Mhammed Ettioutioui

Analyses de la conjecture de Syracuse : Démonstration

We give in this paper an analysis of differentes prorieties of Syracuse sequence. And we give a proof of the Syracuse conjecture by considering the inverse sequence and modular studies.

Résumé

Nous donnons dans cet article, nous étudions des différents aspects de la suite de Syracuse. Et nous donnons une démonstration de la conjecture de Syracuse en faisant appel à une suite inverse et en utilisant une représentation modulaire.

L'itération de cette suite en partant d'une valeur initiale N semble aboutir à 1 après un nombre d'étapes, c'est le temps de vol. Ce fait constitue la conjecture de Syracuse qui peut se formuler de la manière suivante :

Conjecture. Pour tout entier N , il existe n ∈ N tel que S n (N ) = 1.

Elle a été étudiée de différentes manières. On renvoi à l'article de Luc-Olivier Pochon et Alain Favre [P-F] qui est très fourni et constitue une référence extrêmement complète sur les différentes pistes et avancées sur ce sujet. Lire aussi l'article de Jean-Paul Delahaye [D] tout aussi complet et qui contient des références et une histoire des différentes recherches. On va considérer une formulation inverse en définissant les fonctions :

f (n) = 2n g(n) = n-1
Lemme. Pour tout entier N , il existe n ∈ N tel que S n (N ) = 1 est équivalent au fait que l'ensemble E 0 soit égal à l'ensemble N.

Remarque. La condition de parité ou non est remplacée par une condition sur l'application g moins contraignante.

On démontre :

Théorème. L'ensemble E 0 est égale à N.

Les études qui suivent ne servent pas qu'a démontrer la conjecture mais aussi à donner des propriétés pour comprendre le comportement des suites de compositions des fonctions f et g ainsi que de la suite de syracuse.

On représente le comportement de f et g sur des graphes modulo 9 qui permettent de voir des aspects qualitatives liés à ces fonctions.

2 Propriétés et résultats.

2.1 Analyse des fonctions f et g.

L'ensemble E 0 est formé de tous les éléments qui sont les images itérées de 1 par les applications f et g lorsque c'est possible. On commence par un exemple :

1 f -→ 2 f -→ 4 f -→ 8 f -→ 16 g -→ 5 f -→ 10 f -→ 20 f -→ 40 g -→ 13 13 f -→ 26 f -→ 52 g -→ 17 f -→ 34 g -→ 11 f -→ 22 g -→ 7.
Ceci peut être écrit sous la forme :

7 = g • f • g • f • g • f • f • g • f • f • f • g • f • f • f • f (1)
une suite de composition de f et g. Un autre exemple de construction dans laquelle les chiffres entre parenthèses (i; j) désignent respectivement i la classe du nombre en base de quatre et j la classe du nombre en base de trois. Quand le nombre obtenue est un multiple de trois, la suite continue par multiplications successives par 2. On part du nombre 7.

(3;1) On donne ici une liste propriétés pour comprendre les fonctions f et g et permettre de visualiser la formulation inverse, l'ensemble E 0 et la suite de Syracuse :

7 f -→ (2;2) 14 f -→ (0;1) 28  g (1;2) 9 f -→ (0;2) 56 f -→ (0;1) 112  g (1;1) 37  g (0;0) 12 f -→ (0;2) 224 f -→ (0;1) 448  g (1;2) 149 f -→ (0;
1. ∀k ∈ N * , f (k) (1) = 2 k ∈ E 0 . 2. ∀n ∈ N * , si [[1; n]] ⊂ E 0 alors ∀k ∈ N * , 2 k l|∀l ∈ [[1; n]] ⊂ E 0 . 3. ∀k ∈ N * , on a : g(4 k ) = 4 k -1 3 = 4 k -1 4-1 = k-1 i=0 4 i ∈ E 0 . 4. Si k ∈ N, alors on a : g(4 k ) ≡ k[3]
5. Si p et q sont deux entiers, alors on a :

g(pq) = pg(q) + g(p).

6. Si p est un entier, on considère sa décomposition en n facteurs premiers (n ≥ 2) :

p = n i=1 p i
On a donc :

g(p) = n l=1 l i=1 p i g(p l+1 ) 7. Si n ≡ 1[3] et n ∈ E 0 , alors on a : g(4n) = n + g(n) ∈ E 0 . 8. Si n ≡ 2[3] et n ∈ E 0 , alors on a : g(8n) = 2n + g(2n) ∈ E 0 . 9. Si n ∈ E 0 alors ∀k ∈ N * , on a f (k) (n) = 2 k n ∈ E 0 f (k) est la k ème itération de f c'est-à-dre f (k) = f • ... • f 10. ∀p et n ∈ N * , on a : p compositions de g .. • g • g • g • ...(n) = 1 3 p (n + 1 2 ) - 1 2
11. ∀p, q et n ∈ N * , on a :

p compositions de a et q compositions de b ... • g • f • ... • f • g • f • ... (n) ≤ p+q compositions de f f • ... • f (n)
12. Plus précisément ∀p, q et n ∈ N * , et p ≥ q on a :

2 p-1 3 q n ≤ p compositions de a et q compositions de b ... • g • f • ... • f • g • f • ...(n) ≤ 2 p 3 q n
Dans ce cas les crochets désignent la partie entière du nombre.

13. Si n ∈ E 0 et n ≡ 1[3], alors ∀k ∈ N * , on a :

g(4 k n) = 4 k n-1 3 = 4 k n-4 k +4 k -1 3 = 4 k g(n) + g(4 k ) ∈ E 0 . 14. Si n ∈ E 0 et n ≡ 2[3], alors ∀k ∈ N * , on a : g(2 2k+1 n) = 2 2k+1 n-1 3 = 2 2k+1 n-2 2k +2 2k -1 3 = 4 k g(2n) + g(4 k ) ∈ E 0 .
15. Si n ∈ E 0 et n est impair, alors g -1 (n) = 3n + 1 appartient à E 0 .

16. Si 2n ∈ E 0 , alors n appartient à E 0 ou 6n + 1 appartient à E 0 . 17. Un autre tableau permet d'avoir d'autres propriétés sur les comportement des fonctions a et b.

Pour m ∈ E 0 , on a :

m 4m-1 3 modulo 3 modulo 4 modulo 6 1 1 1 1 1 4 5 2 1 5 7 9 0 1 3 10 13 1 1 1 13 17 2 1 5 16 21 0 1 3 19 25 1 1 1 Pour m ∈ E 0 et m ≡ 4[9], 4m -1 3 ≡ 5[6].
De ces propriétés, on déduit le théorème suivante :

Théorème. Pour tout k ∈ N, on a :

• si 3k ∈ E 0 alors 4k ∈ E 0 • si 3k + 1 ∈ E 0 alors 4k + 1 ∈ E 0 • si 3k + 2 ∈ E 0 alors 2k + 1 et 4k + 2 ∈ E 0
Démonstration :

• Si 3k ∈ E 0 et en supposant que k est impair (sinon on divise par deux quitte à multiplier par deux à la fin), alors 9k + 1

∈ E 0 . Donc 36k + 4 ∈ E 0 et par suite 12k + 1 ∈ E 0 , finalement 4k ∈ E 0 . • Si 3k + 1 ∈ E 0 alors 12k + 4 ∈ E 0 . Donc 4k + 1 ∈ E 0 . • Si 3k + 2 ∈ E 0 alors 6k + 4 ∈ E 0 . Donc 2k + 1 ∈ E 0 et 4k + 2 ∈ E 0 . □ 2.
2 Étude modulaire et représentation géométrique.

Cette partie est consacrée à une étude qui permet de voir des propriétés supplémentaires des applications f et g. Il s'agit de donner des pistes pour de nouvelles recherches.

On remarque que pour tout entier k ∈ N :

4 9k -1 3 ≡ 0[9] et 4 9k ≡ 1[9]
Et pour tout entier α ∈ [[0; 8]], on a :

4 9k+α -1 3 = 4 9k+α -4 9k + 4 9k -1 3 = 4 9k 4 α -1 3 + 4 9k -1 3 .
Donc, on a le tableau suivant : Cette dernière partie est formée de tous les éléments inversibles dans Z/9Z, donc c'est un groupe.

α 4 α -1 3 4 α -1 3 mod 9 = 4 9k+α -1 3 mod 9 0 0 0 1 1 1 2 5 5 3
Si n ∈ N\3N, alors ṅ ∈ Ċ1 = { 1; 2; 4; 5; 7; 8}. Lemme. L'image de E 0 par l'application φ est égale à Z/9Z.

Démonstration.

On suppose que E 0 n'est égal à N * et on considère le complémentaire de E 0 dans N * .

E = N * \E 0
Les ensembles E 0 et E sont stables pour les applications f et g (pour les éléments dont l'image existe). Si E n'est pas vide, alors il admet un plus petit élément ξ 0 . On a :

[[1; ξ 0 -1]] ⊂ E 0 par définition de ξ 0 .
On a les cas suivants :

• Si ξ 0 = 3k + 1 , alors :

g(ξ 0 ) = ξ 0 -1 3 = k ∈ E et k < ξ 0 .
Donc ξ 0 n'est pas le plus petit élément de E. Et ξ 0 n'est pas congru à 1 mod 3.

• Si ξ 0 = 3k + 2 , alors :

g(2ξ 0 ) = 2ξ 0 -1 3 = 2k + 1 ∈ E et 2k + 1 < ξ 0 .
Donc ξ 0 n'est pas le plus petit élément de E. Et ξ 0 n'est pas congru à 1 mod 3.

• Si ξ 0 = 3k, alors il est impair car sinon sa moitié serait dans E. Donc en posant k = 2m + 1, on a : ξ 0 = 6m + 3, deux cas sont alors possibles :

1. Pour m = 2n donc ξ 0 = 12n + 3.

Il existe une suite (Z p ) (p∈Z) qui dépend de ξ 0 telle que :

...g • f • ... • f...(Z p ) = ξ 0 une composition de f et g un nombre |p| fois, si p ≤ 0 Z p = 2 p ξ 0 si p ≥ 0
Par hypothèse, la suite (Z p ) (p∈Z) donne des termes plus grand que ξ 0 . On peut construire de la même manière une suite parallèle

(Z ′ p ) (p∈Z) qui dépend de ξ ′ 0 = ξ 0 -12 > 1 telle que :    ...g • f • ... • f...(Z ′ p ) = ξ ′ 0 une même composition de f et g, un nombre |p| fois que pour Z p , si p ≤ 0 Z ′ p = 2 p ξ ′ 0 si p ≥ 0
Par construction, la suite (Z ′ p ) (p∈Z) a les mêmes congruences que (Z p ) (p∈Z) , donc elle admet comme minimum ξ ′ 0 . Ces éléments sont alors dans E et de plus, pour tout p ∈ Z, on a : Z ′ p < Z p , Le nombre ξ 0 sous cette forme ne peut pas être le plus petit élément de E. 2. Pour m = 2n + 1 donc ξ 0 = 12n + 9 et posant l = 9n + 7, on a :

g(4l) = 4l -1 3 = 36n + 28 -1 3 = 12n + 9 = ξ 0 avec l < ξ 0 .
En conclusion, l'ensemble E = ∅ et E 0 = N * . □

3 Éxtentions possibles.

• Trouver le nombre d'étape pour atteindre une hauteur donnée.

• Trouver l'altitude maximale. L'étude des propriétés des fonctions f et g peut donner une borne supérieure de l'altitude.

• Trouver le temps de vol. Ce nombre semble être lié à l'altitude maximale.

• On considère deux entiers p et q non nuls et premiers entre eux. La suite de nombres :

S n+1 = Sn p si n ≡ 0[p] qS n + 1 sinon (3)
La question est : « Que peut-on dire de cette suite suivant les couples (p; q) considérés ? » Il semble que le théorème de Bésout intervient dans cette étude.

Annexes

Dans ces annexes, on donne des calculs qui peuvent servir pour avancer sur la compréhension de la suite de Syracuse et voir certaines coïncidences à analyser.

1. On donne ce tableau de temps de vols et les nombres de compositions par f -1 et par g -1 (suite de Syracuse) :

Les compositions par f -1 Les compositions par g On remarque dans ce tableau des répétitions surprenantes qui poussent à analyser de plus près le comportement de la suite pour certains nombres consécutifs.

2. On remarque que les graphes des suites de Syracuse dont les nombres de départs sont 27 et 31 sont des translatés l'un de l'autre. Cette remarque ne concerne pas que ce couple de nombres et donne une idée sur d'autres manières de démontrer le théorème de Syracuse.

  La suite de Syracuse est définie par :S 0 (p) = p ∈ N et S n+1 (p) = Sn(p) 2 si S n (p) est pair 3S n (p) + 1 sinon (1)

  Ces remarques conduisent à une étude modulo 9 (moins triviale que modulo 3) et à une représentation graphique de Z/9Z sur un cercle. L'application f sur N peut être associée à une transformation de Z/9Z qui a plusieurs cycles. Donc induit une action sur cet ensemble. L'application g n'est pas définie sur Z/9Z mais on peut construire des transformations qui lui correspondent.φ : N -→ Z/9Z n -→ ṅLes représentations de φ • f et φ • g sont données par les diagrammes des figures 2.2 et 2.2. (Nous n'allons pas écrire les expressions de ces éléments).

Figure 2 . 1 :

 21 Figure 2.1 : Action de f sur Z/9Z Figure 2.2 : "Action" de g sur Z/9ZOn a trois cycles qui correspondent aux parties de classes stables par l'action de a : Ċ0 = { 0}, Ċ3 = { 3; 6} et Ċ1 = { 1; 2; 4; 5; 7; 8}.

Figure 2

 2 Figure 2.3 : 0 modulo 3 Figure 2.4 : 1 modulo 3 Figure 2.5 : 2 modulo 3

  On voit que les deux graphes sont superposables avec un décalage. Et on peut montrer que pour tout entier n ≥ 1, on a : S n+5 (27) = S n (31). 27) = S 0 (31) = 31.

si n ≡ 1[3] (2)On définit l'ensemble E 0 formé de toutes les images de 1 par les itérations des applications f et g (quand c'est possible). Il est facile de voir l'équivalence suivante :


Juan Olives
email: olivesbanosjuan@gmail.com.

Finite-displacement elastic solution due to a triple contact line

At the line of triple contact of an elastic body with two immiscible fluids, the body is subjected to a force concentrated on this line, the fluid-fluid surface tension. In the simple case of a semi-infinite body, limited by a plane, a straight contact line on this plane, and a fluid-fluid surface tension normal to the plane, the classical elastic solution leads to an infinite displacement at the contact line and an infinite elastic energy. By taking into account the body-fluid surface tension (i.e., isotropic surface stress), we present a new and more realistic solution concerning the semi-infinite body, which gives a finite displacement and a ridge at the contact line, and a finite elastic energy. This solution also shows that Green's formulae, in the volume and on the surfaces, are valid (these formulae play a central role in the theory).

I. INTRODUCTION

Surface properties of deformable bodies have many applications, e.g., in adhesion, coating, thin films and nanosciences. When a deformable body is in contact with two fluids (e.g., liquid and air), it is subjected to the fluid-fluid surface tension along the body-fluid-fluid contact line. The classical solution for the deformation of an elastic body, occupying a semiinfinite space limited by a plane, and subjected to a fluid-fluid surface tension concentrated on a straight line of this plane and normal to the plane, leads to an infinite displacement at this line and an infinite elastic energy. 1 However, the presence of a ridge at the contact line was experimentally observed 2 and then confirmed by other experiments and Fourier transform calculations. 3 In our previous work, 4 by taking into account the body-fluid surface stresses and surface energies, we showed that there are two equilibrium equations at the contact line: (i) the equilibrium of the three surface stresses (with no contribution of the volume stresses); (ii) a scalar equation involving the surface energies, the surface stresses, and the surface strains, which leads to a generalization of the classical Young's equation (only valid for a rigid body) (see Sec. II). The first equation-as an equilibrium of three forces tangent to the three interfaces-implies a finite displacement and the formation of a ridge at the contact line. This equation was then experimentally confirmed. 5 In the present paper, by taking into account the body-fluid surface tension (i.e., isotropic surface stress), we obtain a new solution of the above elastic problem concerning the semi-infinite body, with a finite displacement and a ridge at the contact line, and a finite elastic energy.

II. GENERAL EQUILIBRIUM EQUATIONS

In the previous work Ref. 4 (with additional comments in Ref. 6 and mathematical aspects in Ref. 7), following the general variational method of Gibbs, 8 we gave the thermodynamic definition and properties of the surface stress and obtained the following equilibrium equations for any deformable body:

1) On each body-fluid surface: div σs + ρ s ḡ + σ • n + p n = 0

(1)

σ * s = σ s , (2)
where σ s is the body-fluid surface stress, ρ s the surface mass excess per unit area, ḡ the gravity field, σ the body volume stress, n the unit vector normal to the surface (oriented from the fluid to the body), and p the fluid pressure; σs = ι • σ s , where ι is the natural injection of the tangent plane to the surface T x (S) in the three-dimensional space E (i.e., σs βi = σ αβ s ∂ α x i , with components α and β on the surface, and i in E), and div σs is a special divergence based on the tensorial product of the covariant derivative on the surface and the usual derivative in E, defined in Ref. 7 (i.e., with components: (div σs

) i = ∂ β (σ αβ s ∂ α x i ) + Γ β βγ σ αγ s ∂ α x i ). Eq. (1) has a tangential component div σ s + ρ s ḡt + (σ • n) t = 0 (3)
(in which div σ s is the usual surface divergence and the subscript t indicates the vector component tangent to the surface) and a normal component

l n : σ s + ρ s ḡn + σ nn + p = 0, ( 4
)
where l is the curvature fundamental form on the surface, l n = l • n (i.e., with components:

l i αβ = ∂ αβ x i -Γ γ αβ ∂ γ x i , l n,αβ = l i αβ n i ; the symbol : means double contraction of the indices), ḡn = ḡ • n, and σ nn = (σ • n) • n.
The eigenvalues of l n are the principal curvatures, 1 R 1 and 1 R 2 , of the surface (a curvature being positive when its center is on the side of n). If σ s is isotropic, i.e., σ s = σs I (eigenvalue σs and I the identity), we have

l n : σ s = σs ( 1 R 1 + 1 R 2 ). (5)
Similar equations were previously written, but under some particular assumptions, e.g., the existence of a "surface traction field" 9 or in the special case of elastic bodies. 10,11 2) On the body-fluid-fluid contact line, there are two equations (as previously found in the particular case of the elastic thin plate 12,13 ):

The first one is vectorial (three-dimensional) and corresponds to a line fixed on the body (but the line can move because the body is deformable):

σ bf • ν bf + σ bf • ν bf + γ ff ν ff = 0, (6)
in which the subscripts b, f, and f respectively denote the body and the two fluids, σ bf is the bf surface stress, ν bf the unit vector normal to the contact line and tangent to the bf surface (directed to the inside of bf), idem for σ bf , ν bf , and ν ff , and γ ff is the ff surface tension.

This equation expresses the equilibrium of the three surface stresses acting on the contact line (with no contribution of the volume stresses), and determines the angles of contact ϕ f , ϕ f , and ϕ b , respectively measured in f, f , and b (satisfying

ϕ f + ϕ f + ϕ b = 2π
). This point was then experimentally verified in Ref. 5.

The second equation is scalar and corresponds to a line moving with respect to the body, but fixed in space (i.e., the displacement of the material points of the body, due to the deformation, exactly compensates the displacement of the line with respect to the body, so that the line remains fixed in space):

(σ bf,νν -γ bf ) a νν -(σ bf ,νν -γ bf ) a νν + σ bf ,τ ν (a τ ν -a τ ν ) = 0, (7)
in which τ is a unit vector tangent to the line, σ bf,νν and σ bf,τ ν are respectively the components along ν bf and τ of the bf surface stress acting on the line, idem for σ bf ,νν and σ bf ,τ ν , γ bf and γ bf are respectively the bf and bf surface energies, a νν is the surface stretching deformation, normal to the line, in the bf side, a τ ν the surface shear deformation, parallel to the line, in the bf side, and idem for a νν and a τ ν in the bf side. An equivalent form of this equation is (with the help of Eq. ( 6))

-γ bf + γ bf a r,νν -γ ff cos ϕ f -γ ff sin ϕ f cos ϕ b + a r,νν sin ϕ b + σ bf ,τ ν a r,τ ν = 0, (8)
where a r,νν = a νν aνν and a r,τ ν = a τ ν -aτν aνν .

Note that, for a perfectly rigid body, the above Eqs. ( 6) and ( 7) cannot be written (since they are based on the possible displacement of the material points, due to the deformation) and, in this case, we obtain the unique scalar equation

-γ bf + γ bf -γ ff cos ϕ f = 0, (9)
which is the classical Young's equation (in which γ bf and γ bf are surface energies, and not surface tensions). If the body is deformable, Young's equation is not valid and the valid Eq. ( 8) is the generalization of Young's equation. Indeed, in the rigid body limit, a r,νν = 1, a r,τ ν = 0, and

ϕ b = π (thus, lim ϕ b →π cos ϕ b +1
sin ϕ b = 0), and Eq. ( 8) leads to Young's equation. Also note that, in the case of a very little deformable body, i.e., an almost rigid body, the surface (and volume) stresses become almost infinite, so that the two first terms in Eq. ( 6) are almost infinite and this equation implies that ϕ b is almost equal to π (in order to equilibrate the finite fluid-fluid surface tension), i.e., that there is almost no ridge on the surface. Finally, note that, if the body is a fluid, then σ bf -γ bf I = σ bf -γ bf I = 0 (which implies that σ bf,νν -γ bf = σ bf ,νν -γ bf = σ bf ,τ ν = 0) so that Eq. ( 7) is useless, and Eq. (6) expresses the equilibrium of the three fluid-fluid surface tensions.

Eq. (7) expresses that the variation of surface energy (e.g., increase in bf surface and decrease in bf surface, when the line moves with respect to the body) is equal to the work of the surface stresses acting on the line (due to the displacement of the material points, on each side of the line), when the line remains fixed in space. This equation gives a condition on the components a r,νν and a r,τ ν of the "relative" surface deformation of the bf side with respect to the bf side (we defined the "relative deformation gradient" in Ref. 14).

If σ bf ,τ ν = 0 (which, e.g., occurs if the surface stress is isotropic), it gives the relative surface stretching deformation a r,νν (normal to the line) of the bf side with respect to the bf side.

III. VALIDITY OF GREEN'S FORMULA: NO CONTRIBUTION OF THE VOLUME STRESSES

The preceding theory is based on Green's formula

V σ : Dw dv = - V divσ • w dv - S (σ • n) • w da, ( 10
)
where σ is the volume stress tensor, w a virtual displacement, V a volume of the body in the neighborhood of the contact line, S the boundary surface, n the unit vector normal to the surface and directed towards the interior of the body, dv an element of volume, and da an element of area. With the help of a first example of solution, we showed in Ref. 7 that:

1) Owing to the singularity at the contact line, the components of σ do not belong to the Sobolev space H 1 (V).

2) Nevertheless, Green's formula remains valid, because all the components ∂ j u i and σ ij are either bounded or subjected to the inequality

|∂ j u i |, |σ ij | ≤ c| log r| + d (11)
in V (r is the distance to the contact line; c and d being positive constants), and these inequalities imply that lim ε→0 S(ε)

(σ • n) • w da = 0, (12)
where S(ε) is the boundary surface of a small tubular volume V(ε) of the body, of radius ε, around an element of contact line (V(ε) is bounded by (i) the surface of the body and (ii) a half-cylinder of radius ε around the contact line). The validity of Green's formula is based on Eq. ( 12), which directly expresses that the volume stresses have no contribution at the contact line.

3) The elastic energy is finite (in the neighborhood of the contact line).

These results will be confirmed with the solution given in the present paper.

IV. APPLICATION TO THE SEMI-INFINITE ELASTIC BODY

Let us consider a semi-infinite isotropic elastic body b, occupying the half space x ≥ 0 in the orthonormal frame (Ox, Oy, Oz ), in contact with a fluid f occupying the region x < 0 and y > 0, and another fluid f in the region x < 0 and y < 0. It is subjected to the fluidfluid surface tension γ ff , here denoted σ l , which is a force parallel to Ox and concentrated on the line x = y = 0 (Fig. 1a). Sign convention: σ l > 0 if the direction of the force is opposite to Ox (which is the case of a fluid-fluid surface tension), and σ l < 0 if the direction of the force is Ox (in this case, it is a compression). We suppose that the bf and bf surface stress tensors are isotropic, with the same constant eigenvalue, noted σs > 0 (surface tension), and that the bf and bf surface energies are equal. In this case, Eq. ( 7) only implies that a r,νν = 1 (same surface stretching in the bf side and in the bf side), which will be obviously satisfied owing to the symmetry of the problem with respect to the plane y = 0. [START_REF]Eq. (9) is not valid (since γ bf = γ bf and ϕ f > π/2; see Fig. 1b) and Eq. (8)[END_REF] The other Eq. ( 6) may be written as

σ l = 2 σs cos ϕ,
where ϕ = ϕ b /2 (see Fig. 1b). Clearly, the elastic displacement components u x and u y are only functions of (x, y), u z = 0, and, by symmetry, u x (x, -y) = u x (x, y) and u y (x, -y) = -u y (x, y). In the approximation of small deformations, i.e., when the components of the displacement and their first derivatives are small, we have cos ϕ ≈ ∂ y u x (0, 0+) (derivative at x = 0, y = 0+), and the preceding equation gives

σ l = 2 σs ∂ y u x (0, 0+). ( 13
)
The Eqs. ( 3)-( 4) on the surface x = 0 are (in the absence of gravity and using Eq. ( 5))

(σ • n) t = 0, σs ( 1 R 1 + 1 R 2 ) + σ nn + p = 0
(div σ s = 0, because σ s = σs I and σs is constant on the surface). In the case of small deformations, n is approximately directed along Ox, 1 R 1 + 1 R 2 ≈ ∂ yy u x , and these equations lead to

σ xy = 0, (14)
σs

∂ yy u x + σ xx + p = 0. ( 15
)
Although we consider the case of no fluid pressure on the surface (i.e., p = 0), we will represent the tension σ l concentrated on the contact line as a Dirac distribution of "pressure" p = -σ l δ(y), which allows to include Eq. ( 13) in Eq. ( 15), with the unique equation on the surface σs

∂ yy u x + σ xx = σ l δ(y). (16)
Indeed, integrating this equation on the surface, for -ε ≤ y ≤ ε, and taking ε → 0, gives

2 σs ∂ y u x (0, 0+) + lim ε→0 ε -ε σ xx dy = σ l ,
which leads to Eq. ( 13), because the term involving σ xx is equal to 0, as explained in the preceding section: it is a consequence of Eq. (11) or Eq. ( 12) (which will be confirmed for the solution of the present paper) and expresses that the volume stress σ xx gives no contribution at the contact line.

V. THE ANALYTIC FUNCTIONS F AND H

In the following, z will denote the complex variable x+iy and u the complex displacement u x + iu y (function of the complex variable z). We use the general Kolosov's solution of plane strain elasticity

u(z) = - 1 2µ (k F (z) + z F (z) + G(z)),
where k = -λ+3µ λ+µ = 4ν -3 (λ, µ Lamé's coefficients, ν Poisson's coefficient; -3 < k < -1; see Ref. 16, vol. II, ann. XVI), based on the two analytic functions F and G, which we will write using the new analytic function H = G -zF (i.e., H(z) = G(z) -zF (z)):

-2µ u = k F + H + (z + z) F . ( 17
)
As consequences,

-2µ ∂ x u = k F + H + 2F + (z + z) F -2µ ∂ y u = i(k F -H -(z + z) F ) -2µ ∂ yy u = -(k F + H + (z + z) F ). ( 18
)
Similarly, Kolosov's expressions of the volume stresses may be written as

σ xx + iσ xy = F -H -(z + z) F σ yy -iσ xy = F + H + 2F + (z + z) F . (19)
Owing to the symmetry u(z) = u(z) of our problem, we look for analytic functions F and H with the same property, F (z) = F (z) and H(z) = H(z).

As we expect that σ xx , σ xy , and ∂ yy u x vanish at the infinity of the body, i.e., for |z| → +∞ with x > 0 (this will be confirmed on the final solution), Eqs. ( 14) and ( 16) on the surface x = 0 may be extended at the infinity of the body:

σ xy = 0 on x = 0 and at the infinity, (20) σs

∂ yy u x + σ xx =      σ l δ(y) on x = 0, 0 at the infinity, (21)
i.e., using the above expressions (F -H -(z + z) F ) = 0 on x = 0 and at the infinity,

(F -H -(z + z) F ) + σs 2µ (k F + H + (z + z) F ) =     
σ l δ(y) on x = 0, 0 at the infinity.

Assuming that (z + z) F and ((z + z) F ) vanish at the infinity (which will be confirmed on the final solution) and since z + z = 0 on x = 0, the preceding equations may be written as (F + H ) = 0 on x = 0 and at the infinity,

(F -H + σs 2µ (k F + H )) =      (22)
σ l δ(y) on x = 0, 0 at the infinity.

(

) 23
Using the transformation

z = ω(ζ) = 1 -ζ 1 + ζ and denoting Φ(ζ) = F (ω(ζ)), Ψ(ζ) = H(ω(ζ)),
B the disk |ζ| < 1, and S the circle |ζ| = 1, Eq. ( 22) becomes

(- (1 + ζ) 2 2 (Φ + Ψ )) = 0 on S,
and, since the first member is an harmonic function in B, it leads to

(- (1 + ζ) 2 2 (Φ + Ψ )) = 0 in B.
The analytic function -(1+ζ) 2

2

(Φ + Ψ ) is therefore equal to a real constant a in B, hence We may take this constant equal to 0, since an additive constant in F (or in H) only produces an additive constant in u but does'nt change the derivatives of u and the stress tensor (according to Eqs. ( 17)-( 19)):

F + H = real const. a (for z ≥ 0). ( 24
) (F + σs 4µ (k -1)F - a 2 ) =     
Ξ(ζ) = - (1 + ζ) 2 2 Φ -α (1 + ζ) 3 2 (Φ + 1 + ζ 2 Φ ) - a 2 , α = σs 4µ (1 -k) > 0,
(Ξ(ζ)) = 1 2π π -π 1 -|ζ| 2 |ζ -e iθ | 2 σ l δ(θ) dθ = σ l 2π 1 -|ζ| 2 |ζ -1| 2 = σ l 2π 1 + ζ 1 -ζ in B,
F -αF - az 2 -β log z = 0 for z ≥ 0 (z = 0). ( 25
)

VI. THE SOLUTION

By extension of the known solution of the differential equation (25) when z is a real variable (see Ref. 18, chap. IV, § 2, n • 3), we obtain the general solution of this equation when z is a complex variable, as

F (z) = b e z/α -e z/α P (z),
where P (z) is a primitive in C -R -of the function e -z/α ( a 2α z + β α log z), satisfying P (z) = P (z), and b a real constant (since F (z) = F (z)). We then find

P (z) = - a 2 e -z/α (z + α) -β(e -z/α log z -Ei(- z α )),
where Ei is the "exponential integral" function, 19 which we here define as the primitive of the function e z z in C -R + , which coincides with the function x →

x -∞ e t t dt when x ∈ R * -. This gives (after addition of the constant -β log α)

F (z) = b e z/α + a 2 (z + α) + β(log z α -e z/α Ei(- z α )).
From this expression, we obtain

(z + z) F = b α 2 (z + z) e z/α - β α z + z z ( z α e z/α Ei(- z α ) + 1), (26)
which shows that b must be equal to 0, in order to satisfy the first assumption preceding Eqs. ( 22)-( 23), because z α e z/α Ei(-z α ) + 1 → 0 at the infinity, for z > 0 (proved in the Appendix). Note that (with b = 0)

(z + z) F = - β α 2 z + z z ( z α e z/α Ei(- z α ) + 1) + β α z + z z 2
also tends to 0 at the infinity, satisfying the second assumption. Moreover, according to Eqs. (19), ( 18) and (24),

σ xx + iσ xy = (2F -a) -(z + z) F = (- 2β α e z/α Ei(- z α )) -(z + z) F , σ yy -iσ xy = (2F -a) + 2a + (z + z) F , -2µ ∂ yy u = -(k F -F + (z + z) F ),
which shows that σ xx , σ xy , and ∂ yy u tend to 0 and σ yy tends to 2a at the infinity (because e z/α Ei(-z α ) ∼ -α z : see Appendix). The assumptions preceding Eqs. ( 20)-( 21) are thus satisfied. In the following, we take a = 0, in order that all the components of the stress tensor vanish at the infinity (including σ z z = ν(σ xx + σ yy )). According to H = -F (from Eq. ( 24), taking the integration constant equal to 0), the final solution is then

-2µ u = k F -F + (z + z) F , where F (z) = β(log z α -e z/α Ei(- z α )). ( 27
)
The preceding solution holds for α > 0. Note that, if α = 0 (i.e., σs = 0), we obtain the solution

-2µ u = k F -F + (z + z) F , with F (z) = β log z, (28)
from Eq. ( 25) (with a = 0, for the same reason as above), which is exactly the classical Flamant's solution. 1

VII. DISPLACEMENTS AND STRESSES

From Eqs. ( 18)-( 19) (with H = -F ) and the above expression of F (Eq. ( 27)), we also have

-2µ ∂ x u = k F + F + (z + z) F -2µ ∂ y u = i(k F + F -(z + z) F ) σ xx + iσ xy = F + F -(z + z) F σ yy -iσ xy = F + F + (z + z) F , with
F (z) = - β α e z/α Ei(- z α ) F (z) = - β α 2 (e z/α Ei(- z α ) + α z ). ( 29
)
Introducing the function E(z) = log z -e z Ei(-z), we may write

u(z) = β 2µ (-k E( z α ) + E( z α ) - z + z α E ( z α )), (30
Since e z z = 1 z + ∞ n=1 z n-1 n! , we have Ei(z) = log(-z)+ ∞ n=1 z n n n! +const. in C-R + , the constant being equal to the Euler constant γ because Ei(x) = x -∞ e t t dt when x ∈ R * -: Ei(z) = γ + log(-z) + ∞ n=1 z n n n! in C -R + .
Then, in C -R -,

E(z) = -γ e z + (1 -e z ) log z -e z ∞ n=1 (-z) n n n!
tends to -γ and

(z + z)E (z) = -e z (z + z) log z -e z (z + z)(γ + ∞ n=1 (-z) n n n! )
tends to 0, when z → 0, which leads to the finite limit for the displacement

lim z→0 u(z) = - β 2µ (1 -k)γ = - γ π σ l µ (1 -ν). (31)
From the above expressions of F (z), F (z), and Ei(z), we obtain

α β F (z) = -γ -log z α + ε 0 (z) = -γ -log r α -iθ + ε 0 (z), - α β (z + z)F (z) = z + z z + ε 0 (z) = 1 + e -2iθ + ε 0 (z),
where z = re iθ , r > 0, -π < θ < π, and ε 0 (z) denotes any expression of z which tends to 0 when z → 0. Thus,

2µα β ∂ y u = (1 -k)θ + sin 2θ + i((1 + k)(γ + log r α ) -1 -cos 2θ) + ε 0 (z), (32)
which shows that lim z→0, θ constant

∂ y u x = β 2µα ((1 -k)θ + sin 2θ) = σ l σs ( θ π + sin 2θ 4π(1 -ν) ), lim z→0 ∂ y u y =      +∞ if σ l > 0 -∞ if σ l < 0. (33)
Obviously, the infinite limit of ∂ y u y (and, below, ∂ x u x ) is in contradiction with the assumption of small deformations and, strictly speaking, the solution is not valid for z close to 0.

The solution may rather be considered as a mathematical description of the singularity at z = 0. Note that, on the surface x = 0, with y > 0 (i.e., θ = π/2), the above equation shows that ∂ y u x tends to σ l 2 σs when y → 0, y > 0, in agreement with the above Eq. ( 13). Since the initial surface x = 0 is, after deformation, represented by the function y → u x (0, y), there is thus a finite displacement and the formation of a ridge at z = 0 on the surface, as shown in Fig. 2a. In this Fig. 2, the displacements u x (0, y) and u y (0, y) on the surface, obtained from Eq. ( 27), are represented, for ν = 0.3. For comparison, the displacements of the Flamant's solution (Eq. ( 28)) are shown in Fig. 3. Note that, except for the presence of a ridge at the contact line, the behavior on the surface of our solution (Fig. 2) cannot be compared with the results of Ref. 3 which concern a thin film 0 ≤ x ≤ h of elastic body with a zero-displacement at x = h.

In a similar way, we have which shows that lim z→0

2µα β ∂ x u = (1 + k)(γ + log r α ) + 1 + cos 2θ + i(-(1 -k)θ + sin 2θ) + ε 0 (z), (34)
∂ x u x =      +∞ if σ l > 0 -∞ if σ l < 0, lim z→0, θ constant ∂ x u y = σ l σs (- θ π + sin 2θ 4π(1 -ν) ), (35)
and immediately gives the limits of the strain tensor components

ε xx = ∂ x u x , ε yy = ∂ y u y ,
and

ε xy = 1 2 (∂ x u y + ∂ y u x ): lim z→0, θ constant ε xy = σ l σs sin 2θ 4π(1 -ν) . (36)
We also obtain

α β (σ xx + iσ xy ) = -2(γ + log r α ) + 1 + cos 2θ + i sin 2θ + ε 0 (z), α β (σ yy -iσ xy ) = -2(γ + log r α ) -1 -cos 2θ -i sin 2θ + ε 0 (z), (37)
which shows that

lim z→0 σ xx =      +∞ if σ l > 0 -∞ if σ l < 0, lim z→0 σ yy =      +∞ if σ l > 0 -∞ if σ l < 0, lim z→0, θ constant σ xy = σ l σs µ 2π(1 -ν) sin 2θ. ( 38
)
Since the inequality |1 + z e z Ei(-z)| ≤ 2 |z| for z > 0 (proved in the Appendix) may be continuously extended in z ≥ 0, z = 0 (let us recall that z → Ei(-z) is analytic in

C -R -), then e z Ei(-z) ∼ -1
z when |z| → +∞, z ≥ 0, and we may write (from Eqs. ( 27) and ( 29))

1 β F (z) = log r α + iθ + ε ∞ (z), 1 β (z + z)F (z) = - z + z z z α e z/α Ei(- z α ) = 1 + e -2iθ + ε ∞ (z),
where ε ∞ (z) denotes any expression of z which tends to 0 when |z| → +∞, z ≥ 0. Thus,

2µ β u = (1 -k) log r α -1 -cos 2θ + i(-(1 + k)θ -sin 2θ) + ε ∞ (z), which shows that lim |z|→+∞ u x =      +∞ if σ l > 0 -∞ if σ l < 0, lim |z|→+∞, θ constant u y = σ l 4πµ (2(1 -2ν)θ -sin 2θ). ( 39
)
Clearly, the infinite limit of u x is due to the assumption of a force σ l applied on the whole infinite line x = y = 0 (the z axis). Note that, on the surface x = 0, with y > 0 (i.e., θ = π/2), the preceding equation shows that u y tends to σ l 4µ (1 -2ν) when y → +∞ (see Fig. 2b). Also note that, in the limit case ν → 1 2 (i.e., k + 1 → 0), the expression of u (Eq. ( 27)) shows that u y = 0 on the surface x = 0.

From the expression of F (z) in Eq. (29), F (z) tends to 0 and, from Eq. ( 26 The dependence on ν is shown in Eq. ( 30): at z/α fixed, (2µ/β)u linearly varies with k,

i.e., with ν. In particular, on the surface x = 0,

2µ β u x (0, y) = 4(1 -ν) (E( iy α )) 2µ β u y (0, y) = 2(1 -2ν) (E( iy α )),
which shows that all the functions y/α → (2µ/β)u x (0, y) and y/α → (2µ/β)u y (0, y) are similar to those of Fig. 2, i.e., only multiplied by 1-ν 1-0.3 for u x , and by 1-2ν 1-0.6 for u y . In the volume x ≥ 0, the displacements u x and u y are represented in Fig. 4, for ν = 0.25.

In Fig. 4a, we observe that u x increases when the distance to (x, y) = (0, 0) increases. In the xy plane of Fig. 4b, the curves u y = constant show asymptotic directions. There are two lines where u y = 0: the straight line y = 0 (obviously) and a curved line, with an intersection of the two lines at x 0 > 0, y = 0. If x remains constant, x > x 0 , and y increases from 0 to positive values, we observe that u y decreases from 0 to a negative value and then increases to positive values (after crossing the value 0). These observations are consistent with Eq. (39) which gives the asymptotic value u y,∞ of u y as a function of the direction θ: (2µ/β)u y,∞ = 2(1 -2ν)θ -sin 2θ, which decreases from 0 to a negative value and then increases to the positive value π(1 -2ν) (after crossing the value 0), when θ increases from 0 to π/2. In fact, the asymptotic direction(s) θ of the curve (2µ/β)u y = c (constant) is (are) given by the equation c = 2(1 -2ν)θ -sin 2θ. Thus, for y ≥ 0, the curves (2µ/β)u y = c with c > 0 have one asymptotic direction and those with c ≤ 0 have two asymptotic directions, which is consistent with Fig. 4b. Note that these results concerning u y are consistent with the experiments and Fourier transform calculations of Ref. 20-Fig. 3a-d, in the neighborhood of the contact line. is σs ∂ yy u x = -σ xx , for x = 0 and y = 0, which satisfies the inequality of Eq. (42); see Sec. IV and Eq. ( 16)).

IX. CONCLUSIONS

When an elastic body is in contact with two immiscible fluids (e.g., an elastic gel in contact with a liquid and the air), it is subjected to the fluid-fluid surface tension acting on the body-fluid-fluid triple contact line. This force concentrated on the contact line produces a singularity on the elastic body. In the simple case of a semi-infinite body, bounded by a plane, and a force concentrated on a straight line of this plane and normal to the plane, the classical solution of Flamant 1 leads to an infinite displacement at this contact line. In the present paper, a new solution with a finite displacement at the contact line is obtained, by introducing a surface tension (i.e., isotropic surface stress) at the surface of the semi-infinite body and applying the general equilibrium equations (on the surface and at the contact line) of the previous work Ref. 4. Using Kolosov's approach of plane strain elasticity, in the complex plane, with a new function H (Sec. V), and the theory of analytic functions, we are led to a differential equation (Eq. ( 25)), the solution of which gives the explicit expression of the displacement u (Eq. ( 27)), and then the expressions of the derivatives of the displacement and the strain and stress components (Eq. ( 29)). The displacement u is proportional to β/(2µ) = σ l /(4πµ) (σ l the force or fluid-fluid surface tension concentrated on the contact line, µ the Lamé elastic shear modulus) and (2µ/β)u only depends on the coordinates (x/α, y/α) and ν (α = σs (1 -ν)/µ, σs the surface tension at the surface of the body, ν the Poisson's coefficient). Thus, α and β/(2µ) are the characteristic elastocapillary lengths for this solution. The displacement is finite and continuous at the contact line, with the formation of a ridge (see Eq. (31) and Fig. 2a). At the contact line, the xx and yy strain and stress components become infinite, but their xy components have different finite limits when approaching the contact line under different directions (Eqs. (33), (35), (36), and (38)).

Moreover, all the stress components and the derivatives of the displacement vanish at the infinity of the body (Eqs. (40)-( 41)). The components u x and u y of the displacement, as functions of the coordinates (x, y), are shown in Fig. 4. Moreover, in the neighborhood of the contact line, the derivatives of the displacement are either bounded or present a logarithmic divergence (Eq. (42); as in the example of solution of Ref. 7), which implies that the components u i of the displacement belong to H 1 (V) and H 1 (S) (V being a volume of the body and S a portion of surface of the body, in the neighborhood of the contact line). This is a crucial point for the physical validity of the solution because it implies that the elastic energy is finite and both Green's formulae in the volume and on the surfaces (Eqs. ( 43) and (44), which play a central role in the theory of Ref. 4) are valid. Note that, in the classical Flamant's solution, the components u i do not belong to H 1 (V) and the elastic energy is infinite.

FIG. 1 .

 1 FIG. 1. Semi-infinite elastic body b in contact with two fluids f and f , and subjected to the fluid-fluid surface tension σ l normal to the surface x = 0 of the body and concentrated on the line x = y = 0 (a). At the surface of the body, there is a constant surface tension σs . (a) Before deformation; (b) after deformation.

1 2 σ

 2 l δ(y) on x = 0, 0 at the infinity, i.e., using the variable ζ, (Ξ(ζ)) = σ l δ(θ) on S, where

  and ζ = e iθ , -π ≤ θ < π (δ(θ) = 1 2 δ(y) because, when x = 0, dθ = -2 dy at y = 0). The function (Ξ(ζ)) being harmonic in B, we apply Poisson's formula (see Ref. 17, (23.61.11.2))

  function Ω(ζ) = Ξ(ζ) -β 1 + ζ 1 -ζ is therefore equal to a pure imaginary constant in B, and thus equal to 0 (owing to the symmetry property of Ω, Ω( ζ) = Ω(ζ), due to the same property of Φ): z ≥ 0 (and z = 0), which implies that F -αF -az 2 -β log z is a real constant (since F (z) = F (z), implying the same property for the preceding expression; we use the usual logarithm defined in C -R -).

  ) which shows that the displacement u is proportional to β/(2µ) and (2µ/β)u only depends on z/α and k (i.e., ν). Thus, α = σs (1 -ν)/µ and β/(2µ) = σ l /(4πµ) are the characteristic elastocapillary lengths for this solution. Similarly, (2µα/β)∂ x u and (2µα/β)∂ y u only depend on z/α and k, and (α/β)σ xx , (α/β)σ yy , and (α/β)σ xy only depend on z/α.

FIG. 2 .

 2 FIG. 2. Displacements u x (a) and u y (b) on the surface x = 0 as functions of y, for ν = 0.3. The graph u x (a) represents the form of the surface after deformation. Notations: α = σs (1 -ν) µ , β = σ l 2π .

FIG. 3 .

 3 FIG. 3. The classical Flamant's solution: displacements u x (a) and u y (b) on the surface x = 0 as functions of y, for ν = 0.3. Notation: β = σ l 2π . The graph u x (a) represents the form of the surface after deformation. At x = y = 0, u x is infinite and u y is discontinuous.

  ) (with b = 0), (z + z) F (z) tends to 0 when |z| → +∞, z ≥ 0, which implies that ∂ x u and ∂ y u tend to 0 when |z| → +∞, (40) and σ xx , σ xy , σ yy , and σ z z tend to 0 when |z| → +∞ (41) (as mentioned in Sec. VI).

s

  Appendix A: Expression of Ei(z) when z < 0 Since |e zt | = e xt , the function t → e zt t is integrable in [1, +∞[, for x = z < 0, and we notef (z) = -+∞ 1 e zt t dt. If z = x < x 0 < 0, then | e zt t | = e xt t < e x 0 t t which is integrable for t ∈ [1, +∞[, which implies that f is analytic in z < x 0 (see Ref. 22, (13.8.6)(iii)) for all x 0 < 0, hence f is analytic in z < 0. Moreover, for z = x ∈ R * -, f (x) =ds = Ei(x). The two analytic functions f and Ei are then equal in z < 0, i.e., Ei(z) = -for z < 0, where Γ z is the path:t ∈ [1, +∞[ → v = zt ∈ C.By integration by parts (e z(t-1) e -z Ei(z)| ≤ 2 |z| for z < 0, and then Ei(z) ∼ e z z when |z| → +∞, z < 0, or, if z > 0, |1 + z e z Ei(-z)| ≤ 2 |z| for z > 0, and Ei(-z) ∼ -e -z z when |z| → +∞, z > 0.

As ν increases, the distances between the curves of Fig. 4a increase, which means that (at 2µα/β constant) the gradient of u x decreases, at each point (x/α, y/α). In Fig. 4b, as ν increases to 0.5, x 0 /α (abscissa of the intersection of the straight line y = 0 and the curved line u y = 0) decreases to 0. Moreover, if θ 0 denotes the asymptotic direction of the curved line u y = 0, i.e., 2(1 -2ν)θ 0 = sin 2θ 0 , 0 ≤ θ 0 ≤ π/2, clearly θ 0 increases from 0 to π/2 when ν increases from 0 to 0.5. When ν reaches the limit value 0.5, this curved line u y = 0 becomes the straight line x = 0. |x + iy| < r 0 , 0 < z < l 0 (l 0 positive constant), the preceding inequalities imply that all the components ∂ j u i and σ ij belong to L 2 (V) (because r(log r) 2 and r| log r| are integrable on [0, r 0 ]). Since u is continuous (with a finite limit at z = 0), this shows that the components u i of the displacement belong to the Sobolev space H 1 (V) (as a consequence, the components

VIII. THE DISPLACEMENTS BELONG TO

). This point is crucial for the physical validity of the solution because it implies that:

1) The elastic energy

2) Green's formula

is valid, because the components w i (as u i ) belong to H 1 (V), the components σ ij belong to L 2 (V), and the components (divσ) i belong to L 2 (V) (owing to the equilibrium equation divσ = 0; note that the derivatives ∂ l σ ij generally do not belong to L 2 (V) and the components σ ij do not belong to H 1 (V)) (see Theorem 4.4.7 in Ref. 21).

Let us recall that, in the classical Flamant's solution, u is infinite at the contact line, the derivatives ∂ j u i (and σ ij ) do not belong to L 2 (V), and the elastic energy is infinite.

In the theory presented in Ref. 4, we also used Green's formula on the surface

(with the notations of Sec. II and (D s w) βi = ∂ β w i ), where S is a portion of the bf (or bf ) surface adjacent to the contact line, C the boundary curve, ν the unit vector normal to this curve, tangent to the surface, and directed towards the interior of S, da an element of area, and dl an element of length. The derivatives ∂ α u i , being bounded or subjected to the inequality of Eq. ( 42), belong to L 2 (S) (because (log r) 2 and | log r| are integrable on [0, r 0 ]), so that the components u i belong to H 1 (S). Here also, applying the same Theorem as above, Green's formula is valid, because the components w i (as u i ) belong to H 1 (S), the components σs βi belong to L 2 (S) (since σs is constant on the surface), and the components (div σs ) i belong to L 2 (S) (the tangential component is equal to 0 and the normal component


Ph.D Shueh-Ying Liao
email: shueh-ying.liao@ens-paris-saclay.fr

廖學盈 Pud-Ps

Construction of Cilin Lexical Database: Exploring Metaphor and Metonymy in Classical Chinese Vocabulary

Keywords: literary database, Cílín Diǎnyì, Chinese word construction, metonymy, metaphor

published or not. The documents may come

I. Introduction

The Cílín Diǎnyì 詞林典腋 is often regarded as a teaching resource in classical literature or as a reference material for writing. Within the realm of traditional sinology, it falls under the category of reference books (lèishū 類書). Cílín Diǎnyì categorizes words, word groups, phrases, or fine sentences based on themes, juxtaposing them in pairs for easy reference. This systematic arrangement aids scholars in constructing words and forming sentences based on established patterns. The examples found within Cílín Diǎnyì have undergone repeated validation through the creative practices of countless poets, making them quintessential to classical Chinese poetry themes and word constructions. However, perhaps due to its primarily utilitarian nature, Cílín Diǎnyì is often relegated to the role of an instructional tool and receives limited attention from literary researchers. Its wealth of lexical constructions is often regarded merely as a collection of diverse, old examples. This study aims to prepare a project for the construction of a relational database for Cílín Diǎnyì, with the goal of uncovering the inherent word construction patterns within Cílín Diǎnyì and the concealed literary aesthetic principles embedded within these patterns.

Structure of Cílín Diǎnyì

Cílín Diǎnyì encompasses a total of 31 thematic « categories » (mén 門) and 625 « subcategories » The compilation logic can be summarized as follows: In accordance with this, when examining the original text, « 太乙|長庚 » constitutes the first word pair with right-left parallelism, and « 五緯|三台 » forms the second word pair, both adhering to this pattern. When a right-left word pair parallelism results in the separation of sentence pairs or word pairs across two pages, it is modified to upper and lower correspondences. For example, the first instance of a three-character word pair in the leftmost vertical row of the column is considered an upper and lower correspondence. Sentence pairs are arranged below word pairs and continue downward. In the absence of sentence breaks, parallelism may appear, and sentence segmentation is determined based on meaning or tonal patterns.

For this study, the electronic version of Cílín Diǎnyì provided by Taiwanese linguist HSU André Chang-Mo (許長謨) was used for word and sentence pairs and word construction annotations. The initial observations, experiments, and analysis were conducted using two-character words as the basis.

Origin of the Cílín Diǎnyì Electronic Version

The electronic version of the Cílín Diǎnyì originates from the research conducted by Professor HSU in recent years and has found applications in language education and cultural promotion. Professor HSU, along with his research team, which included graduate students, undertook tasks such as inputting entries, identifying example pairs, annotating word constructions, and verifying éditions. For sentence pairs, which also vary from two-character to nine-character words, the initial characters of French numerals are used to encode the « starting sentence » and « corresponding sentence » in accordance with the character length of the entries.

II. Exploration Strategy

In the construction of the database, emphasis is placed not only on data completeness but also on data correlation and modeling, ensuring that real-world linguistic evidence is well reflected in the structure of the database. Therefore, if we wish for the database to assist in the exploration of word construction patterns and aesthetic principles, rather than being a mere repository of textual data, we need to interpret and elaborate on the relationships within the original data and establish a model for queries and responses.

The word construction patterns and aesthetic principles investigated in this study are not explicitly recorded within the Chinese characters themselves. However, by objectifying the compilation and usage logic of word examples and clearly annotating the role and function of each character within this logic, it becomes possible to reconstruct the formalities of word construction and aesthetic standards. Hence, we have selected the following aspects for investigation:

1. The quantity and proportion of word construction types.

2. The differentiation power of word construction morphemes.

3. The network of nodes for word construction morphemes.

4. The function and position of word construction morphemes.

These investigations aim to reveal the underlying connections between vocabulary, structure, and categories.

III. Exploration Methods

The specific exploration methods are as follows: 1) Word construction is categorized using HSU's classification method [START_REF] Zheng | 姿與言: 詩國革命新論[END_REF] , and the quantity and proportion of word constructions are tallied. 5 In the survey conducted, character positions within words were used to distinguish categories. This approach emphasizes the significance of character order in classifying words and reveals structural patterns and relationships within the dataset. t-SNE was employed to visualize how positional differences impact word distribution in a lower-dimensional space, providing valuable insights for linguistic analysis and category characterization based on character positions. This methodology uncovers hidden linguistic patterns, offering a deeper understanding of lexical organization. Each entry in Cílín Diǎnyì consists of two to nine Chinese characters, with most forming a single morpheme. Some entries have characters that cannot be separated and are composed of multiple characters, collectively forming a single morpheme. The labeling of characters follows these guidelines:

When extracting each character from the example : 1) Phonetic information is annotated, with tone distinctions converted to tonal patterns. If there are multiple pronunciations, they are separated by the « | » symbol.

2) The three-dimensional position of the character within the example (sentence pair or word pair) is marked. This includes the length of characters in the example, the upper-lower sequence of the entry (0 for upper, 1 for lower), and the character's position within the example (starting with 0 for the first character, 1 for the second, and so on).

3) The category of the example in which the character is located is noted. 4. The HSU's word construction model for the pair of entries in the example are labeled.

漢字 聲母 韻母 平仄 開合 等呼 上下 對序 詞內 字序 例對 字長 詞門 詞類 詞例 詞構 太 透 泰 仄 開 一 0 0 4 【天文門】01TW 【天文總】01TW00 太乙|長庚 MH
An alternative approach is to employ a vocabulary management method using a relational database, with Chinese characters as the axis, creating a master The table for the constituent components of each category is as follows :

Category Differentiation

Each category is treated as a collected sample, with the word examples within the category serving as features. Subsequently, the characters contained in the examples are divided into two groups: the previous character in one group and the subsequent character in another. Based on TF-IDF (Term Frequency-Inverse Document Frequency) normalized word frequency, the distances between samples are calculated using the t-SNE (t-Distributed Stochastic Neighbor Embedding) algorithm, and then projected into a two-dimensional plane. Finally, the concentration and dispersion of sample data points in the analyzed graph are observed.

The higher the Silhouette Score, indicating more concentration of data points within the same category, and the lower the Davies-Bouldin Index, signifying greater dispersion of data points between different categories, the more balanced the differentiation of characters' positions. This is reflected in a higher Quality Ratio defined in this work.

In the following, « C » represents chinese character, lowercase « q » (qián 前) represents the previous character, and lowercase « h » (hòu 後) represents the subsequent character. The key positions that differentiate characters based on « category » and « subcategory » for various word constructions are as follows: In summary, for both « category » and « subcategory », the key positions for differentiation are as follows: the subsequent character is key in subject-predicate (SP) and modifier-head (MH); the previous character is key in verb-object (VO). In coordination (CO) structures, the subsequent character distinguishes « category », while the previous character distinguishes « subcategory ».

When shifting analysis from « category » to « subcategory » within the same word construction and at the same character position, notable differences in clustering strength (Silhouette Score) are observed. For instance, in modifier-head (MH), the clustering strength for « modifier (Cq) » increased by 42%, whereas « head (Ch) » only increased by 36% :

Chinese Characters Semantic Network

In this section, we explore the creation of a network with Chinese characters as nodes. Each

Chinese character, accompanied by its quantitative attributes, such as its position in a word pair (upper/lower), its position in the pair's sequence (previous/subsequent), and the number of characters in the pair (4, 6, 8…18), serves as a « node » in the network. The connections, or « directed edges », between nodes are determined by the order of characters in word examples. For example, in the word pair « 太乙|長庚 », the directed edges connect characters in this sequence: «

太-乙-長-庚 ».
Positional information for each character is recorded to keep track of where they appear within different word pairs or sentence pairs. This information ensures that each character's connections are unique to their specific context. We calculate the number of nodes that can function as previous character(s) and those that can serve as subsequent character(s). Additionally, we assess the level at which they can extend to form complete words. This analysis helps us understand the connection capabilities of each node within the network.

The top ten characters with the most generative capacity and their respective word construction positions are as follows:

« 三-1-0-4 » 6 , « 雨-0-1-4 », « 月-0-0-4 », « 風-1-0-4 », « 雲-1-0-4 », « 風 -0-0-4 », « 雲-0-0-4 », « 月-0-1-4 », « 石-0-1-4 »
, and « 玉-0-0-4 ». These characters play a significant role in forming various word structures within the network.

Distribution of Chinese Characters' Functions and Positions within Word Constructions

Building upon the analyses conducted earlier, we've identified the crucial positions that differentiate categories within each word construction, such as the subsequent characters in modifier-head structures. Additionally, based on the second analysis, we've pinpointed potential nodes that facilitate the generation of additional word entries within each word construction. 7 For example, nodes like « 風-0-0-4 », « 雲-0-0-4 », or « 月-0-1-4 » within the node network.

For Chinese characters with stronger connectivity, we delve deeper to examine their distribution ratios across various positions within each word construction, corresponding categories, and 7 The node graph can be queried and filtered based on nodes, connections, categories, positions, or phonetic conditions.

semantic classes. This allows us to understand the typical distribution and category correspondence for each character within word constructions. For instance, an investigation into the distribution of the character « 風 » across different word constructions reveals that « 風 » can be used across various word constructions, particularly in modifier-head (MH), verb-object (VO), and subject-predicate (SV) structures. In modifier-head structures, it predominantly occupies the lower part and frequently serves as the subsequent character in categories with strong differentiation. In verb-object structures, it also tends to be in the lower part and often serves as the subsequent character in categories with weaker differentiation. In subject-predicate structures, it mainly appears as the previous character and does not play a role as the subsequent character, with weaker category differentiation. In summary, despite its widespread usage across various word constructions, « 風 » primarily assumes a non-central role within the word construction: it was verified that subject-predicate and verb-object structures, respectively with the predicate and the verb as key characters, have a higher discriminative power in terms of categories. Consequently, in structures containing verbs, « 龜 » does not show strong category-differentiating power.

條件:('風', 'VO',
In summary, « 龜 » has not functioned as a verb and is mostly employed as a modifier in modifier-head structures. While the central role in these structures is not its strength, it is not insignificant either. Therefore, innovating by changing the position of « 龜 » in modifier-head structures is not easy. Conversely, in subject-predicate or verb-object structures, successfully altering « 龜 » grammatical roles can yield unique effects. For example, using « 龜 » as a verb implies « to hide », resulting in the derived word « 龜著 » (to hide) in the affix/derivation structure.

An example usage might be: When he got scared, he would run back home and hide (他一害怕就

跑回家龜著).
This investigation clarifies the similar and distinct conditions required for generating new words and creating new meanings. Phonetics can also be used as search criteria to explore new words or meanings generated from similar phonetic structures, investigating analogies or logical connections between new words or meanings and phonetic patterns.

V. Conclusion and Discussions

This article extensively explores the intricate relationship between Chinese lexical constructions and themes, emphasizing the dynamic nature of language. It particularly highlights how Chinese characters, as morphemic components, show variations in their positions, functions, meanings, and poetic nuances within different word constructions.

In conclusion, the results of this investigation reveal that when isolated from their word constructions, Chinese characters represent mere morphemes or partial components of morphemes and do not inherently define their parts of speech.

Analyzing the distribution of the same character on either side of a central word's core is essentially an analysis of its ability to create metonymic transformations within morphemes or morphemic components. For instance, in modifier-head structures, characters that can be interchanged (i.e., functionally interchangeable) to form words of different categories provide an example of such associations 8 . This kind of association stems from everyday experiences and corresponds to a poet's ability to employ metonymies to touch (chùwù 觸物) things. Analyzing within word constructions, when the same position, morpheme, or morphemic component can be replaced while maintaining the same word category, it delves into the metaphorical structure of a word. The example pairs in the word forest highlight the metaphoric logic behind words within the same theme. Such analogies arise from rational analysis and correspond to the poet's ability to comprehend (yuánlǎn 圓覽) in poetic metaphors. Viewing a poet's approach to word formation from this perspective helps understand how in poetry, despite being from different backgrounds (wù sūi hú yuè 物雖胡越), things can eventually harmonize (hé zé gān dǎn 合則肝膽) 9 .

Under the constraints of classical metrics, classical Chinese poetry is actually required to be created at least on the level below phrases. Therefore, it's challenging for poetry to have a strict sense of sentence structure: we simply use the concept of everyday sentences to interpret the formal sentences in poetry as a temporary measure. In fact, a complete sentence in poetry, a paragraph defined by a self-contained meaning and distinct from other meanings, often spans several formal sentences. Moreover, throughout the entire poem, it's as if the entire poem is a single word, and all the Chinese characters are morphemes, like a series of language gestures to pass over different

(

  lèi 類), with word examples organized from shorter to longer character sequences. These examples are arranged in word pairs or sentence pairs, juxtaposed side by side, considering tonal patterns and bearing semantic similarities. Such characteristics facilitate the examination of the correctness of each example sentence pair, word pair, and individual word.

Figure 1 :

 1 Figure 1: Cílín Diǎnyì compiled and supplemented by TANG Wenlu 湯文璐 in the Qing Dynasty, included in the Augmented Compilation of Rhymes and Verses (Zēngbǔ Shīyùn Hébì 增補詩韻合璧)

  Pair) 6 Entry (Upper Pair) 5 Entry (Lower Pair) 8 Entry (Upper Pair) 7

  2) Chinese characters within word constructions are divided into central and non-central characters. The T-SNE (t-Distributed Stochastic Neighbor Embedding) method is then used to test the positions of characters that maximize the differentiation of categories in each word construction. 5 3) Each pair of examples is considered as directional edges, with the characters contained in the example pair regarded as nodes, forming a node network. The number of connections in and out of each character node and their hierarchy is calculated. 4) The typical functions and positions of each character within the word construction are computed.

4

  Hsu, C. -M. 許長謨 (2010). 詞彙與語義之義證. In 漢語語言結構義證 (p. 125). 里仁書局, 臺北. Categories are as follows: 1) Simple Words: 聯綿 (LM, lianmian/binome) and 狀聲 (ON, onomatopoeia); 2) Compound Words: 並列 (CO, coordination), 偏正 (MH, modifier-head), 主謂 (SP, subject-predicate), 動賓 (VO, verb-object), and 動補 (VC, verb-complement); 3) Complex Words: 重疊 (DP, duplication) and 衍生 (AF, affix/derivation) 4) Location (CL, location), Proper Noun (PN, proper noun), Verb-Verb (VV, verb-verb), and Pivotal/Embedded (BY, baoyun/pivotal/embedded).

Category) show the

  most diverse word example lengths. In each category, two-character, three-character, and four-character word examples are predominant, with a fair representation of six-character examples. Five-character, seven-character, and other parallel structures are also present. In terms of word constructions, apart from Jìyìmén 技藝門 (Art Category), Lǐyímén 禮儀門 (Etiquette Category), and Zhíguānmén 職官門, which contain a substantial number of verb-object structures in their examples, the remaining categories are primarily composed of modifier-head structures in their word examples.

Figure 2 :

 2 Figure 2: Diversity of Word Examples

Figure 4 .

 4 Figure 4. t-SNE Test for Modifier-Head Structure

Figure 5 .

 5 Figure 5. Chinese Characters with More than Thirty Layers of Sequential Node Connections

9

  These four terms are used by Liu Xie [劉勰] (465-521) in his famous work Wénxīn Diāolóng 文心雕龍 to create a concluding envoi at the end of the Bǐ-Xīng 比興 chapter, which explains how the metaphoric and metonymic forms encompass semantic categories.8 By altering the node settings and removing directional information from the nodes while filtering for the situation where the subsequent character in the upper pair can be exchanged with the previous character in the lower pair, you can identify all the Chinese characters whose upper and lower character positions can be interchanged : 三-重, 雲-鶴, 別-鶴, 梅-欺, 槐-綠, 市-槐, 清-香, 月-落, 明-月, 月-桂, 夜-月, 繅-風, 吹-風, 搖-風, 從-風, 星-流, 星-繁, 星-編, 停-雲, 雲-飛, 暖-風, 綺-霞, 日-觀, 滴-雨, 煙-磨, 洞-雲, 天-行, 管-絃, 弓-張, 賓-鴻, 紫-蓋, 飛-高, 花-黃, 銀-黃, 合-璧, 振-玉, 玉-碎, 屑-玉, 玉-縷, 玉-白, 烏-金, 紫-金, 楓-江, 燕-玉, 飛-龍, 山-飛, 飛-鸞, 破-鏡, 珠-連, 垂-珠, 寶-珠, 國-香, 雪-香, 珠-綴, 散-珠, 文-龍, 織-蒲, 翔-鸞, 盤-龍, 作-解, 凝-烟, 簪-花, 白-花, 吟-秋, 木-落, 沙-眠, 濃-陰, 紫-苔, 戀-蝶, 戲-蝶, 盤-鴉, 水-面, 梅-熟, 化-女, 化-雀, 紅-蓼, 殘-紅, 紅-醉, 帶-露, 三-春, 祖-竹, 嘯-竹, 乳-花, 柳-舒, 柳-細, 柳-穿, 紅-輕, 流-翠, 丹-書, 埋-雪, 草-黃, 新-詩, 冰-彈, 綠-葉, 紅-苞, 寒-燈, 舞-鶴, 停-針, 白-薤, 立-竹, 塢-竹, 毛-竹, 徂-暑, 交-衣, 薦-酒, 吟-蛩, 密-葉, 子-白, 駕-鶴, 怨-鶴, 炭-獸, 奔-崖, 中-虛, 馬-馳, 衣-錦, 孫-桐, 化-魚, 狐-疑, 丹-心, 池-鳳, 棲-鳳, 調-鼎, 調-鹽, 勁-節, 批-鱗, 細-腰, 螭-蟠, 身-輕, 心-虛, 翅-薄.semantic categories within a poetic body 10 . The word construction pattern is thus the key to access a poem's vision.11 Since poetic parallelism also relies on the compositional structure of classical verse, the question arises whether the combination of phonetics and the metaphorical structure of example pairs can generate analogical relationships. Can the pronunciation's place and manner of articulation of phonemes be related to the metonymic mechanism of morphemes or morphemic components within word constructions to produce such wide-ranging yet nuanced connections? As we continue to build the Cilin Lexical Database based on the reconstructed model of classical morphemic relationships, these questions, accompanied by the analysis process, will gradually lead us to systematic explanations.Building on our investigation, word embedding practices in Chinese studies prompt a crucial reevaluation. It's evident that our understanding of Chinese word meanings calls for a shift from surface-level word embedding to a deeper exploration of morphemes in text mining, including the consideration of their positional meanings. Current word embedding techniques limit our understanding to the surface, hindering an in-depth grasp of the complexities in the Chinese language. Annotating morphemes-examining phonemes, lexemes, and their positions and positional meanings-promises a more profound comprehension, overcoming the constraints of word-centric methods. 11 In the abstract of Liu, C.-L. [劉昭麟], Chang, W.-T., Chu, C.-T., & Zheng, T.-Y. (2023), Machine learning and data analysis for word segmentation of classical Chinese poems: Illustrations with Tang and Song examples, published in Digital Scholarship in the Humanities, the authors discuss the word's crucial role in accessing a Chinese classical poem.

  While Professor HSU has published certain content from this electronic version in books and research reports, the complete dataset has not been publicly available until now. This electronic version, which has undergone years of revision and compilation by Professor HSU, primarily, but not exclusively, references the following historic editions of Cílín Diǎnyì:

	editor		title		acronym	purpose
	湯祥瑟		詩韻合璧		合	used for cross-referencing
	湯文璐		詩韻全璧		全	used for cross-referencing
	余照		增廣詩韻集成		集	used for cross-referencing
	盧元駿		詩詞曲韻總檢		檢	served as the manuscript reference source
	length example	struct_1 n°struct_2 struct_3 cross-reference
	3	魚信斷/雁書遲	4908	SP	MH	《全》雁書還《檢》《集》雁聲遲

They transformed the traditional printed manuscript into a digital text file, accompanied by essential reference annotations. Additionally, a spreadsheet-style version was produced, offering basic search and filtering functions.

The electronic version of the Cílín Diǎnyì has gradually taken on the structure of a relational database by incorporating several crucial metadata elements. Through the calculation and encoding of entry lengths, this dataset can provide information on syllable patterns within word examples. In the case of word pairs, ranging from two-character words to nine-character words, numerical codes are used, and the upper and lower components are separated by a « / ». For example, « upper pair / lower pair » represents the structure of word pairs:

Investigation Results 1. Variety of Word Examples When

  categorizing word construction types based on the total number of examples, they are ranked in the following order: modifier-head, verb-object, subject-predicate, and coordination. The remaining word construction types have very few examples. The categories of word constructions under Qìyòngmén 器用門 (Craftsmanship Category), Dìlǐmén 地理門 (Geography Category), and Guǒpǐnmén 果品門 (Fruit Category) show the highest diversity in word construction types.

							上下	詞內	例對
	漢字 聲母 韻母 平仄 開合 等呼				詞門	詞類	詞例	詞構
							對序	字序	字長
	1	5	2422	1	0	1	0	0	4	1	1	1	4
	By managing in this manner, it becomes possible to differentiate characters with the same font that,
	in fact, originate from different contextual backgrounds: 1) « Quantitative Features » can be used to
	investigate the grammatical functions of Chinese characters, such as the length of characters in
	examples, upper-lower sequence, and character order within words. 2) « Qualitative Features » can
	be employed to explore the semantic attributes of Chinese characters, including semantic category,

table. Other information is organized in separate tables, with numerical references linking them to the master table. This approach can effectively maintain vocabulary consistency and optimize search efficiency. semantic subcategory, word examples, word constructions, or phonetic patterns such as initials, finals, tonal patterns, and combinations of open and closed syllables. Currently, this study only examines cases involving two-character words.

IV.

Table 1 :

 1 Word Construction Quantity in Each Category (Two-Character Words)

	門類	MH	SP	VO	CO	DP	VC	LM	PN	VV	ON	詞構種類
	【器用門】19QY	291	12	90	14	4	7	2	8	1	0	9
	【地理門】03DL	193	56	90	19	4	2	2	1	0	0	8
	【果品門】26GP	174	27	41	4	2	6	1	1	0	0	8
	【天文門】01TW	130	71	88	9	6	2	1	0	0	0	7
	【音樂門】08YY	67	8	31	2	1	0	1	2	0	0	7
	【服飾門】20FS	209	8	51	16	1	12	0	11	0	0	7
	【草木門】24CM	243	53	91	11	7	1	2	0	0	0	7
	【百花門】25BH	312	59	102	20	5	4	0	4	0	0	7
	【飛禽門】27FQ	137	22	95	7	4	8	0	0	0	1	7
	【走獸門】28ZS	84	10	65	1	1	2	1	0	0	0	7
	【昆蟲門】30KC	66	16	48	4	2	2	1	0	0	0	7
	【人倫門】09RL	93	11	80	10	2	13	0	0	0	0	6
	【形體門】12XT	68	24	40	12	3	11	0	0	0	0	6
	【武備門】14WB	44	9	35	5	0	1	0	1	0	0	6
	【技藝門】15JY	55	10	70	5	2	6	0	0	0	0	6
	【珍寶門】17ZB	40	3	26	4	0	3	0	2	0	0	6
	【宮室門】18GS	166	10	67	12	0	4	0	1	0	0	6
	【飲食門】21YS	178	13	63	8	2	0	0	4	0	0	6
	【布帛門】23BB	45	7	17	4	0	0	1	2	0	0	6
	【外編】31WP	90	13	20	4	4	10	0	0	0	0	6
	【時令門】02SL	145	53	66	3	0	1	0	0	0	0	5
	【帝后門】04DH	43	8	21	8	1	0	0	0	0	0	5
	【職官門】05ZG	68	6	47	9	0	0	0	1	0	0	5
	【治政門】06ZZ	29	6	30	2	0	1	0	0	0	0	5
	【禮儀門】07LY	19	1	29	2	0	0	0	0	1	0	5
	【人事門】10RS	105	29	86	15	0	2	0	0	0	0	5
	【文事門】13WS	76	14	32	4	0	2	0	0	0	0	5
	【外教門】16WJ	66	7	30	3	0	0	0	2	0	0	5
	【菽粟門】22SL	72	19	33	4	4	0	0	0	0	0	5
	【鱗介門】29LJ	57	4	35	1	0	1	0	0	0	0	5
	【閨閣門】11GG	37	8	27	3	0	0	0	0	0	0	4
	Total Quantity	3402	597	1646	225	55	101	12	40	2	1
	Total Percentage (%)	55.94	9.82	27.07	3.7	0.9	1.66	0.2	0.66	0.03	0.02

Bǎihuāmén 百花門 (Flower Category), Guǒpǐnmén 果品門 and Zhíguānmén 職官門 (Official

Table 2 :

 2 Word Construction Ratios in Each Category (Two-Character Words)

	門類	MH (%)	SP (%)	VO (%)	CO (%)	DP (%)	VC (%) LM (%)	PN (%) VV (%)	ON (%)
	【天文門】01TW	42.35	23.13	28.66	2.93	1.95	0.65	0.33	0	0	0
	【時令門】02SL	54.1	19.78	24.63	1.12	0	0.37	0	0	0	0
	【地理門】03DL	52.59	15.26	24.52	5.18	1.09	0.54	0.54	0.27	0	0
	【帝后門】04DH	53.09	9.88	25.93	9.88	1.23	0	0	0	0	0
	【職官門】05ZG	51.91	4.58	35.88	6.87	0	0	0	0.76	0	0
	【治政門】06ZZ	42.65	8.82	44.12	2.94	0	1.47	0	0	0	0
	【禮儀門】07LY	36.54	1.92	55.77	3.85	0	0	0	0	1.92	0
	【音樂門】08YY	59.82	7.14	27.68	1.79	0.89	0	0.89	1.79	0	0
	【人倫門】09RL	44.5	5.26	38.28	4.78	0.96	6.22	0	0	0	0
	【人事門】10RS	44.3	12.24	36.29	6.33	0	0.84	0	0	0	0
	【閨閣門】11GG	49.33	10.67	36	4	0	0	0	0	0	0
	【形體門】12XT	43.04	15.19	25.32	7.59	1.9	6.96	0	0	0	0
	【文事門】13WS	59.38	10.94	25	3.12	0	1.56	0	0	0	0
	【武備門】14WB	46.32	9.47	36.84	5.26	0	1.05	0	1.05	0	0
	【技藝門】15JY	37.16	6.76	47.3	3.38	1.35	4.05	0	0	0	0
	【外教門】16WJ	61.11	6.48	27.78	2.78	0	0	0	1.85	0	0

Table 2 :

 2 Word Construction Ratios in Each Category(Two-Character Words)

	門類	MH (%)	SP (%)	VO (%)	CO (%)	DP (%)	VC (%) LM (%)	PN (%) VV (%)	ON (%)
	【珍寶門】17ZB	51.28	3.85	33.33	5.13	0	3.85	0	2.56	0	0
	【宮室門】18GS	63.85	3.85	25.77	4.62	0	1.54	0	0.38	0	0
	【器用門】19QY	67.83	2.8	20.98	3.26	0.93	1.63	0.47	1.86	0.23	0
	【服飾門】20FS	67.86	2.6	16.56	5.19	0.32	3.9	0	3.57	0	0
	【飲食門】21YS	66.42	4.85	23.51	2.99	0.75	0	0	1.49	0	0
	【菽粟門】22SL	54.55	14.39	25	3.03	3.03	0	0	0	0	0
	【布帛門】23BB	59.21	9.21	22.37	5.26	0	0	1.32	2.63	0	0
	【草木門】24CM	59.56	12.99	22.3	2.7	1.72	0.25	0.49	0	0	0
	【百花門】25BH	61.66	11.66	20.16	3.95	0.99	0.79	0	0.79	0	0
	【果品門】26GP	67.97	10.55	16.02	1.56	0.78	2.34	0.39	0.39	0	0
	【飛禽門】27FQ	50	8.03	34.67	2.55	1.46	2.92	0	0	0	0.36
	【走獸門】28ZS	51.22	6.1	39.63	0.61	0.61	1.22	0.61	0	0	0
	【鱗介門】29LJ	58.16	4.08	35.71	1.02	0	1.02	0	0	0	0
	【昆蟲門】30KC	47.48	11.51	34.53	2.88	1.44	1.44	0.72	0	0	0
	【外編】31WP	63.83	9.22	14.18	2.84	2.84	7.09	0	0	0	0

Table 3 :

 3 Testing of Chinese Character Category Differentiation Positions

			Silhouette Score		Davies-Bouldin Index		Quality Ratio
			(SS, range from -1 to 1)	(DBI, range from 0 to 1)		(SS/DBI)
	Type	Pos.	Cat	Sub-cat	Diff.	Cat	Sub-cat	Diff.	Cat	Sub-cat	Diff.
		Cq	0,08	0,5	0,42	0,63	0,7	0,07	0,12	0,71	0,59
	MH	Ch	0,29	0,65	0,36	0,33	0,61	0,28	0,87	1,07	0,2
		Cq	0,1	0,6	0,50	0,51	0,67	0,16	0,19	0,9	0,71
	SP	Ch	0,47	0,73	0,26	0,13	0,59	0,46	3,7	1,24	-2,46
		Cq	0,28	0,74	0,46	0,32	0,52	0,2	0,86	1,41	0,55
	VO	Ch	0,05	0,63	0,58	0,52	0,68	0,16	0,1	0,92	0,82
		Cq	0,28	0,89	0,61	0,44	0,45	0,01	0,63	1,98	1,35
	CO	Ch	0,31	0,86	0,55	0,38	0,53	0,15	0,84	1,62	0,78

  1, 1, 4), 數量: 33, 比率: 18.97%, 詞條:逐雨|追風...... 條件:('風', 'VO', 0, 1, 4), 數量: 13, 比率: 7.47%, 詞條:因風|伴月 占風|觀日...... 數量: 27, 比率: 15.52%, 詞條:雨剪|風梭 日嫩|風嬌...... 條件:('風', 'SP', 0, 0, 4), 數量: 16, 比率: 9.20%, 詞條:風脆|雨香 風勁|雲濃...... 條件:('風', 'MH', 1, 1, 4), 數量: 32, 比率: 18.39%, 詞條:穀雨|梅風 十雨|五風...... 條件:('風', 'MH', 1, 0, 4), 數量: 21, 比率: 12.07%, 詞條:月姐|風姨 清白|風流...... 條件:('風', 'MH', 0, 0, 4), 數量: 18, 比率: 10.34%, 詞條:風腳|雨拳 風駟|雷鞭...... 條件:('風', 'MH', 0, 1, 4), 數量: 9, 比率: 5.17%, 詞條:鷹風|鶴雲 濕風|陰雲......

	條件:('風', 'VC', 1, 0, 4), 數量: 1, 比率: 0.57%, 詞條:雨折|風掀
	條件:('風', 'SP', 1, 0, 4), 條件:('風', 'CO', 1, 0, 4), 數量: 2, 比率: 1.15%, 詞條:儒雅|風神 富貴|風流
	條件:('風', 'CO', 0, 0, 4), 數量: 2, 比率: 1.15%, 詞條:風流|搖曳 風渙|隨雷
	For example, let's consider the character « 龜 » as the subject of investigation, limiting the focus to
	four-character word examples:
	條件:('龜', 'VO', 1, 1, 4), 數量: 1, 比率: 5.88%, 例對:棲鵠|麗龜
	條件:('龜', 'SP', 1, 0, 4), 數量: 1, 比率: 5.88%, 例對:魚戲|龜遊
	條件:('龜', 'MH', 1, 0, 4), 數量: 6, 比率: 35.29%, 例對:鼈石|龜山 椿算|龜齡 鶴眼|龜文 鳳彩|龜文 貝錦|龜紗 馬蹄|龜背
	條件:('龜', 'MH', 0, 0, 4), 數量: 5, 比率: 29.41%, 例對:龜齡|馬齒 龜甲|蛛絲 龜殼|繭頭 龜甲|鶴文 龜甲|龍文
	條件:('龜', 'MH', 1, 1, 4), 數量: 2, 比率: 11.76%, 例對:砥柱|寶龜 銅鶴|神龜
	條件:('龜', 'MH', 0, 1, 4), 數量: 1, 比率: 5.88%, 例對:洛龜|岡鳳
	條件:('龜', 'CO', 1, 0, 4), 數量: 1, 比率: 5.88%, 例對:熊虎|龜蛇
	It is evident that the character « 龜 » most frequently appears as a modifier in the « four-character
	modifier-head pairs » in the lower part (35.29%) and as a modifier in the upper part (29.41%). It is
	relatively less common for « 龜 » to function as the headword in modifier-head pairs, with
	occurrences of 11.76% and 5.88%, respectively. Additionally, « 龜 » is observed in structures
	containing verbs, specifically in subject-predicate and verb-object structures, appearing only in the
	lower part of these structures and serving as the subject or object. Earlier, using the T-SNE method,

D (deux: 2), T (trois:

3), Q (quatre: 4), C (cinq: 5), S (six: 6), P (sept: 7), H (huit: 8), N (neuf: 9). The exception is for « 7 », which is represented by the letter « P » to avoid confusion with « 6 », which uses the letter « S ».

For the sentences pairs, we found TS, TH, TC, SS, SQ, SP, SC, QS, QQ, QP, QH, QC, PS, PQ, HQ, HC, CT, CS, CQ, CC, SQQ, QQQQ, and QCC.

The « 三 » represents Chinese characters, « 1 » indicates being in the lower pair the example, « 0 » signifies being the previous character in the pair below the example, and « 4 » represents a two-character pair with a total word length of four Chinese characters. All other nodes are interpreted in a similar manner.

Zheng, Y.-Y. [鄭毓瑜]. (2017). 姿與言: 詩國革命新論. 麥田出版, 臺北. The author emphasizes the categorical association of words organized like functional organisms within one poetic body, including the universe, nature, people, the environment (p.67), and all physical and physiological activities of a poet when they pronounce words (pp.45-46).