Coverage and Connectivity Issues and Challenges[END_REF] represent flow conservation constraints. Once a solution is determined, variables x ij define a possible communication tree among the deployed sensors. It is as if we considered that each sensor sent one unit of data, thus all variables x ij > 0 define the quantity of data that traverses arc (i, j). In the worst case, the data of all deployed sensors are sent through a single arc. It follows that the value that each variable x ij can assume is upper bounded by min |N |, k∈K n k . The value of M can be set to this value,i.e., M = min |N |, k∈K n k .

We highlight that the proposed model can couple the case where a target has to be covered by at least S > 1 sensors. Indeed, it is sufficient to introduce the parameter S in the right-hand-side of constraint [START_REF] Laturkar | Coverage Optimization Techniques in WSN using PSO: A Survey International Journal of Computer Applications[END_REF] for the point that needs to be covered by the S sensors (possibly all points).

Computational results

In what follows, we analyze the behaviour of the proposed mathematical model, referred in the sequel as GMc, on randomly generated instances.

We have considered also the case in which the number of sensors is minimized. In particular, the objective function [START_REF] Al-Karaki | Gawanmeh The Optimal Deployment, Coverage, and Connectivity Problems in Wireless Sensor Networks[END_REF] has been modified in i∈N k∈K y k i . The model with the latter objective function is referred in the sequel as GMs.

The models have been implemented in Java language and solved by CPLEX 12.51. The tests were carried out on an Intel(R) core(TM) i7-4720HQ CPU 2.60GHz 8GB RAM machine.

The main aim of this section is to present the numerical results obtained and to show the bi-objective nature of the considered problem, with the respect to the minimization of the overall cost, i.e., function [START_REF] Al-Karaki | Gawanmeh The Optimal Deployment, Coverage, and Connectivity Problems in Wireless Sensor Networks[END_REF] and the minimization of the number of deployed sensors (i.e., i∈N k∈K y k i ). We suppose that the higher the sensing range, the higher the cost. On the one hand, when minimizing the number of sensors, those with higher sensing range are chosen, on the other hand, sensors with low sensing range are preferred because they have lower cost than those with high sensing range. The numerical results confirmed the conflicting nature of the two objectives. A sensitivity analysis on the number of available typologies of sensors is also carried out. Finally, we evaluate the effectiveness of the solutions from a communication point of view.

We deploy a varying number of nodes at position < x, y > such that X min ≤ x ≤ X max and Y min ≤ y ≤ Y max in a field with varying size.

The results are collected on instances generated by considering the following characteristics:

-X min = Y min = 0 -X max = Y max ∈ {10, 12, 14, 20} -n k = 100, ∀k ∈ K -each point i ∈ N has integer coordinates (X i , Y i ) in the defined field
The testing phase has been carried out by considering four scenarios that differ from each other for the dimension of the (square) field, i.e., (10, 10), [START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF][START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF], [START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF][START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF], and (20, 20). The instances are characterized by the following parameters:

-|K| ∈ {2, 3, 4, 5} -r k randomly chosen in the interval [2, 6] -R k = R * r k with R = max{1, αf }, α ∈ [0, 1], f = √ 3 meaning that 1 ≤ R k r k ≤ √ 3 -c k = r 2 k -each point h ∈ C has coordinates associated to the centre of the cell. Each corner is a point in N .
the sink is located at coordinate (0, 0)

We follow the scientific literature to generate the instances. The dimensions of the field and the sensing range values refer to [START_REF] Elloumi | Rovedakis Optimization of wireless sensor networks deployment with coverage and connectivity constraints[END_REF], whereas, the connectivity range values satisfy the property described in [START_REF] Khoufi | Survey of Deployment Algorithms in Wireless Sensor Networks: Coverage and Connectivity Issues and Challenges[END_REF]. Indeed, R k r k ≤ √ 3 guarantees that the full coverage does not imply the full connectivity. Fixing the number of typologies of sensors, we generate 5 different configurations varying the seed for the randomly generated parameters.

Numerical results

In this section, we analyze the results obtained by solving GMc, aimed at minimizing the overall cost, and the model GMs in which the objective function, to be minimized, represents the number of deployed sensors (i.e., i∈N k∈K y k i ).

First of all, we analyze the efficiency of the proposed model and give some insights on the conflicting nature of the two criteria, i.e., cost and number of sensors. Secondly, we analyze the solutions obtained by both GMc and GMs under a communication point of view. In particular, we investigate how good the network topology built by the sensor deployment is for routing protocols, in terms of path diversities and energy consumption.

Models evaluation. The results obtained with MGs and MGc are shown in Table 1. We report, for each dimension of the field and number of typologies of sensors |K|, the number of deployed sensors under column #sensors, the cost, i.e., the value of (1), under column cost, the execution time of CPLEX for solving the instances in seconds under column time. We impose a time limit of 1800 seconds to the solver for solving each instance, for both models. The superscript near to the time indicates the number of instances that are not solved to optimality within the imposed time limit. We notice that each line of Table 1 reports average results over the five instances generated by varying the seed.

Row AVG reports the average results on the instances with the same dimension of the field, row AVG2 shows the average results over the instances with the same value of |K|, and row AVG3 reports the average results over all instances. The results reported in Table 1 highlight that the model GMc is more difficult to solve than GMs. Indeed, 13, 17, 20, and 20 instances with the field (10, 10), [START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF][START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF], [START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF][START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF], and (20, 20) are not solved to optimality by the solver within the time limit, respectively. We highlight that 20 instances are considered for each field dimension. Thus, more than a half of instances are not solved for each field dimension. For the biggest ones, the optimal solution is not available for all the instances. It follows that solving GMc requires 3.17 times higher computational time than that needed by the solver to handle GMs, on average (see row AVG3).

Comparing the performance of the two models varying the dimension of the field, we observe that the higher the field dimension the lower the difference in term of computational effort between GMc and GMs. Indeed, GMs is 897.53, 23.44, 8.89. and 1.02 times faster than GMc for the fields (10, 10), [START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF][START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF], [START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF][START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF], and (20, 20), respectively. It is worth noting that these results are highly influenced by the imposed time limit. Indeed, GMs is able to obtain the optimal solution for all the instance, excluding the one with field [START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF][START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF] and the 9 with field (20, 20). In addition, solving GMs requires, on average, 1.39, 70.00, 202.38, and 1771.00 seconds. Thus, a high increase in the computational effort is observed. This trend is not evident for GMc since the time limit imposes an upper bound on the computational cost. The same conclusion can be drawn from the results obtained varying the values of |K|. In this case, GMc is 2, 86, 2.98, 3.52, and 3.34 times slower than GMs for |K| equals to 2, 3, 4, and 5, respectively.

Table 1 highlights the conflicting nature of the two objective functions, i.e., the minimization of the cost and the minimization of the number of sensors.

On the one hand, GMs provides solutions with 6.00 deployed sensors and cost equals to 179.91, on average. On the other hand, the solutions obtained by solving GMc are characterized by 15.08 deployed sensors with a cost of 131.85, on average. Thus, GMc provides solutions that are 1.56 less expensive than those obtained with GMs, but they require 2.51 times more sensors than those deployed by GMs. This is an expected trend since the higher the coverage r, the higher the cost. Thus, when the number of sensors is minimized, those with higher values of r are preferred with a consequent increase in the cost. On the other hand, when the cost is minimized, sensors with low values of r are chosen. This implies the use of a higher number of such sensors, in order to guarantee the coverage of the field.

Referring to the number of available typologies of sensors |K|, we do not observe a clear trend of the difference, in terms of cost, between GMc and GMs (see row AVG2 for each value of |K| of Table 1). Indeed, the solutions obtained by solving GMs are, on average, 1.65, 1.00, 1.36, and 1.52 times more expensive that those obtained by solving GMc for |K| equals to 2, 3, 4, and 5, respectively. This behaviour is due to the fact that GMs does not provide the optimal solution for almost all the instances with field (20, 20). This implies that the number of sensors deployed is not minimized, then the cost tends to decrease (by the conflicting nature of the two objectives). As a matter of this fact, if we consider only the fields (10, 10), [START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF][START_REF] Rebai | Snoussi Exact methods for sensor deployment problem with connectivity constraint in wireless sensor networks[END_REF], and [START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF][START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF], the difference in terms of cost between GMc and GMs increases when |K| increases. Indeed, GMs provides solutions that are 1.29, 1.29, 1.37, and 1.57, times more expensive than those provided by GMc for |K| equal to 2, 3, 4, and 5, respectively.

Considering the number of deployed sensors, the higher the value of |K|, the higher the difference between the two models (see row AVG2 for each value of |K| of Table 1). Indeed, GMs provides solutions with a number of deployed sensors that is 1.65, 2.82, 2.86, and 3.34 times lower than that deployed by the solutions obtained by GMc, on average, for values of |K| equals to 2, 3, 4, and 5, respectively. Topology evaluation. The set of sensors yet deployed aims to efficiently cover and monitor the full area and to constitute a connected network. This property ensures that the base station can communicate with every single sensor and vice-versa. However, this communication is multihop and relies on different routing protocols. In this section, we evaluate the properties of the topologies obtained in the previous sections globally and with regards to the communication and routing costs. We have compared the topologies obtained previously using a home-made C simulator.

Figure 1 shows the average number of nodes selected by each model for several numbers of available typologies of sensors. We can observe that as expected, the number of nodes selected increases with the area size. Naturally, the solutions obtained with GMs present less nodes than observed for those provided by GMc but also (not shown on the figure), it tends to select only one kind of sensors, the ones with the longer ranges, unlike the GMc that selects several types of sensors to cover the full area at a better cost.

Figure 2 shows the average degree of nodes, i.e., the average number of nodes a node can directly communicate with. It tends to increase with the size of the area. This is due to the fact that the area is bounded and to cover all single small area in the border, we need to increase the density over these points. The node degree increases quicker with GMs than with GMc, which makes GMc potentially more scalable. The mean degree obtained with GMs is very low (between 1 and 2 neighbors in average), which also implies a single communication path between two nodes. This can be a cause of unreliability since all traffic will take the same path and provokes potential bottlenecks. Each node represents a point of failure on the path since there is no possible alternative path.

Figure 3 shows the average length edge. Obviously, since there are less nodes in the solution provided by GMs than in GMc over a same area, the edges are longer in average with GMs. Thus, they are thus more energy-consuming.

Figures 4 and5 show some topology examples obtained by model GMc and GMs, respectively, for different number of sensor typologies. In particular, we depict the solutions to the instances with field dimension (20, 20) and [START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF][START_REF] Renzaglia | Multi-UAV Visual Coverage of Partially Known 3D Surfaces: Voronoi-based Initialization to Improve Local Optimizers Computing Research Repository[END_REF] for GMc and GMs, respectively. The edges represent the possible communication among the deployed sensors. Each color corresponds to a typology of sensor.

Although the deployment cost could be reduced with GMs since there are less sensors to deploy, the solutions obtained with the GMc ensure a better cost operation. First, it has been designed to provide the lowest operation cost but also, the topologies that GMc generates provide a better basis for routing data between the different nodes, offering more routing options and lower energy consumption for data transmission.

Conclusions and future work

In this paper, we have addressed a generalized deployed sensors problem with coverage and connectivity constraints, considering different typologies of sensors. We have formulated the problem as a mixed integer program and we provide a deep evaluation of the solutions obtained, analyzing the efficiency and the effectiveness with respect to the deployment cost. We have compared the solutions obtained with those determined by minimizing the number of deployed sensors. Since the cost increases with the increase of the sensing range, the objectives of minimizing the cost and that of minimizing the number of sensors are in conflict one with the other. Finally, we have assessed the obtained sensor deployment from a quality of communication topology perspective. The numerical results suggest that the solutions obtained when the cost is minimized present a high potential in terms of quality of communication. To the best of our knowledge, we are the first to propose such a study. Our model can be easily extended to ensure a k-coverage of every point (each point of the grid is in sensing range of at least k different sensors). As a future work, we will investigate further the impact of the so-built topology on routing utilisation together with the global cost (deployment and hardware cost together with software cost).

Fig. 1 :

 1 Fig. 1: Average number of nodes varying the field dimension.

Fig. 2 :

 2 Fig. 2: Average node degree varying the field dimension.

Fig. 3 :

 3 Fig. 3: Average edge length varying the field dimension.

5 Fig. 4 :

 54 Fig. 4: Topologies obtained with GMc for different numbers of sensor typologies.

5 Fig. 5 :

 55 Fig. 5: Topologies obtained with GMs for different numbers of sensor typologies.

  17. Z. Yu, G. Shan, G. Xu and X. Duan Method of multi-sensor optimal deployment for area coverage. International Conference on Electronics Technology (ICET), Chengdu, China, 2018.

Table 1 :

 1 Average results obtained with GMc and GMs, varying the dimension of the field and the number of available typologies of sensors.

			GMc		GMs
		|K| #sensors cost	time #sensors cost	time
		2	8.80 54.00 953.36 2	3.20 75.40	1.28
	(10, 10)	3 4	6.40 53.00 749.78 2 10.20 54.00 1800.00 5	2.60 71.00 2.20 79.20	1.08 1.42
		5	8.20 43.80 1494.98 4	2.20 79.20	1.80
	AVG		8.40 51.20 1249.53 13	2.55 76.20	1.39
		2	9.80 76.20 1433.52 3	4.60 110.80	23.82
	(12, 12)	3 4	8.20 77.40 1528.67 4 11.80 75.40 1800.00 5	3.60 100.80 3.00 99.60	57.80 71.88
		5	10.80 68.60 1800.00 5	3.20 104.20	126.51
	AVG		10.15 74.40 1640.55 17	3.60 103.85	70.00
		2	14.20 110.20 1800.00 5	5.20 124.40 387.56 1
	(14, 14)	3 4	11.60 104.00 1800.00 5 12.00 117.40 1800.00 5	4.60 130.60 4.40 158.40	112.57 173.58
		5	13.80 101.00 1800.00 5	4.20 151.20	135.83
	AVG		12.90 108.15 1800.00 20	4.60 141.15 202.38 1
		2	24.40 284.20 1800.00 5	21.60 554.20 1684.00 4
	(20, 20)	3 4	36.40 388.80 1800.00 5 22.60 262.60 1800.00 5	11.40 319.60 1800.00 5 10.20 367.20 1800.00 5
		5	32.00 239.00 1800.00 5	9.80 352.80 1800.00 5
	AVG		28.85 293.65 1800.00 20	13.25 398.45 1771.00 19
		2	14.30 131.15 1496.72 15	8.65 216.20 524.16 5
	AVG2	3 4	15.65 155.80 1469.61 16 14.15 127.35 1800.00 20	5.55 155.50 492.86 5 4.95 176.10 511.72 5
		5	16.20 113.10 1723.75 19	4.85 171.85 516.04 5
	AVG3		15.08 131.85 1622.52 70	6.00 179.91 511.20 20


Shuang-Ren Zhao

Quantization/normalization of the mutual energy principle

Keywords: Poynting, Maxwell, Schrödinger, Self-energy, Mutual energy, Mutual energy flow, Reciprocity theorem, Radiation, Newton's third law, Action, Reaction, Advanced wave, Photon, Electron, Wave and particle duality, Huygens Principle, Quantization, Normalization

quantization/normalization of the mutual energy principle

Introduction

Action at a distance and the absorber theory

The theory about advanced wave became most interesting work for this author. This author noticed the absorber theory of Wheeler and Feynman [1,2]. The absorber theory is based on the action-at-a-distance [9,11,6]. In the absorber theory, any current source sends half retarded wave and half advanced wave. For a source we only notice the source sends the retarded wave, we did not notice it also sends the advanced wave. Some one will argue that if in the same time the source sends the retarded wave, it also sends the advanced wave, the source loss the energy from the retarded wave and acquire the energy from the advanced wave, and hence, it doesn't send any energy out. However we all know that the source can send the energy out. This means the absorber theory also has some thing which is not self-consistence. This is also the reason that the absorber theory has not been widely accepted. But any way, the absorber theory accepts the advanced wave as a real wave instead of some virtual wave. This author is inspired by this a lot. The transactional interpretation of John Cramer has introduced the advance wave to the whole quantum mechanics [START_REF] Cramer | The transactional interpretation of quantum mechanics[END_REF][START_REF] Cramer | An overview of the transactional interpretation[END_REF].

Stephenson offered a good tutorial about the advanced wave [START_REF] Stephenson | The relevance of advanced potential solutions of maxwell's equations for special and general relativity[END_REF].

The mutual energy theorems

W.J. Welch introduced a reciprocity theorem in arbitrary time-domain [START_REF] Welch | Reciprocity theorems for electromagnetic fields whose time dependence is arbitrary[END_REF] in 1960 (this will be referred as Welch's reciprocity theorem in this article). In 1963 V.H. Rumsey mentioned a method to transform the Lorentz reciprocity theorem to a new formula [START_REF] Rumsey | A short way of solving advanced problems in electromagnetic fields and other linear systems[END_REF], (this will be referred as Rumsey's reciprocity theorem). In early of 1987 Shuang-ren Zhao (this author) has introduced the concept of mutual energy and the mutual energy theorem [START_REF] Shuang Ren Zhao | The application of mutual energy theorem in expansion of radiation fields in spherical waves[END_REF] (this will be referred as Zhao's mutual energy theorem). In the end of 1987 Adrianus T. de Hoop introduced the time domain cross-correlation reciprocity theorem [START_REF] Adrianus | Time-domain reciprocity theorems for electromagnetic fields in dispersive media[END_REF], (this will be referred as Hoop's reciprocity theorem). Welch's reciprocity theorem is a special case of the Hoop's reciprocity theorem.

Assume there are two current sources J 1 and J 2 . J 1 is the current of a transmitting antenna. J 2 is the current of a receiving antenna. The field of J 1 is described as E 1 and H 1 . The field of the current J 2 is E 2 and H 2 . Assume J 2 has a some distance with J 1 . Hoop's reciprocity theorem can be written as,

- ∞ t=-∞ V1 J 1 (t) • E 2 (t + τ )dV = ∞ t=-∞ V2 E 1 (t) • J 2 (t + τ )dV (1)
if τ = 0, we have,

- ∞ t=-∞ V1 J 1 (t) • E 2 (t)dV = ∞ t=-∞ V2 E 1 (t) • J 2 (t)dV (2)
This is Welch's reciprocity theorem. The Fourier transform of Hoop's reciprocity theorem can be written as,

- V1 J 1 (ω) • E 2 (ω) * dV = V2 E 1 (ω) • J 2 (ω) * dV (3)
Where "*" is the complex conjugate operator. In this article if the variable t is applied in a formula, it is in time-domain. If ω is applied, it is in Fourier frequency domain. This is the Rumsey's reciprocity theorem and is also Zhao's mutual energy theorem. Hence this 4 theorems can be seen as one theorem in different domain: time-domain or Fourier domain. Shuang-ren Zhao noticed that this theorem is an energy theorem, hence the two fields in the formula must all physic waves. The other author referred the theorem as reciprocity theorem, as a reciprocity theorem, it can be a mathematical theorem. One of the two fields can be virtual instead of real. If it is virtual even it is an advance wave, that can be easily accepted. If the two fields are all real as the mutual energy theorem required, we must first accept the advanced wave. The advanced wave are not obey the traditional causality consideration.

The mutual energy flow theorems

This author introduced the mutual energy flow theorem 30 years later than the mutual energy theorem. The mutual energy flow theorem is following,

- ∞ t=-∞ V E 2 (t) • J 1 (t)dV dt = ∞ t=-∞ " Γ (E 1 (t) × H 2 (t) + E 2 (t) × H 1 (t)) • ndΓdt = ∞ t=-∞ V2 E 1 (t) • J 2 (t)dV dt (4)
Here

-´∞ t=-∞ ˝V E 2 (t) • J 1 (t)dV dt is the energy offered by the current J 1 (t) of emitter. ´∞ t=-∞ ˝V2 E 1 (t) • J 2 (t)dV dt is the energy received by the absorber. ´∞ t=-∞ ! Γ (E 1 (t) × H 2 (t) + E 2 (t) × H 1 (t))
• ndΓdt is the energy flow goes through any surface Γ. Γ is at the place between the source and the sink. The mutual energy flow theorem is stronger than the mutual energy theorem. It tells us how the energy flow goes from the the source to the sink.

The mutual energy principle and self-energy principle

This author also introduced the mutual energy principle and the self-energy principle. Which can overcome the difficulty that the Maxwell equations, which conflict with the energy conservation law. The mutual energy flow theorem for two charges in an empty space are following,

- " Γ (E 1 × H 2 + E 2 × H 1 ) • ndΓ = V (E 1 • J 2 + E 2 • J 1 )dV + V (E 1 • ∂D 2 + E 2 • ∂D 1 + H 1 • ∂B 2 + H 2 • ∂B 1 )dV (5)
where ∂ = ∂ ∂t . From the mutual energy principle the mutual energy flow theorem and mutual energy theorem Eq.( 54) can be derived.

It should be noticed that, mutual energy theorem is actually energy conservation law for a space has only two charges, one is emitter and another is absorber. This is artificial space in which there is only two charges and nothing else. If the mutual energy principle is the axiom of the electromagnetic field theory, the mutual energy theorem is derived as an energy conservation law. Or we can say the energy conservation law can be derived through the mutual energy principle. The mutual energy theorem can also be derived through Maxwell equations, but it is only can be derived as a energy theorem: or mutual energy theorem, not a energy conservation law. Since there is also the self-energy, self-energy flow in the space. This author introduced the self-energy principle, self-energy principle tell us that the self-energy flow does not transfer any energy. The self-energy flow is canceled by the time-reversal self-energy flow. After introducing the self-energy principle, we can also prove that the mutual energy theorem is an energy theorem (energy law).

The problem of the mutual energy flow theorem

The mutual energy flow theorem and the mutual energy theorem is energy conservation law. The total energy of the mutual energy flow has very good property because it does not change for any surfaces between the source and the sink. It has the shape very thin in two ends (source and sink) and very thick in the middle between the two ends. Hence, it looks particle in the place of source and sink. These properties are all properties a photon or a particle needs. However there is still a problem that is the energy of the photon will decrease with the distance between the two ends. The farther the distance between the source and the sink, the smaller the energy of the photon.

This article try to solve this problem. The quantization or normalization method for the mutual energy principle is introduced.

2 Quantization/normalization of the mutual energy principle

The mutual energy flow theorem can be written as,

- ∞ t=-∞ V E 2 (t) • J 1 (t)dV dt = ∞ t=-∞ " Γ (E 1 (t) × H 2 (t) + E 2 (t) × H 1 (t)) • ndΓdt = ∞ t=-∞ V2 E 1 (t) • J 2 (t)dV dt (6)
J 1 (t) and J 2 (t) are the current intensity of the emitter and absorber, assume that this is a constant. E 1 (t) is the electric field of the retarded wave and E 2 (t) the advanced wave which are decrease with the distance between the emitter and the absorber:

r = ||x 2 -x 1 ||.
x 1 is position of the emitter and x 2 is the position of the absorber. Hence, we have,

J 1 (t), J 2 (t) ∼ const (7)
E 1 (t) ∼ 1 r J 1 (t) (8)
E 2 (t) ∼ 1 r J 2 (t) (9)
lim r→∞ ∞ t=-∞ V2 E 1 (t) • J 2 (t)dV dt ∼ lim r→∞ 1 r = 0 (10) lim r→∞ ∞ t=-∞ V E 2 (t) • J 1 (t)dV dt ∼ lim r→∞ 1 r = 0 (11)
We know a photon has constant energy, hv = const. where h is Planck constant. v is the frequency. In order to make,

lim r→∞ ∞ t=-∞ V2 E 1 (t) • J 2 (t)dV dt = const (12)
where const take a constant value which is the energy of a photon const = hv. where h is Planck constant, v is the frequency of the light. We need either

J 1 (t) ∼ r, J 2 (t) ∼ r (13) E 1 (t) ∼ 1 r J 1 (t) = const (14) E 2 (t) ∼ 1 r J 2 (t) = const (15)
This means when the distance between the two charge increased, the current of the charge intensity needs to increase. Or

J 1 (t), J 2 (t) ∼ const (16) E 1 (t) ∼ J 1 (t) (17)
E 2 (t) ∼ J 2 (t) (18)
This tell us that the electromagnetic fields do not decrease with the distance r. This author thought the above both are possible. First option let us see Eq.(16-18). This author assumes that if the waves (retarded wave and the advanced wave) all collapse back in all directions, but the mutual energy flow still keep a energy channel from the source to the sink. Hence, the waves has collapse back in all other direction, the waves amplitude will increase at the directions of the the energy channel. This make the field amplitude not decrease at the direction of along the energy channel. The energy channel is at the direction of the mutual energy flow.

The self-energy flow collapse back has no any price to the photon system. Hence, if the energy of a photon becomes smaller, it can send more times until the energy of a whole photon is sent out. This also can be simulated by increase the current J 1 and J 2 , hence, we can let

J 1 ∼ r, J 2 ∼ r (19)
The wave will looks like it is propagate inside a wave guide. The shape of the wave guide is very thin in the two ends of wave guide. It is very wide in the middle between the two ends. Hence, we can have,

||E 1 (x 1 )|| = ||E 1 (x 2 )|| ||E 2 (x 1 )|| = ||E 2 (x 2 )|| (20)
It is clear that there is,

||E 1 (x 3 )|| < ||E 1 (x 2 )|| ||E 2 (x 3 )|| < ||E 2 (x 2 )|| (21)
Here x 3 is a point between x 1 and x 2 . || • || is the norm of the vector. But the energy go through each stream line should be still equal. This author assume that the wave sends according to Maxwell equations which is a wave decrease with the distance r, since only in one direction which is direction of the absorber there is the advanced wave, which leads the retarded wave go along that direction. All the waves of self energy flow collapse back to the source and re-send out. The waves which collapse back are time-reversal waves. When the wave is re-sent out, only in the direction of the absorber can produce a mutual energy flow. hence the value of the energy flow is increased in the direction of absorber. The retarded wave in the mutual energy flow goes alone the direction of absorber. The advanced wave goes to the direction of the emitter. Hence, the waves does not decrease with the distance r.

This can be seen as a normalization process, the whole mutual energy of the mutual energy flow,

∞ t=-∞ " Γ (E 1 (t) × H 2 (t) + E 2 (t) × H 1 (t)) • ndΓdt = const = hv (22)
The mutual energy flow calculated through the retarded wave and the advanced wave in the space produced a photon which is decrease with the distance between the source and think. After the normalization the photon energy must equal to hv. Where h is the Planck constant. v is the frequency of the photon.

Since after this normalization a photon with correct energy is produced we also call this is a quantization process.

It seams the energy has collapse from all space to the absorber, but actually the self-energy flow collapse back through the time-reversal wave. After the self-energy is collapsed back the energy is re-sent out many many times, only at the direction of the absorber produced the mutual energy flow is produced. The mutual energy flow need to be increased to a whole photon.

Microscopic Maxwell equations, Macroscopic Maxwell equations

This author assume the solutions of the microscopic Maxwell equation, which are the retarded wave and the advanced wave is the wave decrease with the distance r. The electromagnetic field for a photon that is quantized electromagnetic field, which is not decrease with the distance r. The macroscopic wave which is the wave average the photon energy from an absorber to a 4πr 2 area, and hence, also reduce with the distance r. Hence, the macroscopic wave and microscopic wave are all retarded wave and advanced wave satisfy the Maxwell equations, the amplitude of the wave reduce with the distance r. How ever, the amplitude of the quantized wave is not reduce with the distance r.

The result of Maxwell equations

E mic = ||E mic || ∼ 1 r (23)
E mac = ||E mac || ∼ 1 r (24)
E quantized = ||E quantized || ∼ cost ( 25
)
E mic is the microscopic wave. E mac is the macroscopic wave. E quantized is quantized wave.

Figure 1: A photon does not reduce the amplitude of the electromagnetic field, but if average each photon to the whole area 4πr 2 , the energy will decrease with the fact 1 r 2 , and the field will decrease with the distance fact 1 r .. is because a statistics average. E mac is the field of the Maxwell equations. The field sends in one direction (one of a absorber) is E quantized = ||E quantized || ∼ const. However, the probability this absorber in that direction receive a photon is decrease with the area of the sphere with the radio equal to r. If we consider the probability

E mac = P (a)E quantized ∼ 1 r (26)
where P (a) is the probability to whole sphere. Hence, in the electromagnetic field theory, we often have the field which decreases with the distance r that is because the statistic average effect. Hence, the Maxwell equations still can be established on the base of statistic average. The field we obtained from Maxwell equations is,

E mac = 1 T T t =0 E(t)dt ∼ 1 r (27)
The field of the Maxwell equation is still correct. The only thing need to know is that it is average result to time. Figure 1 shows the photon is send at different directions. The photon energy is not decrease with the distance. But the field of Maxwell which is a average on the whole sphere with area 4πr 2 . This leads the averaged field decrease with the distance of distance r.

This author still assume that in the beginning the emitter and the absorber send the broadcast wave which is the wave decrease with distance r, and satisfies the Maxwell equations. Assume that the emitter sends the retarded wave, assume that the absorber sends advanced wave. This wave all belong to broadcast wave which is decrease the amplitude with distance r. If the two waves do not synchronized, the wave is returned to its source or think throguh the timereversal waves corresponding to the two broadcast waves. If the two waves are synchronized, the advanced wave become the lead wave to the retarded wave, which increase the amplitude of both the retarded wave and the advanced wave in the direction of energy channel. The two waves do not decrease with the distance between the two charges (emitter and the absorber). However, if we average this kind wave in all direction, we can obtained the original wave of the Maxwell equations offered for single charge, which decrease with the distance r.

Hence, we can say that the advanced wave produced a wave guide to the retarded wave. The wave of this synchronized wave are wave in a guide. Waveguide wave is focalized broadcast wave. This focalization process can be referred as quantization of the the mutual energy principle. After this quantization, the energy of the mutual energy flow does not decrease with the distance r = ||x 2x 1 ||. (1) The photon energy does not decrease with distance between emitter and absorber and (2) the photon energy are equal at any surfaces between the emitter and absorber are two most important properties of the photon.

The original mutual energy principle

The original mutual energy principle related to the Maxwell equations should be seen also the result of the statistic average.

From that we obtained the result the lager the distance between the emitter to the absorber, the smaller the energy of the photon. This result is not correct, but it is good enough to a macroscopic result. In the macroscopic situation how big the energy of a photon is no sense. If the photon is small, with more photon will have the same effect. The number of photon is also no since to the macroscopic situation. Hence in the macroscopic wireless wave situation, there are infinite photons, Maxwell equations and original mutual energy principle all still can be applied.

The mutual energy principle batter than the Maxwell equations at the point it overcome the problem of the conflict the Maxwell equations with the energy conservation law.

The mutual energy flow theorem

The mutual energy flow theorem still can be established after the quantization of the mutual energy principle. The reason is that the mutual energy principle is also established on a wave guide according to the original theorem, after the quantization that is just like the field restricted on the wave guide. A original derivation of the mutual energy principle and mutual energy flow theorem are still effective.

3 Huygens principle

Huygens sources

Considering,

E 1 (t) × H 2 (t) • n = E 1 (t) • H 2 (t) × n = E 1 (t) • J h2 (t) (28)
E 2 (t) × H 1 (t) • n = n•E 2 (t) × H 1 (t) = n × E 2 (t) • H 1 (t) = K h2 • H 1 (t) (29)
we can defined the Huygens source,

σ 2 = J h2 (t) = H 2 (t) × n K h2 (t) = n × E 2 (t) (30)
The inner product becomes,

(ξ 1 , ξ 2 ) = ∞ t=-∞ dt " Γ (E 1 (t) • J h2 (t) + H 1 (t) • J h2 (t))dΓ (31) n•E 1 (t) × H 2 (t) = n × E 1 (t) • H 2 (t) = K h1 (t) • H 2 (t) (32)
E 2 (t) × H 1 (t) • n = E 2 (t) • H 1 (t) × n = E 2 (t) • J h1 (33)
The Huygens source is,

σ 1 = J h1 (t) = H 1 (t) × n K h1 (t) = n × E 1 (t) (34)
The inner product becomes,

(ξ 2 , ξ 1 ) = ∞ t=-∞ dt " Γ (E 1 (t) × H 2 (t) + E 2 (t) × H 1 (t)) • ndΓ = ∞ t=-∞ dt " Γ (H 2 (t) • K h1 (t) + E 2 (t) • J h1 (t))dΓ (35)
In the above, we have choose ξ 1 as the retarded wave, ξ 2 as the advanced wave.

(

ξ 2 , ξ 1 ) = (ξ 2 , σ 1 ) = (σ 2 , ξ 1 ) (36)
where,

(ξ 2 , σ 1 ) ≡ ∞ t=-∞ dt " Γ (H 2 (t) • K h1 (t) + E 2 (t) • J h1 (t))dΓ (37) (ξ 1 , σ 2 ) ≡ ∞ t=-∞ dt " Γ (E 1 (t) • J h2 (t) + H 1 (t) • J h2 (t))dΓ (38)
From the above we can see the field ξ 1 and ξ 2 can be replaced as the corresponding Huygens sources σ 1 and σ 2 .

Figure 2: The source is inside the volume V F , the sink is inside of the volume V I . F is the surface of the volume V F , I is the surface of the volume V I . B is the surface between the source and the sink. .

Generalized Huygens principle

In the quantum mechanics there is the bra ξ| and the ket |ξ , which can be used as the definition for the inner product, i.e.,

ξ 1 |ξ 2 ≡ ξ 1 ||ξ 2 ≡ (ξ 1 , ξ 2 ) ≡ ∞ t=-∞ dt " Γ (E 1 (t) × H 2 (t) + E 2 (t) × H 1 (t)) • ndΓ (39)
where

ξ 1 ≡ [E 1 , H 1 ] (40)
ξ 2 ≡ [E 2 , H 2 ] (41)
The mutual energy flow theorem can be written as,

ξ F |ξ F I = ξ B F |ξ B I = ξ I F |ξ I (42)
ξ B F is the field at B produced by the source at F . The subscript F is used to express the final point or the place of the sink. The superscript B is the place of the field. Here B is the surface between the I and F . I is the initial source place. The mutual energy flow theorem and the mutual energy theorem together can be rewritten as,

- ∞ t=-∞ VI J I (t) • E I F (t)dV = ξ F |ξ F I = ξ B F |ξ B I = ξ I F ||ξ I = ∞ t=-∞ VF E F I (t) • J F (t)dV (43)
The details, can be found in Figure 2. In the above ξ F I is field at F , which is produced by the source I. ξ F is the field at F produced by the source at F . ξ F |ξ F I is the inner product at the surface F . ξ B F |ξ B I is a inner product at the surface B. ξ I F |ξ I is a inner product at the surface I. We use F to express the surface of the volume V F . I is the surface of the volume V I . B is any surface between I and F . We assume V I is the source, V F is the sink.

The above mutual energy flow theorem is also suitable to quantum mechanics, can be written as

Ψ b |Ψ b a = Ψ c b |Ψ c a = Ψ a b |Ψ a (44) Ψ c b |Ψ c
a is the inner produce on the surface c. Ψ b |Ψ b a is the inner product on the surface b Ψ a b |Ψ a is the inner product on the surface a. In the following discussion we only use electromagnetic field as example, the result is also suitable to the quantum mechanics.

We have known that in quantum mechanics there is,

i |q i q i | = 1 (45)
In our situation, our inner product is at the surface B. That is a integral on the surface B and, hence, we can rewritten the above formula as,

i |ξ Bi ξ Bi | = 1 (46)
Substitute the above formula to the mutual energy flow theorem Eq.( 43), the mutual energy flow theorem can be written as,

ξ F |ξ F I = i ξ B F |ξ Bi ξ Bi |ξ B I = ξ I F |ξ I (47)
Considering the mutual energy flow theorem about the surface B and F we have,

ξ B F |ξ Bi = ξ F |ξ F Bi = ξ F |G F B |ξ Bi (48)
In the above ξ B F |ξ Bi is a inner product at the surface B. ξ F |ξ F Bi is a inner product at F . The first equal sign is because the mutual energy flow theorem, the inner product can be moved from the place of B to the place F . In the second equal sign, we have considered the definition of

G F B , hence, |ξ F Bi = G F B |ξ Bi . Where G F B is the gain from |ξ Bi to |ξ F Bi . G F
B is a operator, or matrix. Hence, we have,

ξ F |ξ F I = i ξ F |ξ F Bi ξ Bi |ξ B I = i ξ F |G F B |ξ Bi ξ Bi |ξ B I = i ξ F |G F Bi |ξ Bi ξ Bi |G Bi I |ξ I = ξ I F |ξ I ( 49
)
The above formula actually has problem, G F B is still the gain operator, but it actually is the gain operator from |ξ Bi to |ξ F I . And hence can be written as

G F
Bi . G F Bi actually is the matrix element. In the above, insert the unit operator

i |ξ Bi ξ Bi | = 1, AB = j A|ξ j ξ j |B = j A k j |ξ j ξ j |B j i = j A k j B j i ( 50
)
The two matrix AB after insert the unit operator become the matrix element expression. Hence, Eq(49) can be rewritten as, From the above Eq.( 49), we still have also,

ξ F |ξ F I = i ξ F |G F Bi |ξ Bi ξ Bi |G Bi I |ξ I = ξ I F ||ξ I (
i |ξ Bi ξ Bi | = 1 (52)
It should be noticed that, in Eq.( 47), the unit operator Eq.( 46) is inserted to the inner product ξ B F |ξ B I . In Eq.( 49) to obtained the unit operator Eq.( 52) we have to apply the mutual energy flow theorem

ξ B F |ξ Bi = ξ F |ξ F Bi . In other side, ξ I F | = ξ F |T I F | = ξ F |T B F T I B | (53)
Where T I F is the gain to the left vector ξ F |, Hence,

ξ I F |ξ I = ξ F |T B F T I B |ξ I ( 54
)
Insert i |ξ Bi ξ Bi | = 1 to the above we obtain,

ξ I F |ξ I = i ξ F |T B F |ξ Bi ξ Bi |T I B |ξ I = i ξ F |T Bi F |ξ Bi ξ Bi |T I Bi |ξ I = ξ F |ξ F I (55)
This is the Huygens principle for the advanced wave.

Application of the Huygens principle to more surfaces

The received energy at the final point F is

ξ F ||ξ F I ( 56
)
Substitute the following 3 formulas

i |ξ Ai ξ Ai | = 1 (57) i |ξ Bi ξ Bi | = 1 (58) i |ξ Ci ξ Ci | = 1 (59)
to the above formula we have,

ξ F |ξ F I = ξ F |G F I ξ I = ξ F G F C G C B G B A G A I ξ I = kji ξ F |G F C |ξ Ck ξ Ck |G C B |ξ Bj ξ Bj |G B A |ξ Ai ξ Ai |G A I |ξ I = kji ξ F |G F Ck |ξ Ck ξ Ck |G Ck Bj |ξ Bj ξ Bj |G Bj Ai |ξ Ai ξ Ai |G Ai I |ξ I ( 60
)
In the above, we have considered that,

ξ F I = G F I ξ I (61)
where G F I is the gain from ξ I to ξ F I , it is clear that we have,

G F I = G F C G C B G B A G A I ( 62
)
There is also the similarly formula for the advanced wave.

The amplitude of the field is not change alone the streamline

According to the assumption of this article, the amplitude of the field alone the streamline will not change, hence we have,

G F I = exp(-iHT ) (63)
||G F I || = 1 (64)
G F I = G F C G C B G B A G A I = exp(-i N i=1
HδT )

Here H is the Hamilton. This formula is the theory base of the path integral. In the path integral the wave does not decrease with distance. When this author work out the "updated version of the path integral", the streamline integral introduced by this author. In that time this author also noticed that Feynman path integral does not decrease with distance. In the beginning this author thought that it is a mistake of Feynman. Now we know that Feynman is correct at this point. In the future this author will correct the streamline integral using the quantization of the mutual energy principle. After that the wave will not decrease with distance along the streamline. This offers a theory basis to the path integral. This concept can be further wider to that the wave can be calculated at any path, on which the amplitude of the wave does not change. Any paths differ than energy streamline will have very smaller effect and will cancel each other, hence Feynman define the path integral on all paths still can obtained correct result. However, this author has only proved that the electromagnetic field does not decrease along the streamline. Feynman widen it to all paths, the electromagnetic field does not decrease along any paths that has not been proved as correct.

Numerical calculation

The retarded wave for the photon inside the nature wave guide can be calculated according to the boundary condition there is only one source and one sink in the space. The retarded wave started from the source end at the sink. The space between the source and the sink is the nature wave guide. The wave inside the wave guide should satisfy the Maxwell equations. The advanced wave can be calculated exact same way. Hence, the retarded wave and the advanced wave are same. Perhaps the finite element method can be applied for this kind of calculation.

The author introduce an approximate method, we need to obtained the stream line of the photon, which can be obtained by the traditional mutual energy flow theorem. That means we first calculate the retarded wave sending from the emitter to free space, the amplitude of the wave is calculated according to ∼ 1 r . Then calculated the advanced wave sending from absorber, the amplitude of the wave is calculate according to ∼ 1 r . The mutual energy flow is calculated through the retarded wave and the advanced wave in free space. This way we calculate the stream line from the source to the sink. It should be largely correct (perhaps it is exactly correct, the author has not proved it). This way even the energy of photon is not correct but we obtained the streamline.

When we obtained the streamline, we have known that this kind of calculation the mutual energy flow will decrease with the distance. The mutual energy flow is equal to,

∞ t=-∞ V E 1 (t) • J 2 (t)dV dt (66) If J 2 (t) ∼ const, E 1 (t) ∼ 1 r , ´∞ t=-∞ ˝V E 1 (t) • J 2 (t)dV dt ∼ 1 r .
According to the mutual energy flow theorem the mutual energy flow

´∞ t=-∞ ! Γ (E 1 (t) × H 2 (t) + E 2 (t) × H 1 (t)) • ndΓdt ∼ 1
r . In order to make the mutual energy flow do not decrease according to 1 r We increase the J 2 (t), J 2 (t) according to

J 1 (t) := J 1 (t) r, J 2 (t) := J 2 (t) r (67)
This means we use J 1 (t) r to replace J 1 (t). Use J 2 (t) r to replace J 2 (t), This way,

∞ t=-∞ " Γ (E 1 (t) × H 2 (t) + E 2 (t) × H 1 (t)) • ndΓdt = const (68)
The mutual energy flow will keep do not change when the distance r increase.

In this way, the E 1 (t), H 1 (t) and E 2 (t), H 2 (t) can be calculated using the currents of the point charges J 1 (t) and J 2 (t). The stream line and the energy flow intensity can be calculated according to Eq.(68).

The collapse of the wave

In quantum mechanics, there is the concept of wave function collapse. I generally agree that wave is collapsed. However the wave function collapse cannot described by any mathematical formula. There are also many different wave collapse. Fist there are collapse in the measure time that means wave is collapsed when the particle is measured, this belong to Copenhagen interpretation. John Cramer introduced the concept of the continuous collapse, which means the wave continuously collapses to the light ray. According the John Cramer's transactional interpretation, the retarded wave and the advanced wave both collapse to the light ray. This author suggest that the both wave collapse instead of the light ray, but the mutual energy flow. The mutual energy flow is more concrete concept for light ray. the light ray looks a line but the shape of the mutual energy flow is thin in the two ends and thick in the middle between the two ends.

Since the the collapse cannot be described. This author introduced the time-reversal waves. We assume that the retarded wave and the advanced wave collapse back with the time-reversal waves. Here this author means the selfenergy flow all collapse back. Hence retarded wave and the advanced wave will be resend out from the emitter and the absorber. The mutual energy flow will Hence the wave collapse in consist of two processes one is the the two selfenergy flows for the retarded wave and the advanced wave collapse back to the emitter and absorber, the second is the mutual energy flow accumulatively increase. This two process together build the wave collapse process. The all this two processes can be mathematically described. Hence we can offer a wave collapse process described mathematically.

This author also introduced a possibility that the mutual energy flow collapse back with anti-mutual energy flow. Especially in case an absorber obtained a photon energy that is not enough as a whole photon, it will be returned through time-reversal mutual energy flow. In this situation, stimulated radiation, the absorber or emitter in the neighbor will have big chance to send the retarded wave or advanced wave of a photon. All half photon, and the partial photon will collapse back, the energy is resend out. W have applied this method to explain why there is not half photon or partial photon.

This author introduced the concept of time-reversal waves and time-reversal mutual energy flow which is anti-photon all still correct. Figure 3 shows the collapsed waves. The black line shows the wave collapse. The red line further tell us the collapse is through a collapse back (time-reversal wave for the selfenergy flow or time-reversal mutual energy flow).

Conclusion

We have know the photon is a wave package, that means the shape of photon looks like a pollywog, that means it very thin at the two ends and thick in the middle between the two ends. The two ends are the position of emitter and the absorber. This author looks a theory can produce a wave package looks like pollywog from the concept of the retarded wave and the advanced wave. John Cramer introduced the method of superposition. The retarded wave superposed with the advanced wave. However the superposition cannot produced a wave package which does not send energy out at infinite big sphere. This author looks function which has 0 values every where but not 0 only at the link line from emitter position to the absorber position. This function consist of the retarded wave from the emitter and the advanced wave from absorber. This author found the mutual energy flow satisfies this requirement. The mutual energy flow is starting from the emitter and ends at the absorber. The mutual energy flow theorem guarantees that the energy go through any surface between the emitter to the absorber is all equal. Mutual energy flow actually is also the the product of the superposition of the retarded wave and the advanced wave. However it is the energy flow of superposed fields of the advanced wave and the retarded wave subtract the energy flows of the retarded wave and the advanced wave themselves. The mutual energy is also the surface inner product of the retarded wave and the advanced wave.

However, the free space electromagnetic field decrease with the distance between the source point and the field point. This leads the mutual energy flow also decrease with the distance between the two charges the emitter and the absorber. The photon is not decrease with the distance between the emitter and the absorber.

In order to overcome this difficulty, this author normalized the mutual energy flow. So the mutual energy flow should always the energy of one photons. This normalization process ask the field of the retarded field are equal at the emitter position and the absorber position. It is same to the advanced wave. That means the field of the photon is not decrease with distance. This can be understand by the superposition of the retarded wave and the advanced wave, the two wave is superposed only at the linkage line between the emitter and the absorber, hence, the field are become much strong in that direction. This is also because the retarded wave and the advanced wave are lead wave of each other, the wave has no other direction can go. The energy flow must be one photons energy.

The inner product of the retarded wave and the advanced wave (the mutual energy flow) can guarantee that the energy flow only goes from the source to the sink. Hence, this author choose the mutual energy flow to describe the photon. The mutual energy flow has very good property that is it is equal at any surface between the emitter to the absorber. This property is the most important property the photon needs. However, the mutual energy flow decrease when the distance between the emitter and absorber increase. It is known that the photon energy is a constant, hence, this author normalized the mutual energy flow. A wave which does not decrease in amplitude is applied to calculate the mutual energy flow.

In this article this author introduced the concept of quantization of the mutual energy, the mutual energy flow, mutual energy principle. According this theory, the emitter (source) sends retarded wave and the absorber (sink) sends advanced wave. Both emitter and absorber send the waves randomly. If there is no an advanced wave synchronized to the retarded wave, bother waves are returned through the time-reversal waves. These waves are broadcast waves, the amplitude of the broadcast wave decrease with the distance r. In case there is advanced wave synchronized with the retarded wave, the both waves the retarded wave and the advanced wave increase the amplitude on the direction alone the line linked the emitter and the absorber. All energy will go through the nearby of the line links both charges. The amplitude of the wave will not decrease alone the energy streamline. This process can be seen as a focalization process of the traditional Maxwell theory and the mutual energy principle. This focalization process make the mutual energy flow between the emitter and the absorber as a constant even the distance between the emitter to the absorber increased. This result make the mutual energy flow become a quantum or photon. Through statistical average process to the quantization of the mutual energy principle, a traditional mutual energy principle will obtained. The Maxwell theory can also be obtained through the statistical average process to the quantization of the mutual energy principle.

The traditional mutual energy principle is the foundation of the quantization of the mutual energy principle. The traditional mutual energy principle offers a consist theory without conflict to the law of the energy conservation. After a small step of the quantization, a final new theory for the electromagnetic fields is produced.

  51) This is the Huygens principle for the retarded wave. In the surface B, |ξ Bi is the unit Huygens source, ξ Bi |ξ Bi I = ξ Bi |G Bi I |ξ I is the value of the Huygens source, |ξ Bi ξ Bi ||ξ Bi I is the Huygens source. ξ F |G F B |ξ Bi ξ Bi |ξ Bi I is the contribution of the Huygens source to the surface F . i ξ F |G F B |ξ Bi ξ Bi |ξ B I is all contributions of the Huygens sources to the surface F .

Figure 3 :

 3 Figure 3: Waves collapse are through two process (1) collapse back(time-reversal self-energy flow) (2) The mutual energy flow accumulatively increases. .


Lukman A Akanbi

Emmanuel R Adagunodo

Amos David

Integrating Document Usage with Document Index in Competitive Intelligence Process

Keywords: Content Analysis and Indexing Methods Document usage, document representation, document index, usage modelling, decision problem, attribute value pair

The amount of information in term of documents, available to users as a result of information retrieval process for the purpose of resolution of decision problems is a major factor that determines whether economically viable decisions would be made or not. Various works in the literature had addressed the challenges of representing the documents with key terms (generated from the document) as well as the variations in the meaning of each key terms. In this work, a document representation scheme that is based on the key terms generated from the documents and their usage was developed. To realize this document representation scheme, a computational model for capturing document usage was designed with the use of attribute value pair technique of document annotation. The document usage model designed was applied in the development of a Competitive Intelligence based Document Usage Creation and Exploration system that is currently under development. A preliminary evaluation of the document usage model based on cosine similarity function between user query and documents set was carried out. The result obtained shows that representing documents in terms of their usage can enhance the quality of information search results as documents that would hitherto be considered not relevant to user query are found to be ranked very relevant based on previous usages.

INTRODUCTION

In information retrieval for the purpose of solving a decision problem, the amount of information in terms of documents as well as detailed information about their previous usage that are available to users is a major factor that influences result of the problem resolution. Okunoye et al. in [START_REF] Okunoye | Amtea: Tool for creating and exploring annotations in the context of economic intelligence (Competitive Intelligence)[END_REF] identified that discovery of relevant information for solving decision problems is pivotal to making right decision and that the information has to be sought, collected, processed with a view to elicit knowledge relevant to the problem as well as representing both the collected information and elicited knowledge in a form that will facilitate information reuse.

The modern information retrieval (IR) models (i.e. Boolean, Vector and Probabilistic) employ terms in the documents to produce the index terms to represent the documents in the document space [START_REF] Baeza-Yates | Modern Information Retrieval[END_REF], [START_REF] Manning | AnIntroduction to information Retrieval[END_REF]. This documents representation technique based on the terms available in the document is employed by most systems that provide search facility.

Therefore such systems always return relevant documents based on the terms found in the query. However, experience has shown that there are relevant documents whose terms are orthogonal to the terms available in the query submitted by the user. The reason for this situation is express in the hypothesis that every search activity for information is associated with a decision problem (DP). This problem may be expressed explicitly or is implicit to the information seeker. It is however believed that, keeping track of the kind of DP that documents have been used to resolve would ease the process of resolving similar problems in the future.

There are various works in the literatures that had addressed the challenge of representing the documents with key terms as well as the variations in the meaning of each key term (for example, [START_REF] Deerwester | Indexing by Latent Semantic Analysis[END_REF] [START_REF] Gupta | Context Based Indexing in Search Engines using Ontology[END_REF]. However, the concept of document usage has not been given attention. A document usage model based on attribute-value-pair (AVP) technique of document annotation proposed in [START_REF] Okunoye | Design and Development of a Model for Generating and Exploiting Annotation in the Context of Economic Intelligence[END_REF] is therefore presented in this work.

The rest of the paper is organized as follows: Competitive intelligence is described in section two; document annotation and its relevant to competitive intelligence is described in section three. In section four, the document usage model and the architecture of the CI-DUCE system are presented. In section five the evaluation of the model and the preliminary results are discussed and section six concludes the work.

COMPETITIVE INTELLIGENCE

As a special case of application of IR process, CI is a systematic process of information gathering, processing, analysis and decomposition, which is conducted within the context of the external environment of an organization's activities. This process is with the major goal of supplying the right information, at the right moment, and in the correct structure, to the right person, in order to support the best decision possible [START_REF] Trigo | Using competitive intelligence as a strategic tool in a Higher Education context[END_REF]. The Strategic and Competitive Intelligence Professionals (SCIP) in [START_REF]About SCIP, Strategic and Competitive Intelligence Professional[END_REF] defines CI as the legal and ethical collection and analysis of information regarding the capabilities, vulnerabilities, and intentions of business competitors. The information legally and ethically collected is to be used by the decision makers in an organization to support them in arriving at the best possible decision. Dishman and Calof in [START_REF] Dishman | Competitive intelligence: a multiphasic precedent to marketing strategy[END_REF] described CI as a process involving the gathering, analyzing and communicating of environmental information to assist in strategic decisionmaking, and as such it is the fundamental basis of the strategic decision-making process. Camelo et al. [START_REF] Camelo | Competitive Intelligence and the Development of Corporate Universities[END_REF] pointed out that the CI process should aim to provide information on technologies and general commercial tendencies in order to facilitate the decision making process, helping to achieve the strategic goal of the company. According to Odumuyiwa [START_REF] Odumuyiwa | Collaborative Information Seeking in the Competitive Intelligence Process[END_REF], CI is linked to various similar concepts such as economic intelligence, business intelligence and knowledge management and it is a process that embodies decision making. CI as an information process is made up of the following phases [START_REF] David | Relevance Information in Economic Intelligence[END_REF]: i) identification of decision problem ii) transformation of decision problem to information search problem iii) identification of relevant sources of information iv) collection of relevant information v) analysis of collected information to extract indicators for decision making vi) interpretation of indicators vii) decision making

According to [START_REF] Camelo | Competitive Intelligence and the Development of Corporate Universities[END_REF], in CI process, the time needed for execution is of crucial importance, as in case of short or medium term work, decision making depends on vast quantity of information and this precise lack of time available to process an unpredictable volume of information, with selection of only that which is useful for decision making can reasonably be seen as one of the main challenges encountered in CI.

The focus of the present work is on how to improve items (iii) and (iv) of the CI process listed above in order to enhance the process of decision making in the shortest time possible thereby addressing the concern raised in [START_REF] Camelo | Competitive Intelligence and the Development of Corporate Universities[END_REF]. These two items is basically an IR process.

DOCUMENT ANNOTATION

According to [START_REF] Ogilvie | Retrieval using Document Structure and Annotations[END_REF], Annotation can be seen as simply information about the document, assigned by a process or human, after the original creation of the document, such as information provided by a named-entity tagger or an automatic layout analysis tool. This information may be less certain than structural markup, as the process or person creating the annotations may not know the author's original intent. Annotation is also described as a tool or technique used to describe a set of information. These set of information could range from title of a document to a section of a document and/or the whole chapter in a book. As identified in [START_REF] Robert | AMIE: An Annotation Model for Information Research[END_REF] annotation tools are becoming very important for document interpretation. Its importance as a tool in information interpretation can be seen with its popularity in text processors like Microsoft word and Adobe Acrobat that integrate features that enable users to annotate document. More than the normal use of annotation for document interpretation, annotation tool can be designed to assist in information research [START_REF] Robert | AMIE: An Annotation Model for Information Research[END_REF].

The traditional domain of document annotation covers the annotation of arbitrary textual documents, or parts of them, which can be manual (i.e. performed by one or more people), semi-automatic (based on automatic suggestions), or fully automatic. Manual annotation tools allow users to add annotations to web pages or other resources, and share these with others. Automatic tools can perform similar annotations (such as named-entity recognition) without manual intervention [19].

The meaning of annotation can vary significantly when considered in a specialized context. For example, in historical and religious scholarship, annotations can provide contextual detail about primary sources, or describe interpretations or differing perspectives. Whereas in the legal and governmental domains, annotations often provide references to relevant instances of an abstract concept in practice, such as court decisions associated with particular statutes, but the term can also refer to quite lengthy documents about specific cases [START_REF] Macmullen | Annotation as Process, Thing, and Knowledge: Multi-domain studies of structured dataannotation[END_REF].

From the fore going, one thing stand out about the concept of annotation, i.e. extracting or adding information to an information object of interest. This work is therefore a specific application of the annotation concept to add usage to an information object and made such usage part of document representation. This is believed to enhance the process of DP resolution in CI environment. Some of the examples of uses of annotation in CI process are discussed in the next subsection.

Some Applications of Document Annotation

In [START_REF] Robert | AMIE: An Annotation Model for Information Research[END_REF], Robert and David presented annotation model that will assist in information research for decision making. The Annotation Model for Information Exchange (AMIE) was reported to have been conceived with the objective of information sharing and reuse. The use of annotation process as an indexing approach that allows the users to identify and regroup documents or their sub-elements under a particular use context was employed in [START_REF] Maghreb | Open System for Indexing and Retrieving Multimedia Information[END_REF]. [START_REF] Okunoye | Dynamic Knownledge Capitalization through Annotation among Economic Intelligence Actors in a Collaborative Environment[END_REF] is a work on knowledge capitalization through annotation among economic intelligence actors in a collaborative environment that uses annotation concept to develop a model for eliciting knowledge among CI actors.

Another interesting work that made use of annotation concept is the knowledge and information management (KIM) system presented in [START_REF] Kiryakov | Semantic annotation, indexing, and retrieval[END_REF]. The work introduces the authors' vision for a holistic architecture for semantic annotation, indexing, and retrieval of documents with regard to extensive semantic repositories. Another similar work is [START_REF] Castells | An Adaptation of the Vector-Space Model for Ontology-Based Information Retrieval[END_REF], which proposed a model for the exploitation of ontology-based knowledge bases to improve search over large document repositories. Their approach includes an ontology-based scheme for the semiautomatic annotation of documents and a retrieval system. The retrieval model is based on an adaptation of the classic vector-space model, including an annotation weighting algorithm, and a ranking algorithm.

THE DOCUMENT USAGE MODEL

The document usage is modelled as a quadruple that comprises of the following components. Users represented as U, document object represented as D, decision problem represented as P and the environment represented as E. The U component of the equation 1 has the attributes that includes id, status, exp and estab. The id is a unique identity allocated to the user during the course of registering to use the system. The status is the current position of the user in the establishment or organisation. The exp is the attribute of the user that captures the experience of the user. This experience is considered in respect of the number of years the user had spent, participating in the core business of the establishment. Note that this number of year is not necessarily the same as the number of year spent in the establishment. For example, somebody moving from one company to another company in the same line of business cannot be said to be novice in the business activity of the new company. Such a user already had cognate experience on the core business of the company and as such the year of experience in the previous company is also considered. The users' establishment information such as type of establishment (private or public) and name of the establishment are captured with the estab attribute of the user component of the document usage model. In other word, for clarity purpose the user attribute could be represented as a tuple as shown in equation 2.

U = {id, status, exp, estab} 2

The decision problem represented as P in the DU equation is an attribute used to capture the description of the decision problem by the user. It comprises the stake's object, signal and hypothesis. This representation of the decision problem with the stake attribute is inspired by the work on explanation and modelling of decision problem presented in [START_REF] Bouaka | Développement d'un modèle pour l'explicitation d'un problème décisionnel: un outil d'aide à la decisiodans un contexte d'intelligence economique[END_REF]. The stake object is used to capture the user's description of the decision problem at hand, while the stake signal and stake hypothesis allows the user to supply the level of information he has about the problem and what the establishment stands to gain or loss respectively. The P component of the document usage model can therefore be describe as a tuple as shown in equation 3. The environment (E) component of DU equation describes the environment in respect of the source of the document. It takes either of the two values internal or external. If the establishment that is creating the usage for a document is the producer of that document, then the value of environment attribute of the document usage model is internal. For example, in the case of a head of a unit in a hospital that uses medical records of patience in the hospital to examine the performance of a group of drug expended in the hospital over a period of time. The E attribute's value will be external if the source of the document is outside of the establishment.

The flow of activity in the process of usage creation is shown in figure 1. The document usage model presented in this paper is being implemented as part of a Competitive Intelligence base Document Usage Creation and Evaluation (CI-DUCE) system that is currently being developed. Therefore the flowchart in figure 1 is a part in the whole system design.

From the figure, when the user log in to create usage for a particular document, the system determine whether the logged user is an information watcher or not, if the user is an information watcher (i.e. person charged with the responsibility of information search and system update in a CI system), then the user is prompted to select the user that has authorised the creation of usage for the selected document.

The user then supplies appropriate values for the decision problem attribute and the environment attribute. Finally the document usage index is updated accordingly. If the user however is not Information watcher, then the system captures the logged user information and allows the user to go straight to supply the values for the decision problem and environment attributes.

Document Usage Model Evaluation

As stated in [START_REF] Manning | AnIntroduction to information Retrieval[END_REF], the representation of a set of documents as vector in a common vector space is known as the vector space model (VSM) and is fundamental to a host of information retrieval operations ranging from scoring documents on a query, document classification and document clustering. To evaluate the document usage model presented in this work, the evaluation was based on the VSM of the classic IR model which has been adjudged to outperform the Boolean and Probabilistic models of IR [START_REF] Baeza-Yates | Modern Information Retrieval[END_REF].

Two document representations were considered for the evaluation. The first representation is based on the key terms in the document (i.e. document index generated based on the terms in the document). This representation is the basis of document index for the existing search engines. The second representation is based on the terms in the document and the usage index (i.e. term index + usage index). Each decision problem (i.e. user query terms) is then compare with the document collection to calculate the similarity between it and the documents in the collection. Figure 2 shows the formal description of two document representation schemes.

As demonstrated in [START_REF] Berry | Understanding Search Engines: Mathematical Modelling and Text Retrieval[END_REF], a document collection comprised of n documents which are indexed by m terms can be represented as an m x n term-by-document matrix A. The n (column) vectors representing the n documents forms the columns of the matrix. Thus, the matrix element aij is the weighted frequency at which term i occurs in document j. Using the VSM, the columns of A are interpreted as the document vectors, and the rows of A are considered the term vectors.

Data for Usage Model Evaluation

Data for evaluation of the document usage model was obtained from 20 postgraduate students of the Obafemi Awolowo University, Nigeria. The data was collected with the use of questionnaire and interview. The respondents were asked to describe their research work in less than 80 words. The descriptions were used to form the stake object Obj of the decision problem attribute P. Title of six documents that the respondents found very useful to their work were also collected. The documents titles and the DP description are shown in Figure 3 for one of the respondents. From the figure, d1 through d6 represent the documents and dp represents the DP. After the pre-processing of the documents to extract the key terms, the document collection represented with only the key terms in document is shown in figure 4 and the document collection represented based on document usage model (i.e. document representation based on terms in the document and the document usage) is shown in figure 5. These two document collection are transform to document vectors as shown in equations 7 and 8.

To calculate the similarity between the documents collection represented as term-by-document matrix in equations 7 and 8, the DP stake object represented as dp in figure 3 is also represented in form of a vector in the document collection space after pre-processing task to extract the key terms. The resulting vector is shown in equation 9. The similarity between the decision problem and the documents in the document collection is determined by calculating the cosine of angle between decision problem vector P (equation 9) and the document vectors (equations 7 and 8). dj is defined as the jth document vector (i.e. the jth column of the term-by-document matrix A), then the cosines between the decision problem vector P and the n = 6 document vectors are defined by equation 10.

Term based document index

A document is represented as a vector 𝑑 ⊂ 𝐷 where 𝐷 is the vector of all the documents in the document collection space. 𝑑 is a subset of 𝐷 . The result of the cosine similarity analysis between the DP and each of the documents in the document vector space in both cases is shown in Table 1. The usual practice in the literature is to set a threshold at which document will be considered as relevance to the query and return ranked base on the similarity value. Note that the higher the similarity value, the closer the document and the query (DP in this case).

𝐷 = (d 1 , d 2 , d 3 d 4 ….. d n ) d i = document i in document collection D. 𝑑 = (
From the result in Table 1, for the term-based index, documents d4 and d5 would not be part of the returned document for a search based on the terms in the DP because their cosine similarity is 0.00000 each.

However in the term+usage based index, documents d4 and d5 not only have non-zero value for the cosine similarity, their values are even greater than that of some documents that were considered similar (i.e. relevant) to the DP based on termbased index. The is due to the level of involvement of the document in the problem resolution process even though the terms in the document may not correlate to any term in the DP description. The result in Tables 1 is depicted graphically in Figure 6. It is clear from the figure that the similarity between documents and DP increased drastically from an average value of 0.10000 for term-based to 0.80000 for usage based index. Although, for this analysis, only the titles of the documents were considered, the result will also follow the same pattern when the full texts in the documents are considered.

𝑨 = 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0

1 1 0 0 1 1 0 0 0 0 1 0 … … … . … … … . … … … .
… … … . 0 0 0 0 0 0 0 0 0 1 0 0 Where j = 1 …….. n (n= number of document in the document collection space). t = the number of terms in the vector space.

CONCLUSION

The IR process component of the CI process is very crucial to speedy and easy resolution of DPs facing an organisation. This work has sought to keep track of what documents have been used for and incorporate it into the document representation scheme to enhance the quality of IR process at a later time. The document usage model presented is used to preserve the effort of the decision makers (users) in discovering relevant documents for the resolution of DPs. The result obtained shows that representing documents in terms of their usage can enhance the quality of information search results as documents that would hitherto be considered not relevant to user query are found to be ranked very relevant based on previous usages.

The idea presented in this work can be used to develop a Decision Support System (DSS) for various industries that need to resolve one problem or the other. For example, the concept can be used to develop DSS for various units in the hospitals. For example, medical doctors can use the system to keep track of the process of treating patients (from diagnosis to drug prescription). The system can later be used by another doctor that have similar case at hand, it could also be used by the administrators to examine how doctors are faring in the hospital. A pharmacy unit of the hospital could use the system to keep track of the kind of medical conditions each drug is used to address. This will enhance the quality of decisions to be made on each drug when there is need for such.

  DU = {U, P, D, E} 1 Where DU = Document usage U = Users attributes. P = Decision problem. D = Document E = Environment. R = User's rate of document relevance to DP.

P

  of the document usage model contains information about the document to which usage is to be created. It comprise of document identity (id), and three document information elements namely document title (doc_t), document abstract or summary (doc_abs) and the rest part of the document (d_body). Worthy of mention at this point is the fact that only text documents are considered in this work. The findings in this work will be extended to other form of documents. Equation 4 presents the document attribute of the usage model as a tuple. D = {id, doc_t, doc_abs, d_body} 4

Figure 1 :

 1 Figure 1: Flowchart of Document Usage Creation Module

Figure 5 :

 5 Figure 5: Documents Representation based on terms in the document and usage

Figure 4 :

 4 Figure 4: Documents Representation based on only terms in the document

Figure 3 :

 3 Figure 3: Sample Documents and DP for Analysis

Figure 6 :

 6 Figure 6: Similarity between DP and Documents

Formal description of the Key term based and Usage based document Index

	t 1 , t 2 , t 3 , t 4 ….. t n ) t i 𝑑 = terms in the document index Term + usage based document index Two components to represent document D. a) d = (t 1 , t 2 , t 3 , t 4 ….. t n ), t i D b) decision problem stake object represented as vector 𝑃 containing terms k i used to describe the problem. 𝑃 = (k 1 , k 2 , k 3 , k 4 , …. k n ) New document representation dr is given by dr =(d . P) =( t 1 , t 2 , t 3 , t 4 ….. t n . k 1 , k 2 , k 3 , k 4 , …. k n ). Figure 2: Start Information Watcher No Log in Capture Logged User data Usage Specify Decision Problem Yes -stake object -stake signal -stake hypothesis Document User Select Usage Select document
	Select Document Source
	Environment
	-internal
	-external
	Update Document
	Usage Index
	Exit
	Stop

Table 1 : Result of Cosine Similarity Analysis between DP and Documents

 1

		Term-based	Term + Usage
	Documents	Index	Based Index
	d 1	0.23094	0.86141
	d 2	0.07454	0.76714
	d 3	0.25820	0.97373
	d 4	0.00000	0.84515
	d 5	0.00000	0.79057
	d 6	0.25820	0.84853

ACKNOWLEDGEMNET

This work was supported by the French Government Scholarship through the French Embassy in Nigeria. The Obafemi Awolowo University is hereby acknowledge for fully supporting this research effort.


Lia Lamacque

Florian Sabin

Thierry Ameglio

Guillaume Charrier

Stéphane Herbette

Variations of the stem diameter as an indicator of plant mortality under extreme drought in lavender

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Variations of the stem diameter as an indicator of plant mortality under extreme drought in lavender

Lia Lamacque 1,2 , Florian Sabin 2 , Thierry Améglio 2 , Guillaume Charrier 2 , Stéphane Herbette 2

1 Iteipmai, Station 26 Domaine de La Vesc, 26740 Montboucher-sur-Jabron ; 2 Université Clermont Auvergne, INRA, PIAF, F-63000 Clermont-Ferrand, France

Context:

Percentage Loss of Diameter as a « point of non-return »

French lavender crops suffer from a high rate of mortality due to drought events and we need to improve our capacity to monitor and predict plant mortality under drought.

Aims:

• Find a new accurate mortality index (or point of non-return) based on continuous measurement of stem diameter.

• Investigate the mechanisms of drought-related mortality in lavender PépiPIAF dendrometer on a lavender

Electrolyte leakage (%)

Key messages and perspectives

• PLD max is a point of non return in lavender species

• Is PLD max a relevant threshold in other species ?

• PLD is a new indicator that could help to anticipate lavender death in the field

• Cavitation occurs mainly before cell lysis under drought

Fig 4 .

 4 Fig 4. PLD-course of PLC (dotted line) and EL (black line) in a dehydrating L.a.The first grey dotted vertical line is the PLD 50h (PLC-driven), the red vertical line is the PLD 50 (Fig.3), the last grey vertical line is the PLD 50c (EL-driven) and the red dashed vertical line is the meanPLD max . This experiment was performed in climatic chamber at constant temperature (25°C) over ca. 10 days.

Fig 1 .

 1 Fig 1. Example of continuous diameter variation during the growing season for a control (green) and a droughted plant (blue): two successive irrigation stops followed by rewatering.