Didier Henrion

Alessandro Rudi

Solving moment and polynomial optimization problems on Sobolev spaces *

Keywords: Polynomial optimization, semidefinite optimization, moment problems, harmonic analysis. AMS MSC: 90C23, 90C22, 44A60, 42-08

Using standard tools of harmonic analysis, we state and solve the problem of moments for positive measures supported on the unit ball of a Sobolev space of multivariate periodic trigonometric functions. We describe outer and inner semidefinite approximations of the cone of Sobolev moments. They are the basic components of an infinite-dimensional moment-sums of squares hierarchy, allowing to solve numerically non-convex polynomial optimization problems on infinite-dimensional Sobolev spaces, with global convergence guarantees.

Introduction

The moment-SOS hierarchy, also known as the Lasserre hierarchy, was originally introduced in the early 2000s to solve globally finite-dimensional polynomial optimization problems (POP) [START_REF] Lasserre | Moments, positive polynomials and their applications[END_REF][START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF][START_REF] Blekherman | Semidefinite optimization and convex algebraic geometry[END_REF][START_REF] Nie | Moments and polynomial optimization[END_REF]. Then it was extended to polynomial differential equations and their optimal control, see [START_REF] Henrion | The Moment-SOS Hierarchy -Lectures in Probability[END_REF] for a recent overview of applications and more references. The main technical ingredients on which the moment-SOS hierarchy relies are sums of squares (SOS) representations of positive polynomials (the so-called Positivstellensätze) [START_REF] Marshall | Positive polynomials and sums of squares[END_REF] and its dual problem of moments [START_REF] Schmüdgen | The moment problem[END_REF] providing conditions satisfied by moments of a positive measure supported on a finite-dimensional set. These conditions are truncated to finite degrees, yielding a converging hierarchy of semidefinite optimization problems of increasing size that can be solved numerically using interior-point algorithms [START_REF] Nesterov | Interior-point polynomial algorithms in convex programming[END_REF][START_REF] Ben-Tal | Lectures on modern convex optimization[END_REF].

After more than two decades of research in polynomial optimization, the application range of the moment-SOS hierarchy is now being extended to challenging nonconvex nonlinear optimization problems formulated on infinite-dimensional functional spaces, e.g. problems of calculus of variations or partial differential equations. The moment-SOS hierarchy has been recently extended to reproducible kernel Hilbert spaces for polynomial optimization [START_REF] Rudi | Finding global minima via kernel approximations[END_REF] or optimal transport [START_REF] Muzellec | Nearoptimal estimation of smooth transport maps with kernel sums-of-squares[END_REF]. Measures supported on infinite-dimensional spaces arise naturally as relaxed controls for infinitedimensional optimization [START_REF] Fattorini | Infinite dimensional optimization and control theory[END_REF]. Ambrosio's superposition principle [START_REF] Ambrosio | Transport equation and Cauchy problem for non-smooth vector fields[END_REF] whose finitedimensional Euclidean version was used in [START_REF] Henrion | Korda Convex computation of the region of attraction of polynomial control systems[END_REF] to prove convergence of the moment-SOS hierarchy for approximating the region of attraction of polynomial differential equations, has been extended to infinite-dimensional Hilbert or Banach spaces [START_REF] Ambrosio | Lecture notes on the DiPerna-Lions theory in abstract measure spaces[END_REF]. Measures on infinite-dimensional spaces are also used in fluid dynamics, see e.g. [START_REF] Fjordholm | On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations[END_REF] or more recently [START_REF] Rosa | Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations[END_REF] which presents itself as an infinite-dimensional extension of the finite-dimensional SOS setup of [START_REF] Tobasco | Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems[END_REF]. The solution of the moment problem for measures supported on infinite-dimensional spaces is more technically involved than its finite-dimensional counterpart, see [START_REF] Infusino | The full infinite dimensional moment problem on semi-algebraic sets of generalized functions[END_REF][START_REF] Schmüdgen | On the infinite-dimensional moment problem[END_REF] and references therein. The recent reference [START_REF] Henrion | Infinite-dimensional moment-SOS hierarchy for nonlinear partial differential equations[END_REF] shows however that the heat equation with polynomial nonlinearities can be solved numerically with the infinite-dimensional moment-SOS hierarchy, with convergence guarantees provided by a recent solution of the moment problem on nuclear spaces [START_REF] Infusino | Moment problem for algebras generated by a nuclear space[END_REF].

The present paper aims at contributing to the numerical solution of the infinitedimensional moment problem in a functional analytic framework which makes its analysis as well as its numerical implementation as simple as possible. We use basic tools from harmonic analysis to state and solve the moment problem on the Sobolev space of periodic multivariate trigonometric functions. This allows us to construct an infinite-dimensional moment-SOS hierarchy to solve various kinds of Sobolev POPs, namely non-convex POPs on Sobolev spaces, with global convergence guarantees.

In order to keep this paper as short and elementary as possible, we do not describe here potential applications of the moment-SOS hierarchy for solving non-linear calculus of variations problems, or optimal control problem involving non-linear partial differential equations. Such applications are certainly very promising, and they will be reported in further communications.

The outline of the paper is as follows. In Section 2 we state our Sobolev moment problem. In Section 3 we reformulate our Sobolev moment problem as a moment problem in the Fourier coefficients. In Section 4 we propose inner and outer semidefinite approximations of the Sobolev moment cone. This allows us to solve different types of Sobolev POPs with an infinite-dimensional moment-SOS hierarchy in Section 5. Concluding remarks and potential extensions are mentioned in Sections 6 and 7.

Sobolev moment problem

Consider the space of Sobolev functions on the n-dimensional unit torus T n whose derivative up to order m are square integrable:

H m (T n ) := {f : T n → C : f 2 H m (T n ) < ∞}
where the norm is defined as:

f 2 H m (T n ) := |a|≤m T n D a f (x) 2 dx with a ∈ N n , |a| = n i=1 a i and D a = ∂ |a| ∂x a 1 1 ...∂x an k .
Let c 0 (Z n ) denote the set of sequences consisting of a finite number of elements of Z n , allowing repetitions. Let us consider the closed bounded unit ball of H m (T n )

B := {f ∈ H m (T n ) : f H m (T n ) ≤ 1}.
Let µ be a measure supported on B, and let a ∈ c 0 (Z n ). The moment of µ of index a is defined as:

y a := B m a (f ) dµ(f ) with m a (f ) := a∈a f, e a
the monomial of f of index a, for the scalar product

f, e a H m (T n ) :=

T n f (x)e a (x)dx, e a (x) := e -2πi a,x R n .

Note that since a ∈ c 0 (Z n ), it follows that m a (f ) is the product of finitely many coefficients. The algebraic degree is d a := # a, the cardinality of a, i.e. the number of terms in the product defining m a (f ). The harmonic degree is the integer δ a := max a∈a max i=1,...,k |a i |.

Example 1. Let n = 2. The empty set a = {} = ∅ indexes the mass B dµ(f ), a = {(0, 0)} indexes the first degree moment B f, e (0,0) dµ(f ), a = {(0, 0), (0, 0)} indexes the second degree moment B f, e (0,0) 2 dµ(f ), a = {(1, 0), (0, -1), (0, -1)} indexes the third degree moment B f, e (1,0) f, e (0,-1)

2 dµ(f ), etc.

We can now state the problem addressed in this paper.

Sobolev moment problem: Let N ∈ N. Given an index set A ∈ c 0 (Z n ) N and a vector (y a ) a∈A ∈ C N , find a measure µ on B such that

y a = B m a (f ) dµ(f ), for all a ∈ A. (2)
Since we defined a monomial in Sobolev space, we can also define a polynomial p as a linear combinations of monomials:

p(f ) := a∈spt p p a m a (f )
where the support spt p ⊂ c 0 (Z n ) is the index set of coefficients in the monomial basis, the algebraic degree is d(p) := max a∈spt p d a and the harmonic degree is δ(p) := max a∈spt(p) δ a .

Fourier embedding

Let us reformulate our moment problem in the space of Fourier coefficients. The following results are classical [START_REF] Adams | Sobolev spaces[END_REF].

Define the Fourier transform F :

L 2 (T n ) → ℓ 2 (Z n ), f → c where c := (c a ) a∈Z n and c a := f, e a H m (T n ) is the Fourier coefficient of index a ∈ Z n of f . The adjoint of F is the inverse Fourier transform F * : ℓ 2 (Z n ) → L 2 (T n ), c → f = c, e ℓ 2 = a∈Z n c a e -a
where e := (e a ) a∈Z n .

Proposition 1. The space H m (T n ) admits an equivalent norm in terms of the Fourier basis:

f 2 H m (T n ) = a∈Z n w a c 2 a , c = F f and w a := (1 + a, a ) m , a ∈ Z n .
Define the diagonal operator W :

ℓ 2 (Z n ) → ℓ 2 (Z n ), (c a ) a∈Z n → (c a w a ) a∈Z n .
Proposition 2. The space H m (T n ) is isomorphic to ℓ 2 via the operator F * W . Namely, defining f c := F * c for any c ∈ ℓ 2 (Z n ), we have

H m (T n ) = {f c : c ∈ ℓ 2 (Z n )}, f c 2 H m (T n ) = c 2 W := k∈Z n w k c 2 k .
From this it follows that for any f ∈ H m (T n ) there exists a unique c ∈ ℓ 2 (Z n ) such that f = F * W c. Now define the set of Fourier coefficients

E := {F f : f ∈ B} ⊂ ℓ 2 (Z n ). Proposition 3. E is compact. Moreover B is one to one to E and in particular B = {F * c : c ∈ E}. Proof. E is the linear image of the closed unit ball B of H m (T n ). By definition E = {F F * W c : c ∈ ℓ 2 (Z n )} = W ℓ 2 (Z n ). The compactness of E is obtained by noting that 1/w a → 0 as |a| → ∞.
Geometrically, E is an ellipsoid in the space of Fourier coefficients, with coordinates bounded by 1/w a for any a ∈ Z n .

Let

ν := F * # µ (3)
denote the pushforward measure of µ through F * . For any a ∈ c 0 (Z n ), the moment y a of µ can then be expressed as a moment of ν in the space of Fourier coefficients:

y a = B m a (f ) dµ(f ) = E c a dν(c), c a := a∈a c a . (4

)

Proposition 4. There is a solution to the Sobolev moment problem (2) on the ball B if and only if there is a solution to the Fourier moment problem

y a = E c a dν(c), ∀ a ∈ A (5)
on the ellipsoid E.

Proof. Since the ball B is one to one to the ellipsoid E and in particular

B = {F * c : c ∈ E}, then for any f ∈ B there exists a c ∈ E s.t. f, e a H m (T n ) = F * c, e a ℓ 2 = w a c a .
Conversely, for any f ∈ B there exists a unique c ∈ E such that f = F * W c.

Semidefinite approximations of the Sobolev moment cone

Given an index set A ⊂ c 0 (Z n ) N , let us define the cone of Sobolev moments

C(A) := {(y a ) a∈A : y a = B m a (f ) dµ(f ) for some µ supported on B} ⊂ C N .
The Sobolev moment problem introduced previously is the membership oracle for C(A), namely: does a given vector belong to this cone ? Despite being convex and finite-dimensional, the cone C(A) is difficult to manipulate directly. It must be approximated by linear sections and projections of a finite-dimensional convex cone on which optimization can be carried out efficiently, namely the semidefinite cone.

Outer approximations

We can construct outer approximations, or relaxations of C(A), by projecting finitedimensional spectrahedra (i.e. linear sections of the semidefinite cone) of increasing size. Let Π A :

ℓ 2 (Z n ) → C N ,
{ k q * k q k : q k ∈ C[c] r,ρ }
and the quadratic module

Q r,ρ := {p = s 0 + s 1 (1 -c 2 W ) : s 0 , s 1 ∈ Σ r,ρ }.
Given a sequence y = (y a ) a∈Z n ∈ ℓ 2 (Z n ), define the linear functional ℓ y : C[c] → C, p(c) := a p a c a → ℓ y (p) := a p a y a . Let d A := max a∈A d a denote the algebraic degree of A, and let ρ A := max a∈A δ a denote the harmonic degree of A. Finally, define the following cone

C out r,ρ (A) := Π A {y : ℓ y (p) ≥ 0 for all p ∈ Q r,ρ } . Proposition 5. For any r ≥ d A and ρ ≥ ρ A , C out r,ρ (A) is a semidefinite representable outer approximation of C(A).
Proof. To prove the outer approximation claim, let us a vector y ∈ C(A) and let us prove that y ∈ C out r,ρ (A). Since y ∈ C(A), from Proposition 4 there exists a measure ν supported on E such that y a = E c a dν(c). In particular, for any real valued polynomial p(c) = a p a c a which is non-negative on E, vector y is such that ℓ y (p) = a p a y a = a p a E c a dν(c) = E p(c)dν(c) is nonnegative. In particular this holds for polynomials of the form p = q * 0 q 0 + q * 1 q 1 (1c 2 W ), and hence y ∈ C out r,ρ (A). To prove the semidefinite representability claim, observe that the quadratic form ℓ y : C[c] → R, q → ℓ y (q * q) can be expressed as a Hermitian matrix linear in y. Nonnegativity of the quadratic form is therefore equivalent to positive semidefinitess of a matrix which is linear in y, i.e. as a linear matrix inequality. Testing nonnegativity of ℓ y (p) for all p ∈ Q r,ρ amounts to testing non-negativity of q → ℓ y (q * q) and q → ℓ y ((1c 2 W )q * q). These quadratic forms are finite dimensional, so it follows that testing membership in C out r,ρ (A) amounts to testing membership in the projection of a spectrahedron, a finite-dimensional linear slice of the semidefinite cone.

Proposition 6. C out ∞,∞ (A) = C(A).
Proof. According to Putinar's Positivstellensatz -see e.g. [START_REF] Laurent | Sums of squares, moment matrices and optimization over polynomials[END_REF]Thm. 3.20] or [19, Thm. 2.14], every polynomial p which is strictly positive on E can be written as p = s 0 + s 1 (1c 2 W ) for s 0 and s 1 sums of Hermitian squares of polynomials. So the closure of the quadratic module Q r,ρ coincides with the cone of polynomials that are non-negative on E. Proposition 7. For any r ≥ r A and ρ ≥ ρ A , the Hausdorff distance d H between C(A) and C out r,ρ (A) is bounded as follows

d H (C(A), C out r,ρ (A)) ≤ 9 (2ρ A + 1) n r 2 A r 2 . ( 6
)
Proof. Any polynomial in C[c] r,ρ can be written as p(c) = w * φ(c), for some w ∈ C N and with φ : C N → C K , where N is the number of Fourier coefficients up to harmonic degree ρ for functions on the torus T n , i.e. N = (2ρ + 1) n , while K is the number of Chebyshev polynomials up to degree r that we can build on C N , i.e. K = r+N +1 r . This representation allow us to identify C out r,ρ (A) with K s with s = r and C(A) with K s and s = r from [START_REF] Bach | Exponential convergence of sum-of-squares hierarchies for trigonometric polynomials[END_REF] and the use their Corollary 1 (note that we do not have a scale factor 1/(2r + 1) d ).

Denote by y(µ) ∈ C K the vector of all the moments of a measure µ supported on the N Fourier coefficients up to algebraic degree r. By applying Corollary 1 of [START_REF] Bach | Exponential convergence of sum-of-squares hierarchies for trigonometric polynomials[END_REF], we have that for any y ∈ C out r,ρ there exists a measure µ supported on the N Fourier coefficients, such that

Π (r A ) r ( y -y(µ)) Fro ≤ 9Nr 2 A r 2 ,
where Π

(r A ) r is a diagonal matrix such that (Π (r A ) r
) a,a = 1 iff a is the index of a Chebyshev polynomial with algebraic degree less or equal to r A , otherwise it is (Π

(r A ) r ) a,a = 0. The proof is concluded by noting that C(A) = {Π (r A ) r y(µ) | µ ∈ M ρ },
where M ρ is the set of measures supported on Fourier coefficients with maximum harmonic degree ρ.

Inner approximations

Another approach consists of expressing measure ν as being absolutely continuous with respect to some reference measure γ whose moments can be easily calculated, e.g. the Gaussian measure on ℓ 2 (Z n ) [START_REF] Bogachev | Gaussian measures[END_REF].

Let

C inn r,ρ (A) := {(y a ) a∈A : y a = E c a p(c)dγ(c) for some p ∈ Q r,ρ }.

Proposition 8. For all r, ρ ∈ N, C inn r,ρ (A) is a semidefinite representable inner approximation of C(A).

Proof. The Radon-Nikodým derivative p of ν = pγ with respect to γ is constrained to the quadratic module Q r,ρ , so that ν is non-negative on E. The proof is concluded by observing that y and p are related linearly, with p belonging to the semidefinite representable set Q r,ρ .

Proposition 9. C inn ∞,∞ (A) = C(A).
Proof. As the proof of Proposition 6, according to Putinar's Positivstellensatz, elements of Q r,ρ can approximate as closely as desired any polynomial nonnegative on E, i.e. we can construct a sequence p r,ρ ∈ Q r,ρ such that pp k W → 0 when r, ρ → ∞. It follows that for all a ∈ A, E c a p r,ρ (c)dγ(c) → E c a p(c)dγ(c) = y a when r, ρ → ∞.

5 Solving Sobolev POPs

Harmonic Sobolev POP

We are now fully equipped to solve a harmonic Sobolev POP (polynomial optimization problem) of the form

p * := inf f ∈B p(f ) (7)
where

p(f ) = a∈A p a m a (f )
is a given Hermitian polynomial in the indeterminate f ∈ B, of support A := spt p. Problem ( 7) is called harmonic because the harmonic degree δ(p) is finite, and the problem does not involve harmonics of degrees higher than δ(p). Note that the infimum in ( 7) is always attained, since it is a finite-dimensional problem and B is bounded.

Harmonic Sobolev POP ( 7) is equivalent to the linear problem Theorem 1. For all r ≥ r ′ ≥ d(p) and ρ ≥ ρ ′ ≥ δ(p), it holds

p out r ′ ,ρ ′ ≤ p out r,ρ ≤ p out ∞,∞ = p * = p inn ∞,ρ ≤ p inn r,ρ ≤ p inn r ′ ,ρ ′ .
Proof. It follows readily by applying Propositions 5, 6, 8 and 9.

Example

Consider the harmonic Sobolev POP

p * = min f ∈B f, e 0 4 H 0 (T ) + ( f, e 1 2 H 0 (T ) -1/4) 2
on B ⊂ H 0 (T ), i.e. n = 1, m = 0 and harmonic degree ρ = 1. Observe that the function to be minimized is non-convex in f .

The harmonic Sobolev POP is equivalent to the harmonic Fourier POP

p * = min c -1 ,c 0 ,c 1 c 4 0 + (c 2 1 -1/4) 2 s.t. c 2 -1 + c 2 0 + c 2 1 ≤ 1.
Note that the Fourier coefficient c -1 does not appear in the objective function, and hence without loss of generality it can be set to zero. Alternatively, it may be desirable to penalize the higher degree Fourier coefficients with a quadratic regularization term.

With the outer moment-SOS hierarchy, at algebraic relaxation degree r = 2, we obtain the two global minimizers c * -1 = 0, c * 0 = 0, c * 1 = ±1/2 and the corresponding functions f * (x) = ±e -2πix /2 achieving the global minimum p * = p out 2,1 = 0.

Algebraic Sobolev POP

Another class of POP on Sobolev functions is

p * = inf f ∈B L(p(f, D a 1 f, . . . , D a l f )) ( 9
)
where p is a given real valued multivariate polynomial of degree d p of a function f ∈ B and its derivatives D a j f , a j ∈ N n , j = 1, . . . , l and L : L ∞ (T n ) → R is a given bounded linear functional. Coefficients of p are bounded functions in x. For example

L(p(f )) = T n (p 1 (x)f (x) + p 2 (x) Df (x) 2 2 )dσ(x) ( 10
)
where σ is a given probability measure on T n and p 1 , p 2 are given real polynomials of x.

Note that contrary to harmonic problem [START_REF] Berlinet | Reproducing kernel Hilbert spaces in probability and statistics[END_REF], the non-linearity hits directly the function value f (x) and its derivatives, and hence problem (9) generally involves infinitely many harmonics. Still, problem ( 9) is called algebraic because p is a finite degree polynomial.

The following result guarantees that the non-linear functional defined above is well defined on H m (T n ) with m large enough. (Ω) then D a f is a Lipschitz function for any a satisfying |a| ≤ m. Since now D a j f ∈ L q (Ω) for any q ∈ [1, ∞] (due to the boundedness of Ω), the desired result is obtained by applying the Hölder inequality.

Let us express the objective function of (9) as a polynomial function of c, the Fourier coefficients of f . Indeed, if f = c, e ℓ 2 = a∈Z n c a e -a then a monomial of degree d ∈ N writes

f d = c, e d ℓ 2 = a 1 ,a 2 ,...,a d ∈Z n c a 1 c a 2 • • • c a d e -(a 1 +a 2 +...+a d )
and it follows that

L(f d ) = a 1 ,a 2 ,...,a d ∈Z n c a 1 c a 2 • • • c a d z -(a 1 +a 2 +...+a d )
where z a := L(e a ) is the moment of index a ∈ Z n of linear functional L. Similarly, successive derivatives of f will be expressed as linear functions of c, and hence polynomials of these derivatives will be multivariate polynomials of c. Overall, the objective function is a polynomial q in infinitely countably many variables with finite algebraic degree d(q) = d p and infinite harmonic degree δ(q) = ∞. Algebraic Sobolev POP (9) can therefore be written equivalently as the algebraic Fourier POP

p * = inf c∈E q(c) = a∈Z n q a c a .
In order to apply the moment-SOS hierarchy, we reformulate this POP as a linear problem

p * := min ν∈P (E) E q(c)dν(c)
on P (E), the set of probability measures on the Fourier ellipsoid E. In turn, this is equivalent to the infinite-dimensional linear problem

p * := min y∈C(Z n ) a∈Z n q a y a s.t. y ∅ = 1
on the full cone of moments

C(Z n ) := {(y a ) a∈Z n : y a = E c a dν(c) for some ν supported on E} ∈ ℓ 2 (Z n ).
In contrast, the harmonic Sobolev POP of the previous section was reformulated as the finite-dimensional linear problem (8) on a truncated cone of moments.

As in the previous section, for every finite algebraic resp. harmonic degrees r and ρ, we can define the outer approximations

C out r,ρ (Z n ) := Π A {(y a ) a∈Z n : ℓ y (q) = a∈Z n
q a c a ≥ 0 for all q ∈ Q r,ρ } and inner approximations

C inn r,ρ (Z n ) := {(y a ) a∈Z n : y a = E c a q(c)dγ(c) for some q ∈ Q r,ρ } such that C inn r,ρ (Z n ) ⊂ C(Z n ) ⊂ C out r,ρ (Z n ) and asymptotically C inn ∞,∞ (Z n ) = C(Z n ) = C out ∞,∞ (Z n
), but these are now infinite-dimensional cones that must be truncated to be manipulated numerically.

Given algebraic resp. harmonic degrees r and ρ, let us consider the finite-dimensional linear problem p * r,ρ := min y∈C(Ar,ρ) a∈Ar,ρ q a y a s.t. y ∅ = 1 on the finite-dimensional cone of moments C(A r,ρ ) indexed by

A r,ρ := {a ∈ Z n : d a ≤ r, δ a ≤ ρ}.
We can design an outer moment-SOS hierarchy of lower bounds p out r,ρ := min y∈C out r,ρ (Ar,ρ) a∈Ar,ρ q a y a as well as an inner moment-SOS hierarchy of upper bounds p inn r,ρ := min y∈C inn r,ρ (Ar,ρ) a∈Ar,ρ q a y a for increasing algebraic resp. harmonic relaxation degrees r and ρ. Our convergence result then follows immediately from the above considerations.

Theorem 2. For finite r, ρ it holds p out r,ρ ≤ p * r,ρ ≤ p inn r,ρ . Asymptotically it holds p out ∞,∞ = p * = p inn ∞,∞ .

Example

Consider the algebraic Sobolev POP

p * = inf f ∈B T (f (x) 2 -1/2) 2 dσ(x)
where σ is the Dirac measure at 0 on B ⊂ H 0 (T ), i.e. m = 0 and n = 1. Observe that the function to be minimized is non-convex in f .

Since f (0) = a∈Z c a , the moments z a of the linear functional in the objective function are equal to one for all a ∈ Z so the problem can be written as the algebraic Fourier POP p * = inf c∈E q(c)

with q(c) = 1 4 - a 1 ,a 2 ∈Z c a 1 c a 2 + a 1 ,a 2 ,a 3 ,a 4 ∈Z c a 1 c a 2 c a 3 c a 4 .
With the outer moment-SOS hierarchy, at algebraic relaxation degree r = 2 and harmonic relaxation degree ρ = 0, we obtain the two global minimizers c * 0 = ± √ 2/2 and the corresponding functions f * (x) = ± √ 2/2 achieving the global minimum p * = p out 2,0 = 0. Note that for such problems it may be desirable to penalize the higher degree Fourier coefficients with a quadratic regularization term.

Kernel Sobolev POP

While an algebraic Sobolev POP generally requires an infinite number of Fourier coefficients to be expressed, actually there exists a better basis, based on kernel methods [START_REF] Aronszajn | Theory of reproducing kernels[END_REF], where the problem admits a representation in terms of a finite number of coefficients. Since H m (T n ) is a reproducing kernel Hilbert space when m > n/2, there exists a kernel function k : T n × T n → R such that k(x, y) = k(y, x), k(•, x) ∈ H m (T n ) for any x, y ∈ T n and more importantly, we have the reproducing property: for any f ∈ H m (T n ) and any x ∈ T n , the following holds

f (x) = f, k(•, x) H m (T n ) .
In particular, for the case of H m (T n ) the kernel is known in closed form in terms of the Bessel function of the second kind, see [START_REF] Berlinet | Reproducing kernel Hilbert spaces in probability and statistics[END_REF]Sec. 7.4]. Then the powerful and fundamental result in machine learning known as the Representer Theorem [START_REF] Schölkopf | A generalized representer theorem[END_REF] holds.

Theorem 3. The Sobolev POP min f ∈H m (T n ) p(f (x 1 ), . . . , f (x l )) (11)
for a given polynomial p is equivalent to the finite-dimensional POP

min w∈R l p( c 1 , w R l , . . . , c n , w R l ) (12)
where c j := (k(x i , x j )) i=1,...,l , j = 1, . . . , l. The first problem admits a solution and only if the second problem admits a solution, and both problems have the same value. In particular, denoting by f * the solution of the first problem and w * the solution of the second problem, we have

f * (•) = l j=1 w * j k(•, x j ).
More generally, the Sobolev POP

min f ∈H m (T n ) p( g 1 , f H m (T n ) , . . . , g l , f H m (T n ) ) (13)
for a given polynomial p and given g j ∈ H -m (T n ) is equivalent to the finite-dimensional POP [START_REF] Fialkow | The truncated K-moment problem: a survey. Operator theory: the state of the art, 25-51[END_REF] where c j = ( g i , g j H m (T n ) ) i=1,...,l , j = 1, . . . , l and

f * = l i=1
w * j g j .

Note that POP ( 11) is a particular case of POP ( 13) corresponding to the choice g j (.) = k(., x j ) since f, k(., x j ) H m (T n ) = f (x j ), j = 1, . . . , l.

Theorem 3 implies that for kernel Sobolev POPs of the form (13), we can apply the standard finite-dimensional moment-SOS hierarchy [START_REF] Lasserre | Moments, positive polynomials and their applications[END_REF] with convergence guarantees.

Note that Theorem 3 also holds when p is any continuous function which is bounded below (not necessarily a polynomial), for any measurable space X beyond T n , and any space of functions on X that is a reproducing kernel Hilbert space, for example any Sobolev space H m (X) where X ⊆ R n is a domain with locally Lipschitz boundary and m > n/2.

Example

Revisiting Example 5.2.1, since the objective function is (f (0) 2 -1/2) 2 , i.e. p(t) = (t 2 -1/2) 2 and x 1 = 0, l = 1 in Sobolev POP [START_REF] Fattorini | Infinite dimensional optimization and control theory[END_REF], it can be expressed equivalently as the univariate POP min w 1 ∈R (w 2 1 k(0, 0) 2 -1/2) 2 whose solutions are w 1 = ± √ 2 2k(0,0) , corresponding to the following minimizers f * (x) = ± √ 2k(x,0) 2k(0,0) .

Solution recovery

When solving infinite-dimensional calculus of variations of control problems, we may be faced with truncated moment problems on Sobolev spaces with an increasing number of Fourier coefficients. When the number of Fourier coefficients goes to infinity, we know that there is a single representing measure, i.e. the infinitedimensional moment problem is determinate.

Proposition 11. A mesure µ supported on B is uniquely determined by its infinitedimensional sequence of moments (y a ) a∈c 0 (Z n ) .

Proof. The sequence of moments (y a ) a∈c 0 (Z n ) in (5) exists and is unique with respect to ν, since ν is a measure defined on E which is a compact Hausdorff set, and the function m a (c) is a monomial in the coefficients c. So we can apply the Stone-Weierstrass theorem. To conclude, note that ν and µ are in one-to-one relation via the invertible linear map F , recall (3).

Given a sequence of moments, we may want to recover the representing measure on B. In the finite-dimensional case, the Christoffel-Darboux kernel can be used to approximate the support of a measure given its moments [START_REF] Lasserre | The Christoffel-Darboux kernel for data analysis[END_REF]. It would be interesting to extend this kernel to Sobolev spaces.

Conclusion

In this paper we address and solve numerically the moment problem for measures supported on the unit ball of a Sobolev space. We describe how the finitedimensional moment-SOS hierarchy can be extended to this infinite-dimensional setup, allowing to solve numerically polynomial optimization problems on Sobolev spaces while preserving approximation and convergence guarantees.

All our developments are done for a specific basis of complex exponentials (1), but similar results could be achieved for any basis with good approximation properties for the Sobolev space H m (T n ) or other reproducing kernel Hilbert spaces, as highlighted in Section 5.3.

Our approach can also be generalized with the exact same construction to other spaces like Sobolev spaces on general domains, Besov or Triebel-Lizorkin spaces and more generally quasi-Banach spaces where there exists a Schauder basis with reasonable approximation properties.

Finally, applications of these techniques and the infinite-dimensional moment-SOS hierarchy to the approximation of solutions of nonlinear calculus of variations problems or optimal control involving non-linear partial differential equations remain to be investigated.

  y → (y a ) a∈A denote the projection map onto the subspace indexed by A. We use the Fourier embedding of the previous section to express the Sobolev moment cone as the Fourier moment cone C(A) = {(y a ) a∈A : y a = E c a dν(c) for some ν supported on E} ⊂ C N . Let C[c] denote the set of complex polynomials of the indeterminate c. Elements of C[c] can be expressed as linear combinations of monomials p(c) = a∈spt(p) p a c a with algebraic degree d(p) := max a∈spt(p) d a and harmonic degree δ(p) := max a∈spt(p) δ a . We are particularly interested in Hermitian polynomials, i.e. elements of C[c] with values in R. Given r, ρ ∈ N, let C[c] r,ρ := {p ∈ C[c] : d(p) ≤ r, δ(p) ≤ ρ} and define the cone of Hermitian polynomial sums of squares Σ r,ρ :=

p

  * := min µ∈P (B) B p(f )dµ(f ) on P (B), the set of probability measures on the Sobolev ball B. Using the Fourier embedding, harmonic Sobolev POP (7) is equivalent to the harmonic Fourier POP p * := min c∈E p(c) and the linear problem p * := min ν∈P (E) E p(c)dν(c) on P (E), the set of probability measures on the Fourier ellipsoid E. In turn, this is equivalent to the linear problem p * := min y∈C(A) a∈A p a y a s.t. y ∅ = 1 (8) on the cone of moments C(A). Therefore we can design a moment-SOS hierarchy of lower bounds p out r,ρ := min y∈C out r,ρ (A) a∈A p a y a as well as a moment-SOS hierarchy of upper bounds p inn r,ρ := min y∈C inn r,ρ (A) a∈A p a y a for increasing algebraic resp. harmonic relaxation degrees r ≥ d(p), ρ ≥ δ(p).

  Proposition 10. The functional of Sobolev POP (9) is bounded when f ∈ H ) where s = max j=1,...,k |a j |.Proof. By the Sobolev embedding theorem[START_REF] Adams | Sobolev spaces[END_REF] in a bounded set Ω with Lipschitz boundary in R n , when f ∈ H

	s+n/2+1 2 (T n m+n/2+1
	2

This work was co-funded by the European Union under the project ROBOPROX (reg. no. CZ.


domain: phys chim chim.cris

text:
Lachkar

M Idboumlik

M Kadiri

N Hamdi

M Driouch

A F I Ngopoh

I Lakkab

E-E Bendeif

M Sfaira

B El Bali

M Lachkar

Prof. Dr Abdelkader Zarrouk
email: azarrouk@gmail.com

Synthesis of novel hybrid decavanadate material (NH 4 ) 2 (H 2 en) 2 {V 10 O 28 }.4H 2 O: Characterization, anticorrosion and biological activities

Keywords: Decavanadate hybrid material, Structural analysis, Corrosion inhibition, Biological activity

Synthesis of novel hybrid decavanadate material (NH4)2(H2en)2V10O28.4H2O: Characterization, anticorrosion and biological activities

Introduction

Making up to 150 ppm of the Earth's core, and comprises 0.014% of the Earth's crust, vanadium (V) is an extensively dispersed element generally occurring at low concentrations in about 65 minerals; almost 15×10 -3 g is detected in a 75 Kg heavy human body [START_REF] Tracey | Vanadium Chemistry, biochemistry, pharmacology and practical applications[END_REF][START_REF] Crans | Does anion-cation organization in Na+containing X-ray crystal structures relate to solution interactions in inhomogeneous nanoscale environments: Sodium-decavanadate in solid state materials, minerals, and microemulsions[END_REF]. Vanadium has three stable oxidation states: V +3 , V +4 , and V +5 , which have crucial biological significance, especially vanadates containing V +5 . The oxidation states -1 and +2 are unstable forms. V, as Mo and W, are oxidized to give polyoxometalates (POMs), which are anionic clusters based on oxocomplexes of transition metals; they have attracted considerable attention by virtue of having a variety of domains including catalysis, biology, materials science [START_REF] Aureliano | Decavanadate in vitro and in vivo effects: Facts and opinions[END_REF]. The polyoxovanadates (POVs) exhibit a large variety of structural features, ranging from metavanadates chains [VO 3 - ] n , oxides layered [V 4 O 10 ], to polyanions in a compact state [V 10 O 28 ] 6-, depending on the synthesis conditions, especially medium concentration and pH [4]. Interestingly, Aureliano et al. reported on the decrease of the V +5 -coordination upon pH increases of the solutions aiming the crystals synthesis [START_REF] Aureliano | Characterization of decavanadate and decaniobate solutions by Raman spectroscopy[END_REF]. Indeed, for pH ≈ 6, one get a yellow solution containing essentially decavanadate core (H n V 10 O 28 ) n-6 (n = 0-4). For pH ≈ 6, a yellow solution containing essentially decavanadate core (H n V 10 O 28 ) n-6 (n = 0-4) is gotten. Increasing pH and changing concentration conditions produce other polyoxovanadates, with [V 2 O 7 2-] and [V 4 O 12 4-]; whereas a decrease in ph eventually leads to the creation of additional decavanadate forms: [HV 10 O 28 ] 5-, [H 2 V 10 O 28 ] 4 - , [H 3 V 10 O 28 ] 3-, [H 4 V 10 O 28 ] 2- [START_REF] Jagoba | Hybrid Polyoxometalates : Synthesis , Crystal Structures , Thermostructural Behavior and Anchoring to Tailored Polymeric Surfaces Jagoba Martín Caballero[END_REF]. The growing interest of these polyanions is due to their important biological activities such as interactions with lipid interfaces, insulin enhancing, cell death and probing muscle contraction [START_REF] Aureliano | Decavanadate (V10 O28) 6and oxovanadates: Oxometalates with many biological activities[END_REF].

Recently, V-based compounds have been investigated for corrosion inhibition of Al alloys as a suitable alternative to Chrome [START_REF] Kendig | Corrosion inhibition of aluminum and aluminum alloys by soluble chromates, chromate coatings, and chromate-free coatings[END_REF][START_REF] Buchheit | Corrosion Science: A Retrospective and Current Status in Honor of Robert P. Frankenthal[END_REF]. However, due to the relatively high solubility of vanadium oxides in aqueous solutions, the area has received little study [START_REF] Guan | Corrosion protection of aluminum alloy 2024-T3 by vanadate conversion coatings[END_REF][START_REF] Trypuć | Solubility investigations in the NH4Cl+NaVO3+NH4VO3+NaC l+H2O system at 303 K[END_REF]. In an investigation of the release kinetics and protection effectiveness of vanadate-based pigments in epoxy-coated AA2014-T6 panels, Smith et al. [START_REF] Smith | Proceedings of Eurocorr[END_REF] found that vanadates are effective pigments for safeguarding Al alloys. A later investigation by Iannuzzi et al. concluded that in aqueous solutions, at neutral to basic pH range, vanadates gave the best performance as inhibitor for Aluminum [START_REF] Iannuzzi | Aluminum Alloy Corrosion Inhibition by Vanadates[END_REF]. Based on these findings, we tested our new hybrid vanadate (NH 4 ) 2 (H 2 en) 2 {V 10 O 28 }.4H 2 O as corrosion inhibitor for mild steel in both acidic medium 1 M HCl and neutral solution of 3% NaCl. Our choice aims to select the appropriate medium, and to study further the mechanism of corrosion inhibition by potentiodynamic polarization (PP) and electrochemical impedance spectroscopy (EIS) measurements. We hereby report on the synthesis, via a soft chemistry route, of the title compound. We discussed its spectroscopic and thermal features correlated to its crystal structure. Moreover, we reported on its corrosion inhibition, bactericidal inhibition and antioxidant activities.

Experimental, Materials and Methods

Synthesis and crystallization of (NH 4 ) 2 (H 2 en) 2 {V 10 O 28 }.4H 2 O

All of the reagents and solvents used for the reported researches were purchased from Sigma-Aldrich and employed without additional purification. Ammonium metavanadate (NH 4 )VO 3 (300 mg, 2.5 mmol) diluted in 20 mL distilled water and kept at 90 °C with continuous agitation.

The obtained light-yellow solution was allowed to cool to 50 °C and 80 mg (0.4 mmol) of CuCl 2 .4H 2 O and 30 mg (0.5 mmol) of ethylenediamine were added before adjusting the brown color solution to pH 5 by adding HNO 3 acid. The mixture was further stirred about 30 min before to be filtered and kept at room temperature. Orange crystals rose after two weeks, which are suitable for X-Rays single crystal analysis. They have been washed with an ethanol-water (80-20%) mixture and dried at ambient conditions.

Single crystal X-ray diffraction measurements and structural determination

The single crystal X-ray diffraction measurements were acquired at 100 (2) K on a Bruker D8 Venture diffractometer, supplied with a two-dimensional CMOS detector (Bruker PHOTON100), using the Mo K -radiation (= 0.71073 Å). For the unit cell determination, the Bruker APEX package software [START_REF]CrysAlis CCD and CrysAlis RED[END_REF] was employed, along with data reduction and analytical absorption adjustments. Direct techniques and consecutive Fourier difference syntheses were used to solve the appropriate structure in the space group P-1, and it was refined against F2 using weighted full-matrix least-squares methods in the SHELXL97 software [START_REF] Sheldrick | Crystal structure refinement with SHELXL[END_REF]. The non-H atoms were polished in an anisotropic manner. The WinGX software package [START_REF] Farrugia | WinGX suite for small-molecule single-crystal crystallography[END_REF] was used to do all computations. The structural graphics were drawn using DIAMOND program [START_REF] Brandenburg | Crystal Impact GbR[END_REF]. Table 1 illustrates the crystal data, data gathering, and structural refinement information. The atomic coordinates are reported in Table 2. Table 3 and Table 4 show selected bond lengths and angles, as well as specifics of hydrogen bonding geometries. The CIF data file of the investigated compound may be found in the Cambridge Structural database under the number: CCDC 2104907. Copies of the published material can be acquired, free of charge, on application to the Director, CCDC, 12 Union Road, Cambridge CB2 1EZ, UK, fax: +44 (0)1223 336033 or Email: deposit@ccdc.cam.ac.uk Equivalent atoms are generated via symmetry transformations: #1 -x,-y+1,-z+1. Table 3 Selected bond lengths (Å) for complex (H2en)2(NH4)2{V10O28}.4H2O.

= -14→14 k = -17→17 l = -18→18 R[F 2 > 2(F 2 )]/ wR(F 2 )/S 0.023/0.066/1.13
V(4)-O(11) 1.6109(7) V(5)-O(6) 2.2302(7) V(4)-O(4) 1.8413(7) V(5)-V(3)#1 3.0684(2) V(4)-O(10) 1.8427(7) V(5)-V1 3.0986(2) V(4)-O(12) 1.8837(7) V1-O(5) 1.6086(7) V(4)-O(7)#1 2.0444(7) V1-O(4) 1.8202(7) V(4)-O(6) 2.3155(7) V1-O(3) 1.8643(7) V(4)-V1 3.0508(2) V1-O(14) 1.8819(7) V(4)-V(2)#1 3.0712(2) V1-O(2) 2.0524(7) V(4)-V(3) 3.1048(2) V1-O(6) 2.3082(7) V(2)-O(7) 1.6881(7) V1-V(3) 3.0790(2) V(2)-O(2) 1.6948(7) V(3)-O(9) 1.6170(7) V(2)-O1 1.8962(7) V(3)-O(3) 1.7925(7) V(2)-O(8) 1.9500(7) V(3)-O(10) 1.8433(7) V(2)-O(6) 2.0952(7) V(3)-O(8) 1.9756(7) V(2)-O(6)#1 2.1310(7) V(3)-O1#1 2.0197(7) V(2)-V1 3.0686(3) V(3)-O(6) 2.2347(6) V(2)-V(4)#1 3.0712(2) V(3)-V(5)#1 3.0684(2) V(5)-O(13) 1.6155(7) O(8)-V(5)#1 2.0153(7) V(5)-O(12) 1.8170(7) O1-V(3)#1 2.0198(7) V(5)-O(14)
1.8171( 7) O(7)-V(4)#1 2.0444(7) V(5)-O1

1.9972( 7) O(6)-V(2)#1 2.1309 [START_REF] Aureliano | Decavanadate (V10 O28) 6and oxovanadates: Oxometalates with many biological activities[END_REF] To generate equivalent atoms, symmetry transformations are employed: #1 -x,-y+1,-z+1. Table 4 Hydrogen bonding network in the framework of complex (H2en)2(NH4)2{V10O28}.4H2O. Equivalent atoms are generated via symmetry transformations:

#1 -x,-y+1,-z+1 #2 -x+1,-y+2,-z+1 #3 -x,-y+2,-z+2 #4 x,y,z+1 #5 -x,-y+1,-z+2 #6 -x+1,-y+2,-z+2 #7 -x+1,-y+1,-z+1 #8 x+1,y,z #9 x,y,z-1

Infrared spectroscopy measurements

At room temperature, the FT-IR spectra of a powder formed of grounded crystals of the title compound was measured on a VERTEX 70 FTIR Spectrophotometer with a resolution of 4 cm -1 in the range 4000-400 cm -1 . The sample was analyzed in powder form (ATR technique).

Thermal analysis measurements

The thermogravimetric analysis was used to investigate the compond's thermal stability (TGA) measurements were performed as in the previous case. TGA data were collected using an SDT-Q600 analyzer from TA instruments. The sample is heated from room temperature to 600 °C at a rate of 10 °C min -1 . Samples were measured in open platinum crucibles with air flow.

Electrochemical measurements

Solutions for electrochemical tests were prepared by dissolving the new hybrid compound of decavanadate (NH 4 ) 2 (H 2 en) 2 (V 10 O 28 ).4H 2 O as corrosion inhibitor in two electrolytic media.

The hydrochloric acid solution was obtained by dilution of concentrated acid (37%, d= 1.19).

The second electrolyte was 3% of NaCl, which had a purity ≥ 99% by dissolution in distillated water. The working mild steel (MS) electrode was rectangular with the following chemical composition 0.370% C, 0.230% Si, 0.680% Mn, 0.016% S, 0.077% Cr, 0.011% Ti, 0.059% Ni, 0.009% Co, 0.160% Cu and the remainder iron (Fe). The working electrode was coated by a resin to delimitate the surface area exposed to the electrolyte solution of 1 cm 2 , and was abraded using 120 to 2000 grit emery papers, rinsed with distilled water, degreased in acetone, and freshly washed with bi-distilled water.

The electrochemical experiments were performed in the conventional three-electrode cell using a platinum counter electrode (CE) and Ag/AgCl 3M KCl as reference electrode. All these experiments were conducted by the BioLogic SP-150 device, controlled with analysis software (Ec-Lab® V11.20). For potentiodynamic polarization (PP) measurements, the potential was scanned at a sweep rate of 1 mV s -1 in bearing potential range of -250 mV to +250 mV versus the open circuit potential. Electrochemical impedance spectroscopy (EIS) measurements were performed at corrosion potential in the frequency range of 100 kHz to 100 mHz and an AC voltage amplitude of 10 mV peak to peak. The working electrode was immersed in the test solution for 30 minutes prior to the PP and EIS measurements to establish a steady state potential. All experiments were repeated, minimum in triplicate, and the temperature of cell was controlled by a thermostatic water bath.

UV-visible

The electronic spectrum was carried out by using a Camspec, M 550 Double Beam scanning UV-visible spectrophotometer using a quartz cell of 1 cm length with a wavelength range of 200-800 nm -1 .

Surface Morphology

Surface evaluations were performed on mild steel sample in order to better understand their surface morphology, using ESEM Quanta 200 FEI scanning electron microscopy (SEM).

Evaluation of antimicrobial activity in vitro

The microdilution technique was used to determine the examined compound's minimum inhibitory concentration (MIC) and minimum bactericidal concentration (MIB). The process entails preparing twofold dilutions of the antimicrobial agent in a liquid growth medium spread in a 96-well microtiter plate. After dilution of the standardized microbial suspension to a scale of 0.5 McFarland, the cells were injected with a microbial inoculum made in the same medium.

The 96-well microtiter plate is incubated under appropriate conditions depending on the microorganism investigated once it has been well mixed. Upon 20 hours of incubation at 37 °C, 15 L of 0.015 percent resazurin was added and incubated for a further 2 hours to detect the color changes [START_REF] Balouiri | Methods for in vitro evaluating antimicrobial activity: A review[END_REF].

Antioxidant activity

Free radical scavenging

The antioxidant capacity of the material was determined using the 1,1-diphenyl-2-picrylhydrazyl (DPPH) procedure reported by Brand-Williams et al. [START_REF] Brand-Williams | Use of a free radical method to evaluate antioxidant activity[END_REF]. First, 0.1 mL of the complex with different concentrations was added to 3.9 mL of a methanol solution of DPPH (1.01×10 -2 mM). The mixture was incubated in the dark for 30 minutes before measuring absorbance at 517 nm. A mixture of 0.l mL methanol and 3.8 mL DPPH solution was employed as a control. The decrease in absorbance shows the tested sample's ability to scavenge free radicals. Based on the percentage of DPPH radicals scavenged, the scavenging activity was determined using Equation 1 below:

% Inhibition = [(AB -AS)/AB] × 100 (1)
where AB represents the absorbance of the control reaction (which contains all reagents except the test compound), and AS denotes the absorbance of the test compound. As a positive control, butylatedhydroxytoluene (BHT) was employed.

Ferric reducing power assay (FRAP)

The reducing power of the novel complex was determined according to the approach outlined by Oyaizu et al. [START_REF] Oyaizu | Studies on products of browning reaction. Antioxidative activities of products of browning reaction prepared from glucosamine[END_REF]. 0.5 mL of varied concentrations (0-4000 g/mL) were mixed with 2.5 mL of phosphate buffer (2.5 mL, 0.2 M, pH 6.6) and potassium ferricyanide (2.5 mL, 1 percent) in 1 mL of distilled water, at 50 °C, the whole solution was incubated. 2.5 mL of 10% (w/v) trichloroacetic acid, 2.5 mL of distilled water, and 0.5 mL of a 0.1 percent (w/v) FeCl3 solution were added to the mix after 20 minutes of incubation. A UV-visible spectrophotometer was used to measure absorbance at 700 nm against a blank, and the compound's reduced power was compared to that of gallic acid, which was employed as a reference standard.

Phosphomolybdenum assay (PM)

According to Prieto et al. [START_REF] Prieto | Spectrophotometric quantitation of antioxidant capacity through the formation of a phosphomolybdenum complex: Specific application to the determination of vitamin E[END_REF], the Phosphomolybdenum Assay (PM) or Total Antioxidant Capacity (TAC) was conducted. 3 mL molybdate reagent (acid sulfuric (0.6 M), sodium phosphate (28 mM), and ammonium molybdate (4 mM) were mixed with 0.3 mL sample at various concentrations (250 -500 -1000 g/mL). For 90 minutes, the mixture was kept at 95°C.

The absorbance was measured at 695 nm, and the results were represented as (g mL -1 ) ascorbic acid equivalents. As a positive control, glycerin was employed.

Results and discussion

Structure and crystal packing

The asymmetric unit of (NH 4 ) 2 (H 2 en) 2 {V 10 O 28 }.4H 2 O (Figure 1) contains one-half decavanadate anion [V 10 O 28 ] 6-, as denoted hereafter, one ethylenediammonium cation (H 2 en) 2+ , one ammonium cation (NH 4 ) + and two H 2 O molecules. The decavanadate unit [V 10 O 28 ], of D 2h symmetry, is similar to the group found in hummerite [START_REF] Hughes | The crystal structure of hummerite, KMg(V5O14).8H2O: bonding between the [V10O28] 6structural unit and the {K2Mg2(H2O)16} 6+ interstitial complex[END_REF] or in pascoite [START_REF] Swallow | The crystal structure of pascoite, Ca3V10O28.16H2O[END_REF]. [START_REF] Brown | The Bond-Valence Method: An Empirical Approach to Chemical Structure and Bonding' in Structure and Bonding in Crystals[END_REF], where R is the V-O distance and s is the bond number, was used to assess the likelihood of protonation sites in the decavanadate anion. The computed values vary between 1.683 to 2.051 for all oxygen atoms, as shown in Table 5, omitting protonation of the decavanadate anion.

Infrared spectroscopy

The infrared spectrum of (NH 4 ) 2 (H 2 en) 2 {V 10 O 28 }.4H 2 O is depicted in Figure 3 and Table 6 lists the suggested assignments for several significant bands. Based on the decavanadate-based coordination polymers description [START_REF] Jin | Synthesis and Crystal Structure of Decavanadate-Based Coordination Polymers[END_REF], the stretching mode of the terminal V-O bonds may be shown in the spectrum at 930 cm -1 , where two strong bands can be seen. The bands at 508 and 448 cm -1 are attributed to the symmetric V-O-V bridge, whereas the characteristics at 810, 722 cm -1 are related to the bridging anti-symmetric V-O-V vibrations. The presence of the organic ethylenediammonium cation is confirmed by the stretching C-N bands at 1043, 1079 and 2419 cm -1 , as by the vibrational bands at 3160-3030 cm -1 and 1600 cm -1 that correspond, respectively, to N-H bending and the H-N-H scissoring. The bands 2757, 2932 and 2117 cm -1 are related to the C-H stretching [START_REF] Marchewka | Ethylenediammonium dication: H-bonded complexes with terephthalate, chloroacetate, phosphite, selenite and sulfamate anions. Detailed vibrational spectroscopic and theoretical studies of ethylenediammonium terephthalate[END_REF]. The broad band at 1404 cm -1 might be assigned to the N-H bending vibration of the ammonium cation. Further, the presence of water molecules is confirmed by the 3517-3238 cm -1 and 1807 cm -1 bands [START_REF] Putrevu | Decavanadate with a novel coordination complex: Synthesis and characterization of (NH4)2[Ni(H2O) 5(NH3)]2(V10O28) •4H2O[END_REF].

Thermal analysis

The thermal decomposition of (NH 4 ) 2 (H 2 en) 2 {V 10 O 28 }.4H 2 O was performed under air atmosphere and exhibits a total weight loss of 25% (Figure 4). The mass loss observed in the 100-500 °C temperature range corresponds to three consecutive weight losses around 100, 160 and 260 °C. The initial phase weight loss 12.2 % correlates to the water molecules' departure, ammonium ion and ½O 2 (calcd. 10.8 %). Next step corresponds to the loss of the ethylenediamonium organic cation (exp. 9.1 %, calcd. 10.4 %) and further weight loss beyond 260 °C relates to decavanadate cluster decomposition. Actually, the calculated value of the loss weight of the last step agrees to the degradation of decavanadate cluster and the formation of (V 2 O 5 ) and (V 3 O 7 ) (exp. 78.7 %, calcd. 78.1 %) [START_REF] Omri | Novel decavanadate cluster complet (HImz)12(V 10O 28) 23H 2O: Synthesis, characterization, crystal structure, optical and thermal properties[END_REF]. According to TGA analysis, and to confirm the nature of the final product after calcination, temperature dependent in situ XRD data were acquired at room temperature and 450 °C, as shown in Figure 5. The pattern at RT was quite similar to the pattern that had been simulated using the coordinates from the single-crystal investigation of (NH 4 ) 2 (H 2 en) 2 {V 10 O 28 }.4H 2 O. However, at 450 °C the structure changes, while keeping the crystallinity of the sample and presenting a different product. The 450 °C XRD pattern showed the characteristic peaks of V 2 O 5 and V 3 O 7 as well [START_REF] Chalotra | Oxygen deficient V2O3: A stable and efficient electrocatalyst for HER and high performance EDLCs[END_REF], approving the TGA calculation results, with a small unidentified diffraction peak.

Speciation vs. corrosion inhibition

Changes in vanadate speciation with pH

Because of a lack of understanding of speciation, the complexity of vanadate hydrolysis in aqueous solutions, and the occurrence of metastable equilibria, the results of known investigations on corrosion inhibition by vanadates are clouded. The major goal of this part is to figure out which vanadate oligomer provides the best mild steel corrosion resistance in two aggressive solutions (1 M HCl and 3 percent NaCl). We used pH measurements, ultravioletvisible spectroscopy, PP and EIS measurements, and scanning electron microscopy initially.

According to the literature, small changes in solution pH may have a large effect on vanadate speciation [START_REF] Ralston | Electrochemical Evaluation of Constituent Intermetallics in Aluminum Alloy 2024-T3 Exposed to Aqueous Vanadate Inhibitors[END_REF][START_REF] Ralston | Corrosion Inhibition of Aluminum Alloy 2024-T3 by Aqueous Vanadium Species[END_REF]. (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O is a colorless complex when dissolved in 1 M HCl, the solution has a pH of 0.315. Whereas, when dissolved in 3% NaCl, the solution turned yellow and its pH is of 7.6, as shown in Figure 6.

Simulated (100 K) § V 2 O 5 ¨V3 O 7 ªunidentified Experimental (R.T) § § § ¨ § § § § § § ä § 2

UV-Visible spectroscopy measurements

According to Aureliano, the presence of [V 10 O 28 ] 6-anions is responsible for the yellow color of the solution [START_REF] Aureliano | Decavanadate: A journey in a search of a role[END_REF]. Actually, the yellow color of solutions containing V 5+ (3d 0 ) ions are due to electrons transfer from the oxygen bonding orbitals to the vacant vanadium d orbitals. These charge transfer bands move towards the UV-region when the crystal field splitting of d orbitals decreases [START_REF] Valente | Metal solution precursors: Their role during the synthesis of MoVTeNb mixed oxide catalysts[END_REF]. Electronic absorption spectra of were taken with a UV-Vis spectrometer within the range [200-

800 nm] at room temperature of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O in both 1 M HCl and 3% NaCl at the same concentration of 5×10 -4 M. As shown in Figure 7, the 280 nm (black line) and 391 nm (red line) bands are attributed to the V 5+ tetrahedrally coordinated in monovanadates and octahedrally in decavanadates, respectively [36][37][START_REF] Zhetcheva | Synthesis and spectral characterization of a decavanadate/chitosan complex[END_REF].

EIS measurements of vanadate solution

The

Corrosion morphology of mild steel in vanadate solution

To observe the action of the two solutions on the topography surface of MS, scanning electron microscopy was used. Figure 9 shows the SEM micrographs of MS specimen exposed to 1 M HCl and 3% NaCl during 24 h, in which the complex (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O was dissolved. The effects of decavanadates are dramatic on MS surface; very large alterations on the surface are evident, indicating severe corrosion of specimen (Figure 9a). Rather Figure 9b indicates that in the presence of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O in clear solution, the rate of corrosion is suppressed, as can be seen from the decrease of the corroded area. The surface is covered by a thin layer of monovadates molecules, which effectively controls the dissolution of MS. Monovanadates were only found in clear metavanadate solutions, but decavanadates were always present in yellow solutions. The latter, which contained 3 percent NaCl at pH = 7.6, showed no appreciable inhibition, and any rise in decavanadate content was only harmful. The clear solution containing monovanadates, on the other hand, demonstrated substantial inhibition.

Electrochemical study of the clear solution

PP measurements

The PP measurements represent an essential method to gather details about the kinetics of corrosion system being investigated. It can be concluded from these results (Figure 10) that both anodic iron dissolution and cathodic hydrogen evolution reaction were inhibited after the addition of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O in 1 M HCl solution. The inhibition of these reactions was more pronounced on increasing the concentration of the title complex, ranging from 1×10 -6 to 5×10 -4 mol L -1 . From cathodic branches, a broad linearity is clearly shown in the absence as well as in the presence of different concentrations of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O. This behavior confirms that the reduction of hydrogen takes place at the mild steel surface by charge transfer mechanism [START_REF] Rice-Evans | Structure-antioxidant activity relationships of flavonoids and phenolic acids, Free Radic[END_REF]. In the other side, the anodic branches exhibit almost an absence of Tafelian behavior and the inhibition mode remains dependent on the electrode potential. In addition, the title complex influences the anodic reaction only at over potentials more negative than -300 mV Ag/AgCl . Over this potential, the title compound assigns no inhibiting effect, and thereby the anodic branches recorded in the presence of the title compound joined the blank. This can be explained by desorption of the studied complex "molecules" from the formed film. The desorption potential is a common term for this potential [START_REF] Wang | Anodic desorption of inhibitors I. The phenomenon of anodic desorption of inhibitors[END_REF].

As stated above, the anodic dissolution of mild steel is not activation controlled. Therefore, according to the shape of the anodic curves, it will be erroneous to extract the anodic Tafel slopes ß a in order to evaluate the corrosion densities [START_REF] Aourabi | Phenolic fraction of Ammi visnaga extract as environmentally friendly antioxidant and corrosion inhibitor for mild steel in acidic medium[END_REF]. As a result, by extrapolating solely the right-hand side of Tafel cathodic to the corrosion potential, corrosion current densities can be determined. The extraction of electrochemical parameters from the polarization curves was carried out using two methods: Tafel, and Stern & Geary (S&G), which use these curves differently, depending on the range potentials E corr ± 250 mVAg/AgCl, E corr ± 25 mVAg/AgCl, respectively. The values of i corr , E corr , cathodic Tafel slopes ß c and η% issued from linear Tafel segments of cathodic curves and R p obtained by S&G method are regrouped in Table 7. The data were fitted by using Tafel Fit analysis tool based on Linear Square method used along with EC-Lab software. In this case the inhibition efficiency derived from Tafel method signified by η Tafel %, is calculated using the Equation 2, as follows:

100 % /          - = corr inh corr corr Tafel i i i  (2)
where: i corr/inh and i corr represent the corrosion current density values, with and without (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O, respectively.

The inhibition efficiency of S&G method, given by η S&G %, is calculated from the polarization resistance, using the following Equation 3:

100 % / / &          - = inh p p inh p G S R R R  (3)
where: P R and inh P R / are the polarization resistances of the MS without and with addition of the studied complex, respectively. and Srinivansan that the change in the values of ß c can be ascribed to the thickening of the electric double layer due to the adsorbed inhibitor molecules [START_REF] Jo'm | Elucidation of the Mechanism of Electro-Lytic H Y D R O G E N Evolution By the Use of H-T Separation Factors[END_REF].

It can be concluded that the inhibiting efficiencies calculated by the S&G method show the same trend as those determined by extrapolating the Tafel cathodic straight lines to the corresponding corrosion potentials.

EIS measurements

To confirm the results extracted from PP measurements and to acquire more information on the inhibition mechanism, obtained for the MS electrode after 30 min of immersion at the corrosion potential in the 1 M HCl with and without (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O, Figure 11 shows the impedance diagrams, given in both Nyquist and Bode coordinates. Figure 13 gives a detailed comparison between the simulations with the two circuits mentioned above. Furthermore, the obtained results have a physical meaning. R s represents the solution resistance, R ct is the charge transfer resistance and CPE dl represents the capacitance of the high frequency semicircle that can be attributed to the charge transfer process. R f represents the resistance of the adsorbed inhibitor and CPE f is the capacitance of the inhibitor film due to the inhibitor's adsorption onto the MS surface.

(R s , R ct ,Q dl ,R f , Q f ).
2

The electrochemical impedance parameters for MS in 1 M HCl in the presence of different 2 . Qdl and Qf in μF s n-1 cm -2 .

11 with 1 = (H2en)2(NH4)2{V10O28}.4H2O. 12 13 14 Table 8 shows clearly that R ct and Q dl values have opposite trend at the whole concentration 15 range. A large R ct is associated with a slower corroding system [START_REF] Growcock | Time-Resolved Impedance Spectroscopy of Mild Steel in Concentrated Hydrochloric Acid[END_REF][START_REF] Lebrini | Polyphosphate derivatives of guanidine and urea copolymer: Inhibiting corrosion effect of Armco iron in acid solution and antibacterial activity[END_REF]. Furthermore, the 16 decrease in the Q dl with increase in (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O concentration may be 17 attributed to the formation of a protective layer on MS surface [START_REF] Keddam | Reaction Model for Iron Dissolution Studied by Electrode Impedance: II . Determination of the Reaction Model[END_REF]. The increase of n dl value 18 after addition of the title compound in the corrosive solution can corroborate this assumption.

19 Indeed, the value of n dl grows as well 0.8871, when compared to that obtained in pure 1 M HCl 20 of 0.8232. This can be attributed to a certain decrease in the initial surface homogeneity 21 resulting from the adsorption of the title complex " molecules" on the most active adsorption 22 centers at MS interface [START_REF] Growcock | Time-Resolved Impedance Spectroscopy of Mild Steel in Concentrated Hydrochloric Acid[END_REF]. The resistance R f has high values compared to R ct values (the total 23 resistance R p is dominated by R f and increases significantly with the (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O concentration. However, the values of Q f have the same trend as those of R f in the whole concentration range, while almost no change is observed in the value of n f with lower than those of n dl , probably related to the energy dissipation in the adsorbed layer [START_REF] Popova | Inhibitive properties of quaternary ammonium bromides of N-containing heterocycles on acid mild steel corrosion. Part II: EIS results[END_REF].

It is clearly evident that increasing the concentration of the inhibitor enhances R p , and thus improves the efficiency of the inhibition until reaching their maximum value at 5×10 -4 M to 92%. This value suggests that inhibitors function by adsorption at the metal/solution interface, leading to a protective film on the metal surface.

Biological activities

Antimicrobial tests

The new hybrid decavanadate compound (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O was evaluated for antibacterial efficacy against two Gram-(Pseudomonas aeruginosa ATCC27653 and Escherichia coli CIP5412) and two Gram+ (Bacillus subtilis ILP1428B, Staphylococcus aureus CIP543154) strains. Table 9 shows MIC (Minimum Inhibitory Concentration) and MBC (Minimum Bactericidal Concentration) by ppm. The compound showed the highest inhibition activity against P. aeruginosa (625 ppm), B. subtilis (1250 ppm), and E. coli with 2500 ppm, although there was no substantial action against S. aureus CIP. Table 9 The inhibition activity and the bactericidal activity of complex 1. On the other hand, the compound displayed a lower bactericidal activity with B. subtilis (5000 ppm), and it was not determined (ND) for P. aeruginosa and E. coli CIP. All these results show a weak activity compared to similar decavanadate complexes [START_REF] Chen | Preparation, characterization and antibacterial activity of chitosan-Ca3V10O28 complex membrane[END_REF], chitosan-Ca 3 V 10 O 28 , and (NH 4 ) 6 V 10 O 28 , that manifest good bactericidal activity with S. aureus and E. coli (Table 10).

MIC

Antioxidant tests

The DPPH test is one of the most often used procedures for determining antioxidant activity.

Accordingly, an antioxidant reacts with DPPH free radical and reduces it to DPPH-H form and consequently, the absorbance decreases. The degree of discoloration reveals the antioxidant compound's scavenging capacity as well as its hydrogen-donating ability [START_REF] Shekhar | Antioxidant Activity by DPPH Radical Scavenging Method of Ageratum conyzoides Linn. Leaves[END_REF]. As seen in Figure 14, the title compound exhibited a moderate free radical scavenging activity compared with BHT thus, a concentration of 4000 µg/mL inhibited 35.3 % compared with BHT (78 %). In general, the antioxidant activity is expressed in terms of IC 50 , which is the concentration of the antioxidant that decreases the 50 % of DPPH [START_REF] Scherer | Antioxidant activity index (AAI) by the 2,2-diphenyl-1-picrylhydrazy l method[END_REF]. However, in this study the 50 % of DPPH inhibition by (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O wasn't achieved; thus, the concentrations used in this study were not sufficient to inhibit the IC 50 of inhibition. The ferric reducing power test FRAP was used to evaluate the title compound's reducing power. This latter measures the reducing potency of extract and standard antioxidants; thus, higher absorbance indicates higher reducing potency. The mechanism involved in this essay is based on a single-electron transfer system [START_REF] Huang | The chemistry behind antioxidant capacity assays[END_REF]. As depicted in Figure 15, there was a significant difference between the reducing power of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O and the gallic acid even with high concentrations. In the majority of times, we use different methods to evaluate the antioxidant activity. Therefore, The FRAP test was chosen because it is rapid and easy to carry out, and the reaction is repeatable and linearly related to Biskup et al. [START_REF] Biskup | Antioxidant activity of selected phenols estimated by ABTS and FRAP methods[END_REF]. The phosphomolybdenum test measures total antioxidant activity and provides a direct estimate of an antioxidant's reducing power. Particularly, this test is based on the reducing capacity of the Mo(VI) to Mo(V) by an antioxidant and the formation of a green complex of Mo(V) at an acidic pH and higher temperature of 95 °C. A seen in Figure 16, and comparing with gallic acid, the complex revealed a moderate antioxidant capacity similar to the previous results obtained using the DPPH test. We believe that no other authors have investigated the antioxidant properties of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O. Our findings demonstrated the moderate antioxidant activity of the title compound confirmed by three validated antioxidants tests.

Conclusion

The hybrid ethylenediamine/V 10 O 28 compound was elaborated applying simple instrumental equipment, affordable reactants and the use of water as a solvent, which conforms to the concept of "green chemistry". N─H···O and O─H···O hydrogen bonds connect the decavanadate anions, ethylenediammonium cations, ammonium cations, and water molecules to create a complex three-dimensional network of (H 2 en) 2 {V 10 O 28 }.4H 2 O. The IR reveals the presence of the VO6 octahedron, the ethylenediammonium and ammonium cations, and the water molecule, which matches the single-crystal X-ray structural investigation results. The thermogravimetric study reveals that the decomposition of the hybrid vanadate occurs in three stages, with the loss of the organic moiety, ammonium cation, water molecules, and decavanadate degradation.

Electrochemical measurements confirmed that yellow decavanadates. solutions in 3% NaCl exhibited no significant inhibition efficiency and increasing decavanadate concentration was unfavorable. Besides clear metavanadate solutions in 1 M HCl containing monovanadate exhibited strong inhibition. In addition to vanadate ions, a variety of other atoms including Nitrogen and Oxygen have also been intercalated with the surface gallery to promote the anticorrosive properties. In addition, it should be noted that the inhibiting efficiencies calculated from EIS data follow the same trend as those obtained from Tafel and LPR methods, though some differences, reflecting the approximations on which each of them is based. The biological activity of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O shows mediocre antimicrobial and antioxidant activities.

Fig. 1 .

 1 Fig. 1. ORTEP image shows the asymmetric unit of (NH4)2(H2en)2{V10O28}.4H2O with non-H atom displacement ellipsoids drawn at a probability level of 70%.

Fig. 2 .

 2 Fig. 2. Illustration of the framework of (NH4)2(H2en)2{V10O28}.4H2O showing hydrogen bonds between the clusters.

1 Fig. 3 .

 13 Fig. 3. Infrared spectrum of (H2en)2(NH4)2{V10O28}.4H2O.

Fig. 4 .

 4 Fig. 4. Thermographic curve of (H2en)2(NH4)2{V10O28}.4H2O.

Fig. 5 .

 5 Fig. 5. The simulated, experimental and the thermal treatment at 450°C XRD pattern of (NH4)2(H2en)2{V10O28}.4H2O.

Fig. 6 .

 6 Fig. 6. Photograph of (H2en)2(NH4)2{V10O28}.4H2O dissolved in 3% NaCl (Left) and 1 M HCl (Right).

Fig. 7 .

 7 Fig. 7. UV-visible spectra of (H2en)2(NH4)2{V10O28}.4H2O, measured at 25 ºC, in HCl (5×10 -4 M) (black) and NaCl (5×10 -4 M) (red).

Fig. 8 .

 8 Fig. 8. Nyquist plots of the mild steel matrix in: (a) 1 M HCl, (b) 3% NaCl with 5×10 -4 M of (H2en)2(NH4)2{V10O28}.4H2O at 25 °C.

Fig. 9 .

 9 Fig. 9. SEM micrographs of the MS surface after 24 h immersion: (a) in 3% NaCl at 5×10 -4 M of (H2en)2(NH4)2{V10O28}.4H2O. (b) in 1 M HCl at 5×10 -4 M of the title complex.

Figure 10 Fig. 10 .

 1010 Fig. 10. PP curves of mild steel in 1 M HCl at 25 °C and different concentrations of (H2en)2(NH4)2{V10O28}.4H2O.

Fig. 11 .

 11 Fig. 11. Mild steel Nyquist and Bode graphs in 1 M HCl without and with various amounts of (H2en)2(NH4)2{V10O28}.4H2O at 25 degrees Celsius. with 1 = (H2en)2(NH4)2{V10O28}.4H2O.

18 Fig. 12 .

 1812 Fig. 12. Equivalent electric circuit models.

  with the EEC { R 1 + CPE 2 / [R 2 + CPE 3 / R 3 ]} fitting curve obtained with the EEC { R 1 + CPE 2

Fig. 13 .

 13 Fig. 13. Nyquist and Bode diagrams simulated by {Rs+CPEdl/Rct} and {Rs+CPEdl/(Rct+CPEf/Rf)} at different concentrations of (H2en)2(NH4)2{V10O28}.4H2O. with 1 = (H2en)2(NH4)2{V10O28}.4H2O.

Fig. 14 .

 14 Fig. 14. Free radical scavenging activity vs. BHT of (H2en)2(NH4)2{V10O28}.4H2O. with 1 = (H2en)2(NH4)2{V10O28}.4H2O.

Fig. 15 .

 15 Fig. 15. Ferric reducing power vs. gallic acid of (H2en)2(NH4)2{V10O28}.4H2O. with 1 = (H2en)2(NH4)2{V10O28}.4H2O.

Fig. 16 .

 16 Fig. 16. Total antioxidant capacity of (H2en)2(NH4)2{V10O28}.4H2O vs. gallic acid. with 1 = (H2en)2(NH4)2{V10O28}.4H2O.

Table 1

 1 Structure refinement, data collection, and crystal data of (NH4)2(H2en)2{V10O28}.4H2O.

	Chemical formula	CH9N1.50O8V2.50
	Mr	297.45
	Crystal system, space group	Triclinic, P\-1
	Temperature (K)	100
	a, b, c (Å)	8.4975(2), 10.2550(3), 10.9165(3)
	, ,  (°)	103.950(2), 98.466(2), 113.978(3)
	V (Å 3 ), Z	810.67(4), 4
	F(000)	588
	Radiation type / ( Å)	Mo K  0.71073
	Crystal sizes	?
	 (mm -1 )	2.87
	R(int)	0.033
	No. measured reflections	40050
	No. independent reflections	8433
	reflections with I ≥ 2 (I)	7918
	max, min	37.9°, 2.0°
	Reciprocal space	h

Table 2

 2 Isotropic or equivalent isotropic displacement parameters and fractional atomic coordinates (Å 2 ) for

	(NH4)2(H2en)2{V10O28}.4H2O.
		x	y	z	U(eq)
	V(4)	6591	81531	74111	61
	V(2)	18141	49001	47561	41
	V(5)	11981	53491	75541	51
	V1	40301	79811	69381	71
	V(3)	12241	77031	46231	61
	O(8)	13641	59211	35551	61
	O1	13531	39201	60261	61
	O(12)	5831	67871	83131	71
	O(9)	12001	85751	35731	101
	O(14)	35601	66541	79181	71
	O(2)	39701	62091	55341	71
	O(7)	18801	35011	36081	71
	O(3)	35471	85801	55041	81
	O(13)	11901	44991	86231	101
	O(4)	31261	90681	79041	71
	O(6)	10331	63231	59491	51
	O(5)	61681	89861	75171	111
	O(10)	5951	87261	59361	71
	O(11)	1731	92721	84121	101
	N(3)	6771	82431	108551	81
	N(2)	34321	57831	18061	111
	N1	70581	82321	48811	111
	C1	59191	79461	35841	111
	C(2)	52151	63051	27191	111
	O(1W)	29781	28171	8171	151
	O(2W)	41041	80731	6851	131

Table 5

 5 Bond number (s) calculations for all the oxygen atoms in the {V10O28} 6-anion. 2+ cations. {V 10 O 28 } 6-anion behaves as reported in[START_REF] Boşnjaković-Pavlović | Crystallographic statistical study of decavanadate anion based-structures: Toward a prediction of noncovalent interactions[END_REF]. Table4summarizes the significant H-Bonds linking the {V 10 O 28 } 6-anions to the interstitial entities (H 2 en) 2+ , (NH 4 ) + and H 2 O.

	Atom	Σs	Atom	Σs
	O1	1.853	O(8)	1.787
	O(2)	1.825	O(9)	1.683
	O(3)	1.826	O(10)	1.728
	O(4)	1.789	O(11)	1.735
	O(5)	1.732	O(12)	1.778
	O(6)	2.051	O(13)	1.699
	O(7)	1.764	O(14)	1.693

Figure 2 illustrates the cohesion of adjacent [V 10 O 28 ] 6-structural units through weak electrostatic interactions engaged with the interstitial contents, i.e., ethylenediamine (C 2 H 8 N 2 ) referred to as "en", water molecules "H 2 O" and ammonium "NH 4 + ". The strong O-H•••O and N-H•••O interactions with the (H 2 en) 2+ and H 2 O entities are primarily mediated by the O(3), triply connected, and O(2), double-linked oxygen atoms, while the terminal oxygen (O t ) is involved mainly in the weakest N-H•••O interactions with (NH 4 ) + . Each {V 10 O 28 } 6-anion is surrounded by two (H 2 en)

Table 6 .

 6 Infrared bands and assignments for complex 1.

	Band	̴ 930		810, 717, 532	3160-3030; 1600	1043, 1079 1807; 3517-3238	1404
	(cm -1 )
	Assignment The		Asymmetric	N-H (en) bending	The	H-O-H bending	N-H (NH4)
		stretching	and symmetric	H-N-H scissoring	stretching	O-H stretching	bending
		of	the	stretching		C-N
		terminal	vibrations of
		V-O		V-O-V

Table 7

 7 Electrochemical parameters obtained from the PP curves of the MS immersed in 1 M HCl at different concentrations of 1, and at 25 °C.

			Tafel fit			Rp fit
	Electrode/electrolyte	Ecorr	icorr	ßc	ηTafel	Ecorr	RP	ηS&G
	interface	mVAg/AgCl	µA cm -2	mV dec -1	%	mVAg/AgCl	Ω cm 2	%
	MS / 1 M HCl	-428.8	1439.9	-156.8	-	-428.7	23.5	-
	MS / 1×10 -6 M of 1	-437.7	792.60	-165.6	44.9	-437.5	60.6	61.2
	MS / 1×10 -5 M of 1	-440.4	281.06	-123.1	80.5	-443.1	84.7	72.3
	MS / 5×10 -5 M of 1	-425.2	187.27	-114.5	87.0	-425.6	132.0	82.2
	MS / 1×10 -4 M of 1	-436.8	182.90	-112.3	87.3	-435.8	180.0	86.0
	MS / 5×10 -4 M of 1	-427.1	70.89	-146.1	95.1	-428.4	433.0	94.6
	with 1 = (H2en)2(NH4)2{V10O28}.4H2O.

Table 7

 7 shows that the corrosion current density i corr decreases considerably in the presence of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O with rise of concentrations and reaches a minimum value of 70.89 µA cm -2 at 5×10 -4 M. However, the R p values have an opposite trend with i corr .However, there is no definite trend in the shift of E corr values at different concentrations of the complex. The corrosion potential of the MS shifts, in the range, from -11 to 3 mV relative to the uninhibited solution. Therefore, it has been proposed from the literature that if the E corr shift upon addition of the inhibitor is greater than 85 mV relative to the corrosion potential of the blank[START_REF] Ferreira | Evaluation of the inhibitor effect of L-ascorbic acid on the corrosion of mild steel[END_REF], then the inhibitor can be classified as either cathodic or anodic type and if the shift is less than 85 mV, then the inhibitor can be considered as mixed type. Consequently, the results obtained indicate that (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O acts as mixed type inhibitor with predominant cathodic effectiveness. Also, the cathodic Tafel line values ß c , change with the addition of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O suggesting that the mechanism of hydrogen evolution reaction is changed in the presence of the title compound. This behavior is explained by Bockris

Consequently, the inhibition efficiency, calculated from Tafel and S&G methods, increases to rich 95.1% and 94.6%, respectively.

3

  concentrations of (H 2 en) 2 (NH 4 ) 2 {V 10 O 28 }.4H 2 O are reported in Table8. The values used to 4 determine the efficiency of the inhibitor are the values of the polarization resistance calculated 5 by the following equation (R P = R ct + R f ). The inhibition efficiency is calculated based on

	6
	7	Equation 5:
	8		EIS	%	=	   	R	p	/ R inh p	/	-inh	R	p	   		100	(5)

Table 8 9

 8 EIS parameters and inhibition efficiency for mild steel in 1 M HCl at different concentrations of 1.

	10
	Electrode/electrolyte interface RS	Qdl	ndl	Rct	Qf	nf	Rf	Rp	𝜼EIS% χ 2 /|Z|
	MS / 1 M HCl	0.7765 280.0 0.8232 30.17 512.3	0.6501 3.792	33.96	-	0.336
	MS / 1×10 -6 M of 1	0.8182 166.5 0.8641 38.17 708.7	0.5295 45.66	83.83	60	0.255
	MS / 1×10 -5 M of 1	0.8010 155.6 0.8774 44.07 765.8	0.5088 63.40	107.78 68	0.259
	MS / 5×10 -5 M of 1	0.7971 148.7 0.8743 51.34 905.3	0.4928 98.90	150.24 77	0.255
	MS / 1×10 -4 M of 1	0.7899 137.9 0.8816 59.90 998.1	0.5085 182.80	242.70 86	0.269
	MS / 5×10 -4 M of 1	0.8201 121.1 0.8871 83.36 1069	0.5067 335.10	418.46 92	0.248
	Rs, Rct, Rf and Rp in Ω cm

Table 10 Antibacterial

 10

	activity of (NH4)2(H2en)2{V10O28}.4H2O, (NH4)6V10O28, and chitosan-Ca3V10O28.
			Antibacterial, MIC
	Microorganisms		(ppm)
		(NH4)2(H2en)2{V10O28}.2H2O (NH4)6V10O28	Chitosan-Ca3V10O28
	Staphylococcus aureus No activity	50	12.5
	Escherichia coli	2500	50	12.5
	with 1 = (H2en)2(NH4)2{V10O28}.4H2O.

Acknowledgements

We are grateful for measurement time on the X-ray diffraction platform PMD²X of the Institute Jean Barriol (University of Lorraine, France). The authors are also very indebted to the members of LIMAS laboratory for carrying out the electrochemical studies in terms of enhancing the anticorrosion properties of decavanadate.

Rs CPE

Declaration of competing interest

The authors state that they have no known competing financial interests or personal ties that might have influenced the research presented in this paper.


Frank Daerden
email: frank.daerden@aeronomie.be

Lori Neary

Sébastien Viscardy

Justin Erwin

Séverine Robert

Shohei Aoki

Yannick Willame

Ann Carine Vandaele

Stephen Lewis

James Holmes

Aul Streeter

Manish Patel

Francisco González-Galindo

Miguel Angel Lopez-Valverde

José Juan Lopez-Moreno

Marco Giuranna

Giuseppe Sindoni

Francesca Altieri

Giancarlo Bellucci

Franck Lefèvre

Geronimo Villanueva

Michael J Mumma

Michael D Smith

Giuseppe Etiope

James Whiteway

R Todd

Michael J Wolff

Robert E Novak

Sophie Bauduin

Jacek W Kaminski

Ian R Thomas

Bojan Ristic

Cédric Depiesse

Jon Mason

Atmospheric model support for NOMAD on ExoMars/TGO

Mars atmospheric models have become increasingly important for the support of space missions to Mars and for the interpretation of the obtained observations. The NOMAD ("Nadir and Occultation for MArs Discovery") spectrometer suite on board the ExoMars Trace Gas Orbiter (TGO) has been designed to investigate the composition of Mars' atmosphere, with a particular focus on trace gases, clouds and dust, and started its science operations in April 2018. Within the NOMAD science team, a wealth of state-of-the-art models and related expertise has been incorporated to support and validate NOMAD in terms of (1) observation planning, (2) providing a priori for retrievals, (3) interpretation of observations and derivation of new science results, and (4) data assimilation. This abstract provides an overview of the modeling capability on the NOMAD science team and addresses how models can contribute to fulfill the main science objectives of the mission.

Introduction

The NOMAD team includes 3 of the most advanced Mars General Circulation Models (GCMs) to date: (1) the GEM-Mars model [1,2], (2) the UK version of the LMD model [3,4], and (3) the LMD model [3,5]. Besides the parameterizations for atmospheric dynamics and physics, necessary to represent the atmospheric circulation and thermodynamic state, the applied GCMs uniquely contain modules for atmospheric chemistry, which is crucial to understand and interpret the observations by NOMAD.

Planning of observations

The complexity of TGO's operations, the observational constraints, and the specific capabilities of the various instruments and channels, impose a dedicated planning of observations for optimal science return. Atmospheric models can help in this planning by indicating which times, seasons and geolocations are of special interest, e.g. where current knowledge of atmospheric processes is known to be poor and requires specific observations. This may relate to the water cycle, photochemical cycles, dust storms etc. In the case of detection of special events, e.g. for methane, atmospheric models can provide new forecasts to support the mid-term observation planning.

A priori information for retrieval

At BIRA-IASB the GEM-Mars model has been applied to prepare atmospheric profiles to be used as a priori information in the retrieval algorithms. This work is presented in detail in an accompanying abstract at this conference by Erwin et al.

Interpretation of observations and new science results

TGO was designed to provide a refined search for atmospheric trace gases. Besides advancing the detection limit for many species and creating inventories of them, the other main science objectives of TGO are (1) to understand the atmospheric processes that involve the detected trace gases, and (2) search for sources and sinks of the detected trace gases. This is where model support is vital, as it directly relates theoretical processes (reactions, sources, sinks, …) to a 3D+time atmospheric state, that can be compared to the (sparse) set of observations. The differences between model and data provide direct insight in the plausibility of the imposed processes, and provide suggestions in how to modify them if necessary. In the case of detection of methane releases, the models can provide information on the source location by either doing an ensemble of forward simulations [Viscardy et al., this conference] or by calculating back trajectories [6].

Data assimilation

Data assimilation is a technique that allows to combine the dense 3D+time (theoretical) information from models with the sparse set of actual observations, in order to provide the most complete set of information of the atmospheric state and composition. The UK team has considerable expertise on data assimilation for Mars [7] and will extend this work with data assimilation of atmospheric state and chemical composition from the NOMAD observations. Data assimilation is also envisaged using the GEM-Mars model.

Acknowledgements

The NOMAD experiment is led by the Royal Belgian Institute for Space Aeronomy (IASB-BIRA), assisted by Co-PI teams from Spain (IAA-CSIC), Italy (INAF-IAPS), and the United Kingdom (Open University). This project acknowledges funding by the Belgian Science Policy Office (BELSPO), with the financial and contractual coordination by the ESA Prodex Office (PEA 4000103401, 4000121493), by MICIIN through Plan Nacional (AYA2009-08190 and AYA2012-39691), by the Italian Space Agency through grant 2018-2-HH.0, as well as by UK Space Agency through grant ST/P000886/1. US investigators were supported by the National Aeronautics and Space Administration. The research was performed as part of the "Excellence of Science" project "Evolution and Tracers of Habitability on Mars and the Earth" (FNRS 30442502).

The NOMAD Team

Scientific team: Vandaele, Ann Carine; Lopez Moreno, Jose Juan; Bellucci, Giancarlo; Patel, Manish; Allen, Mark; Alonso-Rodrigo, Gustavo; Altieri, Francesca; Aoki, Shohei; Bauduin, Sophie; Bolsée, David; Clancy, Todd; Cloutis, Edward; Daerden, Frank; D'Aversa, Emiliano; Depiesse, Cédric; Erwin, Justin; Fedorova, Anna; Formisano, Vittorio; Funke, Bernd; Fussen, Didier; Garcia-Comas, Maia; Geminale, Anna; Gérard, Jean-Claude; Gillotay, Didier; Giuranna, Marco; Gonzalez-Galindo, Francisco; Hewson, Will; Homes, James; Ignatiev, Nicolai; Kaminski, Jacek; Karatekin, Ozgur; Kasaba, Yasumasa; Lanciano, Orietta; Lefèvre, Franck; Lewis, Stephen; López-Puertas, Manuel; López-Valverde, Miguel; Mahieux, Arnaud; Mason, Jon; Mc Connell, Jack; Mumma, Mike; Nakagawa, Hiromu, Neary, Lori; Neefs, Eddy; Novak, R.; Oliva, Fabrizio; Piccialli, Arianna; Renotte, Etienne; Robert, Severine; Sindoni, Giuseppe; Smith, Mike; Stiepen, Arnaud; Thomas, Ian; Trokhimovskiy, Alexander; Vander Auwera, Jean; Villanueva, Geronimo; Viscardy, Sébastien; Whiteway, Jim; Willame, Yannick; Wilquet, Valérie; Wolff, Michael; Wolkenberg, Paulina -Tech team: Alonso-Rodrigo, Gustavo; Aparicio del Moral, Beatriz; Barzin, Pascal; Beeckman, Bram; BenMoussa, Ali; Berkenbosch, Sophie; Biondi, David; Bonnewijn, Sabrina; Candini, Gian Paolo; Clairquin, Roland; Cubas, Javier; Giordanengo, Boris; Gissot, Samuel; Gomez, Alejandro; Hathi, Brijen; Jeronimo Zafra, Jose; Leese, Mark; Maes, Jeroen;Mazy, Emmanuel; Mazzoli, Alexandra; Meseguer, Jose; Morales, Rafael; Orban, Anne; Pastor-Morales, M; Perez-grande, Isabel; Queirolo, Claudio; Ristic, Bojan; Rodriguez Gomez, Julio; Saggin, Bortolino; Samain, Valérie; Sanz Andres, Angel; Sanz, Rosario; Simar, Juan-Felipe; Thibert, Tanguy