Philippe Cieutat
email: philippe.cieutat@uvsq.fr

ON BOCHNER'S ALMOST-PERIODICITY CRITERION

Keywords: 2020 Mathematic Subject Classification: 35B10, 35B40, 42A75, 47H20 Bochner almost periodicity, periodic function, almost periodic function, asymptotically almost periodic function, nonlinear semigroup, periodic dynamical system

We give an extension of Bochner's criterion for the almost periodic functions. By using our main result, we extend two results of A. Haraux. The first is a generalization of Bochner's criterion which is useful for periodic dynamical systems. The second is a characterization of periodic functions in term of Bochner's criterion.

Introduction

The almost periodic functions in the sense of Bohr have been characterized by Bochner by means of a compactness criterion in the space of the bounded and continuous functions [START_REF] Bochner | A new approach to almost periodicity[END_REF][START_REF] Bochner | Continuous mappings of almost automorphic and almost periodic functions[END_REF]. The Bochner's criterion plays an essential role in the theory and in applications. We give a new almost-periodicity criterion for functions with values in a given complete metric space which is useful to study the almost periodicity of solutions of dynamical systems governed by a family of operators with a positive parameter. This criterion is an extension of Bochner's criterion. Then Haraux gave a generalization of Bochner's criterion [9, Theorem 1], called a simple almost-periodicity criterion which is useful for periodic dynamical systems. From our result, we deduce an extension of this criterion. We also obtain an extension of an other result of Haraux which characterizes the periodic functions in terms of the Bochner's criterion [START_REF] Haraux | Sur les trajectoires compactes de systèmes dynamiques autonomes[END_REF]. In the same spirit, we treat the asymptotically almost periodic case.

We give a description of this article, the precise definitions will be given in Section 2. Throughout this section (X, d) is a complete metric space. An almost periodic function u : R → X in the sense of Bohr is characterized by the Bochner's criterion which is the following: u is bounded and continuous, and from any real sequence of real numbers (τ n ) n , there exists a subsequence (τ φ(n) ) n such that the sequence of functions (u(t + τ φ(n) )) n is uniformly convergent on R. In Section 3, we give two extensions of Bochner's criterion. First u : R → X is an almost periodic if and if only if in the Bochner's criterion, we impose that the terms of the sequence of real numbers (τ n ) n are all positive. Second u : R → X is an almost periodic if and if only if in the Bochner's criterion, the convergence of the subsequence of functions (u(t + τ φ(n) )) n is uniform only on [0, +∞). These improvements are useful to study the almost periodicity of solutions of an evolution equation governed by a family of operators with a positive parameter, in particular for a C 0 -semigroup of linear operators or more generally, for an autonomous dynamical system (nonlinear semigroup).

From our extension of Bochner's criterion, we give new proofs which are direct and simpler on known results on the almost periodicity of solutions of autonomous dynamic systems.

Haraux gave a generalization of Bochner's criterion the called a simple almost-periodicity criterion [START_REF] Haraux | A simple almost-periodicity criterion and applications[END_REF]Theorem 1]. This criterion makes it possible to choose in the Bochner's criterion, the sequence of real numbers (τ n ) n in a set of the type ωZ which is very useful for periodic dynamical systems. From our extension of Bochner's criterion, in Section 4, we deduce an improvement of this result. An asymptotically almost periodic function u : R + → X is a perturbation of almost periodic. A such function is characterized by a property of the type of the Bochner's criterion. In the same spirit, we extend this characterization of asymptotically almost periodic functions. Then we apply these results to study the almost periodicity of solutions of periodic dynamical systems.

Bochner's criterion can also be expressed in terms of the relative compactness of the set {u(• + τ ); τ ∈ R} in a suitable set of continuous functions. A periodic function is a special case of almost periodic function. A direct consequence of [START_REF] Haraux | Sur les trajectoires compactes de systèmes dynamiques autonomes[END_REF]Proposition 2] given by Haraux characterizes a periodic function in terms of the Bochner's criterion. This characterization is the following: u : R → X is continuous is periodic if and if only if the set {u(• + τ ); τ ∈ R} is compact. In Section 5, By using our improvement of Bochner's criterion, we give an extension of the Haraux's characterization of periodic functions. We will also give a result on asymptotically periodic functions of the type of Haraux result described above. Then we apply these results to study the periodicity of solutions of autonomous dynamical systems.

Notation

Let us now give some notations, definitions and properties which will be used. Throughout this section (X, d) is a complete metric space. R, Z and N stand respectively for the real numbers, the integers and the natural integers. We denote by R + := {t ∈ R; t ≥ 0}. Let E be a topological space. We denote by C(E, X) the space of all continuous functions from E into X. When J = R or J = R + , we denote by BC(J, X) the space of all bounded and continuous functions from J into X equipped with the sup-distance, denoted by

d ∞ (u, v) := sup t∈R d(u(t), v(t)) when J = R and d ∞,+ (u, v) := sup t≥0 d(u(t), v(t))) when J = R + for u, v ∈ BC(J, X). The metric spaces (BC(R, X), d ∞ ) and (BC(R + , X), d ∞,+ )) are complete.
We now give some definitions and properties on almost periodic, asymptotically almost periodic functions with values in a given complete metric space.

A subset D of R (respectively of R + ) is said to be relatively dense if there exists > 0 such that D ∩ [α, α + ] = ∅ for all α ∈ R (respectively α ≥ 0). A continuous function u : R → X is said to be almost periodic (in the sense of Bohr) if for each ε > 0, the set of ε-almost periods:

P(u, ε) = τ ∈ R ; sup t∈R d(u(t + τ ), u(t)) ≤ ε is relatively dense in R.
An almost periodic function u has its range u(R) relatively compact, that is its closure denoted by cl (u(R)) is a compact set of (X, d). We denote the space of all such functions by AP (R, X). It is a closed metric subspace of (BC(R, X), d ∞ ). An almost periodic function u is uniformly recurrent, that is there exists a sequence of real numbers (τ n ) n such that lim n→+∞ sup t∈R d(u(t + τ n ), u(t)) = 0 and lim n→+∞ τ n = +∞. To see that consider the Bohr's definition of u ∈ AP (R, X), then the set of 1 n -almost periods satisfies P(u, 1 n )∩[n, +∞) = ∅, for each integer n > 0. A useful characterization of almost periodic functions was given by Bochner. The Bochner's criterion which may be found in [START_REF] Levitan | Almost periodic functions and differential equations,Translated from the Russian[END_REF]Bochner's theorem,p. 4] in the context of metric spaces. Before to cite this criterion, we need to introduce the translation mapping of a function of BC(R, X). For τ ∈ R and u ∈ BC(R, X), we define the translation mapping T τ u ∈ BC(R, X) by T τ u(t) = u(t + τ ) for t ∈ R.

Theorem 2.1 (Bochner's criterion). For u ∈ BC(R, X), the following statements are equivalent.

i) u ∈ AP (R, X). ii) The set {T τ u; τ ∈ R} is relatively compact in (BC(R, X), d ∞ ).
Haraux gave a generalization of Bochner' criterion the called a simple almost-periodicity criterion [9, Theorem 1] which is useful for periodic dynamical systems. Theorem 2.2 (Haraux's criterion). Let D be a relatively dense subset of R. The following statements are equivalent for u ∈ BC(R, X).

i) u ∈ AP (R, X). ii) The set {T τ u; τ ∈ D} is relatively compact in (BC(R, X), d ∞ ).
Periodic functions, which are a special case of almost periodic functions, are also characterized in terms of Bochner's criterion. This criterion is a direct consequence of a result of Haraux.

Theorem 2.3. [8, Consequence of Proposition 2] The following statements are equivalent for u ∈ BC(R, X). i) u is periodic.

ii) The set {T τ u; τ ∈ R} is a compact set of (BC(R, X), d ∞ ).

For some preliminary results on almost periodic functions with values in a given complete metric space, we refer to the book of Levitan-Zhikov [START_REF] Levitan | Almost periodic functions and differential equations,Translated from the Russian[END_REF] and in the special case of Banach spaces to the book of Amerio-Prouse [START_REF] Amerio | Almost periodic functions and functional equations[END_REF].

The notion of asymptotic almost periodicity was first introduced by Fréchet [START_REF] Fréchet | Les fonctions asymptotiquement presque-périodiques continues[END_REF] in 1941 in the case where X = C. A continuous function u : R + → X is said to be asymptotically almost periodic if there exists v ∈ AP (R, X) such that lim

d(u(t), v(t)) = 0.
An asymptotic almost periodic function u has its range u(R + ) relatively compact. We denote the space of all such functions by AAP (R + , X). It is a closed metric subspace of (BC(R + , X), d ∞,+ ). An asymptotic almost periodicity function u : R + → X is characterized by u ∈ AP P (R + , X) if and only if u ∈ C(R + , X) and for each ε > 0, there exists M ≥ 0 such that the τ ≥ 0 ; sup To prove this characterization, Ruess and Summers use results from the paper [START_REF] Ruess | Compactness in spaces of vector valued continuous functions and asymptotic almost periodicity[END_REF] by the same authors. For τ ≥ 0 and u ∈ BC(R + , X), we define the translation mapping T + τ u ∈ BC(R + , X) by T + τ u(t) = u(t + τ ) for t ≥ 0. Theorem 2.4. [15, a part of Theorems 1.2 & 1.3] Let (X, d) be a complete metric space. For u ∈ BC(R + , X), the following statements are equivalent.

i) u ∈ AAP (R + , X).
ii) The set {T + τ u; τ ≥ 0} is relatively compact in (BC(R + , X), d ∞,+ ). For some preliminary results on asymptotically almost periodic functions, we refer to the book of Yoshizawa [START_REF] Yoshizawa | Stability theory and the existence of periodic solutions and almost periodic solutions[END_REF] in the case where X is a finite dimensional space, to the book of Zaidman [START_REF] Zaidman | Almost-periodic functions in abstract spaces[END_REF] where X is a Babach space and to Ruess and Summers [START_REF] Ruess | Asymptotic almost periodicity and motions of semigroups of operators[END_REF][START_REF] Ruess | Minimal sets of almost periodic motions Math[END_REF][START_REF] Ruess | Compactness in spaces of vector valued continuous functions and asymptotic almost periodicity[END_REF] in the general case: X is a complete metric space.

An improvement of Bochner's criterion

An almost periodic function is characterized by the Bochner's criterion, recalled in Section 2. Our main result is an extension of Bochner's criterion. Then we deduce new proofs which are direct and simpler on known results on the solutions of autonomous dynamic systems. Before to state our extension of Bochner's criterion, we need to introduce the restriction operator R :

BC(R, X) → BC(R + , X) defined by R(u)(t) := u(t) for t ≥ 0 and u ∈ BC(R, X). Theorem 3.1. Let (X, d) be a complete metric space. For u ∈ BC(R, X) the following statements are equivalent. i) u ∈ AP (R, X). ii) The set {T τ u; τ ≥ 0} is relatively compact in (BC(R, X), d ∞ ).
iii) The set {R(T τ u); τ ∈ R} is relatively compact in (BC(R + , X), d ∞,+ ).

In our results, the compactness and the relative compactness of a set often intervene. To prove them, we will often use the following result whose proof is obvious. Recall that a set A of a metric space (E, d) is relatively compact if its closure denoted by cl (A) is a compact set of (E, d).

Lemma 3.2. Let E be a set, (G 1 , d 1 ) and (G 2 , d 2 ) be two metric spaces. Let u : E → G 1 and v : E → G 2 be two functions. Assume there exists M > 0 such that ∀x 1 , x 2 ∈ E, d 1 (u(x 1 ), u(x 2 )) ≤ M d 2 (v(x 1 ), v(x 2 )).
Then the following statements hold.

i) If the metric space (G 1 , d 1 ) is complete and v(E) is relatively compact in (G 2 , d 2 ), then u(E) is relatively compact in (G 1 , d 1 ). ii) If v(E) is a compact set of (G 2 , d 2 ), then u(E) is a compact set of (G 1 , d 1 ). Proof of Theorem 3.1. i) =⇒ iii).
It is obvious by using the Bochner's criterion and the continuity of the restriction operator R.

iii) =⇒ ii). The set u(R) = {R(T τ u)(0); τ ∈ R} is relatively compact in X as the range of {R(T τ u); τ ∈ R} by the continuous evaluation map at 0 from BC(R + , X) into X. By assumption,

H := cl ({R(T t u) ; t ∈ R}) is a compact set of (BC(R + , X), d ∞,+ ).
For all τ ≥ 0, we define φ τ : H → X by φ τ (h) = h(τ ). The functions φ τ are 1-Lipschitz continuous and for each t ∈ R, the set {φ τ (R(T t u)) = u(τ + t) ; τ ≥ 0} is included in the relatively compact set u(R). By density of {R(T t u) ; t ∈ R} in H and the continuity of φ τ , it follows that {φ τ (h) ; τ ≥ 0} is relatively compact in X for each h ∈ H. According to Arzelà-Ascoli's theorem [11, Theorem 3.1, p. 57], the set {φ τ ; τ ≥ 0} is relatively compact in C(H, X) equipped with the sup-norm denoted by d C . From the density of {R(T t u) ; t ∈ R} in H and the continuity of φ τ , we deduce that for τ 1 and τ 2 ≥ 0, sup

h∈H d(φ τ 1 (h), φ τ 2 (h)) = sup t∈R d (φ τ 1 (R(T t u)), φ τ 2 (R(T t u))) = sup t∈R d (u(τ 1 + t), u(τ 2 + t)) = sup t∈R d (T τ 1 u(t), T τ 2 u(t)), then d C (φ τ 1 , φ τ 2 ) = d ∞ (T τ 1 u, T τ 2 u). From Lemma 3.2, it follows that {T τ u; τ ≥ 0} is relatively compact in the complete metric space (BC(R, X), d ∞ ) since {φ τ ; τ ≥ 0} is also one in (C(H, X), d C ). ii) =⇒ i). For τ 1 , τ 2 ≥ 0, d ∞ (T τ 1 u, T τ 2 u) := sup t∈R d(u(τ 1 + t), u(τ 2 + t)). Replacing t by t -τ 1 -τ 2 in the upper bound, we get d ∞ (T τ 1 u, T τ 2 u) = d ∞ (T -τ 1 u, T -τ 2 u). Then the set {T τ u; τ ≤ 0} = {T -τ u; τ ≥ 0} is relatively compact in BC(R, X) since {T τ u; τ ≥ 0} is also one. Therefore the set {T τ u; τ ∈ R} is relatively compact in BC(R, X) as the union of two relatively compact sets in BC(R, X). According to Bochner's criterion, u ∈ AP (R, X).
The connection between the almost periodicity of a solution of a dynamical system and its stability is well known (see the monograph by Nemytskii & Stepanov [13,Ch. 5]. This weakened form of Bochner's criterion: Theorem 3.1 makes it possible to obtain direct and simpler proofs on these questions. Let us start by recalling some definitions on dynamical systems.

A dynamical system or nonlinear semigroup on a complete metric space (X, d) is a one parameter family (S(t)) t≥0 of maps from X into itself such that i) S(t) ∈ C(X, X) for all t ≥ 0, ii) S(0)x = x for all x ∈ X, iii) S(t + s) = S(t) • S(s) for all s, t ≥ 0 and iv) the mapping S(•)x ∈ C([0, +∞), X) for all x ∈ X.

For each x ∈ X, the positive trajectory of x is the map S(•)x : R + → X. A function u : R → X is called a complete trajectory if we have u(t + τ ) = S(τ )u(t), for all t ∈ R and τ ≥ 0.

We will need a notion of Lagrange-type stability to ensure that a solution with a relatively compact range is almost periodic. Recall that (S(t)) t≥0 is equicontinuous on a compact set K of X, if forall ε > 0, there exists δ > 0, such that Corollary 3.3. Let (S(t)) t≥0 be a dynamical system on a complete metric space (X, d) and u be a complete trajectory such that u(R) is relatively compact. Then u is almost periodic if and only if (S(t)) t≥0 is equicontinuous on cl (u(R)) the closure of u(R).

∀x 1 , x 2 ∈ K, d(x 1 , x 2 ) ≤ δ =⇒ sup t≥0 d(x 1 , x 2 ) ≤ ε.
Proof. Let us denote the compact set K := cl (u(R)). It follows by density of u(R) in K and the continuity of S(t), that {S(t)x; t ≥ 0} ⊂ K for each x ∈ K. According to Arzelà-Ascoli's theorem, (S(t)) t≥0 is equicontinuous on K if and only if (S(t)) t≥0 is relatively compact in C(K, X). From Theorem 3.1, we have u ∈ AP (R, X) if and only if {T τ u; τ ≥ 0} is relatively compact in BC(R, X). Then it remains to prove that (S(t)) t≥0 is relatively compact in C(K, X) equipped with the sup-norm if and only if {T τ u; τ ≥ 0} is relatively compact in (BC(R, X), d ∞ ). This results from the following equalities, for τ 1 and τ 2 ≥ 0, sup

t∈R d (T τ 1 u(t), T τ 2 u(t)) = sup t∈R d (S(τ 1 )u(t), S(τ 2 )u(t)) = sup x∈K d (S(τ 1 )x, S(τ 2 )x)
and Lemma 3.2.

Remark 3.4. a) The condition of equicontinuity required by Corollary 3.3 is satisfied by a bounded dynamical system : d (S(t)x 1 , S(t)x 2 ) ≤ M d (x 1 , x 2 ) for some M ≥ 1 and in particular for a C 0 semigroup of contractions. In this case, the almost periodicity of a complete trajectory u having a relatively compact range results from Corollary 3.3. We can also obtain this result with the implication iii) =⇒ i) of Theorem 3.1 and the inequality sup

t≥0 d(R(T τ 1 u)(t), R(T τ 2 u)(t)) = sup t≥0 d (S(t)u(τ 1 ), S(t)u(τ 2 ) ≤ M d(u(τ 1 ), u(τ 2 )) for τ 1 , τ 2 ∈ R.
b) For a bounded C 0 -semigroup (S(t)) t≥0 , the main result of Zaidman [START_REF] Zaidman | On relatively compact trajectories of semigroups of class C 0[END_REF] asserts that a positive trajectory u with relatively compact range satisfies a condition called the generalized normality property in Bochner's sense, without concluding that u is almost periodic. This condition is nothing but hypothesis iii) of Theorem 3.1, so u is almost periodic.

Using Theorem 2.4, we give a new proof which is direct and simpler of the following result which can be found in [15, Theorem 2.2, p. 149].

Corollary 3.5. Let (S(t)) t≥0 be a dynamical system on a complete metric space (X, d) and u be a positive trajectory such that u(R + ) is relatively compact. Then u is asymptotically almost periodic if and only if (S(t)) t≥0 is equicontinuous on cl (u(R + )).

Proof. The proof is analogous to that of Corollary 3.3, using Corollary 2.4 instead of Theorem 3.1 and by replacing R by R + and AP (R, X) by AAP (R + , X).

An improvement of Haraux's criterion

Haraux gave a generalization of Bochner'scriterion [9, Theorem 1], the called a simple almost-periodicity criterion which is useful for periodic dynamical systems. From our main result, Theorem 3.1, we deduce an extension of the Haraux's criterion, recalled in Section 2. In the same spirit, we extend the well-known characterization of asymptotically almost periodic functions. To end this section, we give an exemple of application on a periodic dynamical system.

We give an extension of the Haraux's criterion (see Theorem 2.2). Recall that we denote by R the restriction operator R : BC(R, X) → BC(R + , X) defined by R(u)(t) := u(t) for t ≥ 0 and u ∈ BC(R, X).

Corollary 4.1. Let (X, d) be a complete metric space. For u ∈ BC(R, X) the following statements are equivalent.

i) u ∈ AP (R, X).
ii) The set {T τ u; τ ∈ D} is relatively compact in (BC(R, X), d ∞ ) where D be a relatively dense subset of R + .

iii) The set {R(T τ u); τ ∈ D} is relatively compact in (BC(R + , X), d ∞,+ ) where D be a relatively dense subset of R. iii) =⇒ ii). To establish this implication, using Theorem 3.1, it suffices to show that assertion iii) implies that {R(T τ u); τ ∈ R} is relatively compact in BC(R + , X). The proof of this last implication is a slight adaptation of those of those of the Haraux's criterion given in [9, Theorem 1]. A similar proof will be detailed in the following result as there will be technical issues. To demonstrate that {R(T τ u); τ ∈ R} is relatively compact, it suffices in the proof of ii) =⇒ i) of Corollary 4.3 to take = 0 and replace D) is a relatively dense subset of R. According to Haraux's criterion, we have u ∈ AP (R, X).

{T + τ u; τ ∈ D} by {R(T τ u); τ ∈ D}. ii) =⇒ i). For τ 1 , τ 2 ≥ 0, d ∞ (T τ 1 u, T τ 2 u) = sup t∈R d(u(τ 1 + t), u(τ 2 + t)). Replacing t by t -τ 1 -τ 2 in the upper bound, we get d ∞ (T τ 1 u, T τ 2 u) = d ∞ (T -τ 1 u, T -τ 2 u). Then the set {T τ u; τ ∈ -D} = {T -τ u; τ ∈ D} is relatively compact in BC(R, X) since {T τ u; τ ∈ D} is also one. Therefore the set {T τ u; τ ∈ D ∪ (-D)} is relatively compact in BC(R, X). Moreover D ∪ (-
We extend Theorem 2.4, the well-known characterization of asymptotically almost periodic functions. For τ ∈ R + and u ∈ BC(R + , X), we define the translation mapping T + τ u ∈ BC(R + , X) by T + τ u(t) = u(t + τ ) for t ≥ 0. Corollary 4.3. Let (X, d) be a complete metric space and let D be a relatively dense subset of R + . For u ∈ BC(R + , X) the following statements are equivalent.

i) u ∈ AAP (R + , X).
ii) The set {T + τ u; τ ∈ D} is relatively compact in (BC(R + , X), d ∞,+ ). Remark 4.4. To establish implication ii) =⇒ i), by using Theorem 2.4, it suffices to prove that assertion ii) implies that {T + τ u; τ ≥ 0} is relatively compact in (BC(R + , X), d ∞,+ ). The proof of this last implication is an adaptation of those of the Haraux's criterion. But contrary to the proof of implication iii) =⇒ ii) in Corollary 4.1, there are technical issues. These technical difficulties come from the fact that when D is a relatively dense subset in R + , the sets D and [t -, t] can be disjoint for some 0 ≤ t ≤ . For this reason we give the complete proof of this implication.

Proof of Corollary 4.3. i) =⇒ ii). It is a consequence of Theorem 2.4.

ii) =⇒ i). We will prove that assumption ii) implies {T + τ u; τ ≥ 0} is relatively compact in BC(R + , X), then we conclude by using Theorem 2.4. The subset D being relatively dense in R + , there exists > 0 such that D ∩ [α, α + ] = ∅ for all α ≥ 0.

• We prove that u is uniformly continuous on [ , +∞). Let us fix ε > 0. By assumption the set {T + τ u; τ ∈ D} is in particular relatively compact in C([0, 2 ], X), then it is uniformly equicontinuous on [0, 2 ], that is there exists δ > 0 such that • We prove that {T + τ u; τ ≥ } is relatively compact in BC(R + , X) . Let (t n ) n be a sequence of real numbers such that t n ≥ . We have D ∩ [t n -, t n ] = ∅ for each n ∈ N, since t n -≥ 0, then there exist τ n ∈ D and σ n ∈ [0, l] such that t n = τ n + σ n . By compactness of the sequences (σ n ) n in [0, ] and (T + τn u) n in BC(R + , X), it follows that lim n→+∞ σ n = σ and lim

s 1 , s 2 ∈ [0, 2l], |s 1 -s 2 | ≤ δ =⇒ sup τ ∈D d(u(s 1 + τ ), u(s 2 + τ )) ≤ ε. ( 4
n→+∞ sup t≥0 d(u(t + τ n ), v(t)) = 0 (up to a subsequence). From the following inequality sup t≥0 d(u(t n + t), v(σ + t)) ≤ sup t≥0 d(u(τ n + σ n + t), u(τ n + σ + t) + sup t≥0 d(u(τ n + t), v(t))
and the uniform continuity of u, we deduce that lim

n→+∞ sup t≥0 d(u(t n + t), v(σ + t)) = 0. Then {T + τ u; τ ≥ } is relatively compact in BC(R + , X).
• We prove that u ∈ AAP (R + , X). The function u is uniformly continuous on R + , since u is continuous on [0, ] and uniformly continuous on [ , +∞). Then the map û : R + → BC(R + , X) defined by û(τ ) = T + τ u for τ ≥ 0 is continuous, consequently the set {T + τ u; 0 ≤ τ ≤ } is relatively compact in BC(R + , X). The set {T + τ u; τ ≥ 0} is relatively compact in BC(R + , X) as the union of two relatively compact sets. According to Theorem 2.4, u ∈ AAP (R, X).

Using corollaries 4.1 and 4.3, we give a proof which is direct and simpler of the following result which can be found in [START_REF] Haraux | Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes[END_REF][START_REF] Haraux | A simple almost-periodicity criterion and applications[END_REF][START_REF] Haraux | Systèmes dynamiques dissipatifs et applications[END_REF]. Before we recall some definitions on process.

A process on a complete metric space (X, d) according to Dafermos [START_REF] Dafermos | Almost periodic processes and almost periodic solutions of evolution equations. Dynamical systems[END_REF] is a two parameter family U (t, τ ) of maps from X into itself defined for (t, τ ) ∈ R × R + and such that i) U (t, 0)x = x for all (t, x) ∈ R × X, ii) U (t, σ + τ ) = U (t + σ, τ ) • U (t, σ) for all (t, σ, τ ) ∈ R×R + ×R + and iii) the mapping U (t, •)x ∈ C([0, +∞), X) for all (t, x) ∈ R×X.

For each x ∈ X, the positive trajectory starting of x is the map U (0, •)x : R + → X. A function u : R → X is called a complete trajectory if we have u(t

+ τ ) = U (t, τ )u(t) for all (t, τ ) ∈ R × R + . A process U is said ω-periodic (ω > 0) if U (t + ω, τ ) = U (t, τ ) for all (t, τ ) ∈ R × R + . A process U is said bounded if we have for some M ≥ 1 for all (τ, x 1 , x 2 ) ∈ R + × X × X d (U (0, τ )x 1 , U (0, τ )x 2 ) ≤ M d (x 1 , x 2 ).
Corollary 4.5. [START_REF] Haraux | Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes[END_REF][START_REF] Haraux | A simple almost-periodicity criterion and applications[END_REF], [10, Théorème 6.4.6, p. 84] Let U be a ω-periodic process on a complete metric space (X, d). If U is bounded, then the following statements hold. i) If u is a complete trajectory of U such that u(-ωN) is relatively compact, then u is almost periodic.

ii) If u is a positive trajectory of U such that u(ωN) is relatively compact, then u is asymptotically almost periodic.

Proof. i) The process U is ω-periodic, then we have u(nω) = U (-mω, (n+m)ω)u(-mω) = U (0, (n + m)ω)u(-mω) for all n, m ∈ N. From the boundedness assumption on U , we deduce that d (u(nω), u(mω)) ≤ M d (u(-mω), u(-nω)), then u(ωN) is relatively compact since u(-ωN) is also one, therefore u(ωZ) is relatively compact. From assumptions on the process U , it follows that for all n, m ∈ Z,

sup τ ≥0 d (u(τ + nω), u(τ + mω)) ≤ M d (u(nω), u(mω)) . (4.2)
From Lemma 3.2, {R(T nω u); n ∈ Z} is relatively compact in (BC(R + , X), d ∞,+ ) since u(ωZ) is also one in (X, d). We conclude with Corollary 4.1 by setting D = ωZ.

ii) For all n, m ∈ N, (4.2) holds on the positive trajectory u, then from Lemma 3.2, {T + nω u; n ∈ N} is relatively compact in (BC(R + , X), d ∞,+ ) since u(ωN) is also one in (X, d). We conclude with Corollary 4.3 by setting D = ωN.

Bochner's criterion in the periodic case

Periodic functions are a special case of almost periodic functions. Haraux gave a characterization of periodic functions in terms of Bochner's criterion which is recalled in Section 2. This criterion is a direct consequence of [START_REF] Haraux | Sur les trajectoires compactes de systèmes dynamiques autonomes[END_REF]Proposition 2]. Haraux established a general result [8, Théorème 1] implying as a special case a characterization of periodic functions and the fact that any compact trajectory of a one-parameter continuous group is automatically periodic.

In this section, we give an extension of this characterization of periodic functions in the spirit of the main result of this article. We also treat the asymptotically periodic case. Then we apply these results to study the periodicity of solutions of dynamical systems.

Recall that we denote by R the restriction operator R : BC(R, X) → BC(R + , X) defined by R(u)(t) := u(t) for t ≥ 0 and u ∈ BC(R, X).

Corollary 5.1. Let (X, d) be a complete metric space. For u ∈ BC(R, X) the following statements are equivalent.

i) The function u is ω-periodic (ω > 0).

ii) The set {T τ u; τ ≥ 0} is a compact set of (BC(R, X), d ∞ ).

iii) The set {R(T τ u); τ ∈ R} is a compact set of (BC(R + , X), d ∞,+ ).

Proof. i) =⇒ ii).
From assumption, it follows that the function τ → T τ u from R into BC(R, X) is continuous and ω-periodic. Then the {T τ u; τ ≥ 0} = {T τ u; 0 ≤ τ ≤ ω} is a compact set of (BC(R, X), d ∞ ) as the range of a compact set by a continuous map.

ii) =⇒ i). For τ 1 , τ 2 ∈ R, d ∞ (T τ 1 u, T τ 2 u) := sup t∈R d(u(τ 1 + t), u(τ 2 + t)), we get d ∞ (T τ 1 u, T τ 2 u) = d ∞ (T -τ 1 u, T -τ 2 u). Then the set {T τ u; τ ≤ 0} = {T -τ u; τ ≥ 0} is compact in BC(R, X) since {T τ u; τ ≥ 0}
is also one. Therefore the set {T τ u; τ ∈ R} is a compact set of BC(R, X) as the union of two compact sets in BC(R, X). According to Theorem 2.3, u is periodic. i) =⇒ iii). It is obvious by using Theorem 2.3 and the continuity of the restriction operator R.

iii) =⇒ i). By using Theorem 2.3, we have to prove that K := {T τ u; τ ∈ R} is a compact set of (BC(R, X), d ∞ ). As consequence of Theorem 3.1 and Bohner's criterion, the set K is relatively compact in (BC(R, X), d ∞ ) and the function u is almost periodic.

It remains to prove that K is closed in (BC(R, X), d ∞ ). Let (τ n ) n be a sequence of real numbers such that lim n→+∞ d ∞ (T τn u, v) = 0. Let us prove that v = T τ u for some τ ∈ R. By continuity of the operator R, we have lim

n→+∞ d ∞,+ (R(T τn u), R(v)) = 0. By assumption, the set {R(T τ u); τ ∈ R} is in particular closed in (BC(R + , X), d ∞,+ ), then R(v) = R(T τ u) for some τ ∈ R, that is ∀t ≥ 0, v(t) = T τ u(t).
(5.1) We have to prove that (5.1) holds on the whole real line. The function T τ u is almost periodic as translation of an almost periodic function and v is also one as uniform limit on R of almost periodic functions. Let us denote by φ : R → R the function defined by φ(t) := d(T τ u(t), v(t)). The function φ is almost periodic [START_REF] Levitan | Almost periodic functions and differential equations,Translated from the Russian[END_REF]Property 4,p. 3 & 7,p.6]. An almost periodic function is uniformly recurrent, then there exists a sequence of real numbers such that lim n→+∞ τ n = +∞ and lim n→+∞ φ(t + τ n ) = φ(t) for all t ∈ R. From (5.1), it follows φ(t) = 0 for all t ≥ 0, so we deduce that φ(t) = lim n→+∞ φ(t + τ n ) = 0 for all t ∈ R.

Then v(t) = T τ u(t) for all t ∈ R. This ends the proof.

According Theorem 2.4, if the set {T + τ u; τ ≥ 0} is relatively compact in BC(R + , X), then the function u is asymptotically almost periodic. We now give an answer to the question what can be said about the function u when {T + τ u; τ ≥ 0} is a compact set of BC(R + , X). For u ∈ BC(R + , X), we say that u is ω-periodic (ω > 0) on [t 0 , +∞) for some t 0 ≥ 0 if u(t + ω) = u(t) for all t ≥ t 0 .

Corollary 5.2. Let (X, d) be a complete metric space. For u ∈ BC(R + , X) the following statements are equivalent. i) There exists t 0 ≥ 0 such that u is ω-periodic on [t 0 , +∞).

ii) The set {T + τ u; τ ≥ 0} is a compact set of (BC(R + , X), d ∞,+ ). Remark 5.3. Let u be a function which satisfies condition i) of Corollary 5.2.

i) Let us denote by v ∈ C(R, X) the ω-periodic function satisfying u(t) = v(t) for t ≥ t 0 . A such function v exists and is unique, v is defined by v(t) = u(t -[ t-t 0 ω ]ω) where [ t-t 0 ω ] denotes the integer part of t-t 0 ω . ii) The function u is a special case of asymptotic almost periodic function where the almost periodic function v is periodic and d(u(t), v(t)) = 0 for t ≥ t 0 .

Proof of Corollary 5.2. i) =⇒ ii). Let us denote by v the function defined in Remark 5.3. By Corollary 5.1 and the periodicity of v, we have

{R(T τ v); τ ≥ t 0 } = {R(T τ v); τ ∈ R} is a compact set of (BC(R + , X), d ∞,+ ).
First, we have T + τ u = R(T τ v) for τ ≥ t 0 , then {T + τ u; τ ≥ t 0 } is a compact set. Second, the function u is uniformly continuous on R + , then the function from R + to BC(R + , X) defined by τ → T + τ u is continuous. Then the set {T + τ u; 0 ≤ τ ≤ t 0 } is compact.

Therefore the set {T + τ u; τ ≥ 0} is a compact set of BC(R + , X) as the union of two compact set.

ii) =⇒ i). As consequence of Theorem 2.4, the function u is asymptotically almost periodic, that is lim t→∞ d(u(t), v(t)) = 0 for some v ∈ AP (R, X). An almost periodic function is uniformly recurrent, then there exists a sequence of real numbers (t n ) n such that lim n→+∞ t n = +∞ and lim n→+∞ v(t + t n ) = v(t) for all t ∈ R. We deduce that ∀t ∈ R, lim n→+∞ u(t + t n ) = v(t).

(5.2)

First we prove that v is periodic. For t ∈ R, τ 1 , τ 2 ≥ 0, we have for n enough large d(u(t + t n + τ 1 ), u(t + t n + τ 2 )) ≤ sup s≥0 d(u(s + τ 1 ), u(s + τ 2 )). From (5.2), it follows that sup t∈R d(v(t + τ 1 ), v(t + τ 2 )) ≤ sup s≥0 d(u(s + τ 1 ), u(s + τ 2 )) for each τ 1 and τ 2 ≥ 0. According to Lemma 3.2, {T τ v ; τ ≥ 0} is a compact set of (BC(R, X), d ∞ ) since {T + τ u ; τ ≥ 0} is also one in (BC(R + , X), d ∞,+ ). As consequence of Corollary 5.1, the function v is periodic.

Second we prove that: ∃t 0 ≥ 0 such that ∀t ≥ 0, v(t) = u(t + t 0 ). By compactness of {T + τ u; τ ≥ 0}, there exists a subsequence (T

+ t φ(n ) u) n such that lim n→+∞ d ∞,+ (T + t φ(n ) u, T + t 0 u) = 0 for some t 0 ≥ 0. From (5.2) we deduce that R(v) = T + t 0 u, that is v(t) = u(t + t 0 ) for all t ≥ 0.
Then u(t) = v(t -t 0 ) for each t ≥ t 0 where the function v(• -t 0 ) is periodic on R. Now we give an example of application on dynamical systems of Corollary of 5.1 and Corollary of 5.2. For the definition of a dynamical system, see above Corollary 3.3 in Section 3.

Corollary 5.4. Let (S(t)) t≥0 be a dynamical system on a complete metric space (X, d).

i) If u is a positive trajectory, then u is periodic on [t 0 , +∞) for some t 0 ≥ 0 if and only if u(R + ) is a compact set and (S(t)) t≥0 is equicontinuous on u(R + ).

ii) If u is a complete trajectory, then u is periodic if and only if u(R) is a compact set and (S(t)) t≥0 is equicontinuous on u(R).

iii) There exists a complete trajectory which is periodic if and only if there exists a positive trajectory u such that u(R + ) is a compact set and (S(t)) t≥0 is equicontinuous on u(R + ). Remark 5.5. Thus under the assumption of equicontinuity, a complete trajectory of a dynamical system with a compact range is necessarily periodic, although there are almost periodic functions with a compact range, which are not periodic. An example of such function is given by Haraux in [START_REF] Haraux | Sur les trajectoires compactes de systèmes dynamiques autonomes[END_REF].

Proof of Corollary 5.4. i) Remark that if u is a positive trajectory which is periodic on [t 0 , +∞) for some t 0 ≥ 0, then first u(R + ) is compact and second u ∈ AAP (R + , X) (see Remark 5.3). As consequence of Corollary 3.5, the set (S(t)) t≥0 is equicontinuous on u(R + ). Reciprocally assume the positive trajectory u is such that (S(t)) t≥0 is equicontinuous on the compact set u(R + ). It remains to prove that the positive trajectory u is periodic on [t 0 , +∞) for some t 0 ≥ 0. For each x ∈ u(R + ), the map S(•)x is continuous and satisfies S(t)x ∈ u(R + ) for each t ≥ 0. Then the map S(•)x is bounded and continuous , so the map Φ : u(R + ) → BC(R + , X) with Φ(x) = S(•)x is well-defined. The continuity of Φ results of the equicontinuity of (S(t)) t≥0 on u(R + ). Then the set Φ(u(R + )) = {Φ(u(τ )) ; τ ≥ 0} is a compact of BC(R + , X). Moreover Φ(u(τ ))(t) = S(t)u(τ ) = u(t + τ ) for t and τ ≥ 0, then Φ(u(τ )) = T + τ u, so {T + τ u ; τ ≥ 0} is a compact set of BC(R + , X). According to Corollary 5.2, the function u is periodic on [t 0 , +∞) for some t 0 ≥ 0.

ii) The proof of ii) is similar to that of i) by using Corollary 3.3 instead of 3.5, Corollary 5.1 instead of Corollary 5.2 and by replacing the map Φ : u(R + ) → BC(R + , X) with Φ(x) = S(•)x by the map Φ : u(R) → BC(R + , X). This permits to prove that the set {Φ(u(τ )) = R(T τ u); τ ∈ R} is a compact set of (BC(R + , X), d ∞,+ ).

iii) If v is a complete trajectory which is periodic, then v(R) is compact and according to ii), (S(t)) t≥0 is equicontinuous on v(R). So the restriction u of v on R + is a positive trajectory such that u(R + ) = v(R + ) = v(R) since v is periodic, then (S(t)) t≥0 is equicontinuous on the compact set u(R + ). Reciprocally, assume that u is a positive trajectory such that (S(t)) t≥0 is equicontinuous on the compact set u(R + ). According to i), u is ω-periodic on [t 0 , +∞) for some t 0 ≥ 0. Let us denote by v the function defined in Remark 5.3. For t ≥ s, there exists n 0 ∈ N such that s + n 0 ω ≥ t 0 . The function v is ω-periodic and u is a positive trajectory satisfying u(τ

) = v(τ ) for τ ≥ t 0 , then v(t) = v(t + n 0 ω) = u(t + n 0 ω) = T (t -s)u(s + n 0 ω) = T (t -s)v(s + n 0 ω) = T (t -s)v(s)
for t ∈ R and n enough large. Then v is a periodic complete trajectory.

Remark 5.6. Under i) of Corollary 5.4, one can have t 0 > 0, that is the positive trajectory u is not the restriction of a periodic complete trajectory. For example, consider the bounded dynamical system (S(t)) t≥0 on L 1 (0, 1) defined by (S(t)x)(s) =    x(s -t) if t < s < 1 0 if 0 < s < t for x ∈ L 1 (0, 1) and 0 < t < 1. For t ≥ 1, we set S(t) = 0. Then all positive trajectories have a compact range and the alone complete trajectory is the null function. Thus all positive trajectories are not the restriction of a periodic complete trajectory except the null function.

Not all dynamical systems have this pathology, some systems are such that if two positive trajectories have the same value at the same time, then they are equal. If we consider such systems, we get more refined results from Corollary 5.4.

A dynamical system (S(t)) t≥0 has the backward uniqueness property if any two positive trajectories having the same value at t = t 0 ≥ 0 coincide for any other t ≥ 0. This property is equivalent to S(t) ∈ C(X, X) is injective for each t ≥ 0. We say that a positive trajectory u is extendable to a periodic complete trajectory, if there exists a periodic complete trajectory such that its restriction on R + is u.

Corollary 5.7. Let (S(t)) t≥0 be a dynamical system on a complete metric space (X, d). Assume that (S(t)) t≥0 has the backward uniqueness property. i) If u is a positive trajectory, then u is periodic on R + if and only if u(R + ) is a compact set and (S(t)) t≥0 is equicontinuous on u(R + ). In this case the positive trajectory u is extendable to a periodic complete trajectory v.

ii) If v is a complete trajectory, then v is periodic if and only if v(R + ) is a compact set and (S(t)) t≥0 is equicontinuous on v(R + ).

Proof. i) The direct implication results of i) of Corollary 5.4. For the reciprocal implication we use i) of Corollary 5.4. Then the positive trajectory u is ω-periodic on [t 0 , +∞) for some t 0 ≥ 0. Let us denote by v ∈ C(R, X) the ω-periodic function satisfying u(t) = v(t) for t ≥ t 0 (see Remark 5.3). The restriction of v on R + and u are two positive trajectories having the same value at t = t 0 (t 0 ≥ 0). From the backward uniqueness property, we have u(t) = v(t) for t ≥ 0, then u is periodic on R + . By build, v is periodic and as in the proof of iii) of Corollary 5.4, we deduce that v is a complete trajectory.

ii) The direct implication results of ii) Corollary 5.4, since v(R + ) = v(R). For the reciprocal implication, we consider v a complete trajectory such that v(R + ) is a compact set and (S(t)) t≥0 is equicontinuous on v(R + ). Then the restriction u of the complete trajectory v on R + is a positive trajectory such that u(R + ) is compact and (S(t)) t≥0 is equicontinuous on u(R + ). According to i), u is ω-periodic on R + . Let us denote by w ∈ C(R, X) the ω-periodic function satisfying u(t) = w(t) for t ≥ 0. As in proof of iii) Corollary 5.4, we deduce that w is a complete trajectory. Fix T > 0. The two maps ṽ and w : R + → X defined by ṽ = v(•, -T ) and w = w(•, -T ) are two positive trajectories having the same value at t = T . From the backward uniqueness property, we have ṽ = w, that is v(t) = w(t) for t ≥ -T . Since T is arbitrary, then v(t) = w(t) for each t ∈ R where w is a periodic complete trajectory. This proves that v is a periodic complete trajectory.

  t→∞

  t≥M d(u(t + τ ), u(t)) ≤ ε is relatively dense in R + [15, Theorems 1.3]. In the context of metric spaces, Ruess and Summers give a characterization of asymptotically almost periodic functions in the spirit of Bochner's criterion.

Using Theorem 3 . 1 ,

 31 we give a new proof which is direct and simpler of the following result which can be found in [10, Theorem 4.3.2, p. 51] or partly in [12, Markov's theorem, p. 10].

Remark 4 . 2 .

 42 Our main result, Theorem 3.1 is obviously a particular case of Corollary 4.1. But to present our results, it was easier to start with Theorem 3.1. To prove Corollary 4.1, we use Haraux's criterion and Theorem 3.1. Proof of Corollary 4.1. i) =⇒ iii). It is a consequence of Theorem 3.1.

  .1) Let t 1 , t 2 be two real numbers such that t 1 , t 2 ≥ and |t 1 -t 2 | ≤ δ. We can assume without loss of generality that ≤ t 1 ≤ t 2 ≤ t 1 + . We have D ∩ [t 1 -, t 1 ] = ∅ since t 1 -≥ 0, then there exists τ ∈ D such that 0 ≤ t 1 -τ ≤ t 2 -τ ≤ 2l. Taking account (4.1), we deduce that d(u(t 1 ), u(t

2 )) = d(u((t 1 -τ ) + τ ), u((t 2 -τ ) + τ )) ≤ ε. Hence u is uniformly continuous on [ , +∞).


Mahmoud Shahrokhi
email: m.shahrokhi@uok.ac.ir

Alain Bernard
email: alain.bernard@ec-nantes.fr

A fuzzy accident risk analysis approach for a concurrent engineering platform

Keywords: Risk analysis, concurrent engineering, fuzzy logic, virtual reality, augmented reality

This paper uses the fuzzy theory basics to assess accident risk that may damage a target (e.g., humans, assets, or the environment). Converting a target's position to a membership degree in a fuzzy danger zone (FDZ) helps calculate risk indices. Using FDZs normalizes the effects of different kinds of hazards, similarly visualizes them, and distinguishes the impact of a threat on various types of targets. This paper presents a related mathematical formulation. The proposed approach evaluates the accident risks by simulating industrial activities in CAD, virtual reality, and augmented reality when using a concurrent engineering platform. This approach can calculate the risk index during a human task simulation and through real-time human interaction with a virtual machine during safety analysis and training. The results also are credible to activate alarm systems according to the operator's limbs place during work. The model provides a normalized and similar scale for various risks (e.g., biological, chemical, and physical hazards) and computes the effect of danger on different target types.

Introduction

Using the three digital prototypes in CAD, virtual reality, and augmented reality creates realistic visualizations, renderings, and animations of the complex systems. It plays an essential role in concurrent engineering platforms, where various disciplines use these models to predict industrial systems' behavior before constructing and employing them. The conventional risk analyses approaches are inconvenient for using these models to perform quantitative risk analyses. This research develops a fuzzy approach for doing risk analysis in a concurrent engineering context by using geometric shapes for facilitating collaboration and sharing risks information by visualizing the risk entities and preventing inconsistencies.

Background

Dangers are the potential of things or situations to cause harm, and "risk" is an indicator for evaluating their importance by aggregating the severity and the likelihood of their possible consequences [START_REF] Jo | Dynamic management of human error to reduce total risk[END_REF], [START_REF] Polet | Modelling border-line tolerated conditions of use (BTCU) and associated risks[END_REF]. A danger creates hazards (e.g., a toxic substance that makes a toxification hazard). A danger zone (DZ) is a hypothetical area where the threat can potentially cause harm [START_REF] Gallego | Lightning risk assessment using fuzzy logic[END_REF]. The hazard magnitude denotes the parameters to explain the potential of the hazard for harming the target. It is the value of physicals (e.g., speed and temperature), chemical (e.g., the amount and density of one or several hazardous materials), biological or conceptual parameters (e.g., the probability or risk of an accident). Figure 1 shows the hazard magnitude variation around a hypothetical hazard object indicated by a red star. The horizontal axis (x-axis )represents the locations around the danger source, and the vertical axis shows the intensity of the danger in its neighborhoods. The dashed line in this figure shows the hazard magnitude variation as a mathematical function of the one-dimensional geometric coordinates. In this figure, the traditional danger zone is shown with a red line and includes every point where the assumed intensity of the danger is more than an acceptable level (𝐻). Some examples of hazard amplitude are the intensity of hazardous radiation around a radioactive material, the high-temperature air around a fire source, the explosion blast wave, or the toxic gas concentration during a poisonous gas leak. It can even be an understanding of the concept of collision risk around a dangerous moving object.

In this way, a conventional DZ is a crisp (classic) set of points in the workplace, illustrated as follows:

𝑍 = {𝑥|𝐻(𝑥) > 𝐻} 𝑥 ∈ 𝑋 (1)

3

The literature review

This chapter introduces the basic concepts and a review of previous applications of 3D models in risk analysis.

Bernard and Hasan considered a danger zone as one of the workplace entities, characterized by its name and origin [START_REF] Bernard | Working Situation Model for Safety Integration during Design Phase[END_REF]. Hasan and et al. considered DZ one of the system's safety modeling entities described by its name, nature, size, and source [START_REF] Hasan | Integrating safety into the design process: elements and concepts relative to the working situation[END_REF]. Pouliquen and et al. use it for estimating the severity of danger, in real-time, in the virtual reality context for a sheet-pressing operation [START_REF] Pouliquen | Virtual hands and virtual reality multimodal platform to design safer industrial systems[END_REF]. In recent years, very little research has been done on risk analysis in 3D using the concept of a danger zone. Alba and et al. proposed a distributed real-time anti-collision approach that uses hazardous areas to create the ability to prevent collisions with other objects on the robot, based on accurate 3D simulation of a manufacturing cell [START_REF] Alba | A real time distributed approach to collision avoidance for industrial manipulators[END_REF]. Yi-Hao and Yet-Pole construct a three-dimensional fire risk analysis technique for common building fires by simulating the thermal radiation hazards [START_REF] Yi-Hao | Development of a 3D risk analysis technique for fire disaster and its application on accident simulation of a public site[END_REF]. Cao and et al. studied flame propagation behaviors in explosion venting and the quantitative relationship of the maximum flame front length, and they calculated the variation of the temperature field in explosion venting [START_REF] Yi-Hao | Development of a 3D risk analysis technique for fire disaster and its application on accident simulation of a public site[END_REF].

4

The proposed model

According to fuzzy set theory, not only can an object be inside or outside a set, but it can also have a degree of membership. By assigning membership between 0 and 1 to each point, Shahrokhi and Bernard introduce a "fuzzy space" to distribute risk around a hazard factor [START_REF] Shahrokhi | A fuzzy approach for definition of dangerous zone in industrial systems[END_REF]. It explains how each location has a degree of belonging to the fuzzy danger set. Shahrokhi and Bernard measured instantaneous risk indices for each body part for simulation sequences by measuring body parts' membership in a Gaussian FDZ [START_REF] Shahrokhi | Modelling of the occupational hazards entities in a general model of products processes and resources[END_REF].

The proposed model description

This paper proposes a mathematical risk analysis model using the concept of a fuzzy danger zone to consider several types of targets with a different values. The effects of hazards vary on the different types of targets. The proposed model is developed based on the following assumptions:

1. There is a dangerous source that produces a hazard. 2. A hazard barrier reduces some part of the hazard magnitude.

3. There are several types of targets. 4. The hazard has different effects on different kinds of targets. 5. There is a danger zone for each type of target. 6. Each target has a target protective barrier that reduces the danger for the target. 7. The exposure time is known and limited. 8. The exposure mode is known and similar for all target types. 9. The targets are immovable during the exposure time. 10. The hazard and target attributes are constant and unchanged over the exposure time.

The proposed model formulation

The 𝑘 ) The value of damage (loss of the value) of the target 𝑡 𝑗 𝑘 caused by accident, based on the target place By using fuzzy sets notifications, an FDZ is an ordinary fuzzy set such as 𝑍 ̃, defined by its domain (x) and membership function 𝜇 𝑍 ̃(𝑥) as [START_REF] Zimmermann | Fuzzy set theory-and its applications[END_REF]:

𝑍 ̃= ∫ 𝜇 𝑍 ̃(𝑥) 𝑥 𝑋 (2)
In this way, the value of the membership function 𝜇 𝑍 ̃(𝑥) is a continuous function that indicates the degree of membership of point x in a fuzzy set 𝑍 ̃, which can take any value between 0 and 1, as follows:

𝜇 𝑍 ̃: 𝑥 → [0,1] 𝑥 ∈ 𝑋 (
For example, Fig. 5 shows an FDZ that assigns a membership degree to each point's coordinates in mono-dimensional space.

Fig 5. A demonstration of a fuzzy mono-dimensional DZ

Assigning 0 membership to a point means that the hazard magnitude is equal to or less than the maximum hazard acceptable level. Setting 1 to a place indicates that the danger for the target in that place is equal to or greater than a fatal or destructive threshold level. The effect of protective barriers describes their power to neutralize the danger as the proportion of reduction of the danger amplitude (𝜑). The values 0 and 1for 𝜑 indicate an unprotected and a fully protected target, respectively. The values between 0 and 1 concern a protective barrier that diminishes some part of the hazard. Therefore, the FDZ for a protected target is defined as follows:

𝜇 𝑍 ̃𝑗 𝑘 (𝑥 𝑡 𝑗 𝑘 ) = { 0, ( 1
Where ℎ 𝑗 (𝐻) maps the hazard magnitude (or other hazard attributes) to its harmfulness for the protected target on a scale between 0 and 1.

If it would be possible to define target vulnerability function 𝑑 (𝜇 𝑍 ̃𝑗 𝑘 ) for calculating the expected damage, based on 𝜇 𝑍 ̃𝑗 𝑘 , then, by defining function D as follows, we can directly estimate the target damage based on the target location:

𝐷 𝑡 𝑗 𝑘 (𝑥 𝑡 𝑗 𝑘 ) = 𝑑 (𝜑 𝑗 𝑘 (𝜇 𝑍 ̃𝑗 𝑘 (𝑥 𝑡 𝑗 𝑘 ))) 𝑥 𝑡 𝑗 𝑘 ∈ 𝑋 (9)
Where 𝜑 𝑗 𝑘 apply the effect of the target barrier on the fuzzy danger zone.

Numerical example

Here, an accident scenario is presented in a one-dimensional workplace space to illustrate the model application. The objective is to assess the degree of inhalation damage of 4 targets of two types exposed to the toxic cloud of carbon dioxide for 10 minutes. Before the exposure, the ventilation system reduces 15% of the leaked gas concentration (𝜑 𝑘 ℎ = 0.15). The following table presents the CO2 concentration in different workplace locations before and after ventilation operation. Suppose two different types of targets exposing to high CO2 concentrations. Table 3 illustrates the accident analysis process and results according to the exemplified scenario. human limbs) according to their type and location. Also, the following limits the application of the model."

1. The model lacks validation. 2. Using this approach requires developing more complex risk analysis applications. 3. Model development requires a lot of data on the nature of risk, the effect of distance on reducing it, and the effect of risk on the various types of targets expressed by mathematical formulas. 4. The model ignores the effect of the time mode of exposure of the target to the danger. 5. The model assumes that the risk factor and a goal location remain constant during the exposure. 6. The model applies to hazards whose severity depends on the target distance from them. More efforts are necessary to integrate complementary risk analysis methods such as FMEA and fault trees into CAD platforms. Supposing a linear effect of barrier on the hazard magnitude assumes that it neutralizes a part of the hazard. Using a more realistic function for the actual impact of the safety measures improves the model. This approach is only applicable to dangers with known spatial effects. The model supposes that danger zone attributes and the target location are fixed and recognized during the exposure. Further research can discuss combining the probability and fuzzy danger zone concepts and combining several FDZs.

Conclusion

This paper provides an overview of definitions of a DZ as an essential risk analysis concept. It supports applying the fuzzy set theory for modeling DZs as a meter of spatial danger variation. It presents a mathematical model to calculate a real-time risk index during three-dimensional simulations of activities. Using this approach helps perform the risk and design analyses concurrently and improves multidisciplinary communication. Future research may consider the effects of multiple dangers, variations of the target exposure time and place.

Fig 1 .

 1 Fig 1. Representation of a mono-dimensional DZ, as a classic set of points

  model uses the following symbols: k Target type index (𝑘 ∈ 𝐾) 𝐽 𝑘 Set of targets of type k 𝜑 𝑘 ℎ The portion of the reduced hazard magnitude by the hazard safety barrier for the target type k 𝜑 𝑗 𝑘 A fuzzy hedge describing the ability of the protective target barrier to reduce the magnitude of the hazard for the target 𝑡 𝑗 𝑘 𝑡 𝑗 𝑘 Target j, (jth target) of target type k, (𝑘 ∈ 𝐾, 𝑗 ∈ 𝐽 𝑘 ) 𝑥 𝑡 𝑗 𝑘 Coordinates of the jth target of target type k X Workplace universe of discourse 𝐻(x) The magnitude of the hazard in point x 𝐻 𝑗 𝑙 Maximum acceptable level of the hazard for target j 𝐻 𝑗 ℎ Minimum mortal (destructive) level of the hazard for target j 𝑍 ̃𝑘 Fuzzy danger zone produced by the hazard for the target type k 𝜇 𝑍 ̃𝑘 𝐻 (𝐻(x)) The membership degree FDZ (Hazard severity), in point x, according to the hazard amplitude for target type k 𝜇 𝑍 ̃𝑗 𝑘 (𝑥 𝑡 𝑗 𝑘) Membership degree of presence of protected target j in the fuzzy danger zone 𝑍 ̃𝑘, according to the target position (final hazard severity) 𝑑 (𝜇 𝑍 ̃𝑗 𝑘 ) Target damage, the vulnerability function of the target of type k, which calculates the expected percentage target damage, based on 𝜇 𝑍 ̃𝑗 𝑘 𝑊 𝑡 𝑗 𝑘 The value of the target 𝑡 𝑗 𝑘 𝐷 𝑡 𝑗 𝑘 (𝑥 𝑡 𝑗

3 )Fig 2 .

 32 Fig 2. A demonstration of a mono-dimensional DZ, as an α-cuts of an FDZ

Fig 3 .Table 1 .Fig. 4

 314 Fig 3. The relationship between the hazard magnitude and the membership degree of the FDZ for a specific target type For example, table 1 illustrates the physiological tolerance time for various carbon dioxide concentrations (percent by Volume (% V/ ( : Table 1. Physiological tolerance time for different carbon dioxide concentration The concentration of Carbon Dioxide in Air (% V/V) (𝑯(x)) Maximum Exposure Limit (Minutes) 0.5 Undefined 1.0 Undefined 2.0 480 1.5 60 3.0 20 4.0 10 5.0 7 6.0 5 7.0 Less than 3 Fig. 4 may present the relationship between the concentration of carbon dioxide in the air and the degree of membership in the FDZ for 10 minutes' exposure of target type k:

Fig 4 .

 4 Fig 4. The degree of membership in the FDZ for the exemplified scenario

Table 2 .

 2 Reduction of the CO2 concentration by using the hazard barrier, in the exemplified scenario

	Coordinates of the place(x)	𝑯(x) (percent by Volume) Immediately After air after the accident ventilation
	4	8.05	7
	5	6.9	6
	8	5.75	5

Table 3 .

 3 The accident analysis process and results, according to the exemplified scenario This paper aims to integrate risk analysis into a 3D design platform based on using geometric profiles to illustrate the spatial risk distribution. Using an FDZ in a CAD model visualizes the normalizing effects of different kinds of hazards. For example, while simulating a toxic spill accident, each FDZ gives a typical picture of a gradual decrease in the mass concentration as the cloud moves away from the epicenter of the evaporation, which can be compared with created FDZ from an explosion accident, at another point, at the same time. It can be applied to calculate the risk index during a human task simulation and through real-time human interaction with a virtual machine during safety analysis and training. The results also are credible to activate alarm systems according to the operator's limbs place during work. The use of fuzzy danger zone has the following advantages:1. Visualize a representative danger zone in 3D platforms and calculate the membership degree in an FDZ as a hazard index using computerized calculation methods during a simulation. 2. Create a normalized and similar scale for various risks (e.g., biological, chemical, and physical hazards). 3. Compute the effect of danger on different types of targets. 4. Calculate a momentary risk index for each specific target (e.g., different

	Target	Target type	value ($)	Target	Target place	magnitude	Hazard	severity	Hazard	effect of	safety barrier	The target	severity	Final hazard	function (%)	vulnerability	Target	damage	Target
	𝒕 𝒋 𝒌	k	𝑾 𝒕 𝒋 𝒌	𝒙 𝒕 𝒋 𝒌 𝑯(x) 𝝁 𝒁 ̃𝒌 𝑯 (𝑯) 𝝋 𝒋 𝒌 (𝝁) 𝝁 𝒁 ̃𝒋 𝒌 (𝒙 𝒕 𝒋 𝒌 ) 𝒅 (𝝁 𝒁 ̃𝒋 𝒌 ) (%)	Monetary unit
	𝑡 1 1		10 6	4		7	1	(𝜇) 1.2	1	100𝜇 𝑍 ̃1 1 100		10 6
		1
	𝑡 2 1		10 8	5		6	0.9	(𝜇) 1.4	0.86	100𝜇 𝑍 ̃2 1 86 8.6 × 10 7
	𝑡 1 2 𝑡 1 2	2	10 6 10 7	5 8		6 5	0.9 0.7	(𝜇) 2 (𝜇) 1.5	0.81 0.59	70𝜇 𝑍 ̃𝑗 𝑘 56.7 5.67 × 10 5 70𝜇 𝑍 ̃𝑗 𝑘 41.3 4.13 × 10 6
	5	Discussion and Limitations


Philippe Sarrazin
email: philippe.sarrazin@ujf-grenoble.fr

Keywords: stereotype lift, stereotype threat, gender, motor performance, heart rate reactivity, self-confidence. Stereotype lift in motor performance 3 Investigating

Achievement gaps between social groups may result from stereotype threat effects but also from stereotype lift effects -the performance boost caused by the awareness that an outgroup is negatively stereotyped. We examined stereotype lift and threat effects in the motor domain and investigated their mediation by task involvement and self-confidence. Physiological (heart rate reactivity) and self-reported indices were used to examine these questions. Males and females performed a balance task about which negative stereotypes about either males or females were given. No gender information was given in a third (control) condition. Results showed no stereotype threat but a stereotype lift effect, participants performing significantly better after negative outgroup stereotypes were explicitly linked to performance on the balance task compared to the other conditions. Concerning males, this effect was mediated by higher self-confidence and task involvement. The implications of these results for understanding the gender inequalities in sports and physical activities are discussed.

Other studies suggest that stereotype threat is not the only process by which stereotypes can lead to social inequalities in achievement. While having a deleterious effect on their targets, negative stereotypes may also cause inequities by improving the performance of individuals not targeted by the negative stereotypes-a phenomenon called stereotype lift [START_REF] Walton | Stereotype lift[END_REF]. Evidence of stereotype lift effects has appeared for non-stereotyped targets used as control participants in many studies that documented stereotype threat. Specifically, stereotype lift effects occur when non-stereotyped participants perform better after a negative stereotype about an outgroup is made salient compared to when it is not made salient (e.g., [START_REF] Croizet | Stereotype threat undermines intellectual performance by triggering a disruptive mental load[END_REF][START_REF] Spencer | Stereotype threat and women's math performance[END_REF][START_REF] Steele | Stereotype threat and the intellectual test performance of African Americans[END_REF] see [START_REF] Walton | Stereotype lift[END_REF] for a meta-analysis). To explain the stereotype lift effect, [START_REF] Walton | Stereotype lift[END_REF] suggested that negative outgroup stereotypes exert their impact by encouraging downward social comparisons with a denigrated outgroup (e.g., Fein & Spencer, Stereotype lift in motor performance 4 1997). When comparing themselves with a socially devalued outgroup, people may experience an elevation in their self-confidence and motivation to succeed, which may, in turn, improve their performance. At this writing, no work has yet provided support for these assumptions about what mediates stereotype lift. The current study was designed to examine stereotype lift and threat effects with a particular focus on potential mediators of the process by which negative stereotypes can enhance or decrease performance of individuals.

The Mediational Processes of Stereotype Lift and Threat Effects

Although much more research has investigated how stereotype threat exerts its deleterious effect on performance, the evidence of what mediates this effect is mixed. For example, some studies have found a partial mediation of stereotype threat by anxiety (e.g., [START_REF] Spencer | Stereotype threat and women's math performance[END_REF], performance expectations (e.g., [START_REF] Cadinu | Stereotype threat: The effect of expectancy on performance[END_REF] and self-handicapping strategies (e.g., [START_REF] Keller | Blatant stereotype threat and women's math performance: Selfhandicapping as a strategic means to cope with obtrusive negative performance expectations[END_REF], but no one construct has emerged as a strong reliable mediator. One reason for the paucity of data is that many studies used only self-report methods to capture negative affective states or cognitions. There are at least two potential limitations to this approach. First, participants may be concerned with self-presentation strategies and motivated to appear invulnerable to stereotypes (e.g., [START_REF] Bosson | When saying and doing diverge: The effects of stereotype threat on self-reported versus non-verbal anxiety[END_REF]. Second, stereotype threat processes may be unconscious (e.g., [START_REF] Croizet | Extending the concept of stereotype threat to social class: The intellectual underperformance of students from low socioeconomic backgrounds[END_REF], so that participants may not be able to accurately report the proposed mediational construct.

Recent studies have attempted to overcome these limitations by using indirect measures to capture the mediators of stereotype threat. For example, [START_REF] Schmader | Converging evidence that stereotype threat reduces working memory capacity[END_REF] used a dual-processing task to show that stereotype threat was mediated by a reduction in working memory capacity and Bosson and colleagues (2004) reported that stereotype threat was mediated by non-verbal anxiety behaviors. Physiological indices of threat have also been used as indirect measures. Some studies showed that a threatening situation may Stereotype lift in motor performance 5 increase blood pressure [START_REF] Blascovich | African Americans and high blood pressure: The role of stereotype threat[END_REF][START_REF] Scheepers | When the pressure is up: The assessment of social identity threat in low and high status groups[END_REF] and Croizet and colleagues (2004) showed that stereotype threat effects were mediated by increased cognitive load assessed by heart rate (HR) variability. Their covert and preconscious nature makes physiological indices especially useful for measuring how people respond to the salience of negative stereotypes in a performance situation.

The current study examined whether stereotype lift and threat effects would also manifest themselves through physiological indices, more particularly HR responses. An increase in HR has been interpreted in previous studies as indicating greater effort [START_REF] Wright | Effort determination of cardiovascular response: An integrative analysis with applications in social psychology[END_REF] or task engagement -psychological involvement in a task produced by striving toward a self-relevant goal -(for a detailed description of the relationship between HR and effort/engagement, see Blascovich & Tomaka, 1996;[START_REF] Wright | Effort determination of cardiovascular response: An integrative analysis with applications in social psychology[END_REF], both referring to the idea of involvement in the task. HR reactivity appears thus as particularly relevant to test the assumptions of Walton & Cohen (2002) relative to the role of task engagement in the mediation of stereotype lift. Furthermore, this index may also mediates stereotype threat as previous studies showed that threatened individuals may reduce their effort as a selfhandicapping strategy for coping with stereotype threat [START_REF] Stone | Battling doubt by avoiding practice: The effects of stereotype threat on selfhandicapping in white athletes[END_REF].

Using physiological mediators of stereotype lift and threat permits triangulation on multiple mediators when collected with self-report measures of self-confidence and anxiety.

Thus, this study used both physiological and self-reported indices to investigate the mediators of stereotype lift and threat effects.

Gender Stereotypes and Motor Performance

Another contribution of the current study was to document stereotype lift and threat effects in the context of examining gender differences in motor performance. Gender inequities in motor performance, especially in the domain of sport, are well documented.

Indeed, studies showed that males are superior to females in many motor tasks, especially those that involve strength or speed (e.g., Bois, Sarrazin, Brustad, Trouilloud, & Cury, 2002;[START_REF] Eccles | Gender differences in sport involvement: Applying the Eccles' expectancy-value model[END_REF]Thomas & French, 1985). Although gender inequalities in sports are commonly thought to be the result of biological differences, they may also be caused by the social expectancies and self-beliefs related to stereotype processes. Previous research indicates that sports and physical activities are considered as a masculine domain in western countries (e.g., [START_REF] Czisma | Sports stereotypes and gender[END_REF][START_REF] Koivula | Sport participation: differences in motivation and actual participation due to gender typing[END_REF][START_REF] Riemer | Gender typing of sports: an investigation of Metheny's classification[END_REF].

Thus, examining gender differences in motor performance from a stereotype influence perspective may provide new insights into the phenomena [START_REF] Stone | Dual cues and processes determine how women experience stereotype threat in sports[END_REF].

Previous studies on stereotype threat in sports have documented racial differences in performance when racial stereotypes were made salient while performing a golf-putting task [START_REF] Stone | Battling doubt by avoiding practice: The effects of stereotype threat on selfhandicapping in white athletes[END_REF][START_REF] Stone | Stereotype threat effects on Black and White athletic performance[END_REF]. However stereotype lift has never been demonstrated in the athletic domain. Given that it is not clear whether stereotype effects are the same on cognitive and motor performance [START_REF] Beilock | Stereotype threat and sport: Can athletic performance be threatened?[END_REF], more studies are needed in this area to investigate the pervasiveness of stereotype lift across domains.

The Present Research

The goals of the current research were (1) to examine stereotype lift and threat effects on the motor performance of male and female targets, and (2) to measure both self-report and physiological mediational processes of these effects. Although many sports are traditionally classified as inappropriate for women (e.g., boxing), others are considered as inappropriate for men (e.g., dancing) (e.g., [START_REF] Koivula | Sport participation: differences in motivation and actual participation due to gender typing[END_REF]. Thus, we were interested in comparing targets' reactions when negative stereotypes about males and females were salient during a novel motor performance, in this case, a balance task. It was hypothesized that negative stereotypes about males and females could be activated with a balance task insofar as it belongs to the motor domain and is thus related to sports. Many studies reported the existence of shared stereotypes according to which "masculine" sports are those requiring strong athletic abilities and "feminine" sports are those requiring fine and precise motor skills (e.g., [START_REF] Ignico | Development and verification of a gender-role stereotyping index for physical activities[END_REF][START_REF] Riemer | Gender typing of sports: an investigation of Metheny's classification[END_REF]. Thus, the poor ability of females on this task could be attributed to the common belief that women have less athletic ability than men, and the poor ability of males could be attributed to the belief about the poor movement finesse of men. Performing a balance task is then a stereotype-relevant context in which both stereotype lift and threat may occur.

In order to investigate stereotype lift and threat processes, gender stereotypes were activated in an explicit manner; that is, male and female participants were told that either males or females are well known to perform poorly on the balance task. We reasoned that in order for stereotype lift to occur on a novel task, targets could not be expected to infer that their group is superior to another group. The processes assumed to underlie stereotype lift, such as downward social comparison and enhanced feelings of self-confidence, would be more systematically activated if participants were told about the outgroup's inferiority on the task.

In accordance with stereotype lift theory, it was predicted that targets would perform better on the balance task when they were told that an outgroup was known to perform poorly on the task as compared to when they were not provided with group-performance information. This stereotype lift effect would be mediated by higher self-reported selfconfidence and task involvement indicated by an increase in HR 1 (e.g., [START_REF] Wright | Effort determination of cardiovascular response: An integrative analysis with applications in social psychology[END_REF]. In accordance with stereotype threat theory, it was predicted that targets would perform lower on the balance task when they were told that their ingroup was known to perform poorly on the task as compared to when they were not provided with groupperformance information. This stereotype threat effect would be mediated by lower self-Stereotype lift in motor performance 8 reported self-confidence, higher self-reported anxiety and lower task involvement indicated by a decrease in HR.

Method Participants and Design

Participants were 66 undergraduate students (33 males and 33 females) enrolled in the sport sciences program at the University of Grenoble (France). They participated in a 2 (sex: males vs. females) X 3 (test frame manipulation: males inferior vs. females inferior vs. noinformation control) between-subjects design. These students were recruited on a voluntary basis by announcements made during sport psychology classes. The data of three participants (one male and two females) were discarded because of technical problems with the stabilometer, leaving thirty-two males and thirty-one females in this experiment.

Procedure

Participants completed the procedures individually. The experimenter explained to them that they would complete some questionnaires and perform a motor test of balance. A cardio frequency meter was then attached to the participants' chest, who were asked to relax for a few seconds while their baseline HR was recorded.

Then participants were assigned to one of three test frame conditions. Two conditions involved the activation of a blatant negative stereotype by emphasizing the poor balance ability of males or females. In the "males inferior" ("females inferior") condition, participants were told that: "The goal of this experiment is to compare balance in males and females (females and males). Previous studies showed that males (females) have problems keeping their balance compared to females (males), and we are trying to understand why they have these difficulties". After the instructions, participants watched videotape showing a female (male) performing the motor task very well for five seconds.

Stereotype lift in motor performance 9

In the control condition, participants were told that: "We are trying to understand the psychological and physiological processes involved in performance on this task". After the instructions, participants watched the two videotapes watched in the two other conditions.

Then participants completed a pre-performance questionnaire packet that contained a manipulation check and the pre-competitive state anxiety and confidence scales. Thirty seconds prior to the beginning of the test, the experimenter started the cardio frequency meter recording HR continuously. After participants completed the motor test they were fully debriefed.

Dependent measures

Manipulation check

Participants were asked to indicate on a 1 (strongly disagree) to 7 (strongly agree) scale whether "generally, females perform higher than males" and whether "generally, males perform higher than females". A difference score was computed by subtracting the male inferior score from the female inferior score such that a positive value would indicate that females were perceived as performing better than males, and a negative value would indicate that males were perceived as performing better than females. The order of these two items was counterbalanced.

Athletic performance test

The task was a stabilometer motor task that requires participants to maintain their balance in a standing position on a platform rotating around a central pivoting shaft.

Participants began the task with one side of the platform touching the floor. The goal was to maintain balance as long as possible on the platform. Time in balance was defined as the time during which the platform did not touch the floor and was automatically measured with a stopwatch linked to the stabilometer by an electrical system. Each time the platform touched the floor, participants were asked to start another trial, with the platform putting down on the Stereotype lift in motor performance 10 floor. The test period lasted four minutes and the performance score corresponded to the mean time of the three best trials.

State anxiety and self-confidence

The Revised Competitive State Anxiety Inventory-2 (R-CSAI-2, [START_REF] Cox | Measuring anxiety in athletics: The revised competitive state anxiety inventory-2[END_REF] was administered to the participants. This inventory is composed of three subscales measuring self-confidence, somatic and cognitive anxiety. Seven items measured somatic anxiety (e.g., "I feel tense in my stomach"). Five items measured cognitive anxiety (e.g., "I am concerned about performing poorly") and five items measured self-confidence (e.g., "I am confident about performing well"). Participants responded on a 1 (not at all) to 7 (very much) scale.

Heart Rate HR was recorded with an ambulatory device (cardio frequency meter Polar s610i, Polar Electro Oy, Finland, 2000). The Polar 610i is a lightweight instrument strapped on a belt linked to the participant by three electrodes on the chest. It is equipped with a microprocessor that measures the time lapse between two heartbeats with millisecond precision. The data were recorded by the Precision Performance Polar 3.02.007 program (Polar Electro Oy, Finland, 2001) that stored every five seconds the mean HR of the last five seconds. In the next analyses HR was taken into account at six different times in addition to the baseline value: thirty seconds before the beginning of the test period, when the test began and at each minute of the test.

Results

Preliminary Analyses

Before testing the primary hypotheses, we examined whether participants correctly perceived the gender differences on the task. A 2 (sex) X 3 (test frame manipulation) between-subjects analysis of variance (ANOVA) showed only a significant main effect of the Stereotype lift in motor performance 11 test frame manipulation on the perceived gender difference score, F (2, 57) = 235.97, p < .001. Comparisons showed that females were perceived as performing higher than males in the males inferior condition (M = -4.82, SD = 0.33) compared to the control condition (M = 0.05, SD = 0.36), F (1, 57) = 99.78, p < .001. Moreover, males were perceived as performing higher than females in the females inferior condition (M = 5.36, SD = 0.33) compared to the control condition, F (1, 57) = 118.89, p < .001. Thus, the data indicated that the manipulations had the intended effects on the perceived gender differences on the task.

Individuals may be susceptible to stereotype effects only when the task is selfrelevant, and the self-relevance of a task may be reflected by task engagement (e.g., [START_REF] Blascovich | Predicting athletic performance from cardiovascular indexes of challenge and threat[END_REF]. We thus confirmed that participants were engaged in the task by testing against zero HR reactivity scores computed by subtracting the baseline resting value from the pre-performance value (e.g., Blascovich et al., 2004). Only pre-performance HR was examined because an HR increase during the task may indicate engagement but also the metabolic demands of the task. Results showed that pre-performance HR increases were significantly greater than zero in the three conditions (ts > 4.37, ps < .001), indicating that participants were engaged thirty seconds before the task begins.

Balance Performance

The means of performance and the potential mediators according to sex and test frame manipulation are reported in Table 1.

In order to test the primary hypotheses, a 2 (sex) X 3 (test frame manipulation) ANOVA was performed on the measure of balance performance. The results showed only a significant sex-by-test frame manipulation interaction effect, F (2, 57) = 3.43, p = .04, η² = .11. Within-gender comparisons across conditions showed that males in the females inferior condition tended to perform higher (M = 24.26) than in the control condition (M = 9.87), F Stereotype lift in motor performance 12

(1, 57) = 3.67, p = .06, although this difference did not reach significance. However, males in the males inferior condition (M = 11.96) did not perform significantly lower than in the control condition, F (1, 57) = .07, ns. Results also showed that females in the males inferior condition performed significantly higher (M = 29.81) than in the control condition (M = 11.62), F (1, 57) = 5.54, p = .02. However, females in the females inferior condition (M = 15.07) did not perform lower than in the control condition, F (1, 57) = .20, ns. Thus, both males and females showed a stereotype lift effect when the balance task was framed as a measure of a negative outgroup stereotype but no stereotype threat effect when the balance task was framed as a measure of a negative ingroup stereotype.

Self-Confidence

The internal reliability of the self-confidence subscale was satisfactory (α = .74). We then combined the items to create an index that was subjected to the ANOVA analysis. A significant sex-by-test frame manipulation interaction effect on self-confidence was found, F (2, 57) = 3.66, p = .03, η² = .11. Within-gender comparisons showed that males were more self-confident in the females inferior condition (M = 3.95) than in the control condition (M = 3.18), F (1, 57) = 4.06, p = .05. However, males were not less self-confident in the males inferior condition (M = 3.07) than in the control condition, F (1, 57) = .08, ns. Selfconfidence of females was not different across conditions (Fs < 1), meaning that the test frame manipulation did not influence the self-confidence of females.

Anxiety

The internal reliability of the somatic anxiety (α = .89) and cognitive anxiety (α = .87) subscales was satisfactory. We then combined the items to create two indices which were subjected to the ANOVA analysis. No main or interactive effects were found on the somatic and cognitive anxiety (Fs < 1), meaning that the test frame manipulation did not influence these variables.

Stereotype lift in motor performance 13

Physiological Activity

Preliminary Analyses

The 2 X 3 ANOVA showed no main or interactive effects of sex and test frame manipulation on baseline HR, Fs < 1.21, ns. Reactivity scores were then computed for each time interval (e.g., [START_REF] Blascovich | African Americans and high blood pressure: The role of stereotype threat[END_REF][START_REF] Croizet | Stereotype threat undermines intellectual performance by triggering a disruptive mental load[END_REF].

HR Reactivity

Given that HR during the task may indicate not only engagement but also the physical exertion required by the task, significant interaction effects were expected primarily on preperformance HR reactivity. We then performed separate ANCOVA analyses for each HR reactivity score with baseline HR as a covariate to further control for relative baseline levels.

The analyses revealed a significant sex-by-test frame manipulation interaction effect on preperformance HR reactivity, F (2, 56) = 3.66, p = .03, η² = .10. Within-gender comparisons showed that the pre-performance HR reactivity of males tended to be significantly higher in the females inferior condition (AjM = 18.77) than in the control condition (AjM = 10.18), F

(1, 56) = 3.52, p = .06. Pre-performance HR reactivity of males in the males inferior condition (AjM = 6.82) was not lower than in the control condition, F < 1. Results also showed that the pre-performance HR reactivity of females tended to be higher in the males inferior condition (AjM = 15.75) than in the control condition (AjM = 7.84), F (1, 56) = 2.77, p = .10, although this trend was not significant, and that pre-performance HR reactivity of females in the females inferior condition (AjM = 10.84) was not lower than in the control condition, F < 1.

In addition, no other effects were found on the other time intervals, meaning that the test frame manipulation did not affect HR reactivity of participants while performing the test.

Mediational Analyses

The previous analyses showed that although the test frame manipulation affected identically the performance of males and females, it affected the self-confidence and preperformance HR reactivity of males only. To examine whether these two variables mediated the stereotype lift effect for males, we compared the self-confidence and pre-performance HR reactivity of males in the lift and control conditions using contrast coding. Specifically, we assigned codes of 1, 0 and -1 to males in the females inferior, males inferior and control conditions respectively. The intercorrelations between the variables for males and females are indicated in Table 2. To further control for relative baseline levels we regressed preperformance HR reactivity score onto its baseline value and we took into account this residual in the subsequent analyses.

Several regression analyses were performed following the procedure advocated by [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations[END_REF]. As depicted in Figure 2, the test frame manipulation significantly predicted self-confidence (ß = .35, p < .05) and tended to predict pre-performance HR reactivity (ß = .31, p = .08), showing that males in the lift condition were more self-confident and tended to be more involved before the beginning of the task compared to the two other conditions. More importantly, when the potential mediators and the test frame manipulation were included in the same equation, the direct pathway from the test frame manipulation to performance (ß = .39, p = .03) dropped to nonsignificance (ß = .15, ns) whereas the effects of self-confidence and HR on performance were significant (ß = .33, p = .04 and ß = .39, p = .02, respectively). These results showed that self-confidence and pre-performance HR reactivity mediated independently the test frame manipulation -performance relationship. 2

Discussion

This research was aimed to study stereotype lift and threat effects in a context of motor performance and to examine physiological and self-reported mediators of these effects.

Stereotype lift in motor performance 15

The analyses of the balance performance data revealed the predicted stereotype lift effect: Males and females performed significantly better after negative outgroup stereotypes were explicitly linked to performance on the balance task compared to when negative stereotypes were not made salient (e.g., [START_REF] Croizet | Stereotype threat undermines intellectual performance by triggering a disruptive mental load[END_REF][START_REF] Spencer | Stereotype threat and women's math performance[END_REF]. The data support the hypothesis that stereotype lift can emerge when non-targets are provided with explicit downward social comparison information about another group's performance on the task. They provide an important extension of previous research by demonstrating significant stereotype lift effect on a nonacademic motor task with both males and females.

Another unique finding in the current data is the evidence of processes that mediate stereotype lift. Among males, the stereotype lift effect was mediated by their involvement in the task and by their level of self-confidence. The data suggest that the explicit downward comparison with a devalued outgroup improved males' self-confidence and task involvement, and these variables contributed to the observed performance boost. Although this study supports the main assumptions of [START_REF] Walton | Stereotype lift[END_REF], the gender difference in mediation was not expected. It is possible that the activated stereotypes about males and females were different in accessibility. Given that motor performance is a male-oriented domain (e.g., [START_REF] Koivula | Sport participation: differences in motivation and actual participation due to gender typing[END_REF], the belief that women have less athletic ability than men may be more accessible than the belief about the poor movement finesse of men. The difficulty for females to rely on a strong stereotype could explain why they were not more self-confident and involved in the task in the males inferior condition. Future research is necessary to uncover the processes that mediated the lift effect observed for females in the current research.

It is also important to note that the mediation of stereotype lift for males occurred before the beginning of the task. Indeed self-confidence was measured a few minutes before the task and only pre-performance HR reactivity mediated the performance. This is consistent Stereotype lift in motor performance 16 with research by [START_REF] Stone | Battling doubt by avoiding practice: The effects of stereotype threat on selfhandicapping in white athletes[END_REF], who noted that stereotype threat might start its detrimental effects in the days, hours or minutes prior to the critical performance. Thus, [START_REF] Stone | Battling doubt by avoiding practice: The effects of stereotype threat on selfhandicapping in white athletes[END_REF] showed that threatened participants coped with the potential threat by engaging in proactive defensive reactions like behavioral self-handicapping before their performance began. Our results suggest that the possibility of outperforming another group may also be anticipated and begin once the negative stereotype about the outgroup is linked to the upcoming performance. The extra self-confidence and energy mobilization may have prepared the organism to meet expected task demands, enabling participants to be immediately efficient while performing the task (e.g., [START_REF] Wright | Task difficulty, cardiovascular response, and the magnitude of goal valence[END_REF]. Nevertheless, as noted earlier, when measured during the task, the cardiac response may have indicated not only involvement in the task but also other variables like the physical exertion required by the task to maintain balance. This unclear meaning of HR did not allow us to examine the mediating role of task involvement during the task. Whereas the current data indicate that stereotype lift is mediated by increased self-confidence and task involvement before the performance begins, the processes that mediate stereotype lift during the task are still unknown.

Are our findings evidence for what Shih and colleagues called the stereotype susceptibility effect [START_REF] Shih | Stereotype susceptibility: Identity salience and shifts in quantitative performance[END_REF], defined as the performance boost caused by activation of a positive ingroup stereotype? In contrast to stereotype susceptibility, stereotype lift is triggered by a negative outgroup stereotype. In other words, whereas stereotype susceptibility focuses on groups targeted by positive stereotypes, stereotype lift focuses on groups that are non-stereotyped [START_REF] Walton | Stereotype lift[END_REF]. In this study, the instructions emphasized the poor ability of one particular group (thus activating a negative stereotype), suggesting that the ability of the other group was normal and not particularly high on the balance dimension. Thus the findings offer more support for the stereotype lift than for the stereotype susceptibility hypothesis. 3 This distinction is important because Stereotype lift in motor performance 17 activating a negative outgroup stereotype implies ingroup advantage indirectly rather than directly [START_REF] Walton | Stereotype lift[END_REF]. This activation is thus unlikely to threaten performance by creating concern about failure to meet high expectations held for one's group -phenomenon known as "choking under pressure" -as it may sometimes occur when positive ingroup stereotypes are activated [START_REF] Cheryan | When positive stereotypes threaten intellectual performance: the psychological hazards of "model minority" status[END_REF][START_REF] Smith | A stereotype boost or choking under pressure? Positive gender stereotypes and men who are low in domain identification[END_REF].

Finally, this study reported no stereotype threat effect. One explanation is that the balance task was not perceived as a difficult test. Previous research has shown that the salience of negative stereotypes is more likely to impact performance when the task is difficult (e.g., [START_REF] O'brien | Stereotype threat and arousal: Effects on women's math performance[END_REF]. It is possible that the act of balancing is not sufficiently challenging to induce the processes that underlie stereotype threat responses.

However, that a lift effect did emerge suggests that perhaps threat and lift responses are asymmetrically related to each other. That is, people may experience enhanced confidence and task engagement, and perform better, when positive stereotypes are salient, regardless of the difficulty of the task. No prior research has examined the role of task difficulty in stereotype lift effects and this will be a fruitful direction for future research.

Another possible explanation of this absence of stereotype threat is stereotype reactance -an engagement in behaviors that are counter to those prescribed by the stereotype. Based on the psychological reactance theory [START_REF] Brehm | A theory of psychological reactance[END_REF], the stereotype reactance theory [START_REF] Kray | Battle of the sexes: Gender stereotype confirmation and reactance in negotiations[END_REF] states that reactance occurs when people perceive limitations to their ability to perform, in this case when negative ingroup stereotypes are linked to performance on a task. Kray and colleagues suggested that what differentiates an engagement in threat or reactance behaviors is the nature of the activation of stereotypes.

When a negative ingroup stereotype is explicitly linked to performance on a task, people become aware of the stereotype and may try to defeat it. Although the performance of participants when the negative ingroup stereotype was explicitly linked to their performance Stereotype lift in motor performance 18 did not significantly differ from that of the control condition, it tended to be higher. The explicit statement about the poor performance of their group on the balance task may have motivated participants to try to defeat the negative characterization. Their performance did not significantly differ from that of the control condition maybe because some participants responded to the explicit negative stereotype with threat whereas others responded with reactance. This suggests that there may be individual difference variables, like selfengagement in the performance domain [START_REF] Stone | Battling doubt by avoiding practice: The effects of stereotype threat on selfhandicapping in white athletes[END_REF][START_REF] Stone | Stereotype threat effects on Black and White athletic performance[END_REF]) social anxiety or self-confidence, that determines whether people feel threatened or challenged when confronted with a negative explicit stereotype about their group. We performed follow-up analyses to examine whether the relationship between the activation of a negative ingroup stereotype and performance was moderated by pre-performance heart rate reactivity, anxiety or self-confidence, but these analyses did not show any significant interactions. Investigating personal and contextual factors that distinguish threat and challenge responses when explicit negative stereotypes are linked to performance represents an important direction for future research.

Whereas the nature of stereotype activation (explicit vs. implicit) may moderate stereotype threat effects, [START_REF] Walton | Stereotype lift[END_REF] reported that it should not moderate stereotype lift effects. This could offer another explanation for why only stereotype lift effects were found in this study.

To conclude, this study supports the idea that situations activating gender stereotypes may induce gender differences in achievement, and that the gender appropriateness of motor tasks may be socially constructed. More importantly, this research shows that these differences are not only due to the debilitating effects of stereotype threat but also to the performance boosts of stereotype lift. This could suggest one way in which stigmatized individuals are doubly handicapped by stereotypes: even if they managed to cope with a threatening stereotype, they may still underperform relative to non-stereotyped targets who can experience a performance boost. In the sports area where many activities are sex-typed, the gender gaps existing in achievement and motivation can be the result of the activation of gender stereotypes. Footnotes 1. Blood pressure (BP) indexes were also collected along with HR. However the meaning of BP is unclear, interpreted as an indicator of threat in some studies (e.g., [START_REF] Blascovich | African Americans and high blood pressure: The role of stereotype threat[END_REF] and effort in others (e.g., [START_REF] Wright | Effort determination of cardiovascular response: An integrative analysis with applications in social psychology[END_REF]. Moreover the analyses showed no effect of test frame manipulation on BP indexes, so they were dropped from this study.

2. When self-confidence and pre-performance heart rate reactivity were included in separate analyses, they both significantly predicted performance (β = .38, p = .03 and β = .44, p = .01, respectively), whereas the relationship between test frame manipulation and performance was not significant (β = .26, ns for both analyses).

3. As suggested by an anonymous reviewer, examining the correlation between each manipulation check item and the performance of subjects placed in the lift conditions could shed light on whether the boost effect derives from negative stereotypes about the outgroup, or positive stereotypes about the ingroup. That is, performance in the lift condition might correlate better with the item assessing negative expectancies about the outgroup than positive expectancies about the ingroup. Unfortunately, the correlations were non significant, probably because the samples were too small (n = 11 in both lift conditions).

Stereotype lift in motor performance 26

Figure Captions Figure 1 .

 Captions1 Figure Captions Figure 1. Mediation of stereotype lift effect by self-confidence and heart rate reactivity for males.

Table 1 .

 1 Means of Performance and the Potential Mediators According to Sex and Test FrameManipulation.

	Test frame manipulation

Table 2 .

 2 Intercorrelations Between the Variables for Males and Females

	Variable	1	2	3	4	5	6	7
	1. Test frame manipulation	-	0.38*	0.16	0.14 -0.02	0.23	0.00
	2. Performance	0.42*	-	0.16	-0.02 -0.13	0.16	0.08
	3. Self-confidence	0.44*	0.48**	-	-0.39* -0.31	0.07	0.19
	4. Cognitive anxiety	0.12	-0.15	-0.03	-	0.69*** -0.22	0.20
	5. Somatic anxiety	0.11	-0.22	-0.12	0.59***	-	-0.05	0.30
	6. Pre-performance HR	0.45*	0.52** 0.23 -0.14	-0.16	-	0.31
	7. Mean HR during test	0.33	0.23	0.01	0.36*	0.25	0.24	-