Ambroise Vest
email: ambroise.vest@math.unistra.fr

OBSERVATION OF VIBRATING SYSTEMS AT DIFFERENT TIME INSTANTS

Keywords: Mathematics Subject Classification. Primary 93B07; Secondary 42C99, 35L05 obsevability inequality, wave equation, Fourier series, Diophantine approximation

In this paper, we obtain new observability inequalities for the vibrating string. This work was motivated by a recent paper by A. Szijártó and J. Hegedűs in which the authors ask the question of determining the initial data by only knowing the position of the string at two distinct time instants. The choice of the observation instants is crucial and the estimations rely on the Fourier series expansion of the solutions and results of Diophantine approximation.

Introduction

Let q be a nonnegative number. The small transversal vibrations of a string of length π fixed at its two ends satisfy 1 (1)

     y ′′ -y xx + qy = 0
in R × (0, π), y = 0 in R × {0, π}, y(0) = y 0 , y ′ (0) = y 1 in (0, π).

The obtaining of observability inequalities for the vibrating string and for oscillating systems in general has been the object of many works. Indeed, observability being dual to controllability (cf. D. L. Russell [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF]), it is often a starting point to obtain controllability results (see e.g., J.-L. Lions [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF], A. Haraux [START_REF]Quelques méthodes et résultats récents en théorie de la contrôlabilité exacte[END_REF]). A useful tool to obtain such inequalities is the Fourier series expansion of the solutions (cf. V. Komornik and P. Loreti [START_REF] Komornik | Fourier Series in Control Theory[END_REF]).

Among all the different ways to observe the system (1), pointwise observation has been widely studied (see e.g., J.-L. Lions [START_REF] Lions | Poinwise control of distributed systems[END_REF], A. Haraux [START_REF] Haraux | Remarques sur la contrôlabilité ponctuelle et spectrale de systèmes distribués[END_REF]). It consists in getting estimations of the form (y 0 , y 1 ) I ≤ c y(•, ξ) O .

The main difficulties are the choice of the norms for the initial data • I as for the observation • O and the choice of a strategic point ξ in the domain. These particular points can be characterized by some of their arithmetical properties (see e.g., A. G. Butkovskiy [START_REF] Butkovskiy | Certain control problems in distributed systems[END_REF], V. Komornik and P. Loreti [START_REF] Komornik | Multiple-point internal observability of membranes and plates[END_REF]).

Following a recent paper of A. Szijártó and J. Hegedűs [START_REF] Szijártó | Observation problems posed for the Klein-Gordon equation[END_REF], we focus on a pointwise-in-time observation. Such type of observation seems to have been studied at first by A. I. Egorov [START_REF] Egorov | On the observability of elastic vibration of a beam[END_REF] and L. N. Znamenskaya [START_REF] Znamenskaya | State observability of elastic vibrations of a string under boundary conditions of the first kind[END_REF]. Given two norms, one for the initial data • I and one for the observation • O , the objective is to find two times t 0 and t 1 such that [START_REF] Bugeaud | Approximation by Algebraic Numbers[END_REF] (y 0 , y 1 ) I ≤ c( y(t 0 ) O + y(t 1 ) O )

From a practical point of view, such an inequality means that only knowing the position of the whole system at two different instants, we are able to recover the initial data y 0 and y 1 .

Definition 1.1. A pair (t 0 , t 1 ) of real numbers such that the observability inequality [START_REF] Bugeaud | Approximation by Algebraic Numbers[END_REF] holds is called a strategic pair (for (2)). 2The main idea of this paper is the following : depending on how the quantity t 0 -t 1 π is approximable by rational numbers, such pointwise-in-time observability inequalities hold. The main tools are the explicit expansion of the solutions in Fourier series and classical results of Diophantine approximation.

Let us describe the organization of the paper and state (informally) the main results.

In section 2, after recalling the definition of adapted functional spaces to study the well-posedness of (1), we reformulate the observation problem in this setting. These spaces, denoted by D s (s ∈ R), correspond essentially to the domain of -∆ s/2 . Then, we may chose two real numbers r and s such that

• I = • D s and • O = • D r .
In section 3, we investigate the observation of the classical string (i.e. q = 0). We prove (see Theorem 3.3) the following result :

Assume that r -s ≥ 1. Then, there exist strategic pairs. Moreover, if the inequality is strict, then almost all pairs are strategic. This result is optimal in the sense that there cannot be any strategic pair if r -s < 1.

In section 4, we prove that the difference r -s (see Theorem 4.1) can be reduced by adding further observations.

In section 5, we focus on the loaded string (i.e. q > 0). First we recall the main result of [START_REF] Szijártó | Observation problems posed for the Klein-Gordon equation[END_REF] in Theorem 5.1, which states essentially that if (t 0 -t 1 )/π is a rational number along with another hypothesis, then (t 0 , t 1 ) is a strategic pair with r -s = 1. After analyzing the occurence of such pairs under the above hypotheses in Proposition 5.2, we use another method to obtain new observability inequalities. We can state the following result (see Theorem 5.3):

Assume that r -s = 1. If (t 0 , t 1 ) is a strategic pair for the classical string, then it is also a strategic pair for the loaded string provided that q is sufficiently small. Finally, in section 6, we extend our method to the vibrating beam and rectangular plates.

Problem setting and notations

Let us recall the construction of some useful functional spaces related to the above problem (see e.g., [10, pp. 7-11], [1, pp. 335-340]). The functions sin(kx), k = 1, 2, . . . form an orthogonal and dense system in L 2 (0, π). We denote by D the vector space spanned by these functions and for s ∈ R, we define an euclidean norm on D by setting

∞ k=1 c k sin(kx) 2 s := ∞ k=1 k 2s |c k | 2 .
The space D s is defined as the completion of D for the norm . s . Then, D 0 coincide with L 2 (0, π) with equivalent norms and more generally, it is possible to prove that for s > 0,

D s = f ∈ H s (0, π) : f (2j) (0) = f (2j) (π) = 0, ∀ 0 ≤ j ≤ s -1 2 .
Identifying D 0 with its own dual, D -s is the dual of D s . For example,

D 0 = L 2 (0, π), D 1 = H 1 0 (0, π) and D -1 = H -1 (0, π) with equivalent norms.
Now, we recall a well-posedness result for the problem (1) via an expansion of the solutions in Fourier series. We set

ω k := k 2 + q, k = 1, 2, . . . Proposition 2.1. Let s ∈ R.
For all initial data y 0 ∈ D s and y 1 ∈ D s-1 , the problem (1) admits a unique solution

y ∈ C(R, D s ) ∩ C 1 (R, D s-1 ) ∩ C 2 (R, D s-2 )
given by

(3) y(t, x) = ∞ k=1 (a k e iω k t + b k e -iω k t ) sin kx,
where the complex coefficients a k and b k satisfy3 (4)

y 0 2 s + y 1 2 s-1 ≍ ∞ k=1 k 2s (|a k | 2 + |b k | 2 ).
The observability problem that we are going to investigate in this paper is the following. Given two real numbers r and s such that s ≤ r, we ask wether or not there are two times t 0 and t 1 such that (5)

y 0 s + y 1 s-1 ≤ c( y(t 0 ) r + y(t 1 ) r )
for a positive constant c, independent of the initial data (y 0 , y 1 ) ∈ D r × D r-1 .

3. Observability of the classical string (q = 0)

In this paragraph, we assume that q = 0 in the problem [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]. The following statement transforms the observation inequality [START_REF] Courant | Methods of Mathematical Physics[END_REF] to a problem of Diophantine approximation.

Proposition 3.1. The pair (t 0 , t 1 ) is strategic if and only if there is a positive constant c such that4 (6)

k(t 0 -t 1 ) π ≥ c k r-s , k = 1, 2, . . .
For the proof, we need the following Lemma 3.2. Set x ∈ R. We have

| sin kx| ≍ kx π , k = 1, 2, . . . Proof of Lemma 3.2.
We follow the proof of [START_REF] Komornik | Multiple-point internal observability of membranes and plates[END_REF]Lemma 2.3]. Denoting by m the nearest integer from kx/π,

| sin kx| = | sin(kx -mπ)| = sin kx π -m π .
We notice that |kx/π-m|π ≤ π/2. Hence, using the estimations (2/π)|t| ≤ | sin t| ≤ |t| which hold for |t| ≤ π/2, we have

2 π kx π -m π ≤ sin kx π -m π ≤ π kx π -m , i.e. 2 kx π ≤ | sin kx| ≤ π kx π .
Proof of Proposition 3.1. Using the Fourier series expansion (3) of the solutions of (1) and the estimation (4), we observe that the left member in ( 5) is equivalent 5to

∞ k=1 k 2s (|a k | 2 + |b k | 2 )
and the right member is equivalent to

∞ k=1 k 2r (|a k e ikt0 + b k e -ikt0 | 2 + |a k e ikt1 + b k e -ikt1 | 2 ).
Therefore, the observability inequality (5) holds if and only if there exist a positive constant c ′ such that for all k = 1, 2, . . . and all complex numbers a and b,

k 2s (|a| 2 + |b| 2 ) ≤ c ′ k 2r (|ae ikt0 + be -ikt0 | 2 + |ae ikt1 + be -ikt1 | 2 ). (7)
Now, for all k, we consider the linear maps T k in C × C (endowed with its usual euclidean norm) defined by

T k (a, b) := (ae ikt0 + be -ikt0 , ae ikt1 + be -ikt1 ).
Hence, the estimation [START_REF] Falconer | Fractal Geometry : Mathematical Foundations and Applications[END_REF] holds for all k if and only if all the T k are invertible and there exists a positive constant c ′′ independent of k such that

1 T -1 k ≥ c ′′ k r-s .
The determinant of T k equalling 2i sin k(t 0 -t 1 ), we deduce that all the T k are invertible if and only if (t 0 -t 1 )/π is irrational. In that case, their inverses are given by

T -1 k (a, b) = 1 2i sin k(t 0 -t 1 ) (e -ikt1 a -e -ikt0 b, -e ikt1 a + e ikt0 b)
and a computation of their norms yield

T -1 k = 1 + | cos k(t 0 -t 1 )| √ 2| sin k(t 0 -t 1 )| Thus, 1 T -1 k ≍ | sin k(t 0 -t 1 )| ≍ k(t 0 -t 1 ) π .
The first estimation follows from the expression of T -1 k while the second estimation is a consequence of the Lemma 3.2. We observe that if (6) holds, then (t 0 -t 1 )/π must be irrational and that ensures that all the T k are invertible. The proof is complete. In the following lemma, we gather some classical results of Diophantine approximation. 6 For a real number α, we set If α < 1 then the set E 1 defined in the Lemma 3.4 is empty. Indeed, if we suppose that x ∈ E α , then for sufficiently large k, kx ≥ 1/k with is in contradiction with a theorem of Dirichlet (see [4, p.4]) that asserts that if x is irrational, then the inequality kx < 1/k as infinitely many solutions in k.

E α := {x ∈ R : ∃c > 0 : kx ≥ ck -α , k = 1, 2, . . .}.
If α ≥ 1, then we use the Lemma 3.4. One can notice that the set of pairs (t 0 , t 1 ) ∈ R 2 such that (t 0 -t 1 )/π ∈ E α has full (resp. zero) Lebesgue measure or Hausdorff dimension in R 2 if E α has full (resp. zero) Lebesgue measure or Hausdorff dimension in R. For the Lebesgue measure, this results from Fubini's theorem. For the Hausdorff dimension, this is a consequence of its behaviour with a product of sets and its invariance by a bi-Lipschitz transformation (see e.g., [START_REF] Falconer | Fractal Geometry : Mathematical Foundations and Applications[END_REF]). 6 The results concerning the Lebesgue measure are due to A. Khinchin and the result concerning the Hausdorff dimension is due to V. Jarník Remarks.

• The assertion (a) of Corollary 3.3 can be seen as an optimality result. Indeed, it means that with only two observations, the difference r -s between the orders of the Sobolev norms in the inequality ( 5) must be at least 1. • One cannot obtain such estimations with only one observation. Indeed, let t 0 ∈ R. Then, the function y(t, x) = sin(t -t 0 ) sin(x) is a solution to (1) with y(0) = 0 or y ′ (0) = 0, but y(t 0 ) = 0. • If the pair (t 0 , t 1 ) is strategic, then, having only access to the two observations i.e. the position of the string at times t 0 and t 1 , we can recover the initial data y 0 and y 1 using the expansion in Fourier series of y(t 0 ) and y(t 1 ) and the applications T -1 k . Moreover, the observability inequality ensures a "continuity property" in this reconstruction process. Indeed, if two sets of observations are close, then the two sets of associated initial data must be close too.

• In the same way, if r -s ≥ 1, we can obtain estimations of the form

y 0 s + y 1 s-1 ≤ c( y ′ (t 0 ) r-1 + y(t 1 ) r ), y 0 s + y 1 s-1 ≤ c( y(t 0 ) r + y ′ (t 1 ) r-1 ), y 0 s + y 1 s-1 ≤ c( y ′ (t 0 ) r-1 + y ′ (t 1 ) r-1 ),
• Applying the Hilbert Uniqueness Method (see [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF], [START_REF] Komornik | Exact Controllability and Stabilization. The Multiplier Method[END_REF]), it is possible to prove the following exact controllability result : let 0 < t 0 < t 1 < T such that the observability inequality (5) holds with r = 0 and s = -1. Then, given initial data (y 0 , y 1 ) ∈ D 2 × D 1 , we can find two control vectors v and w in D 0 such that the solution (that can be defined rigorously) of the inhomogeneous problem 7      y ′′ -y xx = δ(t -t 0 )v + δ(t -t 1 )w in (0, T ) × (0, π), y = 0 on (0, T ) × {0, π}, y(0) = y 0 , y ′ (0) = y 1 in (0, π).

satisfy y(T ) = y ′ (T ) = 0.

With more observations

In this paragraph, we still assume that q = 0 in (1). In section 3, we have seen that with only two observations, it is necessary that r -s ≥ 1 in the estimation [START_REF] Courant | Methods of Mathematical Physics[END_REF]. In this paragraph, we prove that adding other observations, it is possible to reduce the difference r -s.

Theorem 4.1. Let t 1 , t 2 , . . . , t n ∈ R with n ≥ 2, r ∈ R and set s := r -1/(n -1).
Assume that among the (t i -t j )/π, 1 ≤ i, j ≤ n, we can extract n -1 elements τ 1 , . . . , τ n-1 that belong to a real algebraic extension of Q of degree n and such that 1, τ 1 , . . . , τ n-1 are linearly independent over Q. Then, there exists a positive constant c such that

y 0 s + y 1 s-1 ≤ c( y(t 1 ) r + . . . + y(t n ) r )
for all initial data (y 0 , y 1 ) ∈ D r × D r-1 .

The proof relies on the following 8 7 δ is the Dirac mass in 0. 8 The second inequality in the Lemma will only be used in section 6. Lemma 4.2 ([4, p. 79]). Let x 1 , . . . , x n be numbers that belong to a real algebraic extension of Q of degree n + 1 such that 1, x 1 , . . . , x n are linearly independent over Q. Then, there exists a positive constant c, only depending on x 1 , . . . , x n , such that max kx j ≥ ck -1/n , k = 1, 2, . . .

and k 1 x 1 + k 2 x 2 + . . . k n x n ≥ c(max |k j |) -n , (k 1 , . . . , k n ) ∈ Z n \ {(0, . . . 0)}.
Proof of Theorem 4.1. Adapting the method described in the proof of Theorem 3.1, it is sufficient to obtain the estimation

n p=1 |ae iktp + be -iktp | 2 ≥ ck -2 n-1 (|a| 2 + |b| 2 ),
where c is a positive constant, independent of a, b ∈ C and k ∈ N * . With no loss of generality, we can assume that τ p = (t 1 -t p+1 )/π for p = 1, . . . , n -1. We have

n p=1 |ae iktp + be -iktp | 2 = n p=2 ( 1 n -1 |ae ikt1 + be -ikt1 | 2 + |ae iktp + be -iktp | 2 ) ≥ c 1 n p=2 (|ae ikt1 + be -ikt1 | 2 + |ae iktp + be -iktp | 2 ) ≥ c 2 n p=2 | sin k(t 1 -t p | 2 (|a| 2 + |b| 2 ) ≥ c 3 n p=2 k(t 1 -t p ) π 2 (|a| 2 + |b| 2 ) ≥ c 3 max k(t 1 -t p ) π 2 (|a| 2 + |b| 2 ) ≥ c 4 k -2/(n-1) (|a| 2 + |b| 2 )
for all k = 1, 2, . . ., with positive constants c 1 , c 2 , c 3 , c 4 independent of a, b ∈ C. The numbers 1, (t 1 -t 2 )/π, . . . , (t 1 -t n )/π are independent over Q. In particular the numbers (t 1 -t p )/π, p = 1, . . . , n are irrational numbers. This ensures that some corresponding linear transformations on C × C (see the proof of Theorem 3.1) are invertible and implies the second inequality. The third inequality is a consequence of Lemma 3.2 while the last inequality results from Lemma 4.2.

Remark. Formally, letting the number of observations tend to +∞, setting r = 0 and T > 0, we recover an internal observability result :

y 0 2 0 + y 1 2 -1 ≤ c T 0 π 0 |y(t, x)| 2 dx dt.
5. Observability of the loaded string (q > 0)

In this paragraph, we assume that q > 0 in [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF] and that r and s are two real numbers such that r -s = 1. First, let us recall the Theorem 5.1 (A. Szijártó and J. Hegedűs, [17, Theorem 1 p.4]). Let t 0 and t 1 be real numbers such that

(8) t 0 -t 1 π ∈ Q and (9) sin((t 0 -t 1 ) k 2 + q) = 0, k = 1, 2, . . .
Then, (t 0 , t 1 ) is an strategic pair.

Are such hypotheses easily satisfied? We can answer this question with the following Proposition 5.2. The set of strategic pairs satisfying the hypotheses (8) and ( 9)

is dense in R 2 .
Proof. It is sufficient to prove that for each real number τ and each δ > 0, there exists a real number τ ′ satisfying the three conditions : |τ -τ ′ | < δ, τ ′ ∈ πQ and sin τ ′ k 2 + q = 0 for all k = 1, 2, . . . First, we notice that sin(ζ k 2 + q) = 0 if and only if ζ k 2 + q ∈ πZ. Now, we distinguish three cases.

1. If q is an irrational number. The set πQ being dense in R, there exists a number τ ′ ∈ πQ such that |τ -τ ′ | ≤ δ. Moreover, τ ′ can be written τ ′ = (a/b)π with a ∈ Z and b ∈ N * relatively primes. Assume that there exist k ∈ N * and n ∈ Z such that

τ ′ k 2 + q = nπ ⇐⇒ a b k 2 + q = n.
Then, q = n 2 b 2 a 2 -k 2 ∈ Q, which is in contradiction with our assumption on q.

2. If q is an integer. We recall that if (a/b)π ∈ πQ, then, sin((a/b)π k 2 + q) = 0 if and only if (a/b) k 2 + q ∈ Z. Moreover, the quantity k 2 + q is either an integer or an irrational number (depending on the fact that k 2 + q is a square or not). For sufficiently large k, k 2 + q cannot be an integer. Indeed,

k 2 + q = k 1 + q k 2 = k 1 + q 2k 2 + o 1 k 2 = k + q 2k + o
1 k and this is not an integer for sufficiently large k. Hence, for such k, it is an irrational number and so is (a/b) k 2 + q. Now, let τ ′′ := (a/b)π ∈ πQ such that

|τ ′′ -τ | < δ 2 .
We are going to perturb a little bit the rational number (a/b) in order to construct a number τ ′ such that the sine neither vanish. From the above discussion, the quantity k 2 + q can take at most a finite number of integer values when k varies. We denote them by x 1 , . . . , x N (if it does not take any integer value, then it is always an irrational number and we can take τ ′ = τ ′′ ). Let p be a prime number that is not a divisor of any of the numbers x 1 , . . . , x N . For sufficiently large n,

π (p n -1)a p n b -τ < δ.
and p n does not divide a. Now, two cases are possible. If k 2 + q is not an integer, then it is an irrational number and (p n -1)a p n b k 2 + q ∈ Z.

On the other hand, if k 2 + q is an integer, then k 2 + q = x l for one l ∈ {1, . . . , N } and (p n -1)a

p n b k 2 + q = (p n -1)ax l p n b ∈ Z.
because p n does not divide (p n -1)ax l . Finally,

τ ′ := (p n -1)a p n b π
satisfies the three expected conditions.

3. If q is a rational number but not an integer. Then, we can write q = c/d, where c and d are integers. Hence,

τ k 2 + q = τ k 2 + c d = τ k 2 + cd d 2 = τ d k 2 d 2 + cd
and we are lead back to the case where q is an integer. Now, we give another method to obtain an observability result for the loaded string.

Theorem 5.3. Let (t 0 , t 1 ) be a strategic pair for the classical string i.e. [START_REF] Komornik | Exact Controllability and Stabilization. The Multiplier Method[END_REF] | sin(k(t

0 -t 1 )| ≥ c k , k = 1, 2, . . .
for a suitable positive constant c. Then, it is also a strategic pair for the loaded string, provided that q is sufficiently small.

Remark. This result can be viewed as a complementary result to Theorem 5.1 since the hypothesis [START_REF] Komornik | Exact Controllability and Stabilization. The Multiplier Method[END_REF] implies that (t 0 -t 1 )/π is irrational; hence (8) cannot hold.

Proof. Applying the method described in the proof of Proposition 3.1, a necessary and sufficient condition for estimation (5) to hold true is

(11) | sin(ω k (t 0 -t 1 ))| = | sin( q + k 2 (t 0 -t 1 ))| ≥ c ′ k , k = 1, 2, . . . ,
where c ′ is a positive constant, independent of k.

Comparing the quantities | sin ω k (t 0 -t 1 )| and | sin(k(t 0 -t 1 ))|, we will find a sufficient condition that implies [START_REF] Komornik | Fourier Series in Control Theory[END_REF]. Let us estimate the difference

| sin( q + k 2 (t 0 -t 1 )) -sin(k(t 0 -t 1 ))|.
For a fixed k ∈ N * , we consider the application f k , defined for x ≥ 0 by

f k (x) := sin( k 2 + x(t 0 -t 1 )).
We have

|f ′ k (x)| = | cos( √ k 2 + x(t 0 -t 1 ))||t 0 -t 1 | 2 √ k 2 + x ≤ |t 0 -t 1 | 2k .
From the triangle inequality and the mean value theorem,

|f k (0)| -|f k (q)| ≤ |f k (q) -f k (0)| ≤ |t 0 -t 1 |q 2k .
Hence,

| sin( q + k 2 (t 0 -t 1 ))| ≥ | sin(k(t 0 -t 1 ))| - |t 0 -t 1 |q 2k ≥ c k - |t 0 -t 1 |q
2k and these estimations are satisfied for all k = 1, 2, . . . Thus, if the quantity [START_REF] Komornik | Multiple-point internal observability of membranes and plates[END_REF] c ′ := c -|t 0 -t 1 |q 2 is positive, the estimation (5) is true. A sufficient condition is [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF] q < 2c |t 0 -t 1 | .

Remarks.

• The inequality ( 13) can be rewritten more precisely as

9 q < 4 |t 0 -t 1 |(K((t 0 -t 1 )/π) + 2)
.

Indeed, from the proof of Lemma 3.2 and classical results of Diophantine approximation (see [START_REF] Shallit | Some facts about continued fractions that should be better known[END_REF]), the hypothesis (10) holds if and only if the the number (t 0 -t 1 )/π is badly approximable by rational numbers so that its partial quotients are bounded i.e. K((t 0 -t 1 )/π) is finite. Moreover,

| sin k(t 0 -t 1 )| ≥ 2 k t 0 -t 1 π ≥ 2 (K((t 0 -t 1 )/π) + 2)k .
• It is possible to avoid a restriction on the size of the potential q. Set ξ := (t 0 -t 1 )/π ∈ R \ Q and ν(ξ) := lim inf k→+∞ k kξ .

If ξ is badly approximable, then ν(ξ) > 0. Moreover, if ξ ′ is an irrational number such that its partial quotients coincide with those of ξ from a certain rank, then ν(ξ ′ ) = ν(ξ) (see [4, p. 11]). Let us construct a strictly decreasing sequence of irrational numbers by setting ξ 0 = ξ and

ξ n+1 = ξ n 1 + ξ n = 1 1 + 1/ξ n .
We can assume that 0 < ξ < 1 so that its development in continued fraction has the form Thus, for all n, ν(ξ n ) = ν(ξ) > 0 and the sequence (ξ n ) converges to zero. Now, from the definition of ν(ξ) and the Lemma 3.2, we obtain, for k sufficiently large,

ξ = ξ 0 = [0; a 1 ,
| sin kπξ n | ≥ 2 kξ n ≥ 2 ν(ξ) k .
Hence, going back to the relation ( 12), if we choose n sufficiently large so that 2ν(ξ) -ξ n πq 2 > 0 and if we assume moreover that sin(ω k πξ n ) = 0, k = 1, 2, . . . , then, choosing t 0 and t 1 such that t 0 -t 1 = πξ n , the observability inequality holds.

6. Extension of the method to beams and plates 6.1. Observability of a hinged beam. The small transversal vibrations of a hinged beam of length π satisfy ( 14)

     y ′′ + y xxxx = 0 in R × (0, π), y = y xx = 0 in R × {0, π}, y(0) = y 0 , y ′ (0) = y 1 in (0, π).
Using the same spaces D s as for the vibrating string, we have the

y 0 2 s + y 1 2 s-2 ≍ ∞ k=1 k 2s (|a k | 2 + |b k | 2 ). (16)
In this case, the observability problem turns to the following one : given two real numbers r and s such that s ≤ r, we are looking for two times t 0 and t 1 such that (17)

y 0 s + y 1 s-2 ≤ c( y(t 0 ) r + y(t 1 ) r )
for a positive constant c, independent of the initial data (y 0 , y 1 ) ∈ D r × D r-1 . Again, such a pair (t 0 , t 1 ) will be called an strategic pair.

Proposition 6.2. The pair (t 0 , t 1 ) is strategic if and only if there is a positive constant c such that k 2 (t 0 -t 1 ) π ≥ c k r-s , k = 1, 2, . . . The observability problem can be stated exactly as in the previous paragraph. In other words, we are looking for pairs (t 0 , t 1 ) satisfying the estimation [START_REF] Szijártó | Observation problems posed for the Klein-Gordon equation[END_REF].

From the expression of the eigenvalues,

λ m,n ≍ m 2 + n 2 .
Moreover, an adaptation of Lemma 3.2 yields | sin λ m,n (t 0 -t 1 )| ≍ λ m,n (t 0 -t 1 ) π .

Hence, setting θ 1 := (π(t 0 -t 1 ))/a 2 , θ 2 := (π(t 0 -t 1 ))/b 2 and α := (r -s)/2, and applying the same method as for the vibrating string, we get the

Theorem 3. 3 .

 3 (a) If r -s < 1, there cannot be any strategic pair. (b) If r -s = 1, the set of strategic pairs has zero Lebesgue measure and full Hausdorff dimension in R 2 . (c) If r -s > 1, the set of strategic pairs has full Lebesgue measure in R 2 .

Lemma 3 . 4 (

 34 [4, pp. 120-121], [2, p. 104], [7, p. 142]). (a) If α = 1, then E α has zero Lebesgue measure and full Hausdorff dimension in R. (b) If α > 1, then E α has full Lebesgue measure in R. Proof of Theorem 3.3. The result is a consequence of the Proposition 3.1 and results of Diophantine approximation.

Proposition 6 . 1 .

 61 Let s ∈ R. For all initial data y 0 ∈ D s and y 1 ∈ D s-2 , the problem (14) admits a unique solution y ∈ C(R, D s ) ∩ C 1 (R, D s-2 ) ∩ C 2 (R, D s-4 ) given by (15) y(t, x) = ∞ k=1 (a k e ik 2 t + b k e -ik 2 t ) sin kx, where the complex coefficients a k and b k satisfy

Theorem 6. 3 . 2 . 6 . 2 .y|c m,n | 2 . 6 . 4 . 2 s + y 1 2 s

 326226422 (a) If r-s = 2, there is a set of strategic pairs that is infinite, has zero Lebesgue measure and full Hausdorff dimension in R 2 .(b) If r -s > 2, there is a set of strategic pairs that has full Lebesgue measure in R Observability of a hinged rectangular plate. Let a and b be positive real numbers and Ω = (0, a) × (0, b) ⊂ R 2 the rectangular domain whose boundary is denoted by Γ. The small transversal vibrations of a hinged plate whose shape is delimited by′′ + ∆ 2 y = 0 in R × Ω, y = ∆y = 0 in R × Γ, y(0) = y 0 , y ′ (0) = y 1 in Ω.The eigenvalues of the operator -∆ with Dirichlet boundary conditions are (see e.g.,[START_REF] Courant | Methods of Mathematical Physics[END_REF])λ m,n = m 2 π 2 a 2 + n 2 π 2 b 2 , m, n = 1, 2, . . .with associated eigenfunctions e m,n (x, y) = sin mxπ a sin nyπ b , m, n = 1, 2, . . . These functions form an orthogonal and dense system in L 2 (Ω). For s ∈ R, we define D s as the completion of the vector space spanned by the functions e m,n for the euclidean norm ∞ Proposition Given y 0 ∈ D s and y 1 ∈ D s-2 , the problem (18) has a unique solution y ∈ C(R, D s ) ∩ C 1 (R, D s-2 ) ∩ C 2 (R, D s-4 ), whose expansion in Fourier series is y(t, x) = ∞ m,n=1 (a m,n e iλm,nt + b m,n e -iλm,nt )e m,n (x, y), where the complex coefficients a m,n and b m,n satisfy y 0 (|a m,n | 2 + |b m,n | 2 ).

  a 2 , a 3 , . . .]. Therefore, 1/ξ 0 = [a 1 ; a 2 , a 3 , . . .] and 1 + 1/ξ 0 = [1 + a 1 ; a 2 , a 3 , . . .], whence ξ 1 = [0; 1 + a 1 , a 2 , a 3 , . . .] and by recurrence ξ n = [0; n + a 1 , a 2 , a 3 , . . .].

9 K(x) denotes the largest partial quotient in the continued fraction of x, i.e. if the development in continued fraction of x is given by x = [a 0 ; a 1 , a 2 , . . .], then K(x) := sup k≥1 a k .

In particular, this notion depends on the the norms in the left and right members.

A ≍ B means that there are two positive constants c 1 and c 2 such that c 1 B ≤ A ≤ c 2 B.

If x is a real number, x denotes the distance between x and the nearest integer.

in the sense of the symbol ≍

Proposition 6.5. The pair (t 0 , t 1 ) is strategic if and only if there is a positive constant c such that

We give sufficient conditions for (19) to hold. First case: particular domains. We assume that there exists a positive integer

Therefore, setting θ := θ 2 , the estimation (19) simplifies in

We have already seen that if r -s ≥ 2, the above estimation holds for some choices of t 0 and t 1 . More precisely the Theorem 6.3 remains true in this case. Second (general) case. It is not always possible to uncouple the expression m 2 θ 1 + n 2 θ 2 as we did in the first case. Nevertheless, we can use some results on the approximation of linear forms by rationals. Theorem 6.6.

(a) Assume that r -s = 4. If t 0 and t 1 are real numbers such that θ 1 and θ 2 belong to a real algebraic extension of Q of degree 3 and 1, θ 1 , θ 2 are linearly independent over the rationals, then (t 0 , t 1 ) is an strategic pair. (b) Assume that r -s > 4. Then, almost all (in the sense of the Lebesgue measure) couples (t 0 , t 1 ) are strategic.

Proof. The assertion (a) is a direct consequence of the Theorem 6.5 and the second estimation of the Lemma 4.2. the assertion (b) is a consequence of the Theorem 6.5 and of a generalization of the Lemma (3.4) (see [2, p. 24]).


Slavkov Dragan

Hajdukovic
email: dragan.hajdukovic@physac.me

A Cyclic Universe alternatively dominated by matter and antimatter

It was recently suggested that what we call dark matter and dark energy, can be explained as the local and global effects of the gravitational polarization of the quantum vacuum by the immersed Standard Model matter. This result appears as the consequence of the working hypothesis that by their nature quantum vacuum fluctuations are virtual gravitational dipoles. Here, we argue that, as a consequence of the same hypothesis, we may live in a cyclic universe with cycles alternatively dominated by matter and antimatter. Hence, we have related matterantimatter asymmetry in the Universe with the cyclic nature of the Universe. At least mathematically there is no the initial singularity, there is no need for the cosmic inflation and there is an amusing explanation of the matterantimatter asymmetry: our Universe is dominated by matter because the previous cycle was dominated by antimatter (and the next cycle would be dominated by antimatter again).

Introduction

According to our best knowledge (the Standard Model of Particles and Fields), particles and antiparticles are always created in pairs. However, as far as we know, our Universe is dominated by matter. How can a symmetric process of creation (matter and antimatter created in the same quantity) give rise to an asymmetric result (one of them dominates the universe)? It is one of the greatest mysteries in physics, astrophysics and cosmology. Apparently, in the primordial Universe, something has forced this matterantimatter asymmetry. The mainstream hypothesis is that it is caused by CP violation, but please do not be misled by the known CP violation; in order to create such an enormous asymmetry it must be an unknown type of CP violation, many orders of magnitude stronger than the known one! It is an open question if such an extreme CP violation existed in the early Universe.

We propose a radically different scenario based on the working hypothesis that by their nature quantum vacuum fluctuations are virtual gravitational dipoles. The simplest and the most elegant realization of this hypothesis is, if particles and antiparticles have the gravitational charge of the opposite sign (which will be tested on ongoing experiments AEGIS [START_REF] Kellerbauer | Proposed antimatter gravity measurement with an antihydrogen beam[END_REF], ALPHA [START_REF] Charman | Description and first application of a new technique to measure the gravitational mass of antihydrogen[END_REF] and GBAR [START_REF] Perez | The GBAR experiment: gravitational behavior of antihydrogen at rest[END_REF] at CERN); of course nature may surprise us with a different realization of the gravitational dipoles-like behavior of the quantum vacuum.

Before we continue, let us mention major consequences [START_REF] Hajdukovic | Is dark matter an illusion created by the gravitational polarization of the quantum vacuum[END_REF][START_REF] Hajdukovic | Quantum vacuum and dark matter[END_REF][START_REF] Hajdukovic | Quantum vacuum and virtual gravitational dipoles: the solution to the dark energy problem?[END_REF][START_REF] Hajdukovic | The signatures of new physics, astrophysics and cosmology?[END_REF][START_REF] Hajdukovic | Virtual gravitational dipoles: The key for the understanding of the Universe[END_REF][START_REF] Hajdukovic | What if quantum vacuum fluctuations are virtual gravitational dipoles?[END_REF], 10] of our hypothesis. First, the hypothesis permits tconsidering the well-established Standard Model matter (i.e. matter made from quarks and leptons interacting through the exchange of gauge bosons) as the only matter-energy content of the Universe; of course a content immersed in quantum vacuum "enriched" with virtual gravitational dipoles. What we call dark matter and dark energy, can be explained as the local and global effects of the gravitational polarization of the quantum vacuum by the immersed Standard Model matter (the polarized quantum vacuum acts as an effective gravitational charge). In addition, the gravitational charge density of the quantum vacuum is zero, which is the simplest solution to the cosmological constant problem; a tiny, effective gravitational charge density might appear as the result of the immersed Standard Model matter.

In the present paper we enlarge this astonishing series of consequences. We may live in a cyclic Universe with cycles alternatively dominated by matter and antimatter; with each cycle beginning with a

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28th Texas Symposium on Relativistic Astrophysics Geneva, Switzerland --December 13-18, 2015 macroscopic size and the accelerated expansion. Consequently, there is no singularity, no need for an inflation field, and there is an elegant explanation of the matter-antimatter asymmetry in the universe: our universe is dominated by matter because the previous cycle of the Universe was dominated by antimatter (and the next cycle would be dominated by antimatter again). Of course, when a theory is in question, experiments and astronomical measurements always have the last word, but so far, there is no other competing theory which apparently has the potential to explain the whole spectrum of phenomena with a single hypothesis. Time will show if it is a way out of the greatest crisis in the history of physics.

In section 2 we address the question What would be the outcome of an eventual Big Crunch? The surprising answer is that a Big Crunch of our Universe would end at a macroscopic size with extremely fast conversion of matter to antimatter; it may look like a new Big Bang but it is not a Big Bang because it starts with a macroscopic initial size and hence without singularity and without need for an inflation field of unknown nature.

In section 3 we argue that during the expansion of the Universe quantum vacuum converts from a cosmological fluid with negative pressure to nearly pressureless fluid. According to the cosmological field equations it means that the accelerated expansion converts to the decelerated one.

What would be outcome of a Big Crunch

If quantum vacuum fluctuations are virtual gravitational dipoles we can borrow a result from quantum electrodynamics, which is valid for virtual electric dipoles (for instance electron-positron pairs). The key point is that quantum vacuum fluctuations can be converted into real particles; something can be created from apparently nothing. A virtual electron-positron pair (and in principle any charged particleantiparticle pair) can be converted to a real one by an external field which, during their short lifetime, can separate particle and antiparticle to a distance of about one reduced Compton wavelength. For a constant acceleration g (which corresponds to a constant electric field) the particle-antiparticle creation rate per unit volume and time, can be written as

m cr m n cr m m m c g mc g g n n g g c dtdV dN     2 1 2 2 4 , ; exp 1                         (1)
which is the famous Schwinger formula [START_REF] Hobson | General Relativity an Introduction for Physicists[END_REF].

The Schwinger mechanism is valid only for an external field that has the tendency to separate particles and antiparticles. Consequently, Eq. ( 1) can be used for the gravitational field, only if, particles and antiparticles have gravitational charge of the opposite sign. Hence, if our hypothesis is correct, an extremely strong gravitational field (stronger than the critical acceleration cr g in ( 1)) would create a huge number of particle-antiparticle pairs from the physical vacuum. It is important to underline that virtual pairs are spatially separated and converted into real pairs by the expenditure of the external field energy. Let us note that according to (1) the critical accelerations for creation of electron-positron and protonantiproton pairs are respectively

               2 2 2 2 2 2 2 2 2 2 d sin 1 d d d    d r kr r t R t c s (2)
where k=+1, k=-1 and k=0 correspond respectively to closed, open and flat Universe.

The dynamics of the above space-time geometry is entirely characterised by the scale factor R(t) which is the solution of the Einstein equation

                 2 2 2 3 8 3 3 4    (3)
The cosmological field equations can be solved only if we know the content of the Universe. According to the Standard Model the content of the Universe are two well-established cosmological fluids: non-relativistic Standard Model matter (usually called pressureless matter or dust) and radiation, respectively described by (Note that index 0 denotes the present day value):

2 4 0 3 0 0 3 1 , 0 , c p R R and p R R r o r r m m                      (4)
The third fluid proposed by the Standard Model is quantum vacuum, but the use of this fluid is blocked by the cosmological constant problem; that is why other hypothetical fluids (dark matter, dark energy…) have been proposed. Following our hypothesis the full content of the universe can be modelled with only three fluids, two fluids described by (4), plus quantum vacuum "enriched" with virtual gravitational dipoles; we will come back to this in section 3. With a conversion rate as in the first two numerical columns of Table 1, the matter of our Universe can be transformed into antimatter of the next cycle of the universe in a tiny fraction of second! With conversion rate as in third numerical column, the time needed for conversion is of the order of the Planck time. If the rate is as in the fourth column, conversion takes a tiny fraction of the Planck time. Apparently, if quantum vacuum fluctuations are gravitational dipoles, matter of our Universe survive the trek to singularity; much before singularity it would be converted to antimatter of a new cycle of the Universe. Hence, there is a very simple and beautiful qualitative picture of the expected phenomena. An extremely strong gravitational field would create a huge number of particle-antiparticle pairs from the physical vacuum; with the additional feature that matter tends to reach toward singularity while antimatter is violently "ejected" farther and farther from singularity. The amount of created antimatter is equal to the decrease in the mass of the collapsing matter Universe. Hence, the quantity of matter decreases while the quantity of antimatter increases in the same amount; the final result might be conversion of nearly all matter into antimatter. If the process of conversion is very fast, it may look like a Big Bang starting with a macroscopic initial size many orders of magnitude greater than the Planck length.

The gravitationally polarized quantum vacuum as cosmological fluid

The results from the previous section have physical importance only if there is a future transition from expansion to contraction of our Universe. According to current mainstream cosmology expansion would be eternal; there is no mechanism to stop expansion. However, if our hypothesis is correct, the expansion may convert to contraction [10] (see also my articles in this proceedings devoted to dark matter and dark energy).

The key point is that our hypothesis permits replacing the galactic halo of dark matter by the halo of the gravitationally polarized quantum vacuum. Namely, in the limit of zero external gravitational field quantum vacuum may be considered as a fluid of randomly oriented gravitational dipoles (Figure 1); both the total gravitational charge and the gravitational polarization density g P (i.e. the gravitational dipole moment per unit volume) are zero. However, the random orientation of virtual dipoles may be broken by Consequently, if quantum vacuum fluctuations are virtual gravitational dipoles, the quantum vacuum in the Universe is a cosmological fluid with the sum of all gravitational charges equal to zero; however, because of the existence of the halos of non-random oriented dipoles, it is a fluid with a large effective gravitational charge. Roughly speaking there is a maximum size of the halo for each massive body, galaxy or cluster of galaxies. Simply, after a characteristic size, the random orientation of dipoles dominates again. However, because of proximity of other bodies, many halos are smaller than the maximum size. A halo of the maximum size can be formed only if other bodies (competing in the polarization of the quantum vacuum) are sufficiently far away. This behaviour is schematically presented in Figure 3.

Hence, with the expansion of the universe, galactic halos must increase in size and the content of the effective gravitational charge caused by the polarization of the quantum vacuum. However, it is not an endless increase (see Figure 4); once the halos have reached their maximum size they remain unchanged with the further expansion. Now, the key point is that there is a period in which the size of the individual galactic halos and the total effective gravitational charge of the quantum vacuum increase with the expansion, which means that the polarized quantum vacuum behaves as a fluid with negative pressure, and, according to the first of cosmological equations [START_REF] Perez | The GBAR experiment: gravitational behavior of antihydrogen at rest[END_REF] this is exactly what is necessary for the accelerated expansion of the Universe. Namely, this field equation tell us that the expansion of the Universe can be accelerated (

0  R  
) only if there is a cosmological fluid with sufficiently big negative pressure, so that the sum in the equation is negative.

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28th Texas Symposium on Relativistic Astrophysics Geneva, Switzerland --December 13-18, 2015

However the accelerated expansion is only the first part of the story. The second part is that during the expansion of the Universe quantum vacuum converts from a cosmological fluid with negative pressure to nearly pressureless fluid (this is a consequence of the fact that increase of the effective gravitational charge is not endless). According to the cosmological field equations it means that the accelerated expansion converts to the decelerated one; apparently there is a mechanism to prevent an eternal expansion.

Conclusion

We have suggested physical mechanisms which can prevent both the collapse of the Universe to singularity and the eternal expansion. The phase of expansion of the Universe may be followed by contraction; a cyclic Universe may be the way in which nature works. The mystery of matter-antimatter asymmetry may be inherent part of the cyclic nature of the Universe.

.

  The Einstein tensor  G is determined by FLRW, but in order to solve the Einstein equation we must know the Energy-momentum tensor  T . Key point: The Energy-momentum tensor is approximated by the energy momentum tensor of a perfect fluid characterised at each point by its proper density  and pressure p . If the cosmological fluid consists of several distinct components denoted by n, the final results are cosmological field equations [13]:

Table 1 :.

 1 Acceleration and particle-antiparticle creation rate for different scale factors R In order to get an idea about the outcome of an eventual Big Crunch, we may use the gravitational version of the Schwinger mechanism (1) together with the first of cosmological equations (3); more precisely we use R g    in order to calculate the particle-antiparticle creation rate (1). TodayR   has asmall value of the order of However (and it is crucial for our arguments), in the primordial Universe acceleration R   may be extremely big as can be seen from Table1. It is useful to point out that the size of the Solar System and the size of the Sun.

Figure 1 .

 1 Figure 1. Randomly oriented gravitational dipoles (in absence of an external gravitational field)

Figure 2 .

 2 Figure 2. Halo of non-random oriented gravitational dipoles around a body (or a galaxy) with baryonic mass M b

Figure 3 .

 3 Figure3. The effective gravitational charge of a body (blue line) increases from the "bare" charge measured at its surface to a constant maximum charge at a large distance from the body. Other bodies can prevent the effective gravitational charge from increasing above a limit presented by the red line

Figure 4 .

 4 Figure 4. The effective gravitational charge of the quantum vacuum in the Universe depends on the scale factor R(t). Note that with the expansion of the Universe the polarized quantum vacuum converts from a cosmological fluid with negative pressure to a pressureless fluid!

Acknowledgments:

We thank the "Hipotekarna Banka" in Montenegro for the sponsorship of our participation on the 28th Texas Symposium on Relativistic Astrophysics.


Sylvie Huet
email: sylvie.huet@irstea.fr

Guillaume Deffuant Irstea
email: guillaume.deffuant@irstea.fr

In the Leviathan Model, the average opinion is higher when vanity is weaker with lower valued agents

In the Leviathan model, agents form opinions about each other. During dyadic meetings, they directly influence each other, or talk about peers through gossip. Speakers highly valued by their listeners are more influential. Moreover, they are subject to a vanity process; if an agent feels undervalued, then he decreases his opinion about the despising agent and on the contrary he increases his opinion about a compliment-giver agent. This model produces several different patterns depending on the parameter values, but the average opinion is always negative whatever the emerging pattern. As a consequence, the gossips tend to be mostly negative. Since this is not in complete accordance with the social-psychology literature indicating that both mostly negative and positive gossip can be observed, we investigate the changes to operate to obtain also mostly positive gossips. We found that when the vanity process is less triggered by agents held in low esteem, the average opinion significantly increases. We check that with this modification, the model still yields the same patterns.

Introduction

The recently proposed Leviathan model [START_REF] Deffuant | The Leviathan model: Absolute dominance, generalised distrust and other patterns emerging from combining vanity with opinion propagation[END_REF] considers individuals forming and maintaining opinions about each other, including themselves through processes of opinion propagation and vanity. The opinion propagation includes a part which can be interpreted as gossiping. From its simple virtual dynamics of individual interaction emerges a collection of patterns such as "dominance", "equality", "hierarchy", "elite" or "crisis". One striking emerging effect is that in these patterns, the average opinion is always negative [START_REF] Deffuant | The Leviathan model: Absolute dominance, generalised distrust and other patterns emerging from combining vanity with opinion propagation[END_REF]. As a consequence, gossiping is mainly negative. At first, this sounds in accordance with the observations that gossip is more often negative [START_REF] Wert | A social comparison account of gossip[END_REF]. But a recent study surveying a group of working nurses in a hospital shows the opposite [START_REF] Ellwardt | Who are the objects of positive and negative gossip at work? A social network perspective on workplace gossip[END_REF]: negative gossip is quite rare in this group.

This paper aims at modifying the Leviathan model in order to get a more positive average opinion and consequently more positive gossips. With this aim, we tested two modifications of the vanity process, which, in the original model, applies with all interlocutors:

 Vanity is weaker with lower agents: An agent is subject to weaker vanity when discussing with an agent that he considers lower than himself;

 Vanity is weaker with higher agents: An agent is subject to weaker vanity when discussing with an agent that he considers higher than himself. We found that when the vanity is weaker with lower agents, the average opinion tends to be significantly more positive. Moreover, the same patterns as in the original model are observed. Two of them show a highly positive opinion on average. However one pattern has probably been modified in deep by the change vanity process.

The paper is organised as follows. We start by a description of the dynamics and the emerging properties of the model. We describe the impact of vanity process modifications on the number of positive opinions and on the patterns. We then propose some theoretical explanations of the main result. Finally, we discuss our result and identify some complementary studies to make.

2

The model and the experimental design

The agent dynamics of the Leviathan model, inspired by Hobbes [START_REF] Hobbes | Léviathan. Traité de la matière, de la forme et du pouvoir ecclésiastique et civil[END_REF] and more recent studies from social-psychologists [5-10] [11] is presented in the next subsection. The following subsection describes the different patterns emerging from the interactions [START_REF] Deffuant | The Leviathan model: Absolute dominance, generalised distrust and other patterns emerging from combining vanity with opinion propagation[END_REF].

2.1

The Leviathan model: The agent dynamics

We consider a set of N agents, each agent i is characterised by her list of opinions about the other agents and about herself: (ai,j)1 ≤i,j≤N . We assume ai,j lies between -1 and +1 , or it is undefined (equal to nil) if the agent i never met j and nobody has talked to i about j yet. At initialisation, we suppose that the agents never met, therefore all their opinions are undefined. When opinions change, we always keep them between -1 and +1, by truncating them to -1 if their value is below -1 after the interaction, or to +1 if their value is above +1. The individuals interact in uniformly and randomly drawn pairs (i, j) and at each encounter, we apply two processes: the opinion propagation and vanity. We follow the people's interactions considering a time range called iteration. We assume one iteration, i.e. one time step t →t + 1, is N/2 random pair interactions (each individual interacts N times on average during one iteration).

We now describe the processes with more details.

Opinion propagation with highly valued agents being more influential

The strength of the propagation of opinion is ruled by a parameter ρ multiplied by a coefficient p i,j . This function implements the hypothesis that if i has a high opinion of j, then j is more influential on i. It is a logistic function (with parameter σ) of the difference between the opinion of i about j (a i,j ) and the opinion i about herself (a i,i ). If a i,j = nil (j is unknown to i), we assume that i has no opinion because he has not met or hear about j. At the first meeting, we suppose that the a priori about j is neutral and we set a i,j ← 0. Let us also observe that, at the initialisation, an agent has no opinion about herself thus we also set a i,i ← 0 at the first discussion. Then we compute the propagation function p i,j , which rules the intensity of the opinion propagation from j to i :

       / exp 1 1 ii ij ij a a p    
p i,j tends to 1 when a i,j -a i,i is close to 2 (i values j higher than herself), and tends to 0 when it is close to -2 (i values j lower than herself). When σis small, p ij rapidly changes from 0 to 1. When σis large, this change is progressive.

This propagation coefficient of influence is computed to determine the influence a speaker has onto a listener about the opinion the listener has of oneself, of the speaker and of their peers.

Vanity and opinion influence between protagonists due to direct contact

Let us assume that agents i and j have been drawn. During their first meeting, agent i and agent j don't know each other and their opinions are nil. Then, they instantaneously become 0 which is the neutral opinion. This initiates the meeting dynamics between i and j.

Then, i and j talk about themselves: i talks about herself and j, while j talks about herself and i. This direct exchange called face-to-face implies influence of each of them on what they think about themselves and the other, and a vanity process applied only by the listener to the talker.

This vanity process expresses that agents tend to reward the agents that value them more positively than they value themselves and to punish the ones that value them more negatively than they value themselves. The vanity equation considering the reaction of i to what j says about i has the following form:

)) , ( Random (

        ii ji ij a a v
This vanity is added to the influence i received from j regarding what she thinks about j to compute the updated opinion of i about j: a ij .The agent i compares her selfopinion a ii to the opinion j tells about her a ji . If the perceived opinion of the other (j) is higher than her self-opinion, i increases her opinion of j (reward). Else i decreases her opinion of j (punishment). This particular reaction of i aims to increase or decrease the future influence of j in order to protect or enhance herself. The parameter ω rules the importance of the vanity process.

The influence of j on the opinion i has on i and j is controlled by the propagation coefficient p i,j (see 2.1.1) and a parameter ρ.

The modification of i's opinion of i is assumed as simply depending on the difference between the opinion of i about herself and the opinion of j about i. The modification of i's opinion of j is assumed on the difference between the opinion of i about j and the opinion of j about j (influence) and on the vanity v ij .

Moreover, an agent i has no direct access to the opinions of another one (j) and can misunderstand her. To take into account this difficulty, we consider the perception of the agent i as the value a jz more or less a uniform noise drawn between - and + ( is a model parameter). This random addition then corresponds to a systematic error the agents make regarding the others' opinions. That can be seen as a noise that distorts the perception that i has about j 's opinions. The parameter δ rules the amplitude of this noise.

During the interaction, the procedures face-to-face(i,j) and face-to-face(j,i) are successively applied. The face-to-face can be formally described in pseudo-codes as follows (example given for face-to-face(i,j):

Face-to-face(i,j) if a ii = nil, a ii ← 0 if a ij = nil, a ij ← 0 )) , ( Random (          ii ji ij ii ii a a p a a ij ij jj ij ij ij v a a p a a       
)) , ( Random (

  

The gossip follows the face-to-face.

Gossip: individuals discuss about those they know

During an encounter, we have seen that agent j propagates to i her opinions about herself (j) and about i. It also propagates to i her opinions about k agents randomly chosen among her acquaintances through gossip (or about all her acquaintances if there are fewer than k).

The agent i modifies her opinion about the individual z that j talked using the same "influence mechanism" than the one used in the face-to-face. She applies the propagation coefficient p i,j multiplied by the influence parameter ρ to the difference between what j told about z and what she thinks of z.. More formally, the process can be written in pseudo-code as follows:

Gossip(i, j) Repeat k times: Choose randomly z taking into account a jz ≠ nil, z ≠ j If a iz = nil, a iz ← 0 )) , ( Random (          iz jz ij iz iz a a p a a

Summary

Finally, the model has 6 parameters:  the number of individuals;  , the reverse of the sigmoidal slope of the propagation coefficient;  k, the number of acquaintances an individual talked about during a meetingthey are randomly chosen among her acquaintances;

 , maximum intensity of the noise when someone is alluded to;  , the coefficient of the vanity process;

 ρ, the parameter controlling the intensity of the coefficient of the influence.

The following algorithm describes one iteration: N/2 random pairs of individuals are drawn, with reinsertion, and we suppose that each individual influences the other during the encounter. The update is synchronous: every opinion changes occurring during a meeting are computed on the same value of opinions taken at the beginning of a pair meeting.

Repeat N/2 times: Choose randomly a couple (i,j) Save the opinions which are going to change in temporary variables to ensure the update during the i and j meeting is synchronous Face-to-face(i,j) Face-to-face(j,i) Gossip(i,j) Gossip(j,i)

The Leviathan model: Emerging properties

This section aims to remind what we know from emerging properties of the Leviathan model. We have already said a bias to negativity of the whole population emerges from the agent dynamics: the average opinion is negative, sometimes very strongly [START_REF] Deffuant | The Leviathan model: Absolute dominance, generalised distrust and other patterns emerging from combining vanity with opinion propagation[END_REF]. This is due to a difference between the opinion an agent has of herself and the average opinion of this agent in the population. This difference is driven by the noise during the conservation and the assymetrical property of the influence (the agents are more influenced by the ones they value high than by the others). It corresponds to a bias of oneself for herself which is called positivity bias [START_REF] Hoorens | Self-enhancement and superiority biases in social comparison[END_REF]. In other words, everyone thinks themselves better on average than the others. Such a bias can be amplified by the vanity for example, leading to a negative average opinion as an emerging process. Indeed the difference for agents between their opinion of themselves and the average opinion of them in the population leads them to be always desappointed in the interactions. Then, because of the vanity process, they continuously punish their interlocutors by decreasing their opinion of them. Such a punishment leads to a negative average opinion for the whole population. This is typically what occur in the crisis or the equality patterns we present now.

Indeed the dynamics not also reproduces biases but shows a striking variety of patterns representing structures of opinions. This variety can be described by five main patterns and several of them often take place in a single simulation. This section describes them shortly.

In a first set of patterns, when the opinion propagation is strong (high value of ), the agents tend to all have the same opinion about each agent (the differences are due to the randomness). If we call, for sake of simplicity, reputation 1 the average opinion about an agent, the distribution of reputations gives a good description of these patterns. There are generally more negative than positive reputations. For some patterns the number of reputations decreases progressively when getting more positive, giving the idea of a hierarchy. For other patterns, there is a single or a couple of agents that have a strongly positive reputation, while all the others have a very negative one. Agents with a positive reputation can be identified as leaders. These consensual leaders characterise two patterns emerging from the dynamics: (1) the absolute dominance or (2) a multiple-leaders hierarchy. There is one pattern without leaders: (3) the "crisis" in which each agent has a very negative opinion of all the others and of herself.

The second set of patterns emerges when vanity is strong. In these patterns there is no consensus about each agent who can be highly valued by some agents and lowly valued by others; we identified two main patterns "equality" and "elite". In (4) equality, each agent has a positive opinion about herself; she is connected by strong positive mutual opinions with a small set of agents and has very negative opinions about all the others. All agents have a similar number of positive (and negative) links. The (5) elite pattern shows two categories of agents: the elite and second category agents. The elite agents have a positive self-opinion and are strongly supported by a friend, but they have a very negative opinion of all the other elite agents and of all the second category agents. The second category agents have a very negative self-opinion, they have a very negative opinion of all the other second category agents and their opinion about the elite agents is moderately positive.

Modifying vanity for getting more positive opinions

This section describes our study. It begins by presenting our hypothesis and the experimental design allowing the study by simulations. A following subsection relates to the relevance of our hypothesis regarding the cause of the sign of gossiping while a next one investigates how the change in vanity impacts the emerging structures of opinions. A last subsection gives some explanations about our results.

Hypothesis and experimental design

In the original model, the vanity process does not depend on the relative value of the interlocutors; a flattery or an offense coming from a highly valued agent has the same effect as if it came from a lowly valued agent. It was choice for simplicity because it seems that the difference of value can weaken or strengthen the vanity, depending on the circumstances. We know the coupling of such a function to the influ-ence function with a sigmoidal coefficient based on the level of esteem for the talker implies a strong tendency of people to have a negative opinion of others.

We investigate the behaviour of the model with different variants of the vanity: the first one corresponds to the original model (same vanity process whatever the value difference); in the second one, vanity is weaker when discussing with lower valued agents; and in the third one vanity is weaker when discussing with higher valued agents :

-Constant vanity (CV) means that the vanity is applied to everyone with the same strength (as in the original model) ; -Weaker vanity for lower valued (WLV) means that the strength of the vanity is weaker when discussing with lower valued agents; this is obtained by multiplying the vanity equation by the sigmoid function p ij ; -Weaker vanity for higher valued (WHV) means that vanity is weaker when discussing with higher valued agents; this is obtained by multiplying the vanity equation by the function 1p ij . We now present the experimental design as well as how some values are measured and aggregated to build indicators of the dynamics.

The model includes 6 parameters and it is difficult to make an exhaustive study in the complete parameter space. In addition to the study of coefficient ɷ, we decided to study with more attention the influence of parameters k and σ because they have been poorly studied previously. We also decided to vary the noise δ, but not as much as k and σ. Also ρ varies poorly since it has been tested in details in [START_REF] Deffuant | The Leviathan model: Absolute dominance, generalised distrust and other patterns emerging from combining vanity with opinion propagation[END_REF]. In particular, we know from this first study that in the plane defined by parameters ρ and  the transitions between patterns take place on right lines of origin (0,0). We fix N, the number of agents to 40, in order to make tractable results of our study and =1. We vary the other parameters as follows:

 k , the number of acquaintances about which the pair of agents discuss in the opinion influence takes the values 1, 2 , 3, 4, 5, 10, 15, 20, 25 and 30;  δ, the intensity of noise disturbing the evaluation of other's opinions takes two different values: 0.1 and 0.3;  σ, ruling the slope of the logistic function determining the propagation coefficients takes the values 0, 0.01, 0.02, 0.06, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8;  ρ, ruling the intensity of the opinion propagation coefficient takes three values: 0.05, 0.5, 1;

For each set of parameter values, we run the model for 201 000 iterations (one iteration corresponding to N/2 random pair interactions), and we repeat this for 30 replicas. From iteration 30 000 to 200 000 every 10000 iterations, we measure a group of values allowing us to make conclusions about the impact of on the vanity variants on the average opinion and on the patterns. The measured values over times of 30 replicas are averaged to form one indicator. The indicators are presented in more details in in the next sections.

3.1

Average proportion of positive opinions

To study how modifications of vanity impact the number of positive gossip corresponding to positive opinions, we consider the average proportion of positive opinions over the 30 replicas for each set of parameters. Figure 1 shows this indicator averaged over the tested values of δ and ρ, for the various values of k and σ and the three variants of vanity process, on the left for constant vanity, in the middle for weaker vanity with higher valued agents (WHV), on the right weaker for lower valued agents (WLV). From these figures, we notice that for WLV the proportion of positive values is much higher than for the other variants and for some values of k and σ, the positive opinions represent more than 50 % (thus a majority) of the opinions while for other vanity variants, they always represent a minority (less than 25 %). Indeed we see onto the two left graphs that the average rate is mainly from 0 to 0.125 (in very light blue), or at most from 0.125 to 0.25 (in light blue) for low values of k and/or high values of σ. Differently, on the right graph (weaker vanity with lower valued) we observe the average rate of positive opinions is often between 0.37 to 0.5 (light green), especially for large values of k and σ, and goes up to [0.5,0.625] for low values of σ (in dark green). The results averaged over every other tested parameter values do not change the conclusion we can draw: for the weaker vanity with lower valued, the number of positive opinions is significantly higher.

These experiments suggest that the weaker vanity with lower valued produces populations where positive opinions are a majority or a minority while they are always a minority when vanity is constant or weaker with higher valued agents. The next section is dedicated to the impact of the changes of the vanity process on the patterns.

Constant vanity

Weaker vanity with higher valued

Weaker vanity with lower valued The Leviathan model includes a dynamics of the esteem based on a coupling between a vanity process and an opinion propagation process. The two processes occur during a direct experience a listener and a speaker have of each other during a meeting. The opinion propagation rules also the indirect experience listeners and speakers have of the others through gossiping. Both these mechanisms make people changing their opinions of each other. The original Leviathan model exhibits a strong tendency of opinions to be negative, and hence a strong dominance of negative gossiping Whereas the literature in social psychology has shown that gossiping can be either majorly negative [START_REF] Wert | A social comparison account of gossip[END_REF] or positive [START_REF] Ellwardt | Who are the objects of positive and negative gossip at work? A social network perspective on workplace gossip[END_REF] without explaining why.

We investigate variants of the vanity process that could lead to more positive average opinions and we observe that majority weaker vanity with lower valued agents leads to much larger proportions of positive opinions than with the other variants of vanity, whereas the opinions can still be majorly negative for some parameters. On the contrary, a constant vanity or a weaker vanity with higher valued agents always leads to a majority of negative opinions.

From these observations, we conclude that with a weaker vanity with lower valued agents, the positivity bias (self-opinion higher than the average opinion on the agent) has a weaker effect and does not always leads to a global negativity. It is then, among the tested form for vanity, the only one in accordance with both of the sociopsychological cited studies on gossip. This is also consistent with those about self-esteem values mainly showing that low self-esteem people do not truly dislike themselves; they view themselves positively, just less positively than do high self-esteem people [START_REF] Baumeister | Self-Presentational Motivations and Personality Differences in Self-Esteem[END_REF].

Dynamic patterns of behaviour of the population emerging from this model [START_REF] Deffuant | The Leviathan model: Absolute dominance, generalised distrust and other patterns emerging from combining vanity with opinion propagation[END_REF] have been identified. They are still diagnosed for each of the different vanity variant we consider. With a weaker vanity with lower-valued agents, the hierarchy pattern, is significantly positive on average and shows strong variations. This makes this pattern richer than it was from the original model in which it remains majorly negative. Despite their average level of positivity close in every vanity dynamics conditions, the dominance and elite patterns can be very positive with this vanity variant. We suspect nevertheless that the dominance and elite patterns could be modified with this vanity variant and this would require deeper investigation.

We do not claim that changing vanity is the only way for observing both more positive than negative gossips. Indeed, the positivity bias and a tendency to a global negativity exist also in absence of vanity when the dynamics is only driven by the propagation of opinions. Investigating the change of biases in this case is one of our future goals.

More generally, we think that this model can suggest new views on current issues in social psychology, as well as being improved by incorporating more results from this science.

Fig. 1 .

 1 Fig. 1. Average proportion of positive opinions (over time and replicas) for the three variants of the vanity process. On abscissa are values of the number of acquaintances gossiped about k and on ordinate are the tested values for the sigmoid parameter σ. The meaning of the colours is given at the bottom of each graph under the abscissa

Reputations is understood in this paper as a consensus, inspired by the Emler's definition of reputation[START_REF] Emler | A social psychology of reputation[END_REF]: "consensus among knowledge informations as to the attributes of targets". We do not claim modelling "reputations" and other more relevant studies are more relevant for this issue than ours.

Impact of vanity modifications on patterns of opinions

We consider a new indicator: the average over all the agents of the difference between their maximum and their minimum opinions. We call this measure "dispersion of opinions". It varies from 0 to 2. Figure 2a on the left shows that the opinion dispersion is smaller for large ρ (ie propagation strength) whatever the vanity variant and higher for small value of ρ=0.05. A large noise, such as δ=0.3, tends to increase the dispersion but the effect of ρ remains, especially for weaker vanity with lower valued agents (in green). This confirms it still exists two different areas that we can distinguish by the level of consensus about each agent and then probably our two different sets of patterns (see 2.2 for more details). We applied the diagnosis presented in [START_REF] Deffuant | The Leviathan model: Absolute dominance, generalised distrust and other patterns emerging from combining vanity with opinion propagation[END_REF] to define if already identified emerging patterns of opinions are still observable with the new vanity variants. Figure 2b on the right presents the average part of positive opinions of the diagnosed patterns. Firstly we observe that the five patterns are still diagnosed whatever vanity variant. Secondly this is clear the "hierarchy" is more positive for weaker vanity with lower valued agents and shows strong variations in its proportion of positive opinions. The other structures remain almost unchanged by the vanity variant but "elite" and "dominance", show strong variations of the proportions of positive opinions for the variant WLV, which would require more investigation. However, this is out of the scope of this paper.

An explanation of the impact of the various forms of vanity

In order to get an intuition about the effect of the various forms of vanity, we consider the simple case σ = 0 (the sigmoid function p ij is then either 0 or 1) for which we can clearly identify who is going to be punished or rewarded, and k=0. Then, at the first meeting when everyone has an opinion valued at 0, the following table n°1 describes what occur and how the value of a ij changes due to the vanity (d is a positive number comprised between 0 and δ) ; we suppose that the opinion of i about j is 0, and of i about himself is also 0. Table n°1. Evolution in the face-to-face of the opinion of i about j (a ij ) for various forms of vanity and σ = 0, k=0; d is a positive number comprised between 0 and δ Vanity heuristic

These observations point out why weaker vanity with lower valued agents causes agents to see each other more positively than the other variants of vanity. In any case, the lowest opinion of i of j is higher for this variant since it is (d1-d2) while it is -d2 for the other variants.

Overall, at the very beginning, dispersion of opinions is smaller. More generally, the constant vanity process leads to cycles of mutual opinion increase or decrease that are stopped only by the limits of opinion values (+1 or -1) in the original model. When vanity is weaker with lower valued agents, the cycles of mutual opinion decrease are strongly attenuated when one of the protagonists is much lower than the other. This can often stop the usual escalations driving many agents to get very low opinions (close to -1).

Moreover, the initial analysis of the Leviathan model shows that the average negative opinion is due to tendency of the agents to have a higher self-opinion than the average opinion on them. Indeed, this difference leads them to be disappointed on average by the opinions of the others, and thus to decrease their opinion about them by vanity. When the vanity is weaker for lower-valued agents, this process is attenuated for such agents.

Therefore, overall the opinions tend to be less negative. .


ENDOLUMINAL MR RECEIVER COIL BASED ON ELECTRO-OPTICAL CONVERSION AND ACTIVE OPTICAL DECOUPLING

Isabelle Saniour 1 , Anne-Laure Perrier 2 , Reina Aydé 1 , Gwenaël Gaborit 2,3 , Lionel Duvillaret 3 , and Olivier Beuf 1 An active optical-base decoupling system was designed and built. Once illuminated, photodiodes provide a sufficient DC current that ensure coil decoupling during RF transmission phase. The basic idea is to ensure a DC current to the PIN diode by using photodiodes (Figure 2b). Decoupling efficiency was evaluated by inserting into a cylindrical phantom filled with a 5 g/L of saline water solution three different endoluminal coils : reference coil , optically-decoupled endoluminal coil and a non-decoupled endoluminal coil (without a PIN diode).Then images taken by the body coil were compared (Figure 4). The body coil was used as a transceiver so endoluminal coil were considered to be constantly decouple.

Results:

The graph in the figure 3 indicates that the experimental and the theoretical values are matching. This graph shows a good linearity of the results and a dynamic range of the input power exceeding 100 dBm. And the magnetic field is ranging between 0.3 pT at -104 dBm (electronic noise) to 2.10 5 pT at 14 dBm. In addition, the optical decoupling was verified. The figure 4 shows that both reference and optical decoupled coils have a comparable behavior. While the coil without decoupling circuit shows a clear artifact due to B1 concentration. Discussion: Using a Ti:LiNbO3 waveguide, the NMR signal detected by the receiver coil was converted into an optical signal and measured after being optically processed. Results demonstrated both an excellent linearity and sensitivity. Besides, the concept of optical decoupling was demonstrated without SNR or B1 uniformity penalties. Based on these results, a fully optical endoluminal receiver coil was designed by combining the two described systems (Figure 5). Such a coil may provide very useful information about bowel diseases and human gastrointestinal wall layers withoutthe fear of heating biological tissues.

References:

[1] O.

1

  Université de Lyon, CREATIS, CNRS UMR 5220, Inserm U1044, INSA-Lyon, Université Lyon 1, Villeurbanne, France, 2 Université de Savoie, IMEP-LAHC, UMR 5130, Le Bourget-du-Lac, France, 3 KAPTEOS, Sainte-Hélène-du-Lac, France Target audience: Researchers working on endoluminal receiver coils for MRI. Introduction: The use of endoluminal coil in magnetic resonance imaging (MRI) located close to the region of interest enhances image spatial/temporal resolution using the local signal-to-noise ratio (SNR) gain. Wall bowel analysis could be then performed and wall layer could be distinguished for a better diagnosis and characterization of inflammation and lesions [1]. Conventionally, coaxial cables connecting the MRI console and the receiver endoluminal coil are used to transmit the NMR signal and the DC bias current used for active decoupling. However, patient safety can be compromised by heating of tissues located in proximity of these cables. In fact, the electric field accompanying the radiofrequency (RF) magnetic field B1 induces high-frequency currents along the metallic wire and thus increases the specific absorption rate (SAR) by the tissues lying nearby [2]. To ensure full patient safety, an optical fiber was chosen as an alternative solution to transfer optically the RMN and the DC decoupling signal. Both electrical signals are converted into optical signals: the conversion of NMR signal is based on Pockels effect ensured by an electro-optical (EO) crystal which its refractive indexes change according to an applied electric field. While DC signal is converted by the use of optoelectronic devices. The electro-optical (EO) conversion and the active optical-base decoupling were demonstrated and proven separately in previous works [3], [4]. In the following, the results of these previous works are presented and a novel type of endoluminal receiver coil combining both optical transmission and active optical decoupling was designed. Methods: The conversion of NMR signal is ensured by associating the endoluminal coil to an EO Ti:LiNbO3 waveguide. Figure 1 illustrates the experiment performed on an optical bench. Firstly, the body coil generates a RF electrical power Pin ranging from -101 dBm to 14 dBm. The B1 of RF waves is detected by the endoluminal receiver coil. This coil was designed to resonate at a frequency of 128.2 MHz corresponding to the proton's resonant frequency at 3T. Therefore, an electromotive force ε is applied to the waveguide and in turn induced an electrical field Ez lying between the electrodes and proportional to B1. The refractive indexes of the waveguide vary linearly with the induced Ez (Pockels effect). At the same time, a linearly polarized laser beam (pigtailed DFB laser, λ = 1550 μm) is emitted toward the waveguide. Hence the polarization state of the laser beam varies according to Ez. Then the modulation of polarization state is treated, converted into an electrical power by a fast photodiode and amplified. The output power Pout is visualized on a spectrum analyzer.

Figure2.

  Figure2. Equivalent circuit of the interventional receiver coil. a) Galvanicdecoupled coil circuit. b) Opticallydecoupled coil circuit.

Figure1.

  Figure1. Schematic set up for the experiment of EO conversion and optical transmission.

Figure 4 .

 4 Figure 4. Axial images of a phantom obtained using GRE sequence with: a) A galvanic-decoupled coil. b) An optically-decoupled coil. c) A coupled coil.

Figure 3 .

 3 Figure 3. The output power in function of the input power and the corresponding magnetic field (top axis). The dotted line represents the electronic noise threshold.

Figure 5 .

 5 Figure 5. Schematic set up for a fully optical endoluminal receiver coil. The blue link represents the optical fiber.