P G Grinevich

R G Novikov
email: novikov@cmap.polytechnique.fr

Moutard transforms for the conductivity equation *

Keywords: Darboux-Moutard transforms, Conductivity equation, Integrability, Generalized analytic functions. Mathematical Subject Classification 35Q79, 35Q60, 35J15, 35C05, 30G20

We construct Darboux-Moutard type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux-Moutard type transforms for generalized analytic functions.

In addition, at least, some of the Darboux-Moutard type transforms of the present work admit direct extension to the conductivity equation in multidimensions. Relations to the Schrödinger equation at zero energy are also shown.

Introduction

We consider the two-dimensional isotropic conductivity equation:

div σ(x)∇u(x) = 0, x = (x 1 , x 2 ), x ∈ D ⊆ R 2 , ( 1
)
where D is an open bounded domain in R 2 . This equation arises in different physical context; see, for example, [START_REF] Landau | Theory of Elasticity[END_REF], [START_REF] Landau | Electrodynamics of Continuous Media[END_REF]. In particular, in electrical problems σ(x) is the elecrical conductivity in D, and u(x) is the electric potential. In solid state thermal problems σ(x) is the heat conductivity, and u(x) is the temperature.

In the present work we show that the conductivity equation ( 1) admits Moutard-type transforms, going back to [START_REF] Moutard | Sur la construction des équations de la forme 1 z ∂ 2 z ∂x∂y = λ(x, y) qui admettenent une intégrale générale explicite[END_REF]. Such transforms were successfully used in studies of integrable systems of mathematical physics and differential geometry, in spectral theory and in complex analysis; see, for example, [START_REF] Matveev | Darboux transformations and solitons[END_REF], [START_REF] Nimmo | Superposition principles associated with the Moutard transformation: an integrable discretization of a 2+1dimensional sine-Gordon system[END_REF], [START_REF] Yu | Darboux transformation for the modified Veselov-Novikov equation[END_REF], [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF], [START_REF] Novikov | Two-dimensional von Neumann-Wigner potentials with a multiple positive eigenvalue[END_REF], [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and Möbius geometry[END_REF], [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF], [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms, and holomorphic maps[END_REF], [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF], [START_REF] Matuev | The Moutard transformation of twodimensional Dirac operators and the conformal geometry of surfaces in four-space[END_REF], [START_REF] Novikov | Moutard type transformation for matrix generalized analytic functions and gauge transformations[END_REF], [START_REF] Novikov | Darboux-Moutard transformations and Poincaré-Steklov operators[END_REF]. In particular, the present article can be considered as a direct continuation of our recent works [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF] on Moutard-type transforms for the generalized analytic functions. In turn, works [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF] were stimulated by [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and Möbius geometry[END_REF].

We recall also that equation [START_REF] Allaire | Analyse asymptotique spectrale d'un problme de diffusion neutronique[END_REF] is the continuity equation

∂ x 1 I 1 + ∂ x 2 I 2 = 0, x ∈ D, (2)
for the current density I, I(x) = σ(x)∇u(x), I = (I 1 , I 2 ), x ∈ D.

Assuming that D is simply connected, we consider also the stream function v for I, where

∂ x 2 v = I 1 , -∂ x 1 v = I 2 , x ∈ D, (4)
(see, for example, [7]), where integration constant may depend on the particular situation.

The stream function v satisfies the following equation:

div σ -1 (x)∇v(x) = 0, x = (x 1 , x 2 ), x ∈ D ⊆ R 2 . (5)
In literature equation ( 5) is sometimes considered as the conjugate equation to (1); see, for example, [START_REF] Chaabi | A boundary value problem for conjugate conductivity equations[END_REF].

In particular, in the present work we use the fact that equation (1) can be written as a reduction of the following two-dimensional Dirac equation (see [START_REF] Brown | Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions[END_REF]):

∂ z 0 0 ∂ z - 0 q q 0 ψ 1 ψ 2 = 0 in D, (6)
where

∂ z = 1 2 (∂ x 1 -i∂ x 2 ) , ∂ z = 1 2 (∂ x 1 + i∂ x 2 ) , q = q(x), ψ j = ψ j (x), j = 1, 2, x = (x 1 , x 2 ). (7)
We recall that if u satisfies (1), then

ψ 1 = σ 1/2 ∂ z u, ψ 2 = σ 1/2 ∂ z u, (8)
satisfy [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF], where

q = - 1 2 ∂ z log(σ), q = - 1 2 ∂ z log(σ). (9)
We use also that ( 6) is equivalent to the following equation:

∂ z ψ = qψ in D, (10)
which is the basic equation of the generalized analytic functions theory (see [START_REF] Vekua | Generalized Analytic Functions[END_REF]). More precisely:

(i) if ψ 1 , ψ 2 satisfy (6), then ψ + = 1 2 (ψ 1 + ψ 2 ) and ψ -= 1 2i (ψ 1 -ψ 2 ) solve (10); (ii) if ψ + , ψ -satisfy (10), then ψ 1 = ψ + + iψ -, ψ 2 = ψ + + iψ -solve (6).
The property that ψ 1 , ψ 2 and q in (6) admit representations (8), ( 9) implies a non-trivial reduction of equation [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF]. The compatibility of this reduction with the Moutard-type transforms from [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF] is established in the present article.

The main results of the present work are given in Sections 3 and 4. In these Sections we construct and study Moutard-type transforms for the twodimensional conductivity equation [START_REF] Allaire | Analyse asymptotique spectrale d'un problme de diffusion neutronique[END_REF]. In addition, in Section 4 we show that, at least, some of these Moutard-type transforms admit direct extension to the conductivity equation in multidimension. Besides, in Section 5 we continue studies of the Moutard-type transforms constructed in Sections 3 and 4 in the framework of relations between the multidimensional conductivity equation and the Schrödinger equation at zero energy.

Note that, for the case of bounded domain D, possible natural analytical assumptions on q(x), σ(x) are as follows:

q ∈ L p (D), ( 11
) σ ∈ W 1,p (D), ( 12
) 0 < σ 0 ≤ σ(x) ≤ σ 1 < +∞, ( 13
)
where p > 2. Actually, assumption [START_REF] Matuev | The Moutard transformation of twodimensional Dirac operators and the conformal geometry of surfaces in four-space[END_REF] is essential in the standard theory of generalized analytic functions, see [START_REF] Vekua | Generalized Analytic Functions[END_REF], and assumption ( 13) is essential in the standard mathematical theory of the conductivity equation for the ellipticity.

Note also that the Moutard-type transforms constructed in the present work permit us, in particular, to study equation ( 1) not only under assumptions [START_REF] Moutard | Sur la construction des équations de la forme 1 z ∂ 2 z ∂x∂y = λ(x, y) qui admettenent une intégrale générale explicite[END_REF], [START_REF] Nimmo | Superposition principles associated with the Moutard transformation: an integrable discretization of a 2+1dimensional sine-Gordon system[END_REF], but also for singular conductivities σ, not satisfying [START_REF] Nimmo | Superposition principles associated with the Moutard transformation: an integrable discretization of a 2+1dimensional sine-Gordon system[END_REF], in a way similar to the approach used in [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF] to study generalized analytic functions with contour singularities.

Simple Moutard transforms for generalized analytic functions

Following [START_REF] Vekua | Generalized Analytic Functions[END_REF], [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF], we consider the pair of conjugate equations of the generalized analytic function theory:

∂ z ψ = q ψ in D, (14)
∂ z ψ + = -q ψ+ in D, (15)
where ∂ z , ∂ z are defined in (7),

z = x 1 + ix 2 , z = x 1 -ix 2 , D is an open simply connected domain in C ∼ = R 2 , q = q(z) is a given function in D.
In addition, in this article the notation f = f (x) = f (z) does not mean that f (z) is holomorphic function in z unless it is explicitly specified. Next, as in [START_REF] Vekua | Generalized Analytic Functions[END_REF], [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF], we associate with a pair of functions ψ, ψ + , satisfying [START_REF] Novikov | Two-dimensional von Neumann-Wigner potentials with a multiple positive eigenvalue[END_REF], [START_REF] Novikov | Moutard type transformation for matrix generalized analytic functions and gauge transformations[END_REF], respectively, the following imaginary-valued potential ω ψ,ψ + defined by:

∂ z ω ψ,ψ + = ψψ + , ∂ z ω ψ,ψ + = -ψψ + in D, (16)
where the pure imaginary integration constant may depend on the particular situation. We recall that the compatibility of ( 16) follows from ( 14), [START_REF] Novikov | Moutard type transformation for matrix generalized analytic functions and gauge transformations[END_REF]. Let f , f + be some fixed solutions of equations ( 14), ( 15), respectively, with given q. Then a simple Moutard-type transform M = M q,f,f + for the pair of conjugate equations ( 14), ( 15) is given by the formulas (see [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF]):

q = Mq = q + f f + ω f,f + , (17)
ψ = Mψ = ψ - ω ψ,f + ω f,f + f, ψ+ = Mψ + = ψ + - ω f,ψ + ω f,f + f + , (18)
where ψ, ψ + are arbitrary solutions of ( 14) and ( 15). The point is that the functions ψ, ψ+ defined in ( 18) satisfy the conjugate pair of Moutard-transformed equations (see [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF]):

∂ z ψ = q ψ in D, (19)
∂ z ψ+ = -q ψ+ in D, (20)
where q is defined in [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF]. In addition (in the simplest case), if D is a simply connected open bounded domain with C 1 -boundary, q satisfies (11) and f, f + ∈ W 1,p (D), and ω f,f + = 0 in D ∪ ∂D, then the transformed coefficient q satisfies (11) as well as the initial q (see [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]). On the other hand, the Moutard-type transforms [START_REF] Taimanov | Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces[END_REF], [START_REF] Taimanov | The Moutard transformation of two-dimensional Dirac operators and Möbius geometry[END_REF] permit to create and remove contour singularities in q, ψ, ψ + , see [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF]- [START_REF] Grinevich | Moutard transform approach to generalized analytic functions with contour poles[END_REF].

3 Reductions to the two-dimensional conductivity equation

In this Section we construct simple Moutard-type transforms for the conductivity equations ( 1), ( 5) as reductions of Moutard-type transform for the generalized analytic functions; see Section 2.

In this Section we assume that D is an open simply connected domain in

C ∼ = R 2 .
Lemma 1 A regular complex-valued function q(z) admits representation [START_REF] Landau | Electrodynamics of Continuous Media[END_REF] in D with a positive σ(z) if and only if

∂ z q(z) = ∂ z q(z), z ∈ D. (21)
This statement follows directly from the property that ( 21) is the compatibility condition for [START_REF] Landau | Electrodynamics of Continuous Media[END_REF] and from the formula

σ(z) = σ(z 0 ) exp   -2 z z 0 q(ζ)dζ + q(ζ)d ζ   , where σ(z 0 ) > 0.
In addition, in Lemma 1, for bounded D, the regularity assumption [START_REF] Matuev | The Moutard transformation of twodimensional Dirac operators and the conformal geometry of surfaces in four-space[END_REF] on q corresponds to the regularity assumptions ( 12), ( 13) on σ.

Let us define the following two special solutions f + R and f + I of equation [START_REF] Novikov | Moutard type transformation for matrix generalized analytic functions and gauge transformations[END_REF], where q is given by ( 9) with a regular positive σ:

f + R = σ(z), f + I = i σ(z) . ( 22
)
Lemma 2 A regular complex-valued function ψ(z) satisfies equation ( 14) with q(z) given by ( 9) with a positive σ(z) if and only if there exists a realvalued solution u(z) of ( 1) such that

ψ(z) = σ 1/2 (z)∂ z u(z), ψ(z) = σ 1/2 (z)∂ z u(z). (23)
In addition,

u = -iω ψ,f + I , (24)
where f + I is defined in [START_REF] Yu | Darboux transformation for the modified Veselov-Novikov equation[END_REF], ψ is defined in (23), ω ψ,ψ+ is defined via [START_REF] Novikov | Darboux-Moutard transformations and Poincaré-Steklov operators[END_REF].

Note also that

ψ = 1 2 σ -1/2 (I 1 -iI 2 ), I 1 = σ ∂u ∂x 1 , I 2 = σ ∂u ∂x 2 , (25)
where ψ, u are the functions of (23), and I is the current for the conductivity equation ( 1), see formula (3).

Lemma 3 For the conductivity equation ( 1) with regular positive σ the following formula holds:

v = -iω ψ,f + R , ( 26
)
where v(z) is the stream function associated via (4), [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms, and holomorphic maps[END_REF] with real-valued u(z) satisfying (1), f + R is defined in [START_REF] Yu | Darboux transformation for the modified Veselov-Novikov equation[END_REF], ψ is defined in (23).

Lemmas 2, 3 are proved in Section 6.

Theorem 1 Let q(z) be given by ( 9) in D with a positive regular σ(z). Let the transform q → q, ψ → ψ be defined by:

q = Mq = q + f f + ω f,f + , ψ = Mψ = ψ - ω ψ,f + ω f,f + f, (27)
where ψ denotes an arbitrary solution of ( 14), f is a fixed solution of equation ( 14),

f + = f + R or f + = f + I
, where f + R and f + I are defined in [START_REF] Yu | Darboux transformation for the modified Veselov-Novikov equation[END_REF]. Then ψ satisfies the Moutard-transformed equation [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF], and q admits the representation

q = - 1 2 ∂ z log(σ), (28)
where

σ =        -σ ω 2 f,f + R if f + = f + R , -σω 2 f,f + I if f + = f + I . (29)
In addition, the following Moutard-transformed conjugate pair of conductivity equations holds:

div σ∇ũ = 0 in D, (30)
div σ-1 ∇ṽ = 0 in D, (31)
where ũ = -iω ψ, f + I , f + I = i √ σ , (32) ṽ
= -iω ψ, f + R , f + R = √ σ. ( 33
)
The following scheme summarizes the Moutard-type transforms for the conductivity equations (1), [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms, and holomorphic maps[END_REF] given in Theorem 1:

σ (29) {f,f + } / / σ (28) / / q, σ, u (9),(23) / / q, ψ (27) {f,f + } / / q, ψ, (34)
σ, ψ (32),(33) / / ũ, ṽ.

The point is that each step in scheme (34) is given by quadratures. Theorem 1 is proved in Section 6.

Remark 1

In Theorem 1 we have the following two important cases:

(i) If ω f,f + has no zeroes in D, then σ arising in (29) is a regular positive function in D. If D is bounded, then ω f,f + can be always defined without zeroes by an appropriate choice of integration constant.

In addition, in a similar way with the simplest case of Section 2, if D is a simply connected open bounded domain with C 1 -boundary, σ satisfies ( 12), ( 13), f ∈ W 1,p (D), then the transformed coefficient σ satisfies ( 12), ( 13).

(ii) If ω f,f + has zeroes in D, then σ arising in (29) is non-negative and has either zeros or poles in D. In these singular cases the standard methods for solving the conductivity equation (30) does not work; but these both singular cases are interesting and relevant for physical problems. The point is that the Moutard-type transform of Theorem 1 generating σ simultaneously provides a method for solving equations (30), (31).

Simple Moutard transforms for the multidimensional conductivity equation

Let M I and M R denote the Moutard-type transforms of Theorem 1 for f + = f + I and f + = f + R , respectively. In the next Theorem we give an explicit local realization of the transform M I for conductivity equation ( 1) and an explicit local realization of the transform M R for the conjugate equation [START_REF] Grinevich | Generalized analytic functions, Moutard-type transforms, and holomorphic maps[END_REF]. However, the action of M I on the solutions of ( 5) and M R on the solutions of ( 1) is non-local and requires one quadrature.

Theorem 2 Suppose that f + = f + I in Theorem 1 in Section 3. Then the Moutard-type transform of Theorem 1 for the two-dimensional conductivity equation ( 1) is reduced to the transform

σ → σ = M I σ = u 2 1 σ, ( 35
) u → ũ = M I u = u -1 1 u,
where u 1 = -iω f,f + I , u(x) is an arbitrary real solutions of (1), and ũ(x) satisfies the Moutard transformed conductivity equation (30).

Suppose that f + = f + R in Theorem 1 in Section 3. Then the Moutardtype transform of Theorem 1 for the conjugate two-dimensional conductivity equation ( 5) is reduced to the transform

σ → σ = M R σ = v -2 1 σ, ( 36
) v → ṽ = M R v = v -1 1 v, where v 1 = -iω f,f + R , v(x)
is an arbitrary real solution of ( 5), and ṽ(x) satisfies the Moutard transformed conjugate conductivity equation (31).

Due to formulas (24), (26) in Lemmas 2, 3, functions u 1 (x), v 1 (x) in Theorem 2 are fixed real solutions of ( 1), ( 5), respectively.

Note that the Moutard-type transform (35) admits a direct extension to the conductivity equation

div σ(x)∇u(x) = 0, x = (x 1 , x 2 , . . . , x d ), x ∈ D ⊆ R d , (37)
in dimension d ≥ 1 (and, in particular, in dimension d = 3). where u denotes an arbitrary solution of (37), and w is a fixed solution of (37). Then the following Moutard-transformed conductivity equation holds:

div σ(x)∇ũ(x) = 0, x ∈ D ⊆ R d . ( 39
)
Theorems 2, 3 are proved in Section 6.

The following result shows that subsequent application of Moutard-type transforms from Theorem 3, and, as a corollary, subsequent application of transforms from Theorem 2 of the type M I , only, or of the type M R , only, does not generate new more complicated transforms.

Proposition 1

The following composition formula holds:

M σ,ũ 2 • M σ,u 1 = M σ,u 2 , where σ = M σ,u 1 σ, ũ2 = M σ,u 1 u 2 , ( 40
)
where M = M σ,w denotes the Moutard-type transform defined by (38), u 1 , u 2 are arbitrary fixed solutions of (37).

In particular, from (38), (40) it follows that

M σ,ũ 2 • M σ,u 1 = id, if u 2 = 1. ( 41
)
Proposition 1 is proved in Section 6.

On the other hand, subsequent application of transforms from Theorem 2 of the type M I and then M R (or M R and then M I ) yields already new transformations.

Examples 1) Theorem 3 implies, for example, that the multidimensional conductivity equation (37) with σ = w 2 , where w is a real harmonic function in D, is integrable in the sense that all its solutions u are of the form u = w -1 φ, where φ is an arbitrary real harmonic function in D.

2) Theorem 2 implies, for example, that the two-dimensional conductivity equation ( 1) in an open simply connected domain D with σ = w -2 , where w is a real harmonic function in D, is integrable in the sense that all its solutions u are of the form

u(x) = (42) = - x x 0 w(ξ)∂ ξ 2 φ(ξ) -φ(ξ)∂ ξ 2 w(ξ) dξ 1 -w(ξ)∂ ξ 1 φ(ξ) -φ(ξ)∂ ξ 1 w(ξ) dξ 2 + c,
where φ is an arbitrary real harmonic function in D, x, x 0 ∈ D, x 0 is fixed, c is an arbitrary real constant. Here we used formulas (36) and formulas (3), (4).

3) Theorem 2 also implies, for example, that the two-dimensional conductivity equation ( 1) in an open simply connected domain D with σ = w -2 u 2 1 , where w is a real harmonic function in D, u 1 is given by (42) with fixed φ = φ 1 and c = c 1 is integrable in the sense that all its solutions U are of the form U = u -1 1 u, where u is given by ( 42). Here we used formulas (35) with σ and u of Example 2.

For simplicity, in Examples 1-3 one can assume that w and u 1 have no zeroes in D, but, formally, these examples are also valid if w or u 1 have zeroes in D.

4) More generally, applying subsequently transforms of Theorem 2, where applications of M R are followed by applications of M I and vice verse, we obtain a very large class of integrable in quadratures two-dimensional conductivity equations in a open simply connected domain D.

Relations to the Schrödinger equation

It is well-known that the substitution

u = σ -1/2 ψ, (43)
reduces the multidimensional conductivity equation (37) to the Schrödinger equation at zero energy:

-∆ψ(x) + Q(x)ψ(x) = 0, x ∈ D ⊆ R d , d ≥ 1, (44)
where

Q(x) = ∆ σ 1/2 (x) σ 1/2 (x) . ( 45
)
Proposition 2 a) The potential Q defined by ( 45) is invariant with respect to the transforms σ → σ defined in (38). b) In the two-dimensional case the potential Q defined by ( 45) is not invariant with respect to the transforms σ → σ defined in (36). In addition, Q is not invariant with respect to transform σ → σ -1 relating the coefficients in the conjugate pair (1), ( 5) of the two-dimensional conductivity equations.

Note that Proposition 2 is in a good agreement with the fact, that no nontrivial Moutard-type transform is known for the Schrödinger equation (44) in dimension d ≥ 3, whereas the the Schrödinger equation (44) in dimension d = 2 admits a quite rich family of Moutard transforms; see, for example, [START_REF] Moutard | Sur la construction des équations de la forme 1 z ∂ 2 z ∂x∂y = λ(x, y) qui admettenent une intégrale générale explicite[END_REF], [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF].

Item a) of Propositions 2 is proved by the following calculation:

Q = ∆ σ1/2 ) σ1/2 = ∆ wσ 1/2 wσ 1/2 = w∆ σ 1/2 wσ 1/2 + ∆w)σ 1/2 + 2 ∇w ∇σ 1/2 wσ 1/2 = = Q + σ∆w + ∇w ∇σ wσ = Q + div σ∇w wσ = Q. ( 46
)
Here we used, in particular, that w satisfies the multidimensional conductivity equation (37). Item b) of Propositions 2 follows from Examples 1 and 2 given in Section 4 and the fact that w -1 is not harmonic, in general, for harmonic w (where σ = w 2 in Example 1 and σ = w -2 in Example 2 with initial σ ≡ 1).

Finally, consider the equations

-div σ(x)∇u(x) + Q 1 (x)u(x) = 0, x ∈ D ⊆ R d , (47)
-div σ(x)∇w(x) + Q 2 (x)w(x) = 0, x ∈ D ⊆ R d . (48)
Then (at least formally)

-div σ(x)∇ũ(x) + q(x)ũ(x) = 0, x ∈ D ⊆ R d , (49)
where

σ(x) = w 2 (x)σ(x), ũ(x) = w -1 (x)u(x), q(x) = w 2 (x)(Q 1 (x)-Q 2 (x)), (50)
w(x) is a fixed solution of (48), u(x) is an arbitrary solution of (47). For particular σ, Q 1 , Q 2 , w such a result can be found in the literature, see e.g. [START_REF] Vanninathan | Homogenization of eigenvalue problems in perforated domains[END_REF], [START_REF] Allaire | Analyse asymptotique spectrale d'un problme de diffusion neutronique[END_REF]. This result for σ = const > 0, Q 1 = Q 2 , reduces the Schrödinger equation at zero energy (47) to the conductivity equation (37). In particular, this permits to construct multidimensional integrable conductivity equations from multidimensional Schrödinger equations integrable at zero energy. In addition, in dimension d = 2 a very large class of Schrödinger equations integrable at zero energy can be constructed via classical Moutard transform (see, e.g. [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF], [START_REF] Novikov | Two-dimensional von Neumann-Wigner potentials with a multiple positive eigenvalue[END_REF], [START_REF] Novikov | Darboux-Moutard transformations and Poincaré-Steklov operators[END_REF]). This approach for constructing integrable conductivity equations will be developed elsewhere.

6 Proofs of Lemmas 2, 3, Theorems 1-3 and Proposition 1

Proof of Lemma 2. If σ is a real-valued regular positive function, u is a real-valued regular function, and q, ψ are defined by ( 9) and (23), respectively, then it is known that ψ satisfies ( 14) if and only if u satisfies (1); see Introduction. This can be also verified by a direct calculation. Conversely, suppose that q is defined by ( 9) with a regular real-valued positive σ, and ψ satisfies [START_REF] Novikov | Two-dimensional von Neumann-Wigner potentials with a multiple positive eigenvalue[END_REF]. Define u by (24). It remains to verify that (23) holds. This verification uses ( 16), [START_REF] Yu | Darboux transformation for the modified Veselov-Novikov equation[END_REF] and consists of the following:

∂ z u = ∂ z (-iω ψ,f + I ) = -iψf + I = -iψ i √ σ = ψ √ σ , (51)
∂ z u = ∂ z (-iω ψ,f + I ) = -i -ψ f + I = iψ -i √ σ = ψ √ σ . ( 52
)
Lemma 2 is proved. Proof of Lemma 3. Formulas (4) defining v can be rewritten as

∂ z v = - i 2 (I 1 -iI 2 ), ∂ z v = i 2 (I 1 + iI 2 ). ( 53
)
Using formulas [START_REF] Novikov | Darboux-Moutard transformations and Poincaré-Steklov operators[END_REF], definition f + R in [START_REF] Yu | Darboux transformation for the modified Veselov-Novikov equation[END_REF] and formulas (25) one can see that (53) is equivalent to (26).

Lemma 3 is proved.

Proof of Theorem 1. The statement that ψ satisfies Moutard-transformed equation [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF] was proved in [START_REF] Grinevich | Moutard transform for the generalized analytic functions[END_REF].

If q is defined by (28) with σ defined by (29), then:

q = - 1 2 ∂ z log(σ) = - 1 2 ∂ z log(σ) + ∂ z log(ω f,f + R ) = (54) = q + f f + R ω f,f + R = q + f f + R ω f,f + R if f + = f + R , q = - 1 2 ∂ z log(σ) = - 1 2 ∂ z log(σ) -∂ z log(ω f,f + I ) = (55) = q - f f + I ω f,f + I = q + f f + I ω f,f + I if f + = f + I .
Here, we used that f + R is real-valued and f + I is imaginary-valued. In fact, calculations (54), (55) prove representations (28), (29) for q in (27). Formulas (30), (32) and (31), (33) follow directly from the Moutardtransformed equation [START_REF] Taimanov | On the Moutard transformation and its applications to spectral theory and Soliton equations[END_REF], the representation (28) and Lemmas 2 and 3.

This completes the proof of Theorem 1 under the assumption that ω f,f + has no zeroes in D.

Proof of Theorem 2. The first of formulas (35) follows directly from (29). In view of formulas (30), (32), the proof of the second of formulas (35) consists of the verification that the following identity holds:

(u 1 ) -1 u = -iω ψ, f + I , (56)
where f + I is defined in (32), ψ is defined in (27), where ψ is defined in (23), f + = f + I , and f and u 1 are related by ( 23), (24) with ψ = f , u = u 1 . In

Theorem 3

 3 Let σ be a real positive regular function in D, where D is an open domain in R d , d ≥ 1. Let the transform σ → σ, u → ũ be defined by σ = Mσ = w 2 σ, (38) ũ = Mu = w -1 u,

Acknowledgments. We thank Grégoire Allaire for drawing our attention to the articles [21], [1], which use a reduction of equation (47) to equation (49) via (48), (50).

The main part of the work was fulfilled during The first author was partially supported by the Russian Foundation for Basic Research, grant 17-51-150001 "Quasilinear equations, inverse problems, and their applications". The second author was partially supported by PRC 1545 CNRS/RFBR: " Équations quasi-linéaires, problèmes inverses et leurs applications".

turn, (56) can be rewritten as:

where

where we used (24) for ψ, u and for ψ = f , u = u 1 . We have:

where we used (23) for ψ, u and for ψ = f , u = u 1 . Thus, (57) holds. Formula (58) follows, for example, from reality of u 1 , √ σ, u. This completes the proof of (35), at least if u 1 has no zeroes in D. Formulas (36) are proved in similar way, at least if v 1 has no zeroes in D.

Theorem 2 is proved.

Proof of Theorem 3. We have that:

∂ j σ(x)∂ j ũ(x) = w∂ j σ∂ j u) -u∂ j σ∂ j w , j = 1, . . . , d.

Therefore, div σ(x)∇ũ(x) = w div σ∇u) -u div σ∇w = 0. (63) Theorem 3 is proved, at least is w has no zeroes in D.

Proof of Proposition 1

We have:

Remark 2 Formally, the proofs of Theorems 1-3 and Proposition 1 remain valid if ω f,f + , u 1 , v 1 , w, u 2 have zeroes in D, but a proper analytic picture requires subsequent investigations in these cases.


Ivan Morandi

Francesco Bronzino

Renata Teixeira

Srikanth Sundaresan
email: srikanth@icsi.berkeley.edu

Service traceroute: Tracing Paths of Application Flows

Traceroute is often used to help diagnose when users experience issues with Internet applications or services. Unfortunately, probes issued by classic traceroute tools differ from application traffic and hence can be treated differently by routers that perform load balancing and middleboxes within the network. This paper proposes a new traceroute tool, called Service traceroute. Service traceroute leverages the idea from paratrace, which passively listens to application traffic to then issue traceroute probes that pretend to be part of the application flow. We extend this idea to work for modern Internet services with support for identifying the flows to probe automatically, for tracing of multiple concurrent flows, and for UDP flows. We implement command-line and library versions of Service traceroute, which we release as open source. This paper also presents an evaluation of Service traceroute when tracing paths traversed by Web downloads from the top-1000 Alexa websites and by video sessions from Twitch and Youtube. Our evaluation shows that Service traceroute has no negative effect on application flows. Our comparison with Paris traceroute shows that a typical traceroute tool that launches a new flow to the same destination discovers different paths than when embedding probes in the application flow in a significant fraction of experiments (from 40% to 50% of our experiments in PlanetLab Europe).

Introduction

Internet services and applications rely on highly distributed infrastructures to deliver content. When applications stop working or when their performance degrades, service providers and more sophisticated users often resort to traceroute to narrow down the likely location of the problem. Traceroute issues probes with increasing TTL to force routers along the path towards a destination to issue an ICMP TTL exceeded message back to the source, which iteratively reveals the IP addresses of routers in the path [START_REF] Jacobson | traceroute[END_REF].

Traceroute, however, may fail to reveal the exact path that a given application flow traverses. For example, Luckie et al. [START_REF] Luckie | Traceroute Probe Method and Forward IP Path Inference[END_REF] have shown that depending on the traceroute probing method (ICMP, UDP, and TCP) the set of reached destinations and discovered links differ. The authors explain these differences by the presence of middleboxes in the path such as load balancers and firewalls that make forwarding decisions based on flow characteristics. These results imply that diagnosing issues on application flows must ensure that traceroute probes have the same characteristics as the application's packets.

This paper develops a traceroute tool, called Service traceroute, to allow discovering the paths of individual application flows. Service traceroute passively listens to application traffic to then issue probes that pretend to be part of the application flow. Some traceroute tools (for instance, paratrace [START_REF] Kaminsky | Parasitic Traceroute via Established TCP Flows & IPID Hopcount[END_REF], TCP sidecar [START_REF] Sherwood | Touring the internet in a tcp sidecar[END_REF], and 0trace [START_REF] Edge | Tracing behind the firewall[END_REF]) already enable probes to piggyback on TCP connections. These tools observe an active TCP connection to then insert traceroute probes that resemble retransmitted packets. TCP sidecar was developed for topology mapping, whereas paratrace and 0trace for tracing past a firewall. As such, they lack the support for tracing paths of modern application sessions, which fetch content over multiple flows that change dynamically over time. First, these tools provide no means to identify the set of application flows to trace. They require as input the destination IP address and the destination port to detect the target application flow. Second, they trace one target application flow at a time. Finally, these tools lack the support for tracing application flows using UDP as transport protocol, which are increasing thanks to the adoption of QUIC protocol [START_REF] Langley | The quic transport protocol: Design and internet-scale deployment[END_REF].

Our work makes the following contributions. First, we develop and implement Service traceroute ( §2), which we release as open source software. Service traceroute is capable of identifying application flows to probe and of tracing the paths of multiple concurrent flows of both TCP and UDP flows. For example, a user may simply specify trace 'Youtube' and Service traceroute will identify Youtube flows and then trace all of their paths. Service traceroute is configurable to cover a large variety of Internet services.

Our second contribution is to conduct the first thorough evaluation of the effect of embedding tracetoute probes within application flows. One issue with this approach is that we may hurt application performance. Our evaluation shows that in the vast majority of cases, Service traceroute has no side-effect on the target application ( §4). Finally, we compare Service traceroute with 0Trace, which also embeds probes within a target application flow, and with Paris Traceroute, which launches a new flow for probing ( §5). Our comparison with Paris traceroute shows that when we launch a new flow with traceroute probes we observe a different path in around 40% to 50% of paths depending on the application. This difference reduces considerably for the majority of applications when we run Paris traceroute with the same flow ID as the target application flow, which shows that differences are mostly due to middleboxes that make forwarding decisions per flow. These results highlight the need for Service traceroute, which automatically identifies the flow IDs of the target application to create probes.

Tool Design and Implementation

Service traceroute follows the same high-level logic as paratrace or 0trace. Given a target application flow, which we define as the application flow whose path we aim to trace, Service traceroute proceeds with two main phases. The first phase is the passive observation of a target application flow to define the content of the probes. Then, the second phase involves active injection of TTL-limited probes within the application flow. The main difference is that Service traceroute identifies the flows to trace automatically and supports tracing paths traversed by multiple application flows concurrently. The user can either directly specify the set of target application flows or simply describe a high-level service (e.g., Youtube). Service traceroute will then trace paths traversed by all the flows related to the target service. This section first describes the two phases focusing on the new aspects of Service traceroute to allow per service tracing and then presents our implementation. Library and command-line versions of Service traceroute, together with the scripts to perform data analysis are available as open source projects [START_REF]Service Traceroute[END_REF].

Observation of target application flow

Service traceroute passively observes traffic traversing a network interface to search for packets with the flow-id of the target application flows. 3 Service traceroute takes a set of target application flows as input, in contrast with previous tools which can only trace the path traversed by one single application flow. Users can either explicitly specify one or more target application flows or they can simply specify a service. Service traceroute uses a database of signatures of known services to inspect DNS packets in real-time and identify flows that match the target service. We release the DB as open source, so users can contribute to add or update the signatures in the database [START_REF]Service Traceroute[END_REF]. We define as signature the set of domains and IP addresses corresponding to a specific service. For instance, 'google.com' or the corresponding IP addresses can be used in the signature to detect Google services. Our current database has signatures for popular video streaming services such as Netflix, Youtube, and Twitch. We identify web flows simply from the domain or the host name given as input. For additional flexibility, it is possible to add domains and IP addresses via command line parameters or through the library API.

Path tracing

Only once it identifies a packet belonging to the target application flow, Service traceroute will start the tracing phase. This phase works as classic traceroute implementations sending probes with increasing TTL, but Service traceroute creates a probe that takes the form of an empty TCP acknowledgement that copies the 5-tuple of the flow as well as its sequence number and acknowledgement number (similar to paratrace and 0trace). We rely on the flow-id plus the IPID field to match issued probes with the corresponding ICMP responses. We note that this is sufficient to correctly identify probes even when tracing multiple concurrent target application flows. The maximum number of concurrent target application flows varies based on the used configuration as the IPID field is dynamically sliced based on the number of probes that have to be generated. For example, with traceroute standard parameters, i.e. maximum distance of 32 and 3 packets per hop, Service traceroute can trace paths of more than 600 target application flows in parallel.

Service traceroute stops tracing when the target application flow closes to avoid any issues with middleboxes (which may interpret probes after the end of the connection as an attack) and also to reduce any network and server overhead. In contrast to prior tools that only support TCP, we add support for UDP. In this case, we create probes with empty UDP payload, but with the same 5-tuple flow-id as the target application flow. Given UDP has no explicit signal of the end of the flow (like the FIN in TCP), we stop tracing if the flow produces no further packets (either received or sent) after a configurable time interval.

Implementation

We implement Service traceroute in Go and release command-line and library versions. The command-line version is useful for ad-hoc diagnosis, whereas the library allows easy integration within monitoring systems. The library version of Service traceroute outputs a json data structure that contains the discovered interfaces with the observed round-trip-time values. For the command line version, Service traceroute shows the results of each trace in the traceroute format, i.e., the list of hops with the corresponding round-trip times.

Service traceroute is configurable to adapt to different applications. It includes three types of probing algorithms that capture the tradeoff between tracing speed and network overhead. PacketByPacket sends only one probe at a time. HopByHop sends a configurable number of probes with the same TTL at a time (3 by default). Concurrent sends all probes to all hops at once. Given that Service traceroute requires the target application flow to be active during tracing, some applications with short flows (e.g., Web) require the higher overhead of Concurrent to complete all the probes within the flow duration. Service traceroute also allows configuring the number of probes for each TTL, the inter-probe time, and inter-iteration time (i.e., the time between packets with different TTL) to further control the tradeoff between tracing speed and overhead. Finally, Service traceroute allows to specify three types of stop conditions: the maximum distance from the source, the maximum number of non-replying hops, like Paris Traceroute, or explicit stop points in the form of IP addresses. The stop condition is particularly important for Service traceroute because the destination host will never respond with an ICMP error message as probes are part of the target application flow.

Following extensive calibration tests (reported in Section 3), we set Service traceroute to use as default the Concurrent mode, together with a maximum distance of 32 and 3 probes per hop.

Evaluation Method

We design our evaluation around two questions. First, does Service traceroute affect the target application flows? Service traceroute injects new packets within the application flow. Although the majority of these packets will be discarded before they reach the servers, a few probe packets will reach the end-host and can potentially affect the target application flows. Second, do paths discovered with Service traceroute differ from those discovered by other traceroute methods? One assumption of our work is that paths taken by classic traceroute probes may not follow the same paths as the packets of the target application flows. We present a preliminary evaluation to help answer these questions, where we use Service traceroute to trace paths of target application flows corresponding to the two most popular Internet services: Web and video. We compare our results with that of Paris traceroute [START_REF] Augustin | Avoiding Traceroute Anomalies with Paris Traceroute[END_REF] and 0Trace [START_REF] Edge | Tracing behind the firewall[END_REF].

Web. We select the top-1000 Alexa webpages on April 14 2018 as target web flows.

Video. We focus on two popular video streaming services: Twitch and YouTube. We select Twitch videos on their homepage where Twitch shows dynamically a popular live streaming video. While for YouTube, we select 20 random videos from the catalogue obtained after arbitrarily searching with the keyword "4K UHD". With YouTube, we evaluate both TCP and QUIC.

Calibration. We run extensive experiments to calibrate Service traceroute for these two applications varying the probing algorithm and the number of probes per hops [START_REF] Morandi | Service traceroute: Tracing paths of application flows[END_REF]. Our results (not shown for conciseness) indicate that the best settings to maximize the fraction of completed traceroutes while minimizing the probing overhead is different for video and Web. For video, we use the HopByHop probing algorithm with a timeout of 2 seconds to wait for ICMP replies, whereas for web we use the Concurrent probing algorithm. For both, we set the maximum distance to 32 and the number of probes per hop to 3.

Comparison with Paris traceroute. We select Paris traceroute because its Multipath Detection Algorithm (MDA) [START_REF] Veitch | Failure Control in Multipath Route Tracing[END_REF] can discover with high probability all paths between the source and the destination in case there is a load balancer in the path. This allows us to disambiguate whether the differences we may encounter between Paris traceroute and Service traceroute are because of load balancing or some other type of differential treatment. We evaluate four versions of Paris traceroute with MDA enabled using the three protocols ICMP, UDP, and TCP as well as Paris traceroute to trace a single path with the same 5-tuple as the target application flow.

Comparison with 0Trace. We select 0Trace as it implements the idea of embedding probes in a target application flow and it has a working implementation. 4 0Trace, however, requires as input the flow to probe, which is hard to know in advance. We used Service traceroute's DNS resolution to detect the flow to probe and then launch 0Trace. Unfortunately, the download time for web pages is extremely short and our script was too slow to detect the target application flows and then run 0Trace. Hence, for this comparison we focus only on Twitch and Youtube. This experience illustrates the challenge of running 0Trace in practice and highlights the importance of integrating flow identification in Service traceroute.

Experiment setup. We run our tests during 30 days in July 2018 from 27 PlanetLab nodes in Europe. 5 Experiments for video and Web are similar. We first launch Service traceroute, then we start streaming a video or downloading a webpage, once that is done we run the four versions of Paris traceroute and 0Trace back-to-back. Then, we stream again the same video or download the same webpage without Service traceroute. We have run a total of 459 videos, 153 for Twitch and 306 equally split between YouTube with TCP and with QUIC, and 1000 Web experiments. All datasets collected in our experiments are available [START_REF]Service Traceroute[END_REF].

Data representativeness. Webpages in the Alexa top-1000 list are often hosted on CDNs [START_REF] Scheitle | A long way to the top: Significance, structure, and stability of internet top lists[END_REF]. This choice guarantees that we cover the large majority of locations hosting web content (i.e., all major CDN providers), but we may fail to capture the behavior of smaller webpages. Another bias comes from our choice of PlanetLab nodes as it is well known that they are mostly connected via academic networks [START_REF] Banerjee | The interdomain connectivity of planetlab nodes[END_REF][START_REF] Spring | Using planetlab for network research: myths, realities, and best practices[END_REF] and hence may fail to capture the behavior of commercial ISPs. Even with these limitations, our European-scale evaluation is useful to determine whether or not Service traceroute affects the application flows of popular services (top-1000 Alexa as well as Twitch/Youtube). It is also useful to shed some light on whether there are differences between paths discovered with Service traceroute and more traditional traceroute paths. The generalization of these results to vantage points located in other areas of the Internet and to other services would require larger scale experiments.

Side Effects of Service traceroute

This section evaluates whether Service traceroute affects target application flows. Firewalls or servers may mistakenly interpret too many duplicated packets within a flow as an attack or losses, which in turn may cause application flows to be blocked or achieve lower throughput. Although the idea of piggybacking traceroute probes within application flows has been around for approximately a decade, there has been no prior evaluation of whether it can hurt target application flows. TCP sidecar evaluates the intrusiveness of their method, but only by measuring the number of abuse reports [START_REF] Sherwood | Touring the internet in a tcp sidecar[END_REF]. 5 Service traceroute failed to run on PlanetLab US nodes due to compatibility issues.

PlanetLab US nodes use an old Linux distribution (Fedora 2), which lacks many required tools and libraries necessary to run our program.

Metrics

We select different metrics to measure properties of target application flows. Flow duration refers to the time between the first and the last packet of a flow.

For TCP, we measure the time from the server SYN to the first FIN or RST.

For UDP, we measure the time from the first and the last packet coming from the server. We compute the average throughput of a target application flow as the total application bytes divided by the flow duration. In addition to these metrics, which we can compute for both TCP and UDP flows, we have three other TCP specific metrics: the number of resets, which capture the number of target application flows closed by resets; window size is the difference between the minimum and the maximum TCP window size of the server for an application flow; and the number of retransmissions is the number of retransmission from the server per application flow.

Aborted flows

We first study whether Service traceroute causes flows to be aborted. We have seen no video sessions that ended with resets in our experiments. Even though our analysis is only from PlanetLab vantage points in Europe, we believe that this result will hold more generally for both Twitch and Youtube as these type of large video providers deploy multiple versions of the same software across servers/caches [START_REF]Netflix Open Connect Overview[END_REF][START_REF]Google Cloud Overview[END_REF]. Any differences will depend on middleboxes placed either close to the clients or in the path towards the service. Our results for webpage downloads are also encouraging, we see no aborted flows. In some preliminary experiments we observe resets for three of the top-1000 websites only with Service traceroute. Our manual analysis suggests that either some firewall close to the website or the web server itself is resetting the flows due to the duplicate packets.

Flow performance

We next evaluate whether Service traceroute affects flow performance. Figure 1 presents the cumulative distribution function of the flow duration in seconds with and without Service traceroute. We present eight curves: two for video sessions over TCP both for Twitch and Youtube, two for Youtube sessions over UDP, and two for all web page downloads. We see that the distributions with and without Service traceroute are mostly the same. Similarly, our analysis (omitted due to space constraints) shows that the distributions of average throughput, TCP window size, and retransmissions are similar with and without Service traceroute. We conclude that Service traceroute has no effect on the performance of target application flows.

Comparison with traceroute tools

The key motivation for building Service traceroute is that we must send probes within the target application flow to discover its path. Although Luckie et al. have observed different paths depending on the traceroute method (UDP, ICMP, or TCP), no prior work has studied how often piggybacking traceroute probes within application flows will discover different paths. This section compares Service traceroute with different traceroute probing methods using Paris traceroute, which discover all paths between a source and destination in the presence of load balancing, and 0Trace, which also piggybacks probes inside an application flow.

Metrics

We select two metrics to compare the discovered paths. The path length captures the distance from the source to the last hop that replies to probes. For Paris traceroute, we take the length of the longest path in case of multiple paths. The path edit distance captures the edit distance between the path discovered with Service traceroute and that discovered with another traceroute (0Trace or Paris traceroute). The edit distance is the minimum number of operations (insertion, deletion, and substitution) needed to transform one string (in our case, one path) into the other. In case Paris traceroute returns multiple paths, we select the one with the smallest edit distance. This allows us to focus on the best case. We treat empty hops (marked with a * ) as any other character. When we observe differences between paths, we analyze where the differences are in the path: origin AS, middle of the path, or destination AS. We map IPs to ASes using the RIPEstat Data API [START_REF]RIPEstat Data API[END_REF]. The location where the two paths diverge help us understand the placement of middleboxes.

Path lengths

We study the length of paths discovered with Service traceroute, Paris traceroute MDA (TCP, ICMP, and UDP), and 0Trace. The comparison of path lengths helps capture which versions of traceroute discover more hops. For application diagnosis it is important that the tool reveals most (hopefully all) of the path, so that we can identify issues in any parts of the path. Figure 2 presents the cumulative distribution functions of path length for each service: Web, Twitch, and Youtube (UDP and TCP). We see that for all three services, probing with TCP and UDP discovers less hops. The Web results confirm Luckie et al. [START_REF] Luckie | Traceroute Probe Method and Forward IP Path Inference[END_REF]'s analysis from ten years ago, which showed that UDP probes cannot reach the top Alexa web sites as probes correspond to no active flow. Service traceroute discovers longer paths for all three services. ICMP and 0Trace discover paths that are almost as long as those discovered by Service traceroute. The next sections characterize the path edit distance and the location of path differences to shed light on the causes of the differences we observe in path length.

Path differences when tracing with different flow IDs

This section studies the differences in paths discovered by Service traceroute versus by other traceroutes in the most typical case, i.e., when traceroute starts a new flow and picks the port numbers with no knowledge of the target application flow ID. We compare with Paris traceroute MDA using TCP, UDP, and ICMP.

Figure 3 presents the cumulative distribution functions of path edit distance between Service traceroute and Paris traceroute for Web, Twitch, and Youtube (UDP and TCP). A path edit distance of zero corresponds to the case when the Paris traceroute output contains the path discovered by Service traceroute. We see that even though we select the closest path in Paris traceroute's output to compute the edit distance, the path discovered with Service traceroute only matches that discovered by Paris traceroute MDA in about 55% of the webpage downloads, 50% of the Twitch sessions, and almost 75% of the Youtube streaming sessions. When paths discovered by Service traceroute differ from paths discovered by Paris traceroute, the edit distance is relatively high as the vast majority of paths towards these services is less than 15 hops long. In general, Twitch has longer paths (up to 30 hops) and Youtube shorter (up to 20 hops), which explains the differences in the values of edit distance we observe. For Twitch, UDP discovers paths that are the most similar to Service traceroute's paths, whereas for both Web and Youtube, ICMP leads to the most similar paths.

Path differences when tracing with same flow ID as application

One possible explanation for the differences we observe in the previous section is the fact that the flow ID of Paris traceroute probes is different than the ID of the target application flow, which can trigger different forwarding decisions in middleboxes that act per flow. In this section, we compare Service traceroute's output with Paris traceroute when it uses the same flow ID as the target application flow. Note that in this case Paris traceroute still runs after the target application flow finishes, we get the correct flow ID based on the Service traceroute's run just before in order to guarantee a complete match Figure 4 compares the path discovered by Service traceroute with that discovered by Paris traceroute when using the exact same flow ID as the target application flow. In this case, Paris traceroute discovers the same path as Service traceroute more often than when probing with MDA: about 65% of Twitch sessions, 91% of Youtube sessions, and 93% of web downloads. This result shows that issuing probes with a different flow ID than that of the target application flow causes most of the differences we observe in the previous section. The remaining differences are due to three possible causes: (i) path changes that might occur between the runs of Service traceroute and Paris traceroute; (ii) per-packet load balancing; or (iii) middleboxes (such as application-layer proxies or firewalls) that track the state of TCP connections and may hence drop packets after connections are terminated. In fact, in our initial testing we noticed cases of probes not generating any ICMP response if issued after the target application flow finishes. We further examine the paths for Twitch to shed light on the reasons for the large fraction of paths that are different between Paris traceroute and Service traceroute. It is unlikely to have routing changes for about 45% of paths and we verified that there are no middleboxes dropping our probes (which would appear as stars). Thus, we conjecture that the differences are likely due to per-packet load balancing, but we must run further experiments to verify this conjecture.

We also compare Service traceroute with 0Trace. Unfortunately, due to how web browsers loop across a large number of different ports, both Twitch and Youtube often change port numbers between consecutive runs. Given that we launch a new video session to probe with 0Trace, the result is that Service traceroute and 0Trace often issue probes with different flow IDs. This issue biases our experiments, and hence we omit these results (available at [START_REF] Morandi | Service traceroute: Tracing paths of application flows[END_REF]).

Location of path divergence points

To help explain our results we study the location of the points where Service traceroute's and Paris traceroute's paths diverge, which we call the divergence point. Table 1 shows the fraction of experiments with divergence points at the origin AS, the middle of the path, and the destination AS. We conduct this analysis to help explain the results in the previous sections, but we note that the findings are heavily biased by our vantage points and destinations. For the three applications, when comparing with MDA most of the divergence points are in the middle (from 19% for Youtube to above 40% for Twitch). Given the middle contains more hops it is not too surprising that it also contains more divergence points. When using Paris traceroute with the same flow ID, however, the percentage of divergence points in the middle decreases substantially to less than 2% for Web and Youtube and to 24% for Twitch. This result indicates that divergence points in the middle mostly correspond to middleboxes that perform per-flow forwarding. Paris traceroute's MDA discovers all possible interfaces for every hop of the path and we compare the closest path MDA finds to Service traceroute's output, but MDA often uses different flow IDs than the target application flow and hence it may not get the exact same sequence of hops end-to-end. For Paris traceroute with the same flow ID, we observe more divergence points at the origin, which may indicate path changes. The only exception is Twitch, which still has around 24% of divergence points in the middle. Our analysis of these divergence points shows that half of them are within a single ISP: GTT Communications (AS 3257).

Related Work

Since Jacobson's original traceroute tool [START_REF] Jacobson | traceroute[END_REF], a number of new versions have emerged with different features and with new methods for constructing probes (e.g., Paris traceroute [START_REF] Augustin | Avoiding Traceroute Anomalies with Paris Traceroute[END_REF][START_REF] Veitch | Failure Control in Multipath Route Tracing[END_REF] and tcptraceroute [START_REF] Torren | Tcptraceroute-a traceroute implementation using tcp packets. man page[END_REF]). All these traceroute versions have a drawback for the goal of diagnosing a target application flow because they start a new flow to send probes. As such, middleboxes may treat them differently than the target application flow. Service traceroute avoids this issue by piggybacking traceroute probes within active application flows. This idea was first introduced in paratrace [START_REF] Kaminsky | Parasitic Traceroute via Established TCP Flows & IPID Hopcount[END_REF], which is no longer available, and then re-implemented in 0trace [START_REF] Edge | Tracing behind the firewall[END_REF] with the goal of tracing through firewalls and in TCP sidecar [START_REF] Sherwood | Touring the internet in a tcp sidecar[END_REF] for reducing complaints of large-scale traceroute probing for topology mapping. Unfortunately, none of these tools is actively maintained. Service traceroute adds the capability of automatically identifying application flows to trace by a domain name, of tracing UDP flows as well as of tracing multiple concurrent flows that compose a service. We release both a commandline and a library version as open source. Furthermore, we present an evaluation of the side-effects of piggybacking traceroute probes within application traffic as well as of its benefit by comparing the differences with Paris traceroute and with 0Trace. Our characterization reappraises some of the findings from Luckie et al. [START_REF] Luckie | Traceroute Probe Method and Forward IP Path Inference[END_REF], which show that the discovered paths depend on the protocol used in the probes. Their study, however, includes no traceroute tools that piggyback on application flows.

Conclusion

In this paper we present Service traceroute, a tool to trace paths of flows of modern Internet services by piggybacking TTL-limited probes within target application flows. Our evaluation of paths to popular websites and video services from PlanetLab Europe shows that Service traceroute's probing has no effect on target application flows. Moreover, a typical traceroute tool that launches a new flow to the same destination discovers different paths than when embedding probes in the application flow in a significant fraction of experiments (from 40% to 50% of our experiments) as our comparison with Paris traceroute shows. When we set Paris traceroute's flow ID to that of the target application flow, the resulting paths are more similar to Service traceroute's. Identifying the flow ID to probe, however, is not trivial. Modern applications rely on a large pool of servers/ports. Even to run 0Trace, which implements the same idea of piggybacking probes in the application flow, we had to rely on Service traceroute's functionality to identify target application flow IDs to probe. In future work, we plan to add the support of IPv6 to Service traceroute. We further plan to perform a larger scale characterization of results of Service traceroute across a wide variety of services and a larger set of globally distributed vantage points.

Fig. 1 .

 1 Fig. 1. Flow duration of target application flows with and without Service traceroute

Fig. 2 .

 2 Fig. 2. Length of paths discovered with different versions of traceroute.

Fig. 3 .

 3 Fig. 3. Path edit distance between Service traceroute and Paris Traceroute.

Fig. 4 .

 4 Fig. 4. Path edit distance between Service traceroute and Paris traceroute using the same flow ID.

Table 1 .

 1 Location of divergence points [% of all flows]

		Web Pages (TCP)	Twitch (TCP)	Youtube (TCP and UDP)
	Configuration	Origin Middle Dest. Origin Middle Dest. Origin Middle	Dest.
	MDA UDP	7.33 39.82 4.92 0.41 50.56 0.64 12.49 19.52	3.15
	MDA TCP	15.13 34.37 2.86 4.28 49.17 0.51 15.67 19.13	3.92
	MDA ICMP	9.11 19.04 17.44 7.35 44.50 1.99 6.81 19.95	1.55
	PT same flow ID 4.06 1.81 1.03 8.43 24.72 2.18 4.08 0.87	4.61

We use the traditional 5-tuple definition of a flow (protocol, source and destination IP, as well as source and destination port).

To make 0Trace work on PlanetLab nodes, we had to replace the library to issue probes from dnet to scapy.

Acknowledgements This work was supported by the ANR Project N o ANR-15-CE25-0013-01 (BottleNet), a Google Faculty Research Award, and Inria through the IPL BetterNet and the associate team HOMENET.


Abboudi Maher

Sabine Ae

Surget Matallana

Jean-Franc Ae

Rontani ¸ois

Richard Ae

Sempe

Fabien ´ae

Joux

M Abboudi

J.-F Rontani

Á R Sempe

S M Surget

Á F Joux

Physiological Alteration of the Marine Bacterium Vibrio angustum S14 Exposed to Simulated Sunlight During Growth

Growth experiments on the marine bacterium Vibrio angustum S14 were conducted under four light conditions using a solar simulator: visible light (V), V ? ultraviolet A (UV-A), V ? UV-A ? UV-B radiation, and dark. Growth was inhibited mainly by UV-B and slightly by UV-A. UV-B radiation induced filaments containing multiple genome copies with low cyclobutane pyrimidine dimers. These cells did not show modifications in cellular fatty acid composition in comparison with dark control cultures and decreased in size by division after subsequent incubation in the dark. A large portion of the bacterial population grown under visible light showed an alteration in cellular DNA fluorescence as measured by flow cytometry after SYBR-Green I staining. This alteration was not aggravated by UV-A and was certainly due to a change in DNA topology rather than DNA deterioration because all the cells remained viable and their growth was not impaired. Ecological consequences of these observations are discussed.

Introduction

Marine bacteria are the main contributors in the transfer of carbon from the dissolved phase into the particulate phase (bacterial biomass production) and in the biogenic production of CO 2 (bacterial respiration) [START_REF] Azam | The ecological role of water-column microbes in the sea[END_REF]. In surface waters, solar radiation may have detrimental effects on bacteria by reducing DNA and protein synthesis, exoenzymatic activity and respiration (for a review see Ref. [START_REF] Jeffrey | Ultraviolet radiation effects on bacteria and viruses in marine ecosystems[END_REF]). Solar ultraviolet radiation (UV-R) reaching the Earth's surface constitutes the most damaging part of solar incident radiation for organisms. UV-B (280-320 nm) and UV-A (320-400 nm) cause distinct but overlapping damages. Exposure to UV-B mainly has a direct effect on DNA by inducing dimerization of DNA bases and blocking DNA transcription and replication. The effects of exposure to UV-A, and to a lesser visible light (400-700 nm), are mainly indirect, by generation of reactive oxygen species (ROS) which interact with DNA, proteins, and lipids to induce damages that may be lethal or mutagenic [START_REF] Elasri | Study of the response of a biofilm bacterial community to UV radiation[END_REF]. However, bacteria can efficiently repair UV-induced DNA damage by (i) photoreactivation after activation of the photolyase enzyme by light in the range of 380-430 nm and (ii) ''dark'' DNA repair mechanisms including nucleotideexcision repair, SOS-error-prone repair, and postreplication recombinational repair.

Adaptative responses of bacteria to grow under solar radiation are of great interest to understand how bacteria can face the challenge of synthesizing DNA under damaging photons. Although extensive research has been carried out on the effect of solar radiation on bacteria, most of it has been conducted to measure the resistance of cells exposed in a nongrowing state (e.g., Ref. [START_REF] Joux | Marine bacterial isolates display diverse responses to UV-B radiation[END_REF]). The capacity of heterotrophic bacteria to grow under light stress has been much less studied [START_REF] Berney | Adaptation to UVA radiation of E. coli growing in continuous culture[END_REF] and, to our knowledge, never for marine bacteria. The objective of this study was to determine how different parts of solar radiation might affect the growth, viability, cell size, DNA content, DNA damage, and fatty acid (FA) contents of the marine bacterium Vibrio angustum S14 [START_REF] Kjelleberg | How do non-differentiating bacteria adapt to starvation?[END_REF].

Materials and Methods

Bacterial Strain and Growth Experiments Under Simulated Solar Exposure V. angustum S14 (UNSW, Sydney, Australia) was grown in an artificial seawater medium supplemented with 3 mM D-glucose (ASW-G) [START_REF] Eguchi | Responses to stress and nutrient availability by the marine ultramicrobacterium Sphingomonas sp. strain RB2256[END_REF] at 30°C with orbital shaking at 130 rpm. Cylindrical quartz flasks with a flat top containing 200 ml of ASW-G and a magnetic stirrer were inoculated with 2 ml of an overnight preculture. Culture flasks were then exposed to simulated sunlight produced by a solar-filtered (1.0 air mass) 1000-W xenon arc lamp (Oriel Corp., Stratford, CT, USA) giving an optical output irradiance of 250 W m -2 PAR, 19.3 W m -2 UV-A, and 0.85 W m -2 UV-B as measured with a UV/visible RAMSES spectroradiometer (TriOS, Germany). Four light treatments were studied simultaneously: (i) full radiation (visible?UV-A?UV-B; culture flask covered with acetate cellulose [50% transmission at 280 nm] to remove residual UV-C), (ii) visible light and UV-A (culture flask covered with Mylar film [50% transmission at 320 nm]), (iii) visible light (culture flask covered with OP3 [50% transmission at 400 nm]), and (iv) dark (culture flask covered with black tape). The four culture flasks were maintained at 30°C by partial submersion in a water bath connected to a cryothermostat with continuous magnetic stirring (*100 rpm). Culture flasks were irradiated during 12 h under the solar simulator (i.e., until the stationary phase) and then maintained in the dark for 12 h. Growth was monitored by optical density (OD) measurements at 433 nm. Colony-forming units (CFU) were determined at T 0 , T 12 h , and T 24 h by plating appropriate dilutions in triplicate on VNSS media [START_REF] Eguchi | Responses to stress and nutrient availability by the marine ultramicrobacterium Sphingomonas sp. strain RB2256[END_REF]. CFU were counted after incubation at 30°C during 24 h.

Flow Cytometry and Epifluorescence Analysis Samples fixed with formaldehyde (2% final concentration) were analyzed at T 0 , T 6 h , T 12 h , and T 24 h on a flow cytometer (FACSCalibur; Becton Dickinson, San Jose, CA, USA) after staining with the nucleic acid dye SYBR Green I (Molecular Probes Inc., Eugene, OR, USA) [START_REF] Abboudi | Effects of photochemical transformations of dissolved organic matter on bacterial metabolism and diversity in three contrasting coastal sites in the Northwestern Mediterranean Sea during summer[END_REF]. Bacteria were detected on a plot of green fluorescence (515-545 nm) versus right-angle light scatter (SSC), using the green fluorescence as the threshold parameter. Fluorescence beads (1.0 lm; Polysciences Inc., Warrington, PA, USA) were added to each sample analyzed to normalize SSC and green fluorescence. Cells were also observed with an epifluorescence microscope (AX70; Olympus Provis) after staining with 4',6-diamidino-2-phenylindole (DAPI; 2.5 lg ml -1 ; Sigma).

Cyclobutane Pyrimidine Dimers Analysis CPDs were quantified at T 0 , T 12 h , and T 24 h by an enzyme-linked immunosorbent assay (ELISA). DNA was isolated from 20 ml of culture using the Wizard genomic DNA purification kit (Promega) and quantified using the Quant-iT PicoGreen dsDNA assay kit (Molecular Probes, Invitrogen). One hundred nanograms of each DNA sample was added to 1% protamine sulfate-treated wells in 96-well ELISA plates. The primary antibody against CPD (MC-062; Kamiya Biomedical Company) and the secondary antibody, a horseradish peroxidase conjugate (HRP rabbit anti-mouse IgG; Invitrogen Molecular Probes), was diluted 1:1000 and 1:2000, respectively, in 0.05% Tween-20/ 0.25% bovine serum albumin (BSA)/19 phosphate-buffered saline blocking solution. Photoproducts were quantified at OD 492 nm 30 min after enzyme reaction with O-phenylenediamine dihydrochloride (Sigma) after stopping the reaction with H 2 SO 4 (2 M). OD was calibrated using standards (0-1000 CPD/Mb DNA) of UV-B-irradiated calf thymus DNA (Sigma) for which CPDs were quantified by high-performance liquid chromatographytandem mass spectrometry [START_REF] Douki | Individual determination of the yield of the main-UV induced dimeric pyrimidine photoproducts in DNA suggests a high mutagenicity of CC photolesions[END_REF].

Fatty Acid Analysis Bacteria sampled at T 12 h under dark and V?UV-A?UV-B conditions were saponified [START_REF] Rontani | GC/MS structural characterization of fatty acids of marine aerobic anoxygenic phototrophic bacteria[END_REF] and total FA fractions were then silylated and analyzed by gas chromatography-mass spectrometry (GC-MS). GC-MS analyses under electron ionization (GC-EIMS) were carried out with an HP 6890 gas chromatograph connected to an HP 5973 mass spectrometer. Structural assignments were based on interpretation of mass spectrometric fragmentations and confirmed by comparison of retention times and mass spectra with those of authentic compounds, when these were available.

Results and Discussion

Effect of UV-B, UV-A, and Visible Light on Bacterial Growth According to the OD, the maximum growth rate of V. angustum in darkness was *1 h -1 (Fig. 1). The bacterial cultures reached the stationary phase after 12 h for all light conditions. During the midlog phase (T 6 h ) there were significant differences in OD measured under the different light conditions (dark[visible[V?UV-A[V?UV-A?UV-B) (Mann Whitney U-test, p \ 0.05), suggesting gradual inhibition of growth due to the different band parts of sunlight. However, in the stationary phase (T 12 h ), the cell yield was significantly lower only for the V?UV-A?UV-B condition (-18%) compared to the other conditions (p\0.05). During the dark period, the OD declined slightly for all conditions except for V?UV-A?UV-B, for which OD was stable. Cell enumerations measured by flow cytometry showed patterns different from those of the OD. First, an inhibition at midlog phase was observed only for the V?UV-A and V?UV-A?UV-B conditions (-65% and -89% compared to the dark condition, respectively; p \ 0.05). Second, the cell number continued to increase during the dark period (between T 12 h and T 24 h ) for all conditions. This increase was more pronounced in V?UV-A?UV-B cultures (?190%) compared to the others (?63% to 86%; p\0.05). The measurement of OD may differ from that of flow cytometry because it combines both cell number and cell size (e.g., the same OD can be obtained with more cells but with a smaller size; see below).

Viability remained high in all cultures and not significantly different from 100% at T 12 h (Table 1). During the prolongation of the stationary phase in the dark, a slight decrease in the percentages of viable cells was detected in visible and V?UV-A cultures, however, these variations must be interpreted with caution because of the imprecision in the methods used to enumerate CFU and total cells. Low quantities of CPDs (close to the detection limit of the method) were detectable in bacteria exposed to V?UV-A?UV-B at T 12 h and T 24 h , but not for the other conditions, highlighting the importance of UV-B in the induction of this DNA damage (Table 1).

The sunlight intensity used in this study corresponds to *60% of the maximal solar intensity that we can measure in summer in the Mediterranean region, however, the cumulative dose over the 12 h of irradiation was similar to that measured daily in the same region [START_REF] Abboudi | Effects of photochemical transformations of dissolved organic matter on bacterial metabolism and diversity in three contrasting coastal sites in the Northwestern Mediterranean Sea during summer[END_REF]. The faculty of V. angustum S14 to grow under the irradiance used in this study emphasizes the high resistance of this strain to light stress, and the low level of CPDs accumulated during its growth under full sunlight suggests active DNA repair activity in this bacterium. The quantity of CPDs measured in these cells corresponds to the levels of residual damage in marine bacteria present at the surface before sunrise [START_REF] Boelen | Diel patterns of UVBR-induced DNA damage in picoplankton size fractions from the Gulf of Aqaba, Red Sea[END_REF]. Photoreactivation may be particularly important in V. angustum S14 to repair DNA damage as already reported for other vibrio species [START_REF] Joux | Marine bacterial isolates display diverse responses to UV-B radiation[END_REF][START_REF] Liltved | Influence of liquid holding recovery and photoreactivation on survival of ultraviolet-irradiated fish pathogenic bacteria[END_REF].

Changes in Cellular DNA Fluorescence According to the DNA fluorescence analyzed by flow cytometry, bacteria in the inocula were equally composed of cells containing one and two fully replicated genomes (Fig. 2a). After 12 h of culture in darkness (stationary phase), the bacterial population was composed of cells containing two and four genomes (Fig. 2b). During the subsequent 12 h of culture, multigenomic cells divided by decreasing in size (from 1.7 to 1.4 lm in length) to give bacteria with a DNA content similar to that of cells observed in the inocula (data not shown). This process, called reductive division, results from the fact that initiated rounds of DNA replication and cell division are completed in the absence of further growth upon entry of cells into the stationary phase [START_REF] Nystro ¨m | Stationary-phase physiology[END_REF] and has been reported for other marine Vibrio [START_REF] Smigielski | Genes responsible for size reduction of marine Vibrios during starvation are located on the chromosome[END_REF].

For cultures under visible light we observed a profound alteration in DNA fluorescence at T 12 h (Fig. 2c). A similar alteration was observed in V?UV-A culture (Fig. 2d), suggesting the predominant role of visible light in this alteration. According to the fluorescence of cells containing a single genome observed in the inocula and in the cultures at T 24 h in darkness, we defined a threshold below which cells were called DNA-. The proportion of DNA-increased continuously in cultures under the visible condition, to reach 45% at T 24 h (Fig. 3c). The V?UV-A condition showed a similar pattern, whereas V?UV-A?UV-B presented fewer DNA-cells at T 12 h (10%) and T 24 h (25%) (Fig. 3c). The lower staining observed in cells grown under visible and V?UV-A conditions cannot be due to a modification of the membrane permeability for the fluorescent dye, because cells were fixed with formaldehyde, or to a degradation of DNA, because all the cells were viable. We hypothesize that growth under high irradiance (mainly due to visible light) could induce a change in DNA topology that could have an impact on fluorochrome fixation on DNA. In bacteria, supercoiling of the circular chromosome is under the control of several topoisomerases that maintain the optimum conformation of the DNA for processes like transcription and DNA replication [START_REF] Luttinger | The twisted 'life' of DNA in the cell: bacterial topoisomerases[END_REF]. The decrease in DNA negative supercoiling has been observed in Escherichia coli after visible light irradiation and treatment with hydrogen peroxide [START_REF] Horiuchi | Relaxation of supercoiled plasmid DNA by oxidative stresses in Escherichia coli[END_REF][START_REF] Weinstein-Fischer | Escherichia coli response to hydrogen peroxide: a role for DNA supercoiling, Topoisomerase I and Fis[END_REF]. Yet there is no evidence that relaxation of DNA is sufficient to induce a change in in vivo SYBR fluorescence, and further studies are required to test this hypothesis. These studies may help clarify the role of light and oxidative stress in the heterogeneity of DNA fluorescence revealed by flow cytometry in bacterial communities of aquatic ecosystems after fluorescent nucleic acid staining [START_REF] Bouvier | A comparative study of the cytometric characteristics of High and Low nucleicacid bacterioplankton cells from different aquatic ecosystems[END_REF].

Cell Filamentation Cells grown under visible or V?UV-A light or in the dark showed similar sizes when examined by epifluorescence microcospy (Table 1) and similar SSC measured by flow cytometry (Fig. 3a) at T 12 h and T 24 h . In contrast, we observed filamentous bacteria with variable lengths for the V?UV-A?UV-B condition after 12 h of growth (Table 1). These cells showed a mean SSC and a mean DNA fluorescence 2 and 1.5 times higher, respectively, than those of bacteria cultivated in darkness (Fig. 3a,b). According to the fluorescence of cells with one genome, filaments contained a mean of eight genomes copies. After the dark period, these filamentous cells decreased in size (Table 1) and in cellular DNA fluorescence (Fig. 3b). Cell filamentation is a well-known process induced by the SOS response after different physical or chemical treatments that damage DNA (e.g., UV-R) or inhibit DNA replication (e.g., nalidixic acid, thymine starvation) [START_REF] Janion | Some aspects of the SOS response system-a critical survey[END_REF]. The SOS response is a strategy employed by bacterial cells to inhibit cell division while damaged DNA is repaired, thereby preventing the transmission of deleterious mutations to daughter cells. Here we observed that filamentation was induced by UV-B but not by UV-A or visible light. Induction of filamentous bacteria in aquatic ecosystems has already been reported for some bacterial species as a mechanism for grazing resistance [START_REF] Ju ¨rgens | The potential importance of grazingresistant bacteria in planktonic systems[END_REF] but never, to our knowledge, as a response to UV-R. The degree of filamentation certainly depends on the available substrates and the growth rate of bacteria. Because the growth rate of bacteria in aquatic systems is well below that of V. angustum in this study, we can expect that filamentation induced by UV-R is also certainly less important. If this process occurs, but insufficiently to form long filamentation, these bigger bacteria may be more prone to grazing than smaller cells.

Fatty Acid Composition

The FA composition was analyzed in cells after 12 h of growth under V?UV-A?UV-B (filamentous cells) and dark conditions (Table 2). Both cells presented very similar FA profiles, dominated by palmitoleic acid. Saturated (SFA) and monoinsatured (MUFA) FA represented *55% and *35% of the total FA, respectively. The remaining fraction corresponded to unknown compounds and hydroxyacid. Polyunsaturated fatty acids (PUFA) were absent. The FA composition measured in V. angustum is congruent with those measured in other Vibrio species [START_REF] Thompson | Biodiversity of vibrios[END_REF]. The few published papers dealing with the effects of UV-R on the FA composition of marine organisms concern microalgae and they reported contradictory results. Indeed, as the response to UV-R, the authors observed (i) an increase in the proportion of PUFA [START_REF] Lange | Effects of UVB-irradiated algae on life history traits of Daphnia pulex[END_REF][START_REF] Gupta | Impact of UV-B radiation on thylakoid membrane and fatty acid profile of Spirulina platensis[END_REF][START_REF] Liang | Effects of nitrogen source and UV radiation on the growth, chlorophyll fluorescence and fatty acid composition of Phaeodactylum tricornutum and Chaetoceros muelleri (Bacillariophyceae)[END_REF], (ii) no significant differences in the FA profiles [START_REF] Skerratt | Effect of UV-B radiation on lipid content of three Antarctic marine phytoplankton[END_REF], and (iii) an increase in the proportion of SFA [START_REF] Goes | Effect of UV-B radiation on the fatty acid composition of the marine phytoplankter Tetraselmis sp.: relationship to cellular pigments[END_REF][START_REF] Hessen | UV-induced changes in phytoplankton cells and its effects on grazers[END_REF]. Alterations of the FA profiles could result from (i) photosensitized degradation processes induced by sensitizer substances present in these organisms or (ii) changes in the biosynthesis of these compounds as a response to light stress. The stability of the FA composition of V. angustum S14 observed in the present work shows that UV-R cannot induce FA photodegradation in this bacterium, probably due to the lack of adequate photosensitizers, and that biosynthesis pathways of FA are not modified as a response to light stress. This first study on the effects of UV-R on the FA composition of a heterotrophic bacterium must be completed with other bacterial species knowing that important diversity of FA composition exists for these bacteria. It has been observed that changes in FA composition of phytoplankton induced by UV-R could have

Fig. 1

 1 Fig. 1 Growth curves of V. angustum S14 in ASW-G at 30°C under different light treatments. Cell growth was measured as optical density at 433 nm (OD 433 nm ) (a) and cell number measured by flow cytometry (b). The open bar represents the period of simulated sunlight exposure and the black bar represents the period of darkness. Points are represented as the average of at least three independent experimental values; error bars are 1 SD

Fig. 2

 2 Fig. 2 SYBR-Green I fluorescence versus side scatter cytograms of V. angustum S14 in the inocula (a) and after 12 h of growth in darkness (b), visible light (c), visible?UV-A light (d), and visible?UV-A?UV-B light (e). DNA-area circumscribes the cells with an apparent DNA content less than one genome

Fig. 3

 3 Fig. 3 Change in relative side scatter (a), DNA green fluorescence (b), and percentages of DNA-cells (c) during the growth of V. angustum S14 under the different light treatments. The open bar represents the period of simulated sunlight exposure and the dark bar represents the period of darkness. Points represented are the average of at least two independent experimental values; error bars are 1 SD

Table 1

 1

	Changes in length, width, percentage colony forming units (CFU), and	Light treatment	Incubation time (h)	Length (lm)	Width (lm)	% CFU a	CPDs (/Mb DNA)
	cyclobutane pyridine dimers (CPDs) during growth in different light treatments: data	All Dark	0 12	1.9 (0.4) 1.7 (0.4)	1.2 (0.2) 1.0 (0.2)	104 (46) 91 (12)	0 0
	are the average (standard		24	1.4 (0.3)	1.1 (0.2)	68 (3)	0
	deviation) of at least two	Visible	12	1.7 (0.5)	1.0 (0.2)	102 (2)	0
	independent experiments		24	1.5 (0.4)	1.0 (0.2)	94 (15)	0
		V?UV-A	12	1.7 (0.6)	0.9 (0.2)	107 (2)	0
			24	1.5 (0.5)	1.0 (0.2)	73 (3)	0
		V?UV-A?UV-	12	42 (36.6)	1.2 (0.2)	77 (11)	24.3 (11.4)
	a Expressed as a percentage of	B	24	4.8 (9.2)	1.1 (0.3)	74 (22)	8.7 (12.4)
	total counts

DNA -

Table 2

 2 Fatty acid (FA) composition of V. angustum S14 after 12 h of growth in the dark or under visible?UV-A?UV-B light: results expressed as the percentage of total FA (range) from two independent experiments

	Main FA	Light treatment
		Dark	V?UV-A?UV-B
	Palmitoleic acid (C 16:1 )	50.6 (49.3-51.9)	50.1 (47.5-52.7)
	Palmitic acid (C 16:0 )	26.1 (20.4-31.8)	30.3 (27.0-33.7)
	3-OH-C 12:0	9.4 (5.9-12.8)	7.9 (4.5-11.4)
	Vaccenic acid (C 18:1 )	4.7 (3.5-5.8)	3.6 (3.0-4.2)
	Myristic acid (C 14:0 )	4.2 (4.1-4.3)	3.7 (3.1-4.3)
	Lauric acid (C 12:0 )	3.4 (1.9-4.9)	2.9 (1.1-4.7)
	Unknown	1.7 (1.1-2.4)	1.5 (0.8-2.2)
	Total saturated FA	55.3 (55.8-57.7)	53.7 (50.5-56.9)
	Total monounsaturated FA	33.7 (26.4-41.0)	36.9 (31.2-42.7)

Acknowledgments We thank T. Douki for determination CPDs by HPLC-MS/MS for standard samples. We are grateful to N. West for improvement of the language. This study was funded by the CNRS-INSU through the UVECO and ATIPE projects and by a fellowship from the Syrian Atomic Energy Commission to M.A.