Fatima Aboud

Didier Robert
email: didier.robert@univ-nantes.fr

Asymptotic expansion for nonlinear eigenvalue problems

In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is

.

Introduction

Let us introduce the following family of differential operators, L(λ) = -△ x + (P (x)λ) 2 (1.1)

where △ x is the Laplace operator in R d x , λ is a complex parameter, P is a polynomial of degree m ≥ 2 such that the leading homogeneous part P m of P satisfies P m (x) > 0 for every x ∈ R d \{0} (in other words we say that P is a positive-elliptic polynomial). Such family of operators play an important role when studying analytic smoothness of solutions of differential operators with multiple characteristics (see [START_REF] Helffer | Semiclassical analysis of a nonlinear eigenvalue problem and analytic hypoellipticity[END_REF] and references there). They also appear in the theory of damped oscillations in mechanics [START_REF] Friedman | Nonlinear Eigenvalue Problems[END_REF][START_REF] Krein | On the mathematical principles in the linear theory of damped oscillations of continua I[END_REF]. The question we want to adress here is: "does there exist λ ∈ C and u in the Schwartz space S(R d ), u = 0, such that L(λ)u = 0?" In [START_REF] Chanillo | Nonlinear eigenvalues and analytic hypoellipticity[END_REF] the authors have proven existence of non trivial solutions for 1 ≤ d ≤ 3, assuming that m is large enough for d = 3. After, Helffer-Robert-Wang proved in [START_REF] Helffer | Semiclassical analysis of a nonlinear eigenvalue problem and analytic hypoellipticity[END_REF] the following result.

Theorem 1.1 Assume that d is even and that P is a positive-elliptic polynomial of degree m ≥ 2. Then there exist λ ∈ C and u ∈ S(R d ), u = 0, such that L(λ)u = 0.

The proof given in [START_REF] Helffer | Semiclassical analysis of a nonlinear eigenvalue problem and analytic hypoellipticity[END_REF] shows that there exist an infinite number of such eigenvalues [START_REF] Robert | Nonlinear Eigenvalues Problems[END_REF] located in the half-plane {λ ∈ C, ℜλ ≥ 0}. But it is not known if the generalized eigenfunctions span all the Hilbert space L 2 (R d ), excepted for d = 1 [START_REF] Pham The Lai | Sur un problème aux valeurs propres non linéaire[END_REF]. For d odd , d ≥ 3, m ≥ 2, the problem of existence of non zero solutions is still open and it was conjecture in [START_REF] Helffer | Semiclassical analysis of a nonlinear eigenvalue problem and analytic hypoellipticity[END_REF] that such solutions exist whatever the dimension d.

In this paper we prove that this is true for every elliptic polynomial if d = 3 and for large classes of elliptic polynomials for d = 5, 7. We also discuss a numerical approach to prove that some coefficient in a semi-classical trace formula is not zero. For d ≥ 9 we conjecture that this coefficient is not zero hence there exist an infinite number of nonlinear eigenvalues.

This work was supported by the program ANR 08-BLAN-0228, NONAa, Research French Ministry.

nonlinear eigenvalue problems

In this section we recall some known properties concerning nonlinear eigenvalue problems. For more details we refer to [START_REF] Gohberg | Introduction à la théorie des opérateurs non nécessairement auto-adjoints[END_REF][START_REF] Markus | Introduction to the spectral theory of polynomial operator pencils[END_REF][START_REF] Robert | Nonlinear Eigenvalues Problems[END_REF]. Let us consider the quadratic family of operators L(λ) = L 0 + λL 1 + λ 2 where L 0 , L 1 are operators in an Hilbert space H. L 0 is assumed to be self-adjoint, positive, with a domain D(L 0 ) and L 1 is

√ L 0 -bounded. Moreover L -1/2 0
is in a Schatten class C p (H) for some real p > 0. The following results are well known.

Theorem 2.1 L(λ) is a family of closed operators in H. λ → L -1 (λ) is meromorphic in the complex plane. The poles λ j of L -1 (λ), with multiplicity m j , coïncide with the eigenvalues with the same multiplicities, of the matrix operator A L in the Hilbert space

H × D(L 1/2 0 ), with domain D(A L ) = D(L 0 ) × D(L 1/2 0 )
where

A P = 0 I -L 0 -L 1 . (2.2)
Let us denote Sp[L] the eigenvalues of A L (which coïncide with the poles of L -1 (z)).

Remark 2.2 It may happens that Sp[L] is empty.

The following one dimensional example is interesting and was discussed in [START_REF] Pham The Lai | Sur un problème aux valeurs propres non linéaire[END_REF][START_REF] Christ | Analytic hypoellipticity, representation of nilpotent groups and non-linear eigenvalue problem[END_REF][START_REF] Christ | Examples of analytic non-hypoellipticity of ∂b[END_REF].

L m,g (λ) = - d 2 dx 2 + (x m -λ) 2 + gx m-1 . (2.3)
For every m ≥ 2, m even, L m,0 has infinity many eigenvalues but L m,m has no eigenvalue. The last statement is a consequence of the factorization

L m,m (λ) = (x m -λ + d dx )(x m -λ - d dx ).
So, we can compute all solutions for the equation L m,m (λ)u = 0 and see that a non-null solution u is never bounded on R.

But if m is odd, L m,m (λ)u = 0, has infinity many eigenvalues on the imaginary axis [START_REF] Christ | Examples of analytic non-hypoellipticity of ∂b[END_REF].

On the other side there exist sufficient general conditions to have Sp[L] = ∅ [START_REF] Gohberg | Introduction à la théorie des opérateurs non nécessairement auto-adjoints[END_REF][START_REF] Markus | Introduction to the spectral theory of polynomial operator pencils[END_REF]. Unfortunately these conditions are not fulfilled for our example

L(λ) = -△ x + (P (x) -λ) 2 when d ≥ 2.
The following formula appears for the first time in [START_REF] Kh | Trace formula for polynomial operators and its applications[END_REF] and will be very useful for our purpose.

Theorem 2.3 For k large enough (k ∈ N, k > p) and for z ∈ C\Sp[L],
we have

Tr(A L -z) -k-1 = -1 k! Tr[ d k dz k (L(z) -1 L ′ (z)], (2.4)
where each above operators are trace class.

Using Lidskii Theorem [START_REF] Gohberg | Introduction à la théorie des opérateurs non nécessairement auto-adjoints[END_REF] and (2.4), we get

λ∈Sp[L] m λ (λ -z) -k-1 = -1 k! Tr[ d k dz k L(z) -1 L ′ (z)) ]. (2.5)
where m(λ) is the multiplicity of the eigenvalue λ.

As it was nicely remarked in the paper [START_REF] Chanillo | Nonlinear eigenvalues and analytic hypoellipticity[END_REF], a sufficient condition for Sp[L] = ∅ is that the r.h.s in (2.5) is not zero. To check this property a natural method is to introduce parameters and use semiclassical analysis. In [START_REF] Helffer | Semiclassical analysis of a nonlinear eigenvalue problem and analytic hypoellipticity[END_REF] the authors also use Lidskii theorem and semi-classical analysis on the matrix system A L . Here we consider more directly the scalar family of operators L(z) where computations are easier even if the dependence in z is nonlinear.

Semiclassical parametrix

For simplicity we assume here that P is homogeneous of degree m ≥ 2 and P (x) > 0 for x ∈ R d , x = 0. By the scaling transformation x = τ 1/m y with = τ -(m+1)/m and z = λ τ we can see that L(λ) is unitary equivalent to the semiclassical Hamiltonian τ 2 L(z) where

L(z) = -2 △ x + (P (x) -z) 2 . (3.6)
L(z) is the -Weyl operator with the symbol L(z, x, ξ) = ξ 2 + (P (x)z) 2 . For semiclassical analysis tools and -Weyl quantization we refer to [START_REF] Robert | Autour de l'approximation semi-classique[END_REF]. Here we use the notation Ĥ for the -Weyl quantization of the symbol H or for convenience, Ĥ = Op w (H).

Using semiclassical operator calculus, we can construct a good parametrix for L(z) -1 for z ∈ Λ where Λ is the sector

Λ = {z ∈ C, |z| ≥ r 0 , π/2 + δ < arg(z) < 3π/2 -δ}; r 0 > 0, δ > 0. Theorem 3.1 There exists a semiclassical symbol K ( ) (z), z ∈ Λ, 0 < < 1, such that K ( ) (z; x, ξ) ≍ j≥0 2j K 2j (z; x, ξ), L(z) -1 = Op w (K (z)).
(3.7)

Moreover the asymptotic expansion has the following meaning: for every N ≥ 1 we have

L(z).Op w   0≤j≤N 2j K 2j (z)   = I + 2N +2 Op w R ( ) 2N (z)
where the symbol R ( ) 2N (z) satisfies the following estimates : for every α, β ∈ N d we have

∂ α x ∂ β ξ R ( ) 2N (z; x, ξ) ≤ C(N, α, β) µ(x, ξ) 2m + |z|µ(x, ξ) m µ(x, ξ) 2m + |z| 2 µ(x, ξ) -2N -|α|-|β|
(3.8) for every α, β ∈ N d , where C(N, α, β) is uniform in z ∈ Λ and where µ(x, ξ) = (1 + |x| 2m + |ξ| 2 ) 1/2m . sketch of proof. The method to get such result is standard and was used many times to construct parametrix of elliptic pseudo-differential operators [START_REF] Seeley | Complex powers of an elliptic operators, Singular Integrals[END_REF]. Usually the z-dependence is linear but here it is quadratic. Moreover here we need accurate estimates for the remainder term in the product of pseudodifferential operators depending on parameters. The necessary estimates for R ( ) 2N (z; x, ξ) are established using the technics coming from the papers [START_REF] Dauge | Weyl's formula for a class of pseudodifferential operators with negative order on L 2 (R n )[END_REF][START_REF] Bouzouina | Uniform semiclassical estimates for the propagation of quantum observables[END_REF]. An other difficulty here is that we shall need to compute the symbols K 2j for j large enough. This computations are not easy, so we have to be explicite as far as possible. Using the product formula for -pseudodifferential operators, we get at the initial step:

K 0 (z; x, ξ) = 1 L(z; x, ξ) = 1 |ξ| 2 + (P (x) -z) 2 (3.9)
and the induction formula

K 2j = -K 0   0≤ℓ≤j-1 |α|+|β|=2(j-ℓ) Γ(α, β)∂ α ξ ∂ β x L(z)∂ β ξ ∂ α x K 2ℓ   (3.10)
where Γ(α, β) = (-1) |β| 2 2(j-ℓ) α!β! . Let us compute K 2 and K 4 .

K 2 = L 2 (z) L 3 (z) + L 3 (z) L 4 (z) , L 2 (z) = (P (x) -z)△P (x) + |∇P (x)| 2 , L 3 (z) = -2[(P (x) -z)D 2 P (x)ξ • ξ + (∇P (x) • ξ) 2 + (P (x) -z) 2 |∇P (x)| 2 ],
where D 2 P (x) is the Hessian matrix of P in variable x. Now using (3.10) we have

K 4 = -K 0 |β|=4 Γ(0, β)∂ β x L(z)∂ β ξ K 0 + |α|=2 Γ(α, 0)∂ α ξ L(z)∂ α x K 2 + |β|=2 Γ(0, β)∂ β x L(z)∂ β ξ K 2 ) . (3.11)
By induction on j, we easily get that

K 2j (z; x, ξ) = j+1≤k≤3j Q 2j k (x, P -z, ξ) L(z; x, ξ) k+1 , (3.12
)

Q 2j k (x, P -z, ξ
) is a polynomial in ((Pz), ξ), with a total degree ≤ k -2, with coefficients depending on derivatives of P (x).

The following lemma will be useful later. Let us denote val [Q 2j k ], the valuation of Q 2j k as a polynomial in Pz, ξ. Let us recall the definition of valuation. Denote by I the ideal with generators

ξ 1 , • • • , ξ d , P -z, in the ring C ∞ (R ξ × R x ). If Q ∈ C ∞ (R ξ × R x ), val[Q] is the biggest integer p such that Q ∈ I p . Lemma 3.2 We have val[Q 2j k ] ≥ 2(k -1 -2j), for 2j + 2 ≤ k ≤ 3j, and j ≥ 1.
Proof. This is easily proved by induction on j, using (3.12) and the following formula. Let Q and L be smooth functions in R n , a multiindex α ∈ N n , then we have

∂ α Q L k+1 = C(µ j , γ k ) ∂ α-γ Q(∂ γ1 L) µ1 • • • (∂ γ ℓ L) µ ℓ L µ+k+1 (3.13)
where in the sum we have the conditions,

γ j ∈ N n , µ j ∈ N, γ ≤ α, µ 1 +• • • µ ℓ = µ, µ 1 |γ 1 | + • • • + µ ℓ |γ ℓ | = |γ|. ⊓ ⊔ Remark 3.3
The parametrix computed above is enough to get qualitative informations. Quantitative informations are much more difficult to get except for the first orders (j = 0, 1). When j is larger it is not so easy to compute explicitely the terms Q 2j k (x, Pz, ξ).

Remark 3.4 It is not difficult to extend the above results when the elliptic polynomial P (x) has lower terms: P = P m + P m-1 + • • • P 1 + P 0 where P j is homogeous with degree j and P m (x) > 0 for x ∈ R d \{0}. Then we have

P (τ 1/m y) = τ P (ε) (y)
with ε = τ -1/m = 1/(m+1) and P (ε) (y) = P m (y) + εP m-1 (y) + • • • + ε m P 0 (y). So P (ε) is a uniform elliptic family of polynomials and we can easily see that the constructions in (3.8) are uniform in the small parameter ε.

A trace formula

Recall that Sp[L] denote the generalized eigenvalues of the quadratic family L(z), m λ is the multiplicity of the eigenvalue λ.

Let f an holomorphic function in Λ such that |f (z)| ≤ C(1 + |z|) -µ , ∀z ∈ Λ. (4.14)
For our applications we shall choose f (z) = (z + λ) -µ , for a suitable parameter λ ∈ C. Let be Γ a complex contour in Λ defined as follows.

Γ = {re ±iθ0 , r ≥ r 0 } ∪ {r 0 e iθ , θ 0 ≤ θ ≤ 2π -θ 0 },
where r 0 > 0 and π 2 < θ 0 < π.

Proposition 4.1 Assume that µ > d(m+1) m . Then f (A L ) is a

trace class operator and we have

Tr(f(A L ) = λ∈Sp[L] m λ f(λ) = Tr[ Γ L(z) -1 L′ (z)f(z)dz], (4.15)
where Γ F (z)dz = 1 2iπ Γ F (z)dz (contour integral in the complex plane). Proof. This a direct consequence of the Cauchy integral formula and Theorem 2.4. ⊓ ⊔ Theorem 4.2 For f as above, for every d ≥ 1 we have in the semiclassic regime ց 0, modulo O( +∞ ),

λ∈Sp[L] m λ f (λ) ≍ j≥0 C (d) 2j (f ) 2j-d . (4.16) If d is odd, C (d) 0 (f ) = 0 (4.17)
and for d even,

C (d) 0 (f ) = 2(-1) d/2 (2π) -d R 2d f (P (x) + |η|)dxdη. (4.18)
For the other terms (j ≥ 1) we have the following qualitative information

C (d) 2j (f ) = 0≤k≤nj R d A 2j,k (x)f (k) (P (x))dx, (4.19)
where A 2j,k (x) are polynomials in ∂ γ x P (x), |γ| ≤ 2j and n j depends on j.

Moreover if d is odd, then C (d) 2j (f ) = 0 for d ≥ 4j + 1 .
Proof. The asymptotic expansion (4.16) is a direct consequence of (3.8) and usual properties of trace operation for Weyl quantization. Let us compute C (d) 0 (f ). We have the integral formula:

C (d) 0 (f ) = - Γ 2(P (x) -z) |ξ| 2 + (P (x) -z) 2 f (z)dzdξ dx,
where dx = (2π) -d dx. By the residue theorem we get

C (d) 0 (f ) = R d ×R d [f (P (x) + i|ξ|) + f (P (x) -i|ξ|)]dξ dx.
For a > 0 we have

R d ×R d (f (P (x) + a|ξ|)dξ dx = a -d R d ×R d (f (P (x) + |ξ|)dξ dx.
So by analytic extension and evalution at a = i we get formula (4.17) and (4.18). In particular we see that for d even, there exists f satisfying (4.14) such that C (d) 0 (f ) = 0.

For j ≥ 1, using (3.12), we have

C (d) 2j (f ) = Γ j+1≤k≤3j 2(P (x) -z)Q 2j k (x, P (x) -z, ξ) L(z; x, ξ) k+1 f (z)dzdξ dx. (4.20)
Let us now prove that C (d) 2j (f ) = 0, for 4j + 1 ≤ d. To do that it is convenient to introduce the following integral, for u > 0, v > 0,

J k,ν f (u, v) = Γ (v -z) ν (u + (v -z) 2 ) k+1 f (z)dz. (4.21)
We have easily

J k,ν f (u, v) = (-1) k k! ∂ k ∂u k J 0,ν f (u, v). (4.22)
And using the residue theorem, we get (f ) = 0 for d ≥ 4j + 1, we shall prove that each term in the sum (4.20) vanishes, after integration in z and ξ. Suppose first that j + 1 ≤ k ≤ 2j + 1. We have

J 0,ν f (u, v) = i ν-1 u (ν-1)/2 2 (-1) ν+1 f (v + i √ u) + f (v -i √ u) . ( 4
Q 2j k (x, P (x) -z, ξ) = ν,γ R ν,γ (x)(P (x) -z) ν ξ γ . Hence Γ 2(P (x) -z)Q 2j k (x, P (x) -z, ξ) L(z; x, ξ) k+1 f (z)dzdξ
is a sum of integrals like

I k ν (f )(x, ξ) = Γ (P (x) -z) ν L(z; x, ξ) k+1 f (z)dz.
By integration by parts in z we have

I k ν+1 (f ) = ν 2k I k-1 ν-1 (f ) - 1 2k I k-1 ν (f ′ ). (4.24)
So, we can assume that ν = 0. But we have

I ℓ 0 (g)(x, ξ) = (-1) ℓ ℓ! ∂ ℓ ∂u ℓ J 0,ν g(u, P (x))| u=|ξ| 2 .
So we have

I ℓ 0 (g)(x, ξ) = O(|ξ| 2-2ℓ
) near ξ = 0. Now we remark that for ℓ ≤ 2j + 1 and d ≥ 4j + 1 we have ℓ < d 2 + 1, hence ξ → I ℓ 0 (g)(x, ξ) is integrable and, using the analytic dilation argument already used for j = 0, we get I ℓ 0 (g)(x, ξ) = 0, hence Γ 2(P (x)z)Q 2j k (x, P (x)z, ξ) L(z; x, ξ) k+1 f (z)dzdξ = 0. Now, assume that 2j + 2 ≤ k ≤ 3j. Using Lemma.3.2, we have

Q 2j k (x, P (x) -z, ξ) = ν+|γ|≥2(2k-1-2j) R ν,γ (x)(P (x) -z) ν ξ γ .
As above, we integrate by parts in z to have the possibility to put ν at 0 and then we use ξ γ to decrease the order of the singularity in ξ as far as possible

(integrability near ξ = 0) of Γ Q 2j k L k+1 dz.
We conclude by the analytic dilation argument. ⊓ ⊔ So, we have proven that in odd dimension d, C

(d) 2j (f ) = 0 if 2j ≤ d-1
2 . We conjecture that the next following terms are not 0; more precisely we claim: Conjecture: For every j ∈ N, j ≥ 1, there exists f satisfying (4.14) such that we have we have

C (4j-1) 2j (f ) = 0, and C (4j-3) 2j (f ) = 0 (4.25)
In the following sections we shall check this conjecture for d = 1, 3 and we shall compute analytic formula for C

(5) 4 (f ) and C

4 (f ). Unfortunately, these analytic expressions have many terms and it is not obvious that C (d) 4 (f ) = 0 for d = 5, 7, for every elliptic polynomial P . We shall see that this is true for convex polynomials for d = 7 and satisfying a technical condition if d = 5. Moreover we get, using numerical computations for particular non-convex polynomials P , that C (f, P (ε) ) = 0, for d ≥ 4j + 1 and for every ε small enough. Assume now that d = 4j 0 -3 or d = 4j 0 -1, j 0 ≥ 1. Then using a Taylor expansion in ε, computed for ε = 1/(m+1) , we get

C 2j (f, P ( 1/(m+1) ) ) ≍ k≥0 γ k k/(m+1) , (4.26) in particular γ 0 = C (d)
2j0 (f, P m ) which is supposed to be not 0, as we have explained before.

Estimate the density of eigenvalues

First of all let us remark that the nonlinear spectrum Sp[ L] of L is included in the two quarters {z ∈ C, ℜ(z) ≥ 0, ±ℑ(z) > 0}. On one side, it is easy to see that if λ ∈ R and L(λ)u = 0 then u = 0. On the other side, if ℜ(λ) < 0 and L(λ)u = 0, computing ℑ( L(λ)u, u ) we conclude that u = 0.

Let us denote by

N (R) = #{z ∈ Sp[ L]; |z| ≤ R} and N (R) = N =1 (R).
Proposition 5.1 For every real µ, µ > d(m + 1)/m, there exists

C k > 0 such that N (R) ≤ C k R µ -d , ∀R ≥ 1, ∀ ∈]0, 1]. (5.27) If C (d) 2j (f ) = 0 with d > 2j, then for every r > 0, ε > 0 there exists c ε,r > 0 such that N (r -ε ) ≥ c ε,r -δ , ∀ ∈]0, 1],
(5.28)

where δ = d -2j. Moreover if j = 0 (d even) then the estimates is valid with ε = 0. So that, in even dimension, for every R > 0, N (R) behaves like -d .

Proof. The proof of (5.27) is a direct consequence of Weyl-Ky-Fan inequality [START_REF] Robert | Nonlinear Eigenvalues Problems[END_REF]. We first remark that for every ε > 0 there exists R ε > 0 such that if

-π/2 -ε ≤ arg z ≤ π/2 + ε, |µ| ≥ R ε then we have |t + z| 2 ≥ (1 -ε)(t 2 + |z| 2 ).
Let us choose f (λ) = (λ + t) -µ with k large enough (µ > d(m + 1)/m) and t > 0. We apply (4.16) to get the following inequalities

C 1 -δ ≤ z∈Sp[ L] (t + z) -µ ≤ z∈Sp[ L] |t + z| -µ ≤ C 2 z∈Sp[ L] (t + |z|) -µ
But for every µ, µ 1 , large enough, such that µµ 1 is large enough, we have

z∈Sp[ L] |z|≥R (t + |z|) -µ ≤ R -µ1 z∈Sp[ L] (t + |z|) µ1-µ
We choose now R = r -ε to get

N (r -ε ) ≥ |µ|≤R (1 + |µ|) -k ≥ c ε,r -δ

⊓ ⊔

The above results concern the semi-classical regime. Now we give estimates for = 1 and high energy regime Corollary 5.2 For R ր +∞ we have

N (R) = O(R d(m+1)/m ). If C (d) 2j (f ) = 0 with d -2j > 0, then for every ε > 0 there exits c ε > 0 such that c ε R δ(m+1)/m-ε ≤ N (R)
If j = 0, the estimate is true with ε = 0 and c 0 > 0.

1-d and 3-d cases

In this section we prove the following result. Theorem 6.1 For d = 1, 3, there exits f satisfying (4.14) such that for every m ≥ 2, we have

C (d) 2 (f ) = 0. More precisely, we have C (1) 2 (f ) = - 1 16 R f (3) (P (x))P ′ (x) 2 dx (6.29) C (3) 2 (f ) = - 1 48π R 3 f ′ (P (x))|∇P (x)| 2 dx (6.30)
We can choose f (λ) = (λ + t) -µ with µ > d(m + 1)/m and t > 0.

Proof. We compute with the explicite form we got before for K 2 . We have, for d = 1, 3,

C d 2 (f, x) = -2 Γ (P -z)H 2 dz,
where

H 2 = R d K 2 dξ. But we have H 2 = (P -z) d-5 △P b 3 -2b 4,1 + (P -z) d-6 |∇P | 2 b 3 -2b 4 -2b 4,1 , hence C (1) 2
(f ) = 1 2π 8 3 b 4,1 - 4 3 b 3,0 + 2 3 b 4 R f (3) (P (x))P ′ (x) 2 dx. (6.31)
C (1) 2 (f ) = - 1 16 R f (3) (P (x))P ′ (x) 2 dx (6.32) C (3) 2 (f ) = 2 (2π) 3 4b 4,1 -2b 4 -2b 3 R 3 f ′ (P (x))|∇P | 2 (x)dx (6.33) and C (3) 2 (f )
= -

1 48π R 3 f ′ (P (x))|∇P | 2 (x)dx (6.34)

⊓ ⊔

We have seen that for d odd, d ≥ 5, C

5-d and 7-d cases

We have to compute in more details the term K 4 from (3.11). Recall that we have

C (d) 4 (f ) = -2 R d x Γ (P -z) R d ξ K 4 (z; x, ξ)dξ f (z)dz dx. (7.35)
We have to compute the following three integrals, depending on x ∈ R d and z ∈ C.

I (1) β = Γ(0; β) R d 1 |ξ| 2 + (P (x) -z) 2 ∂ β ξ 1 |ξ| 2 + (P (x) -z) 2 dξ; |β| = 4, I (2) α = Γ(α, 0) R d 1 |ξ| 2 + (P (x) -z) 2 ∂ α ξ (|ξ| 2 )∂ α x K 2 dξ; |α| = 2, (7.36)
I (3) β = Γ(0; β) R d 1 |ξ| 2 + (P (x) -z) 2 ∂ β ξ K 2 dξ; |β| = 2. (7.37) (7.38)
Using the new variable η such that ξ = (Pz)η (plus an analytic extension), we get

I (1) β = a(β) (P (x) -z) 8-d , a(β) = 1 16β! R d 1 1 + |η| 2 ∂ β η 1 1 + |η| 2 dη. (7.39) a(β) = 0 only when β = (β 1 , • • • , β d ) is such that β j = 4 and β k = 0 for k = j or β j = β k = 2, j = k and β ℓ = 0 if ℓ = j, ℓ = k.
In the first case a(β) = a 1 and in the second case a(β) = a 2 where

a 1 = 1 96 R d (4η 2 1 -(1 + |η| 2 )) 2 (1 + |η| 2 ) 6 dη, (7.40
)

a 2 = R d η 2 1 η 2 2 (1 + |η| 2 ) 6 dη. (7.41)
It is convenient to introduce the following notations.

b j = R d dη (1 + |η| 2 ) j , b j,k = R d η 2k 1 dη (1 + |η| 2 ) j , b j,k,ℓ = R d η 2k 1 η 2ℓ 2 dη (1 + |η| 2 ) j . (7.42)
where j, k, ℓ ∈ N are such that the integrals are finite. Of course these integrals can be computed with the Euler beta and gamma special functions (see appendix for more explicite expressions). So we have a 1 = 1 6 b 6,2 -1 3 b 5,1 + 1 96 b 4 and a 2 = b 6,1,1 . Using integration by parts, in x or in ξ, we get the following formulas

C 4 (f ) =
C 4,2 (f ; x) = 2 |α|=2 Γ(α, 0) Γ R d ∂ α ξ (|ξ| 2 )∂ α x P -z L(z) K 2 dξ f (z)dz, (7.46) C 4,3 (f ; x) = 2 Γ (P -z)∂ β x L(z)I (3)
β f (z)dz. (7.47)

Now we compute each term. After elementary but tedious computations we get the following results, using the notations:

∂ j = ∂ ∂ xj , ∂ 2 j = ∂ 2 ∂ 2 xj , ∂ 2 j,k = ∂ 2 ∂ 2 xj ,x k (7.48) a 1 = 1 6 b 6,2 - 1 3 b 5,1 + 1 96 b 4 (7.49) a 2 = b 6,1,1 (7.50)
For d=5 , we have 3) (P ), (7.52)

C 4,1 (f ; x) = -20f (P ) a 1 1≤j≤5 ∂ 4 j P + a 2 j<k ∂ 2 j ∂ 2 k P , (7.51) +8f ′ (P ) a 1 1≤j≤5 (∂ 2 j P ) 2 + a 2 j<k (∂ 2 j,k P ) 2 , C 4,2 (f ; x) = E 1 2 f ′ (P ) - E 2 4 f ′′ (P ) + E 3 12 f (
C 4,3 (f ; x) = G 1 2 f ′ (P ) - G 2 4 f ′′ (P ) + G 3 12 f (3) (P ). (7.53)
where

E 1 = (△P ) 2 b 4 -2b 5 -2b 5,1 + 4b 6,1 , (7.54)
E 2 = |∇P | 2 △P b 4 + 12b 6 -10b 5 -2b 5,1 + 16b 6,1 -16b 7,2 , (7.55)
E 3 = |∇P | 4 20b 6 -6b 5 -16b 7 + 16b 7,1 -12b 6,1 , (7.56
)

G 1 = (△P ) 2 12b 6,1 -2b 5 -16b 7,2 j (∂ 2 j P ) 2 (7.57) -32b 7,1,1 j =k (∂ 2 j P )(∂ 2 k P ) -16b 7,1,1 j =k (∂ 2 j,k P ) 2 , G 2 = △P |∇P | 2 24b 6,1 -16b 7,1 -4b 5 + 4b 6 -32b 7,2 j (∂ j P 2 )(∂ j P ) 2 -32b 7,1,1 j =k (∂ 2 j P )(∂ k P ) 2 -64b 7,1,1 j =k (∂ 2 j,k )(∂ j P )(∂ k P ), (7.58)
G 3 = |∇P | 4 12b 6,1 -16b 7,1 -2b 5 + 4b 6 -16b 7,2 j (∂ j P ) 4 -48b 7,1,1 j =k (∂ j P ) 2 (∂ k P ) 2 . (7.59)
Finally, we get

C 4 (f ; x) = A 0 (x)f (P (x)) + A 1 (x)f ′ (P (x)) + A 2 (x)f ′′ (P (x)) + A 3 (x)f (3) (P (x)), (7.60) A 0 (x) = -20a 1 j ∂ 4 j P -20a 2 j<k ∂ 2 j ∂ 2 k P, (7.61) A 1 (x) = 8(a 1 -b 7,2 ) j (∂ 2 j P ) 2 + 4(a 2 -2b 7,1 ) j =k (∂ 2 j,k P ) 2 (7.62) + (△P ) 2 2 b 4 -4b 5 -2b 5,1 + 16b 6,1 -16b 7,1,1 j =k (∂ 2 j P )(∂ 2 k P ), A 2 (x) = |∇P | 2 △P 4 14b 5 -16b 6 -b 4 + 2b 5,1 -40b 6,1 + 32b 7,1 (7.63) +8b 7,2 j (∂ 2 j P )(∂ j P ) 2 + 8b 7,1,1 j =k (∂ 2 j P )(∂ k P ) 2 +16b 7,1,1 j =k (∂ 2 j,k P )(∂ j P )(∂ k P ), A 3 (x) = |∇P | 4 12 24b 6 -8b 5 -16b 7 - 4 3 b 7,2 j (∂ j P ) 4 -4b 7,1,1 j =k (∂ j P ) 2 (∂ k P ) 2 . (7.64)
Using explicit computations (see Appendix), we get

A 0 (x) = π 3 24 j ∂ 4 j P - π 3 48 j<k ∂ 2 j ∂ 2 k P, (7.65)
A 1 (x) = - 11π 3 480 j (∂ 2 j P ) 2 + π 2 480 j =k (∂ 2 j,k P ) 2 (7.66) - π 3 160 (△P ) 2 - π 2 240 j =k (∂ 2 j P )(∂ 2 k P ), A 2 (x) = π 3 96 ∇P | 2 △P + π 3 160 j (∂ 2 j P )(∂ j P ) 2 (7.67) + π 2 480 j =k (∂ 2 j P )(∂ k P ) 2 + π 2 240 j =k (∂ 2 j,k P )(∂ j P )(∂ k P ), A 3 (x) = - π 3 576 |∇P | 4 - π 3 960 j (∂ j P ) 4 - π 3 960 j =k (∂ j P ) 2 (∂ k P ) 2 . (7.68)
For d=7 we have

C 4 (f ; x) = A 0 (x)f (P (x)) + A 1 f ′ (P (x)), (7.69) A 0 (x) = |∇P | 2 △P 2 14b 5 -16b 6 -b 4 + 2b 5,1 -40b 6,1 + 32b 7,1 (7.70) +16b 7,2 j (∂ 2 j P )(∂ j P ) 2 + 16b 7,1,1 j =k (∂ 2 j P )(∂ k P ) 2 +32b 7,1,1 j =k (∂ 2 j,k P )(∂ j P )(∂ k P ), A 1 (x) = |∇P | 4 2 24b 6 -8b 5 -16b 7 -8b 7,2 j (∂ j P ) 4 -24b 7,1,1 j =k (∂ j P ) 2 (∂ k P ) 2 . (7.71)
Using explicit computations as for d = 5 (see Appendix), we get

A 0 (x) = π 4 120 j (∂ 2 j P )(∂ j P ) 2 + π 3 360 j =k (∂ 2 j P )(∂ k P ) 2 + π 3 180 j =k (∂ 2 j,k P )(∂ j P )(∂ k P ), (7.72) A 1 (x) = - π 4 240 |∇P | 4 - π 4 240 j (∂ j P ) 4 - π 3 240 j =k (∂ j P ) 2 (∂ k P ) 2 . (7.73)
We should like to use these formulas with f (λ) = (λ + t) -µ , µ > d(m + 1)/m and t > 0, to prove that C We assume d ≥ 3 and 2jq > 1. We have The following computations has been performed by Guy Moebs, Research Engineer, Laboratoire Jean-Leray, CNRS-University of Nantes.

The method used to compute multi-dimensional integrals is Monte-Carlo, with a cut-off of the domain to reduce it in a bounded domain fitting with the behaviour of the polynomial P . In each example, 100 simulations are computed with at least 10 9 events.

  As we shall see in the next section, the property C (d) 2j (f ) = 0 gives easily a lower bounds on the density of eigenvalues. Remark 4.3 Following Remark 3.4 we can extend our results to polyhomogeneous polynomials P = P m + P m-1 + • • • P 1 + P 0 . To follow the dependence in the coefficients, we note C 2j (f, P ) the coefficient C 2j (f ) with polynomial P . In particular we have C (d) 2j

  = 0. So we have to compute C (d) 4 (f ) for d = 5, 7.

C 4

 4 (f ; x) = C 4,1 (f ; x) + C 4,2 (f ; x) + C 4,3 (f ; x)

7 . 2 P

 72 ) = R d C 4 (f ; x) dx = 0 (d = 5, 7). With f like above, we can see easily that C (d) 4 (f ) = 0 for the following polynomials: 1 P (x) = 1≤j≤d α j x m j , α j > 0, 1 ≤ j ≤ d, d = 5, (x) = 1≤j,k≤d α j,k x j x k , is a positive-definite quadratic form, d = 5, 7 3 d = 7 and P is convex. 4 d = 5, P is convex and satisfies the inequalities 1≤j<k≤5 convex polynomials, we can check that C (d) 4 (f ) = 0, d = 5, 7, for many examples with numerical computations, supporting our conjecture that for every elliptic polynomial P , C (d) 4 (f ) = 0, if d = 5, 7(see Appendix). For d = 9, 11, it seems difficult to compute C (d)6 (f ) by hand. We need more help from symbolic and numerical computations to check our conjecture.

8 appendix 8 . 1

 881 Formulas for b j,k,ℓ

1 ( 1 - 8 . 2

 1182 t) y-1 dt = Γ(x)Γ(y) Γ(x + y) So computing in polar coordinates, b j (d) = π d/2 Γ(jd/2) Γ(j)Now, by elementary computations we get easilyb j,1 (d) = 1 d (b j-1 (d)b j (d)) (8.77) b j,2 (d) = B(5/2, j -Numerical computations for C 4 (f )

Example. 1 d

 1 = 5, polynôme P (x) =

  .23) From (4.22) and (4.23) we can compute J k,ν f (u, v).

	(d) 2j To prove that C

Example.2 d = 7, polynôme P (x) =


Amir Zidi

Amna Bouhana
email: amnabouhana@yahoo.fr

Afef Fekih
email: afef.fekih@louisiana.edu

Mourad Abed

Amna Bouhana

Afef Fekih

Abed Mourad

Personalization of Itineraries search using Ontology and Rules to Avoid Congestion in Urban Areas

Keywords: Personalization of itineraries, Ontology, Rules, Urban freight transport. 

There is a relatively small amount of research covering urban freight movements. Most research dealing with the subject of urban mobility focuses on passenger vehicles, not commercial vehicles hauling freight. However, in many ways, urban freight transport contributes to congestion, air pollution, noise , accident and more fuel consumption which raises logistic costs, and hence the price of products. The main focus of this paper is to propose a new solution for congestion in order to improve the distribution process of goods in urban areas and optimize transportation cost, time of delivery, fuel consumption, and environmental impact, while guaranteeing the safety of goods and passengers. A novel technique for personalization in itinerary search based on city logistics ontology and rules is proposed to overcome this problem. The integration of personalization plays a key role in capturing or inferring the needs of each stakeholder (user), and then satisfying these needs in a given context. The proposed approach is implemented to an itinerary search problem for freight transportation in urban areas to demonstrate its ability in facilitating intelligent decision support by retrieving the best itinerary that satisfies the most users' preferences (stakeholders).

INTRODUCTION

Urban Freight transport has often been neglected in urban traffic management. Though it is very important for the business and life of a city, commercial traffic has not received much attention in the transport planning process.

The last decades have witnessed tremendous efforts aimed at creating and developing a new generation of intelligent freight transportation systems with the objective of controlling and optimizing congestion, fuel consumption, and environment problems and increasing the safety of passenger and goods in urban areas.

The question of efficiency for urban freight transport systems is considered one of the most complicated matters because it involves many actors such as shippers, carriers, receivers, public authorities and citizens. Further, it requires a significantly large and robust information system to manage and insure the operation of freight transportation in order to reduce the complexity of urban freight transport and improve its performance. Thus, a better organization of urban freight traffic can reduce urban congestion, atmospheric pollution and noise while improving security issues for people and goods. One way to overcome some of the problems of urban areas is to propose some solutions to the congestion problem.

Solving this problem requires the availability of reliable and relevant information system about different transportation means used to insure the transportation of goods in urban areas. However, the existence of several actors in city logisics and several operators managing these various transportation means as well as the diversity of information sources and user profiles make the task of designing a homogeneous information system a challenging one. The problem is no longer the availability of information but rather the ability to select the relevant information that meets the precise needs and interests of various stakeholders (residents, retailersauthorities, supplier, carrier) in order to select the best itinerary that minimizes the combination of sets of criteria that satisfy the user needs and interests and overcome the problem of congestion thus improving the operation of distribution network in urban areas.

Hence, personalization in itinerary search has recently become a major goal for intelligent freight transport systems. As a result, several approaches for personalized information systems in the field of transportation, especially for passengers, were recently developed to provide users with information that satisfies their preferences [START_REF] Letchner | Trip router with individualized preferences (TRIP): incorporating personalization into route planning[END_REF][START_REF] Akasaka | Personalized pedestrian navigation system with subjective preference based route selection[END_REF]Onisawa, 2008, Nadi et al., 2011;[START_REF] Bouhana | A Proposal of Personalized itinerary Search Methods in the Field of Transport[END_REF][START_REF] Bouhana | An integrated Case-Based Reasoning and MCDM technique for personalized itinerary search[END_REF]. Note that all the above mentioned approaches used classical methods of personalization such as content-based or collaborative-based methods.

In recent years, new paradigms for personalization of applications have emerged such as the Ontology concept [START_REF] Shi | Ontology-based personalized retrieval in support of reminiscence[END_REF]) (Olivera et al., 2013) [START_REF] Riano | An ontology-based personalization of health-care knowledge to support clinical decisions for chronically ill patients[END_REF]. Ontology allows for the sharing of a large group of information sources or knowledge bases, knowledge organization, and interoperability among complex systems. Ontologies are a level of description of the knowledge of a system. The use of ontology offers many advantages such as determining the appropriate criteria for the user and context modeling. In decision problems, the ontology-based approach is employed to incarnate the role of experts or experienced practices in personalized evaluation processes.

In the literature, only few studies have applied ontologybased approaches to the field of transportation. In Niarki et al., 2009, the authors developed a new approach for personalized route planning search in public transport based on the combination of the AHP methods and the Ontology. Olivera et al., 2013 used ontology to generate personalized user interfaces for transportation interactive. However, all the above mentioned studies have only focused on the personalization of information for passengers in public transport and neglected freight transportation in inter-city or urban areas.

In order to overcome those limitations, we propose in this work to provide a new ontology-based personalization approach for freight transportation in urban areas which satisfies the preferences of all city logistics' actors. The main objective of this research is to design new itinerary search system in order to improve the distribution process of goods in urban areas and also to reduce their consequences such as congestion, fuel consumption, and environmental impact.

In this work, we present a new technique of personalization to mitigate the problems of congestion and reduce its negative consequences. The main contributions of this paper are as follows:

-The design of a new method of personalized itinerary search for freight transportation in urban areas based on ontology and rules in order to alleviate the problem of congestion and its consequences.

-City logistics ontology is used as the representation formalism to model both content sources and user profiles.

-The personalization tasks are solely carried out based on the inference power of the rules.

The rest of this paper is organized as follows. In section 2 we present our new approach of personalization in city logistics based on ontology and rules. The results of the implementation of the proposed approach to an urban transport problem are illustrated in Section 3. Some concluding remarks and proposed future work are discussed in section 4.

PROPOSED APPROACH

The proposed approach is based on the ontology-based city logistics paradigm GenCLON proposed by [START_REF] Anand | GenCLOn: An ontology for city logistics[END_REF].

Scope of GenCLON

City logistics is the study of the dynamic management and operations of urban freight transport and distribution systems.

The aim is to ensure optimum productivity, reliability and customer service whilst reducing environmental impacts, air pollution emissions, energy consumption and traffic congestion. City logistics is classified as a discipline which can cope with sustainable problems encountered in urban logistics freight transportation. One of its key characteristics is its ability to take into consideration the heterogeneity of stakeholders in the decision process (heterogeneity of information and interest exchanged between different stakeholders (Residents-Retailers-Authorities-Supplier-Carrier). In, [START_REF] Anand | GenCLOn: An ontology for city logistics[END_REF], the authors proposed an ontology based city logistic tool (GenCLOn). The aim of this ontology is to formalize the domain knowledge of the city logistics in order to facilitate communication and share different knowledge with several terminologies and types of decisions made by the different actors in the city logistics such as residents, retailers, authorities, suppliers, carriers.

From a semantic point of view Anand et al., 2012 used the concept of ontology to develop a sort of glossary or a common language in order to facilitate communication and coordination among the different actors involved in City logistics. The objective of this ontology is to capture the maximum of knowledge about the city logistics domain.

In the City logistics the major objectives or interest are economic; environmental and social which can ultimately influence the urban movement pattern (see Fig. 1) Fig. 1. Taxonomy objectives in GenCLOn [START_REF] Anand | GenCLOn: An ontology for city logistics[END_REF] From the above figure, we can conclude that the competitively and the growth of economy in urban area depend on congestion reduction, nuisance reduction, consumption of fuel reduction.

In order to alleviate the problems of congestion and its consequences in urban areas, we propose here a new solution based on the concept of personalization using the city logistics ontology [START_REF] Anand | GenCLOn: An ontology for city logistics[END_REF] and rules in order to optimize a set of objectives in urban freight transportation.

Rules Based Personalization

In general, ontologies concentrate on classification methods, by defining 'classes', 'subclasses' and associating individual resources to such classes, and characterizing the relationships among classes and their instances see Figure .1. Rules, on the other hand, focus on defining a general mechanism aimed at discovering and generating new relationships based on the existing ones.

In our paper, we use rules in order to obtain an efficient application in the field of personalization of itineraries in order to improve the quality of urban freight transport and avoid congestion. In our approach, we define a set of rules using SWRL among GenCLOn [START_REF] Anand | GenCLOn: An ontology for city logistics[END_REF]. The objective of these rules is to satisfy the user's preferences (user's profiles) in a given context.

, Both private and public users or stakeholders are involved in city logistics with different preferences (Fig. 2). For private stakeholders, such as shippers, carriers and receivers, the main objective is to reduce transportation cost and minimize the distance travelled for a given transport mission. For public stakeholders (public authority) and residents, the main objectives are getting better accessibility inside the city with less fuel consumption as well as reduction of the environmental impacts such as pollution, nuisances, etc… Fig. 2. Taxonomy of stakeholders and their objectives Therefore, in order to satisfy the preferences of all the stakeholders, while reducing the negative impacts of freight transportation, personalization of information has a key role in improving the process of itinerary search Using, GenCLOn ontology as a backbone for our new approach for personalization, we develop an ontology that perfectly represents the city logistics domain as well as all the knowledge about the content sources and user profiles. That is, the combination between ontologies and rules can lead to more efficient personalization. A set of DL-safe rules was written, using SWRL. These rules match user's preferences which mainly represent stakeholders' objectives or resources in a given itinerary. The rule is a conjunction of concepts and axioms that constitute knowledge. In Table .1, we cite some axioms which basically illustrate relations between concepts. Based on the organization of stakeholders, several kinds of itinerary patterns can be offered to the user for a better planning. We organized them in taxonomy of itinerary patterns (Fig. 3). In this way, an itinerary pattern is classified as a shipper pattern, a receiver pattern or a shipper-receiver pattern.

Fig.3. Taxonomy of itinerary pattern

Initially, three SWRL rules were written to formalize the taxonomy of itinerary pattern based on stakeholders' objectives to check whether user preferences were satisfied or not.

An example is to express the fact that "if we have an itinerary in which shipper has an objective, then the itinerary pattern (Iitinerary_pattern(i?)) is classified as a shipper pattern (Shipper_pattern(i?). the SWRL description for this rule is depicted in fig. 4 in which, the shipper has a transport cost reduction as an objective. The rest of the rule describes the resources used by the carrier.

SWRL Rule #1

Transport_cost_reduction(?t), Loading( ?l), Carrier(c?), Itinerary_pattern(?i), Receiver( ?e), Shipper(?s), Driver(?d), Road_freight_vehicle( ?r), has_ressource( ?c, ?d), has_ressource( ?c, ?r), has_objective(?s, ?t)->Shipper_pattern(?i) Fig. 4. Rule for shipper preferences

In the second rule (Fig. 5), the itinerary pattern (Iitinerary_pattern(i?)) is classified as a receiver pattern (receiver_pattern(i?) since the receiver has other nuisance reduction as an objective.

SWRL Rule #2

Other_nuisance_reduction(?o), Loading(?l), Carrier(c?), Itinerary_pattern(?i), Receiver( ?e), Shipper(?s), Driver(?d), Road_freight_vehicle(?r), has_ressource( ?c, ?d), has_ressource( ?c, ?r), has_objective( ?e, ?o)->Receiver_pattern(?i) Fig. 5. Rule for receiver preferences For the case where the shipper and the receiver have objectives in the same itinerary, we wrote a third rule (Fig. 6).

SWRL Rule #3

Transport_cost_reduction(?t), Other_nuisance_reduction(?o), Loading(?l), Carrier(c?), Itinerary_pattern(?i), Receiver(?e), Shipper(?s), Driver(?d),

Road_freight_vehicle(?r), has_ressource( ?c, ?d), has_ressource( ?c, ?r), has_objective( ?s, ?t), has_objective( ?e, ?o)->Shipper_receiver_pattern(?i) Fig. 6. Rule for shipper and receiver preferences Consequently, a SWRL rule was defined in order to associate the user with all the itineraries that meet exactly his preferences.

APPLICATION

As mentioned previously, the domain in which the approach was evaluated was urban Freight transport. We use GenCLOn core ontology which contains x concepts. In order to verify the approach and illustrate its application in a real world itinerary search, a real data collection is used to build the first knowledge base. It was retrieved from Paris Open Data.

The interface and the functionality of the application are implemented, as usual for web applications, in JEE (Jsp and Servlet). For the ontology extension and development, we use OWL (Ontology Web Language) in order to ensure the maximum possible expressiveness and the efficient support of reasoning. The definition of rules is implemented in SWRL. Protégé 4 has been selected as the appropriate ontology environment to implement our ontology and its associated rules. We make use of Pellet, an open-source Java based OWL reasoner, which guarantees complete OWL reasoning. The special feature of Pellet is its support for reasoning with DL-safe rules which makes it particularly suitable for our needs.

In the proposed system, we adopt keyword-based interface as it provides a comfortable and relaxed way to query about the end user, in which the user sets the addresses of the shipper and the receiver (Fig 4). The system provides also the user with a configuration to set his preferences, in which he can select the desired criteria to be included in personalized itinerary search.

Fig.7 .User interface

In (Fig. 7) we have the query: "go from Rue Saint Denis (shipper) to Rue Galvani (receiver)" with search criteria: Time less than 30 minutes, vehicle type is truck, fuel consumption less than 0.270, emission CO2 less than 500 co2/km and transportation cost less than 50 euro".

When a user submits a query in which the stakeholders like Results are shown in (Fig. 8), we have a set of personalized results which represent real world itineraries. In this case, the stakeholders like shipper (origin) and receiver (destination) has respectively an economic (Transportation cost) and environmental (Gaz emission) objectives and the type of vehicle chosen is a truck. Then, concepts with labels {"Carrier", "Shipper", "Receiver","Road_freight_vehicle", "Transport_cost_reduction", "Emission_reduction"} and properties with labels {"connect", "origin", "destination", "has_objective", "has_resource"} are selected from GenCLOn. Finally, itineraries are calculated based on Rule#3 which contains the selected concepts and properties.

CONCLUSION

In this paper, we proposed a new solution for improving the process of freight transportation in urban areas, while decreasing the problem of congestion and its consequences. Our approach integrates the concept of personalization in the intelligent freight transportation as a technique to alleviate the problem of congestion in urban areas by providing the users with a personalized itinerary that satisfies exactly their needs and preferences. The proposed approach is based on ontology and SWRL rules. The ontology is used to formalise the representation of various sources of information and to capture knowledge about user preferences in city logistic while the inference power of the rules insures the personalization process.

In our future research, we plan to further optimize the itinerary search by including information about traffic lights and road congestion in real time. This will lead to a dynamic and personalized itinerary search engine for urban freight transport and distribution systems.

Fig

  Fig.8. Query results

Table . 1 Axioms in GenCLOn (Anand et al.2012)

 .

	Axioms
	Carrier has_activity some Loading
	Carrier has_ressource some Driver
	Carrier has_ressource some Road_freight_vehicle
	Receiver has_objective some Other_nuisance_reduction
	Shipper has_objective someTransport_cost_reduction


Simon Coulombe

Marcus Dejardin

Sylvain Luc

Covid or not Covid? Psychological Distress and Entrepreneurial Intentions among Canadian Workers during the Pandemic

Keywords: well-being, psychological distress, entrepreneurial intention, COVID-19. JEL-Codes: D91, I15, L26

Triggering events can be associated with entrepreneurial intentions. Using data from a survey on the mental health of Canadian workers carried out in the context of the COVID-19 pandemic crisis, we specifically test the association between entrepreneurial intentions and psychological distress, along with other individual demographic and personal-level measurements such as risk-aversion, and financial concern. Our results substantiate a positive relationship between entrepreneurial intention and individual psychological distress, as well as financial deterioration measured at the household level. Taken as a whole, our contribution helps to feed a discussion on the links between mental health and entrepreneurship in a process of personal resilience and, more generally, on well-being as a motivational factor in the entrepreneurial process.

Introduction

The COVID-19 pandemic is far more than a health crisis. The pandemic has affected not only societies but also their economies on a large scale. To restrain the spreading of the COVID-19 virus, several governments worldwide have implemented social distancing measures, lockdowns and curfews on their territories, with dreadful economic consequences, particularly on the labor market (OECD, 2020a). Considering Canada, a country to which the empirical analyses in the present paper will refer, by April 2020, 5.5 million workers had been impacted by the crisis, of which 3.0 million lost their job and 2.5 million experienced work absences related to COVID (Statistics Canada, 2020b). According to the Canadian Labour Force Survey, the Canadian national unemployment rate hit 13.7% in May 2020, reaching an unprecedented record (Statistics Canada, 2020b).

To limit the negative economic impact of job losses and the deterioration of financial conditions among those directly affected, governments decided, often concurrently, to adopt temporary policies to support workers and businesses (OECD, 2020b(OECD, , 2020c)). In Canada, the federal government implemented the Canadian Emergency Benefit program, which guaranteed taxable income of up to CAD 2,000 per month for people who had lost their jobs or had suffered loss of income related to COVID-19 (Government of Canada, 2021a). Also, Canada has developed wage subsidy programs to support businesses in their efforts to keep their employees employed (Government of Canada, 2021b). Other complementary measures have also been promulgated by the provincial governments.

Beyond job losses, absences from work and financial deterioration affecting several people, health conditions have also led to more or less substantial changes in working conditions. For some workplaces, the sanitary conditions have created strong constraints on the organization of work. For others, teleworking has become the norm, sometimes requiring workers to work from home, in variable environmental conditions impacting notably work-family balance and the mental health of workers [START_REF] Escudero-Castillo | Furloughs, teleworking and other work situations during the COVID-19 lockdown: Impact on mental well-being[END_REF]. Although in some cases this upheaval in work habits may have contributed to an improvement in the quality of life and well-being of workers, in the majority of cases, it has instead led to a deterioration contributing to psychological distress [START_REF] Evanoff | Work-related and personal factors associated with mental well-being during the COVID-19 response: Survey of health care and other workers[END_REF][START_REF] Stephan | Entrepreneurship during the Covid-19 Pandemic: A global study of entrepreneurs' challenges, resilience, and well-being[END_REF][START_REF] Vanhaecht | COVID-19 is having a destructive impact on health-care workers' mental well-being[END_REF][START_REF] Xiao | Impacts of working from home during COVID-19 pandemic on physical and mental well-being of office workstation users[END_REF][START_REF] Galanti | Work from home during the COVID-19 outbreak: The impact on employees' remote work productivity, engagement, and stress[END_REF]. Such a context of sudden decline in working conditions and workers' quality of life may lead some workers to consider changing their professional career by starting a business or becoming self-employed. According to [START_REF] Shapero | The social dimensions of entrepreneurship[END_REF], an acknowledged classic in entrepreneurship literature, the so-called "entrepreneurial event", i.e. starting a business, can be related to a positive or, more likely, negative triggering antecedent. Interpreting and applying the argument, one comes quite easily to suggest the COVID-19 pandemic crisis might be considered as a possible antecedent of an individual's entrepreneurial intention and behavior. Moreover, even if economic conditions may appear relatively uncertain in times of crises and disrupting events, the latter can also lead to the emergence of new opportunities, which can cause people to seek to exploit them (e.g. [START_REF] Devece | Entrepreneurship during economic crisis: Success factors and paths to failure[END_REF][START_REF] Peris-Ortiz | Entrepreneurship and innovation in a context of crisis[END_REF]. It therefore appears quite relevant to explore in more detail potential entrepreneurial dynamics among workers in this radically disrupted period.

In Canada, without any surprise, COVID-19 initially negatively impacted the selfemployment rate (Statistics Canada, 2021). Although self-employment rates (approximately 15%; Statistics Canada, 2022) were stable from February 2020 to December 2020, the absolute number of self-employed Canadians was overall 6.8% lower in December 2020 than it was before the pandemic. However, there had been a resurgence in the number of self-employed workers in October 2020 (Statistics Canada, 2020a[START_REF] Sjöberg | Emotions and risk perception[END_REF]. This suggests that some workers decided to start (or restart) self-employment ventures despite the pandemic, perhaps by necessity or by opportunity [START_REF] Fairlie | Defining opportunity versus necessity entrepreneurship: Two components of business creation[END_REF]; and it is possible that, at the same period, some workers had intentions to venturing a business even after the state of health emergency was announced and preventing health measures were imposed. These ventures may contribute to workers' resilience and post-shock recovery [START_REF] Monllor | Natural disasters, entrepreneurship, and creation after destruction: A conceptual approach[END_REF]. Important questions must be answered in order to increase our knowledge of selfemployment ventures and intentions in exceptional situations. These questions revolve around an adverse context, that of the pandemic, in which workers, considering their personal situation, may wonder about their future and, more particularly, their professional future. Since pandemic working conditions may have negatively influenced workers' mental health, going as far as individual psychological distress, we propose to explore the latter as possible antecedent of entrepreneurial intention, along with other individual demographic and personal-level measurements such as risk-aversion, and financial concern.

We analyzed data from a survey on mental health carried out among French-speaking Canadian workers between October and November 2020. In the next section, we briefly present some insights derived from previous research on well-being, psychological distress and entrepreneurship. We then explain the methodological aspects of our exploratory research. The fourth section is devoted to the presentation of the results. We then conclude by listing key takeaways, identifying limitations and presenting avenues for future research.

Well-being, psychological distress and entrepreneurship

Well-being is quite difficult to define precisely. It is, however, recognized to be a multidimensional concept [START_REF] Prilleltensky | Assessing multidimensional well-being: Development and validation of the I COPPE scale[END_REF]. Using a broad perspective, Prilleltensky (2012, p. 2) offers a guiding definition: "a positive state of affairs, brought about by the simultaneous and balanced satisfaction of diverse objective and subjective needs of individuals, relationships, organizations, and communities". In line with the World Health Organization's (2020Organization's ( [1948]]) definition of health, well-being is far more than the mere absence of disease as it also includes positive aspects, such as one's opportunities to function effectively on a daily basis and to self-actualize their potential [START_REF] Vittersø | Subjective well-being versus self-actualization: Using the flowsimplex to promote a conceptual clarification of subjective quality of life[END_REF]. Within the realm of psychology, contemporary researchers point out that psychological distress is one aspect that is indicative of negative wellbeing [START_REF] Karademas | Positive and negative aspects of well-being: Common and specific predictors[END_REF][START_REF] Keyes | Mental illness and/or mental health? Investigating axioms of the complete state model of health[END_REF]. It refers to "symptoms of stress, anxiety and depression" (Viertiö et al., 2021, p. 2). Some definitions provided, if we turn now to the treatment that well-being has been given in the entrepreneurship literature, it is clear that its study has gained significant momentum in recent years [START_REF] Wiklund | Entrepreneurship and well-being: Past, present, and future[END_REF]. The majority of contributions have focused on the causes and consequences of well-being on entrepreneurial activity among active entrepreneurs [START_REF] Lerman | The (not so) dark side of entrepreneurship: A meta-analysis of the well-being and performance consequences of entrepreneurial stress[END_REF][START_REF] Wincent | A comprehensive model of entrepreneur role stress antecedents and consequences[END_REF][START_REF] Uy | Joint effects of prior start-up experience and coping strategies on entrepreneurs' psychological well-being[END_REF][START_REF] Lechat | Stressors and satisfactors in entrepreneurial activity: An event-based, mixed methods study predicting small business owners' health[END_REF][START_REF] Stephan | Entrepreneurs' mental health and well-being: A review and research agenda[END_REF][START_REF] Shir | Entrepreneurship and well-being: The role of psychological autonomy, competence, and relatedness[END_REF][START_REF] Abreu | Well-being effects of self-employment: A spatial inquiry[END_REF][START_REF] Wach | Entrepreneurs' stressors and well-being: A recovery perspective and diary study[END_REF]. Recently, some studies have also examined the impact of the pandemic crisis on the mental health of business owners and self-employed people [START_REF] Torrès | Health perception of French SME owners during the 2020 COVID-19 pandemic[END_REF][START_REF] Sjöberg | Emotions and risk perception[END_REF][START_REF] Wolfe | Everybody hurts: Self-employment, financial concerns, mental distress, and well-being during COVID-19[END_REF].

A majority of studies examining the links between entrepreneurship and well-being have considered well-being as a dependent variable [START_REF] Wiklund | Entrepreneurship and well-being: Past, present, and future[END_REF]. And very few were interested in the role of well-being as an antecedent in the premises of entrepreneurial activity (intentions and/or start-up phase). However, one's well-being level could also logically be considered to be an independent variable, i.e. as a psychological factor influencing entrepreneurial action.

Several studies have highlighted the role of positive emotions, which are signs of wellbeing as well as an integral part of its construction [START_REF] Fredrickson | The broaden-and-build theory of positive emotions[END_REF], in several facets of entrepreneurial life [START_REF] Baron | The role of affect in the entrepreneurial process[END_REF] such as in the creativity and innovation processes [START_REF] Baron | The role of affect in the entrepreneurial process[END_REF]Tang, 2011, Perry-Smith and[START_REF] Perry-Smith | In the mood for entrepreneurial creativity? How optimal group affect differs for generating and selecting ideas for new ventures[END_REF], the evaluation of opportunities (Grichnick et al., 2010;[START_REF] Foo | Emotions and entrepreneurial opportunity evaluation[END_REF][START_REF] Foo | Beyond affective valence: Untangling valence and activation influences on opportunity identification[END_REF], entrepreneurial motivations [START_REF] Jia | The impact of positive affect on entrepreneurial motivational outcomes: A self-regulatory perspective[END_REF], or even the perception of risk [START_REF] Sjöberg | Emotions and risk perception[END_REF]. However, while positive emotions are often linked to an increase in the creativity or productivity of individuals [START_REF] Diener | Positive emotions at work[END_REF], negative emotions can also be influential. So, with regard to entrepreneurial intentions and the decision to start business, [START_REF] Doanh | Entrepreneurial behaviour: The effects of the fear and anxiety of Covid-19 and business opportunity recognition[END_REF] have recently showed that fear and anxiety related to COVID-19 have negatively affected the entrepreneurial intentions of Vietnamese university students. Contrastingly, [START_REF] Nikolaev | Dispositional positive and negative affect and self-employment transitions: The mediating role of job satisfaction[END_REF] have reported that people in employment with a natural tendency to experience negative affects (negative dispositional affects), experience more dissatisfaction at work and, for this reason, are more likely to start an entrepreneurial adventure. Similarly, negative dispositional affects have been found to be related to higher likelihood of entering self-employment [START_REF] Wiklund | Entrepreneurship and well-being: Past, present, and future[END_REF].

Our research aims to further the examination of the relations between negative emotions and the antecedents of entrepreneurial activity in the context of COVID-19. In particular, as mentioned earlier, we want to explore if and how psychological distress can be associated with business intentions among workers facing the pandemic context.

Research methodology

Participants and recruitment

Participants were recruited from a survey panel named LEO owned by the marketing firm, Léger Marketing. Participants were informed about the study by Léger Marketing and, if interested to participate, were directed to the 45-minute anonymous study survey on the Qualtrics platform. The marketing firm selected potential participants to invite based on age, gender and region so as to address population representativity.

In order to be eligible to participate in the survey, people had, based on self-reported information, a) to be 18 year old or more, b) be able to read and understand French as the survey was all in French, c) be working at least 20 hours per week at the time of the survey or to have been working at least 20 hours per week before the beginning of the COVID-19 pandemic state of emergency was declared in Canada (i.e., before March 2020). Both wage employed and self-employed people were considered to satisfy this last criterion.

Of the overall sample of 2148 participants who completed the survey, the following groups have been excluded from the present study: a) participants who reported they were self-employed before the COVID-19 pandemic started or b) participants with irrelevant answers to studying entrepreneurial intentions, e.g., a few participants who reported that they had started a business or initiated the process to start a business since the beginning of the pandemic but who also reported that they hadn't have the intention to start a business so far during the pandemic. The rationale for these exclusions was that we wanted to focus the analysis on examining entrepreneurial intentions specifically among those who were, at the time of the survey, wage employed or who had been wage employed before the pandemic, in order to get closer to a group of workers under the influence of the pandemic and explore factors associated with intentions to start a business. The analytical sample was thus composed of 1,372 participants.

Participants from the analytical sample were, on average, 40.4-year-old (SD=11.9) and a vast majority were living in the province of Québec (98.9%). More than 60% were women (60.6%), 39.2% men and 0.2% non-binary or others. About two-thirds (66.3%) were in a relationship (married or civil law). While most were born in Canada (88.3%), 11.7% were immigrants (i.e., not born in the country). Almost half had a university degree (48.2%), 27.5% had a college degree. The remaining (24.3) had completed a high school degree or less. At the time of the survey, 90.4% were wage employed and 9.6% did not have a job in wage employment; of that proportion, 14.4% were, however, self-employed at that same moment (but they were not self-employed before the pandemic, as all participants who were self-employed before the pandemic had been removed from the analytical sample, see previous paragraph). Before the pandemic, in the analytical sample, 94% were wage employed, meaning that a proportion had lost their wage employment. More than a third (35.9%) were parents of children aged less than 18. Risk aversion levels were moderately high as the mean was 3.05 (SD=1.13) on a scale from 1 to 4 (see Covariates' measures below).

Variables and measures

Dependent variable

The dependent variable, i.e., the intention to start or to own a business, was measured using a question adapted from a previous study [START_REF] Luc | Hybrid entrepreneurship: Employees climbing the entrepreneurial ladder[END_REF]. The question presented in French (but translated to English here) read as: "Since the beginning of the COVID-19 crisis last March, have you, alone or with others, intended to start a new business or take over an existing business?". Participants could answer: "Yes", "No" or "I don't know". The variable was recoded to make it binary (Intention; No clearly identified intention) by merging the "No" and "I don't know" categories. Additional questions, using a similar format, were also asked and used for consistency check purposes, pertaining to whether or not participants had concretely initiated the process, during the pandemic, of starting a business, and whether or not they were actually self-employed at the time of the survey, but not before COVID-19 (i.e., meaning they would have started a business since the beginning of the pandemic, and therefore would have had an intention at some point during the pandemic).

Explanatory Variables

Two explanatory variables of interest were measured: psychological distress (as an indicator of negative well-being) as well as financial deterioration. The former was assessed using the validated French version of the Patient Health Questionnaire-4 (PHQ4; [START_REF] Kroenke | An ultra-brief screening scale for anxiety and depression: The PHQ-4[END_REF], which measures the frequency of four anxiety and depression symptoms (e.g., feeling nervous, anxious or on edge; feeling down, depressed or hopeless) over that last two weeks, on a scale from 1 (Not at all) to 4 (Nearly every day). Although it is brief, it has been shown to correlate with other established measures of mental health, to perform almost as good as longer versions of anxiety and depression instruments from the Patient Health Questionnaire, and to be a good screener for common mental health disorders [START_REF] Kroenke | An ultra-brief screening scale for anxiety and depression: The PHQ-4[END_REF]. The average of the four items was used as an index of psychological distress at the time of the survey, treated as continuous in the analysis. The Cronbach's alpha was 0.88, indicating excellent internal consistency.

Financial deterioration was measured using two items. A first item, adapted from [START_REF] Pearlin | The stress process[END_REF], asked participant to rate the financial situation of their household as it was before the COVID-19 pandemic started (i.e., before March 2020) by choosing among four options: comfortable with extra (0), enough but no extra (1), have to cut back (2), and cannot make ends meet (3). The same question was asked a second time, but regarding their current household financial situation. As will be explained below, the first question (before COVID-19 version) was controlled for in the analysis while considering the current situation version, in order to represent financial deterioration in the model.

Covariates

Additional covariates were measured using standard questions so they could be controlled for in the models. This included: gender (recoded as: 1=women and nonbinary; 0=men), year of birth, immigrant status (1=not born in Canada; 0=born in Canada), relationship status (1=in a married or civil law relationship; 0=single), parent of non-adult children status (1=parent of children less than 18 year old; 0: not parent of children that age), highest level of education (recoded as: 1=university degree; 0=college degree or less), wage employment (1=wage employment; 0=not wage employed) at the time of the survey. Risk aversion, which appears important to control for given previous research results suggesting it negatively influences selfemployment entry, was also measured using an index based on participants' answers to three hypothetical income gambles following the questions used by [START_REF] Raffiee | Should I quit my day job? A hybrid path to entrepreneurship[END_REF] and based on the work of [START_REF] Barsky | Preference parameters and behavioral heterogeneity: An experimental approach in the health and retirement study[END_REF]. The questions asked participants if they would accept to gamble, i.e., taking a new job, in scenarios that are moderately, less and more risky in terms of potential positive and negative outcomes. The scoring procedure, provided by [START_REF] Raffiee | Should I quit my day job? A hybrid path to entrepreneurship[END_REF], yields a four-point index from weak (1) to strong (4) risk aversion, treated as continuous.

Analysis

Univariate correlations between all of the study variables were examined, followed by a logistic regression model in which all covariate and explanatory variables of interest were included as predictors of the probability of reporting an entrepreneurial intention during the COVID-19 pandemic. In logistic regression, the logit link function is used to link the probability of the event (self-employment intention in our case) to the linear function of the covariates and explanatory variables.

Results and discussion

Preliminary analysis

Based on descriptive analysis, it appears that only a small subsample indicated they have had the intention to start a business since the beginning of the COVID-19 pandemic (12.3%), with the remaining participants not reporting such intentions (87.7%). With regard to psychological distress ratings, based on published cut-offs [START_REF] Kroenke | An ultra-brief screening scale for anxiety and depression: The PHQ-4[END_REF] for the PHQ4, 48.8% reported none-to-minimal distress (index score between 1.00 and 1.50), 30.5% reported mild levels of distress (between 1.75 and 2.25), 12.8% reported moderate levels of distress (between 2.50 and 3.00) and 8.0%, severe levels of distress (between 3.25 and 4.00). When asked to assess their household financial situation before the pandemic, 11.2 % declared they needed to reduce their expenses or that they were not able to make ends meet. When asked the same, but considering the situation at the time of the survey, this percentage increased to 18.9%.

Table 1 shows the univariate correlations between all the study variables. As reported, the variables most strongly related with entrepreneurial intention were: risk aversion (negative relationship), negative appraisal of the current financial situation (positive relationship), currently being a wage employee (negative relationship), psychological distress (positive relationship) and being an immigrant (positive relationship). Except for current and before COVID-19 appraisal of the household financial situation, which were rightfully strongly correlated (0.63), all correlations among the explanatory variables and controls were below 0.31 and most were much lower, indicating relative independence of the explanatory variables and controls. This supports that multicollinearity would not be an issue when including all these variables as predictors in the logistic regression model. Multicollinearity among the logistic regression predictors was further verified by examining the variance inflation factors (VIFs). None of the VIFs were larger than 1.87, which is indicative of no multicollinearity problems that are severe enough to warrant corrective measures.

Main findings

Table 2 reports the results from the estimated logistic regression model including all explanatory variables of interest as well as covariates as predictors of the likelihood of reporting entrepreneurial intention since the COVID-19 pandemic. All considered covariates are found to be significant predictors, except "being in a relationship" and educational level, the latter being marginally significant.

With regard to the two explanatory variables of interest, psychological distress is found to be a significant positive predictor of entrepreneurial intention, with an increase of one unit on the psychological distress measure being associated with a 1.46 fold increase in the odds of reporting such intention. With regard to financial deterioration, after controlling for one's appraisal of their household situation before the pandemic, negative appraisal of the current situation is found to be positively associated with entrepreneurial intention, with every increase of one unit on the negative financial appraisal scale being associated with a 1.45 fold increase in the odds of reporting such intention. Finally, we can also briefly note that estimates for covariates reveal signs consistent with expectations based on the literature. Year of birth (i.e., being younger) is negatively associated with entrepreneurial intention. This is also the case for being a woman or identifying with another gender identity, and being relatively more riskaverse. Being an immigrant or being parent of children less than 18 years old were associated positively with entrepreneurial intention. Additionally, it is notable -this is important for the discussion that follows-that being currently a wage employee (versus not) enters as well into a negative relationship with entrepreneurial intention.

Discussion

Given that our research is exploratory and the fact that the data are cross-sectional, it is difficult to draw firm findings from our study. Nevertheless, it raises several issues that we discuss in this section.

Psychological distress and entrepreneurship as a resilience process.

As the levels of psychological distress observed in our study are very probably higher than the average rate of the population before COVID-19 [START_REF] Bierman | Social estrangement and psychological distress before and during the COVID-19 pandemic: Patterns of change in Canadian workers[END_REF], it appears reasonable to posit that COVID-19 has increased the perception of psychological distress among the participants of our study. Concomitantly, our findings suggest that psychological distress may positively influence entrepreneurial intention, which could lead to consider thinking about being an entrepreneur and eventually moving towards an entrepreneurial trajectory as a possible resilience path for people in psychological distress. If correct, this reasoning appears to contradict the preliminary results obtained by [START_REF] Doanh | Entrepreneurial behaviour: The effects of the fear and anxiety of Covid-19 and business opportunity recognition[END_REF] on a sample of Vietnamese students for whom fear and anxiety related to COVID-19 negatively influenced entrepreneurial intentions. Nevertheless, individuals in our survey being employed before COVID-19, it is possible that this apparent contradiction can be explained essentially by their conditions of employment and the occurrence of work-family conflicts among the people studied. Several contributions have indeed established a link between job satisfaction and psychological distress (e.g. [START_REF] Amati | Relationship of job satisfaction, psychological distress and stress-related biological parameters among healthy nurses: A longitudinal study[END_REF][START_REF] Ghawadra | Psychological distress and its association with job satisfaction among nurses in a teaching hospital[END_REF][START_REF] Pepe | Psychological distress, job satisfaction and work engagement: A cross-sectional mediation study with a sample of Palestinian teachers[END_REF]. Psychological distress could be associated with the perception of lower job satisfaction, which would in turn stimulate entrepreneurial intentions. This hypothesis is in line with the work of [START_REF] Nikolaev | Dispositional positive and negative affect and self-employment transitions: The mediating role of job satisfaction[END_REF] on the effect of dispositional tendencies to experience negative emotions on entrepreneurial intentions.

Financial deterioration and necessity entrepreneurship

According to our results, the greater the deterioration in the financial situation of the respondent's household, the higher the probability of having entrepreneurial intentions. From this, we may hypothesize that, for some people, the intention to be entrepreneurial is motivated by the desire to make up for a drop in income that threatens the financial situation of their household. For those people, it therefore seems that the chosen trajectory was not to seek another job or to supplement an existing wage income with a complementary job as employee, but to start an entrepreneurial venture. This could be explained by, among other factors, the fact that some of those people have few skills to value on a highly competitive labour market [START_REF] Poschke | Who becomes an entrepreneur? Labor market prospects and occupational choice[END_REF]. Another possible explanation would be that some people have suffered a temporary layoff and are trying in the meantime to entrepreneurially exploit their skills before returning to their job. Yet another possibility would be that, for some of those people, the loss of income being only partial, they are trying to begin a process of work hybridization combining their activity as an employee and as a self-employed worker [START_REF] Luc | Hybrid entrepreneurship: Employees climbing the entrepreneurial ladder[END_REF].

Job loss and opportunity entrepreneurship

Our study suggests that having lost one's wage employment after the state of health emergency has been declared by Canadian public authorities may increase the probability of having an entrepreneurial intention. This result may seem obvious on first reading, but when it is analysed in the light of the other variables used in the model, it is not that obvious. Since we are controlling for financial deterioration, it instead suggests that the effect of job loss, if any, cannot be confused with the worsening of financial conditions. In Canada, people who had lost their jobs or were unable to earn job-related income due to COVID-19 were able to benefit from a financial allowance of up to $2,000 per month. This support measure, called the Canadian Emergency Response Benefit (CERB), made it possible to limit the effects of the loss of professional activity on household income. The CERB might explain the relatively low correlation (-0.13) between the variable "Currently a paid employee" and the variable "Negative financial situation appraisal (current)". The reasoning leads us to infer that there is a net effect of job loss on entrepreneurial intentions and, for most people for whom job loss is increasing the probability of having entrepreneurial intentions, this might be attributed more to the job loss itself than to the financial deterioration it causes. A plausible explanatory hypothesis would be then to consider that for those people, the job loss was a shock that activated a latent desire to start a business [START_REF] Källner | Entrepreneurial motivation and idea generation by displaced employees[END_REF]. In a way, the COVID-19 pandemic and its economic consequences would have triggered the opportunity to be entrepreneur.

Conclusion

Besides the fact that the COVID-19 pandemic has had, and still has, worldwide dramatic economic and social consequences, it is profoundly changing the working conditions of many people. It has shaken the budget of many households and may have affected the mental health of large parts of the population. Given the situation, one can expect various human reactions in various domains. In particular, this disrupted context may act as the impetus for workers to reconsider, whether by choice or by default, their professional occupations. And why not become an entrepreneur?

With this study, we proposed to test psychological distress as a possible antecedent of entrepreneurial intention, along with other individual demographic and personallevel measurements such as risk-aversion, and financial concern. Our findings suggest that psychological distress, job loss and household's financial deterioration positively influence entrepreneurial intention among people that were in employment before the state of health emergency was declared by Canadian public authorities.

Beyond these challenging results, our contribution helps to feed a discussion on the links between mental health and entrepreneurship as a personal resilience process.

In the dualist opportunity/necessity categorization of entrepreneurship motives, the necessity-driven entrepreneur is considered to be "pushed" by the lack of satisfactory job options or by the desire to avoid unemployment, whereas the opportunity-driven entrepreneur is "pulled" by the exploitation of a business opportunity. Our study suggests that this categorization may be too narrow to account for the diversity of motivational factors that could influence the entrepreneurial process and its dynamics.

In particular, our study highlights the importance of advancing research on well-being as a factor influencing the entrepreneurial process.

This study should be considered exploratory. We also recognize some limitations. In particular, results are derived from cross-sectional and retrospective data. They do not allow to compare people's entrepreneurial intention through time. The methodology was carefully chosen based on the type of data available. However, the results obtained cannot strictly confirm the existence of causal links and must be cautiously interpreted. Future studies are requested in this regard [START_REF] Sufyan | Entrepreneurship during the COVID-19 pandemic: A Systematic Literature Review and future research agenda[END_REF]. We can also add that this study has left aside many questions concerning the economic and social policy measures that the situation calls for. Nevertheless, we believe our analysis has provided some important initial empirical findings on the antecedents of entrepreneurial intention that specifically apply during the COVID-19 pandemic.

Table 1 .

 1 Univariate correlations between study variables

	Variables	1	2	3	4	5	6	7	8	9	10	11
	1. Intention to start a self-
	employed business: Yes (vs.
	no clear intention)
	2. Psychological distress	.15***
	3. Negative financial situation	.10***	.16***
	appraisal (before COVID-19)
	4. Negative financial situation	.17***	.31***	.63***
	appraisal (current)
	5. Gender: Women or non-	-.10*** .14***	.06*	.06*
	binary (vs. men)
	6. Year of birth	-.13*** -.22*** .02	-.02	-.22***
	7. Immigrant status	.15***	-.03	.07*	.04	-.06*	-.03
	8. In a relationship	.05	-.11*** -.09**	-.04	-.07**	.08**	.02
	9. Parent of children less than	.11***	-.05	.07**	.08**	-.07*	.00	.10*** .30***
	18
	10. Educational level:	.05	-.07**	-.16***	-.18***	-.02	-.02	.17*** .04	.10***
	University (vs. college or less)
	11. Currently a wage	-.16*** -.12*** -.02	-.13***	-.04	.05	-.02	.01	-.02	.02
	employee (vs. not)
	12. Risk aversion	-.18*** -.10*** -.04	-.11***	.12***	.04	-.11*** .04	.00	.00	.09**

Note: The correlation analysis was conducted with 1,356 participants due to missing values for some of the variables. * p<0.05; ** p<0.01; *** p<0.001.

Table 2 .

 2 Logistic regression model of the predictors of entrepreneurial intention.

	Predictors