Houssam Khalil
email: khalil@math.univ-lyon1.fr

AND Bernard Mourrain
email: mourrain@sophia.inria.fr

Michelle Schatzman

SUPERFAST SOLUTION OF TOEPLITZ SYSTEMS BASED ON SYZYGY REDUCTION

We present a new superfast algorithm for solving Toeplitz systems. This algorithm is based on a relation between the solution of such problems and syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is generated by two elements and the solution of a Toeplitz system T u = g can be reinterpreted as the remainder of a vector depending on g, by these two generators. We obtain these generators and this remainder with computational complexity O(n log 2 n) for a Toeplitz matrix of size n × n.

Introduction

Structured matrices appear in various domains, such as scientific computing, signal processing, . . . They usually express, in a linearize way, a problem which depends on fewer parameters than the number of entries of the corresponding matrix. An important area of research is devoted to the development of methods for the treatment of such matrices, which depend on the parameters defining them.

Among well-known structured matrices, Toeplitz and Hankel structures have been intensively studied [START_REF] Heinig | Algebraic methods for Toeplitz-like matrices and operators[END_REF][START_REF] Kailath | Displacement structure: theory and applications[END_REF]. Nearly optimal algorithms are known for their multiplication by a vector and the solution of linear systems, for such structure. Namely, if A is a Toeplitz matrix of size n, multiplying it by a vector or solving a linear system with A requires Õ(n) arithmetic operations (where Õ(n) = O(n log c (n)) for some c > 0) [START_REF] Bitmead | Asymptotically fast solution of Toeplitz and related systems of equations[END_REF][START_REF] Van Barel | A stabilized superfast solver for nonsymmetric Toeplitz systems[END_REF]. Such algorithms are called superfast, in opposition with fast algorithms requiring O(n 2 ) arithmetic operations.

The fundamental ingredients in these algorithms are the so-called generators [START_REF] Kailath | Displacement structure: theory and applications[END_REF], encoding the minimal information stored in these matrices, and on which the matrix transformations are translated. The correlation with other types of structured matrices has also been well developed in the literature [START_REF] Pan | Structured matrices and polynomials[END_REF][START_REF] Pan | Nearly optimal computations with structured matrices[END_REF], allowing to treat efficiently other structures such as Vandermonde or Cauchy-like structures.

Such problems are strongly connected to polynomial problems [START_REF] Fuhrmann | A polynomial approach to linear algebra[END_REF][START_REF] Bini | Polynomial and matrix computations[END_REF]. For instance, the product of a Toeplitz matrix by a vector can be deduced from the product of two univariate polynomials, and thus can be computed efficiently by evaluation-interpolation techniques, based on FFT. The inverse of a Hankel or Toeplitz matrix is connected to the Bezoutian of the polynomials associated to their generators. Such a construction is related to Gohberg-Semencul formula [START_REF] Gohberg | On the inversion of finite Toeplitz matrices and their continuous analogues[END_REF] (or Trench algorithm [START_REF] Trench | An algorithm for the inversion of finite Toeplitz matrices[END_REF]), which describes the inverse of a Toeplitz matrix in terms of the solution of two specific Toeplitz systems (see also Gohberg-Prupnick formula [START_REF] Gohberg | A formula for the inversion of finite-section Toeplitz matrices[END_REF]).

Most of these methods involve univariate polynomials. So far, few investigations have been pursued for the treatment of multilevel structured matrices [START_REF] Tyrtyshnikov | Fast algorithms for block Toeplitz matrices[END_REF][START_REF] Khalil | Structured and Toeplitz-block-Toeplitz matrices in numeric and symbolic computation[END_REF], related to multivariate problems. Such linear systems appear for instance in resultant or in residue constructions, in normal form computations, or more generally in multivariate polynomial algebra. We refer to [START_REF] Mourrain | Multivariate polynomials, duality, and structured matrices[END_REF] for a general description of multi-structured matrices and their correlations with multivariate polynomials. Surprisingly, these multivariate structure also appear in numerical scheme and preconditionners [START_REF] Khalil | Structured and Toeplitz-block-Toeplitz matrices in numeric and symbolic computation[END_REF]. A main challenge here is to devise superfast algorithms of complexity Õ(n) for the solution of multi-structured systems of size n.

In this paper, we re-investigate the solution of Toeplitz systems T u = g from a new point of view which can be generalized to two-level Toeplitz systems. We correlate the solution of such problems with syzygies of polynomials. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is generated by two elements of degree n and the solution of T u = g can be reinterpreted as the remainder of an explicit polynomial vector depending on g, by these two generators. We give two algorithms, with computational complexity O(n log 2 n), to compute the generators of the module of syzygies. We give finally an algorithm, with computational complexity O(n log 2 n), for the division of the generators by the polynomial vector depending on g. Our new syzygy approach can be connected with Padé approximation method developed in [START_REF] Brent | Fast solution of Toeplitz systems of equations and computation of Padé approximants[END_REF] to compute efficiently particular solutions of Toeplitz linear system. But we replace the computation of generators of structured matrices by the computation of generators of a syzygy module and the solution of the linear system from particular solutions by Euclidean reduction by the generators of the syzygy module.

Let

R = K[x].
For n ∈ N, we denote by K[x] n the vector space of polynomials of degree ≤ n. Let L = K[x, x -1 ] be the set of Laurent polynomials in the variable x. For any polynomial p = n i=-m p i x i ∈ L, we denote by p + the sum of terms with non-negative exponents: p + = n i=0 p i x i and by p -, the sum of terms with strictly negative exponents: p -= -1 i=-m p i x i . We have p = p + + p -. For n ∈ N, we denote by U n = {ω; ω n = 1} the set of roots of unity of order n. For a vector u = (u 0 , . . . , u k-1 ) T ∈ K k , we denote by u(x) the polynomial of degree k -1 given by u

(x) = k-1 i=0 u i x i . Conversely, if v(x) = k-1 i=0 v i x i is a polynomial of degree k -1, we denote by v the vector of length k of coefficients of v(x).
If no confusion arises, we may also use v to denote the polynomial v(x).

Sygygies and Toeplitz matrices

Let T ∈ K n×n be an n × n Toeplitz matrix. Then T is of the following form:

(1)

      t 0 t -1 . . . t -n+1 t 1 t 0 . . . . . . . . . . . . . . . t -1 t n-1 . . . t 1 t 0      
.

Let g = (g 0 , . . . , g n-1 ) ∈ K n be a vector of length n. We are interested in the following problem:

Problem 2.1. Find u = (u 0 , . . . , u n-1 ) ∈ K n such that (2)
T u = g. Definition 2.2. Let E = {1, . . . , x n-1 }, and Π E be the projection of L on the vector space generated by E, along x n , x n+1 , . . . . Definition 2.3. From the matrix T and the vectors g and u we define the following polynomials:

• T (x) = n-1 i=-n+1 t i x i , • T (x) = 2n-1 i=0 ti x i with ti = t i if i < n t i-2n if i ≥ n , • u(x) = n-1 i=0 u i x i , g(x) = n-1 i=0 g i x i .
Notice that T (x) is a Laurent polynomial and that T (x) is a polynomial of degree 2n -1. By construction, we have the following properties:

Proposition 2.4. T = T + + x 2 n T -and T (w) = T (w) if w ∈ U 2 n .
Proof. We can deduce directly, from the definition of T (x) and T (x), that T = T + +x 2n T -. Moreover, since w 2n = 1 and T (x

) = T + (x) + x 2n T -(x), then T (w) = T + (w) + T -(w) = T (w).
According to Proposition 2.1.2 of [START_REF] Mourrain | Multivariate polynomials, duality, and structured matrices[END_REF], we have the following relation between the problem 2.1 and polynomials: Proposition 2.5. We have

T u = g ⇔ Π E (T (x)u(x)) = g(x).
As Π E (T (x)u(x)) is the polynomial T (x)u(x) from which we remove terms of negative degree and of degree ≥ n, then we can write T (x) u(x) as following: Proposition 2.6.

(3)

T (x) u(x) = Π E (T (x)u(x)) + x -n A(x) + x n B(x),
where

A(x) ∈ K[x] n-1 and B(x) ∈ K[x] n-2 .
Proof. By expanding T (x)u(x) we can write

T (x)u(x) = Π E (T (x)u(x)) + (α -n+1 x -n+1 + • • • + α -1 x -1 ) + (α n x n + • • • + α 2n-2 x 2n-2 ) = Π E (T (x)u(x)) + x -n (α -n+1 x + • • • + α -1 x n-1 ) + x n (α n + • • • + α 2n-2 x n-2 ) = Π E (T (x)u(x)) + x -n A(x) + x n B(x)
Therefore, according to Proposition 2.5 and Proposition 2.6, if u is solution of T u = g then there exist two polynomials A(x) and B(x

) in K[x] n-1 such that (4) T (x)u(x) -x -n A(x) -x n B(x) = g(x)
.

By evaluation at the roots ω ∈ U 2n , and since ω -n = ω n and T (ω) = T (ω) for ω ∈ U 2n , we have

T (ω)u(ω) + ω n v(ω) = g(ω), ∀ω ∈ U 2n (ω), where v(x) = -A(x) -B(x) of degree ≤ n -1. Therefore the polynomial T (x)u(x) + x n v(x) -g(x)
is multiple of x 2n -1. We deduce that there exists w(x)

∈ K[x] such that (5) T (x)u(x) + x n v(x) + (x 2n -1)w(x) = g(x)
.

Notice that w(x) is of degree ≤ n -1 because (x 2n -1) w(x) is of degree ≤ 3n -1.
2.1. Syzygies. The solutions of Equation ( 5) is a particular case of the following problem, related to interesting questions in Effective Algebraic Geometry.

Problem 2.7. Given three polynomials a, b, c ∈ R respectively of degree < l, < m, < n, find three polynomials p, q, r ∈ R of degree

< ν -l, < ν -m, < ν -n, such that (6) a(x) p(x) + b(x) q(x) + c(x) r(x) = 0.
The polynomial vector (p, q, r)

∈ K[x] 3 is called a syzygy of (a, b, c). We denote by L(a, b, c) the set of syzygies (p, q, r) ∈ K[x] 3 of (a, b, c), i.e. the solutions of (6). It is a K[x]-module of K[x] 3 and it is called the module of syzygies of (a, b, c). The solutions of Problem 2.7 are L(a, b, c) ∩ K[x] ν-l-1 × K[x] ν-m-1 × K[x] ν-n-1 .
Given a new polynomial d(x) ∈ K[x], we denote by L(a, b, c; d) the set of (p, q, r)

∈ K[x] 3 such that (7) a(x) p(x) + b(x) q(x) + c(x) r(x) = d(x).
Theorem 2.8. For any non-zero vector of polynomials (a, b, c)

∈ K[x] 3 , the K[x]-module L(a, b, c) is free of rank 2.
Proof. By the Hilbert's theorem, the ideal I generated by (a, b, c) has a free resolution of length at most 1 (see [4, chap. 6]), that is of the form:

0 → K[x] p → K[x] 3 → K[x] → K[x]/I → 0.
As I = 0, for dimensional reasons, we must have p -3 + 1 = 0, then p = 2.

Definition 2.9. For a polynomial vector p

= (p 1 , . . . , p k ) ∈ K[x] k , we define deg(p 1 , . . . , p k ) = max(deg(p 1 ), . . . , deg(p k )). Definition 2.10. Assume that deg(p, q, r) ≤ deg(p ′ , q ′ , r ′ ). A µ-base of L(a, b, c) is a basis {(p, q, r), (p ′ , q ′ , r ′ )} of L(a, b, c), with deg(p, q, r) = µ.
We have the following relation between the degrees of the two elements of a basis of L(a, b, c):

Proposition 2.11. Let {(p 1 , q 1 , r 1 ), (p 2 , q 2 , r 2 )} be a basis of L(a, b, c), µ 1 = deg(p 1 , q 1 , r 1 ) and µ 2 = deg(p 2 , q 2 , r 2 ). We have deg(a, b, c) = µ 1 + µ 2 .
Proof. We have

0 → K[x] ν-d-µ1 ⊕ K[x] ν-d-µ2 → K[x] 3 ν-d → K[x] ν → K[x] ν /(a, b, c) ν → 0, for ν ≫ 0.
As the alternate sum of the dimension of the K-vector spaces is zero and

K[x] ν /(a, b, c) ν is 0 for ν ≫ 0, we have 0 = 3 (d -ν -1) + ν -µ 1 -d + 1 + ν -µ 2 -d + 1 + ν + 1 = d -µ 1 -µ 2 .
2.2. The module L( T (x), x n , x 2n -1). Returning to the initial problem, we saw that if u is solution of T u = g then there exist two polynomials v(x) and w(x

) in K[x] n-1 such that (u(x), v(x), w(x)) ∈ L( T (x), x n , x 2n -1; g(x)).
By the proposition 2.11, if (p, q, r) and(p ′ , q ′ , r ′ ) form a basis of L( T (x), x n , x 2n -1) of degree µ 1 and µ 2 respectively then we have µ 1 + µ 2 = 2 n. We are going to show now that in fact L( T (x), x n , x 2n -1) has a n-basis, that is a basis of two elements of degree µ 1 = µ 2 = n: Proposition 2.12. The K[x]-module L( T (x), x n , x 2n -1) has a n-basis.

Proof. Consider the linear map K

[x] 3 n-1 → K[x] 3n-1 (8)
(p(x), q(x), r(x)) → T (x)p(x) + x n q(x) + (x 2n -1)r(x), which 3n × 3n matrix is of the form ( 9)

S :=   T 0 0 -I n T 1 I n 0 T 2 0 I n   ,
where T 0 , T 1 , T 2 are the coefficient matrices of ( T (x), x T (x), . . . , x n T (x)), respectively for the list of monomials (1, . . . , x n-1 ), (x n , . . . , x 2n-1 ), (x 2n , . . . , x 3n-1 ). Notice in particular that T = T 0 + T 2 .

Reducing the first block (T 0 |0| -I n ) by the last block (T 2 |0|I n ), we replace it by the block (T 0 + T 2 |0|0), without changing the rank of S. As T = T 0 + T 2 is invertible, this shows that the matrix S is of rank 3n. Therefore ker(S) = 0 and there is no syzygies in degree n -1.

As the sum 2n = µ 1 +µ 2 , where µ 1 , µ 2 are the degrees of a pair of generators of L( T (x), x n , x 2n -1), and as µ 1 ≥ n and µ 2 ≥ n, we have µ 1 = µ 2 = n. Moreover, L( T (x), x n , x 2n -1) is free of rank 2. Thus there exist two linearly independent syzygies (u 1 , v 1 , w 1 ), (u 2 , v 2 , w 2 ) of degree n, which generate L( T (x), x n , x 2n -1).

A similar result can also be found in [START_REF] Van Barel | A stabilized superfast solver for nonsymmetric Toeplitz systems[END_REF], but the proof much longer than this one, is based on interpolation techniques and explicit computations.

Let us now describe how to construct explicitly two generators (u 1 , v 1 , w 1 ), (u 2 , v 2 , w 2 ) of L( T (x), x n , x 2n -1) of degree n. As T (x) is of degree ≤ 2 n -1 and the map (8) is surjective, there exists (u, v, w) ∈ K[x] 3 n-1 such that [START_REF] Huckle | Computations with Gohberg-Semencul-type formulas for Toeplitz matrices[END_REF] T (x)u(x) + x n v(x) + (x 2 n -1) w = T (x)x n .

We deduce that (u 1 , v 1 , w 1 ) = (x n -u, -v, -w) ∈ L( T (x), x n , x 2n -1). Since there exists (u

′ , v ′ , w ′ ) ∈ K[x] 3 n-1 such that (11) T (x)u ′ (x) + x n v ′ (x) + (x 2 n -1) w ′ = 1 = x n x n -(x 2 n -1), we deduce that (u 2 , v 2 , w 2 ) = (-u ′ , x n -v ′ , -w ′ -1) ∈ L( T (x),
x n , x 2n -1). Now, T and are linearly independent since by construction, The coefficient vectors of x n in (u 1 , v 1 , w 1 ) and (u 2 , v 2 , w 2 ) are respectively (1, 0, 0) and (0, 1, 0), which shows that vectors (u 1 , v 1 , w 1 ), (u 2 , v 2 , w 2 ) ∈ L( T (x), x n , x 2n -1) ∩ K[x] n are linearly independent. Therefore, they form a basis of L( T (x), x n , x 2n -1). Now we can prove our aim theorem:

Theorem 2.13. The vector u is solution of (2) if and only if there exist v(x) and w(x

) in K[x] n-1 such that (u(x), v(x), w(x)) ∈ L( T (x), x n , x 2n -1; g(x))
Proof. If u is solution of (2), we see that there exist v(x)

∈ K[x] n-1 and w(x) ∈ K[x] n-1 such that T (x)u(x) + x n v(x) + (x 2n -1)w(x) = g(x).
Conversely, a solution (u(x), v(x), w(x)) ∈ L( T (x), x n , x 2n -1; g(x))∩K[x] 3 n-1 implies that (u, v, w) ∈ K 3 n is a solution of the linear system:

S   u v w   =   g 0 0   ,
where S is has the block structure [START_REF] Heinig | Algebraic methods for Toeplitz-like matrices and operators[END_REF], so that T 2 u + w = 0 and T 0 u -w = (T 0 + T 2 )u = g. As we have T 0 + T 2 = T , the vector u is a solution of (2), which ends the proof of the theorem.

Computing the inverse of a Toeplitz matrix T is equivalent to computing the first and the last column of T -1 , based on Gohberg-Semencul decomposition (see [START_REF] Heinig | Algebraic methods for Toeplitz-like matrices and operators[END_REF][START_REF] Labahn | Inversion of Toeplitz matrices with only two standard equations[END_REF][START_REF] Huckle | Computations with Gohberg-Semencul-type formulas for Toeplitz matrices[END_REF][START_REF] Kailath | Generalized Gohberg-Semencul formulas for matrix inversion[END_REF] for more details about Gohberg-Semencul decomposition).

We are going to show that the solutions of Equations ( 10) and ( 11) which gives us the n-basis {(u 1 , v 1 , w 1 ), (u 2 , v 2 , w 2 )} is related to the solution of two specific Toeplitz linear systems. Proposition 2.14. Let (u(x), v(x), w(x)) and (u

′ (x), v ′ (x), w ′ (x)) be in K n-1 [x] 3 such that T (x)u(x) + x n v(x) + (x 2 n -1) w(x) = T (x)x n , T (x)u ′ (x) + x n v ′ (x) + (x 2 n -1) w ′ (x) = 1.
Then T u ′ = e 1 and T u = ZT e n , with Z is the lower shift matrix of size n.

Proof. As u ′ (x), v ′ (x), w ′ (x) and 1 are of degree ≤ n-1, then, by Theorem 2.13, T (x

)u ′ (x)+x n v ′ (x)+ (x 2 n -1) w ′ (x) = 1 is equivalent to T u ′ = e 1 (e 1 (x) = 1) and u ′ is the first column of T -1 . We have T (x) = T + (x) + x 2n T -(x), then T (x)u(x) + x n v(x) + (x 2n -1)w(x) = x n T + (x) + x n ((x 2n -1)T -(x) + T -(x)).
Therefore,

T (x)u(x) + x n (v(x) -T + (x)) + (x 2n -1)(w(x) -x n T -(x)) = x n T -(x).
As x n T -(x) is of degree ≤ n -1 and is the polynomial associated with the vector ZT e n , by Theorem 2.13, u is such that T u = ZT e n .

Notice that u is not the last column of T -1 , but we can use u and u ′ to compute it (see [START_REF] Heinig | Algebraic methods for Toeplitz-like matrices and operators[END_REF]). Therefore, defining a n-basis of L( T (x), x n , x 2n -1) from the solution of Equations ( 10) and ( 11) is equivalent to computing the Gohberg-Semencul decomposition of T -1 .

In the following section, we reduce translation of the solution of T u = g to an Euclidean division, based on our decomposition, instead of multiplying g by triangular Toeplitz matrices, based on Gohberg-Semencul decomposition. The advantage of our decomposition is that we can generalized it to two-level problems, which allows us to describe a "Gohberg-Semencul" decomposition of Toeplitz-block-Toeplitz matrices.

Euclidean division

In this section, we show how to obtain the solution vector (u(x), v(x), w(x)) ∈ L( T (x), x n , x 2 n -1; g(x)) ∩ K[x] 3 n from a n-basis of L( T (x), x n , x 2n -1) and a particular solution in L( T (x), x n , x 2 n -1; g(x)).

From Theorem 2.13 we deduce the two following corollaries:

Corollary 3.1. For all g(x) ∈ K n-1 [x], the set L( T (x), x n , x 2n -1; g(x)) ∩ K 3 n-1 [x] has exactly one element.
Proof. As T is invertible, there exists a unique u such that T u = g. From the theorem 2.13, there exists

v(x), w(x) of degree ≤ n -1, such that (u(x), v(x), w(x)) ∈ L( T (x), x n , x 2n -1; g(x)) ∩ K 3 n-1 [x]. The uniqueness is also obvious: if (u ′ (x), v ′ (x), w ′ (x)) ∈ L( T (x), x n , x 2n -1; g(x)) ∩ K 3 n-1 [x], then (u(x), v(x), w(x)) -(u ′ (x), v ′ (x), w ′ (x)) ∈ L( T (x), x n , x 2n -1) ∩ K 3 n-1 [x]
which equal {(0, 0, 0)} (see the demonstration of the proposition 2.12). Then (u(x), v(x), w(x)) = (u ′ (x), v ′ (x), w ′ (x)). Corollary 3.2. Let {(u 1 , v 1 , w 1 ), (u 2 , v 2 , w 2 )} be a n-basis of L( T (x), x n , x 2n -1). Let (p, q, r) be in L( T (x), x n , x 2n -1; g(x)). There exists a unique (u, v, w) ∈ L( T (x), x n , x 2n -1; g(x)) ∩ K 3 n-1 [x] and a unique pair of polynomials p 1 and p 2 such that

  p q r   = p 1   u 1 v 1 w 1   + p 2   u 2 v 2 w 2   +   u v w   .
This decomposition is called the division of (p, q, r) by (u 1 , v 1 , w 1 ) and (u 2 , v 2 , w 2 ).

Proof. From the previous corollary, there exist a unique element in L( T (x), x n , x 2n -1; g(x))∩K 3 n-1 [x], let (u, v, w) be this element. As {(u 1 , v 1 , w 1 ), (u 2 , v 2 , w 2 )} is a n-basis of L( T (x), x n , x 2n -1), and as (p, q, r) -(u, v, w) ∈ L( T (x), x n , x 2n -1), then there exist a unique pair of polynomials unique p 1 and p 2 such that 

 p q r   -   u v w   = p 1   u 1 v 1 w 1   + p 2   u 2 v 2 w 2  
As a consequence of the two corollaries, we have the following important property:

Theorem 3.3. Let {(u 1 , v 1 , w 1 ), (u 2 , v 2 , w 2
)} be a n-basis of L( T (x), x n , x 2n -1), and let g ∈ K n . The remainder of the division of

  0 x n g g   by   u 1 u 2 v 1 v 2 w 1 w 2   is the unique element (u, v, w) ∈ L( T (x), x n , x 2n - 1; g(x)) ∩ K 3 n-1 [x]
, and therefore u is the solution of T u = g.

Proof. The vector

  0 x n g -g   ∈ L( T (x), x n , x 2 n -1; g) is a particular solution. We reduce it by   u 1 u 2 v 1 v 2 w 1 w 2   and obtain   u v w   =   0 x n g g   -   u 1 u 2 v 1 v 2 w 1 w 2   p q , where (u, v, w) ∈ K[x] 3 n-1 ∩ L( T (x), x n , x 2 n -1; g) is the remainder of division. Thus (u, v, w) is the unique vector ∈ K[x] 3 n-1 ∩ L( T (x), x n , x 2 n -1; g).
A way to perform the division is to choose a n-basis {(u 1 , v 1 , w 1 ), (u 2 , v 2 , w 2 )} of L( T (x), x n , x 2n -1) so that the 2 × 2 coefficient matrix of x n in

u 1 (x) u 2 (x) v 1 (x) v 2 (x)
is invertible. In this case we can reduce the polynomial (0, x n g(x)) to reach to a degree < n -1 and we can write in a unique way

0 x n g(x) = p 1 u 1 v 1 + p 2 u 2 v 2 + u v .
By the uniqueness of the remainder in the Euclidean division, we obtain the following proposition:

Proposition 3.4. The first coordinate of the remainder in the division of

0 x n g by u u 2 v 1 v 2
is the polynomial u(x) such that its associated vector u is the solution of T u = g.

So we set the following problem:

Problem 3.5. Given a matrix and a vector of polynomials e(x) e ′ (x)

f (x) f ′ (x) of degree n such that e n e ′ n f n f ′ n is invertible and p(x) q(x)
of degree m ≥ n, find the remainder of the division of p(x) q(x) by e(x) e ′ (x) f (x) f ′ (x) .

We describe here a generalized Euclidean division algorithm to solve problem 3.5.

Let E(x) = p(x) q(x) of degree m, B(x) = e(x) e ′ (x) f (x) f ′ (x) of degree n ≤ m. E(x) = B(x)Q(x)+R(x) with deg(R(x)) < n, and deg(Q(x)) ≤ m -n. Let z = 1 x . We have E(x) = B(x)Q(x) + R(x) ⇔ E( 1 z ) = B( 1 z )Q( 1 z ) + R( 1 z ) ⇔ z m E( 1 z ) = z n B( 1 z )z m-n Q( 1 z ) + z m-n+1 z n-1 R( 1 z ) ⇔ Ê(z) = B(z) Q(z) + z m-n+1 R(z) (12)
with Ê(z), B(z), Q(z), R(z) are the polynomials obtained by reversing the order of coefficients of polynomials E(z), B(z), Q(z), R(z).

(12) ⇒ B(z) -1 Ê(z) = Q(z) + z m+n-1 B(z) -1 R(z) ⇒ Q(z) = B(z) -1 Ê(z) mod z m-n+1
The formal power series B(z)

-1 exists because the constant coefficient of B(z) is invertible. Thus Q(z) is obtained by computing the first m -n + 1 coefficients of B(z) -1 Ê(z), which is obtained by computing W (x) = B(z) -1
, then by multiplying W (x) by Ê(z).

To find W (x) = B(z) -1 we use Newton's iteration. Let f (W ) = B -W -1 . We have

f ′ (W l ).(W l+1 -W l ) = -W -1 l (W l+1 -W l )W -1 l = f (W l ) = B -W -1 l . Thus we set W l+1 = 2W l -W l BW l ,
and W 0 = B-1 0 , which exists. Moreover, we have

W -W l+1 = W -2W l + W l BW l = W (I 2 -BW l ) 2 = (W -W l ) B(W -W l )
Thus W l (x) = W (x) mod x 2l for l = 0, . . . , ⌈log(m -n + 1)⌉.

Proposition 3.6. We need O(n log(n) log(m -n) + m log m) operations to solve problem 3.5.

Proof. We must do ⌈log(m -n + 1)⌉ Newton's iteration to obtain the first m -n + 1 coefficients of B(z) -1 = W (x). And each iteration requires O(n log n) operations (multiplication and summation of polynomials of degree n). Finally, multiplication B(z)

-1 Ê(z) requires O(m log m) operations.
Notice that, for our problem m = n and this algorithm requires O(n log 2 n) arithmetic operations. In the following section, we show how to compute a n-basis in O(n log 2 n) arithmetic operations.

Construction of the generators

The canonical basis of K[x] 3 is denoted by σ 1 , σ 2 , σ 3 . Let ρ 1 , ρ 2 be the generators of L( T (x), x n , x 2n -1) of degree n given by ( 13)

ρ 1 = x n σ 1 -(u, v, w) = (u 1 , v 1 , w 1 ) ρ 2 = x n σ 2 -(u ′ , v ′ , w ′ ) = (u 2 , v 2 , w 2 ),
where (u, v, w), (u ′ , v ′ , w ′ ) are the vectors given in [START_REF] Huckle | Computations with Gohberg-Semencul-type formulas for Toeplitz matrices[END_REF] and [START_REF] Kailath | Generalized Gohberg-Semencul formulas for matrix inversion[END_REF]. We describe two methods for computing (u 1 , v 1 , w 1 ) and (u 2 , v 2 , w 2 ). The first one uses the Euclidean gcd algorithm, the second one is based on the method in [START_REF] Van Barel | A stabilized superfast solver for nonsymmetric Toeplitz systems[END_REF].

We recall firstly the algebraic and computational properties of the well known extended euclidean algorithm (see [START_REF] Zur Gathen | Modern computer algebra[END_REF]): Given p(x), p ′ (x) two polynomials in degree m and m ′ respectively, let

r 0 = p, r 1 = p ′ , s 0 = 1, s 1 = 0, t 0 = 0, t 1 = 1.
and define

r i+1 = r i-1 -q i r i , s i+1 = s i-1 -q i s i , t i+1 = t i-1 -q i t i ,
where q i results when the division algorithm is applied to r i-1 and r i , i.e. r i-1 = q i r i + r i+1 .

Proposition 4.1. Let l ∈ N such that r l = 0. Then r l-1 = gcd(p(x), p ′ (x)).

And more generally we have: Proposition 4.2. For all i = 1, . . . , l we have

s i p + t i p ′ = r i and (s i , t i ) = 1,
and

             deg r i+1 < deg r i , i = 1, . . . , l -1 deg s i+1 > deg s i and deg t i+1 > deg t i , deg s i+1 = deg(q i .s i ) = deg v -deg r i , deg t i+1 = deg(q i .t i ) = deg u -deg r i .
We can now present our algorithm. It can be found in the proof of the following theorem:

Theorem 4.3. By applying the Euclidean gcd algorithm to p(x) = x n-1 T and p ′ (x) = x 2n-1 stopping in degree n -1 and n -2, we obtain ρ 1 and ρ 2 respectively.

Proof. We see that T u = g if and only if there exist a(x) and b(x

) in K[x] n-1 such that T (x)u(x) + x 2n-1 b(x) = x n-1 g(x) + a(x),
where T (x) = x n-1 T (x) is a polynomial of degree ≤ 2n -2. In [START_REF] Huckle | Computations with Gohberg-Semencul-type formulas for Toeplitz matrices[END_REF] and [START_REF] Kailath | Generalized Gohberg-Semencul formulas for matrix inversion[END_REF] we saw that for g(x) = 1 (g = e 1 ) and g(x) = x n T (x) (g = (0, t -n+1 , . . . , t -1 ) T ) we obtain a base of L( T (x), x n , x 2n -1). Notice that T u 1 = e 1 if and only if there exist

a 1 (x) ∈ K[x] n-2 , b 1 (x) ∈ K[x] n-1 such that (14) T (x)u 1 (x) + x 2n-1 b 1 (x) = x n-1 + a 1 (x),
and T u 2 = (0, t -n+1 , . . . , t -1 ) T if and only if there exist a 2 (x)

∈ K[x] n-2 , b 2 (x) ∈ K[x] n-1 such that (15) T (x)(u 2 (x) + x n ) + x 2n-1 b 2 (x) = a 2 (x).
As deg a 1 (x) ≤ n -2 and deg a 2 (x) ≤ n -2, by applying the extended Euclidean algorithm in p(x) = x n-1 T and p ′ (x) = x 2n-1 until we have deg r l (x) = n -1 and deg r l+1 (x) = n -2 we obtain

u 1 (x) = 1 c 1 s l (x), b 1 (x) = 1 c 1 t l (x), x n-1 + a 1 (x) = 1 c 1 r l (x),
and

x n + u 2 (x) = 1 c 2 s l+1 (x), b 2 (x) = 1 c 2 t l+1 (x), a 2 (x) = 1 c 2 r l+1 (x),
with c 1 and c 2 are the highest coefficients of r l (x) and s l+1 (x) respectively. In fact, Equation ( 14) is equivalent to Then as u 1 (x) and 1 c1 s l are unitaries, 1 c1 s l (x) = u 1 (x) which implies that 1 c1 t l (x) = b 1 (x). For the same reasons, we have x n + u 2 (x) = 1 c2 s l+1 (x) and b 2 (x) = 1 c2 t l+1 (x). Finally, T u = e 1 if and only if there exist v(x), w(x) such that [START_REF] Pan | Nearly optimal computations with structured matrices[END_REF] T (x)u(x) + x n v(x) + (x 2n -1)w(x) = 1.

n n-1 n -1 n    n -1                 t -n+1 . . . . . .
As T (x) = T + + x 2n T -= T + (x 2n -1)T -, we deduce that ( 17)

T (x)u(x) + x n v(x) + (x 2n -1)(w(x) + T -(x)u(x)) = 1.
Moreover, we also have

T (x)u(x)-x -n+1 a 1 (x)+ x n b 1 (x) = 1 and x -n+1 a 1 (x) = x n (x a 1 )-x -n (x 2n - 1)x a 1 . Thus (18) T (x)u(x) + x n (b(x) -x a(x)) + (x 2n -1)x -n+1 a(x) = 1.
Comparing ( 17) and ( 18), and as 1 = x n x n -(x 2n -1) we deduce that w( The second method for computing (u 1 , v 1 , w 1 ) and (u 2 , v 2 , w 2 ) is of polynomials and interpolation points. We are interested in computing the coefficients of the canonical basis element σ 1 , σ 2 in this basis. The coefficients of σ 3 can be obtained by reduction of ( T (x) x n ) B(x) by x 2n -1 where

x) = x -n+1 a(x)-T -(x)u(x)+ 1,
B(x) = u(x) u ′ (x) v(x) v ′ (x) ,
where (u, v), (u ′ , v ′ ) are the two first coordinates of the solution of Equations ( 10) and [START_REF] Kailath | Generalized Gohberg-Semencul formulas for matrix inversion[END_REF]. A superfast algorithm for computing B(x) is given in [START_REF] Van Barel | A stabilized superfast solver for nonsymmetric Toeplitz systems[END_REF]. Let us describe how to compute it. By evaluation of (13) at the roots ω j ∈ U 2n , we deduce that (u(x), v(x)) and (u ′ (x), v ′ (x)) are the solution of the following rational interpolation problem:

T (ω j )u(ω j ) + ω n j v(ω j ) = 0 T (ω j )u ′ (ω j ) + ω n j v ′ (ω j ) = 0 , with u n = 1, v n = 0, u ′ n = 0, v ′ n = 1. Definition 4.5.
The τ -degree of a vector polynomial w(x) = (w 1 (x) w 2 (x)) T is defined as τ -deg w(x) := max{deg w 1 (x), deg w 2 (x) -τ } Definition 4.6. A polynomial vector in K[x] 2 is called τ -reduced if the τ -highest degree coefficients are linearly independent.

By construction, the columns of B(x) form a n-reduced basis of the module of polynomial vectors r(x) ∈ K[x] 2 that satisfy the interpolation conditions f j r(ω j ) = 0, j = 0, . . . , 2n -1 with f j = ( T (ω j ), ω n j ) ∈ K 2 . The columns of B(x) are also called a n-reduced basis for the interpolation data (ω j , f j ), j = 0, . . . , 2n -1.

Theorem 4.7. Let τ = n and J be a positive integer. Let λ 1 , . . . , λ J ∈ K and φ 1 , . . . , φ J ∈ K 2 \ {(0, 0)}. Let 1 ≤ j ≤ J and τ J ∈ Z. Suppose that B j (x) ∈ K[x] 2×2 is a τ J -reduced basis matrix with basis vectors having τ J -degree δ 1 and δ 2 , respectively, corresponding to the interpolation data {(λ i , φ i ); i = 1, . . . , j}.

Let τ j→J := δ 1 -δ 2 . Let B j→J (x) be a τ j→J -reduced basis matrix corresponding to the interpolation data {(λ i , φ i B j (λ j )); i = j + 1, . . . , J}.

Then B J (x) := B j (x)B j→J (x) is a τ J -reduced basis matrix corresponding to the interpolation data {(λ i , φ i ); i = 1, . . . , J}.

Proof. For the proof, see [START_REF] Van Barel | A stabilized superfast solver for nonsymmetric Toeplitz systems[END_REF].

When we apply this theorem with λ j = ω j ∈ U 2n as interpolation points, we obtain a superfast algorithm in O(n log 2 n) to compute B(x). See [START_REF] Van Barel | A stabilized superfast solver for nonsymmetric Toeplitz systems[END_REF] for more details.

Conclusion

In this paper, we re-investigate the solution of a Toeplitz system T u = g from a new point of view, by correlating the solution of such a problem with generators of the syzygy module L( T (x), x n , x 2n -1) associated to the Toeplitz matrix T . We show that L( T (x), x n , x 2n -1) is free of rank 2 and that it has a n-basis. We show that finding a n-basis of L( T (x), x n , x 2n -1) is equivalent to computing the Gohberg-Semencul decomposition of T -1 , and we reduce the solution of T u = g to an Euclidean division. We give two superfast algorithms computing a n-basis of L( T (x), x n , x 2n -1) and a superfast algorithm to obtain the solution from this n-basis.

A perspective of this work is to generalize the approach to two-level Toeplitz systems or to Toeplitzblock-Toeplitz matrices and to correlate the basis computation of a multivariate syzygy module to "Gohberg-Semencul" decompositions for Toeplitz-block-Toeplitz matrices.

  deg r l = n -1 then, by Proposition 4.2, deg s l+1 = (2n -1) -(n -1) = n and deg t l+1 = (2n -2) -(n -1) = n -1. By the same proposition, we also have deg s l ≤ n -1 and deg t l ≤ n -2. Therefore, deg u 1 = deg s l and deg b 1 = deg t l .

  which is the part of positive degree of -T -(x)u(x) + 1. This conclude the proof of the proposition.Remark 4.4. The usual Euclidean gcd algorithms are of computational complexity O(n 2 ), but superfast euclidean gcd algorithms use no more then O(n log 2 n) operations, exist. See for example[START_REF] Zur Gathen | Modern computer algebra[END_REF] chapter 11.


David Pontille
email: david.pontille@mines-paristech.fr

Didier Torny
email: torny@ivry.inra.fr

Behind the scenes of scientific articles: defining categories of fraud and regulating cases

Introduction

On March 7, 2012, the blog Retraction Watch 1 announced that Yoshitaka Fujii, Associate Professor of Anesthesiology, had just been fired from the University of Toho for having published nine clinical studies that did not comply with the current ethical guidelines. The blog also announced that one of the journals concerned, Anesthesia & Analgesia, whose subtitle is "the gold standard in anesthesiology," revealed in a communiqué that a study of the integrity of Dr. Fujii's research had been launched in 2010 (Schafer 2012). The journal's American editor-in-chief indicated that 24 other articles could be the subject of scientific fraud. He also stated that in 2000 his journal had published a letter from German researchers presenting a meta-analysis of 47 articles jointly signed by Dr. Fujii (Kranke et al. 2012). These authors underscore the remarkably identical frequency of headaches as side effects of anesthesia in all studies and concluded that only an underlying influence could explain this stupefying consistency.

Dr. Fujii had responded with a short jointly published letter, simply indicating that he had reported empirical observations and that they were expected given the state of knowledge. This response was sufficiently convincing to the International Anesthesia Research Society that Dr. Fujii was able to publish 12 additional articles over the following decade. Yet the German researchers had pursued their meta-analyses, in 2001 showing in Acta Anaesthesiologica Scandinavica that 64% of the clinical data published for an antiemetic originated from Dr. Fujii's team, even though his dosage results diverged substantially from those produced by all of the other laboratories [START_REF] Kranke | The influence of a dominating centre on a quantitative systematic review of granisetron for preventing postoperative nausea and vomiting[END_REF]. Following this publication, the authors alerted the Food and Drug Administration and the Japanese regulatory agency, apparently to no avail.

In March 2010, an editorial in the British journal Anaesthesia described the obstacles to demonstrating falsification and the invention of clinical data, taking as an example the suspicions weighing against Dr. Fujii's data [START_REF] Derry | Fraud or flawed: adverse impact of fabricated or poor quality research[END_REF]. This editorial provoked a large number of responses, leading editorial boards to engage in a reflection on their role in combatting scientific fraud (Wagner 2012). Also, in 2011 when Dr. Fujii submitted an article to the Canadian Journal of Anesthesia, where he had already published 39 articles, the journal opened an inquiry with the University of Toho, which revealed that there was no Committee for the protection of human research subjects, prompting investigations on all of his research production [START_REF] Miller | Expression of concern[END_REF]. At the same time, a meta-analysis of 169 controlled trials cosigned by Dr. Fujii, initiated following the 2010 editorial and verified at length, was published as a special article in Anaesthesia on March 8, 2012. It concluded that the distribution of 28 of the variables published did not correspond to what could be expected, with the probability of occurrence under the hypothesis of independence even reaching 10 -33 for one of them [START_REF] Carlisle | The analysis of 169 randomised controlled trials to test data integrity[END_REF].

Finally, on April 9, 2012, a joint letter from 22 editorial directors was addressed to the leaders of nine institutions that had hosted Dr. Fujii's research and simultaneously published on some of the journals' web sites. It informed them of the publication of this study and announced that the articles would be retracted, unless the institutions involved could provide proof that the articles were trustworthy: "for each study concerned, we request that your institution declare 1/ that the study was conducted as presented in the article, 2/ that you have examined the primary data and verified that they were authentic, and 3/ that the appropriate ethical framework was clearly established for the study" [START_REF] Collectif | Joint Editors-in-Chief Request for Determination 1 Regarding Papers[END_REF]. Answering that call, the Japanese Society of Anesthesiologists led an inquiry and concluded, in its 29 th June 2012 report, that 172 articles jointly signed by Dr Fujii contained falsified data and should be retracted, which is an unequaled record for a single author.

How can one explain that Dr. Fujii was able to publish apparently falsified data for such a long time? Is this an isolated researcher who was particularly skilled in fabricating forgeries corresponding to the expectations of the scientific community, including those of his coauthors2 ? Is this a local institutional problem related to his mandarin status, as in the case of Professor W.S. Hwang, a Korean national hero and fraudulent inventor of human cloning [START_REF] Gottweis | Explaining Hwang-Gate: South Korean Identity Politics between Bionationalism and Globalization[END_REF])? Are we dealing with a disciplinary idiosyncrasy, since two other anesthesiologists have recently undergone similar waves of retracted articles and that 4% of the articles submitted to Anaesthesia have turned out to be plagiarized [START_REF] Yentis | Another kind of ethics: from corrections to retractions[END_REF])? Or should we consider that any academic production is potentially tarnished, implicating all scientific institutions (authors, financers, editors, journals) [START_REF] Ioannidis | Why most published research findings are false[END_REF]?

Two major interpretations are generally contrasted in this type of situation. The first tolls the bell of scientific integrity: it is mainly based on the massive increase of the number of articles retracted (Van Noorden 2011), on the fact that these retractions are strongly correlated with the journals' impact factor [START_REF] Fang | Retracted science and the retraction index[END_REF] or that the retractions for fraud take place in journals with a higher impact factor than the retractions for experimental or calculation errors (Steen 2011a). It also responds to the high frequency of researchers declaring having observed colleagues involved in practices they deem contrary to scientific ethics [START_REF] Fanelli | How many scientists fabricate and falsify research? A systematic review and metaanalysis of survey data[END_REF]. In contrast, the second interpretation ferociously defends tooth and nail a model of self-regulation. It is founded, for example, on the usually brief time between publication and retraction, the generally low number of citations related to these articles [START_REF] Furman | Governing knowledge in the scientific community: Exploring the role of retractions in biomedicine[END_REF], the appearance in some journals of temporary publication bans, and considers the creation of specialized places such as the Abnormal Science and Retraction Watch blogs as extensions of peer review [START_REF] Marcus | The paper is not sacred[END_REF].

Our objective in this article is not to choose between these two positions, which stabilized over the 1980s [START_REF] Broad | Betrayers of the Truth; Fraud and Deceit in the Halls of Science[END_REF][START_REF] Lafolette | Stealing into Print: Fraud, Plagarism, and Misconduct in Science[END_REF]) and were regularly reactivated by other actors with each new case detected. Instead, from a perspective informed by science and technology studies, we wish to establish a general diagnosis on the regulation of publication practices and suggest methods of analysis, by reviewing both the recent cases and others, less recent, that they call to mind. First of all, this paper reviews how the practices deemed to be deviant have been qualifed in the past by distinguishing the categories stemming from data integrity and the conditions of data production from those concerning the relationships between the authors of publications and their content. We will then analyze three specific problems and the institutional responses aiming to prevent them, channel them, and attempt to solve them.

Striking it rich with dubious data

Medical research for the most part has been marked by a progressive rise in ethical demands that strive toward defining the quality of the data produced. The need to collect informed consent from the subjects participating in therapeutic research is a good example, the subject of a vast literature on its concrete manifestations [START_REF] Corrigan | Empty ethics: the problem with informed consent[END_REF]. Similarly, in the construction of all the instruments designed for direct or indirect use in epidemiological studies, protecting the subjects' anonymity is essential to conducting these investigations. These diverse ethical demands frame the conditions of data production: any time compliance is lacking, the reliability of the data is compromised for this reason alone, as seen with the case of Dr. Fujii. But the materials on which the results published are founded can be subject to tampering beyond these regulatory obligations.

Several ways of contravening "good practices" can be identified in the series of affairs discussed herein. The first resides in the selection of relevant data to obtain more coherent results, a smoother curve, a more readable image. R. Fisher, an important statistician, geneticist, and support for eugenics, 80 years after the famous experiments by Gregor Mendel on his peas, showed, for example, that the probabilities obtained were undoubtedly too good to be true and stemmed more from a confirmation bias than from an observation. His research has contributed to designing tools to remove the biases introduced by experimenters, notably in controlled clinical trials [START_REF] Marks | La Médecine des preuves. Histoire et anthropologie des essais cliniques[END_REF]. In this process of selecting relevant data, the question of ill will is secondary to all the possible means of deviating from an experimental, observational, or statistical norm.

The second consists in conscious falsification activities, which can manifest as minor arrangements of experimental material, falsification of data, or manipulation of images. D.

Das, a surgeon working on the protective effects of resveratrol, explaining in fine the probable role played by wine consumption in the French Paradox, was suspected of having massively doctored Western blot images that were the basis of dozens of articles that he published, leading to his being fired from the University of Connecticut after 3 years of inquiry (DeFrancesco 2012). The multiplication of cases raising questions on the falsification of images reveals just how fragile and uncertain are the graphic representations that are held to be tangible components of scientific proof in the end (Lynch & Woolgar 1990), and raises debate on the nature of the data necessary to certify their integrity.

Finally, the third manner stems from fabrication pure and simple. Falsification extends here to all materials and is based on the pure invention of data. The example of the physician and dentist Jon Sudbø comes to mind, who, in 2006, admitted to having literally created 900 patients in a Norwegian cohort, 250 of whom had the same date of birth. This allowed him to demonstrate spectacular effects of taking nonsteroidal anti-inflammatory drugs on oral cancer in heavy smokers and to publish in The Lancet the year before.

The distinction between selection, falsification, and fabrication is not self-evident and instead comes from ex post attribution when cases are revealed after investigations [START_REF] Fanelli | How many scientists fabricate and falsify research? A systematic review and metaanalysis of survey data[END_REF]. As in the affairs concerning Prof. Hwang, when there are no detailed admissions, it is impossible to distinguish between what has been authentically produced in the laboratory and what is instead voluntary or involuntary artifact. If the absence of reproducibility is often considered a clue to falsification, the opposite is not necessarily true: a highly skilled forger can produce a sort of retroplagiarism, arranging or fabricating data that others, through slow scientific work, will have actually produced and analyzed.

Circumventing proof of originality

One of the fundamental principles of scientific activity consists in producing original results, as indicated by the categorization of journal articles that distinguishes "original research" from "reviews", "letters", and "editorials." According to this principle, the recognition of one's peers and the awarding of prizes are strongly related to the priority of original discovery. No importance is supposedly accorded to second place. Among the different ways to get around this requirement, this section will emphasize the question of plagiarism and authorship.

Plagiarism and self-plagiarism

In these conditions, the temptation is sometimes great to cite only partially and sporadically the similar work that has preceded, or even to totally omit its mention. In our era of electronic publications and cut-and-paste, there is no simpler operation than to insert one or several paragraphs written and published earlier by others into an article being written, for example by copying a piece of text from an article on nosocomial infections in Brazil into an article on the same subject in Spain [START_REF] Cisterna | Nationwide sentinel surveillance of bloodstream Candida infections in 40 tertiary care hospitals in Spain[END_REF]. The same clearly holds true for data, figures, and images. At times fraud is not limited to partial plagiarism: at the end of the 1970s, E. Alsabti, an alleged researcher in oncoimmunology, without a doctorate, succeeded in climbing the ladder of an academic career by putting together an impressive list of publications in barely 3 years (Broad & Wade 1982, chap. 3). His tactic was simple: he copied entire articles that had already been published, modified the title, replaced the authors' names with his own, and submitted the manuscript to a less well-known journal. He succeeded in accumulating more than 60 publications in this way, in joining 11 scientific societies and in working in several prestigious American institutions.

Rather than borrowing from others' work, it can be even simpler to dig into one's own data. What researchers and clinicians have long designated as a "pressure to publish" [START_REF] Maddox | Why the pressure to publish?[END_REF]) has turned into the proliferation of articles based on the same research, the spawning of similar results into a variety of texts and the preference for quantity to the detriment of quality [START_REF] Hamilton | Publishing by -and for ? -the numbers[END_REF]). The practice of self-plagiarism has developed to such an extent that it has progressively given rise to terms such as "least publishable unit" [START_REF] Broad | The publishing game: getting more for less[END_REF], "salami slicing" [START_REF] Huth | Irresponsible authorship and wasteful publication[END_REF]), and even "duplicate publication" [START_REF] Susser | Prior, Duplicate, Repetitive, Fragmented, and Redundant Publication and Editonial Decisions[END_REF], and the effects on meta-analyses can be considerable [START_REF] Tramèr | Impact of covert duplicate publication on meta-analysis: a case study[END_REF]. Even if, contrary to plagiarizing others, this censured concealment is interpreted in a variety of fashions, everyone agrees that the basis of the problem lies in the absence of a citation of the original study [START_REF] Freund | Retraction of "Concomitant measurement of copeptin and high-sensitivity troponin for fast and reliable rule out of acute myocardial infarction[END_REF].

Authorship

An alternative manner to bypass the requirements for originality is to maneuver around the edges of authorship. In many domains, the guidelines regulating access to article byline place value on significant intellectual contributions. Yet the suspected articles often carry the name of researchers whose contribution is considered minimal, if not inexistent, and therefore these individuals have a very distant contact with the research conducted and the results published. Since the end of the 1980s, researchers and journal editors have produced, in what is generally called "honorary authorship", three categories of authorship to denounce these situations that they judge unacceptable [START_REF] Rennie | Authorship! Authorship! Guests, ghosts, grafters, and the two-sided coin[END_REF][START_REF] Flanagin | Prevalence of articles with honorary authors and ghost authors in peer-reviewed medical journals[END_REF].

The first designates "guest authorship", which separately treats people whose name is recognized: these are particular names that change the status accorded to an article by their presence alone. The presence of these researchers' names potentially increases the chances of the article publication and its future visibility and thus figures as a sign of quality. This practice is current and arises in many cases. The affair of the cardiologist J.R. Darsee, one of the most closely followed and documented in the 1980s, brought this practice to the light of day: other than the presence of falsified and entirely invented data, the 55 publications that were retracted often carried the name of his mentor, E. Braunwald, who had little knowledge of their content, and who had not been worried during the detailed investigations of the NIH investigators [START_REF] Stewart | The integrity of the scientific literature[END_REF].

Another category, which plays less on this distinction of status or renown, emphasizes more the offer of authorship governed by the principle of reciprocity, as a gift in view of a countergift ("gift authorship"): it is not exceptional that researchers accept to make room for someone so as not to offend a partner making the request, to encourage future collaborations, to maintain good cooperative relations, to thicken a list of publications, or to return the favor [START_REF] Pontille | La signature scientifique. Une sociologie pragmatique de l'attribution[END_REF]. In this context, the above-mentioned case of E. Alsabti, nonetheless revealed an extreme and particular form of gift at the time: some of his articles included the names of fictitious authors whom E. Alsabti, at the beginning of his fall, blamed entirely for the abuses for which he was denounced.

Finally, as with other types of writing, notably literary, the name of the writer is not systematically the name of those who appear on the byline ("ghost authorship"). This is notably the case when a researcher, who does the greatest share of the work and writes all or nearly all an article, allows only his students or less experienced colleagues to be the authors of the article so as to assist them in publishing in a prestigious journal and thus lengthen their publications list.

Institutional regulations

Common reactions in the event of practices deemed deviant consist in reaffirming the general principles at the foundation of scientific integrity: disinterestedness, organized skepticism regularly held as cardinal virtues [START_REF] Merton | Science and technology in a democratic order[END_REF][START_REF] Weed | Epidemiology and virtue ethics[END_REF] and stipulated in charters and professional agreements or through specialized structures within professional or learned societies [START_REF] Coughlin | Invited Commentary: On the Role of Ethics Committees in Epidemiology Professional Societies[END_REF]. These reactions also require more restrictive concrete operations aiming to prevent and limit these phenomena by indirect actions. We will examine here three specific operations relative to responsibility, negative results, and conflicts of interest.

Restore responsibility

In an evaluation system that values individual performance, the increase of the number of authors per article is regularly described. Although guest and gift authorship cast doubt on the attribution of scientific credit, other forms are even more surprising. Some cases, most particularly the above-mentioned J.R. Darsee affair, revealed that some authors were not aware that their name had been adjoined to the suspected publications. The journals consequently took control to defend their self-regulatory function. They sought to return responsibility to the authors in two major steps. The first resulted from the many cases of fraud that shook the 1980s: statements are required, signed by all of an article's producers to ensure that they approve the content and that they indeed intended their name to appear on the article.

The second step played out between 1996 and 2000, following heated debates between researchers, journal editors, and research administrators [START_REF] Biagioli | The instability of authorship: credit and responsibility in contemporary biomedicine[END_REF][START_REF] Pontille | L'auteur scientifique en question: pratiques en psychologie et en sciences biomédicales[END_REF]. They aimed to set up traceability of research operations so as to be able to reattribute responsibility in case of a future problem [START_REF] Torny | Une mémoire pour le futur. La traçabilité comme allocateur de responsabilité[END_REF]. After several conferences and an experimental phase, journals opted for a new procedure, called "contributorship", which promotes transparency: systematically describe the individual contributions in a section of the article specifically designed for this purpose. Before publication, each contributor is now invited to either describe his or her contribution in a written note, or to fill out a form that delineates a taxonomy of research operations. Thus the producers of articles should approve the final version of the text and date their engagement: each one dates and signs the letter or form manually. This procedure reinforces the judicial responsibility of the handwritten signature and forces each contributor to take responsibility for his or her contribution.

Publishing negative results

Whether the result of self-censure or fraudulent practices on the part of researchers, or the practice of journals for visibility or financial reasons [START_REF] Smith | Medical journals are an extension of the marketing arm of pharmaceutical companies[END_REF], publication biases favoring positive studies are well known [START_REF] Simes | Publication bias: the case for an international registry of clinical trials[END_REF][START_REF] Stern | Publication bias: evidence of delayed publication in a cohort study of clinical research projects[END_REF]. The additional effect of this bias is relegating negative results to second-rank journals [START_REF] Kanaan | The value of lesser-impact-factor surgical journals as a source of negative and inconclusive outcomes reporting[END_REF] or even non-publication. This has been demonstrated for antidepressants, for example [START_REF] Turner | Selective publication of antidepressant trials and its influence on apparent efficacy[END_REF], and for a number of substances approved by the Food and Drug Administration between 1998 and 2000 [START_REF] Lee | Publication of Clinical Trials Supporting Successful New Drug Applications: A Literature Analysis[END_REF], to such an extent that certain researchers now take the rate of positive results published as a transdisciplinary indicator of fraud or bias [START_REF] Fanelli | Do pressures to publish increase scientists' bias? An empirical support from US States Data[END_REF] and others speak of "evidence-biased medicine" [START_REF] Melander | Evidence b(i)ased medicineselective reporting from studies sponsored by pharmaceutical industry: review of studies in new drug applications[END_REF].

To solve this problem, two complementary solutions have been proposed. For clinical studies, this means founding academic evaluation of drugs based on applications made to the regulatory agencies rather than on the published literature (Turnet et al. 2012). In addition, certain journals propose systematically publishing negative results for all types of research, such as the Journal of Errology or the Journal of Negative Results in Biomedicine, which has an "unofficial" impact factor of 1.1. These journals consider that unexpected and failed results are important in scientific knowledge, thus returning to one of the founding principles of experimental practice of the seventeenth century, initiated notably by Robert Boyle at the Royal Society of Sciences of London [START_REF] Shapin | Leviathan and the air-pump. Hobbes, Boyle, and the experimental life[END_REF].

Uncover and reduce conflicts of interest

A third question brings together all of the problematic aspects described in the first sections: conflict of interest. This notion was imported into the scientific world in the 1960s to describe the increasing relations between public research and industry, and the problems that this could raise, in particular in expert consultations with health and environmental regulatory agencies [START_REF] Gingras | The emergence and evolution of the expression "conflict of interests" in Science: a historical overview, 1880-2006[END_REF]. It is through this prism of conflict of interest, for example, that the accusations of scientific fraud against Dr. Needleman were interpreted by two of his colleagues while the Environmental Protection Agency had just adopted restrictive measures for the lead industry [START_REF] Rosner | Standing Up to the Lead Industry : An Interview with Herbert Needleman[END_REF]. More generally, in environmental epidemiology, the consequences in terms of regulating results in public health have led the chemical, tobacco, and asbestos industries to not only finance research that was favorable to them [START_REF] Pearce | Corporate influences on epidemiology[END_REF], but also to maintain uncertainty over the long term to fight against stabilized knowledge being established [START_REF] Proctor | Agnotology: The Making and Unmaking of Ignorance[END_REF].

With the ever-increasing meshing and merging of financing from the pharmaceutical industry and academics, the notion of conflict of interest in its contemporary sense truly appeared in the 1980s [START_REF] Gingras | The emergence and evolution of the expression "conflict of interests" in Science: a historical overview, 1880-2006[END_REF]: it was thus repeatedly demonstrated that financing from industry or pharmaceutical laboratories influenced the results produced [START_REF] Abraham | Bias in Science and Medical Knowledge: The Opren Controversy[END_REF][START_REF] Bekelman | Scope and Impact of Financial Conflicts of Interest in Biomedical Research: A Systematic Review[END_REF][START_REF] Friedman | Relationship between conflicts of interest and research results[END_REF][START_REF] Lexchin | Pharmaceutical industry sponsorship and research outcome and quality: systematic review[END_REF], including on subtile modalities such as the favorable presentation of hormone replacement treatments well after their risk-benefit ratio was challenged at the beginning of the 2000s (Fugh-Berman et al. 2001). In this context, Elsevier was denounced for creating six seemingly scientific journals that were actually financed by pharmaceutical laboratories and designed to present compilations of articles that were favorable to them [START_REF] Hansen | Statement From Michael Hansen[END_REF]. Certain forms of ghostwriting also stem from this practice: through specialized companies, the pharmaceutical firms, once the research and articles had been completed, requested researchers who were reputed in their specialty to sign the publications in exchange for consequential financial compensation [START_REF] Mirowski | The Contract Research Organization and the Commercialization of Scientific Research[END_REF][START_REF] Sismondo | Ghost management: how much of the medical literature is shaped behind the scenes by the pharmaceutical industry?[END_REF]).

With reform in mind, institutions and journals began including the financing sources and conflict of interest disclosures as a solution to these problems, with, for example, regular guidelines by the International Committee of Medical Journal Editors beginning in 19933 , even though their implementation took an exceedingly long time in the eyes of their detractors [START_REF] Krimsky | Conflict of interest policies in science and medical journals: editorial practices and author disclosures[END_REF]. Others have suggested radical reforms of complete separation between private interests and public research [START_REF] Schafer | Biomedical conflicts of interest: a defence of the sequestration thesis -learning from the cases of Nancy Olivieri and David Healy[END_REF], a position recently found on expert testimony in the parliamentary discussions after the Mediator French scandal. However, everyone agrees that internal transformations of scientific production, most particularly biomedical research, has made a large number of researchers true entrepreneurs [START_REF] Etzkowitz | Conflicts of interest and commitment in academic science in the United States[END_REF]. The case of A. Potti illustrates these tensions: as soon as his first articles on genomic markers of the sensitivity of different chemotherapies were published in Nature Medicine and the New England Journal of Medicine in 2006, Duke University and his team boasted of "personalized medicine" that they were going to develop so as to attract patients, grants, as well as the large pharmaceutical laboratories, so that they could conduct large-scale trials. But a few years later, Dr. Potti was implicated both from inside the science world by biostatisticians [START_REF] Baggerly | Deriving chemosensitivity from cell lines: Forensic bioinformatics and reproducible research in high-throughput biology[END_REF] and from the outside, through professional letters showing the many "enhancements" that had appeared on his resumé [START_REF] Goldberg | Prominent Duke scientist claimed prizes he didn't win, including Rhodes scholarchip[END_REF], and then by his patients' families attracted by the potentially revolutionary character of his practices, who had sued for having been deceived.

Conclusion

Emphasizing how long these problems have been around, we had a double objective: first, showing the emblematic value of certain cases, truly striking episodes that have produced preventive measures so that these practices would not continue; second, taking stock of the increase in the number of affairs revealed that have led to the retraction of articles as a clue to a "surveillance bias." Just as certain fraudulent practices have become simpler to carry out, the detection and dissemination of behaviors considered problematic has been largely facilitated. For example, journals such as Anaesthesia routinely use plagiarism detection software, but it is also possible to construct tables that juxtapose text from several articles, showing how close they are, putting together a series of an author's variable conflicts of interest, or anonymously denounce image falsification by circulating a video clip4 .

The recurrent stakes in all of these problematic situations revolves around the scientific community's ability to self-regulate. Nearly all of the institutional measures describe herein directly involve academic authors. Yet for the last 15 years, we have seen the journals themselves be targeted, notably in recurrent debates aiming to assess the independence of journal editors [START_REF] Davis | Editorial independence at medical journals owned by professional associations: a survey of editors[END_REF] and to reform peer review [START_REF] Weicher | Peer review and secrecy in the "Information Age[END_REF]. What should be thought of the responsibility of A&A which for years published Dr. Fujii's articles after the first meta-analyses? How should the process leading the Journal of Clinical Microbiology to publish a plagiarized article published a few years early in its own columns be qualified5 ? Like researchers, journals are also subjected to conflicts of interest [START_REF] Friedman | Conflicts of interest and scientific integrity[END_REF][START_REF] Lundh | Conflicts of interest at medical journals: the influence of industry-supported randomised trials on journal impact factors and revenuecohort study[END_REF] and to the imperatives of success, partly founded on their impact factor, that incite them to introduce references motivated solely by journal citation [START_REF] Smith | Journal accused of manipulating impact factor[END_REF], a practice recently qualified as "coercive citations" [START_REF] Wilhite | Coercive Citation in Academic Publishing[END_REF].

Symmetrically, the case of Dr. Fujii reminds us that journals have a tremendous collective organizational power, widely implemented in the authorship and conflict of interest reforms. In addressing their letter to research institutions, journals have placed them before their own responsibility, thus attempting to reverse the burden of proof. However, the distribution of the responsibility between the actors of academia is also played out on another level: the responsibility with regard to patients, the eventual victims of deviant behaviors (Steen 2011b). Indeed, if research having consequences in terms of treatments, prevention modalities, or, more generally, public health is at the heart of conflicts of interest, opening it to nonacademic actors allows new criticisms to be expressed and taken into consideration through judicial and political action on the part of users and citizens.

Let us not forget the brilliant physicist J.H. Schön, covered with prizes and the lead author of eight articles published in Science and seven articles published in Nature between 2000 and 2001, but then fired by Bell Labs and whose dissertation was revoked by the University of Constance.

These guidelines were revised in 2000, and the member journals of the ICMJE took a position on conflicts of interest the following year in a common editorial titled "Sponsorship, Authorship, and Accountability" praising transparency and academic independence. A conflict of interest disclosure form common to all the ICMJE journals was established in 2010.

http://www.youtube.com/watch?v=FXaOqwanWnU

http://retractionwatch.wordpress.com/2011/09/01/should-journals-apologize-to-victims-of-plagiarismmore-on-journal-of-clinical-microbiology-case/

Acknowledgements

We thank Anne Bertrand (EHESS) for her careful reading of the manuscript and her suggestions on an earlier version. Most of the recent affairs studied in this article stem directly or indirectly from their revelation in the blog Retraction Watch, maintained by Adam Marcus and Ivan Oransky. We thank them for their meticulous and nearly daily work assembling this information.


A Lepailleur

O Sorlin

L Caceres

B Bastin

C Borcea

R Borcea

B A Brown

L Gaudefroy

S Grévy

G F Grinyer

G Hagen

M Hjorth-Jensen

G R Jansen

O Llidoo

F Negoita

F De Oliveira

M.-G Porquet

F Rotaru

M.-G Saint-Laurent

D Sohler

M Stanoiu

J C Thomas

Spectroscopy of 26 F to probe proton-neutron forces close to the drip line

Keywords: numbers: 21.10.Dr, 21.60.Cs, 21.60.De, 23.35.+g, 23.20.Lv

A long-lived J π = 4 + 1 isomer, T 1/2 = 2.2(1)ms, has been discovered at 643.4(1) keV in the weakly-bound 26 9 F nucleus. It was populated at GANIL in the fragmentation of a 36 S beam. It decays by an internal transition to the J π = 1 + 1 ground state (82(14)%), by β-decay to 26 Ne, or beta-delayed neutron emission to 25 Ne. From the beta-decay studies of the J π = 1 + 1 and J π = 4 + 1 states, new excited states have been discovered in 25,26 Ne. Gathering the measured binding energies of the J π = 1 + 1 -4 + 1 multiplet in 26 9 F, we find that the proton-neutron π0d 5/2 ν0d 3/2 effective force used in shell-model calculations should be reduced to properly account for the weak binding of 26 9 F. Microscopic coupled cluster theory calculations using interactions derived from chiral effective field theory are in very good agreement with the energy of the low-lying 1 + 1 , 2 + 1 , 4 + 1 states in 26 F. Including three-body forces and coupling to the continuum effects improve the agreement between experiment and theory as compared to the use of two-body forces only.

Introduction.-Understanding the boundaries of the nuclear landscape and the origin of magic nuclei throughout the chart of nuclides are overarching aims and intellectual challenges in nuclear physics research [1]. These are major motivations that drive the developments of present and planned rare-isotope facilities. Studying the evolution of binding energies for the ground and first few excited states in atomic nuclei from the valley of stability to the drip line (where the next isotope is unbound with respect to the previous one) is essential to achieve these endeavours. Understanding these trends and providing reliable predictions for nuclei that cannot be accessed experimentally require a detailed understanding of the properties of the nuclear force [2,3].

In the oxygen isotopes, recent experiments have shown that the drip line occurs at the doubly magic 24 O 16 [4][5][6], as 25,26 O are unbound [7,8]. The role of tensor and three-body forces was emphasized in [9,10] to account for the emergence of the N = 16 gap at 24 O 16 and the 'early' appearance of the drip line in the O isotopic chain, respectively. On the other hand, with the exception of 28 F [11] and 30 F which are unbound, six more neutrons can be added in the F isotopic chain before reaching the drip line at 31 F 22 [12]. One can therefore speculate that the extension of the drip line between the oxygen and fluorine, as well as the odd-even binding of the fluorine isotopes, arise from a delicate balance between the two-body proton-neutron and neutron-neutron interactions, the coupling to the continuum [13] effects and the three body forces [14,15].

The study of 26 F, which is bound by only 0.80 (12) MeV [16], offers a unique opportunity to investigate several aspects of the nuclear force. The 26 F nucleus can be modeled using a simplified single-particle (s.p.) description as a closed 24 O core plus a deeply bound proton in the π0d 5/2 orbital (S π ( 25 F)≃ -15.1(3) MeV [17]) plus an unbound neutron (S ν ( 25 O)≃ 770 +20 -10 keV [7]) in the ν0d 3/2 orbital. This simplified picture arises from the fact that the first excited state in 24 O lies at 4.47 MeV [4,6] and the π0d 5/2 and ν0d 3/2 single particle energies are well separated from the other orbitals. The low-lying 26 F thus arise, to a first approximation, from the interactions of nucleons in the π0d 5/2 and ν0d 3/2 orbits.

J π = 1 + 1 , 2 + 1 , 3 + 1 , 4 + 1 states in
Present experimental knowledge concerning the members of the 26 F is as follows. A J π = 1 + 1 assignment has been proposed in [18] for the ground state of 26 F from the observation that its beta decay proceeds to the J π = 0 + 1 , J π = 2 + 1 states and a tentative J π = 0 + 2 state in 26 Ne. The half-life of 26 F was found to be 10.2±1.4 ms with a P n value of 11±4% [18]. A mass excess ∆M of 18.680(80) MeV was determined for 26 F in [16] using the time-of-flight tech-nique. The J π = 2 + 1 state was discovered at 657 (7) keV [19] from the fragmentation of 27,28 Na nuclei. In addition a charge-exchange reaction with a 26 Ne beam was used in [20] to study unbound states in 26 F. In this reaction, a neutron capture to the νd 3/2 orbital and a proton removal from the πd 5/2 (which are both valence orbitals) are likely to occur leading to the J π = 1 + 1 -4 + 1 states. The resonance observed at 271 (37) keV above the neutron emission threshold [20] could tentatively be attributed to the J π = 3 + 1 in 26 F, as it was the only state of the J π = 1 + 1 -4 + 1 which was predicted to be unbound. With the determination of the binding energies of the J π = 1 + 1 -3 + 1 states, the only missing information is the energy of the J π = 4 + 1 state. In this Letter, we demonstrate that the 4 + 1 state is isomeric and decays by competing internal transition and β decay. Its binding energy is determined and those of the 1 + 1 -2 + 1 states are re-evaluated. The comparison of the measured binding energies of the J π = 1 + 1 -4 + 1 states with two theoretical approaches, the nuclear shell model and Coupled Cluster (CC) theory, provides a stringent test of the nuclear forces, where a large proton-to-neutron binding energy asymmetry is present.

J π = 1 + 1 , 2 + 1 , 3 + 1 , 4 + 1 multiplet in
Experiment.-The 26 F nuclei were produced through the fragmentation of a 77.6 MeV/A 36 S 16+ primary beam with a mean intensity of 2 µAe in a 237 mg/cm 2 Be target. They were selected by the LISE [21] spectrometer at GANIL, in which a wedge-shaped degrader of 1066 µm was inserted at the intermediate focal plane. The produced nuclei were identified from their energy loss in a stack of Si detectors and by their time-of-flight with respect to the GANIL cyclotron radio frequency. The production rate of 26 F was 6 pps with a purity of 22% and a momentum acceptance of 2%. Other transmitted nuclei, ranked by decreasing order of production, were 28 Ne, 29 Na, 27 Ne, 24 O, 22 N and 30 Na. They were implanted in a 1 mm-thick double-sided Si stripped detector (DSSSD) composed of 256 pixels (16 strips in the X and Y directions) of 3×3 mm 2 -each located at the final focal point of LISE. This detector was used to detect the β-particles in strips i, i±1 following the implantation of a radioactive nucleus in a given pixel i. With an energy threshold of ∼80 keV in the individual strips, a β-efficiency of 64(2)% was achieved for 26 F which was implanted at central depth of the DSSSD. The β-efficiency has been determined from the comparison of the intensity of a given γ-ray belonging to the decay of 26 F gated or not on a β-ray. Four clover Ge detectors of the EX-OGAM array [22] surrounded the DSSSD to detect the γ-rays, leading to a γ-ray efficiency of 6.5% at 1 MeV.

The γ-ray spectra obtained up to 2 ms after the implantation of a radioactive nucleus are shown in Fig. 1(a). In this frame the upper (middle) spectrum is obtained by requiring that 26 F (all except 26 F) precedes the detection of a γ ray. A delayed γ-ray transition at 643.4(1) keV is clearly observed after the implantation of 26 F. The bottom spectrum of Fig. 1 (a) is operated in similar condition than the top one, with the additional requirement that γ-rays are detected in coincidence with a β transition. As the 643.4(1) keV is not in coincidence with β particles it must correspond to an internal transition (IT ) de-exciting an isomeric state in 26 F, which has a half-life of 2.2(1) ms (see Fig. 1(b)). This isomer is likely the 4 + state we are searching for. It either decays directly to the 1 + ground state, hereby establishing the 4 + state at 643.4(1) keV. Alternatively, the 643.4(1) keV energy may correspond, but with a weak level of confidence, to the 657 (7) keV state observed in [19]. In this hypothesis, the isomerism of the 4 + state would be due to the emission of a very low energy 4 + → 2 + transition (up to 10 keV to ensure having a long-lived isomer), then followed by the 2 + → 1 + transition. In either case, the excitation energy of the 4 + state lies at approximately 650 (10) keV.

The decay of this 4 + state occurs through a competition between an internal transition (IT ) and β-decay to two states in 26 Ne. The half-lives corresponding to the IT (2.2(1) ms) as well as to the 1499.1(4) keV (2.4(2)ms) and 1843.4(8) keV (2(1)ms) peaks of Fig. 1(c) are the same. These two transitions are seen in mutual coincidences, as well as with the 2017.6(3) keV γ-ray, previously assigned to the 2 + 1 → 0 + 1 transition in 26 Ne in [18]. This establishes two levels at 3516.7(4) keV and 5360.1 (9) keV in 26 Ne as shown in Fig. 2. Following the Gamow-Teller β-decay selection rules the 4 + isomer should mainly proceed to the J π = 4 + 1 state in the vibrator nucleus 26 , which we attribute to the 3516.7(4) keV state.

All other observed transitions in Fig. 1(c) from 26 F belong to the decay of the 1 + ground state, as their halflives differ significantly from that of the 4 + isomeric state. The two γ-ray transitions at 1672.5(3) keV and 1797.1(4) keV were found to be in coincidence with the 2017.6(3) keV transition, but not in mutual coincidence. This establishes two levels at 3690.1(4) keV and 3814.7 (5) keV which have compatible half-lives of 7.7(2) ms, and 7.8(5) ms, respectively. These states presumably belong to the two-phonon multiplet of states J π = 0 + 2 , 2 + 2 , 4 + 1 among which the 3516.7(4) keV one was assigned to J π = 4 + 1 (see above). Using in-beam γ-ray spectroscopy from the fragmentation of a 36 S beam [23], the feeding of the 3516.7(4) keV level was the largest, that of the 3689.8(4) keV state was weaker, while the state at 3814.7(5) keV was not fed. As this method mainly produces Yrast states, i.e. states having the highest spin value in a given excitation energy range, we ascribe J π = 2 + 2 to the state at 3690.1(4) keV, in accordance with [24], and J π = 0 + 2 to the state at 3814.7(5) keV. The fitting of the decay half-lives must include the direct 1 + 1 decay of 26 F as well as the partial feeding from the 4 + 1 → 1 + 1 transitions. This leads to a growth at the beginning of the time spectrum (Fig. 1 (b) for the 1673 keV γ-ray) which depends on the isomeric ratio R and on the internal tran- The β feedings derived from the observed γ-ray intensities are given in Fig. 2. In the β-delayed neutron branch of 26 F to 25 Ne, some levels observed in [18,25,26] are confirmed, while a new state is proposed at 3114.1 (8) keV as the 1413.2(7) keV and 1700.9(4) keV γ-rays are in coincidence and the summed γ-ray energy is observed at 3116(2) keV. A P n value of 16( 4)% (consistent with P n =11(4)% [18]) is extracted for 26 F from the observation of the 979.7 keV γ-ray in the grand-daughter nucleus 25 Na whose branching ratio of 18.1(19)% was determined in [27]. We therefore adopt a mean value of P n =13.50(40)% for 26 F. The proposed level scheme and branching ratios agree relatively well with the shell-model calculation shown on the right side of Fig. 2.

The discovery of this new isomer has an important consequence on the determination of the atomic mass of the 26 F ground state as well on the interpretation of the oneneutron knock-out cross sections from 26 F of Ref. [28]. It is very likely that the measured atomic mass of Ref. [16] corresponds to a mixture of the ground and the isomeric states (unknown at that time). As the 26 F nuclei were produced in the present work and that of [16] in similar fragmentation reactions involving a large number of removed nucleons, we can reasonably assume that the 26 F isomeric ratio is the same in the two experiments. The shift in the 26 F atomic mass as a function of the isomeric ratio R amounts to -6.43 keV/%, which for R=42(8)% yields -270(50) keV.

Discussion.-The comparison between the experimental binding energies of these states can now be made with two theoretical approaches, the nuclear shell model and CC theory. The experimental (calculated) interactions elements arising from the coupling between a d 5/2 proton and a d 3/2 neutron, labeled Int(J), are extracted from the experimental (calculated) binding energies BE as Int(J) = BE( 26 F) J -BE( 26 F free ).

In this expression BE( 26 F f ree ) corresponds to the binding energy of the 24 O+1p+1n system, in which the valence proton in the d 5/2 orbit and the neutron in the d 3/2 orbit do not interact. It can be written as

BE( 26 F free ) = BE( 25 F) 5/2 + + BE( 25 O) 3/2 + -BE( 24 O) 0 + .
Using the relative binding energy of +0.77 +20 -10 MeV [7] between 24 O and 25 O , the measured atomic masses in 25 F and 26 F [16], and the shift in energy due to the isomeric content (see above) it is found that the experimental value of Int( 1) is -1.85 (13) MeV. The values of Int(2)= -1.19 (14) MeV and Int(4)=-1.21 (13) MeV are obtained using the J π = 2 + 1 and J π = 4 + 1 energies of 657 (7)keV and 643.4(1) keV, respectively. A value of Int(3)=-0.49(4) MeV is derived from the energy of the J π = 3 + 1 resonance with respect to the 25 F ground state. In the shell-model calculations of Refs. [29,30], the two-body matrix elements corresponding to interactions in the sd valence space are fitted to reproduce properties of known nuclei. Applying these interactions to nuclei not included in the global fits (such as bound and unbound states in 26 F) implies that shell-model calculations towards the drip lines can be viewed as predictions. Due to the strong coupling to the continuum, and a likely absence of many-body correlations not included in the fits, these interactions may fail in reproducing properties of nuclei like 26 F. Owing to its simple structure, 26 F provides a unique possibility to probe the strength of the proton-neutron interaction close to the drip line. The wave functions of the J π = 1 + 1 -4 + 1 states are composed of mainly (80 -90%) pure π0d 5/2 ⊗ ν0d 3/2 component. By calculating all states in the J π = 1 + 1 -4 + 1 multiplet, it can be seen in Fig. 3 that the J π = 1 + 1 state is less bound than calculated by about 17% (8%) and that the multiplet of experimental states is compressed by about 25% (15%) compared with the USDA (USDB) calculations. This points to a weakening of the residual interactions, which caused the energy splitting between the members of the multiplet.

We have also performed microscopic CC [31,32] calculations for 26 F. This method is particularly suited for nuclei with closed (sub-)shells, and their nearest neighbors. Moreover, CC theory can easily handle nuclei in which protons and neutrons have significantly different binding energies. To estimate the π0d 5/2 -ν0d 3/2 interaction energy (Int(J)), we use CC theory with singles and doubles excitations with perturbative triples corrections [33,34] for the closed-shell nucleus 24 O, the particleattached CC method for 25 O and 25 F [35] and the twoparticle attached formalism for 26 F [36]. We employ interactions from chiral effective field theory [37]. The effects of three-nucleon forces are included as corrections to the nucleon-nucleon interaction by integrating one nucleon in the leading-order chiral three-nucleon force over the Fermi sphere with a Fermi momentum k F in symmetric nuclear matter [38]. The parameters recently established in the oxygen chain [15] are adopted in the present work. We use a Hartree-Fock basis built from N max = 17 major spherical oscillator shells with the oscillator frequency ω = 24 MeV. This is sufficiently large to achieve convergence of the calculations for all isotopes considered. Using two-body nucleon-nucleon forces we get the ground-state energy of 26 F at -173.2 MeV which is underbound by ∼ 11 MeV compared to experiment. However, the relative spectra for the excited states are in fair agreement with experiment (see Fig. 3). In order to account for the coupling to the continuum in 26 F, we use a real Woods-Saxon basis for the ν1s 1/2 and ν0d 3/2 partial waves [39]. The inclusion of continuum effects and threenucleon forces improve the situation, the ground state energy is at -177.07 MeV, and the low-lying spectra is in very good agreement with experiment. The J π = 3 + state in 26 F is a resonance and to compute this state we need a Gamow-Hartree-Fock basis [40]. We are currently working on generalizing the two-particle attached CC implementation to a complex basis. Therefore, the interaction energy of the J = 3 state is not shown in Fig. 3. Consistently with the shell-model calculations described above, a simple picture emerges from the microscopic CC calculations: about 85% of the 1 + -4 + wave functions are composed of 1s0d-shell components, in which configurations consisting of the π0d 5/2 and ν0d 3/2 s.p. states play a major role.

Conclusions.-To summarize, a new J π = 4 + 1 isomer with a 2.2(1) ms half-life has been discovered at 643.4(1) keV. Its isomeric decay to the J π = 1 + 1 ground state and β-decay to the J π = 4 + 1 state in 26 Ne were observed. Gathering the β-decay branches observed from the J π = 1 + 1 and J π = 4 + 1 states, partial level schemes of 26 Ne and 25 Ne were obtained. In addition, the 26 F nucleus is a benchmark case for studying proton-neutron interactions far from stability. The experimental states J = 1 + -4 + arising from the πd 5/2 ⊗ νd 3/2 coupling in 26 9 F are more compressed than the USDA and USDB shell model results. The experimental J π = 1 + 1 , 2 + 1 , 4 + 1 states are less bound as well. These two effects point to a dependence of the effective two-body interaction used in the shell model as a function of the proton-to-neutron binding energy asymmetry. Coupled-cluster calculations including three-body forces and coupling to the particle continuum are in excellent agreement with experiment for the bound low-lying states in 26 F.

Institute for Physics with Exotic Nuclei) under DOE grant number DE-FG02-10ER41700. This research used computational resources of the National Center for Computational Sciences, the National Institute for Computational Sciences, and the Notur project in Norway. The CENBG is greatly acknowledged for the loan of the DSSSD detector. A. L. thanks V. Tripathi for communicating information which we used for calibration purposes.

FIG. 1 :

 1 FIG. 1: (Color online) (a): γ-ray spectra obtained up to 2 ms after the implantation of 26 F (upper spectrum), or after the implantation of any nucleus except 26 F (middle spectrum). The bottom spectrum shows the β-gated γ-rays following the implantation of 26 F. (b) Time spectra between implanted 26 F and γ-rays, from which half-lives were deduced. The 643.4(1) keV and 4 + → 2 + 1 (1499.1(4) keV) transitions have the same half-life, while the one gated on the 2 + 2 → 2 + 1 (1672.5(3) keV) transition has a larger half-life. (c): β-gated γ-ray spectrum following the implantation of 26 F up to 30 ms. Symbols and colors indicate which lines correspond to the β-decay of the 1 + ( ,black) and 4 + ( , red) or to the β delayed-neutron branch ( , blue). The same color codes are used in the decay scheme of Fig. 2. Two lines ( * , green) could not be placed in the decay scheme of 26 F.

FIG. 2 :

 2 FIG. 2: (Color on line) Decay scheme obtained from the decays of the 4 + (red) and 1 + states (black) in 26 F to 26 Ne and 25 Ne (blue). Shell model predictions obtained with the USDB interaction are shown in the right hand side.

FIG. 3 :

 3 FIG. 3: (Color online) Calculated and experimental interaction energies Int(1 -4) in MeV in 26 F. Shell-model calculations are shown in the first column using the USDA or USDB interactions, while the third column shows results obtained with CC calculations. Experimental results are in the center. The thickness of the lines corresponds to ±1σ error bar.

Acknowledgments

This work was partly supported by the Office of Nuclear Physics, U.S. Department of Energy (Oak Ridge National Laboratory), the NSF grant PHY-1068217, the OTKA contract K100835, the grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, PN-II-RU-TE-2011-3-0051 as well as by FUSTIPEN (French-U.S. Theory