Abdelaziz Sameur

Honoré P Yin

Denis Duhamel
email: duhamel@lami.enpc.fr

Vladimir Vilke

Lami

A

Keywords: contact, viscoelastic, analytical, experiment

Introduction

The study of contact problems has its origins in the last quarter of the 19th century when Hertz [START_REF] Hertz | Über die Berührung fester elastischer Körper (On the contact of elastic solids)[END_REF] and Boussinesq [START_REF] Boussinesq | Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques[END_REF] presented solutions to the contact of linear elastic materials.

Several solutions were derived from the solution of Boussinesq, an excellent account of which is given in Galin's book [START_REF] Galin | Contact problems in the theory of elasticity[END_REF] and in Johnson [START_REF] Johnson | Contact mechanics[END_REF]. Later, Sneddon [START_REF] Sneddon | The relation between load and penetration in the axisymetric Boussinesq problem for a punch of arbitrary profile[END_REF] established a solution of the axisymmetric Boussinesq problem which enabled him to deduce simple formulas giving the penetration δ of a punch of arbitrary profile as well as the total force necessary to ensure this penetration.

Then the interest of the viscoelastic contact problem emerged, Graham [START_REF] Graham | The contact problem in the linear theory of viscoelasticity[END_REF] gave an expression for the displacement stress field produced at any point of a linear homogeneous and isotropic viscoelastic half space by an arbitrary time dependent distribution of pressure acting on its boundary. The distribution of normal surface tractions prevailing when a rigid punch of arbitrary profile is pressed against the surface of a viscoelastic half space is determined in terms of a one parameter family of solutions to the corresponding elastic problem. One of the approaches for the resolution of a contact problem of a rigid sphere with a viscoelastic material was suggested by Radok [START_REF] Radok | Viscoelastic stress analysis[END_REF]. It replaces elastic constants by an integral operator corresponding to the viscoelastic stress strain relation in which the radius of the contact area is a monotonically increasing function of time. Later, Hunter [START_REF] Hunter | The Hertz problem for a rigid spherical indenter and a viscoelastic halfspace[END_REF] studied the Hertz's problem in the case of the rebound of a rigid sphere on a viscoelastic half space so that the contact radius increases monotonically to a maximum and then decreases to zero. It was assumed that the distribution of pressure remains elliptic as in the elastic case. Ting [START_REF] Ting | The contact stresses between a rigid indenter and a viscoelastic half space[END_REF] presented a method so that the problem of contact could be solved for an arbitrary contact radius.

Numerical methods for the resolution of the contact were also developed. Webster & Sayle [START_REF] Webster | A numerical model for the elastic frictionless contact of real rough surfaces[END_REF] and Chang & Gao [START_REF] Chang | A simple numerical method for contact analysis of rough surfaces[END_REF] developed a numerical model for the elastic contact of rough surfaces. Younguing & Linquing [START_REF] Younguing | A full numerical solution for the elastic contact of three dimensional real rough surfaces[END_REF] proposed a numerical modelling for an elastic 3D contact of rough surfaces. The authors conclude that the interaction of asperities has a great effect on the calculated results, particularly on the contact deformation. The finite element method was used by Kucharski et al [START_REF] Kucharski | Finite elements model for the contact of rough surfaces[END_REF] who modelled the contact between a sphere and a rigid plane. They obtained a relation between the weight and the surface of contact. Kane et al [START_REF] Kane | Finite element analysis of nonsmooth contact[END_REF] developed robust contact algorithms able to deal with complex contact situations involving several bodies with corners. However these numerical methods lead to time consuming computations. This paper presents a semi analytical model which uses a theory of the interaction potential between a rigid body and an elastic or viscoelatic solid. The aim is to get simple and very fast estimates for the contact force for given body shapes. The contact force can be expressed in terms of the volume of interpenetration, the surface and the perimeter of the base of this volume as well as the mechanical characteristics of surfaces in contact. Then, an approach for the resolution in the case of a rubber block with a viscoelastic behaviour is presented. Radok's approach [START_REF] Radok | Viscoelastic stress analysis[END_REF] is used; it replaces elastic constants by an integral operator corresponding to the viscoelastic stress-strain relation. Experimental validations are made for different shapes of simple rigid bodies (spherical, conical and pyramidal). Finally, a comparison of contact forces in the elastic and viscoelastic case is given.

Classical contact theory

One of the first studies concerning the evaluation of the contact tensions between elastic solids was proposed by Hertz [START_REF] Hertz | Über die Berührung fester elastischer Körper (On the contact of elastic solids)[END_REF]. To express the normal force P versus the indentation, he made the assumptions that the sector of contact is elliptic, that each solid can be regarded as an elastic half space, that there is no friction between the two surfaces in contact and that the surfaces are continuous and no conform. These assumptions lead to the following well known

relation 2 3 δ K P = (1) with 3 4 * R E K = ( 2
)
where P is the normal force applied, R is the equivalent radius of curvature and * E represents the Young's equivalent modulus. The theory of the Hertz's contact is limited to surfaces with smooth and continuous profiles with finite forces everywhere. The problem is different for a surface having edges or corners. The deformation must be sufficiently small in order to be in the field of the linear theory of elasticity.

Consider a cone in contact with a plane surface, the area of contact is supposed to be small compared to the size of the two solids. The deformation is shown in Fig. 1 which presents also the pressure distribution in the contact zone. If the smooth sides of the cone are prolonged beyond the contact surface, the pressures must equal zero at the edges. A classical approach to find the efforts in an elastic half space due to external tractions is given by Boussinesq [START_REF] Boussinesq | Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques[END_REF] and Cerruti [START_REF] Cerruti | [END_REF], who used the potential theory. Love [START_REF] Love | Boussinesq's problem for a rigid cone[END_REF] applied the classical approach of the potential function to the cone and evaluated the contact pressure by the formula

) / ( cosh cot 2 1 ) ( 1 * s a E s p - = α (3)
where s is the distance between the origin O and a point ranging between O and a. Thus the normal force is given by

α π cot 2 1 * 2 E a P = (4)
The quantities α and a are defined in Fig. 1.

Contact between two bodies of complex shape

Interaction potential theory

Suppose that two bodies b 1 and b 2 are in contact at the point O. Oxy is the common tangent plane with axis OZ directed towards the body b 1 (see Fig.

By Ax z z + ≈ - (5)
The points M 1 and M 2 have coordinates (x, y, z 1 ) and (x, y, z 2 ) and belong to the surfaces of the bodies. The values A and B are constant. In general we find an additional term 2Cxy in the last part of expression ( 5) but it can be eliminated by a suitable choice for the axes Ox and Oy.

If the body b 1 moves by a distance δ in the negative direction of axis Oz and if it is assumed that the bodies interpenetrate without deformation, there will be an intersecting domain Γ whose projection on the Oxy plan forms the surface (see Fig. 2b)

{ } δ By Ax x,y σ ≤ + = 2 2 : ( 6
)
with the boundary

{ } δ σ = + = ∂ 2 2 : , By Ax y x ( 7
)
The assumption that the bodies interpenetrate without deformation is an approximation. But this is not too far from reality and this will allow getting the simple expressions that follow. It will also be justified by comparison with the experimental results.

Three characteristics of the intersecting domain will play an important role, namely, the volume V of Γ , the surface S and the perimeter p of σ

∫ ∫ = - - = σ πδ δ AB dy dx By x A V 2 ) ( 2 2 2 (8) ∫ ∫ = = σ πδ AB dy dx S , (9)
) / ( 16

1 2 / 1 B A E A p ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = δ (10) where ) / ( 1 B A E
is the elliptic integral of second species given by

∫ - - = 2 / 0 2 / 1 2 1 ) sin ) / 1 ( 1 ( ) / ( π ϕ ϕ d B A B A E ( 11
)
According to the traditional results of Hertz, in the case where the contact zone is an ellipse of

surface ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≤ + = 1 : ' 2 2 2 2 b y a x x,y σ
, the force P and the coefficients of surfaces are linked by the following relations Love [START_REF] Love | A treatise on the mathematical theory of elasticity[END_REF] ), ( 2

) ( 3 1 3 2 1 k I a P A θ θ + = ∫ ∞ + + = 0 3 2 2 2 1 ) 1 )( 1 ( ) ( z k z dz k I ), ( 2
) ( 3 2 3 2 1 k I b P B θ θ + = ∫ ∞ - + + = 0 3 2 2 2 2 ) 1 )( 1 ( ) ( z k z dz k I ( 12
)
), ( 2

) ( 3 1 2 1 k F a P θ θ δ + = ∫ - = = 2 0 2 2 1 1 1 sin 1 ) ( ) ( π ϕ ϕ k d k F k F where 1 < = a b k , 2 2 1 1 k k - = , i i i E π υ θ 2 1 - = , i =1,2, ) ( 1 k F
is the elliptic integral of first species, A and B can be obtained from Eq. ( 12)

) ( ) ( 2 1 3 k I k I k B A = (13)
It follows that the relation

1 / < = k a b
, which does not depend on the load P, is defined by the geometrical characteristics of surfaces in contact. The force P is given by (see Appendix

A.) 4 / 1 2 / 3 1 4 / 3 2 1 2 / 3 4 / 1 2 1 ) ( )) ( ( ) ( 3 )) ( ) ( ( 2 AB k F k k I k I P θ θ δ + = (14)
The potential U for the interaction of the bodies in contact, is calculated from the expression of virtual works by

4 / 1 2 / 3 1 4 / 3 2 1 2 / 5 4 / 1 2 1 ) ( )) ( ( ) ( 15
)) ( ) ( ( 4 ) ( AB k F k k I k I U P U θ θ δ δ δ δ + = ⇒ ∂ = (15)
Let us represent the relation ( 15) by the form

ξ γ β p S V k f U ) ( = (16)
where V, S and p are defined in the relations ( 8), ( 9) and [START_REF] Kucharski | Finite elements model for the contact of rough surfaces[END_REF]. By comparing the powers of δ and (AB) in the expressions ( 15) and ( 16) of U, we find 2 = β and 2

) 3

( ξ γ + - = . This leads to ) ( 15
) ( 4 ) ( 2 1 2 / ) 1 ( 1 2 θ θ π ξ ξ + = - - k f k f (17)
where

) / ( ) ( ) ( 1 2 / 3 1 4 / ) 1 ( 3 4 / ) 1 ( 2 4 / ) 1 ( 1 1 B A E k F k I I k f ξ ξ ξ ξ - - + = (18)
To study the function

) ( 1 k f , let us express ) ( 1 k I and ) ( 2 k I by the elliptic integrals 2 1 1 1 1 ) ( ) ( ) ( k k E k F k I - - = (19) )) ( ) ( ( 1 ) ( 1 2 1 2 2 k F k k E k k k I - - = (20)
Finally we obtain

) / ( ) 1 ( ) ( ) 1 ( ) ( 1 1 2 / 1 2 2 / ) 1 ( 4 / ) 1 ( 2 1 1 1 4 / ) 1 ( 1 1 1 1 B A E F k k k F E F E k f ξ ξ ξ ξ - - - = - - - + - (21) The function ) ( 1 k f remains bounded if k tends towards 1, since according to Wittaker [16] 4 ) ( lim 1 1 π = → k I k and 2 ) ( lim 1 1 π = → k F k (22) 1 / lim 1 = → B A k and 2 ) 1 ( 1 π = E (23)
Then it follows

ξ ξ π + + → = 1 5 . 0 1 1 2 ) ( lim k f k (24)
The numerical analysis of the function (21) represented in Fig. 3 shows us that the function

8 S p V c U θ θ π + = (25)
The parameter c is a constant which depends only on the geometry of contact surfaces.

According to the preceding development c equals 0.36 for regular contact surfaces and for 5 . 0 = ξ . It is supposed that the potential can be written in the same form (25) in the case of a contact of irregular surfaces. We will identify analytically and by experiments the constant c in order to validate this generalized formula in the case of bodies with arbitrary shapes.

Validation in simple shape cases

For this aim, three surface shapes are considered: the contact is between a plane surface and with spherical, conical and pyramidal rigid surfaces. For each case the relation between the force and the interpenetration will be determined. The constant c is then identified from experimental tests.

To validate and generalize the expression of the potential (25) for any surface shape, let's take the example of a spherical form (see Fig. 4a). Once the expressions of the volume, the surface and the perimeter are known, we can replace then in Eq. ( 25). The expressions of the volume, the surface and the perimeter of the part in contact are given by

) 3 ( 6 2 2 δ δ π + = r V 2 r S π = (26) r p π 2 =
where r is the radius of the base of the volume in contact and δ is the interpenetration of the sphere in the plane surface. They are expressed as function of the angle α by the relations

α sin R r = and ( ) α δ cos 1- = R (27) Suppose that α is very small thus α R r = and 2 2 α δ R = (28)
The relation between r and δ is found by

δ R r 2 2 =
. It is assumed that r << R, thus .

It would also be useful to know the expression of the force in case of a contact between a plane surface and a cone as well as in the case of a contact between a plane surface and a

pyramid. An approximation of c for various surfaces in contact enables one to generalize the contact law (25).

The same method is used for a contact between a conical surface with angleα and a plane surface (see Fig. 4b). The expressions of the volume, the surface and the perimeter of contact are evaluated and replaced in the expression of the potential (25). The volume of the part in contact is evaluated by

δ π 2 3 r V = (31)
The surface of the base in contact is The relation between r and δ is known to be δ α r tg = . So the potential equals

3 2 / 3 * 9 2 15 8 δ α π tg E c U c = (32)
The force P is written as

2 * 2 / 3 45 2 8 δ α π tg E c P c = (33)
In order to find the relation between the interpenetration δ and the radius a, let us take the Boussinesq's equations for a point force on a half space. It follows that the relation of the interpenetration δ can be written under the form

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = s G p s ) 1 ( 2 4 ) ( υ π δ (34) where ) 1 ( 2 υ + = E G
is the shear modulus and υ is the Poisson's ratio. Knowing the pressure repartition (3) given by Love [START_REF] Love | Boussinesq's problem for a rigid cone[END_REF], the interpenetrationδ which is the displacement at the apex of the cone, is written as

∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = a ds s s p s G 0 ) 1 ( 2 ) ( 2 4 1 υ π π δ (35) It becomes G E a 2 ) 1 ( cot 2 * υ α π δ - = (36)
So the relation ( 4) can be written in the form

2 * 2 2 ) 1 ( 8 δ α υ π tg E G P - = . ( 37
)
If the assumption is made that the Young's modulus of the body (1) is very small in front of the Young's modulus of the body (2)

2 1 E E E << =
, we then obtain

( ) ( ) E E E 2 1 2 1 1 1 * 1 υ υ - = - ≈ (38) It follows that * 2 ) 1 ( E G υ - =
, and the relation (36) becomes

Application to a pyramid

After considering the case of a contact between a plane surface with a sphere and a cone, it would be interesting to study the case of a contact between a plane surface and a pyramid (see Fig. 4c). The volume of the part in contact is evaluated by

δ 2 3 1 r V = (40)
The surface of the base in contact is The force is thus written in the form A model of the elastic contact between a plane surface and various surface shapes has been presented. The force P is expressed according to the interpenetrationδ , the geometrical and mechanical characteristics of the bodies in contact, and also by a constant c which was analytically identified for the spherical and conical case. The constants s c , c c and p c will also be identified experimentally and the expressions (30), ( 39), (42) will so be validated.

Extension to materials with a viscoelastic behaviour

Let us consider a pure shear stress, the stress strain relation expressed according to the shear modulus is e G s 2 = . One of the approaches for the resolution of a contact problem between a rigid sphere and a viscoelastic material was suggested by Radok [START_REF] Radok | Viscoelastic stress analysis[END_REF]. His approach replaces elastic constants by an integral operator corresponding to the viscoelastic stress strain relation

∫ ∂ ∂ - = t dt t t e t t t s 0 ' ' ) ' ( ) ' ( ) ( ψ (43)
Let us suppose that the variation of the force P according to the interpenetration δ in the case of elastic contact is written as

γ δ Q G P 2 = ( 44
)
where γ and Q depend on the form of the contact surface. By applying the Radok's method [START_REF] Radok | Viscoelastic stress analysis[END_REF], the expression of the force becomes

∫ - = t dt t dt d t t Q t P 0 ' ) ' ( ' ) ' ( ) ( γ δ ψ (45)
By taking into account the parameters γ and Q defined in Table 1, the expression of the force is finally given by -for a contact between a rigid sphere and a viscoelastic plane

Experimental validation for elastic contacts

In this purpose an experimental device is established. A compression Instron machine, bodies with various shapes (spherical, conical and pyramidal), LabView software for the acquisition of the results and a rubber block are used. Experimental relations between the force and the interpenetration are established for each body in contact with the rubber block. First the Young's modulus of the rubber block is identified by a relaxation test. Knowing the characteristics of the rubber block and the contact bodies, the coefficients s c , c c , p c corresponding to the contact models for the spherical, conical and pyramidal cases will be identified.

Experimental identification of viscoelastic parameters

A simple compression test on the rubber block presented in Fig. 5 is abruptly and quickly carried out by using a compression machine. A deformation of 8% is maintained constant and the evolution of the stress versus the time is recorded. It is noticed an abrupt and rapid increase of the stress to Mpa 0.93 0 = σ just after the load. This deformation is maintained by blocking the crossbar for a long time until the stabilization of the stress which tends towards MPa 0.6 = ∞ σ . The stress versus the time is written under the form of a Prony's series

0 / 1 / 1 / ) ... ( ) ( 1 1 ε σ τ τ τ ∞ - - - - + + + + = - E e A e A e A t t t n t n n n (49)
For the rubber block four characteristics times are found and the results are represented in table 2. After reconstitution of the relaxation curve from characteristics times and amplitudes, in Fig. 6, we notice that there is a good concordance of the two curves. Using less than four characteristic times doesn't allow a good reconstitution of the curve over the whole time range. The static Young's modulus of the block is found to be

MPa E 5 . 7 0 = = ∞ ∞ ε σ

Experimental validation of the elastic contact model

Contact between a rubber block and a sphere

The experimental device is the same as the one used for the relaxation test (see Fig. 7a). The load is carried out at a speed of 0.001 mm/s to simulate a static loading with the Young's . According to Eq. ( 30) and to the experimental results of the contact between the rubber block and the steel ball, it was found

10 . 14 * 2 / 3 3 2 e R E c s = π (50)
Knowing the mechanical and geometrical characteristics of the two bodies in contact, the experimental value from the coefficient s c is deduced and equals 0.34. The theoretical value of the coefficient s c equals 0.36, which is slightly different from the experimental value. This result allows to validate the experiment by recovering the Hertz's result.

Contact between a rubber block and a cone

The same test as the one used for the contact between the rubber block and the sphere is carried out. Two steel cones that have different angles ( °= 60 α and °= 45 α ) are used in order to study the influence of the angle on the distribution of the force. The experimental device is presented in Fig. 7b. The force versus the interpenetration for each cone in contact is given in Fig. 8b. By plotting the logarithmic curve of the force as function of the interpenetration and after having carried out a linear regression for each curves it is found for the cone with The evaluation of the force from the expression (33) adapted for the case of a contact between a plane surface and a cone is given by

2 * 2 / 3 45 2 8 δ α π δ tg E c U P c = ∂ ∂ = (51)

Contact between a rubber block and a pyramid

The expression of the force versus the interpenetration is The coefficients s c and c c were identified experimentally and analytically. p c was identified only experimentally since there is no analytical model for the pyramidal case. In Table 3 the values obtained for various surfaces are presented. The differences between the experimental and the analytical results are quite small. Let's conclude that the value of the coefficient s c equals 0.36. For the conical case the value of the coefficient c c equals 0.45, which is true for any angle α. Considering the absence of a classical analytical theory for the pyramidal shape, we can however conclude that the contact law has the same shape as for the other cases with p c which equals 0.53.

Experimental validation for viscoelastic contacts

The expression of the force according to the interpenetration is put in the form (46), (47) and (48). In the same way as for the case of an elastic contact, experimental tests are carried out in order to validate viscoelastic contact models.

Contact between a rubber block and a steel ball

In this test the same experimental device is used, except that instead of charging at a speed of 0,001 mm/s, the loading is done at a speed of 5 mm/s. The same rubber block and the same steel ball are used so the geometrical and mechanical characteristics are well known. In Fig. 9, we can see the evolution of the force P according to the interpenetration δ for the spherical, conical and pyramidal cases. The curves show that in all cases there is a good concordance between the two results. A comparison between the results obtained in the case of the elastic analytic model and in the case of the viscoelastic analytic model concludes that the value of the force in the case of a viscoelastic contact is higher by 24% than the value of the force in the case of an elastic contact.

Conclusion

The theoretical model of the potential interaction for contacts between regular surfaces was presented and generalized for irregular surfaces. The validation of this model in the case of a contact between a plane surface with a sphere, a cone and a pyramid was made. Then an approach of the solution of a viscoelastic contact problem was seen, where the method of Radok [START_REF] Radok | Viscoelastic stress analysis[END_REF], which consists of replacing elastic constants by an integral operator, was applied.

The viscoelastic behaviour of the material was modeled by a Prony's series. The Young's modulus and the characteristic times of a rubber block were identified. Once these characteristics were identified, tests were carried out in which the rubber block was put into contact with various surfaces (spherical, conical and pyramidal). These tests made it possible to identify the coefficients s c , c c and p c .Thus, we validated the models of elastic contact.

Using the same method, tests of viscoelastic contact at a speed of 5 mm/s with the same rubber block and the various bodies were carried out. The results were compared to the results of a viscoelastic contact model. The results of the model and those of the experiments agree very well. A comparison between the results of the elastic contact model and the results of the viscoelastic contact model shows that for a given value of interpenetration δ , the viscoelastic force is 25% higher than the elastic force for this load speed. All these results validate the simple semi-analytical model developed in this paper for elastic and viscoelastic contacts of bodies with arbitrary shapes.

Appendix A. Calculation of the force P

The force P and the coefficients of surfaces σ and ' σ are linked by the following relations

Love [START_REF] Love | A treatise on the mathematical theory of elasticity[END_REF] ), ( 2

) ( 3 1 3 2 1 k I a P A θ θ + = ∫ ∞ + + = 0 3 2 2 2 1 ) 1 )( 1 ( ) ( z k z dz k I , (A.1) ), ( 2
) ( 3 2 3 2 1 k I b P B θ θ + = ∫ ∞ - + + = 0 3 2 2 2 2 ) 1 )( 1 ( ) ( z k z dz k I , (A.2) ), ( 2
) ( 3 1 2 1 k F a P θ θ δ + = ∫ - = = 2 0 2 2 1 1 1 sin 1 ) ( ) ( π ϕ ϕ k d k F k F , (A.3) where 1 < = a b k , 2 2 1 1 k k - = and i i i E π υ θ 2 1 - = , i =1,2.
From Eqs. (A.1) and (A.2) we obtain

Figure captions

  2.a). The distance between the points M 1 and M 2 of the two bodies in the vicinity of the point O

i E and iυ

  are the Young's modulus and the Poisson's ratio of the body b i . A relation between

  is summarized in the case of a plane contact/sphere by 2 30) expresses the force as function of the interpenetration in the case of a contact between a plane surface and a spherical surface. The constant s c can be determined by identification with the Hertz's law which leads to 36

.

  Fig.8arepresents the evolution of the force versus the interpenetration. By linear regression of the curve we obtained a line that has the following

The values of the coefficients are deducted

  = α 45°. The value of the coefficient c c does not differ much for different angles of the cone and from the analytical value which equals 0.45.

  identify the coefficient p c , the experimental device used for the identification of the coefficients s c and c c is taken again. The pyramid has an angle °= 68 α (see Fig. 7c.). After having plotted the logarithmic curve (see Fig. 8c.) of the force variation versus the interpenetration,

Fig. 1 .

 1 Fig. 1. Contact cone / plane surface.

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. (a) Contact between two bodies of an arbitrary form, (b) Domain of intersection Γ .

Fig. 5 .Fig. 6 .

 56 Fig. 5. Relaxation test.

Fig. 7 .

 7 Fig. 7. Experimental device for a contact between a rubber block and (a) a sphere, (b) a cone,

Fig. 8 .

 8 Fig. 8. Evolution of the force upon the interpenetration for the sphere contact (a), the cone

Fig. 9 .

 9 Fig. 9. Comparison between viscoelastic experimental results and analytic results for the

Fig. 1 .Fig. 2 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 124567 Fig. 1. Contact cone / plane surface.

Table 1

 1 Parameters γ and Q for the various forms of contact surfaces.

	Shapes	Spherical	Conical		Pyramidal
	γ	3/2		2				2
	Q	3 8	R	π 4	tg	α	c p	45 32	α π tg 4 / 3 2

Table 2

 2 Characteristic times and amplitudes for the rubber block.

	i	1	2	3	4
	i τ (second)	33333.33	1110.00	64.94	3.37
	i A (Mpa)	0.58	0.79	1.56	2.29

It follows that the force relation is given by


Jovan Pehcevski
email: jovan.pehcevski@inria.fr

Anne-Marie Vercoustre
email: anne-marie.vercoustre@inria.fr

James A Thom
email: james.thom@rmit.edu.au

Exploiting Locality of Wikipedia Links in Entity Ranking

Information retrieval from web and XML document collections is ever more focused on returning entities instead of web pages or XML elements. There are many research fields involving named entities; one such field is known as entity ranking, where one goal is to rank entities in response to a query supported with a short list of entity examples. In this paper, we describe our approach to ranking entities from the Wikipedia XML document collection. Our approach utilises the known categories and the link structure of Wikipedia, and more importantly, exploits link co-occurrences to improve the effectiveness of entity ranking. Using the broad context of a full Wikipedia page as a baseline, we evaluate two different algorithms for identifying narrow contexts around the entity examples: one that uses predefined types of elements such as paragraphs, lists and tables; and another that dynamically identifies the contexts by utilising the underlying XML document structure. Our experiments demonstrate that the locality of Wikipedia links can be exploited to significantly improve the effectiveness of entity ranking.

Introduction

The traditional entity extraction problem is to extract named entities from plain text using natural language processing techniques or statistical methods and intensive training from large collections. The primary goal is to tag those entities and use the tag names to support future information retrieval. Entity ranking has recently emerged as a research field that aims at retrieving entities as answers to a query. Here the goal is not to tag the names of the entities in documents but rather to get back a list of the relevant entity names. It is a generalisation of the expert search task explored by the TREC Enterprise track [START_REF] Soboroff | Overview of the TREC 2006 Enterprise track[END_REF], except that instead of ranking people who are experts in the given topic, other types of entities such as organisations, countries, or locations can also be retrieved and ranked. For example, the query "European countries where I can pay with Euros" should return a list of entities representing relevant countries, and not a list of entities about the Euro and similar currencies.

The Initiative for the Evaluation of XML retrieval (INEX) has a new track on entity ranking, using Wikipedia as its XML document collection [START_REF] De Vries | INEX 2007 Entity ranking track guidelines[END_REF]. Two tasks are explored by the INEX 2007 entity ranking track: entity ranking, which aims at retrieving entities of a given category that satisfy a topic described in natural language text; and list completion, where given a topic text and a small number of entity examples, the aim is to complete this partial list of answers. The inclusion of the target category (in the first task) and entity examples (in the second task) makes these quite different tasks from the task of full-text retrieval, and the combination of the query and entity examples (in the second task) makes it quite different from the task addressed by an application such as Google Sets 3 where only entity examples are provided.

In this paper, we describe our approach to ranking entities from the Wikipedia XML document collection. Our approach is based on the following principles:

1. A good entity page is a page that answers the query (or a query extended with entity examples). 2. A good entity page is a page associated with categories close to the categories of the entity examples. 3. A good entity page is pointed to by a page answering the query; this is an adaptation of the HITS [START_REF] Kleinberg | Authoritative sources in hyperlinked environment[END_REF] algorithm to the problem of entity ranking. 4. A good entity page is pointed to by contexts with many occurrences of the entity examples. A broad context could be the full page that contains the entity examples, while smaller and more narrow contexts could be elements such as paragraphs, lists, or tables.

Specifically, we focus on whether the locality of Wikipedia links around entity examples can be exploited to improve the effectiveness of entity ranking.

Related work

In this section, we review some related work on link analysis and entity disambiguation and extraction.

Link analysis

To calculate the similarity between a document and a query, most information retrieval (IR) systems use statistical information concerning the distribution of the query terms, both within the document and the collection as a whole. However, in hyperlinked environments, such as the World Wide Web and Wikipedia, link analysis is important. PageRank [START_REF] Brin | The anatomy of a large-scale hypertextual Web search engine[END_REF] and HITS [START_REF] Kleinberg | Authoritative sources in hyperlinked environment[END_REF] are two of the most popular algorithms that use link analysis to improve web search.

We use the idea behind PageRank and HITS in our approach; however, instead of counting every possible link referring to an entity page in the collection (as with PageRank), or building a neighbourhood graph (as with HITS), we only consider pages that are pointed to by a selected number of top-ranked pages for the query. This also makes our link ranking algorithm to be query-dependent (just like HITS), which allows for it to be dynamically calculated at query time.

Cai et al. [START_REF] Cai | Block-level link analysis[END_REF] recognise that most popular linkrank algorithms treat a web page as a single node, despite the fact that the page may contain multiple semantic blocks. Using the visual presentation of a page to extract the semantic structure, they adapted PageRank and HITS to deal with block nodes rather than full page nodes. Nie et al. [START_REF] Nie | Topical link analysis for web search[END_REF] propose a topical link analysis model that formalises the idea of splitting the credit (the authority score) of a source page into different topics based on topical distribution. Our entity ranking approach is based on a similar idea, except that instead of using topics for discrimination we use list-like contexts around the entity examples.

Entity disambiguation and extraction Kazama and Torisawa [START_REF] Kazama | Exploiting Wikipedia as external knowledge for named entity recognition[END_REF] explore the use of Wikipedia as external knowledge to improve named entity recognition, by using the first sentence of a Wikipedia page to infer the category of the entity attached to that page. These categories are then used as features in their named entity tagger. We do not use inferred categories in our approach; instead, we use categories that were explicitly associated with the entity page by Wikipedia authors. Cucerzan [START_REF] Cucerzan | Large-scale named entity disambiguation based on Wikipedia data[END_REF] also uses Wikipedia for entity disambiguation by exploiting (amongst other features) co-references in static contexts such as titles, links, paragraphs and lists. Callan and Mitamura [START_REF] Callan | Knowledge-based extraction of named entities[END_REF] investigate if the entity extraction rules can be dynamically generated. Their rules are based on heuristics exploiting a few pre-defined HTML contexts such as lists and tables. The contexts are weighted according to the number of contained examples; the best contexts are then used to dynamically extract new data. We use pre-defined contexts in our entity ranking approach; however, we also develop a new algorithm that dynamically determines the contexts around entity examples.

ESTER [START_REF] Bast | ESTER: efficient search on text, entities, and relations[END_REF] was recently proposed as a system for searching text, entities and relations. ESTER relies on the Wikipedia links to identify the entities and on the context of the links for disambiguation (using 20 words around the anchor text instead of just the anchor text). This approach primarily focuses on improving the efficiency of the proposed system, while we are more interested in improving the effectiveness of entity ranking.

The Wikipedia XML document collection

Wikipedia is a well known web-based, multilingual, free content encyclopedia written collaboratively by contributors from around the world. As it is fast growing and evolving it is not possible to use the actual online Wikipedia for experiments, and so we need a stable collection to do evaluation experiments that can be compared over time. Denoyer and Gallinari [START_REF] Denoyer | The Wikipedia XML corpus[END_REF] have developed an XML-based corpus based on a snapshot of the Wikipedia, which has been used by various INEX tracks in 2006. It differs from the real Wikipedia in some respects (size, document format, category tables), but it is a very realistic approximation.

Entities in Wikipedia In Wikipedia, an entity is generally associated with an article (a Wikipedia page) describing this entity. For example, there is a page for "The euro . . . is the official currency of the Eurozone (also known as the Euro Area), which consists of the European states of Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, the Netherlands, Portugal, Slovenia and Spain, and will extend to include Cyprus and Malta from 1 January 2008." Fig. 1. Extract from the Euro Wikipedia page every country, most famous people or organisations, places to visit, and so forth. In Wikipedia nearly everything can be seen as an entity with an associated page.

The entities have a name (the name of the corresponding page) and a unique ID in the collection. When mentioning such an entity in a new Wikipedia article, authors are encouraged to link every occurrence of the entity name to the page describing this entity. For example, in the Euro page (see Fig. 1), all the underlined hypertext links can be seen as occurrences of entities that are each linked to their corresponding pages. In this figure, there are 18 entity references of which 15 are country names; more specifically, these countries are all "European Union member states", which brings us to the notion of category in Wikipedia.

Categories in Wikipedia Wikipedia offers categories that authors can associate with Wikipedia pages. New categories can also be created by authors, although they have to follow Wikipedia recommendations in both creating new categories and associating them with pages. When searching for entities it is natural to take advantage of the Wikipedia categories since they would give a hint on whether the retrieved entities are of the expected type. For example, when looking for entities "authors", pages associated with the category "Novelist" may be more relevant than pages associated with the category "Book".

Our entity ranking approach

We are addressing the task of ranking entities in answer to a query supplied with a few examples (task 2). However, our approach can also be used for entity ranking tasks where the category of the target entities is given and no examples are provided (task1).

Our approach to identifying and ranking entities combines: (1) the full-text similarity of the entity page with the query; (2) the similarity of the page's categories with the categories of the entity examples; and (3) the link contexts found in the top ranked pages returned by a search engine for the query.

Architecture

Our entity ranking approach involves several modules and functions that are used for processing a query, submitting it to the search engine, applying our entity ranking algorithms, and finally returning a ranked list of entities. We used Zettair4 as our choice for a full-text search engine. Zettair is a full-text IR system developed by RMIT University, which returns pages ranked by their similarity scores to the query. Zettair is "one of the most complete engines" according to a recent comparison of open source search engines [START_REF] Middleton | A comparison of open source search engines[END_REF]. We used the Okapi BM25 similarity measure which was shown to be very effective on the INEX 2006 Wikipedia test collection [START_REF] Iskandar | Social media retrieval using image features and structured text[END_REF].

The architecture of our approach is described as follows. The topic module takes an INEX topic as input and generates the corresponding Zettair query and the list of entity examples (as one option, the names of the entity examples may be added to the query). The search module sends the query to Zettair and returns a list of scored Wikipedia pages. The link extraction module extracts the links to target entities from a selected number of highly ranked pages, together with the information about the paths of the links (using an XPath notation). The linkrank module calculates a weight for a target entity based on (amongst other factors) the number of links to this entity and the number of entity examples that appear in the context of the link. The category similarity module calculates a weight for a target entity based on the similarity of its categories with that of the entity examples. The full-text IR module calculates a weight for a target entity based on its initial Zettair score. Finally, the global score for a target entity is calculated as a linear combination of three normalised scores coming out of the last three modules.

The above architecture provides a general framework for entity ranking which allows for replacing some modules by more advanced modules, or by providing a more efficient implementation of a module. It also uses an evaluation module to assist in tuning the modules by varying the parameters and to globally evaluate the entity ranking approach.

Score functions and parameters

The global score of an entity page is derived by combining three separate scores: a linkrank score, a category score, and a full-text similarity score.

LinkRank score The linkrank function calculates a score for a page, based on the number of links to this page, from the first N pages returned by the search engine in response to the query. The parameter N has been kept to a relatively small value mainly for performance purposes, since Wikipedia pages contain many links that would need to be extracted. We carried out experiments with different values of the parameter N, by varying it between 5 and 100 with a step of 5, and found that N=20 was a good compromise between performance and discovering more potentially good entities.

The linkrank function can be implemented in a variety of ways; we have implemented a linkrank function that, for a target entity page t, takes into account the Zettair score of the referring page z(p), the number of distinct entity examples in the referring page #ent(p), and the locality of links around the entity examples:

S L (t) = N r=1   z(p r ) • g(#ent(p r )) • lt∈L(pr,t) f (l t , c r |c r ∈ C(p r ))   (1)
where g(x) = x + 0.5 (we use 0.5 to allow for cases where there are no entity examples in the referring page); l t is a link that belongs to the set of links L(p r , t) that point to the target entity t from the page p r ; c r is a context around entity examples that belongs to a set of contexts C(p r ) found for the page p r ; and f (l t , c r ) represents the weight associated to the link l t that belongs to the context c r . The contexts are explained in full detail in sub-section 4.3.

The weighting function f (l r , c r ) is represented as follows:

f (l r , c r ) =      1 if c r = p r (the context is the full page) 1 + #ent(c r ) if c r = e r (the context is an XML element)
Category similarity score To calculate the category similarity score, we use a very basic similarity function that computes the ratio of common categories between the set of categories associated with the target page cat(t) and the set of the union of the categories associated with the entity examples cat(E):

S C (t) = |cat(t) ∩ cat(E)| |cat(E)| (2)
Z score The full-text (Z) score assigns the initial Zettair score to a target entity page. If the target entity does not appear among the initial ranked list of pages returned by Zettair, then its Z score is zero:

S Z (t) =      z(t) if page t was returned by Zettair 0 otherwise (3)
Global score The global score S(t) for a target entity page is calculated as a linear combination of three scores, the linkrank score S L (t), the category similarity score S C (t), and the Z score S Z (t):

S(t) = αS L (t) + βS C (t) + (1 -α -β)S Z (t) (4)
where α and β are two parameters that can be tuned differently depending on the entity retrieval task.

We consider some special cases that allow us to evaluate the effectiveness of each module: α = 1, β = 0, which uses only the linkrank score; α = 0, β = 1, which uses only the category score; and α = 0, β = 0, which uses only the Z score. 5 More combinations of the two parameters are explored in the tuning phase of our approach (section 5).

Exploiting locality of links

The main assumption behind the idea of exploiting locality of links in entity ranking is that references to entities (links) located in close proximity to the entity examples, which typically appear in list-like contexts, are more likely to represent relevant entities than links that appear in other parts of the page. Here, the notion of list refers to grouping together objects of the same (or similar) nature. The aim is therefore to assign a bigger weight to links that co-occur with links to entity examples in such list-like contexts.

Consider the example of the Euro page shown in Fig. 1. Let us assume that the topic is "European countries where I can pay with Euros", and France, Germany and Spain are three entity examples. We see that the 15 countries that are members of the Eurozone are all listed in the same paragraph with the three entity examples. In fact, there are other contexts in this page where those 15 countries also co-occur together. By contrast, although there are a few references to the United Kingdom in the Euro page, it does not occur in the same context as the three examples (except for the page itself).

Statically defined contexts We have identified three types of elements that correspond to list-like contexts in the Wikipedia XML document collection: paragraphs (tag p); lists (tags normallist, numberlist, and definitionlist); and tables (tag table). We design two algorithms for identifying the static contexts: one that identifies the context on the basis of the leftmost occurrence of the pre-defined tags (StatL), and another that uses the rightmost occurrence of the pre-defined tags to identify the context (StatR). We do this to investigate whether the recursive occurrences of the same tag, as often found in many XML documents in the INEX Wikipedia collection, has an impact on the ability to better identify relevant entities.

Consider Table 1, where the links to entity examples are identified by their absolute XPath notations. The three non-overlapping elements that will be identified by the StatL algorithm are the elements p [START_REF] Iskandar | Social media retrieval using image features and structured text[END_REF], p [START_REF] Brin | The anatomy of a large-scale hypertextual Web search engine[END_REF], and normallist [START_REF] Iskandar | Social media retrieval using image features and structured text[END_REF], while with the StatR algorithm p [START_REF] Callan | Knowledge-based extraction of named entities[END_REF] will be identified instead of p [START_REF] Brin | The anatomy of a large-scale hypertextual Web search engine[END_REF] in addition to also identifying the other two elements.

The main drawback of the static approach is that it requires a pre-defined list of element contexts which is totally dependent on the document collection. The advantage is that, once defined, the list-like contexts are easy to identify. Dynamically defined contexts To determine the contexts dynamically, we adapted the concept of coherent retrieval elements [START_REF] Pehcevski | Hybrid XML retrieval: Combining information retrieval and a native XML database[END_REF] initially used to identify the appropriate granularity of elements to return as answers in XML retrieval.

For the list of extracted entities corresponding to entity examples, a Coherent Retrieval Element (CRE) is defined as an element that represents the lowest common ancestor (LCA) of at least two entity examples. To identify the CREs, we sequentially process the list of extracted entity examples by considering every pair of elements, starting from the first element down to the element preceding the last element in the list. For each pair of elements, their LCA is chosen to represent a dynamic context (a CRE). Starting from the first identified CRE, we filter the overlapping elements and end up with a final list of (one or more) non-overlapping CREs that represent the dynamically defined contexts for the page. 6 We refer to this algorithm as DynCRE.

For example, the two dynamic contexts that will be identified for the list of extracted entity examples shown in Table 1 are p [START_REF] Iskandar | Social media retrieval using image features and structured text[END_REF] and normallist [START_REF] Iskandar | Social media retrieval using image features and structured text[END_REF].

Although body [START_REF] Iskandar | Social media retrieval using image features and structured text[END_REF] was also initially identified as a CRE, it was subsequently filtered from the final list since it overlaps with p [START_REF] Iskandar | Social media retrieval using image features and structured text[END_REF] (the first identified CRE).

The main advantage of the dynamic approach is that it is independent of the document collection, and it does not require a pre-defined list of contexts. The possible drawback is that narrow contexts containing only one entity example (such as p [START_REF] Callan | Knowledge-based extraction of named entities[END_REF] in Table 1) are never identified.

Experimental results

We now present results that investigate the effectiveness of our entity ranking approach when using different types of contexts around the entity examples.

Test collection

Since there was no existing set of topics with relevance assessments for entity ranking, we developed our own test collection, which we made available as a training set for other participants in the INEX 2007 entity ranking track. So for these experiments we used our own test collection based on a selection of topics from the INEX 2006 ad hoc track, since most of these topics reflect real-life tasks represented by queries very similar to the short Web queries. We chose 27 topics that we considered were of an "entity ranking" nature, where for each page that had been assessed as containing relevant information, we reassessed whether or not it was an entity answer, and whether it loosely belonged to a category of entity we had loosely identified as being the target of the topic. If there were entity examples mentioned in the original topic these were usually used as entity examples in the entity topic. Otherwise, a selected number (typically 2 or 3) of entity examples were chosen somewhat arbitrarily from the relevance assessments. To this set of 27 topics we also added the Euro topic example that we had created by hand from the original INEX description of the entity ranking track [START_REF] De Vries | INEX 2007 Entity ranking track guidelines[END_REF], resulting in total of 28 entity ranking topics.

We use mean average precision (MAP) as our primary method of evaluation, but also report results using several alternative IR measures: mean of P [START_REF] Callan | Knowledge-based extraction of named entities[END_REF] and P [START_REF] Kleinberg | Authoritative sources in hyperlinked environment[END_REF] (mean precision at top 5 or 10 entities returned), and mean R-precision. We remove the entity examples both from the list of returned answers and from the relevance assessments, as the task is to find entities other than those provided.

Full page context

We used the context of the full page to determine suitable values for the parameters α and β, and also to try out some minor variations to our entity ranking approach (such as whether or not to include the names of the entity examples in the query sent to Zettair).

We calculated MAP over the 28 topics in our test collection, as we varied α from 0 to 1 in steps of 0.1. For each value of α, we also varied β from 0 to (1-α) in steps of 0.1. We found that the highest MAP (0.3570) on this data set is achieved for α = 0.1 and β = 0.8. We also trained using mean R-precision instead of MAP as our evaluation measure, but we also observed the same optimal values for the two parameters.

We used a selected number of runs to carry out a more detailed investigation of the performance achieved by each independent module and by the optimal module combination. We also investigated whether adding names of the entity examples to the query sent to Zettair would have a positive performance impact. The results of these investigations are shown in Tables 2(Q) and 2(QE).

Several observations can be drawn from these results. First, adding names of the entity examples to the query sent to Zettair generally performs worse for all but the linkrank module, for which we see a consistent performance improvement. Second, different optimal values are observed for the two parameters in the two tables, which suggest that adding the entity examples to the query can dramatically influence the retrieval performance. Third, we observe that the best entity ranking approaches are those that combine the ranking evidence from the three modules (runs α0.1-β0.8 for Q and α0.2-β0.6 for QE). With MAP, these two runs perform significantly better (p < 0.05) than the plain Zettair full-text retrieval run, and they are also significantly better than any of the three runs representing each individual module in our entity ranking approach. These results therefore show that the global score (the combination of the three individual scores), optimised in a way to give more weight on the category score, brings the best value in retrieving the relevant entities for the INEX Wikipedia document collection. However, the results also show that using only the linkrank module and the context of the full page results in a very poor entity ranking strategy, which is why below we also experiment with narrow contexts.

Static and dynamic contexts

We now investigate whether using smaller and more narrow contexts has a positive impact on the effectiveness of entity ranking. Tables 3(Q) and 3(QE) show the results of this investigation. These results reflect the case when only the linkrank module (α1.0-β0.0) is used by our entity ranking approach.

As in the case with using the full page context, for all the four runs we observe a consistent performance improvement when names of the entity examples are added to the query sent to Zettair. Importantly, when compared to the baseline (the full page context), we observe a substantial increase in performance for the three runs that use smaller and more narrow contexts, irrespective of the type of query used. These increases in performance are all statistically significant (p < 0.05). However, the type of query sent to Zettair (Q or QE) seems to have an impact on the best performance that could be achieved by these three runs. Specifically, with MAP the StatL run performs best among the three runs when only the topic title is used as an input query (Q), while the StatR run is best when using terms from the topic title and the names of entity examples (QE). In both cases the DynCRE run achieves the best early precision but overall performs worst among the three runs, although the differences in performance between each of the three run pairs are not statistically significant. Implementing narrow contexts in our linkrank module allows for the locality of links to be exploited in entity ranking. By changing the context around entity examples, we would also expect the optimal values for the two combining parameters to change. We therefore varied the values for α and β and re-calculated MAP over the 28 topics in our test collection. For the three runs using narrow contexts we found that the optimal value for α has shifted from 0.1 to 0.2 (in the case of Q), while for the two static runs the optimal α value was 0.3 (in the case of QE). In both cases, the optimal value for β was found to be 0.6. The performances of the three optimal runs were very similar, and all of them substantially outperformed the optimal run using the full page context.

Conclusion and future work

We have presented our entity ranking approach for the INEX Wikipedia XML document collection which is based on exploiting the interesting structural and semantic properties of the collection. We have shown in our evaluations that the use of the categories and the locality of Wikipedia links around entity examples has a positive impact on the performance of entity ranking.

In the future, we plan to further improve our linkrank algorithm by varying the number of entity examples and incorporating relevance feedback that we expect would reveal other useful entities that could be used to identify better contexts. We also plan to carry out a detailed per-topic error analysis, which should allow us to determine the effect of the topic type on entity ranking. Finally, our active participation in the INEX entity ranking track will enable us to compare the performance of our entity ranking approach to those achieved by other state-of-the-art approaches.

Table 1 .

 1 List of links referring to entity examples (France, Germany, and Spain), extracted for the Euro topic.

Table 2 .

 2 Performance scores for runs using the context of the full page, obtained by different evaluation measures. Queries sent to Zettair include only terms from the topic title (Q), or terms from the topic title and the names of entity examples (QE). For each measure, the best performing score is shown in bold.

		P[r]				P[r]
	Run	5	10	R-prec MAP	Run	5	10	R-prec MAP
	Zettair	0.2286 0.2321 0.2078 0.1718	Zettair	0.2000 0.1714 0.1574 0.1427
	α0.0-β0.0 0.2286 0.2321 0.2135 0.1780	α0.0-β0.0 0.2000 0.1714 0.1775 0.1533
	α0.0-β1.0 0.3643 0.3071 0.3151 0.3089	α0.0-β1.0 0.3357 0.2821 0.2749 0.2674
	α1.0-β0.0 0.1571 0.1571 0.1385 0.1314	α1.0-β0.0 0.1857 0.1750 0.1587 0.1520
	α0.1-β0.8 0.4714 0.3857 0.3902 0.3570	α0.1-β0.8 0.3357 0.3286 0.3109 0.3140
	α0.2-β0.6 0.4357 0.3786 0.3751 0.3453	α0.2-β0.6 0.3429 0.3357 0.3362 0.3242
		(Q) Topic title	(QE) Topic title and entity examples

Table 3 .

 3 Performance scores for runs using different types of contexts in the linkrank module (α1.0-β0.0), obtained by different evaluation measures. Queries sent to Zettair include only terms from the topic title (Q), or terms from the topic title and the names of entity examples (QE). For each measure, the best performing score is shown in bold.

		P[r]				P[r]
	Run	5	10	R-prec MAP	Run	5	10	R-prec MAP
	FullPage 0.1571 0.1571 0.1385 0.1314	FullPage 0.1857 0.1750 0.1587 0.1520
	StatL	0.2143 0.2250 0.2285 0.1902	StatL	0.2429 0.2179 0.2256 0.2033
	StatR	0.2214 0.2143 0.2191 0.1853	StatR	0.2429 0.2214 0.2248 0.2042
	DynCRE 0.2214 0.2107 0.2152 0.1828	DynCRE 0.2571 0.2107 0.2207 0.1938
		(Q) Topic title	(QE) Topic title and entity examples

http://labs.google.com/sets

http://www.seg.rmit.edu.au/zettair/

This is not the same as the plain Zettair score, as apart from target entities corresponding to the highest N pages returned by Zettair, the remaining entities are all generated by extracting links from these pages, which may or may not correspond to the ranked pages returned by Zettair.

When the page contains exactly one entity example, the document element (article[START_REF] Iskandar | Social media retrieval using image features and structured text[END_REF]) is chosen to represent a CRE.

Acknowledgements

Part of this work was completed while James Thom was visiting INRIA in 2007.