S Raulot

A Savo

On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain *

Keywords: AMS 2000: 58J50, 35P15 Manifolds with boundary, Differential forms, Dirichlet-to-Neumann operator, Eigenvalues

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain Ω in Euclidean space in terms of the isoperimetric ratio Vol(∂Ω)/Vol(Ω).

Introduction

Let (Ω n+1 , g) be an (n + 1)-dimensional compact and connected Riemannian manifold with smooth boundary Σ. The Dirichlet-to-Neumann operator on functions associates, to each function defined on the boundary, the normal derivative of its harmonic extension to Ω. More precisely, if f ∈ C ∞ (Σ), its harmonic extension f is the unique smooth function on Ω satisfying ∆ f = 0 in Ω, f = f on Σ and the Dirichlet-to-Neumann operator T [0] is defined by:

T [0] f := - ∂ f ∂N
where N is the inner unit normal to Σ. It is a well known result (see [START_REF] Taylor | Partial Differential Equations II[END_REF] for example) that T [0] is a first order elliptic, non-negative and self-adjoint pseudo-differential operator with discrete spectrum

0 = ν 1,0 (Ω) < ν 2,0 (Ω) ≤ ν 3,0 (Ω) ≤ • • • ր ∞.
As Ω is connected, ν 1,0 (Ω) = 0 is simple, and its eigenspace consists of the constant functions. The first positive eigenvalue has the following variational characterization:

ν 2,0 (Ω) = inf Ω |df | 2 Σ f 2 : f ∈ C ∞ (Ω) \ {0}, Σ f = 0 . ( 1
)
The study of the spectrum of T [0] was initiated by Steklov in [START_REF] Steklov | Sur les problèmes fondamentaux de la physique mathématique[END_REF]. We note that the Dirichlet-to-Neumann map is closely related to the problem of determining a complete Riemannian manifold with boundary from the Cauchy data of harmonic functions. Indeed, a striking result of Lassas, Taylor and Uhlmann [START_REF] Lassas | The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary[END_REF] states that if the manifold Ω is real analytic and has dimension at least 3, then the knowledge of T [0] determines Ω up to isometry.

It can be easily seen that the eigenvalues of the Dirichlet-to-Neumann map of the unit ball B n+1 in R n+1 are ν k,0 = k, with k = 0, 1, 2, ... and the corresponding eigenspace is given by the vector space of homogeneous harmonic polynomials of degree k restricted to the sphere ∂B n+1 .

The Dirichlet-to-Neumann operator on forms

In [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF], we extend the definition of the Dirichlet-to-Neumann map T [0] acting on functions to an operator T [p] acting on Λ p (Σ), the vector bundle of differential p-forms of Σ = ∂Ω for 0 ≤ p ≤ n. This is done as follows. Let ω be a form of degree p on Σ, with p = 0, 1, . . . , n. Then there exists a unique p-form ω on Ω such that:

∆ ω = 0

J ⋆ ω = ω, i N ω = 0.
Here ∆ = dδ + δd is the Hodge Laplacian acting on Λ p (Ω) (the bundle of p-forms on Ω) J ⋆ : Λ p (Ω) → Λ p (Σ) is the restriction map and i N is the interior product of ω with the inner unit normal vector field N . The existence and uniqueness of the form ω (called the harmonic tangential extension of ω) is proved, for example, in Schwarz [START_REF] Schwarz | Hodge Decomposition-A method for solving boundary value problems[END_REF]. We let:

T [p] ω = -i N d ω.
Then T [p] : Λ p (Σ) → Λ p (Σ) defines a linear operator, the (absolute) Dirichlet-to-Neumann operator, which reduces to the classical Dirichlet-to-Neumann operator T [0] acting on functions when p = 0. We proved in [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF] that T [p] is an elliptic self-adjoint and non-negative pseudodifferential operator, with discrete spectrum

0 ≤ ν 1,p (Ω) ≤ ν 2,p (Ω) ≤ . . .
tending to infinity. Note that ν 1,p (Ω) can in fact be zero: it is not difficult to prove that KerT [p] is isomorphic to H p (Ω), the p-th absolute de Rham cohomology space of Ω with real coefficients.

The operator T [p] belongs to a family of operators first considered by G. Carron in [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF]. Other Dirichlet-to-Neumann operators acting on differential forms, but different from ours, were introduced by Joshi and Lionheart in [START_REF] Joshi | An inverse boundary value problem for harmonic differential forms[END_REF], and Belishev and Sharafutdinov in [START_REF] Belishev | Dirichlet to Neumann operator on differential forms[END_REF]. In fact, our operator T [p] appears in a certain matrix decomposition of the Joshi and Lionheart operator (see [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF] for complete references). However, one advantage of our operator is its self-adjointness, which permits to study its spectral and variational properties. In particular one has (see [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF]):

ν 1,p (Ω) = inf Ω |dω| 2 + |δω| 2 Σ |ω| 2 : ω ∈ Λ p (Ω) \ {0}, i N ω = 0 on Σ . (2)
For p = 0, . . . , n, we also have a dual operator T

[p] D : Λ p (Ω) → Λ p (Ω) with eigenvalues ν D k,p (Ω) = ν k,n-p (Ω) (for its definition, we refer to [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF]). Here we just want to observe that:

ν D 1,p (Ω) = inf Ω |dω| 2 + |δω| 2 Σ |ω| 2 : ω ∈ Λ p+1 (Ω) \ {0}, J ⋆ ω = 0 on Σ . (3)
In [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF], we obtained sharp upper and lower bounds of ν 1,p (Ω) in terms of the extrinsic geometry of its boundary: let us briefly explain the main lower bound. Fix x ∈ Σ and consider the principal curvatures η 1 (x), . . . , η n (x) of Σ at x; if p = 1, . . . , n and 1

≤ j 1 < • • • < j p ≤ n is a multi-index, we call the number η j 1 (x) + • • • + η jp (x) a p-curvature of Σ. We set: σ p (x) = inf{η j 1 (x) + • • • + η jp (x) : 1 ≤ j 1 < • • • < j p ≤ n} σ p (Σ) = inf{σ p (x) : x ∈ Σ}
and say that Σ is p-convex if σ p (Σ) ≥ 0 that is, if all p-curvatures of Σ are non-negative. For example Σ is 1-convex if and only if it is convex in the usual sense, and it is n-convex if and only if it is mean-convex (that is, it has non-negative mean curvature everywhere).

We then proved that for a compact domain Ω in R n+1 with p-convex boundary, one has

ν 1,p (Ω) > n -p + 2 n -p + 1 σ p (Σ) (4)
for 0 ≤ p < n+1 2 and

ν 1,p (Ω) ≥ p + 1 p σ p (Σ) (5)
for (n + 1)/2 ≤ p ≤ n. The inequality (4) is never sharp, but (5) is sharp for Euclidean balls and actually equality characterizes the ball when p > (n + 1)/2. For all this, and for similar inequalities in Riemannian manifolds we refer to [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF].

In this paper we continue the study of the spectral properties of T [p] . Namely:

• We compute the whole spectrum of T [p] and describe its eigenforms on the unit ball in R n+1 .

• We give a sharp upper bound for the first eigenvalue of T [p] on Euclidean domains, in terms of the isoperimetric ratio Vol(Σ) Vol(Ω) .

It is perhaps worth noticing that in dimension 3 we have the following interpretation of ( 2) and (3) in terms of vector fields. If Ω is a bounded domain in R 3 , then for all vector fields X on Ω which are tangent to the boundary Σ one has:

Ω |divX| 2 + |curlX| 2 ≥ ν 1,1 (Ω) Σ |X| 2 , (6)
with equality iff X is harmonic and its dual 1-form restricts to an eigenform of T [1] associated to ν 1,1 (Ω). Recall that, on a three dimensional Riemannian manifold, the curl of a vector field X is the vector field defined by curlX = ⋆ dX ♯ ♯

where ♯ denotes the musical isomorphism between the tangent space and the cotangent space.

Combining [START_REF] Joshi | An inverse boundary value problem for harmonic differential forms[END_REF] with the estimate (4) gives, for all Euclidean domains with convex boundary:

Ω |divX| 2 + |curlX| 2 > 3 2 σ 1 (Σ) Σ |X| 2 , ( 7
)
where X is any vector field tangent to Σ and σ 1 (Ω) is a lower bound of the principal curvatures of Σ. As a by-product of the calculation in Section 1.2, we will see that [START_REF] Lassas | The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary[END_REF] is almost sharp, because for all vector fields on the unit ball (tangent to the boundary) we have the sharp inequality

B 3 |divX| 2 + |curlX| 2 ≥ 5 3 ∂B 3 |X| 2 (8)
(for the description of the minimizing vector fields for the inequality (8) we refer to Section 2.2).

Similarly, let X be a vector field on a Euclidean domain Ω which is normal to the boundary. Then:

Ω |divX| 2 + |curlX| 2 ≥ ν 1,2 (Ω) Σ |X| 2 , (9)
with equality iff X is harmonic and the Hodge-star of its dual 1-form restricts to an eigenform of T [2] associated to ν 1,2 (Ω). Using (5), we see that, if Σ is mean-convex:

Ω |divX| 2 + |curlX| 2 ≥ 3 2 σ 2 (Σ) Σ |X| 2 . ( 10
)
Note that σ 2 (Σ) = 2H, where H is a lower bound of the mean curvature of Σ. In this situation, our inequality is sharp and is an equality if and only if Ω is a ball in R 3 , in which case the lower bound is 3, and X is a multiple of the radial vector field r ∂ ∂r (see Section 2.2). We end this discussion by remarking that inequalities [START_REF] Lassas | The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary[END_REF] and [START_REF] Savo | On the first Hodge eigenvalue of isometric immersions[END_REF] continue to be true for all bounded domains in three-dimensional Riemannian manifolds with non-negative Ricci curvature.

The spectrum of T [p] on the unit Euclidean ball in R n+1

In this section we compute the spectrum of T [p] on the unit ball B n+1 in R n+1 . Let ∆ (resp. ∆) denote the Hodge Laplacian acting on p-forms of S n (resp. R n+1 ). It will turn out that ∆ and T [p] have the same eigenspaces: so we describe in details the eigenspaces of ∆.

We start from the case p = 0, which is classical. The operator ∆ is simply the Laplacian on functions, and it is a well-known fact that its eigenfunctions are restrictions to S n of homogeneous polynomial harmonic functions on R n+1 . Precisely, let P k,0 be the vector space of all polynomial functions on R n+1 of homogeneous degree k, where k = 0, 1, 2, . . . , and set:

H k,0 = {f ∈ P k,0 : ∆f = 0}.
Then the spectrum of ∆ acting on functions of S n is given by the eigenvalues

µ ′′ k,0 = k(n + k -1), k = 0, 1, 2, . . . ,
with multiplicity M k,0 = dim(H k,0 ) and associated eigenspace J ⋆ (H k,0 ). Now fix an eigenfunction f ∈ J ⋆ (H k,0 ) so that, by assumption, its harmonic extension f is a harmonic polynomial of homogeneous degree k. It is very easy to see that T [0] f = kf : so, f is also a Dirichlet-to-Neumann eigenfunction associated to the eigenvalue k. A standard density argument shows that these are all possible eigenvalues of T [0] . Therefore we have the following well-known result:

Theorem 1. The spectrum of the Dirichlet-to-Neumann operator T [0] on the unit ball in R n+1 consists of the eigenvalues ν ′′ k,0 = k, where k = 0, 1, 2, . . . , each with multiplicity M k,0 = dim(H k,0 ) and associated eigenspace J ⋆ (H k,0 ). Now assume p = 1, . . . , n. The calculation of the spectrum of the Hodge Laplacian on the sphere was first done in [START_REF] Gallot | Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne[END_REF]. We follow the exposition in [START_REF] Ikeda | Spectra and eigenforms of the Laplacian on S n and P n (C)[END_REF].

As the Hodge Laplacian commutes with both the differential and the codifferential, it preserves closed (resp. co-closed) p-forms. Moreover, any exact eigenform is the differential of a co-exact eigenform associated to the same eigenvalue. In the following, we denote by {µ ′ k,p } (resp. {µ ′′ k,p }) the spectrum of the Hodge Laplacian restricted to closed (resp. co-closed) p-forms on the sphere S n .

We can write a p-form on R n+1 as:

ω = i 1 <•••<ip ω i 1 ...ip dx i 1 ∧ • • • ∧ dx ip
and we say that ω is polynomial if each component ω i 1 ...ip is a polynomial function. Now let P k,p be the vector space of polynomial p-forms of homogeneous degree k ≥ 0 on R n+1 and set:

     H k,p = {ω ∈ P k,p : ∆ω = 0, δω = 0}, H ′ k,p = {ω ∈ H k,p : dω = 0} H ′′ k,p = {ω ∈ H k,p : i Z ω = 0}
where Z is the radial vector field

Z = r ∂ ∂r = n+1 j=1 x j ∂ ∂x j . On the boundary, Z = -N . It turns out that H k,p = H ′ k,p ⊕ H ′′ k,p and d : H ′′ k,p → H ′ k-1,p+1 is a linear isomorphism for all k ≥ 1. We set M k,p = dim(H ′′ k,p
) (this number can be computed by representation theory, see Theorem 6.8 in [START_REF] Ikeda | Spectra and eigenforms of the Laplacian on S n and P n (C)[END_REF]).

By the Hodge-de Rham decomposition, any Hodge-Laplace eigenspace splits into the direct sum of its co-exact, exact and harmonic parts. But in the range 1 ≤ p ≤ n the only harmonic forms occur in degree p = n, and are multiples of the volume form of S n . Moreover, for p = n the coexact part is reduced to zero. Then, there is a spectral resolution of L 2 (Λ p (S n )) which consists of the following three types of Hodge-Laplace eigenforms: 1 ≤ p ≤ n -1 and the eigenform ξ is co-exact; 1 ≤ p ≤ n and the eigenform ξ is exact; p = n and ξ is a multiple of the volume form of S n . Correspondingly, we have the following three families of eigenvalues (see [START_REF] Ikeda | Spectra and eigenforms of the Laplacian on S n and P n (C)[END_REF]):

• If 1 ≤ p ≤ n -1 and ξ is co-exact the associated eigenvalues are given by the family

µ ′′ k,p = (k + p)(n + k -p -1) k = 1, 2, . . .
with associated eigenspace J ⋆ (H ′′ k,p ) and multiplicity M k,p . • If 1 ≤ p ≤ n and ξ is exact the associated eigenvalues are given by the family

µ ′ k,p = (k + p -1)(n + k -p) k = 1, 2, . . . with associated eigenspace J ⋆ (H ′ k-1,p
) and multiplicity M k,p-1 . • If p = n and ξ is the volume form of S n we have the associated eigenvalue µ ′′ 1,n = 0. In conclusion, eigenforms of the Hodge Laplacian are suitable restrictions to S n of harmonic, co-closed polynomial forms. In Section 2.1 we will prove that any Hodge-Laplace eigenform of S n is also a Dirichlet-to-Neumann eigenform (associated to a different eigenvalue). This will imply the following calculations.

Theorem 2. Let 1 ≤ p ≤ n -1.
The spectrum of the Dirichlet-to-Neumann operator T [p] on the unit ball of R n+1 is given by the following two families of eigenvalues

{ν ′′ k,p }, {ν ′ k,p } indexed by a positive integer k = 1, 2, . . . :    ν ′′ k,p = k + p with multiplicity M k,p , ν ′ k,p = (k + p -1) n + 2k + 1 n + 2k -1 with multiplicity M k,p-1 .
The eigenspace associated to ν ′′ k,p is J ⋆ (H ′′ k,p ) and consists of co-exact forms. The eigenspace associated to ν ′ k,p is J ⋆ (H ′ k-1,p ) and consists of exact forms. In degree p = n: Theorem 3. The spectrum of the Dirichlet-to-Neumann operator T [n] on the unit ball of R n+1 consists of the eigenvalues:

ν ′′ 1,n = n + 1, with multiplicity one (the associated eigenspace being spanned by the volume form of S n ) and, for k ≥ 1:

ν ′ k,n = (k + n -1) n + 2k + 1 n + 2k -1 , with multiplicity M k,n-1 and associated eigenspace J ⋆ (H ′ k-1,n ).
From this result we deduce that the lowest eigenvalue of T [n] is ν 1,n = n + 1 with multiplicity one for n ≥ 2 and three for n = 1. Indeed, in this last situation, we have ν ′′ 1,1 = ν ′ 2,1 = 2 and the corresponding eigenspace is spanned by the the volume form v of S 1 , dx and dy where d denotes the differential on S 1 . From the above results we obtain: Corollary 4. If ν 1,p denotes the first eigenvalue of T [p] on the unit ball in R n+1 , then:

ν 1,p =      n + 3 n + 1 p if 1 ≤ p ≤ n + 1 2 p + 1 if n + 1 2 ≤ p ≤ n.
The proof of Theorems 2 and 3 splits into two parts. In a first step (see Lemma 6 and 8), we compute the expression of the Hodge Laplacian on p-forms for rotationally symmetric manifolds.

Then in Section 2 we apply these computations to construct the tangential harmonic extension to the unit ball of any p-eigenform of the Hodge Laplacian on S n = ∂B n+1 .

A sharp upper bound by the isoperimetric ratio

As shown in [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF], the existence of a parallel p-form implies upper bounds of the Dirichlet-to-Neumann eigenvalues by the isoperimetric ratio Vol(Σ) Vol(Ω) . These bounds are never sharp, unless p = n. In that case one has, for any (n + 1)-dimensional Riemannian domain Ω:

ν 1,n (Ω) ≤ Vol(Σ) Vol(Ω) , (11)
which is sharp for Euclidean balls. The proof of ( 11) is easily obtained by applying the min-max principle (2) to the test n-form ω = ⋆dE, where E is the mean-exit time function, solution of the problem ∆E = 1 on Ω,

E = 0 on Σ. ( 12
)
If Ω is a Euclidean domain and equality holds in [START_REF] Schwarz | Hodge Decomposition-A method for solving boundary value problems[END_REF], then a famous result of Serrin implies that Ω is a ball. The equality case for general Riemannian domains is an open (and interesting) problem.

In this paper we generalize inequality [START_REF] Schwarz | Hodge Decomposition-A method for solving boundary value problems[END_REF] to any degree p = 0, . . . , n when Ω is a Euclidean domain; in the range p ≥ (n + 1)/2 the estimate is sharp and it also turns out that the ball is the unique maximizer. Namely, we prove:

Theorem 5.
Let Ω be a domain in R n+1 and p = 1, . . . , n. Then:

ν 1,p (Ω) ≤ p + 1 n + 1 Vol(Σ) Vol(Ω) .
Equality holds iff p ≥ (n + 1)/2 and Ω is a Euclidean ball.

We note that the corresponding inequality for the first positive eigenvalue of the Dirichlet-to-Neumann operator on functions:

ν 2,0 (Ω) ≤ 1 n + 1 Vol(Σ) Vol(Ω)
, has been recently proved by Ilias and Makhoul in [START_REF] Ilias | A Reilly inequality for the first Steklov eigenvalue[END_REF], but their approach does not extend to higher degrees.

The Dirichlet-to-Neumann spectrum of the unit Euclidean ball

We first give an expression of the Hodge Laplacian on the unit ball B n+1 in R n+1 . Note that B n+1 = [0, 1] × S n with the metric dr 2 ⊕ r 2 ds 2 n , where ds 2 n is the canonical metric on S n . We consider p-forms on the ball B n+1 of the following type:

ω(r, x) = Q(r)ξ(x) + P (r)dr ∧ η(x),
where η ∈ Λ p-1 (S n ), ξ ∈ Λ p (S n ), and P, Q are smooth functions on (0, 1). We will write for short ω = Qξ + P dr ∧ η.

Then we have: Lemma 6. Let d and δ denote, respectively, the differential and the co-differential acting on S n . Let ω be a p-form as in (13), then:

∆ω = ω 1 + dr ∧ ω 2 ,
where:

ω 1 = Q r 2 ∆ξ -Q ′′ + n -2p r Q ′ ξ - 2P r dη, ω 2 = P r 2 ∆η -P ′ + n -2p + 2 r P ′ η - 2Q r 3 δξ.
For the proof, we refer to Lemma 8 in Section 2.3.

Proof of Theorems 2 and 3

It is enough to show that any Hodge-Laplace eigenform of S n is also a Dirichlet-to-Neumann eigenform, associated to one of the eigenvalues ν ′ k,p or ν ′′ k,p of Theorems 2 and 3. In fact, as the direct sum of all the Hodge-Laplace eigenspaces is dense in L 2 (Λ p (S n )), the list of Dirichlet-to-Neumann eigenvalues we have just found is complete, and the theorems follow.

We now use the classification of the eigenspaces of the Hodge Laplacian done in Section 1.2. It is then enough to prove the following:

Proposition 7.
a) Assume 1 ≤ p ≤ n -1, and let ξ be a co-exact Hodge p-eigenform on S n associated to

µ ′′ k,p = (k + p)(n + k -p -1)
for some k ≥ 1. Then ξ is also a Dirichlet-to-Neumann eigenform associated to the eigenvalue

ν ′′ k,p = k + p.
b) Assume p = n and let ξ be a multiple of the volume form of S n . Then ξ is also a Dirichletto-Neumann eigenform associated to the eigenvalue ν ′′ 1,n = n + 1. c) Assume 1 ≤ p ≤ n and let ξ be an exact Hodge-Laplace p-eigenform associated to

µ ′ k,p = (k + p -1)(n + k -p)
for some k ≥ 1. Then ξ is also a Dirichlet-to-Neumann p-eigenform associated to

ν ′ k,p = (k + p -1) n + 2k + 1 n + 2k -1 .
For the proof, we explicitly determine in all three cases the tangential harmonic extension of ξ using Lemma 6.

Proof of a) Let us compute the tangential harmonic extension ξ of ξ. We let ξ = Qξ with Q = Q(r) to be determined so that ∆ ξ = 0. Note that i N ξ = 0; moreover J ⋆ ξ = ξ whenever Q(1) = 1. As η = 0 = δξ, Lemma 6 gives

∆(Qξ) = µ ′′ k,p Q r 2 -Q ′′ - n -2p r Q ′ ξ hence Qξ is harmonic if Q(r) satisfies: r 2 Q ′′ + (n -2p)rQ ′ -µ ′′ k,p Q = 0. ( 14
)
It is straightforward to check that a solution (at least C 2 ) on [0, 1] is given by Q(r) = r k+p . The tangential harmonic extension of ξ to the ball is then:

ξ = r k+p ξ.
Recall that, by definition, T [p] (ξ) = -i N d ξ. Now:

d ξ = r k+p dξ + (k + p)r k+p-1 dr ∧ ξ.
On the boundary we have r = 1 and dr(N ) = -1, then:

i N d ξ = -(k + p)ξ so that T [p] (ξ) = (k + p)ξ
as asserted.

Proof of b) Note that ξ is co-closed and the associated Hodge-Laplace eigenvalue is µ ′′ 1,n = 0. Proceeding as in a), one finds that the tangential harmonic extension of ξ is ξ = r n+1 ξ and therefore T [n] (ξ) = (n + 1)ξ.

Proof of c)

We first observe that a Hodge-Laplace exact p-eigenform ξ associated to µ ′ k,p is the differential of a co-exact (p -1)-eigenform φ, associated to the same eigenvalue µ ′′ k,p-1 = µ ′ k,p :

ξ = dφ.
From Lemma 6, we see that the p-form Q dφ + P dr ∧ φ is harmonic iff:

         µ ′′ k,p-1 (rP -2Q) = r 3 P ′ + n -2p + 2 r P ′ µ ′′ k,p-1 Q -2rP = r 2 Q ′′ + n -2p r Q ′ . ( 15
)
We observe that, by part a) of the proposition, the (p -1)-form φ is an eigenform for T [p-1] associated to ν ′′ k,p-1 = k + p -1 (note that if p = 1 this fact follows directly from Theorem 1). Now consider the p-form:

η = α k,p r k+p+1 dφ -ν ′′ k,p-1 r k+p dr ∧ φ,
where:

α k,p = (ν ′′ k,p-1 ) 2 µ ′′ k,p-1 = k + p -1 n + k -p .
The p-form η satisfies: ∆ η = 0

J ⋆ η = α k,p dφ, i N η = ν ′′ k,p-1 φ, (16)
where the harmonicity follows from (15) by taking:

P (r) = -ν ′′ k,p-1 r k+p , Q(r) = α k,p r k+p+1 .
Now let φ be the harmonic tangential extension of φ, that is:

∆ φ = 0 J ⋆ φ = φ, i N φ = 0. ( 17
)
As φ is an eigenform for T [p-1] associated to ν ′′ k,p-1 , one has:

∆d φ = 0 J ⋆ d φ = dφ, i N d φ = -ν ′′ k,p-1 φ. (18)
From ( 16) and (18) one sees that the form ω = d φ + η satisfies:

∆ ω = 0

J ⋆ ω = (α k,p + 1) dφ, i N ω = 0.
From the definition of T [p] , we have:

T [p] (α k,p + 1) dφ = -i N d ω. Now: d ω = d η = ν ′′ k,p-1 + α k,p (k + p + 1) r k+p dr ∧ dφ
and, restricted to the boundary:

i N d ω = -ν ′′ k,p-1 + α k,p (k + p + 1) dφ. This means that T [p] ((α k,p + 1) dφ) = ν ′′ k,p-1 + α k,p (k + p + 1
) dφ and so ξ = dφ is a Dirichlet-to-Neumann eigenform associated to the eigenvalue

ν ′′ k,p-1 + α k,p (k + p + 1) α k,p + 1 = (k + p -1) n + 2k + 1 n + 2k -1
as asserted. This ends the proof of the proposition.

On a variational problem for vector fields

Recall the variational problems defined in ( 6) and ( 9). Let B 3 be the unit ball in R 3 . As a consequence of Corollary 4 we have that, for any vector field X on B 3 tangent to ∂B 3 = S 2 :

B 3 |divX| 2 + |curlX| 2 ≥ 5 3 S 2 |X| 2 , ( 19
) because ν 1,1 (B 3 ) = ν ′ 1,1 (B 3 ) = 5 3 .
Let us describe the minimizing vector fields. The eigenspace associated to ν ′ 1,1 is 3-dimensional, spanned by dφ, where φ is a linear function on R 3 . Hence the vector field X attains equality in (19) iff it is dual to the harmonic tangential extension of dφ. Take for example φ = x 1 . As a consequence of the calculation done in the previous section, we see that the harmonic tangential extension of dx 1 to the unit ball in R 3 is, in rectangular coordinates:

ξ = (2 -2x 2 1 + x 2 2 + x 2 3 )dx 1 -3x 1 x 2 dx 2 -3x 1 x 3 dx 3 .
Note that ξ is a polynomial (not homogeneous) 1-form. The dual vector field

X = (2 -2x 2 1 + x 2 2 + x 2 3 ) ∂ ∂x 1 -3x 1 x 2 ∂ ∂x 2 -3x 1 x 3 ∂ ∂x 3
is then a minimizer for the variational problem (19).

On the other hand, as ν 1,2 (B 3 ) = ν ′′ 1,2 (B 3 ) = 3, one sees that, for any vector field normal to S 2 we have:

B 3 |divX| 2 + |curlX| 2 ≥ 3 S 2 |X| 2 . ( 20
)
Now ν ′′ 1,2 is simple and is spanned by the volume form v of S 2 . The tangential harmonic extension of v to the unit ball is

v = x 1 dx 2 ∧ dx 3 + x 2 dx 3 ∧ dx 1 + x 3 dx 1 ∧ dx 2 .
The space of minimizing vector fields for (20) is then one-dimensional, spanned by the dual of ⋆ v, that is, by the radial vector field

Y = x 1 ∂ ∂x 1 + x 2 ∂ ∂x 2 + x 3 ∂ ∂x 3 .

Proof of Lemma 6

Assume that (M n+1 , g) is a rotationally symmetric manifold, that is

M n+1 = [0, ∞) × S n ,
endowed with the Riemannian metric

g = dr 2 ⊕ θ(r) 2 ds 2 n ,
where ds 2 n is the canonical metric on S n and θ is a smooth positive function on (0, ∞). Of course, one gets the space form of curvature -1, 0, 1 when θ = sinh r, r, sin r, respectively. To prove Lemma 6 we will take θ(r) = r. In this setting, any p-form can be split into its tangential and normal components:

ω = ω 1 + dr ∧ ω 2 ,
where ω 1 and ω 2 are forms of degrees p and p -1, respectively. We assume that we can separate the variables, that is:

ω 1 (r, x) = Q(r)ξ(x), ω 2 (r, x) = P (r)η(x),
where ξ ∈ Λ p (S n ), η ∈ Λ p-1 (S n ) and P, Q are radial functions so that ω = Qξ + dr ∧ (P η).

A suitable orthonormal frame

Fix a point (r, x) ∈ M with r ∈ R and x ∈ S n and let ( Ē1 , . . . , Ēn ) be an orthonormal frame on the sphere which is geodesic at x for the canonical metric ds 2 n . If we set Z = ∂/∂r and

E j = θ -1
Ēj for 1 ≤ j ≤ n, it is obvious that the frame (Z, E 1 , . . . , E n ) is g-orthonormal. From the Koszul formula, we compute:

∇ E j E k = -δ jk θ ′ θ Z, ∇ E j Z = θ ′ θ E j , ∇ Z E j = 0, ∇ Z Z = 0 (21)
where ∇ denotes the Levi-Civita connection on (M, g). In particular, we have

∇ E j E k = 0 if j = k and ∇ E j E j = - θ ′ θ Z, j ∇ E j E j = -n θ ′ θ Z. (22)

Computing the divergence and the differential

In this part, we first compute the divergence of a p-form using the orthonormal frame of the previous section. If ω is a p-form on M , then from the definition of the divergence and using the relations (21) and ( 22), we have:

δω(E 1 , . . . , E p-1 ) = - n j=1 E j • ω(E j , E 1 , . . . , E p-1 ) -Z • ω(Z, E 1 , . . . , E p-1 ) -(n -p + 1) θ ′ θ ω(Z, E 1 , . . . , E p-1 )
and

δω(Z, E 1 , . . . , E p-2 ) = n j=1 E j • ω(Z, E j , E 1 , . . . , E p-2 ).
Let φ be a p-form on S n . As the frame ( Ē1 , . . . , Ēn ) of Σ is geodesic at the point x ∈ Σ and the function φ( Ēj 1 , . . . , Ējp ) is constant in the r-direction for any choice of j 1 , . . . , j p , we have:

         n j=1 E j • φ(E j , E 1 , . . . , E p-1 ) = - 1 θ 2 δφ(E 1 , . . . , E p-1 ) Z • φ(E 1 , . . . , E p ) = -p θ ′ θ φ(E 1 , . . . , E p )
Since the vectors E 1 , . . . , E p can be replaced by any set of tangential vectors in the chosen frame, a straightforward calculation using the above equations shows that, for η ∈ Λ p-1 (S n ) and ξ ∈ Λ p (S n ):

       δ(dr ∧ (P η)) = dr ∧ (- P θ 2 δη) -P ′ + (n -2p + 2) θ ′ θ P η δ(Qξ) = Q θ 2 δξ. ( 23
)
For the differential, it is clear that we directly have:

d(dr ∧ (P η)) = -dr ∧ (P dη) d(Qξ) = dr ∧ (Q ′ ξ) + Q dξ. (24)
At this point, using (23) and (24), one proves, after some standard work:

Lemma 8. Let ω = Qξ + P dr ∧ η where η ∈ Λ p-1 (S n ) and ξ ∈ Λ p (S n ). Then:

∆ω = ω 1 + dr ∧ ω 2 ,
where:    Remark 9. Let B R be the geodesic ball of radius R centered at the pole of a rotationally symmetric manifold, that is

B R = {(r, x) ∈ [0, ∞) × S n : r ≤ R}.
One can construct eigenforms of the Dirichlet-to-Neumann operator on B R by solving ordinary differential equations. For example, one sees from the above expression of the Laplacian that the harmonic tangential extension of a co-closed eigenform ξ ∈ Λ p (S n ) for the Hodge Laplacian on the sphere associated with µ ′′ k,p is given by ξ = Qξ, where Q(r) is the smooth function satisfying:

   1 θ n-2p θ n-2p Q ′ ′ = µ ′′ k,p Q θ 2 , Q(R) = 1.
It follows directly from the definition that then

ν ′′ k,p = Q ′ (R)
is an eigenvalue of T [p] . Now take p = n. Then, the only possible value of µ ′′ k,n is zero, corresponding to the volume form of S n , which is parallel hence harmonic. In that case the previous equation reduces to

θ -n Q ′ ′ = 0,
hence, up to multiples, Q ′ (r) = θ n (r). Then:

ν ′′ 1,n (B R ) = θ n (R) R 0 θ n (r)dr = Vol(∂B R ) Vol(B R ) (25)
is an eigenvalue for the Dirichlet-to-Neumann operator on n-forms.

In [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF] we called a domain Ω harmonic if the mean-exit time function E of Ω, solution of the problem [START_REF] Serrin | A symmetry problem in potential theory[END_REF], has constant normal derivative on Σ: we then observed that for a harmonic domain Vol(Σ)/Vol(Ω) is always an eigenvalue of T [n] . A geodesic ball B R , being rotationally invariant, is a harmonic domain because the function E is also rotationally invariant. Therefore the above result (25) is not a surprise.

Proof of Theorem 5

Let Ω be a Riemannian domain with smooth boundary Σ and inner unit normal vector field N . Consider the shape operator of Σ, defined by S(X) = -∇ X N for all X ∈ T Σ. Then S can be extended to a self-adjoint operator S [p] acting on p-forms of Σ by the rule:

S [p] (ω)(X 1 , . . . , X p ) = p j=1
ω(X 1 , . . . , S(X j ), . . . , X p ), for all ω ∈ Λ p (Σ) and for all vectors X 1 , . . . , X p ∈ T Σ.

Let (e 1 , . . . , e n ) be an orthonormal basis of principal directions, so that S(e j ) = η j e j for all j, where η 1 , . . . , η n are the principal curvatures of Σ. Then, for any multi-index 1 ≤ j 1 < • • • < j p ≤ n one has S [p] (ω)(e j 1 , . . . , e jp ) = (η j 1 + • • • + η jp )ω(e j 1 , . . . , e jp ).

In particular, if ω is an n-form: S [n] (ω) = nHω, where H is the mean curvature of Σ.

For later use we observe that, if ξ is a p-form on Ω and L N ξ = di N ξ + i N dξ is its Lie derivative along N , then it follows directly from the definitions that, on Σ, one has (see for example Lemma 18 in [START_REF] Raulot | A Reilly formula and eigenvalue estimates for differential forms[END_REF]):

J ⋆ (L N ξ) = J ⋆ (∇ N ξ) -S [p] (J ⋆ ξ). (26)
The proof of Theorem 5 is based on the following estimates.

Proposition 10. Let ξ be an exact parallel p-form on Ω, with p = 1, . . . , n. If p = 1, then ν 2,0 (Ω)

Ω |ξ| 2 ≤ Σ |i N ξ| 2 . ( 27
)
and if p = 2, . . . , n:

ν 1,p-1 (Ω) Ω |ξ| 2 ≤ Σ |i N ξ| 2 . ( 28
)
Moreover, if equality holds in (27) or (28), then:

S [p] (J ⋆ ξ) = νJ ⋆ ξ,
where ν = ν 2,0 (Ω) when p = 1 and ν = ν 1,p-1 (Ω) when p ≥ 2.

Proof. Inequalities ( 27) and (28) were proved in [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF] and hold, more generally, only assuming that ξ is exact and satisfies dξ = δξ = 0 on Ω. Let us recall the main argument. As ξ is exact, there exists a unique co-exact (p -1)-form θ, called the canonical primitive of ξ, satisfying:

dθ = ξ on Ω,
i N θ = 0 on Σ.

When p = 1, we take θ as the unique primitive of ξ such that Σ θ = 0. We now take θ as test (p -1)-form in the min-max principles (1) and ( 2) and the inequalities (27) and (28) follow after some easy work (see [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF]). Now assume that equality holds: then θ is an eigenform associated to ν, so that i N dθ = -νJ ⋆ θ. This means:

i N ξ = -νJ ⋆ θ on Σ. Differentiating on Σ one gets d Σ i N ξ = -νJ ⋆ ξ, and in turn, as ξ is closed:

J ⋆ (L N ξ) = -νJ ⋆ ξ.
As ξ is parallel, (26) gives J ⋆ (L N ξ) = -S [p] (J ⋆ ξ) and then the assertion follows.

We can now give the proof of Theorem 5. We can assume that p ≤ n -1: in fact, the assertion for p = n is a direct consequence of Theorem 5 and Corollary 6 in [START_REF] Raulot | On the first eigenvalue of the Dirichlet-to-Neumann operator on forms[END_REF]. Let P be the family of unit length parallel vector fields on R n+1 : then P is naturally identified with S n . For V 1 , . . . , V p ∈ P consider the parallel p-form

ξ = V ⋆ 1 ∧ • • • ∧ V ⋆ p
where V ⋆ j denotes the dual 1-form of V j . Note that V ⋆ j is the differential of a linear function, so ξ is also exact. Then, denoting by ν the eigenvalue as in Proposition 10, we have from ( 27) and (28)

ν Ω |ξ| 2 ≤ Σ |i N ξ| 2 .
(29)

We wish to integrate this inequality over all (V 1 , . . . , V p ) ∈ (S n ) p . To that end, we introduce on P = S n the measure:

where ds 2 n is the canonical measure on S n . The normalization is chosen so that, at each point x ∈ R n+1 , and for all tangent vectors X, Y at x one has:

A straightforward calculation using (30) (explicitly done in Lemma 2.1 of [START_REF] Savo | On the first Hodge eigenvalue of isometric immersions[END_REF]) then gives, at each point of Ω (respectively, of Σ):

We now integrate both sides of (29) with respect to (V 1 , . . . , V p ) ∈ (S n ) p . If p = 1 we get:

and if p = 2, . . . , n we get:

, which, after replacing p by p + 1, are the inequalities stated in the theorem.

We now discuss the equality case. If equality holds then, from Proposition 10, we see that

Fix a point x ∈ Σ and an orthonormal frame (e 1 , . . . , e n ) of principal directions at x. Fix a multi-index j 1 < • • • < j p and choose:

At x, one has J ⋆ ξ(e j 1 , . . . , e jp ) = 1 and then, by the definition of S [p] :

the corresponding p-curvature at x. From (31) we then get:

This holds for all multi-indices j 1 < • • • < j p and for all x ∈ Σ: hence, all p-curvatures are constant on Σ, and equal to ν. If p < n, this immediately implies that Σ is totally umbilical, hence it is a sphere. If p = n, we have that the mean curvature of Σ is constant; by the well-known Alexandrov theorem Σ is, again, a sphere. Finally, from Theorem 2, we have that if Ω is a ball, then equality holds for ν 2,0 and for ν 1,p-1 provided that p -1 ≥ (n + 1)/2. The proof is complete.


Athanasios C Pappas
email: apappas@aua.gr

Evangelos Zoidis

Constantinos A Georgiou

Nikolaos Demiris

Peter F Surai

Konstantinos Fegeros

Athanasios ; Pappas

Athanasios C Pappas

A C Pappas

Influence of organic selenium supplementation on the

Keywords: Heavy metals -cadmium, Metals -selenium antioxidant system, cadmium, chicken, copper, iron, selenium, zinc

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Trace elements are minerals that are needed in minute quantities for the proper growth, development, and physiology of the organism. In living animal organisms, trace elements are bound to various ligands in cellular membranes, cytoplasm and blood. They are transported and stored via several proteins including but not limited to albumin, ceruloplasmin, transferrin and metallothioneins which are often non-specific and so may be sites of interaction between trace elements [START_REF] Serra-Majem | Dietary assessment methods for intakes of iron, calcium, selenium, zinc and iodine[END_REF][START_REF] Southgate | Conceptual issues concerning the assessment of nutrient bioavailability[END_REF][START_REF] Strachan | Trace elements[END_REF]). They can be classified as essential including iron (Fe), manganese (Mn), copper (Cu), zinc (Zn) and selenium(Se), probably essential nickel (Ni), vanadium (V) and cobalt (Co) both required in the diet, but that may have toxic effects at supraoptimal concentrations [START_REF] Underwood | The mineral nutrition of livestock[END_REF] and potentially toxic including but not limited to aluminum (Al) arsenic (As), cadmium (Cd), lead (Pb), mercury (Hg) not required in the diet, causing chronic negative effects at low concentrations and lethal effects at high concentrations [START_REF] Bires | Effects on enzymes and the genetic apparatus of sheep after administration of samples from industrial emissions[END_REF][START_REF] Uluozlu | Assessment of trace element contents of chicken products from turkey[END_REF]). The absorption, accumulation and toxicity of each metal may be affected by diverse factors, including interactions with other metals. Indeed, interactions between toxic and essential metals are central to mineral balance and the antioxidant defence system (López-Alonso et al. 2007;Pappas et al. 2010a;Pappas et al. 2010b).

Selenium is a trace element that plays a key role in the antioxidant defence system. It is the integral part of at least 25 selenoproteins and via their actions protects the organism from harmful actions of free radicals [START_REF] Pappas | Selenoproteins and maternal nutrition[END_REF]. Copper and Zn are two of the most abundant trace elements found in the human body and are involved in the metabolism of oxygen and the biochemistry of redox reactions [START_REF] Klotz | Role of Copper, Zinc, Selenium and Tellurium in the Cellular Defense against Oxidative and Nitrosative Stress[END_REF]. The enzyme CuZnsuperoxide dismutase (SOD) catalyzes the dismutation of superoxide, which is constantly formed during aerobic metabolism, to oxygen and hydrogen peroxide. Copper and Zn are joined in cellular defense against oxidants by Se to form a triad of trace elements that are involved in cytosolic antioxidant defense: hydroperoxides, including H 2 O 2 , are reduced to the respective alcohols or water in a reaction that is catalyzed by the selenoenzyme glutathione peroxidase (GPx) with glutathione (GSH) as the electron donor [START_REF] Klotz | Role of Copper, Zinc, Selenium and Tellurium in the Cellular Defense against Oxidative and Nitrosative Stress[END_REF]. Similarly to Cu and Zn, iron's ability to accept or donate electrons makes it central to the catalytic activity of many redox-active enzymes. Iron is an integral part of several classes of enzymes, including cytochromes, enzymes involved in the synthesis of steroid hormones, detoxification of foreign substances in the liver, synthesis of neurotransmitters and DNA synthesis and breakdown [START_REF] Gambling | Iron and copper, and their interactions during development[END_REF]. Cadmium disrupts the antioxidant enzyme system, resulting in increased oxygen radical production, which causes damage to membranous structures such as mitochondria and endoplasmic reticulum [START_REF] Schrauzer | Selenium and selenium-antagonistic elements in nutritional cancer prevention[END_REF]. Cadmium is known to induce the production of nitric oxide (NO) and reactive oxygen species (ROS) such as hydroxyl radicals (OH • ), superoxide anions radicals (O 2

•-) and H 2 O 2 . Furthermore, Cd interferes with antioxidant enzymes, causes enzyme dysfunction and lipid peroxidation [START_REF] Ognjanovic | Cadmium-induced lipid peroxidation and changes in antioxidant defense system in the rat testes: Protective role of coenzyme Q10 and Vitamin E[END_REF].

Essential trace metals may modify health risks from exposure to nonessential toxic metals. A well known example of this is the reduction of Cd toxicity after pre-treatment with Zn or the simultaneous administration of Zn with Cd [START_REF] Jemai | Protective effect of zinc supplementation on blood antioxidant defense system in rats exposed to cadmium[END_REF][START_REF] Webb | The chemistry, biochemistry and biology of cadmium[END_REF]). On the other hand, the ingestion of a diet that is deficient in a particular essential element can enhance the accumulation and toxicity of some toxic metals. Calcium, Fe and Zn deficiency, for instance, enhance susceptibility to Cd and Pb toxicity [START_REF] Goyer | Toxic and essential metal interactions[END_REF][START_REF] Jihen | Protective effects of selenium (Se) and zinc (Zn) on cadmium (Cd) toxicity in the liver and kidney of the rat: Histology and Cd accumulation[END_REF][START_REF] Who | Trace elements in human nutrition and health[END_REF]. Additionally, it has been demonstrated that toxic metals can disrupt trace element metabolism. Cadmium toxicity affects Ca metabolism either by direct toxicity to bone or indirectly from renal toxicity [START_REF] Goyer | Toxic and essential metal interactions[END_REF]Pappas et al. 2010a;[START_REF] Webb | The chemistry, biochemistry and biology of cadmium[END_REF].

Interactions between metals can occur at levels of exposure well below those at which toxicity may be detected. In addition, domestic animals have the advantage that they are an important source of food for humans and good biomonitor species for metal contamination worldwide (Lopez Alonzo et al. 2004;[START_REF] Petersson-Grawe | Cadmium levels in kidneys from Swedish pigs in relation to environmental factors -temporal and spatial trends[END_REF][START_REF] Tuzen | Determination of trace metals in canned fish marketed in Turkey[END_REF].

The present study was part of a project designed to assess the effect of the inclusion of Se in broiler's diet on the gene expression of antioxidant enzymes, trace element interactions and Se impact on fatty acid composition and oxidative stability of lipids in meat. Previously, we evaluated how Se affected gene expression of Se dependent and non-Se dependent antioxidant enzymes (Zoidis et al. 2010). The aim of the present study was to investigate the interactions between Se and some trace elements, both toxic and essential, which are all involved in the antioxidant system. Therefore, the concentration of Se, Cd, Cu, Fe and Zn and their interactions were assessed in blood, liver and breast muscle tissues of 4-and 6 week old broilers fed diets supplemented with Se-yeast. 1). In treatments T2, T3 and T4, the chickens were fed the same basal diet supplemented with additional Se from a yeast source, Sel-Plex ® (Alltech Inc, Nicholasville, KY, USA) to provide 0.15, 0.3 and 3 mg kg -1 Se respectively. Recent findings regarding chemical or enzymatic yeast protein hydrolysis revealed that selenomethionine accounts for 90% of total Se, while selenocystine, γ-glutamylselenocysteine, selenocystathione, methylselenocysteine and other compounds represent the rest of the total Se [START_REF] Schrauzer | Selenium yeast: Composition, quality, analysis, and safety[END_REF].

F

The duration of the study was 6 weeks with housing and care of chickens to conform the guidelines of the Faculty of Animal Science and Aquaculture of the Agricultural University of Athens. The chickens were raised, according to Cobb's management manual, in a house where light and ventilation were controlled. The chickens were fed a starter diet to the 2 nd wk of their life, a grower diet to the 4 th wk and a finisher diet thereafter. Feed and water were provided ad libitum. Each growing phase diet (starter, grower and finisher) contained the same level of added Se which was the appropriate for each experimental treatment (Table 1). The lighting program was 23 hours of light and one hour of dark. Stocking density was according to EU legislation. At weekly intervals, chickens were weighted and the body weight was recorded. Furthermore, feed intake was measured weekly and feed to gain ratio was calculated. At the end of the 4 th and 6 th week, two chickens per pen were sacrificed by neck dislocation and liver, breast muscle and whole blood samples were collected for determination of trace elements concentration. LGC Standards Promochem, Wesel, Germany). The Se recovery procedure was as follows: 4 samples from the same liver tissue (1 g each) were spiked with 250, 500 and 750 µl of Se standard solution of 4 mg kg -1 , respectively. The spiked samples were analyzed after the aforementioned analytical procedure. The recoveries of the procedure used to validate ICP-MS were in the range of 92.1-105.6 % indicating the accuracy of the method. Regarding the NIST-RM, certified and determined trace element concentrations are presented in Table 2.

Statistical analysis

The statistical analysis was performed using SPSS for Windows (Chicago, IL, USA).

The data were analyzed using a multivariate linear model where Se, Cd, Zn, Cu and Fe concentration represented the response variables while treatment, age and tissue were the fixed factors. We tested the hypothesis that the trace element variations are correlated using A particular aspect of our analysis was the comparison of the element correlations before and after running the model. In this way, potential differences (or lack thereof) may be revealed, which could otherwise be confounded due to ignoring the factor influence. The statements of significance presented in this study were based on P ≤ 0.05 unless otherwise stated. In the Tables, the data are presented as the mean ± SE of each of the main effects in turn, pooled for all other main effects. Furthermore, in the results section all statistically significant interactions are presented.

Results

Supplementation of diet with 150, 300 and 3000 µg Se kg -1 significantly (P<0.001) increased the Se concentration (data pooled for age and tissue effects) by a factor of 1.75, 2.1 and 8.6 respectively compared to the Se concentration of broilers fed the Se unsupplemented diet (Table 3). Overall, the concentration of Se in 6-week old broilers was lower compared with its concentration in 4-wk old broilers (data pooled for treatment and tissue effects, Table 3). The significant age X treatment interaction (P<0.001) revealed that Se concentration decreased with age for all four treatments. Tissue type significantly affected Se concentration (P<0.001, data pooled for age and treatment to exacerbate tissue effects, Table 3). Tissue Se concentration in descending order was liver>breast muscle>blood and this pattern was noted for all treatments as indicated by the significant treatment X tissue interactions (P<0.001).

The age X tissue interaction (P<0.001) indicated that Se concentration in blood (pooled for all treatments) decreased as the age of broilers increased (489±12.7 vs. 379±12.7 µg kg -1 ). The same was evident in breast muscle (825±14.2 vs. 756±12.7 µg kg -1 ). However, Se concentration in liver increased as the age of broilers increased (1313±12.7 vs. 1359±12.7 µg kg -1 ). The significant treatment X age X tissue interactions (P<0.001) indicated that, although overall increasing age reduced Se concentration, some treatments for some tissues responded different with increasing age. In detail, for broilers fed diet T2, liver Se concentration did not differ as age progressed (784±25.5 vs. 787±25.5 µg kg -1 ). The same was true for liver Se concentration of broilers fed the T3 diet (869±25.5 vs. 887±25.5 µg kg -1 ). However, liver Se concentration of broilers fed T4 increased with age (3090±25.5 vs. 3271±25.5 µg kg -1 ).

Finally, liver Se concentration of broilers fed T1 reduced with age (115±25.5 vs. 88±25.5 µg kg -1 ).

Selenium supplementation significantly (P<0.001) affected the concentration of Cd in the examined tissues (Table 3). In detail, Cd concentration in broiler tissues (pooled data) of treatments T2, T3, T4 was reduced by 41%, 37% and 13% respectively compared to the concentration of treatment T1. Cadmium concentration in blood was 19.8 times greater (P<0.001) than that in breast muscle while that in liver was 4.9 times greater than that in breast muscle. The present study revealed significant interactions between treatment and tissue regarding Cd accumulation. In detail, Cd concentration in blood of broilers fed diets T2 (21.3±0.86 µg kg -1 ) and T3 (23.8±0.86 µg kg -1 ) was lower than that of broilers fed Se unsupplemented diets (37±0.86 µg kg -1 ). Iron concentration tended (P=0.078) to be increased in the tissues of broilers fed Se supplemented diets (Table 3). Tissue type significantly affected Fe concentration (P<0.001).

In particular, blood was the tissue with the highest concentration of Fe. Iron concentration in blood was 391203±8442 µg kg -1 followed by the concentration in liver (199317±8442 µg kg - 1 ) and breast muscle (5985±8455 µg kg -1 ). Overall, Fe concentration in broilers' tissues did not differ with age, however, the noted age X tissue interaction (P<0.001) revealed that blood Fe concentration decreased as the age increased (421838±11940 vs. 360568±11940 µg kg -1 ).

Furthermore, the treatment X tissue interactions (P<0.05) showed that broilers fed the T2 (425865±16885 µg kg -1 ) and T3 (425702±16885 µg kg -1 ) diet had greater concentration of Fe in the blood than broilers fed the T1 (360502±16885 µg kg -1 ) or T4 (352742±16885 µg kg -1 ) diets.

Zinc concentration in tissues was not affected by Se addition to broiler diets (Table 3).

Zinc concentration was 6.6-fold greater in the liver and 1.4-fold greater in the breast muscle than its concentration in the blood. In detail, Zn concentration in 6-wk old broilers was lower compared with its concentration in 4-wk old broilers.(P=0.034). Furthermore, the age X tissue interactions (P=0.01) indicated that 6-wk old broilers had lower Zn concentration in blood compared to 4-wk old broilers (4204±346 vs. 3571±346 µg kg -1 ). The treatment X age X tissue interactions (P<0.05) indicated that as the age of broilers progressed, the Zn concentration in the blood and liver of broilers fed Se unsupplemented diets decreased (4628±691 vs. 3099±691 µg kg -1 for blood; 4917±691 vs. 4510±691 µg kg -1 for liver).

Copper concentration was not affected by Se addition (Table 3). Liver contained higher (P<0.001) concentration of Cu compared to that of blood and breast muscle. There was a tendency (P=0.064) for 6-wk old broilers to contain more Cu that the 4-wk old ones.

Although Se supplementation did not affect Cu accumulation, the treatment X age X tissue The correlations between the trace elements before and after the application of the model are presented in Table 4. After the application of the statistical model, the discernible points by inspecting the results were that Se and Cu were correlated with all examined trace elements; Zn was correlated with all examined trace elements except for Fe, and Cd correlated with all examined trace elements except for Fe. In addition, the value and significance of some pairwise correlations changed by incorporating the factors' influence (Table 4).

Specifically, Se was negatively correlated with Cd and this relationship became stronger after model application. Similarly, Se correlation with Zn was and remained strongly positive. Likewise, Se correlated positively with Cu and remained so post-model. In contrast, Se and Fe appeared uncorrelated in the descriptive statistics, while in fact they were found to correlate positively when the examined factors were taken into account (Table 4). Hence, this association was masked by ignoring the considered factors.

Cadmium was negatively correlated with Zn and this relationship remained so, albeit weaker after the model application. Furthermore, Cd and Cu were negatively correlated and continued to be so post-model. The application of the model revealed that the seemingly strong correlation between Cd and Fe was spurious as the two elements were in fact uncorrelated (Table 4).

Zinc and Cu were strongly positively correlated and their association remained positive although not as profound. Zinc and Fe remained uncorrelated throughout the analysis.

Finally, Cu and Fe seemed initially uncorrelated, but taking the factors into account exposed a weak negative correlation (Table 4).

Discussion

This study revealed that supplementation of broilers' diet with Se, not only increased Se concentration but also reduced Cd concentration in the tissues. Cadmium is absorbed from the gastrointestinal tract and the lung and is mainly accumulated in liver and kidney where it is bound to metallothionein (MT), a low molecular weight cysteine-rich, intracellular protein with high affinity for both essential and non-essential metals [START_REF] Klaassen | Metallothionein: an intracellular protein to protect against cadmium toxicity[END_REF]. When Cd concentration exceeds the binding capacity of MT, the non-bound Cd causes toxicity possibly due to free radical induction and lipid peroxidation [START_REF] Sarkar | Cadmium-induced lipid peroxidation and the antioxidant system in rat erythrocytes: the role of antioxidants[END_REF]. The administration of Se lowers the toxicity of Cd by shifting the distribution of tissue Cd from MT towards highmolecular-mass proteins [START_REF] Underwood | The mineral nutrition of livestock[END_REF][START_REF] Wangher | Selenium and the brain: A review[END_REF]. The reduction of Cd concentration noted in the present study may be attributed to the formation of Se-Cd complexes. Indeed, addition of Se compounds has been proven an efficient therapy against metal toxicity in mammals [START_REF] Glynn | Influence of sodium selenite on 203 Hg absorption, distribution and elimination in male mice exposed to methyl 203 Hg[END_REF], vegetables [START_REF] Shanker | Study of mercury-selenium (Hg-Se) interactions and their impact on Hg uptake by the radish (Raphanus sativus) plant[END_REF] and fishes (Paulsson and Lundberg 1989). Although only Se-yeast was examined in this study, inorganic

Se has been shown to be effective in reducing the harmful effects of Cd. It has been shown

that treatment of rats with Se prior to Cd administration prevented the effects of Cd, because it bound Cd to low molecular weight, stable, and biologically inactive Cd selenide complexes [START_REF] Ohta | Interactive effects of selenium on chronic cadmium toxicity in rats[END_REF]. According to available data, it seems that the ability of Se to reduce Cd accumulation in organs depends on Se/Cd ratio [START_REF] Sasakura | Biological interaction between transition metals (Ag, Cd and Hg), selenide/sulfide and selenoprotein P[END_REF], their chemical form and the solubility of the formed compounds [START_REF] Feroci | Interactions between different selenium compounds and zinc, cadmium and mercury[END_REF]. [START_REF] Schrauzer | Selenium and selenium-antagonistic elements in nutritional cancer prevention[END_REF] reported that the sequestration of toxic elements by Se represents an efficient detoxification mechanism, but also results in the physiological inactivation and loss of Se. In our study, as the age of broilers progressed, Se concentration in tissues decreased, possibly due to involvement of Se in detoxification mechanisms. Decrease of tissue Se content support the idea of Se-Cd complex formation but its excretion or deposition remains unclear. If Se derivatives are supplemented as a remedy for heavy metal toxicity, they should form insoluble compounds to be excreted. Indeed, early findings by [START_REF] Stowe | Biliary excretion of cadmium by rats: effects of zinc, cadmium, and selenium pretreatments[END_REF] have shown that Se pretreatment increases the rate of biliary excretion of Cd.

In the present study statistically significant positive correlations between Se -Cu and Se -Zn were noted. Indeed, broilers fed Se unsupplemented diets exhibited reduced blood and liver Zn levels as well as reduced liver Cu levels along with age. Similar results were reported in several other studies where Se was positively correlated to Cu and Zn. In detail, it was found that Cu was positively associated with Se in the liver [START_REF] Lopez-Alonso | Interactions between toxic (As, Cd, Hg and Pb) and nutritional essential (Ca, Co, Cr, Cu, Fe, Mn, Mo, Ni, Se, Zn) elements in the tissues of cattle from NW Spain[END_REF]) and in the kidney [START_REF] Blanco-Penedo | Influence of copper status on the accumulation of toxic and essential metals in cattle[END_REF]) of cattle raised in Spain. It is possible that the Se -Cd positive correlation may be related to the function of Se as a regulator of MT synthesis, which may in turn hinder biliary excretion of Cu, leading to its accumulation. Earlier studies in ruminants have also demonstrated that after injection of Se, Cu concentration in the liver increased in animals receiving both normal [START_REF] Hussein | Selenium copper interaction in goats[END_REF] or Cu deficient diets [START_REF] Thomson | Copper and selenium interaction in sheep[END_REF]. However, Kotyzova et al. (2009) reported that in long term Se deficiency of rats an increase of liver Cu concentration was noted. The positive correlation between Se and Zn reported in the present study is in line with reports by [START_REF] Bou | Effect of dietary fat sources and zinc and selenium supplements on the composition and consumer acceptability of chicken meat[END_REF] who noted that Zn supplementation (from 0 mg kg -1 to 600 mg kg -1 ) led to a significant increase in Se content (115 to 213 µg kg -1 ) in chicken meat. In the same study, in line with our results, chicken muscle Zn content decreased with age. This positive Se -Zn correlation could 12 be attributed to MT, the synthesis of which is induced by Zn and other cations [START_REF] Nordberg | Metallothioneins: Historical review and state of knowledge[END_REF][START_REF] Nordberg | Historical perspectives on cadmium toxicology[END_REF]).

In the present study close negative correlations between Cd -Zn and Cd -Cu were noted. These results are in line with previous studies by [START_REF] Elsenhans | Metal-metal interactions among dietary toxic and essential trace metals in the rat[END_REF] where a decrease of rat hepatic Cu concentration was observed after a 2-week administration of Cd.

Similarly, in the plasma of healthy humans, a negative association between Cd and Cu was reported [START_REF] Pasha | Statistical analysis of trace metals in the plasma of cancer patients versus controls[END_REF]. Cadmium shares physiological properties, such as MT-induction, with other elements notably Cu and Zn having similar electron configurations in the outer valence shell of the atom. These elements are thus often mutually antagonistic with exposure to Cd lowering Cu status and exposure to Zn lowering Cd status [START_REF] Underwood | The mineral nutrition of livestock[END_REF]. Although it was initially thought that the only function of MTs was to provide a defense against toxic cations (especially Cd), it is now known that these proteins have a function in regulating normal Cu and Zn metabolism and in protecting cells against damage induced by alkylating agents, oxygen radicals, and ionizing radiation (Lopez-Alonzo et al. 2002). Metallothioneins can also serve as store for Zn and Cd, and be involved in metal transfer [START_REF] Phillips | Effects of cadmium in herbage on the apparent absorption of elements by sheep in comparison to inorganic cadmium added to their diets[END_REF][START_REF] Sharma | Copper status and enzyme, hormone, vitamin and immune function in heifers[END_REF]. The mechanisms by which Cd affects Zn metabolism are not yet well understood but the strong association of Cd and Zn in the examined tissues most likely is related to the fact that the handling of Cd 2+ and Zn 2+ in the tissues intersect at several points. In our study, Cd concentration was greater in 6-wk old broilers compared with its concentration in 4-wk old broilers. The observed Cd accumulation with time is in line with observations by Nriagu et al. (2008) who reported that Cd contents of older cows were much higher than those of younger animals. In other experimental models (i.e. human or cattle) a positive association between Cd and Zn was noted. In detail, in plasma of healthy humans [START_REF] Pasha | Statistical analysis of trace metals in the plasma of cancer patients versus controls[END_REF] ruminants [START_REF] Mills | Copper and zinc status of ewes and lambs receiving increased dietary concentrations of cadmium[END_REF], pigs [START_REF] Poulsen | Zinc and copper as feed additives, growth factors or unwanted environmental factors[END_REF] [START_REF] Blalock | Metallothionein gene expression in rats: tissuespecific regulation by dietary copper and zinc[END_REF][START_REF] Hall | Intestinal metallothionein and the mutual antagonism between copper and zinc in the rat[END_REF]). Both Cu and Zn are integral parts of CuZn-SOD and both these elements together with Se cooperate in the cytosolic antioxidant defense. Copper in excess of cellular needs, mediates free radical production and direct oxidation of lipids, proteins and DNA [START_REF] Parge | Atomic structures of wild-type and thermostable mutant recombinant human Cu, Zn superoxide dismutase[END_REF]). Thus, it seems that a delicate balance between necessity and toxicity for Cu and Zn exists which is achieved both at the cellular level and at the tissue and organ levels [START_REF] Valko | Metals, Toxicity and Oxidative Stress[END_REF].

The present study revealed that in broilers, Fe was negatively correlated with Cu, positively correlated with Se and uncorrelated with Cd or Zn. Several studies have confirmed that Fe and Cu are associated [START_REF] Gambling | Iron and copper, and their interactions during development[END_REF][START_REF] Pasha | Statistical analysis of trace metals in the plasma of cancer patients versus controls[END_REF]. Copper -Fe interactions may occur either at the level of intestinal absorption [START_REF] Gambling | Iron and copper, and their interactions during development[END_REF] or during transportation by the DMT1 (divalent metal transporter 1) [START_REF] Garrick | Parallels and contrasts between iron and copper metabolism[END_REF]. Iron is involved in the antioxidant defense system of the organism by the action of catalase which is a tetrameric heme-containing enzyme, located primarily in the peroxisomes (Zoidis et al.

2010).

Catalase detoxifies H 2 O 2 by catalyzing its breakdown to water and divalent oxygen.

However, ferrous iron is able to trigger oxidations by reducing, as well as by decomposing, previously formed H 2 O 2 [START_REF] Fraga | Iron toxicity and antioxidant nutrients[END_REF]. The finding that Fe is correlated with Se, is in line with previous reports [START_REF] Uluozlu | Assessment of trace element contents of chicken products from turkey[END_REF] and may indicate that occasionally there is a necessity for antioxidant protection against Fe toxicity.

In conclusion, trace elements may interact at the levels of absorption, distribution and retention. Deficiency or supplementation of one of them may alter their balance and trigger antagonistic or synergistic effects.

  The trace element concentrations were determined using inductively coupled plasma mass spectrometry, ICP-MS (Perkin Elmer, Elan 9000, PerkinElmer Life and Analytical Sciences Inc, Waltham, MA, USA). The instrumental parameters of the equipment used were: Nebuliser flow 0.775 L/min, Vacuum pressure 1.5 x 10 -5 torr, Lens Voltage 950 W, Analogue Stage Voltage -1900 V, Pulse Stage Voltage 950 V, Sweeps/reading 20, Readings/replicate 1, Number of replicates 3, Time per sample 83 s. Feed samples were collected before feeding and milled before analysis through a 1 mm sieve (Cyclotec, 1093 sample mill, Tecator, Höganäs, Sweden). Initially, a complete digestion of the samples was performed with a microwave digestion system (CEM, Mars X-Press, NC, USA). Samples (0.5 g) of wet tissue or feed were soaked in 10 ml concentrated HNO 3 (65% w/v, Suprapur, Merck, Darmstadt, Germany). The samples were heated in the microwave accelerated digestion system according to the following program: the power was ramped during 20 min from 100W to 1200W and held for 15 min. The temperature reached a maximum of 200ºC and followed by a cool down cycle for 15 min. Losses of volatile element compounds do not occur as the tubes are sealed during heating. The samples were then filtered with disposable syringe filters (Chromafil, Macherey-Nagel, Duren, Germany) and diluted 50 times with reversed osmosis water (Milli-Q Water Purification Systems, Billerica, MA, USA) before injection in the ICP-MS instrument. Standard solutions used for calibration curves were prepared from high purity standards (Multielenment standard solution, Fluka Analytical, Sigma-Aldrich, St Louis, USA). The analytical procedure was validated using a Se recovery procedure[START_REF] Georgiou | Automated flow injection spectrophotometric determination of para-and meta-substituted phenols of pharmaceutical interest based on their oxidative condensation with 1-nitroso-2-naphthol[END_REF]) and a standard reference material (NIST-RM 8414 -bovine muscle powder-

  .005) showed that Cu concentration in the liver of broilers fed Se unsupplemented diets decreased along with age (4092±123 vs. 3170±123 µg kg -1 ).

  and in the kidney of cattle (Lopez Alonso et al. 2002), a strong positive correlation was observed between Cd and Zn, indicating that the organism and possibly other factors like the mineral status may affect the association of Zn and Cd. Zinc was positively correlated with Cu in the present study. This result is in line with findings by Nriagu et al. (2008) who reported a positive association of Cu and Zn in the liver of free range cattle. Similarly, Blanco-Penedo et al. (2006) reported positive association between Zn and Cu in the kidney of cattle raised in Spain. However, there are reports that indicate that a negative correlation between Cu and Zn may exist. In detail, when dietary Zn levels are very high, tissue Cu concentrations are very low and animals show clinical signs of Cu deficiency or that low levels of Zn in the diet lead to Cu accumulation and Cu toxicity in

  Details of the genotype, nutrition and husbandry of the broilers used in this study have been described previously(Zoidis et al. 2010). In brief, a total of 128 day-old Cobb chickens

	Materials and methods
	Animals, Diets and Design
	(broilers) were used. The chickens were obtained from a commercial hatchery. All chickens
	had the same Se-status because the broiler breeder flock was fed a commercial diet with 0.20
	mg kg -1 Se. There were 4 replicate pens of 4 dietary treatments namely T1, T2, T3 and T4
	randomly allocated in the house. Pen was the experimental unit. There were 8 chickens per
	pen, 32 per treatment. The study was designed considering that a concentration of 0.15 mg Se per kg of diet is recommended by the United States National Research Council (NRC 1994), o r that the maximum allowed Se inclusion level in the United States is 0.3 mg kg -1 (Payne et al.
	2005) and that current total maximum EU authorized level of Se in complete feed is 0.5 mg P kg -1 . In T1, chickens were fed a basal diet without Se supplementation; the only Se present e r e was that of feed ingredients (Table
	R
	e
	v i e
	w
	O n l
	y

  Bartlett's test of sphericity and by calculating the pairwise Pearson correlations. Both methods indicated that the concentrations are indeed correlated, which also reflects biological plausibility. Hence, a multivariate analysis appeared to be necessary. The response variable (Y) was approximately normally distributed as verified by employing both histograms and normality tests. Thus using a multivariate linear model seemed like a reasonable approach.

	The model was:
	Y = intercept + age + tissue + treatment + (age X tissue) + (age X treatment) +
	(tissue X treatment) + (age X tissue X treatment), F r o where Y denotes Se, Cd, Zn, Cu and Fe concentrations.
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	7

  (P=0.001). In detail, 6-wk old broilers had higher Cd concentration in blood compared to 4wk old broilers (38±0.6 vs. 21±0.6 µg kg -1 ). The treatment X age interaction (P<0.001) noted in the present study indicated that the increment of Cd concentration with age differed between treatments. In detail, Cd concentration in tissues of 4-wk vs. 6-wk old broilers fed Cd concentration in the liver of broilers fed diets without added Se increased as the age progressed (5.7±1.23 vs. 16.4±1.23 µg kg -1 for 4-wk vs. 6-wk old broilers respectively).

	diets T1, T2, T3 and T4 was 9.7±0.71 vs. 23.5±0.71, 9.4±0.71 vs. 10.2±0.71, 9.2±0.71 vs.
	11.6±0.71 and 8.9±0.82 vs. 20.1±0.71 µg kg -1 respectively. Furthermore, there was a
	significant (P=0.011) treatment X age X tissue interaction regarding Cd accumulation. In
	detail,	r o F
		P
		e
		e r
		R
		e
		v i e
		w
		O n l
		y

Cadmium concentration was greater in 6-wk old broilers compared with its concentration in 4-wk old broilers. Furthermore, the age of broilers affected the Cd concentration in the tissues as revealed by the age X tissue interactions

  and laboratory animals(Blalock et

	al
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	13

Table 1 .

 1 The alteration of trace element balance may concomitantly affect the antioxidant defense system since several trace elements, like Se, Cu, Fe and Zn, are integral part of various antioxidant enzymes. The present study revealed that Se supplementation, age of broiler and type of examined tissue affect the concentration of trace elements. The level of supplemented Se is important since low or excessive quantities may alter the balance and the correlation to the other elements. Although, significant trace element correlations were reported in this study, they do not necessarily indicate that the level of essential elements, like Se, can determine the amount of toxic metal accumulation in broiler tissues or that toxic metal accumulation in tissues can affect essential element homeostasis. Pappas AC, Georgiou CA, Komaitis Ε, Fegeros K. 2010. Selenium affects the expression of GPx4 and catalase in the liver of chicken. Comp Bioch Physiol B 155:294-300. Composition (g kg -1 ) of control and Se-supplemented diets for chickens

	Future studies may further elucidate trace element interactions.

b Se was added in the form of Sel-Plex (Alltech Inc, Nicholasville, KY, USA) and only in the Se-supplemented diets. c The determined Se content represents pooled data from all three diets (starter, grower and finisher).

Table 2 .

 2 Determined and certified trace element concentrations in the reference material (NIST-RM 8414 -bovine muscle powder)

		Food Additives and Contaminants	Page 22 of 24
	Element Certified value (µg g -1 ) Detected Value (µg g -1 )	Recovery (%)
	Se	0.076 ± 0.010	0.078 ± 0.011	103
	Cd	0.013 ± 0.011	0.016 ± 0.002	123
	Zn	142 ± 14	173 ± 1	122
	Cu	2.84 ± 0.45	3.26 ± 0.08	115
	Fe	71.2 ± 9.2	91.97 ± 1.03	129

http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.uk

Table 3 .

 3 The effect of Se supplementation in the diet of broilers on the concentration (µg kg -1 ) of Se, Cd, Zn, Cu and Fe in blood, liver and breast tissue at 4 th and 6 th week of age The data are presented as means ± SE (n = 4) of the main effect in turn, pooled for all other main effects, based on Multivariate Regression.Means within a column, within a main effect comparison, with different superscripts are different at P< 0.05, unless otherwise stated.

	Factor			Trace Element
	Studied a	Se	Cd	Zn	Cu	Fe
	Age (wk)
	4 th	876 a ±7.65	9.3 a ±0.37	12076 a ±208	1451±37.14	201635±7175
	6 th	831 b ±7.35	16.4 b ±0.36	11454 b ±200	1354±35.68	196034±6893
	Se added
	(µg kg -1 )
	0 (T1)	252 a ± 10.40 16.6 a ±0.50	11935±282	1452±50.46	180568± 9749
	150 (T2)	441 b ± 10.40	9.8 b ±0.50	12031±282	1462±50.46	216623± 9749
	300 (T3)	534 c ± 10.40 10.4 b ±0.50	11313±282	1340±50.46	202732± 9449
	3000 (T4)	2186 d ± 11.20 14.5 c ±0.55	11781±305	1355±54.50	195416±10530
	Tissue
	Blood	434 a ± 9.00	29.7 a ±0.44	3888 a ±244	393 a ±43.70	391203 a ±8442
	Liver	1336 b ± 9.00	7.4 b ±0.44	25766 b ±244	3481 b ±43.70	199317 b ±8442
	Breast	790 c ± 9.55	1.5 c ±0.46	5641 c ±259	332 a ±46.35	5985 c ±8955
	Source of Variation			P Value
	Treatment (Se addition)	<0.001	<0.001	NS	NS	NS (P=0.078)
	Age	<0.001	<0.001	0.034	NS (P=0.064)	NS
	Tissue	<0.001	<0.001	<0.001	<0.001	<0.001
	Treatment X Age	<0.001	<0.001	NS	NS	NS
	Treatment X Tissue	<0.001	<0.001	NS	NS	<0.05
	Age X Tissue	<0.001	<0.001	<0.01	NS	<0.001
	Treatment
	X Age X	<0.001	<0.001	<0.05	<0.01	NS
	Tissue

a

Table 4 .

 4 Correlations between the concentration of Se, Cd, Zn, Cu, Fe prior to and after the application of the statistical model Level of statistical significance of each of the pairwise correlations, *P ≤ 0.05; **P ≤ 0.01;

	Prior to Model Application	Cd	Zn	Cu	Fe
	Se	-0.213*	0.378***	0.356***	-0.112
	Cd		-0.346***	-0.279**	0.654***
	Zn			0.978***	-0.042
	Cu				0.087
	After the Model Application	Cd	Zn	Cu	Fe
	Se	-0.389***	0.461***	0.312**	0.229*
	Cd		-0.249*	-0.236*	-0.189
	Zn			0.390***	0.073
	Cu				-0.267*

***P ≤ 0.001.

http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.ukFood Additives and Contaminants

Acknowledgements

A.C Pappas is grateful to Greek State Scholarship Foundation (IKY) for postdoctoral funding.

We are grateful to NUEVO S.A. (N Artaki, Euboia, Greece) for providing Sel-Plex ® , to E. Komaitis for excellent technical assistance and to A. Kominakis for useful discussions.


Denis Denisov

Seva Shneer
email: shneer@eurandom.tue.nl

Global and local asymptotics for the busy period of an M/G/1 queue

Keywords: Busy period, Busy cycle, Heavy-tailed distributions Mathematics Subject Classification (2000) 60K25, 60G50, 60F10

We consider an M/G/1 queue with subexponential service times. We give a simple derivation of the global and local asymptotics for the busy period. Our analysis relies on the explicit formula for the joint distribution for the number of customers and the length of the busy period of an M/G/1 queue.

Introduction

Let A 1 , A 2 , . . . and B 1 , B 2 , . . . be independent sequences of independent and identically distributed random variables. We call {A i } inter-arrival times and {B i } service times. It is assumed throughout that ρ := E{B 1 }/E{A 1 } < 1, so that the system is stable. Denote also ξ i = B i -A i and S n = n i=1 ξ i . We shall denote by A, B and ξ random variables with the same distributions as A 1 , B 1 and ξ 1 , respectively. We use the standard notation: we denote by M/G/1 the system with exponential inter-arrival times A i and by GI/G/1 the system with arbitrarily distributed A i . The main interest of our paper is the (global and local) asymptotics for the distribution of the length of the busy period

τ = B 1 + • • • + B ν ,
where ν = inf{n : S n < 0} is the number of customers which arrive (and get served) during the busy period.

In [START_REF] De Meyer | On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1[END_REF], it was shown that in the M/G/1 case, if B has a regularly varying distribution, then

P{τ > t} ∼ E{ν}P B > (1 -ρ)t as t → ∞. (1)
This result was generalised in [START_REF] Zwart | Tail asymptotics for the busy period in the GI/G/1 queue[END_REF] to the case of a GI/G/1 queue and under the assumption that the tail P{B > t} is of intermediate regular variation (see the next section for a definition). Later on, it was shown in [START_REF] Baltrūnas | Tail behaviour of the busy period of a GI /GI /1 queue with subexponential service times[END_REF] and [START_REF] Jelenković | Large deviations of square root insensitive random sums[END_REF] that asymptotics (1) hold for the GI/G/1 model in the case when the service-time distribution belongs to another subclass of heavy-tailed distributions. This class includes, in particular, Weibull distributions with parameter α < 1/2. The tails of the distributions considered in [START_REF] Baltrūnas | Tail behaviour of the busy period of a GI /GI /1 queue with subexponential service times[END_REF] and [START_REF] Jelenković | Large deviations of square root insensitive random sums[END_REF] are heavier than e - √ t . As is shown in [START_REF] Asmussen | Sampling at sub-exponential times, with queueing applications[END_REF], the latter condition is crucial for asymptotics (1) to hold.

It was shown in [START_REF] Denisov | Asymptotics for first passage times of Levy processes and random walks[END_REF] that in an M/G/1 queue

P(τ > t) ∼ C P(X t > 0) t as t → ∞, (2)
where C is a constant, X t = N(t) i=0 B it and N is a Poisson process with intensity λ = 1/EA. It was also shown in [START_REF] Denisov | Asymptotics for first passage times of Levy processes and random walks[END_REF] that in the case when the tail of B is heavier than e - √ t (and under some further conditions) the asymptotic equivalence (2) reduces to [START_REF] Asmussen | Asymptotics for sums of random variables with local subexponential behaviour[END_REF].

In this note, we concentrate on the M/G/1 case and have two main goals. The first (and the more important) consists in providing new results on local asymptotics for τ . These results are (not surprisingly) similar to (2):

P τ ∈ (t, t + T ] ∼ P(X t ∈ (0, T ]) λt as t → ∞,
with a fixed T ∈ (0, ∞). Another goal of this paper is to give a simple proof of (1) for the M/G/1 case for some classes of distributions (of service times) with tails heavier than e - √ t . However, our contribution here does not consist only in providing a shorter proof; we believe that our assumptions on the distribution of the service times are close to minimal. In particular, they are satisfied by all the distributions for which relation [START_REF] Asmussen | Asymptotics for sums of random variables with local subexponential behaviour[END_REF] has been proved earlier.

Both aims are achieved by using the explicit formula for the joint distribution of τ and ν (see e.g. [5, (4.63)]):

P(ν = n, τ > t) = ∞ t (λu) n-1 n! e -λu P(B 1 + • • • + B n ∈ du). ( 3
)
The paper is organised as follows. In Sect. 2, we give results on local asymptotics for τ , and in Sect. 3 we present a simple proof of (1) for the M/G/1 queue.

Local asymptotics for the busy period

In this section, we shall present our main results on local asymptotics for τ . Let = (0, T ] and recall that

S n = n i=1 (B i -A i ) and X t = N(t)
i=1 B it. We will start with the following simple result. Theorem 2.1 Assume that the random variable B is absolutely continuous. Then

P{ν = n} = f S n (0) λn ( 4
)
and

f τ (t) = f X t (0) λt . ( 5
)
Here f ξ (x) denotes the density of a random variable ξ at a point x. The next result holds for random variables that are not necessarily absolutely continuous.

Theorem 2.2 Fix a positive and finite value of

T . Assume that P{B 1 ∈ t + }e δt → ∞, as t → ∞, for any δ > 0. Then P{τ ∈ t + } ∼ P{X t ∈ } λt ∼ P{X [t] ∈ } λ[t] as t → ∞,
where [t] denotes the integer part of t.

These asymptotics are not explicit. However, the theorem above reduces the original problem to a particular case of the large-deviations problem for random walks, which is extensively studied in the literature. It turns out that some known largedeviations results (together with the theorem above) yield explicit asymptotics for the distribution of the busy period. Some of these cases will be presented below.

First, we discuss corollaries of the above theorems and then give the proofs. We start by defining two (rather large) classes of distributions.

Definition 2.1 We say that a function f (x) is intermediate regularly varying if lim κ↓1 lim sup x→∞ sup x≤y≤κx f (y + ) f (x + ) -1 = 0. ( 6
)
In particular, (6) holds when f (x) is regularly varying at infinity. Let F (x) = P(B ≤ x) be the distribution function of B and let F (x + ) = F (x + T ) -F (x). We will use the following conditions:

(A) EB < ∞ and F (x + ) is intermediate regularly varying. (B) EB 2 < ∞, sup y≤ √ x F (x -y + ) F (x + ) -1 → 0 and ε (n) ≡ sup x≥2 √ n P{B 1 > √ n, B 2 > √ n, B 1 + B 2 ∈ x + } P(B ∈ x + ) = o(1/n) as n → ∞.
Sufficient conditions for (B) to hold may be derived using, for example, Lemma B.1 and Lemma B.2 of [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF]. In particular, one can see that F (x + ) ∼ e -x β , β < 1/2, as well as F (x + ) ∼ e -ln β x , β > 1, satisfy Condition (B). The following proposition directly follows from Corollary 2.1 of [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF].

Proposition 2.1 Let either Condition (A) or (B) hold. Then

P{S n ∈ } ∼ nP ξ ∈ n|Eξ | + as n → ∞
and hence (since N(t) is a Poisson process)

P{X [t] ∈ } ∼ [t]P X 1 ∈ [t]|EX 1 | + as t → ∞.
For densities, Conditions (A) and (B) can be reformulated as follows:

(A ) EB < ∞ and f B (x) is intermediate regularly varying. (B ) EB 2 < ∞, sup y≤ √ x f (x -y) f (x) -1 → 0 and ε(n) = sup x≥2 √ n x- √ n √ n f (x -y)f (y) dy f (x) = o(1/n) as n → ∞.
Proposition 2.1, Theorems 2.1 and 2.2 immediately allow us to obtain the local asymptotics.

Corollary 2.1 Let either Condition

(A) or (B) hold for some T > 0. Then P{τ ∈ t + } ∼ P(B ∈ t -ρt + ) as t → ∞.
A similar result holds for densities.

Corollary 2.2 Assume that B is absolutely continuous and let one of Conditions

(A ) or (B ) hold. Then f τ (t) ∼ f (t -ρt).
Now we give proofs of the statements presented in this section.

Proof of Theorem 2.1 Put

a = λ -1 = EA, b = EB. Then P{ν = n} = ∞ 0 (λu) n-1 n! e -λu f * n B (u) du = λ -1 ∞ 0 f * n A (u) n f * n B (u) du = λ -1 f * n B-A (0) n .
This implies the first statement of the theorem. We proceed to prove the second statement:

f τ (t) = ∞ n=1 (λt) n-1 n! e -λt f * n B (t) = 1 λt ∞ n=1 P N(t) = n f * n B (t) = 1 λt ∞ n=1 P N(t) = n P{B 1 + • • • + B n ∈ t + du} du = 1 λt P{ N(t) i=1 B i ∈ t + du} du = f X t (0) λt .
Proof of Theorem 2.2 We start by showing that the assumptions of the theorem imply that P{X t ∈ }e δt → ∞ for any δ > 0. Indeed, fix a value of δ > 0 and assume that P{X 1 ∈ t + } ≥ e -δt for sufficiently large t (which is guaranteed by the assumptions of the theorem). Let a = -EX 1 > 0. Fix also ε > 0 and write

P{X t ∈ } ≥ P X t-1 ∈ (-a -ε)t, (-a + ε)t , X t-1 + (X t -X t-1 ) ∈ = (-a+ε)t (-a-ε)t P{X t-1 ∈ du}P{X 1 ∈ -u + } ≥ (-a+ε)t (-a-ε)t P{X t-1 ∈ du}e δu ≥ P X t-1 ∈ (-a -ε)t, (-a + ε)t e -δ(a+ε)t ≥ (1 -ε)e -δ(a+ε)t ,
where the latter inequality follows from the Law of Large Numbers. Since δ > 0 is arbitrary, this implies that P{X t ∈ }e δt → ∞ for any δ > 0 as t → ∞.

We will only prove the asymptotic equivalence P{τ ∈ t + } ∼ P{X t ∈ }/t as t → ∞. The proof of the equivalence P{τ ∈ t + } ∼ P{X [t] ∈ }/[t] may be given following the same lines. Fix any ε > 0. According to Formula (3),

P(τ ∈ t + ) = ∞ n=1 t+T t (λu) n-1 n! e -λu P(B 1 + • • • + B n ∈ du) = ∞ n=1 t+T t P(N (u) = n) λu P(B 1 + • • • + B n ∈ du) = 1≤n≤λt-εt + λt-εt<n≤λt+εt + n>λt+εt (• • • ) ≡ S 1 + S 2 + S 3 .
For the first sum we have:

S 1 ≤ 1 λt 1≤n≤λt-εt t+T t P N(u) = n P(B 1 + • • • + B n ∈ du) ≤ 1 λt P N(t) ≤ λt -εt 1≤n≤λt-εt P{B 1 + • • • + B n ∈ t + } = o e -δt
with some δ > 0 as t → ∞, since N(t) is Poisson distributed with parameter λt. Indeed, it follows immediately from the Chernov bound that for some δ > 0

P N(t) -λt ≥ εt = o e -δt as t → ∞.
It can be shown in a similar way that S 3 = o(e -δt ) as t → ∞. Let us now investigate the asymptotic behaviour (as t → ∞) of the remaining sum:

S 2 = 1 + o(1) 1 λt λt-εt<n≤λt+εt e -λt (λt) n n! × t+T t e -λ(u-t) 1 + u -t t n P(B 1 + • • • + B n ∈ du).
We now consider

e -λ(u-t) 1 + u -t t n = e -λ(u-t)+n log(1+ u-t t ) ≤ exp -λ(u -t) + n u -t t ≤ exp -λ(u -t) + (λ + ε)(u -t) = exp ε(u -t) ≤ exp{εT }.
Here we used the facts that for the sum under consideration n ≤ λ(1 + ε)t and u ≤ t + T , and the inequality log(1 + t) ≤ t. In a similar way, with the help of the inequality log(1

+ t) ≥ t/2, t ≤ 1, one can prove that e -λ(u-t) (1 + u-t t ) n ≥ exp{-εT /2}. Therefore, S 2 ≤ 1 + o(1) e εT λt P X t ∈ , λt -εt ≤ N(t) ≤ λt + εt and S 2 ≥ 1 + o(1) e -εT /2 λt P X t ∈ , λt -εt ≤ N(t) ≤ λt + εt .

Note now that

P X t ∈ , N(t) -λt ≥ εt ≤ P N(t) -λt ≥ εt = o e -δt as t → ∞.
Hence, we have the following upper and lower bounds:

P{τ ∈ t + } ≤ 1 + o(
P{τ ∈ t + } ∼ λ -1 P X 1 ∈ [t]|EX 1 | + = λ -1 P X 1 ∈ [t] -ρ[t] + as t → ∞, since EX 1 = EN(1)EB -1 = ρ -1. Now we should note that P X 1 ∈ [t] -ρ[t] + ∼ P N(1) i=1 B i ∈ [t] -ρ[t] + ∼ EN(1)P B ∈ [t] -ρ[t] + ∼ EN(1)P(B ∈ t -ρt + ) as t → ∞,
by the theorem on local behaviour for randomly stopped sums from [START_REF] Asmussen | Asymptotics for sums of random variables with local subexponential behaviour[END_REF]. Indeed, either Condition (A) or (B) implies that F ∈ S , see [START_REF] Asmussen | Asymptotics for sums of random variables with local subexponential behaviour[END_REF] for this theorem and definitions.

Proof of Corollary 2.2 As is not difficult to see, Condition (A ) implies Condition (A) and Condition (B ) implies Condition (B) for any fixed T . Therefore, Corollary 2.1 implies that

P(τ ∈ t + ) ∼ P(B ∈ t -ρt + ) ∼ Tf B (t -ρt). ( 7
)
The latter equivalence holds since by both Conditions (A ) and (B ) f B is long-tailed, that is,

f B (t + u) ∼ f B (t) as t → ∞ uniformly in u ∈ [0, T ].
Next, it follows from the explicit formula (3) that f τ is longtailed as well and, therefore,

P(τ ∈ t + ) ∼ Tf τ (t). (8)
Combining ( 7) and ( 8), we arrive at the conclusion.

Note that the results of this section hold only for some classes of distributions such that F (x + ) e - √ x . We believe that it is also possible to obtain asymptotics for the distributions with tails lighter than e - √ x , but one should use results for large deviations of random walks other than Proposition 2.1. There are a lot of known results in that direction; we refer the reader to a recent paper of Borovkov and Mogulskii [START_REF] Borovkov | Integro-local and integral theorems for sums of random variables with semiexponential distributions[END_REF] for some new results and a review.

Global asymptotics for the busy period

In this section, we give a short proof of the known result [START_REF] Asmussen | Asymptotics for sums of random variables with local subexponential behaviour[END_REF]. In other words, this section deals with the case T = ∞. Put F (x) = P{B > x}. We consider the same conditions on the distribution of B as those used in the previous section. In the case T = ∞, they transform into:

ε(n) ≡ sup x≥2 √ n P{ξ 1 > √ n, ξ 2 > √ n, S 2 > x} F (x) = o 1 n , n→ ∞.
The following proposition follows from Corollary 2.1 of [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF]. Note that the so-called square-root insensitivity condition F (x) ∼ F (x -√ x) implies thatln F (x) = o( √ x) as x → ∞; hence, the proposition above deals with distributions whose tails are heavier than e - √ x . We now state the main result of this section. We also need the following result.

The proof of this proposition is rather standard (see, for instance, [START_REF] Doney | On the asymptotic behaviour of first passage times for transient random walks[END_REF]) and is thus omitted here.

(

  A) EB < ∞ and F is intermediate regularly varying, i.e. EB 2 < ∞, F (x -√ x) ∼ F (x) and

Proposition 3 . 1

 31 Let either Condition (A) or (B) hold. Then P{S n > 0} ∼ nF n|Eξ | as n → ∞.

Theorem 3 . 1

 31 Let either Condition(A) or (B) hold. Then P{τ > t} ∼ E{ν}P B > (1ρ)t as t → ∞.

Proposition 3 . 2

 32 Let either Condition(A) or (B) hold. Then P{ν > t} ∼ E{ν}P B > E{ξ } t as t → ∞.

  Using the condition e δt P{X t ∈ } → ∞ for any δ > 0 and then letting ε → 0, we arrive at the statement of the theorem.

	1)	e εT λt	P(X t ∈ ) + o e -δt	as t → ∞
	and
	P{τ ∈ t + } ≥ 1 + o(1)	e -εT /2 λt	P(X t ∈ ) + o e -δt	as t → ∞.
	Proof of Corollary 2.1 It follows from Proposition 2.1 and Theorem 2.2 that

Proof of Theorem 3.1 For Condition (B), the estimate from below is a corollary of the CLT and the condition F (x -√ x) ∼ F (x), as x → ∞, a proof may be found in [START_REF] Zwart | Tail asymptotics for the busy period in the GI/G/1 queue[END_REF]. For Condition (A), the estimate from below follows similarly from the Law of Large Numbers.

Therefore, we will concentrate on the estimate from above. We will consider Condition (B). The proof for Condition (A) is similar. It follows from Proposition 3.2 and Condition (B) that

Therefore, there exists a function l(n) ↑ ∞ such that

By the total probability formula,

First,

where that latter equivalence follows from the fact that, when EA 2 < ∞,

as n → ∞. Then,

For u ≥ n -1, probability P(N (u) = n -1) is non-increasing in u. Therefore, P(N (u) = n -1) ≤ P(N (t) = n -1), and this fact implies that

By the Central Limit Theorem for renewal processes,

as t → ∞ and, therefore,

As a result, we have

where the latter follows from Corollary 2.1 of [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF]. For the third term,

as t → ∞ (see ( 9)). Finally, we have

Acknowledgements The authors would like to thank Serguei Foss for drawing their attention to this problem and Onno Boxma for a number of useful discussions and for many suggestions that helped improving the manuscript. The authors are also grateful to the anonymous referees for their thorough reading of the paper and many valuable remarks. The research of both authors was supported by the Dutch BSIK project (BRICKS) and the EURO-NGI project.

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