Teichmüller Spaces

Seddik Gmira

We shall construct Teichmüller spaces alternatively by using quasiconformal mappings, we investigate the case of closed Riemann surfaces of genus g ( 2), and prove the Teichmüller theorem, which states that the Teichmüller space of a closed Riemann surface R of genus g ( 2) is homeomorphic to the open unit ball in the real (6g 6)-dimensional Euclidean space. This is another compacti…cation di¤erent from the Thurston's one [M ]. The base point of the proof is the existence and uniqueness of the extremal quasiconformal mappings called, Teichmüller mappings.

In the case of general Riemann surfaces, Teichmüller's mappings are still uniquely quasiconformal mappings, while formal Teichmüller mappings are not necessarily extremal. The proof of Teichmüller's uniqueness theorem states that a Beltrami coe¢ cient of a Teichmüller mapping is extremal. The original "proof" of Teichmüller's theorem is found in [T ], and the proof in this paper follows that in [B].

Teichmüller Space of a Riemann Surface

Let R be a …xed Riemann surface. For every quasiconformal mapping f : R ! S, where S is another Riemann surface, consider a pair (S; f ). We say that two pairs (S 1 ; f 1 ) and (S 2 ; f 2 ) are equivalent if f 2 f 1 1 is homotopic to a conformal mapping of S 1 onto S 2 . We denote by [S; f ] the equivalence class of (S; f ). The set of all such equivalence classes is the Teichmüller space T (R). Letting id the identity mapping of R, we call [id; R] the base point of T (R).

Teichmüller Spaces of a Fuchsian Group

Fix a Fuchsian model of R acting on the upper half-plane H. We can always assume that is non-commutative, and then the set of all …xed points of fidg contains at least three points. We may assume that each of the points 0; 1; 1 is a …xed point of an element 2 fidg. We always consider the lift e f : H ! H of a quasiconformal mapping f : R ! S which …xes each of 0; 1;and 1. We know, that this lift e f is detemined unique and is the restriction of a quasiconformal mapping of C to H. We call this e f the 1 canonical lift of f with respect to . Using the canonical lift, we have an injection homeomorphism e f : ! P SL (2; R)

de…ned by, e f ( ) = e f e f 1 :Then we have an isomorphism e f of onto another Fuchsian group 1 , and S = H= 1 . We express 1 as e f e f 1 Lemma 1 [S 1 ; f 1 ] = [S 2 ; f 2 ] in the Teichmüller space T (R) if and only if

e f 1 = e f 2
, where e f j is the canonical lift of f j , j = 1; 2. Proof. Supose that [S 1 ; f 1 ] = [S 2 ; f 2 ]. We may assume that S 1 = S 2 = S and f 1 is homotopic to f 2 by composing a suitable conformal mapping of S 1 onto S 2 . A homotopy between f 1 and f 2 is written as a

= e f 1 ( )
Since each of 0; 1; and 1 is …xed by some 2 fidg, then e F 2 …xes also 0; 1; 1. Thus e F 2 is coincident with the canonical lift e f 2 of f 2 with respect to , and hence e f 1 = e f 2 . Conversely, assume that e f 1 = e f 2 = . Then, for every 2 we obtain e f j = ( ) e f j , j = 1; 2: For every t 2 [1; 2] and every z 2 H, letting g z be the geodesic (with respect to the Poincaré metric) connecting e f 1 (z) and e f 2 (z), we denote by f (z; t) the point which divides g z in the ratio t : (1 t).

Then

n e f t (z; 1 t) o 1 t 2
is a homotopy between e f 1 and e f 2 . Thus we have

e f t = ( ) e f t ; t 2 [1; 2]
Hence e f t is projected to a continuous mapping f t : R ! S and we have a homotopy between f 1 and f 2 .

De…nition 1 The reduced Teichmüller space of is the set

T # ( ) = ( e f : e f is a canonical qc mapping of C such that e f ( ) = e f e f 1 is a Fuchsian group )
On the other hand, we de…ne the Teichmüller space T ( ) of the Fuchsian group as follows: let QC ( ) be the set of all canonical quasiconformal mappings ! 2 C such that ! ! 1 are also Fuchsian groups. We say that ! 1 , ! 2 2 QC ( ) are equivalent if ! 1 = ! 2 on R. We denote by [!] its equivalence class. The Teichmüller space of is the set

T ( ) = f[!] : ! 2 QC ( )g Lemma 2 Suppose that R is compact. Two quasiconformal mappings f j : R ! S j (j = 1; 2) satisfy e f 1 = e f 2 if and only if e f 1 = e f 2 on R.
Proof. e f 1 = e f 2 on R implies that for every 2 ; Möbius transformations

(R) = T # ( ) (as sets). Further if R is compact, then T (R) = T ( ). Proof. The mapping [S; f ] 2 T (R) ! e f 2 T # (
) is well de…ned, and injective. Next, for every such e f as in the de…nition of T # (R), we set

1 = e f ( )
. Then e f is projected to a quasiconformal mapping f : R = H= ! H= 1 , and hence determines a point [S; f ] 2 T (R). Thus the original mapping is also surjective, and we have the …rst assertion. The second follows by Lemma 2.

Teichmüller Distance

To introduce a topology on T (R), we de…ne the Teichmüller distance. Let [S; f ] 2 T (R), e f be the complex dilatation of the canonical lift e f with respect to . Then we have We denote by B (H; ) the set of all Beltrami di¤erentials, and we set

B (H; ) 1 = f 2 B (H; ) : k k 1 < 1g Similarly A measurable ( 1; 1)-form = (z) dz=dz on R such that k k 1 = ess sup z2R j (z)j < 1 is called a Beltrami di¤erential on R: We denote by B (R)
the set of all Beltrami di¤erentials on R, and we set

B (R) 1 = f 2 B (R) : k k 1 < 1g
De…nition 3 For any two points

p 1 = [S 1 ; f 1 ], p 2 = [S 2 ; f 2 ] 2 T (R), let F f 1 ;f 2 be the set of all quasiconformal mappings of S 1 onto S 2 which are homotopic to f 2 f 1 1 . The Teichmüller distance on R is given by d (p 1 ; p 2 ) = inf g2F f 1 ;f 2 log K (g)
where K (g) is the maximal dilatation of g Theorem 1 The Teichmüller space T (R) (and hence T # (R)) is complete with respect to the Teichmüller distance. Proof. Let fp n = [S n ; f n ]g 1 n=1 be any Cauchy sequence in T (R) with respect to the Teichmüller distance d. For every " > 0, the de…nition of d implies that we can …nd a su¢ ciently large N " such that for n, m N " , there is a quasiconformal mapping f n;m homotopic to f m f 1 n and satisfying n;m 1 < " where n;m = fn;m . In particular we can …nd a subsequence p n j 1 j=1 and a sequence f n j ;n j+1 1 J=1 of qc mappings such that n j ;n j+1 < 2 j j = 1; 2; 3; :::: Next, let p 0 the base point of T (R). Since fd (p 0 ; p n )g 1 n=1 is a bounded sequence, we may assume that K (f n ) < K for every n with a su¢ ciently large K > 1. Since

K f n j 1 ;n j 1 2 j 1 + 2 j 1 + 4:2 j
for every j, the mapping g j = f n j 1 ;n j f n j 2 ;n j 1 :::::::

f n 1 ;n 2 f n 1
is a quasiconformal mapping on R onto S n j , homotopic to f n and satis…es

K (g j ) K l 1 j=1 1 + 4:2 j
Hence fK (g j )g 1 j=1 is a bounded sequence. We denote by K 1 its supermum. Now, let e g j be the canonical lift of g j with respect to for every j.

Then j = e g j 2 B (H; ) and j 1 k 1 = (1 K 1 ) = (1 + K 1 ) < 1. Also we have 1 2 j j+1 1 j+1 j 1 j j+1 1 = n j ; n j+1 1 < 2 j
for every j. In particular,

j 1 j=1 is a Cauchy sequence in B (H; ). Hence = lim j !1 j exists in B (H; ) and satis…es k k 1 < k 1 Let e
f be the canonical -quasiconformal mapping H. Then we can show that e f 2 QC ( ). Let p = [S; f ] be the point in T (R) determined by e f . Since

tanh d p n j ; p 2 ! j 1 j 1 1 1 (k 1 ) 2 j 1 we see that p n j converges to p. Now, …x a point [R 1 ; f 1 ] in T (R) arbitrarly.
We can de…ne a mapping

[f 1 ] : T (R) ! T (R 1 ) [S; f ] ! S; f f 1 1 with base point [R 1 ; id]. Moreover, we have Proposition 2 The mapping [f 1 ] : T (R) ! T (R 1 ) is an isometrical homeomorphism with respect to the Teichmüller distance Proof. First, since f 1 1 : T (R 1 ) ! T (R) gives the inverse mapping of [f 1 ] , clearly this map is a bijection. Next, for any two points p = [S; f ], q = [S 0 ; g] 2 T (R), the family F f;g = F f f 1 1 ;g f 1 1 . Hence [f 1 ] is an isometry.
We call a mapping such as this

[f 1 ] : T (R) ! T (R 1 ) a translation of the base point [R 1 ; id] of T (R) :
2 Teichmüller Theorems

Holomorphic Quadratic Di¤erentials

In the case of closed Riemann surfaces of genus one, we know that any point in T 1 the Teichmüller space of genus 1, is sent to any other point in T 1 under the projection of an orientation preserving a¢ ne mapping of C. We shall discuss similar "canonical" mappings for the genus g 2.

Let w = f (z) = z+ z ( ; 2 C; j j > j j) be an orientation preserving a¢ ne mapping of C. Applying a suitable rotation to the z-plane, and a suitable rotation and stretching to the w-plane, we can write f (z) in the form

z ! Kx + iy = K + 1 2 z + K 1 2 z; K 1
Hence when we set k = (K 1) = (K + 1), the pull-back of the Euclidean metric jduj 2 on the w-plane under f has the form

K + 1 2 2 jdz + kdzj 2
Upon changing the local parameter z by a conformal mapping z = h (&), we see that jdz + kdzj 2 becomes

jh 0 j 2 d& + k (h 0 ) 2 (h 0 ) 2 d& De…nition 4 A family ' = ' j of holomorphic functions ' j on z j (U j ) for all coordinate neighbourhoods (U j ; z j ) of a Riemann surface R is called a holomorphic quadratic di¤erential on R if it satis…es ' k (z k ) = ' j z jk (z k ) : z 0 jk (z k ) 2 on U j \ U k where z jk = z j z 1 k
We express this formula simply as

' k (z k ) = ' j (z j ) (dz j =dz k ) 2 : We also write ' = ' (z) dz 2 . A holomorphic automorphic form ' (z) of weight 4 with respect to is a holomorphic function ' (z) on H such that ' ( (z)) 0 (z) 2 = ' (z) ; z 2 H, 2
Denote by A 2 (R) the complex vector space of all holomorphic quadratic di¤erentials on R. A holomorphic quadratic di¤erential corresponds to a holomorphic automorphic form of weight 4 with respect to a Fuchsian model of R acting on the upper half-plane H. We denote by A 2 (H; ) the complex vector space of all holomorphic automorphic functions of weight 4 with respect to .

Teichmüller Mappings

As a "locally a¢ ne" quasiconformal mappings of R, we take a mapping f such that for some k, 0 k 1, it satis…es f z = kf z for a suitable local coordinate z around almost every point of R. More precisely, we discuss a quasiconformal mapping f whose Beltrami coe¢ cient f satis…es

f = k ' j'j with a suitable ' 2 A 2 (R).
De…nition 5 Let a positive k < 1 and ' 2 A 2 (R) f0g be given. Then we call a quasiconformal mapping f a formal Teichmüller mapping of R for the pair (k; ') if the Beltrami coe¢ cient f = k'='. Here we regard conformal mappings also as formal Teichmüller mappings which corresponds to the case k = 0 or ' = 0. From here on, writing ' = ' (z) dz 2 : Set

A 2 (R) 1 = ' 2 A 2 (R) : k'k 1 = 2 Z Z R j' (z)j dxdy
Now we have a mapping

T : A 2 (R) 1 ! T (R) ' ! [S; f ]
where f : R ! S = f (R) is a Teichmüller mapping for ' 6 = 0 and f = id for ' = 0.

The main purpose of this section is to show that the mapping T is a surjective homeomorphism. For this purpose we use the Fricke space F g (homeomorphic to R 6g 6 via Fenchel-Nielsen coordinates). We have a bijective continuous mapping : T (R) ! F g and the mapping e T = T :

A 2 (R) 1 ! F g is continuous [IT ].

Teichmüller' s Theorems

The injectivity of the mapping e T follows from the following Teichmüller's uniqueness theorem. A proof of this theorem shall be given at the end of this paper.

Theorem 2 Let f be a Teichmüller mapping for an element ' 2 A 2 (R) 1 and let T (') = [S; f ]. Then every quasiconformal mapping f 1 : R ! S which is homotopic to f satis…es.

f 1 1 f 1
Moreover, the equality holds if and only if f 1 = f . Corollary 1 The mappings T and e T are injective. Proof. It su¢ ces to show the injectivity of T . Assume that T (' 1 ) = T (' 2 ) for ' 1 , ' 2 2 A 2 (R) 1 . Let f j be a Teichmüller mapping for ' j 2 A 2 (R) 1 and T ' j = [S j ; f j ] for j = 1; 2. Then the assumption implies that there is a conformal mapping h : S 1 ! S 2 such that h f 1 is homotopic to f 2 . Thus the theorem gives

f 1 1 = h f 1 1 f 2 1 : Similarly, since h 1 f 2 is homotopic to f 1 , we have f 2 1 f 1 1
Similarly, since h 1 f 2 is homotopic to to f 1 , we conclude that h f 1 1 = f 2 1 which implies that h f 1 = f 2 again by this theorem. In particular

f 1 = f 2 .Thus if ' 1 = 0, then ' 2 = 0. If ' 1 6 = 0, then k' 1 k 1 = k' 2 k 1 , and ' 1 = j' 1 j = ' 2 = j' 2 j a.e on R. Hence ' 1 =' 2 is positive a.e on R. Since ' 1 =' 2 is meromorphic, it should be a constant and then ' 1 = ' 2 . Lemma 3 The image e T (A 2 (R) 1 ) of A 2 (R) 1 is an open set under e T : A 2 (R) 1 ! F g and e
T is a homeomorphism onto its image. Proof. Since e T is continuous, then the last Corolary implies that e T is a continuous bijection. Since A 2 (R) 1 is homeomorphic to R 6g 6 , Brower's theorem on invariance of domains gives the assertion.

Lemma 4 The mapping T :

A 2 (R) 1 ! T (R) is a homeomorphic onto its image. Proof. Since T is injective, it follows that T 1 is well de…ned on E = T (A 2 (R) 1 ). Since T 1 = e T 1 , then T 1 is continuous. To show that T is continuous, …x a point ' 2 A 2 (R) 1 and set p = T (') = [R 1 ; f 1 ] : Consider a translation [f 1 ] : T (R) ! T (R 1
) of the base point, which sends p to the base point [R 1 ; id] of T (R 1 ). Then we know by Proposition 2 that [f 1 ] is a surjective isomerty. We de…ne a mapping

e T 1 = [f 1 ] 1 T 1 : A 2 (R 1 ) 1 ! F g
in the same way as in the case of e T . Since e

T (') = e T 1 (0), Lemma.3 implies that e T 1 1 e T = (T 1 ) 1 [f 1 ] T : A 2 (R) 1 ! A 2 (R) 1
is well de…ned in some neighbourhood of ' and is a homeomorphisme onto its image. Hence, T is continuous at ' if and only if this is also true for T 1 : A 2 (R) 1 ! T (R 1 ) (de…ned similarly to T by using Teichmüller mappings) at the origin. We take any sequence f n g n 1 in A 2 (R) such that k n k 1 ! 0 as n ! 1. Since the maximal dilatation of a Teichmüller mapping for n is equal to (1 + k n k 1 ) = (1 k n k 1 ) for every n, we have

d (T 1 (0) ; T 1 ( n )) log 1 + k n k 1 1 k n k 1 ! 0 as n ! 1
Thus T 1 is continuous at the origin. Since ' is arbitrary, we conclude that T is continuous.

Lemma 5 The spaces T (R) and F g are connected. Proof. Let [S; f ] 2 T (R) be a …xed point and set f = . For every t 0 t 1, let f t be a qc mapping whose Beltrami coe¢ cient is t : We set S t = f t (R). Then we have a continuous curve f[S t ; f t ]g 0 t 1 in T (R) which connects between the base point [R; id] and [S; f ]. Thus T (R) is arcwise connected.

Lemma 6 The mappings T , e T are surjective

T (A 2 (R) 1 ) = T (R) ; e T (A 2 (R) 1 ) = F g
Proof. It su¢ ces to show the assertion for T , since e T = T and :

T (R) ! F g is a bijection [IT ] : By Lemma 3 we get E = T (A 2 (R) 1 ) = 1 e T (A 2 (R) 1 )
which is an open set in T (R). Further Lemma 5 implies that T (R) is connected. Hence the assertion follows if we show that the boundaray @E in T (R) = ;.

Suppose that @E 6 = ;. For any

[S; f ] 2 @E there is a sequence f' n g 1 1 A 2 (R) 1 such that T (' n ) ! [S; f ], and k' n k 1 ! 1 as n ! 1. Let f n be a Teichmüller mapping for ' n . We set T (' n ) = [S n ; f n ].
By the assumption, there is a quasiconformal mapping h n of S n onto S which is homotopic to f f 1 n for every n such that hn 1 ! 0 as n ! 1. In particular, for a suitable k < 1, we get gn 1 k; n = 1; 2; ::::;

where g n = h 1 n f n . On the other hand, since g n is homotopic to f n Theorem 2 impies that gn 1

hn 1 = k'k 1 ! 0 (n ! 1)
This is a contradiction. Thus @E is empty.

As a corollary of Lemma 6, we obtain the following Trichmüller's existence theorem.

Theorem 3 For every quasiconformal mapping f : R ! S, there exists a Teichmüller mapping homotopic to f . Lemma 6 …nishs a proof of Teichmüller's theorem. Namely we have proved the following theorem.

Theorem 4 The mapping T : A 2 (R) 1 ! T (R) is a surjective homeomorphism. In particular, T (R) is homeomorphic to A 2 (R) 1 , and hence to R 6g 6 .

Corollary 2 The representations of the Teichmüller spaces of a closed Riemann surface of genus g 2 T (R), F g and R 6g 6 are homeomorphic to each other.

Proof of Teichmüller' s Uniqueness Theorem

Theorem 2 asserts that deformations of the complex structures with best "e¢ ciecy" are given by applying "local a¢ ne" mappings called Teichmüller mappings. One of the decisive properties of a¢ ne mappings is that they send lines to lines. In the case of Teichmüller mappings, it may be natural to consider geodesics with respect to some metric associated with the given Teichmüller mapping. We consider the "metric" ds 2 = j' (z)j jdzj 2 , where ' = ' (z) dz 2 is the element of A 2 (R) 1 corresponding to the given Teichmüller mapping. This "metric" degenerates at every zero of '. However the notions such as area and length can be introduced with no di¢ culty.

Geometry Induced by a Holomorphic Quadratic Di¤erential

Let ' be …xed in A 2 (R) 1 f0g. If p 0 2 R is not a zero of ', then ' 1=2 = ' (z) 1=2 dz has a single valued holomorphic branch in some neighborhood U of p 0 , and the function is de…ned by

& (p) = Z p p 0 ' 1=2 , p 2 U
gives a holomorphic mapping of U onto C. We call this mapping a 'coordinate around p 0 . Using a '-coordinate, a Teichmüller mapping for ' is represented locally as an a¢ ne mapping F given by

F (&) = & + k&; k = f 1
This follows from the assumption that (Proposition 4)

f = k ' j'j
The projection of a e '-segment to R is called a '-segment. Lemma 7 (Teichmüller) Let h : R ! R be a quasiconformal mapping homotopic to id. Then there is a constant M depending only on R, h and ' such that jh (L)j ' jLj ' M for every '-segment L.

Proof. Let e h be the canonical lift of h with respect to the canonical Fuchsian model of R. Then it su¢ ces to …nd a constant M such that

e h e L e ' jLj e ' M
for every e '-segment e L in H. Lemma 1 gives e h = e h for every 2 .

Hence, letting C z be the e '-geodesic connecting z and e h (z) for every z 2 H, and every 2 we have

C (z) e ' = Z C (z) je ' (u)j 1=2 du = Z Cz je ' ( u)j 1=2 d ( u) = Z Cz je ' ( u)j 1=2 d ( u) = jC z j e ' Since R is compact, then M = 2 n jC z j e ' : z 2 H o is …nite (Note that jC z j e '
is continuous with respect to z). Now, let a e '-segment e L be given with end points z 1 and z 2 . Then the curve C z 1 : e h e L :C

+ M

A prototype of Teichmüller's uniqueness theorem is the following Grötzsch's theorem.

Theorem 5 (Grötzsch) Let f be an a¢ ne quasiconformal mapping of

a closed rectangle R = [0; r] [0; 1] to another rectangle S = [0; s] [0; 1] de…ned by f (z) = z + kz 1 k = Kx + iy
which is a K-quasiconformal on the interior of R, where K = s=r 1 and k = (K 1) = (K + 1) < 1: Then for every homeomorphism f 1 : R ! S which is (1 + k 1 ) = (1 k 1 )-quasiconformal on the interior of R, and maps 0, r, r + i, and i, respecively to 0, s, s + i, and i, it follows that k 1 k Moreover, the equality holds if and only if f 1 = f . Proof. First, we have

s = jf 1 (r + iy) f 1 (iy)j Z r 0 j(f 1 ) x (x + iy)j dx for almost y 2 [0; 1] , we get s Z Z R j(f 1 ) x j dxdy Since (f 1 ) x = (f 1 ) z + f 1 z . Schwarz inequality gives s 2 Z Z R j(f 1 ) z j + f 1 z dxdy 1=2 Z Z R j(f 1 ) z j + f 1 z j(f 1 ) z j f 1 z dxdy: Z Z R j(f 1 ) z j f 1 z dxdy (K 1 r) :s
where

K 1 = (1 + k 1 ) = (1 k 1 ). Hence K 1 K or equivalently that k 1 k.
Further, if the equality holds, then we obtain j(f 1 ) z j + f 1 z = j(f 1 ) z j f 1 z and f 1 z = k j(f 1 ) z j almost every where on R. Thus, f 1 = k a.e on R. This implies that g = f 1 f 1 is 1-qc, and hence conformal on the interior of S. Since g …xes all vertices of S, applying Schwarz're ‡exion principle repeatly we see that g can be extended to an element of Aut (C). Hence, with suitable complex numbers a and b, we have f 1 (z) = af (z) + b:

So, 0 = f 1 (0) = b and s = f 1 (r) = aKr = as. That is f 1 = f .
Thus an a¢ ne mapping as in the Grötzsch's Theorem is extremal. In the case of a closed Riemann surface R, we …nd that any Teichmüller mapping looks very similar to an a¢ ne mapping.

Proposition 4 Fix ' 2 A 2 (R), and let f : R ! S be a Teichmüller mapping for ', and set k = k'k 1 < 1. Then there exists a unique holomorphic quadratic di¤erential on S satisfying the following conditions:

1. If p is a zero of ' of order m, then f (p) is a zero of of the same order.

2. Let p be an arbitrary point of R which is not a zero of and & be a '-coordinate around p. Then there exists a -coordinate ! at f (p) such that ! f = &+k& 1 k : We call these ' and the initial di¤erential and the terminal di¤erential of f respectively. Proof. For a point p 2 R which is not a zero of ', we de…ne a mapping ! = ! p in a neighborhood of f (p) by ! f = &+k& 1 k . Then in some neighborhood U of p, we have

! f = k ' j'j
Since ! f = k'= j'j for every local coordinate ! on f (U ), we see that ! ! 1 is 1-qc, and hence conformal on f (U ). Thus ! is also a local coordinate around f (p). Now for a zero p of ' of order m, we have seen that ' = z m dz 2 with a suitable local coordinate z. De…ne ! as a continuous branch determined by

! f = z (m+2)=2 + kz (m+2)=2 1 k 2=(m+2)
Then we can see similarly that ! is a local coordinate in a neighborhood of f (p). Finally, consider (d! p ) 2 in a neighborhood of every point f (p) such that p is not a zero of ', where ! p = ! as above. Then we can show that these (d! p ) 2 give a single holomorphic quadratic di¤erential on S, which we denote by . From the construction, satisfy 1 and 2. The uniqueness of follows at once from 1 and 2.

Proof of Therem 2 [B], [A]

Assume the assumptions of Theorem 2 are satis…ed. Let be the terminal di¤erential of f . For every p 2 R which is not a zero of ', take a '-coordinate & around p and a -coordinate ! around q = f (p). Consider the horizontal dilatation

(f 1 ; p) = @ (! f 1 & 1 ) @ (0) , & = + i
of f 1 with respect to & and !. Then (f 1 ; p) is de…ned almost every where on R and is a measurable function. Recall that the "horizontal dilatation"

(f; p) = @ (! f & 1 ) @ (0) = 1 + k 1 k = K
for almost every p 2 R. Recalling the proof of Grötzsch's Theorem, we may regard the following inequality as one way to represente the fact that a Teichmüller mapping has the best "e¢ ciency" for deformation of the complex structure of R:

Z Z R (f 1 ; p) d d Z Z R Kd d (1)
where d d = j' (z)j dxdy. Set g = f 1 f 1 and ! = + i . Then the "horizontal dilatation" of g

(g; q) = @ (! g ! 1 ) @ (0)
is de…ned a.e on S, it is measurable and satis…es

K (g; f (p)) = (f 1 ; p) (2) for almost every p 2 R. Since d d = Kd d , we have Z Z R (f 1 ; p) d d = 1 K Z Z R f 1 ; f 1 (q) d d = Z Z S (g; q) d d Hence, the inequality (1) is equivalently to Z Z S (g; q) d d Z Z S d d (3)
Now, we assume the inequality (1), or (3), and prove Theorem 2. First, set

k 1 = f 1 1 , K 1 = (1 + k 1 ) = (1 k 1 ), b f 1 = ! f 1 & 1 , and
J (f 1 ) (p) = b f 1 & (0) 2 b f 1 & (0) 2
Then we get

(f 1 ; p) 2 b f 1 Next, applying Schwarz'inequality to (3), we obtain Z Z S (g; q) 2 d d Z Z S d d Hence, (2) and (4) give Z Z S d d Z Z R (f 1 ; p) K 2 Kd d K 1 K Z Z R J (f 1 ) (p) d d = K 1 K Z Z S d d Thus K 1 K, and hence k 1 k. Finally, if k 1 = k, then both equalities in (4) should hold. Namely b f 1 & + b f 1 & (0) = b f 1 & (0) + b f 1 & (0) and b f 1 & (0) = k b f 1 & (0)
almost every where on R. This implies that f 1 = k'= j'j. Hence g is 1-qc on R. Since g is homotopic to id, the canonical lift of g on H is coincident with id. Then by Lemma 2 we have f 1 = f . Lemma 8 The inequality (3) : R R S (g; q) d d R R S d d holds Proof. First, we assume that 1=2 has a single valued global branch on S, say which is a holomorphic Abelian di¤erential on S. We de…ne the geodesic ‡ow fF t j t 2 Rg on S with respect to the "metric" j j. Set ! = p (q) = Z q p as a -coordinate around p which is not a zero of 1 . We continue the inverse mapping 1 along segments on R in both directions as far as possible. Then we get a locally biholimorphic mapping which is denoted by the same notation

1 p of a domain containing an open interval I p = (u 1 ; u 2 ) of R into S, 1 u 1 u 2 1.
We set H p = 1 p (I p ). This H p is called the -horizontal line passing trough p. When we trace along H p in one direction either H p ends at a zero of , or j!j tends to 1. In particular, restrictingdistance on H p , we can identify H p with a circle or with a subinterval I p R preserving-orientation and length. Let E be the set of p 2 S such that I p is a proper subinterval of R. Since has only a …nite number of zeros, we can say that E has area zero with respect to the "metric" j j. Thus for every p 2 S E, setting l p = 1 p jR , we obtain a mapping l p : R !H p ; l p (0) = p which is locally isometric with respect to the Euclidean metric on R and the metric j j on S. Now we shall de…ne the geodesic ‡ow fF t j t 2 Rg on S with respect to the metric j j. Let E be as before, and set = S E. We de…ne F t by setting F t (p) = l p (t) , t 2 R, p 2

Then we can see that F t is a bijective measurable self-mapping of , which represents the "parallel horizontal translation" by t with respect tocoordinate. In particular for every measurable subset X , it is easy to show that Z Z Ft(X)

d d = Z Z X d d
i.e F t is area-preserving with respect to the "metric" j j. Next, we set (g; q; t) = (g; F t (q)) ; q 2 ; t 2 R Then (g; q; t) is a measurable function on R. Since E = S has area zero with respect to j j, we get for every L Here we set L q = l q ([ L; L]), and hence jL q j = 2L .

Finally applying Teichmüller's Lemma, we conclude from (5) and (6) that Now, in the case where 1=2 has not a simple-valued global branch, we take any local branch of 1=2 and continue it analytically as far as possible. Then we can construct a two-sheeted covering surface e S of S, with a branch point at every zero of of odd order, such that 1=2 becomes a singule-valued (holomorphic Abelian) di¤erential on S, and we apply the argument above to this di¤erential on e S.

  On the other hand, since g is quasiconformal, and hence is ACL(absolutely continuous on lines), we see that Fubini's theorem gives.

& (0) + b f 1 & (0) 2 K 1 J (f 1 ) (p)a.e on R (4)

At a zero of ' of order m 1, it is proved that ' is written in the form ' = z m dz 2 for a suitable local coordinate z on some neighborhood U of p 0 . Hence we see that

cannot be single-valued in any neighborhood p 0 . Consider the metric ds 2 = j' (z)j jdzj 2 . Let e ' 2 A 2 (H; ) be the lift on the upper half-plane H with respect to a Fuchsian model of R. For every piecewise curve C on H, we set

We call this jCj e ' the e '-length of C. Denote by L z 1 ;z 2 the set of all piecwise curves connecting z 1 , z 2 in H. We put

'-geodesic if it satis…es jC 0 j e ' = d e ' (z 1 ; z 2 ) : Now, we describe how a e '-geodesic looks. Assume that there is a e 'geodesic C 0 between z 1 and z 2 in H. For every p 2 C 0 which is not a zero of e ', the length minimality implies that C 0 should be a segment near e & (p) on the &-plane, where e & is a e '-coordinate.

At a zero, p 2 C 0 of order m > 0 C 0 may be broken. However, the angle at p should not be less than 2 = (m + 2)

We call a closed arc L on H a e '-segment if, for every interior point p of L, L is mapped by a e '-coordinate at p to a segment. Then arg e ' (z (t)) z 0 (t) 2 is a constant modulo 2 along any e '-segment L : z = z (t) (0 t 1). Here we consider only closed Riemann surfaces. Then we can easily show that H is complete with respect to this e '-distance. In particular, for any two points of H, there exists a e '-geodesic connecting them. Proposition 3 For any distinct z 1 ; z 2 2 H the e '-geodesic connecting z 1 and z 2 is unique Proof. Let C 1 and C 2 be two e '-geodesics connecting z 1 and z 2 . Then C 1 and C 2 are simple. By replacing z 1 and z 2 with a suitable pair of points if necessaly we may assume without loss of generality that C 1 \ C 2 = fz 1 , z 2 g. Then C 1 [ C 2 bounds a Jordan domain D in H. As has been seen before, C 1 and C 2 consist of a …nite number of e '-segments. We give them an order with respect to the positive orientation of the boundary @D, and denote them by fL j g l j=1 (l 2). Let m j be the order of zeros of e ' at L j \ L j+1 for every j, where L m+1 = L 1 and we allow m j to be zero. Also let j > 0 be the angle between L j and L j+1 . By the argument principle, we can show that

where N is the number of zeros of e ' in D including multiplicity. On the other hand, we have

Further, we know that (2 (m j + 2) j ) = 2 (N + 2)

Since N 0, it follows that j should be not greater than 2 = (m j + 2) for at least three j 0 s. However, j 2 = (m j + 2) for every j not corresponding to z 1 or z 2 , which gives a contradiction.

Corollary 3 A e '-segment L is the unique e '-geodesic connecting its ends points.

The following lemma due to Teichmüller is crucial in the proof of Teichmüller's Theorem. The '-length jLj ' of a curve L on the surface R is de…ned by


Aline Crédeville
email: aline.credeville@umontreal.ca

Dominic Forest
email: dominic.forest@umontreal.ca

Is Visualization Usable for Displaying Web Search Results in an Exploratory Search Context?

Keywords: information visualization, exploratory search, clustering, usability. 1

published or not. The documents may come

Is Visualization Usable for Displaying Web Search

Results in an Exploratory Search Context?

Introduction

The search engine has established itself as an almost exclusive means of searching and browsing the web, and its technical and algorithmic performance is widely recognized. However, the "query-result" paradigm has several interactivity limitations and is only appropriate for a limited spectrum of search taxonomies [START_REF] Schraefel | Building Knowledge: What's beyond Keyword Search?[END_REF]. Indeed, Google's search interface is considered adequate only for 17% of the searches it performs [START_REF] White | Investigating Behavioral Variability in Web Search[END_REF]. Considering the broad spectrum of information search taxonomies [START_REF] White | Investigating Behavioral Variability in Web Search[END_REF], [START_REF] Broder | A Taxonomy of Web Search[END_REF] and information behaviours [START_REF] Wang | Information Behaviour and Seeking[END_REF], a new current in scientific research has emerged in the past years that focuses on the interaction between the user and the system rather than the individual analysis of those entities [START_REF] Cool | A Classification of Interactions with Information[END_REF], [START_REF] Cool | Interactive information retrieval: history and background[END_REF]. Indeed, the searching interface has become a focal point for researchers [START_REF] Marchionini | Toward Human-Computer Information Retrieval[END_REF]- [START_REF] Jansen | The Seventeen Theoretical Constructs of Information Searching and Information Retrieval[END_REF]. For exploratory searching in particular [START_REF] Marchionini | Exploratory Search: from Finding to Understanding[END_REF], new interfaces are being developed for the benefit of relevant exploratory tactics and strategies [START_REF] Wilson | From Keyword Search to Exploration: Designing Future Search Interfaces for the Web[END_REF]. In this respect, visualization of search results appears as a promising device for the information seeking process.

Due to the double interactive and visual-perceptual process it entails [START_REF] Bederson | The Craft of Information Visualization: Readings and Reflections[END_REF], [START_REF] Ware | Information Visualization Perception for Design[END_REF], visualization is characterized by its tendency to amplify the cognitive faculties deployed by the user for knowledge building, analysis and learning processes [START_REF] Card | Readings in Information Visualization: Using Vision to Think[END_REF]. Nevertheless, in the field of information retrieval, the evidence of cognitive amplification is yet to be demonstrated within the exploratory search context.

Thus, our research stands on the crossroads of several challenges. Firstly, although it is known that visualising information aids memorisation and analysis of information in general, the means of implementing visually structured search results remains to be determined. Secondly, the new interactive modalities required by the exploratory research process introduce additional dimensions which extend beyond the reach of typical recall and precision metrics. Thirdly, it has not yet been verified that, when compared to other interfaces designed for exploratory search, visual and interactive presentation of results leads to a clear gain over a textual list-based presentation, based on the same clustering algorithm.

The goal of this research is to assess the gain in terms of usability, from the user's point of view, of exploring search results through a visual layout (as opposed to a textual layout) when performing exploratory searches on the Web. Increasing usability of an interface means reducing the cognitive load associated with its use and simplifying the cognitive, affective and physical processes related to the final task of the user.

Because we believe that interactive visualization reduces cognitive load associated with information treatment processes required by cognitive strategies and research tactics in the context of exploratory information search, we assert that it can benefit the user's investigation and discovery processes beyond the known positive effects of clustered results [START_REF] Kules | Users Can Change their Web Search Tactics: Design Guidelines for Categorized Overviews[END_REF].

Our research aims to evaluate the usability of visualization in the context of exploratory information search on the web. To accomplish this, we wish, through heuristic analysis and controlled experimentation by simulating tasks related to exploratory information search, to:

• Identify the graphical and interactive functional factors of visualization that have an impact on the process of exploratory information searching • Correlate visualization and exploratory information searching in terms of usability on the basis of identifiable search strategies and interactive views, according to either the user's proficiency in a particular field of knowledge or his expertise in information searching • Through interaction between the user and the context-related visualization device, define what represents a gain or an obstacle in the process of exploratory information search, in particular with respect to the user's own strategies and tactics

2

Litterature Review

Exploratory Search

Marchionini [START_REF] Marchionini | Exploratory Search: from Finding to Understanding[END_REF] and White and Roth [START_REF] White | Exploratory Search: Beyond the Query-Response Paradigm[END_REF] have modeled the exploratory search process by providing perspective on the iterative and non-linear dynamics between different cognitive strategies and associated search tactics. This model highlights interactions between different strategies and their associated tactics. Among exploratory search tactics, several behavioural information search models, such as the « berry-picking model » by Bates [START_REF] Bates | The Design of Browsing and Berrypicking Techniques for the Online Search Interface[END_REF], the « sensemaking model» by Dervin [START_REF] Dervin | From the Mind's Eye of the User the Sense-making Qualitative-quantitative Methodology[END_REF], and « information foraging » by Pirolli et Card [START_REF] Pirolli | Information Foraging[END_REF] can be recognized.

Fig. 1. Exploratory search mo 14, reprinted with permission)

The main learning strate [START_REF] Kuhlthau | A Principle of Uncertainty for Information Seeking[END_REF], [START_REF] Kuhlthau | The 'Information Search Process' Revisited: Is the Model still Useful?[END_REF] which consists in ing new information to addr

The process of explorato knowledge, is a typical exam cognitive challenges, the « gine environment is conside Since 2006, an increasin HCIR2010; WISI2010) are with exploratory informatio face are emerging to suppo as mSpace [START_REF] Smith | mSpace.fm[END_REF] and visual i

Search Results Visu

The visualization of search in displaying the results tha and proposing a device that tive views. The visualization of info nal cognitive aid [START_REF] Norman | Things that make us smart[END_REF]. Gra vealing dominant themes or same manner that performin tive load of mental calculati Visualization Usable for Displaying Web Search Results odel based on Marchionini [START_REF] Marchionini | Exploratory Search: from Finding to Understanding[END_REF] (Source: White and Roth [16 egy partly reminds us of Kulthau's more general proc n reducing cognitive and affective uncertainties by prov ress a gap or a weakness in the state of knowledge. ory information, aimed explicitly at the acquisition of n mple of this type of process [START_REF] Schraefel | Building Knowledge: What's beyond Keyword Search?[END_REF]. Thus, with respect to th query-response » paradigm that dominates the search ered inefficient, if not detrimental. ng number of scientific meetings (EESS2006 ; ESI20 e being organized to discuss the main challenges associa on search: interactivity and evaluation. Two types of in ort exploratory information search: faceted interfaces s interfaces such as Carrot2Search [START_REF] Carpineto | A Survey of Web Clustering Engines[END_REF]. ualization results is a device for visualising information that cons at match a user's query in a graphical and dynamic mann t facilitates navigation within these results through inter ormation (fig 2) is an interactive device acting as an ex aphical and interactive representation of results aims at r informational motives that characterize the results, in ng a mathematical calculation in writing relieves the cog ion on the memory. To accomplish this, visualization of search results is designed with the aid of clustering algorithms originating from the field of text searching. The web has and will continue to bring numerous challenges regarding the structuring of search results [START_REF] Carpineto | A Survey of Web Clustering Engines[END_REF]. To this day, unsupervised clustering algorithms (Suffix Tree Clustering, k-means) can be adapted to the dynamics, the heterogeneity and the size of the web. Moreover, this structuring process, with or without downstream visualization, is known to facilitate investigation and discovery within large masses of textual material. As a result of these properties, visualization helps the knowledge crystallization process.

This crystallization process (fig 3) calls for tactics and strategies that are similar to those of exploratory information search. Interactive viewing allows direct manipulation of the graphic representation in order to enable analytical and investigatory processing of results, thus providing an insight into the state of knowledge available on the web on a particular subject.

However, beneath the surface of visualization's generic principles, the application of visualization in the environment of search results remains problematic. Some openly criticize the use of this device on search results given the textual nature of information [START_REF] Hearst | Search User Interfaces[END_REF]. In the commercial sector, two important players have been forced to cease their operations as visual meta-search engines: Kartoo and Grokker. On several occasions, in a general context or with reference to information search, researchers [START_REF] Chen | Top 10 Unsolved Information Visualization Problems[END_REF]- [START_REF] Kerren | Workshop Report: Information Visualization-Human-Centered Issues in Visual Representation, Interaction, and Evaluation[END_REF] report similar findings concerning visualization: challenges include a lack of understanding of the cognitive processes involved, scalability issues associated with the variable size of the information mass to be processed, and difficulties related to categorization and labelling of search results [START_REF] Carpineto | A Survey of Web Clustering Engines[END_REF].

On a more fundamental level, a major obstacle to the establishment of this device is the absence of an authoritative evaluation method [START_REF] Carpendale | Evaluating Information Visualizations[END_REF]. Although the theoretical principle of cognitive amplification is agreed upon, its application is based on a multidimensional analysis, where a deep understanding of the interaction between the user and the system is essential. The question is then: how to evaluate the impact of interactive views on the user experience of information seeking process? Fig. 3. Knowledge crystallization during information visualization process (Source: Card, Mackinlay et Shneiderman [START_REF] Card | Readings in Information Visualization: Using Vision to Think[END_REF] reprinted with permission)

Evaluation of Search Results Visualization

The evaluation of a system aims to assess its capacity to satisfy a user according to chosen criteria [START_REF] Al-Maskari | A review of factors influencing user satisfaction in information retrieval[END_REF]. In the fields of both scientific research and innovation, evaluation serves the role of a barometer that is able to validate the quality of a system with respect to different criteria, such as performance, satisfaction, usefulness or usability. The objective, the context, the task at hand and the stage of maturity of the system are all relevant criteria for selecting a scenario and an evaluative approach [START_REF] Lam | Seven Guiding Scenarios for Information Visualization Evaluation[END_REF]. Traditionally, information retrieval systems were evaluated through performance, a systemcentered criterion, by measuring the system's recall and precision rates [START_REF] Harman | TREC: An Overview[END_REF]. The interdependency between the interactions generated by the system's use and the information seeking process performed by the user makes placing the user at the core of the evaluation process essential.

Evaluating the visualization of search results will address the requirements of the task at hand as well as those of the exploratory search and of the visualization process. To perform this assessment, we aim to take advantage of the shared nature of visualization and information seeking processes, as summarized by Zhang [START_REF] Zhang | Benchmarks and Evaluation Criteria for Information Retrieval Visualization[END_REF], and to gauge the added value of the visualization device. Several evaluation methods are in use, although none have earned general recognition [START_REF] Plaisant | The Challenge of Information Visualization Evaluation[END_REF]. To evaluate the user experience with respect to a given task, it is recommended to assess device usability [START_REF] Lam | Seven Guiding Scenarios for Information Visualization Evaluation[END_REF] In practice, it appears that few usability studies have been conducted to this date [START_REF] Chen | Top 10 Unsolved Information Visualization Problems[END_REF], [START_REF] Freitas | On Evaluating Information Visualization Techniques[END_REF], [START_REF] Valiati | Using Multi-dimensional in-depth Long-term Case Studies for Information Visualization Evaluation[END_REF] and that controlled, user-centric experiments are currently the preferred approach. The latter offer the advantage of precise and generalizable results [START_REF] Carpendale | Evaluating Information Visualizations[END_REF]. Most are focused on method evaluation and measuring the impacts of individual characteristics (gender, spatial abilities, cognitive style, associative memory). However, the heterogeneity of chosen measures, tasks and visualizations thwarts the process of comparing results [START_REF] Julien | Controlled User Evaluations of Information Visualization Interfaces for Text Retrieval: Literature Review and Meta-analysis[END_REF]. In addition, some case studies and longitudinal studies have been performed; their results are very detailed but remain hardly generalizable. Moreover, although not pertaining specifically to the evaluation of visualization of search results, the usability evaluation framework designed by Wilson [START_REF] Wilson | An Analytical Inspection Framework for Evaluating the Search Tactics and User Profiles Supported by Information Seeking Interfaces[END_REF] establishes measures that allow the assessment of interactive features according to the strategies of exploratory research. The last method for evaluating user experience considered in this research focuses on the insight gained by the user from the visualization of the presented information, rather than on the device and its components. North [START_REF] North | Toward Measuring Visualization Insight[END_REF] mentions that these two evaluations are not in opposition but that they are, on the contrary, rather complementary. Indeed, Rester and Pohl's [START_REF] Rester | Methods for the evaluation of an interactive InfoVis tool supporting exploratory reasoning processes[END_REF] study on the evaluation of a visualization device designed to facilitate exploratory research processes confirms this statement.

General Study Design

Our research seeks to evaluate, in terms of usability, the added value of the visualization of search results in the context of exploratory information search on the web. The main hypothesis of this research is that interactive views support the tactics and strategies related to information searching. The evaluation framewor exercises. The first phase a stage process. During the fi ing to heuristics [START_REF] Zuk | Heuristics for information visualization evaluation[END_REF]. The two types of users (experts by simulating an explorator

In practice, the objectiv search results is to demons ganization, interactive displ the user's task in an effici with a high degree of usabi to a minimum. Several obje visualization device. Firstly alization device within and fied. Secondly, the interfac tactics will be compared in

The second phase is of the insights the user receiv the chosen tactics and strate

The system selected for system features a textual a automatic classification alg purposes. Several types of expertise are tested simultaneously to shed light on the various factors known to have an impact on the skills required for the use of information visualization and exploratory information search on the web. Thus, the proposed heuristic analysis will be performed by information visualization experts. The experimental phase calls for two types of users: 15 volunteer users that have demonstrated their proficiency in information handling by successfully completing the Information Search coursework at the Library Sciences School of the University of Montreal, as well as 15 users that have proven skills in a field of knowledge related to the task of information search.

Conclusion

In theory, visualization constitutes an external cognitive aid. It supports memorisation and the processes of analysis and investigation by highlighting patterns and relationships between various pieces of information. These cognitive properties appear to assist the strategies and tactics required by exploratory information search. This research aims to evaluate, in terms of usability, the added value of a device designed for the visualization of search results within an appropriate context. We hope this work will make three types of contribution to evolving knowledge. Theoretically, we seek empirical evidence confirming the cognitive benefits of visualization in the context of exploratory information search. Methodologically, we aspire to devise effective qualitative metrics for evaluating the visual interface for exploratory information search. Lastly, we wish to formulate practical recommendations for successfully integrating visualization into information search interfaces.

Fig. 2 .

 2 Fig. 2. Reference model for information visualization process (Source: Card, Mackinlay et Shneiderman [14] p. 17 reprinted with permission)

Fig. 4 .

 4 Fig. 4. Design of the usability evaluation of search results visualization

Fig. 5 .

 5 Fig. 5. Example of visual clus rot2search engine


A Comparison Study of Clustering or Classification Methods for Search Results

Visualization in Web Search Context

Aline Crédeville École de Bibliothéconomie et de Science de l'Information Université de Montréal aline.credeville@umontreal.ca

Abstract

The amount of information on the Web is steadily growing since its beginning. The number of online information retrieval systems has increased in parallel to this amount of information. These have been designed to help the information seeking process and to perform the user final tasks from his perspective [START_REF] Wilson | Models in Information Behaviour Research[END_REF]. As a result, web search engines are massively used to allow the accomplishment of a large range of environment-dependant and goal-ended tasks [START_REF] Broder | A Taxonomy of Web Search[END_REF][START_REF] Rose | Understanding User Goals in Web Search[END_REF][START_REF] Toms | The Effect of Task Domain on Search[END_REF]. While information retrieval techniques (indexing, organization and ranking) and interactive features have been improved since the last twenty years [START_REF] Baeza-Yates | Modern Information Retrieval[END_REF][START_REF] Manning | Introduction to Information Retrieval[END_REF][START_REF] Shneiderman | Designing the User Interface: Strategies for Effective Human-Computer[END_REF], the on-line information retrieval systems still remain hard to use [START_REF] Borgman | Why Are Online Catalogs Still Hard to Use[END_REF]Markey, 2007a[START_REF] Markey | Twenty-five Years of End-user Searching, Part 2: Future Research Directions[END_REF] and don't fit the cognitive and affective processes of the information searching tasks efficiently [START_REF] Ingwersen | Cognitive Perspectives of Information Retrieval Interaction : Elements of Cognitive IR Theory[END_REF][START_REF] Ingwersen | The turn: integration of information seeking and retrieval in context[END_REF][START_REF] Kuhlthau | Towards Collaboration between Information Seeking and Information Retrieval[END_REF][START_REF] Kuhlthau | The 'Information Search Process' Revisited: Is The Model Still Useful?[END_REF]. The context of the user (professionnal, scholar, or everyday life), his final tasks, his individual differences, and the Kuhlthau's stages of information-seeking process (1991) still need to be taken into account. These design problems generate a high cognitive load because of the growth of affective and cognitive uncertainty [START_REF] Gwizdka | Distribution of Cognitive Load in Web Search[END_REF] which has to be reduced to ease the learning process. Two of the reasons of this uncertainty are, on the one hand, the noise in the considered search results which overhelm the user working memory and, on the other hand, the lack of interactive features which slow down exploration, one of the critical stage of the information-seeking process (Markey 2007a;[START_REF] Kuhlthau | Inside The Search Process: Information Seeking from The User's Perspective[END_REF].

The use of information visualization could bring significant improvements to the design of information retrieval systems. Information visualization is defined as "the use of computer-supported, interactive, visual representations of abstract data to amplify cognition" [START_REF] Card | Readings in Information Visualization: Using Vision to Think[END_REF]. Information visualization, cartography for example, is known to reduce redundancy in data and to facilitate the identification of meaningful patterns through large and multidimensional data [START_REF] Bertin | Sémiologie Graphique: Les Diagrammes, Les Réseaux[END_REF][START_REF] Larkin | Why A Diagram Is (Sometimes) Worth Ten Thousand Words[END_REF][START_REF] Norman | Things That Make Us Smart[END_REF][START_REF] Resnikoff | The Illusion of Reality[END_REF][START_REF] Tufte | Envisioning Information[END_REF]. It has been mainly developed in the information retrieval field as a way to display abstract information in a graphical and logical structured form [START_REF] Card | Readings in Information Visualization: Using Vision to Think[END_REF][START_REF] Chen | Information Visualization: Beyond the Horizon[END_REF]Jin Zhang, 2008) and as a way to interact with information in an information-seeking context [START_REF] Shneiderman | The Eyes Have It: A Task by Data Type Taxonomy for Information Visualizations[END_REF]. In 2000, after ten years of research, the information visualization field has developed largely accepted theoretical foundations. There are yet important issues [START_REF] Burkhard | Visualization Summit 2007: Ten Research Goals for 2010[END_REF][START_REF] Chen | Top 10 Unsolved Information Visualization Problems[END_REF][START_REF] Keller | Visualizing Knowledge and Information: An Introduction[END_REF] to be solved.



The divorce between the logical organization of the abstract information and it representation into an understandable metaphor.  Multidimensional scaling.  The evaluation of usability of visual information retrieval systems [START_REF] Kerren | Workshop Report: Information Visualizationhuman-centered Issues in Visual Representation, In-teraction, and Evaluation[END_REF]Lin, Kerren, & Jiaje Zhang, 2009;[START_REF] Plaisant | The Challenge of Information Visualization Evaluation[END_REF]. Considering these issues of traditional and visual information retrieval, we think that the gap between the ranking structure of search results and their transformation into a meaningful graphical and interactive representation could be bridged with data mining operations. More specifically, classification and clustering algorithms could extract salient structures of the retrieved set of search results in order to shape the visual representation of the results.

In the context of information-seeking with a web search engine, the goals of our research project are the following.

 Identify the organizational factors required to make the graphical representation constructed by the display algorithm a meaningful way to present the retrieved set of search results. More specifically, we seek to answer the following questions.

 What are the constraints imposed by the classification and clustering methods on the possible graphical and interactive visual representation of search results?  What are the parameters to apply for each method of clustering and classification?  Determine which improvements, from the end-user perspective, that are made possible by the visualization of search results, for both the clustering and classification methods. For this goal, the specific questions we want to answer are the following.

 What are the characteristics of the web search strategies enabled by each method of clustering and classification?  What are the graphical and interactive characteristics of the web search strategies enabled by text-listed and visual presentation of search results?  Establish a model of the relations between the logical organization of search results, the graphical and interactive display, the enduser, and the task. To answer these questions, a controlled experimentation is to be conducted according to the framework for Interactive Information Retrieval, designed by [START_REF] Borlund | The IIR Evaluation Model: A Framework for Evaluation of Interactive Information Retrieval Systems[END_REF]. In our experiment, we will compare two Web Information Retrieval Systems (herafter named WIRS); each one tested by a different sample of future librarians and domain experts. The selected end-users will have to execute a simulated search task on the Web. This comparison will take into account the variation of both organizational algorithmic method -classification and clustering -and the textual and visual presentation of the search results. The collected data will consist of the multimedia transactional logs of the web search sessions, semi-controlled user interview, and quantitative measures of subjective relevance assessment. These transactional logs will be used to determine the interactive patterns and deduce the users web search strategies, which are to be confirmed by a semi-controlled user interview. The users will be interviewed about their satisfaction, more specifically on their subjective assessment of the relevance of the graphical and interactive presentation of search results. And, relative relevance and ranked-life relevance will be the quantitative measures to compare the WIRS performance [START_REF] Borlund | The IIR Evaluation Model: A Framework for Evaluation of Interactive Information Retrieval Systems[END_REF].

At Connections 2011, we would like to present in details our research design, the methodological framework used and our preliminaries results.


Alexis Michaud
email: alexis.michaud@univ-paris3.fr

A Measurement from Electroglottography: DECPA, and its Application in Prosody

The present study, drawing on recent research on the use of the derivative of electroglottographic signals, applies to the study of intonation a parameter called DECPA, "Derivative-EGG Closure Peak Amplitude": the amplitude of the peak on the derivative of the EGG signal at glottal closure. A pilot study comparing DECPA, F 0 , global intensity and open quotient (in data from tone languages) suggests that DECPA correlates with pragmatic emphasis (accent). The uses and limitations of this measurement are discussed in relation to the general issue of the measurement of harmonic richness.

Introduction 1.The need for measurements of harmonic richness

It has long been known that vocal effort and perceived loudness are complex phenomena that measurements of global intensity alone cannot capture. The linguistic importance of the spectral slope has been shown for American English and Dutch [START_REF] Sluijter | Spectral balance as a cue in the perception of linguistic stress[END_REF]. As to the linguistic role of spectral tilt, the authors offer a clear-cut interpretation: high-frequency emphasis is a correlate of lexical stress, whereas variations in fundamental frequency (F 0 ) correlate with (intonative) accent, also called "focus target", "sentence accent". But many observations contradict this dichotomy. In Swedish [START_REF] Gobl | Voice source dynamics in connected speech[END_REF], high-frequency emphasis was observed under focus, i.e. a pragmatic phenomenon. Fant et al. report spectral differences under lexical stress, but note that the strengthening of high frequencies is stronger under "emphatic stress" [8:623]. Similarly, [START_REF] Heldner | On the reliability of overall intensity and spectral emphasis as acoustic correlates of focal accents in Swedish[END_REF] uses spectral slope measurements for the detection of what he calls "focus", "focal accent", "accent", "prominence", i.e. phenomena that correspond to [START_REF] Sluijter | Spectral balance as a cue in the perception of linguistic stress[END_REF]'s accent. In English data, an increase in energy in the higher frequencies was found under emphasis [START_REF] Campbell | Loudness, spectral tilt, and perceived prominence in dialogues[END_REF]. Thus, the linguistic importance of harmonic richness is not restricted to the marking of lexical stress, but is a major cue of the supralexical phenomena referred to in the literature as "accent", "focus", "prominence", "emphatic stress".

There is at present no efficient vowel-independent method for measuring the glottal flow wave and the spectral slope, e.g. inverse filtering is vowel-dependent and "theorydependent", resting on certain theoretical assumptions. The argument of the present paper is that an interesting and relatively segment-independent cue can be obtained from electroglottography (hereafter EGG). Though EGG does not provide an estimate of the air flow at the glottis, it is a precise (and noninvasive) measurement of the contact surface of the vocal folds, by means of a high-frequency, low-intensity current passed between two electrodes placed on the neck. The parameters calculated from EGG, however, still vary broadly from author to author, substantiating the critical view of those who point out that EGG does not amount to a measurement of glottal flow, and warn against its unwarranted uses (especially as a tool for medical diagnosis).

New uses of electroglottography

There are however studies (based on high-speed cinematography) showing that parameters corresponding precisely to physiological events can be extracted reliably from EGG, by using the derivative of EGG (hereafter DEGG) : "All of the EGG waveforms exhibit a steeper slope (implying a rapid change in vocal fold contact area) during the glottal closing phase than during the opening phase. This characteristic phenomenon of vocal fold vibration results in a sharp negative pulse in the DEGG waveform at or near the instant of maximum glottal closure" [4:2400] (see figure 1, and [START_REF] Henrich | accepted. On the use of the derivative of electroglottographic signals for characterization of non-pathological voice phonation[END_REF]). The present pilot study is devoted to this sharp pulse in the DEGG signal, which is arguably related to harmonic richness: "Derivative-EGG Closure Peak Amplitude", or DECPA, the amplitude of the peak on the DEGG curve that corresponds to glottal closure. This parameter, pointed out to us by N. Henrich, was first studied in the singing voice, in [5:131-135] (a line of research currently pursued in [START_REF] Castellengo | The "voix mixte" register: a way of managing the transition between the two main laryngeal mechanisms[END_REF]). It is also mentioned in [START_REF] Dromey | Glottal airflow and electroglottographic measures of vocal function at multiple intensities[END_REF], but the authors do not consider DECPA in itself: they calculate the ratio between DECPA and the amplitude of the opening peak. The latter parameter has no known incidence on the resulting sound wave, and our own experiments, which included the measurement of opening peaks, did not bring out any regularity in opening peak amplitude, so it seems more promising to study DECPA separately.

What is the significance of DECPA?

It is known that greater speed of glottis closure results in a less steep spectral slope. DECPA is a measurement at one single point in time for each period: it corresponds to the highest speed in increase of vocal fold contact surface which is reached at the glottis-closure-instant.

DECPA is measurable even for relatively noisy EGG recordings (such as those used in section 3). It is well known that the closing peak is strongly marked on the DEGG signal (by contrast, the opening peak is much less salient, and its detection is impossible in some cases [START_REF] Henrich | Etude de la source glottique en voix parlée et chantée[END_REF]). DECPA can vary by a ratio of 1 to 5 within the same utterance. Its absolute value is not meaningful by itself, as it depends on the recording level set for the EGG signal; what is discussed in the following section is the evolution of this parameter in time, over the whole utterance. DECPA is compared with F 0 , RMS amplitude, and open quotient.

Speech materials

The linguistic data under investigation is from tone languages: Naxi (a Tibeto-Burman language), Mandarin Chinese, and Vietnamese. In these languages, the lexical tones leave a margin of variation, which is used extensively for intonational purposes, in particular reflecting information structure. Languages which thus combine lexical tones and "intonational accents" offer an interesting background for a study of the prosodic marking of informational prominence. In Vietnamese, statistics on 504 target syllables in carrier sentences showed that DECPA is significantly higher under emphasis (see figure 4; details are given in another paper presented at the present conference: "Glottalized and nonglottalized tones under emphasis: open quotient curves remain stable, F 0 curve is modified"). As for Naxi, the materials reported on here are "classical" question-and-answer pairs. The corpus, originally designed for French [START_REF] Delais-Roussarie | The Prosody of Post-focus sequences in French[END_REF], was translated into Mandarin Chinese, and again into Naxi with the help of an informant, retaining only the utterances for which he found natural-sounding equivalents. During the recording session, the Chinese version was read out as a prompt. Cprograms were developed for this study by VU NGOC Tuân (LIMSI-CNRS) to compute F 0 , the open quotient (hereafter O q ), and DECPA.

Predictably, several parameters were found to show differences between prosodically "strong" ("accented") and prosodically "weak" syllables, most notably F 0 , duration, glottis adduction, the consonant-to-vowel ratio, and formant position. The question of the relative importance of individual cues, and of the (notoriously complex) trading relations found between the various cues, is beyond the scope of this presentation. The study will center on the behavior of the Mid and Low tones under focus (Naxi possesses lexical High, Mid and Low tones, and a Rising contour tone).

Results

The Low tone.

In some cases, a DECPA maximum is observed on "focus", a salience not detectable by using F 0 , O q and global acoustic intensity.

Three types of behavior are reported in [START_REF] Michaud | Focus et tons lexicaux: de la variété des paramètres prosodiques. Fonctions et moyens d'expression de la focalisation à travers les langues[END_REF]: (i) intonation overrides the L lexical tone: the resulting F 0 is in the range of a H or M tone (i.e. Low tone under accent is sometimes as high as a nonaccented syllable with lexical High tone); (ii) a strong pharyngeal constriction is exercised and the syllable is devoiced; (iii) in many cases, there is an extra effort which is not signaled only (or even not primarily) by a change in F 0 . This last category will be examined here, pointing out examples in which the focus does not stand out in terms of F 0 or global intensity, whereas it is characterized by a maximum of DECPA.

Figure 2 plots the DECPA and intensity curves for utterance [START_REF] Campbell | Loudness, spectral tilt, and perceived prominence in dialogues[END_REF].

(1) nɯ˩ kʰɑ˩dy˩ tʰv˧ bɯ˧ lɑ˥ ? 2 nd pers. sg.-Beijing-to/(go) out-go-interrogative particule "Are you going to Beijing ?" (context: two people are sitting on a train; answer: "No, I am going to Tianjin.")

The corresponding F 0 and O q curves are found in figure 3. O q gives an indication on the "pressed" quality of voicing and vocal fold adduction [START_REF] Rothenberg | Monitoring vocal fold abduction through vocal fold contact area[END_REF]. Neither O q nor F 0 present any special prominence on the word "Beijing", /kʰɑ˩dy˩/. F 0 on both syllables is lower than on the 2 nd -person pronoun "you"; in terms of F 0 glissando, the verb, /tʰv˧/, is most salient. Neither of the two syllables of "Beijing" stands out in terms of duration either. As for acoustic intensity, measured from the audio, the most intense syllable is the 2 nd -person pronoun, at the beginning of the utterance. By contrast, DECPA is at its maximum on the first syllable of "Beijing", /kʰɑ˩dy˩/, which has greatest pragmatic/ informational importance.

The Mid tone: essentially similar observations.

In utterance (3), the "focus"/"accented syllable" is "seven", /ʂə˞˧/. Its F 0 is lower than that of classifier /tɕɑ˥/. On the curve of intensity, /ʂə˞˧/ does not stand out more than the negation /mə˧wɑ˩/. But, as in example 1, high DECPA values are briefly reached on the focus, /ʂə˞˧/ (see figures 5 and6).

(2) nɯ˩ hu˩mɯ˧kʰv˥ tʂʰwɑ˥ dʐɯ˩ gə˧ xo˧tʂʰə˧ ndzɯ˩ pu˥ tsʰɯ˩ lɑ˥ ?

2

The importance of phonation mechanism

The Naxi examples above are from the speech of a male informant (as are the Vietnamese data of figure 4). Complexities arise when studying DECPA from recordings of female subjects, revealing that one further parameter must be taken into account in interpreting DECPA. This point will be made here by studying data from a female Chinese speaker.

The data consists simply of the sentence He wants this pen, 他要那只笔。ta1 yao4 na4 zhi1 bi3 (3 rd pers. sg.-want, need-deictic-classifier-pen) The two realizations in figures 7-8 correspond to the following interpretations: He wants this pen/ He wants this pen, with accent on different syllables. In figure 7, the accented (or "focused") element is pen, bi3, which carries lexical tone 3: a low-falling-rising tone. It appears that, despite its relatively low F 0 and intensity [intensity is not plotted here], bi3 is the most salient in terms of DECPA, following the pattern observed in section 2. The DECPA valley in the middle of the syllable is due to the fact that, as the speaker reaches the very bottom of her range, phonation is almost interrupted. Figure 8, on the contrary, shows paradoxical results: DECPA is low on the accented/focused word, where high values were expected.

This can be accounted for in light of research by B. Roubeau, which distinguishes four laryngeal mechanisms, on a physiological basis [START_REF] Roubeau | Electroglottographic study of the changes of voice registers[END_REF][START_REF] Roubeau | Control of laryngeal vibration in register change. Vocal Fold Physiology: Acoustic, perceptual, and physiological aspects of voice mechanisms[END_REF][START_REF] Henrich | On the use of electroglottography for characterisation of the laryngeal mechanisms[END_REF]. This four-way distinction is distinct from the classification of phonation types in [START_REF] Catford | Phonation types: The classification of some laryngeal components of speech production[END_REF][START_REF] Laver | The phonetic description of voice quality[END_REF]. Roubeau shows that in speech as well as in singing, phonation can be classified into several "laryngeal registers", or "laryngeal mechanisms": "mechanism 0", or vocal fry; "mechanism I", the most frequent in speech by male subjects, used by female subjects at the lower end of their F 0 range; "mechanism II" (alto/mezzo-soprano/soprano voice), in which the vibrating portion of the vocal folds is smaller (due to compressed arytenoids); and mechanism III, used for the very highest frequencies. It is essential to take the laryngeal mechanism parameter into account in studies of DECPA, as the vibrating portion of the vocal folds is significantly different in these four mechanisms, e.g. DECPA in mechanism II is lower, and shows much less variation, than in mechanism I (a physiological explanation is given in [START_REF] Roubeau | Electroglottographic study of the changes of voice registers[END_REF][START_REF] Roubeau | Control of laryngeal vibration in register change. Vocal Fold Physiology: Acoustic, perceptual, and physiological aspects of voice mechanisms[END_REF]). In our Chinese data, the speaker realizes high pitches in mechanism II, as female speakers are known to do [START_REF] Roubeau | Electroglottographic study of the changes of voice registers[END_REF]; low pitches in mechanism 1; and extra-low pitches, as in the lowest part of tone 3, in mechanism 0 (see annotation to figures 7-8). Three laryngeal registers are thus used in the utterance plotted in figure 8: register 2 during the first three syllables, which have high pitch (DECPA values: medium); register 1 during the fourth syllable, which has lower F 0 (DECPA values: high); register 0 during the last syllable, which is laryngealized throughout (DECPA values: very low), as evidenced by the very irregular glottal cycles. In figure 7, syllables 1 to 4 are in mechanism 2; syllable 5, which has higher pitch than in figure 8, is in mechanism 0 only in the middle (with very low DECPA values), and in mechanism 1 at the beginning and end, where the F 0 values are higher. Similar observations were made with a female Naxi speaker.

General discussion and conclusion : Use and limitations of DECPA in prosodic studies

The limitations of DECPA are:

-Direct comparison can be made neither across speakers nor across recording sessions, as DECPA varies with input level.

-Comparison is only possible within the same laryngeal mechanism. This makes it necessary to determine which mechanism is being used. This is important information, for which no automatic detection procedures have been proposed as yet [START_REF] Henrich | On the use of electroglottography for characterisation of the laryngeal mechanisms[END_REF].

-If there is a double closing peak, which is frequent, no principled answer can be given as yet on which is the "true" DECPA: the first, the higher of the two, or some value computed from the two peaks. (This topic is currently investigated by N. Henrich and C. Gendrot by means of highspeed cinematography.) -Quantifying the relationship between DECPA and spectral slope is complex. Points of reference are missing; e.g., a comparison of spectral slices and DECPA values at different points in the course of a syllable does not offer a reliable basis because of the constantly changing vocal tract shape (and formant frequencies). More generally, it is unlikely that this relationship can be expressed in a universally valid mathematical formula, as DECPA, the maximum speed at closure, is only one of several glottal source parameters. With these reservations, the proposed use of DECPA as part of an experimental setup in linguistic studies of stress, accent and vocal effort is to detect the points of DECPA prominence within an utterance, or across items within a recording session. From our observations, it appears that the linguistic phenomena of accent is better correlated with the DECPA maxima reached (even very briefly) than with DECPA values averaged over syllables. In conclusion, DECPA does not actually amount to the much-wanted measurement of spectral slope, but represents a significant cue in the case of phonation mechanism I.

Resource sharing -Hypertext link

The signals of the examples are currently available at www.cavi.univ-paris3.fr/ilpga/ED/student/stam/index.html. Requests for the C programs for EGG analysis should be sent to Tuan.Vu-Ngoc@limsi.fr.

Acknowledgments

Many thanks to N. Henrich (LAM/CNRS) and T. Vu Ngoc (LIMSI/CNRS) for providing guidance and software. Special thanks also to J. Vaissière, M. Castellengo, K. Kohler, and A. Rialland, C. Mooshammer and C. Gendrot; to Pr. Sun Hongkai and Huang Xing for the invitation to China; to Pr. Kong Jiangping for use of the facilities at his laboratory in Beijing; and to the Naxi and Vietnamese informants and friends. Financial support was given by the Fondation de France and by the Conseil Scientifique of University Paris 3.


M El

On some automatic continuity theorems

Keywords: Uniform topological algebra, saturated locally convex algebra, automatic continuity of homomorphism, weakly regular algebra Mathematics Subject Classification 2010. 46H05, 46H40

We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform topological algebras are extended to a larger class of algebras. Consequences and applications are given.

Introduction

A. C. Cochran introduced, in [START_REF] Cochran | Representation of A-convex algebras[END_REF], the notion of saturated uniformly A-convex algebras. M. Oudadess showed, in [START_REF] Oudadess | V-saturated uniformly A-convex algebras[END_REF], that the class of complete saturated uniformly A-convex algebras (in the sense of Cochran) is empty and introduced a new definition of saturation in uniformly A-convex algebras. In the Banach case, M. Oudadess showed that is a saturated Banach algebra iff is a unital uniform Banach algebra. A. Beddaa extended, in [START_REF] Beddaa | Continuité automatique de morphismes d'algèbres localement convexes[END_REF], the definition of saturation given by M. Oudadess in uniformly A-convex algebras to locally convex algebras with nonempty set of nonzero multiplicative linear functionals.

In the second section, we give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals (Theorems 2.1 and 2.2). We obtain, as a consequence, the following result: Let be a functionally continuous normed algebra with unit. Then is a saturated normed algebra iff is a uniform normed algebra (Corollary 2.4). This improves the above result of M.Oudadess. We also obtain, as a consequence, the following result of A. Beddaa: A uniform topological algebra with unit is a saturated locally multiplicatively convex algebra [START_REF] Beddaa | Continuité automatique de morphismes d'algèbres localement convexes[END_REF]. We show that if is a functionally continuous, saturated locally convex algebra, whose spectrum is equicontinuous, then is a uniform normed algebra (Theorem 2.5). We obtain, as a consequence, the following result of A. Beddaa: The topology of a saturated locally convex Q-algebra is normable [START_REF] Beddaa | Continuité automatique de morphismes d'algèbres localement convexes[END_REF]. At the end of the second section, we introduce the following property: Let be a locally convex algebra with nonempty set of nonzero multiplicative linear functionals, satisfies the property (P) if for all and for every with there exists such that We show that the class of locally convex algebras satisfying property (P) contains both; the class of uniform topological algebras and the class of saturated locally convex algebras. The class of locally convex algebras satisfying property (P) is introduced to give some automatic continuity theorems.

In the third section, we show that some automatic continuity theorems in advertibly complete uniform topological algebras, obtained by A. Beddaa, S. J. Bhatt and M. Oudadess in [START_REF] Beddaa | On automatic continuity of homomorphisms[END_REF], are true if the class of advertibly complete uniform topological algebras is replaced by the larger class of locally convex algebras satisfying property (P) (Theorems 3.1, 3.2 and 3.3). Finally, as an application, we obtain that: If is a weakly regular, functionally continuous, uniform normed algebra with unit, then is the unique uniform norm on (Theorem 3.6). This improves a result of S. J. Bhatt and D. J. Karia [3, Corollary (second affirmation)].

Preliminaries

A topological algebra is an algebra over the complex field which is also a Hausdorff topological vector space such that the multiplication is separately continuous. A locally convex algebra (lc algebra) is a topological algebra whose topology is locally convex. A locally multiplicatively convex algebra [START_REF] Michael | Locally multiplicatively convex topological algebras[END_REF] (lmc algebra) is a topological algebra whose topology is determined by a family of submultiplicative seminorms. For each let the quotient algebra is a normed algebra in the norm . A uniform seminorm on an algebra is a seminorm such that for all Such a seminorm is submultiplicative [START_REF] Dedania | A seminorm with square property is automatically submultiplicative[END_REF]. A uniform topological algebra (uT-algebra) is a topological algebra whose topology is determined by a family of uniform seminorms. A uniform normed algebra is a normed algebra such that for all Let be an algebra and , we denote by the spectrum of and by the spectral radius of A norm on an algebra is an algebra norm if is a normed algebra. An algebra norm on is semisimple [START_REF] Beddaa | On automatic continuity of homomorphisms[END_REF] if the completion of is semisimple. A topological algebra is a Q-algebra [START_REF] Michael | Locally multiplicatively convex topological algebras[END_REF] if the set of quasiinvertible elements is open. For a topological algebra denotes the set of nonzero continuous multiplicative linear functionals on A topological algebra is strongly semisimple if for every , there exists such that A topological algebra is weakly regular [START_REF] Beddaa | On automatic continuity of homomorphisms[END_REF] if given a closed subset there exists a nonzero such that for all A topological algebra is functionally continuous if A topological algebra is weakly -compact-regular [START_REF] Beddaa | On automatic continuity of homomorphisms[END_REF] if given a compact subset , there exists a nonzero such that for all A topological algebra is advertibly complete if a Cauchy net in converges in whenever for some and both converge to 0. Let be an lc algebra with unit such that is a saturated lc algebra [START_REF] Beddaa | Continuité automatique de morphismes d'algèbres localement convexes[END_REF] if the topology of is determined by a family of seminorms such that [START_REF] Beddaa | Continuité automatique de morphismes d'algèbres localement convexes[END_REF] for all (2) for all and with there exist , such that Further if is submultiplicative for all we say that is a saturated lmc algebra. A saturated normed algebra is a normed algebra with unit such that (1) (2) for all with , there exist , such that . [START_REF] Michael | Locally multiplicatively convex topological algebras[END_REF]Proposition 7.5], there exists such that By the previous statement, Theorem 2.1 and the proof of Theorem 2.5, if is a uT-algebra or a saturated lc algebra, then satisfies the property (P).

Uniform topological algebras and saturated locally convex algebras

Let

Remark. There exists a saturated lc algebra which is not a uT-algebra [1, p.131].

Automatic continuity theorems

In this section , we extend some automatic continuity theorems in advertibly complete uniform topological algebras, obtained in [START_REF] Beddaa | On automatic continuity of homomorphisms[END_REF], to lc algebras satisfying property (P). Theorem 3.1. Let be a weakly regular, lc algebra with unit, satisfying (P). Let be an lmc algebra, and let be a one-to-one homomorphism such that (the closure of ) is a semisimple Q-algebra. If is functionally continuous, then is continuous. Theorem 3.5. Let be an lc Q-algebra, be a commutative complete lmc algebra. Let be a one-to-one homomorphism such that is continuous. Then is a Q-algebra.

Proof

Proof. Since is an lc Q-algebra, is continuous at [8, Proposition 13.5], then there exists a continuous seminorm on such that for all Let , is a continuous seminorm on then extends as a continuous seminorm . We have for all . Let for some net in . Let with Since is a commutative complete lmc algebra, we have for all , then is continuous at and so is a Q-algebra by [START_REF] Michael | Locally multiplicatively convex topological algebras[END_REF]Proposition 13.5].

Remark. Theorem 3.5 shows that the hypothesis " is complete and m-convex" is not necessary in [2, Proposition 4].

An algebra norm on an algebra is functionally continuous if is functionally continuous.

Theorem 3.6. Let be a weakly regular, functionally continuous, uniform normed algebra with unit. Let be an algebra norm on

  Let be a saturated locally convex Q-algebra, then is a uniform normed algebra.

	Proof. Let	and	with	there exist	such that
						Let
			, we have		for all	and	Since is Hausdorff and
	is equicontinuous,	is a continuous uniform norm on , then the topology of can be
	defined by the uniform norm
	Corollary 2.6. Proof. is functionally continuous and		is equicontinuous [7, Proposition II.7.1].
	Let	be an lc algebra with		we say that satisfies the property (P) if for all
	and	with		there exists	such that	Let be an lc
	algebra satisfying property (P). Then		Also for
				Hence is commutative and semisimple. Let
	be a uT-algebra without unit. Denote by	the zero map from	to and by the zero map from
	to . Let	and	,
				by [8, Proposition 7.5]. Let	and	with	Since
	is compact
		be an lmc algebra. For each	, let	for all
	and let :	,		, be the natural homomorphism from to	. Let
	with		we may define a multiplicative linear functional on by
	it is clear that		For		let
	denotes the Gelfand transform of
	Theorem 2.1. Let	be a topological algebra with unit The following assertions are equivalent:
	is a uT-algebra;
	(2) The topology is defined by a family		of submultiplicative seminorms such that
	(i)	for all
	(ii) for all	and	with		there exists	such that

Proof. (1) ⇒ (2): By [START_REF] Mallios | Topological algebras, Selected Topics[END_REF]Theorem VIII.5.1], the topology of can be defined by the family of seminorms, , is the set of all compact equicontinuous subsets of

We have for all Suppose that Since the map is continuous and is compact, there exists such that

Clearly, for all i.e.,

⇒

We have for all and then is commutative by (ii