Salim Rao

Nonlinear aggregation of conditional extreme-value index estimators

When an extreme phenomena depends on a covariate, one has to deal with Conditional Extreme Value Analysis. In this paper, we propose a nonlinear aggregation technique to estimate the conditional extreme-value index.

Extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the mean of probability distributions. It seeks to assess the probability of events that are more extreme than any observed prior. Extreme value analysis is widely used in many disciplines, ranging from structural engineering, finance, earth sciences, traffic prediction, geological engineering, etc. For example, EVA might be used in the field of hydrology to estimate the value an unusually large flooding event, such as the 100-year flood. Similarly, for the design of a breakwater, a coastal engineer would seek to estimate the 50-year wave and design the structure accordingly.

Let X be a random variable and F (x) = P (X ≤ x) its distribution function. We can define a new random variable, Y n , as the maximum of n copies of the random variable X: Y n = max{X 1 , X 2 , ..., X n }. Y n is the n-sample maximum of the random variable X. If the events generating the realizations of X are independent, the cumulative distribution of Y n may be expressed as [F (y)] n . Upon definition of a renormalized variable S n = (Y n -b n )/a n , the extreme value theorem establishes that, if

lim n→∞ P (S n < s) = lim n→∞ F n (a n s + b n ) = H(s) (1)
where a n > 0 and b n are normalization constants, then the function H(s) in Eq. ( 1) must be one of the three following types:

• EV1 or Gumbel: H(s) = exp(-exp(-s))

• EV2 or Fréchet:

H(s) = exp(-s -α ) • EV3 or Weibull: H(s) = exp(-|s| α ) 1
The three asymptotic types, EV1-EV3, can be thought of as special cases of a single Generalized Extreme Value distribution (GEV) :

H GEV (s) = exp -1 + γ s -µ σ -1/γ + (2)
where (.) + = max(., 0), µ is the location parameter, σ > 0 is the scale parameter, and γ is a shape parameter. The limit γ = 0 corresponds to the EV1 distribution, γ > 0 to the EV2 distribution (with α = 1/γ) and γ < 0 to the EV3 distribution (with α = -1/γ).

When the studied phenomena depends on a covariate, one has to deal with Conditional Extreme Value Analysis (CEVA). This branch of statistics has become very active these past ten years, the main contributions to this domain are listed below:

• Theoretical issues: [START_REF] Das | Detecting a conditional extreme value model[END_REF][START_REF] Gardes | Functional kernel estimators of large conditional quantiles[END_REF][START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Gardes | Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels[END_REF][START_REF] Gardes | Functional nonparametric estimation of conditional extreme quantiles[END_REF][START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF][START_REF] Beirlant | Local polynomial maximum likelihood estimation for pareto-type distributions[END_REF][START_REF] Ac Davison | Local likelihood smoothing of sample extremes[END_REF][START_REF] Chavez-Demoulin | Generalized additive modelling of sample extremes[END_REF][START_REF] Hall | Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data[END_REF][START_REF] Das | Conditioning on an extreme component: Model consistency with regular variation on cones[END_REF][START_REF] Beirlant | Nonparametric estimation of extreme conditional quantiles[END_REF][START_REF] Abdous | Extreme behaviour for bivariate elliptical distributions[END_REF][START_REF] Wang | Estimation of high conditional quantiles for heavy-tailed distributions[END_REF][START_REF] Wang | Tail index regression[END_REF][START_REF] Rao | A review on conditional extreme value analysis[END_REF] • Quantile regression: [START_REF] Koenker | Quantile regression: An introduction[END_REF][START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF][START_REF] Rosen | Extreme percentile regression[END_REF] • Application to finance: [START_REF] Byström | Managing extreme risks in tranquil and volatile markets using conditional extreme value theory[END_REF][START_REF] Bali | A conditional extreme value volatility estimator based on high-frequency returns[END_REF][START_REF] Ghorbel | Predictive performance of conditional extreme value theory in value-at-risk estimation[END_REF][START_REF] Byström | Extreme value theory and extremely large electricity price changes[END_REF][START_REF] Marimoutou | Extreme value theory and value at risk: application to oil market[END_REF][START_REF] Wang | A model of conditional var of high frequency extreme value based on generalized extreme value distribution[END_REF] Our mail result is the following: Theorem Let γ1 (x), . . . , γK (x) be K consistent estimators of the conditional extreme-value index γ(x). Let w 1 , . . . , w K be K weights summing to one and ψ a continuous bijective function with continuous inverse ψ -1 . Then,

γ(x) := ψ -1 ( K i=1 w i ψ(γ i (x)))
is a consistent estimator of the conditional extreme-value index γ(x).

Proof. The continuity property of ψ implies that ψ(γ i (x)) converges to ψ(γ(x)) in probability. Besides, since K i=1 w i = 1, it follows that i=1 w i ψ(γ i (x)) converges to ψ(γ(x)) in probability as well and the result follows.

Let us highlight that the positivity of the weights in not required. Moreover, this result is valid whatever the sign of γ(x). Finally, it extends the linear aggregation procedure of [START_REF] Rao | Linear aggregation of conditional extreme-value index estimators[END_REF] to the nonlinear case where the function ψ is not the identity.


Oscar Devys
email: oscar.devys@math.univ-lille1.fr

FABER POLYNOMIALS AND SPECTRUM LOCALISATION

come

Introduction

Faber polynomials are now classical objects of study in complex analysis, function theory and approximation theory. We use in this paper Faber polynomials as a basic tool to study the following "spectrum localisation problem" in operator theory: identify conditions under which the spectrum of a Banach space bounded linear operator is included in the interior of a given compact of the complex plane. Other instances of the use of Faber polynomials for various problems in operator theory are [START_REF] Atzmon | Spectral inclusion and analytic continuation[END_REF][START_REF] Badea | Faber-hypercyclic operators[END_REF][START_REF] Badea | Size of the peripheral point spectrum under power or resolvent growth conditions[END_REF][START_REF] Beckermann | Image numérique, GMRES et polynômes de Faber[END_REF][START_REF] Toh | The Kreiss matrix theorem on a general complex domain[END_REF].

Computing or estimating the spectrum of a large matrix, or, more generally, of a bounded linear operator acting on a complex Banach space is an important problem in spectral theory, with possible applications in numerical analysis (see for instance [START_REF] Nevanlinna | Convergence of iterations for linear equations[END_REF]). The classical spectral radius formula ( [START_REF] Conway | A course in functional analysis[END_REF]) r(T ) = lim n→∞ T n 1/n , going back to Arne Beurling and Israel Gelfand, allows to find or estimate the radius r(T ) = sup{|λ| : λ ∈ σ(T )} of the smallest closed disc in the plane centered in the origin and containing the spectrum σ(T ) of the given operator T . The condition r(T ) < 1, corresponding to the exponential stability of T , is an important one for applications. In order to better localise the spectrum and to find inclusions of the form σ(T ) ⊂ K or σ(T ) ⊂ int(K) for some simply connected compact K in the complex plane, the right tool to consider is the asymptotic behaviour of the sequence (F n (T )) n≥0 , where F n is the Faber polynomial of degree n associated with K. We are interested to generalise in this vein (replacing the closed unit disc by more general compacts) several existing results in the literature [START_REF] Mlak | On a theorem of Lebow[END_REF][START_REF] Weiss | Weakly l p -stable linear operators are power stable[END_REF][START_REF] Nikolski | A Tauberian theorem for the spectral radius[END_REF][START_REF] Nikolski | Estimates of the spectral radius and the semigroup growth bound in terms of the resolvent and weak asymptotics[END_REF][START_REF] Van Neerven | Exponential stability of operators and operator semigroups[END_REF]. An interesting feature of our study is the influence 1 of the geometry of the compact and the regularity of its boundary on the results obtained about the localisation of the spectrum.

The present paper is organized as follows. Section 2 is dedicated to the definition and basic properties of Faber polynomials. We present the influence of the regularity of the boundary of the compact on the behaviour of these polynomials and present known applications in operator theory. In Sections 3 and 4 we present two approaches leading to different criteria for the spectrum to be included in the interior of a compact. Some generalisations of results of Mlak ([11]), Weiss ([19]), van Neerven ( [START_REF] Van Neerven | Exponential stability of operators and operator semigroups[END_REF]) and Nikolski ([13, 14]) are given there. We give in Section 5 some explicit estimates based upon the preceding criteria.

Preliminaries on Faber polynomials

2.1. Definition and examples. We present here some classical results about Faber polynomials which will be used later on. We follow the setting of [START_REF] Suetin | Series of Faber polynomials[END_REF]. The reader is referred to [START_REF] Suetin | Series of Faber polynomials[END_REF] or [START_REF] Gaier | Lectures on complex approximation[END_REF] for more details about Faber polynomials and related topics.

Let K be a compact connected subset of the complex plane C, different from a singleton, and whose complement is connected. From the Riemann mapping theorem, we know that there exists a unique conformal map ψ : D

c -→ K c such that ψ(∞) = ∞ and ψ ′ (∞) > 0.
The map ψ has a Laurent expansion for |w| > 1 of the form

ψ(w) = βw + β 0 + β 1 w -1 + • • • + β k w -k + . . .
where β > 0 is the transfinite diameter, or (logarithmic) capacity, of K.

Let φ be the inverse function of ψ. The map φ : K c -→ D c has a Laurent expansion in a neighborhood of infinity of the form

φ(z) = 1 β z + b 0 + b 1 z -1 + • • • + b k z -k + . . .
For n ∈ N, the polynomial part of the Laurent expansion of φ(z) n is called the Faber polynomial of order n and is denoted by F n . The Faber polynomial of order n has degree n and leading coefficient 1/β n . We consider the function ω n defined by the following equation

φ(z) n = F n (z) + ω n (z), (z ∈ K c ).
Then z → ω n (z) is an analytic bounded function on K c which tends to 0 at infinity.

For any R > 1, let Γ R be the analytic Jordan curve {ψ(w

) : |w| = R}. Denote by G R its interior. If z is in G R then we have F n (z) = 1 2iπ Γ R φ(ζ) n ζ -z dζ = 1 2iπ |w|=R w n ψ ′ (w) ψ(w) -z dw.
One can deduce from this relation that the Faber polynomials satisfy the following asymptotic relations

lim n→∞ |F n (z)| 1/n = |φ(z)| for all z / ∈ K and lim n→∞ |F n (z)| 1/n ≤ 1 for all z ∈ K.
It also implies that for any fixed z in G R , (F n (z)) n≥0 is the sequence of the Laurent coefficients of the map

w → ψ ′ (w) ψ(w) -z , |w| > R,
in the neighborhood of the point w = ∞. Therefore the generating function of the Faber polynomials is given by

(2.1) ψ ′ (w) ψ(w) -z = ∞ n=0 F n (z) w n+1 , (|w| > R, z ∈ G R ).
The Faber polynomial of order n of the disk centered at z 0 and of radius R is given by the formula [START_REF] Suetin | Series of Faber polynomials[END_REF]). An analytic function f in a neighborhood of the compact K can be uniquely expanded in a series of Faber polynomials with uniform convergence on a neigborhood of K. That is to say, there exist a neighborhood V of K and complex numbers a n = a n (f ) such that for every z in V we have

F n (z) = ((z -z 0 )/R)
f (z) = ∞ n=0 a n F n (z).
Example 2.2. Let R > 1 and K be the compact delimited by the ellipse with the foci at the points 1 and -1 and semi-axes

a = 1 2 (R + 1/R) and b = 1 2 (R -1/R).
The equation of this ellipse can be written in the form

z = 1 2 Re iθ + 1 Re iθ , 0 ≤ θ < 2π.
Hence, the exterior conformal map associated with K is given by

ψ(w) = 1 2 Rw + 1 Rw .
The corresponding Faber polynomials are

F n (z) = 2 R n C n (z), (n ≥ 1)
, where C n is the Chebychev polynomial of order n, C n (t) = cos(n arccos(t)), (t ∈ [-1; 1]).

We also notice that, for any n ≥ 1, we have

F n (ψ(w)) = w n + 1 R 2n w n .
2.2. Consequences of the regularity of the boundary Γ. If Γ, the boundary of K, is a Jordan curve, its regularity influences the behaviour of the map ψ and the asymptotic behaviour of the sequence of Faber polynomials (F n ) n≥0 on Γ.

A function ρ : R + -→ R + is called a modulus of continuity if it satisfies:

(1) ρ is increasing;

(2) lim t→0 ρ(t) = 0;

(3) ρ is subadditive, i.e.,

ρ(t + s) ≤ ρ(t) + ρ(s).
A modulus of continuity is said to satisfy the Alper condition if

ε 0 ρ(x) x log 1 x dx < ∞,
for some ε > 0.

For a fixed modulus of continuity ρ, a Jordan curve Γ is said to be of class C ρ if it has a parametrization τ : [0; 1) -→ Γ that is differentiable, with τ ′ (x) = 0 of all x ∈ [0, 1) and τ ′ satisfying the continuity condition

|τ ′ (x 1 ) -τ ′ (x 2 )| ≤ ρ(|x 1 -x 2 |), x 1 , x 2 ∈ [0; 1). A Jordan arc is of class C ρ if it is a subarc of a Jordan curve of class C ρ .
The following theorem is a simplified version of [15, Theorem 1.1].

Theorem 2.3 ([15]

). Let Γ be a rectifiable Jordan curve. For z 0 ∈ Γ, assume that Γ has an exterior angle of opening απ at z 0 , with 0 < α ≤ 2, formed by C ρ arcs, where ρ is a modulus of continuity satisfying the Alper condition. Then,

F n (z 0 ) = αφ(z 0 ) n + o(1), as n → ∞.
Remark 2.4. In the above result, the case α = 1 corresponds to a point where the boundary form an arc of regularity C ρ , the (excluded) case α = 0 to an inside pointing cusp and the case α = 2 to an outside pointing cusp.

It is often sufficient to consider ρ(x) = x β for some β > 0. The class C ρ is then the Hölder class C 1,β .

The following definition will be used in the next sections as a basic geometric condition about the regularity of the boundary. We also record for further use the following classical result about analytic Jordan curves and an inequality of Kövari and Pommerenke [START_REF] Kövari | On Faber polynomials and Faber expansions[END_REF].

Theorem 2.7. Suppose that Γ is an analytic Jordan curve. Then ψ has an analytic univalent extension in rD c for some r ∈ (0; 1).

Theorem 2.8. [START_REF] Kövari | On Faber polynomials and Faber expansions[END_REF] If K is convex, then for all z ∈ Γ and for all n ≥ 0,

|F n (z) -φ(z) n | ≤ 1.
2.3. Faber polynomials and operators on a Banach space.

Notation. For a complex Banach space X , we denote by B(X ) the Banach algebra of the bounded linear operators from X into itself and by

I = I X the identity on X . The spectrum σ(T ) of T ∈ B(X ) is defined ([5]) by σ(T ) := {λ ∈ C; (T -λI) is not invertible in B(X )}. Lemma 2.9. Let X be a Banach space. Let T ∈ B(X ) be such that σ(T ) ⊂ K. Then, for λ ∈ C, |λ| > 1, we have (ψ(λ)I -T ) -1 = ψ ′ (λ) -1 ∞ n=0 λ -n-1 F n (T ). Proof. Let λ ∈ C, |λ| > 1. Let R = |λ|. Define f (z) = ψ ′ (λ) ψ(λ) -z , z ∈ G R .
It follows from (2.1) that for every z ∈ G R we have

f (z) = ∞ n=0 λ -n-1 F n (z).
As f is an analytic function on G R , we have from Theorem 2.1 the uniform convergence of the series in any compact subset of G R .

Using σ(T ) ⊂ K, the holomorphic functional calculus gives

(ψ(λ)I -T ) -1 = ψ ′ (λ) -1 ∞ n=0 λ -n-1 F n (T ).
In their article [START_REF] Atzmon | Spectral inclusion and analytic continuation[END_REF], A. Atzmon, A. Eremenko and M. Sodin have proved that for any compact set K ⊂ C with connected complement, there exists a sequence of polynomials (P n ) n≥0 and a positive number r such that if T is a bounded operator on X , its spectrum is included in K if and only if

lim sup n→∞ P n (T ) 1/n ≤ r.
If K is connected, this can be expressed in terms of Faber polynomials.

Theorem 2.10 ( [START_REF] Atzmon | Spectral inclusion and analytic continuation[END_REF]). Let X be a Banach space and T ∈ B(X ). Then σ(T

) ⊂ K if and only if lim sup n→∞ F n (T ) 1/n ≤ 1.

Condition on the weak Faber orbits

Let K be a simply connected compact subset of the complex plane with a non empty interior. The aim of this section is to propose some conditions implying that σ(T ) is included in the interior of K. Such results have been proved in the case when K is the closed unit disc by Mlak [START_REF] Mlak | On a theorem of Lebow[END_REF], Weiss [START_REF] Weiss | Weakly l p -stable linear operators are power stable[END_REF], van Neerven [START_REF] Van Neerven | Exponential stability of operators and operator semigroups[END_REF] and Nikolski [START_REF] Nikolski | A Tauberian theorem for the spectral radius[END_REF][START_REF] Nikolski | Estimates of the spectral radius and the semigroup growth bound in terms of the resolvent and weak asymptotics[END_REF].

Let (F n ) n≥0 be the sequence of Faber polynomials associated with K.

Notation. The set of finitely supported sequences will be denoted C (N) and (e n ) n≥0 will denote the canonical algebraic basis of C (N) . The characteristic function of a set A will be denoted by χ A .

Definition 3.1. A Banach space E ⊂ C N is said to be an admissible sequence space if it satisfies the following properties:

• if |a n | ≤ |b n | for all n ∈ N and (b n ) n≥0 ∈ E, then (a n ) n≥0 ∈ E and (a n ) n≥0 E ≤ (b n ) n≥0 E ; • for all k ∈ N there exists a sequence (a n ) n≥0 ∈ E such that a k = 0.
Proposition 3.2. Let E be an admissible sequence space. Then C (N) ⊂ E.

Proof. Let k ∈ N. There exists a sequence (a n ) n≥0 ∈ E such that a k = 0 by the second condition of admissibility. Then

|χ {k} (n)| ≤ |a k | -1 |a n | for all n ∈ N.
It follows then from the first condition of admissibility that χ {k} belongs to E. By taking finite linear combinations, we get that the finitely supported sequences belong to E.

The following definition extends the corresponding unit-disc analogue from [START_REF] Nikolski | Estimates of the spectral radius and the semigroup growth bound in terms of the resolvent and weak asymptotics[END_REF]. Definition 3.3. Let E ⊂ C N be a Banach space and T ∈ B(X ). The operator T is said to be of weak type E if for every pair (x; x * ) ∈ X × X * we have

F n (T )x; x * n≥0 ∈ E.
Theorem 3.5 below, which is the main result of this section, is a generalisation of the following result of J. van Neerven [START_REF] Van Neerven | Exponential stability of operators and operator semigroups[END_REF].

Theorem 3.4 ( [START_REF] Van Neerven | Exponential stability of operators and operator semigroups[END_REF]). Let X be a Banach space, T ∈ B(X ) and E an admissible sequence space such that for any ζ ∈ T

(1, ζ, ζ 2 . . . , ζ N , 0, 0, . . . ) E -→ N →∞ ∞.
If T is of weak type E, then r(T ) < 1.

Theorem 3.5. Let X be a Banach space, T ∈ B(X ) and E an admissible sequence space such that for any ζ ∈ ∂K

(F 0 (ζ), . . . , F N (ζ), 0, 0, . . . ) E -→ N →∞ ∞.
If T is of weak type E, then σ(T ) ⊂ int(K).

Lemma 3.6. Let E ⊂ C N be a Banach space such that for any k ∈ N, the map

(a n ) n≥0 → a k is bounded. Suppose that T ∈ B(X ) is a weak type E operator. Then sup F n (T )x; x * n≥0 E ; x * ≤ 1, x ≤ 1 < ∞.
Proof. Let us prove that for any x ∈ X , the map

f x :X * → E x * → F n (T )x; x * n≥0 is bounded. Indeed, f x is well defined because T is a weak type E operator. Let (x * n ) n≥0 ⊂ X * , x * ∈ X * and r ∈ E be such that x * n → x * and f x (x * n ) E -→ r.
We infer from the boundedness of the maps (a n ) n≥0 → a k , that for any k ∈ N

[f x (x * n )] k -→ n→∞ r k , where [f x (x * n )] k denotes the k-th element of the sequence f x (x * n ). For any k ∈ N [f x (x * n )] k = F k (T )x; x * n -→ n→∞ F k (T )x; x * = [f x (x * )] k .
Thus f x (x * ) = r and, according to the closed graph theorem [START_REF] Conway | A course in functional analysis[END_REF], f x is bounded. In particular, for any x ∈ X

(3.1) sup F n (T )x; x * n≥0 E ; x * ≤ 1 < ∞.
For any x * ∈ X * let

g x * :X → E x → f x (x * ).
Then, in the same way as for f x , we can prove that, for every x * , the map g x * is bounded. Using (3.1) we obtain that sup

x * ≤1 g x * (x) < ∞,
for an arbitrary x ∈ X . Using the uniform boundedness principle [START_REF] Conway | A course in functional analysis[END_REF] we infer that {g

x * } x * ≤1 is a bounded set. Thus sup F n (T )x; x * n≥0 E ; x * ≤ 1, x ≤ 1 < ∞,
which completes the proof.

Lemma 3.7. Let T ∈ B(X ) be a linear operator acting on the Banach space X . For all λ ∈ ∂σ(T ), all ε > 0 and all N ∈ N there exist x N ∈ X and

x * N ∈ X * satisfying x N = 1, x * N = 1 and | F n (T )x N ; x * N | > |F n (λ)| -ε, n = 0, 1, . . . N.
Proof. Let λ ∈ ∂σ(T ), ε > 0 and N ∈ N. As λ ∈ ∂σ(T ), we have that λ belongs to σ app (T ), the approximate point spectrum of T (cf. [START_REF] Conway | A course in functional analysis[END_REF]Chapter 7.6]). By definition, this means that there exists a sequence (y n ) n≥0 in X such that y n = 1 for all n ∈ N and

T y n -λy n -→ n→∞ 0.
Thus, for each fixed k ∈ N,

lim n→∞ F k (T )y n -F k (λ)y n = 0.
Let n 1 be such that

F k (T )y n 1 -F k (λ)y n 1 < ε, ∀k = 0, 1, . . . N. Let x N = y n 1 and x * N ∈ X * be such that x * N = 1 and x * N ; x N = 1. Then for k ∈ {0, 1, . . . N}, we have | F k (T )x N ; x * N | ≥ | F k (λ)x N ; x * N | -| F k (T )x N -F k (λ)x N ; x * N | > |F k (λ)| -ε.
Proof of Theorem 3.5. From Lemma 3.6 we know that there exists M ≥ 0 such that for any x ∈ X , x * ∈ X * , with x = 1 and x * = 1, we have

F n (T )x; x * n≥0 E ≤ M.
Suppose that σ(T ) int(K), then there exists λ ∈ ∂σ(T )\int(K). From Lemma 3.7 we get that for every ε > 0 and N ∈ N there exist x N ∈ X and

x * N ∈ X * satisfying x N ≤ 1, x * N ≤ 1 and | F n (T )x N ; x * N | > |F n (λ)| -ε, n = 0, 1, . . . N. Therefore, | F n (T )x N ; x * N | ≥ |F n (λ)|χ {0,...N } (n) -εχ {0,...N } (n) for all n. Thus F n (T )x N ; x * N n≥0 E ≥ (F 0 (ζ), . . . , F N (ζ), 0, 0, . . . ) E -ε χ {0,...N } E .
We obtain that for any ε > 0 and N ∈ N

(F 0 (ζ), . . . , F N (ζ), 0, 0, . . . ) E -ε χ {0,...N } E ≤ M.
Hence, for every N ∈ N we have

(F 0 (ζ), . . . , F N (ζ), 0, 0, . . . ) E ≤ M, which gives the desired contradiction if λ ∈ ∂K. If λ / ∈ K, then |F n (λ)| 1/n -→ n→∞ R > 1.
But we know that for any ζ ∈ ∂K, lim sup n→∞ |F n (ζ)| 1/n ≤ 1, and

(F 0 (ζ), . . . , F N (ζ), 0, 0, . . . ) E -→ N →∞ ∞.
Thus, if λ / ∈ K we get from the first condition of admissibility that

(F 0 (ζ), . . . , F N (ζ), 0, 0, . . . ) E -→ N →∞ ∞,
which gives a contradiction.

We now give some illustrating examples; we start with one based on weighted l p spaces.

Corollary 3.8. Let 1 ≤ p < ∞ and let (ω n ) n≥0 be a sequence of non negative numbers. We suppose that for any ζ ∈ ∂K,

∞ n=0 ω n |F n (ζ)| p = ∞.
Let X be a Banach space. Let T ∈ B(X ) be such that for every (x, x * ) ∈ X × X * we have

∞ n=0 ω n | F n (T )x; x * | p < ∞.
Then σ(T ) ⊂ int(K).

Corollary 3.9. Let (ω n ) n≥0 be a sequence of non negative numbers. We suppose that for any ζ ∈ ∂K, lim sup

n→∞ ω n |F n (ζ)| = ∞.
Let X be a Banach space. Let T ∈ B(X ) be such that for every (x, x * ) ∈ X × X * we have sup

n≥0 ω n | F n (T )x; x * | < ∞.
Then σ(T ) ⊂ int(K). Proof of Corollaries 3.8 and 3.9. We have that

lim |z|→∞ |φ(z)| = ∞.
Therefore, there exists a R > 1 such that

σ(T ) ∩ {z : |φ(z)| ≥ R} = ∅.
Thus, we get from Theorem 2.10 that lim sup n→∞ F n (T ) 1/n < R.

Let (ω n ) n≥0 be the positive sequence defined by

ωn = ω n if ω n = 0, R -np if ω n = 0. For every (x, x * ) ∈ X × X * we have ∞ n=0 ωn | F n (T )x; x * | p < ∞.
If p < ∞, then the space

l p (ω) := (a n ) n≥0 ; ∞ n=0 ωn |a n | p < ∞ with the norm: (a n ) n≥0 l p (ω) = ∞ n=0 ωn |a n | p 1/p
is an admissible sequence space and the condition (F 0 (ζ), . . . , F N (ζ), 0, 0, . . .

) l p (ω) -→ N →∞ ∞ is equivalent to ∞ n=0 ωn |F n (ζ)| p = ∞. If p = ∞, then the space l ∞ (ω) = (a n ) n≥0 ; sup n≥0 ωn |a n | < ∞ with the norm (a n ) n≥0 l ∞ (ω) = sup n≥0 ωn |a n |
is an admissible sequence space and the condition

(F 0 (ζ), . . . , F N (ζ), 0, 0, . . . ) l p (ω) -→ N →∞ ∞ is equivalent to lim sup n→∞ ωn |F n (ζ)| = ∞.
It is then sufficient to apply Theorem 3.5 with the space E = l p (ω) to conclude.

We give now another application of Theorem 3.5 based on Orlicz spaces.

Definition 3.11. A function Φ : [0; ∞) -→ [0; ∞) is a Young function if it is convex and it satisfies Φ(x) x -→ x→0 0 and Φ(x) x -→ x→∞ ∞.
For any complex sequence (x n ) n≥0 we set

M Φ (x) = ∞ n=0 Φ(|x n |).
The Orlicz space L Φ is the space of all sequences (x n ) n≥0 such that there exists k > 0 such that M Φ (kx) < ∞.

Theorem 3.12. Let Φ be a Young function, the Orlicz space L φ with the norm

(x n ) n≥0 L Φ = inf{k; M Φ 1 k x ≤ 1}
is a Banach space and if Φ(t) > 0 for all t > 0, then

χ {0,...,n-1} L Φ -→ n→∞ ∞.
For the proof we use the following useful result.

Lemma 3.13 ([18]

). Let ϕ : R + -→ R + be non-decreasing with ϕ(t) > 0 for all t > 0. Then there exists a Young function Φ such that the Orlicz space L Φ contains every sequence (a n ) n≥0 such that Proof. Replacing ϕ by a multiple of ϕ we may assume that ϕ(1) = 1. Let Φ be defined by

Φ(t) = t 0 ϕ(s)ds.
Then Φ is a Young function and if (a

n ) n≥0 is such that n≥0 ϕ(|a n |) < ∞, then the set of indexes n such that |a n | > 1 is finite and if |a n | ≤ 1, then Φ(|a n |) ≤ ϕ(|a n |). Thus we get n≥0 Φ(|a n |) < ∞, so (a n ) n≥0 ∈ L Φ .
But Φ is strictly positive and increasing on (0; ∞), therefore we obtain

χ {0,...,n-1} L Φ = inf k; n-1 k=0 Φ(1/k) ≤ 1 -→ n→∞ ∞.
As Orlicz spaces of sequences are admissible sequence spaces we obtain the following result.

Corollary 3.14. We suppose that (|F n (ζ)|) n≥0 is eventually bounded below by a positive number for every ζ ∈ ∂K. Let T ∈ B(X ) and let ϕ : R + -→ R + be a non-decreasing function with ϕ(t) > 0 for all t > 0. If for every x ∈ X and

x * ∈ X * ∞ n=0 ϕ | F n (T )x; x * | < ∞, then σ(T ) ⊂ int(K).
Remark 3.15. According to Remark 2.6, the above condition about the sequence (|F n (ζ)|) n≥0 is automaticaly satisfied whenever ∂K is an Alper curve with (possible) angles.

Definition 3.16. The function ϕ is said to satisfy the ∆ 2 -condition at 0 if there exist ε > 0 and K > 0 such that for any x ∈ [0; ε], we have ϕ(t/2) ≥ Kϕ(t).

Corollary 3.17. We suppose that (|F n (ζ)|) n≥0 is eventually bounded below by a positive number for every ζ ∈ ∂K. Let T ∈ B(X ), let ϕ : R + -→ R + be a non-decreasing function satisfying the ∆ 2 -condition at 0 with ϕ(t) > 0 for t > 0, and let (ω n ) n≥0 be a sequence of non negative numbers such that

∞ n=0 ω n ϕ(ω n ) = ∞.
If for every x ∈ X and

x * ∈ X * ∞ n=0 ϕ ω n | F n (T )x; x * | < ∞, then σ(T ) ⊂ int(K).
Proof. Adapting the proof of Corollary 3.8, we may suppose ω n to be positive for every n. Replacing if necessary ω n by min(ω n , ε), we may also assume that ω n ∈ (0; ε] for each n. Let Φ be as in Lemma 3.13 and let E be the space

E = (x n ) n≥0 ; (ω n x n ) n≥0 ∈ L Φ , with the norm (x n ) n≥0 E = (ω n x n ) n≥0 L Φ .
The ∆ 2 condition gives that for any t ∈ (0; ε],

Φ(t) = t 0 ϕ(s)ds ≥ t 2 ϕ(t/2) ≥ Kt 2 ϕ(t).
Let k > 1, and m be the integer part of (ln(k)/ ln(2) + 1). Then

n-1 j=0 Φ(ω j /k) ≥ K m 2 m n-1 j=0 ω j ϕ(ω j ).
This proves that χ {0,...,n-1} E -→ n→∞ ∞.

An application of Theorem 3.5 completes the proof.

ϕ(r) (r n F n (T )x; x * ) n≥0 2 -→ r→1 0,
for every x ∈ X and every x * ∈ X * . Then

σ(T ) ⊂ int(K).
Lemma 3.19. Let ϕ be a positive function on [0; 1) such that ϕ(r) tends to infinity and √ 1 -r ϕ(r) tends to 0 as r tends to 1. Then the space E defined by

E = (x n ) n≥0 ; 1 ϕ(r) (r n x n ) n≥0 2 -→ r→1 0 ,
with the norm

(x n ) n≥0 E = sup r∈(0;1) 1 ϕ(r) (r n x n ) n≥0 2 ,
is an admissible Banach function space over N such that

χ {0,...,n} E -→ n→∞ ∞.
Proof. It is easy to see that E is an admissible sequence space. Furthermore, we have

χ {0,...,n-1} E = sup r∈(0;1) 1 ϕ(r) (1, r, r 2 , . . . , r n-1 , 0, 0, . . . ) 2 ,

and

(1, r, r 2 , . . . , r n-1 , 0, 0, . . .

) 2 2 = n-1 k=0 r 2k = 1 -r 2n 1 -r 2 .
Therefore, 1 ϕ(r)

(1, r, r 2 , . . . , r n-1 , 0, 0, . . .

) 2 = 1 ϕ(r) √ 1 -r n √ 1 -r √ 1 + r n √ 1 + r ≥ 1 2 √ 1 -r n ϕ(r) √ 1 -r .
Let M > 0. As √ 1 -rϕ(r) tends to 0 as r tends to 1, there exists

r 0 < 1 such that 1 ϕ(r 0 ) √ 1 -r 0 > 4M.
Let N be such that 1 -r N 0 > 1/2. We get that for each n ≥ N we have χ {0,...,n-1} E ≥ M.

The proof is complete.

Proof of Corollary 3.18. Replacing, if necessary, ϕ(r) by max(ϕ(r); (1 -r) -1/4 ), we may assume that ϕ(r) -→ r→1 ∞. We can then apply Lemma 3.19 and Theorem 3.5 to obtain the inclusion σ(T ) ⊂ int(K).

Condition on the weak resolvent

The purpose of this section is, as in the previous one, to find conditions assuring spectral inclusion in the interior of a given simply connected compact set K. We suppose here that the spectrum of the bounded linear operator under consideration is included in the compact set K. Nikolski [START_REF] Nikolski | A Tauberian theorem for the spectral radius[END_REF] called conditions of this type Tauberian conditions for the spectral radius. The main interest of this approach is that it is well adapted to conditions expressed in terms of the weak resolvent, that is to say on the functions

z / ∈ K → (zI -T ) -1 x; x * ,
where x ∈ X and x * ∈ X * .

We recall that (F n ) n≥0 denotes the sequence of Faber polynomials associated with K and (e n ) n≥0 will denote the canonical algebraic basis of C (N) .

Let R ⊂ C N be a Banach space such that (1) For any k ∈ N, the map (a n ) n≥0 → a k is bounded.

(2) The set of finitely supported sequences

C (N) is dense in R. As C (N) is dense in R, each element ϕ in R * is characterized by the sequence (ϕ n ) n≥0 , where ϕ n = e n ; ϕ R .
This allows us to identify R * with a subset of C N . Via this identification, we can translate the condition (1) as "

C (N) is included in R * ". Proposition 4.1. If ϕ ∈ R * and f ∈ C (N)
, the duality mapping is given by

f ; ϕ R = ∞ n=0 f n ϕ n .
If f ∈ R and ϕ ∈ C (N) ⊂ R * , this equality still holds.

Proof. Let ϕ ∈ R * be finitely supported and let n 0 be such that e n ; ϕ = 0 whenever n ≥ n 0 . From (2), we know that there exists a sequence (g k ) k≥0 of elements in

C (N) such that g k -→ k→∞ f in R.
To prevent any confusion, we will denote by [g k ] n the n-th element of the sequence

g k . Hence, f ; ϕ R = lim k→∞ g k ; ϕ R .
We also have

g k ; ϕ R = n 0 n=0 [g k ] n ϕ n .
Therefore, using the hypothesis (1), we get that [g k ] n -→ k→∞ f n , for every n.

We obtain

f ; ϕ R = lim k→∞ g k ; ϕ R = ∞ n=0 f n ϕ n .
The crucial condition needed for our Tauberian conditions is the following one.

(3) For any ζ ∈ ∂K, the map

V ζ , defined in C (N) ⊂ R * by V ζ : ϕ → n≥0 ϕ n F n (ζ)
is not bounded. Theorem 4.2. Let R ⊂ C N be a Banach space satisfying conditions (1) to (3) for K. Let T ∈ B(X ) be a weak type R operator with σ(T ) ⊂ K. Then, in fact, σ(T ) ⊂ int(K).

Proof. Lemma 3.6 implies that

C := sup F n (T )x; x * n≥0 R ; x * ≤ 1, x ≤ 1 < ∞.
Using Proposition 4.1 we get, for every

ϕ ∈ C (N) ⊂ R * , n≥0 ϕ n F n (T ) = sup n≥0 ϕ n F n (T )x; x * ; x ≤ 1, x * ≤ 1 ≤ sup F n (T )x; x * n≥0 R ; x ≤ 1, x * ≤ 1 ϕ R * ≤ C ϕ R * .
Recall that, according to the spectral mapping theorem [START_REF] Conway | A course in functional analysis[END_REF], we have σ(P (A)) = P (σ(A)) for every polynomial P . Hence if ζ ∈ σ(T ), we have

n≥0 ϕ n F n (ζ) ≤ r n≥0 ϕ n F n (T ) ≤ n≥0 ϕ n F n (T ) ≤ C ϕ R * ,
where r(T ) = sup{|λ|; λ ∈ σ(T )} is the spectral radius of T . We deduce from this inequality that for any ζ ∈ σ(T ), the map V ζ is bounded in C (N) ⊂ R * . This gives us ∂K ∩ σ(T ) = ∅.

Remark 4.3. A weaker version of Corollary 3.8 can be deduced from Theorem 4.2 (one has to suppose that σ(T ) ⊂ K). One can notice that Corollary 3.9 cannot be deduced from Theorem 4.2.

A novelty of the approach of this section is that we can get a generalisation of the following result due to Mlak [START_REF] Mlak | On a theorem of Lebow[END_REF]. [START_REF] Mlak | On a theorem of Lebow[END_REF]). Let H be a Hilbert space and T ∈ B(H). Suppose that r(T ) ≤ 1 and that for any (x, y) ∈ H × H, z → (zI -T ) -1 x; y ∈ H 1 (D).

Theorem 4.4 ([
Then r(T ) < 1.

To extend Mlak's result, we need a generalisation of Hardy spaces of the disc to more general domains. We follow the setting of [START_REF] Duren | Theory of H p spaces[END_REF].

p := (x n ) n≥0 ; f (z) = ∞ n=0
x n z n ∈ H p (D) , with the norm

(x n ) n≥0 = ∞ n=0 x n z n H p (D)
satisfies conditions (1) to (3).

Proof. It is easy to check that conditions (1) and ( 2) are satisfied. Let us prove that the condition (3) is also satisfied.

Suppose on the contrary that condition (3) does not hold. Then there exists

ζ ∈ ∂K such that V ζ is bounded in C (N) ⊂ R * . Let a n = F n (ζ) for every n.
According to the Hahn-Banach theorem, there exists a bounded extension of V ζ to (h p ) * still denoted V ζ . Hence there exists M ≥ 0 such that for every

x * = (x * n ) n≥0 ∈ (h p ) * , |V ζ x * )| ≤ M (x * ) (h p ) * .
But f (r.) H p ≤ f H p for every r ∈ (0; 1) and every f ∈ H p . Thus, for every

(x * n ) n≥0 ∈ (h p ) * , (r n x * n ) n≥0 (h p ) * = sup (r n x * n ) n≥0 ; (x n ) n≥0 ; (x n ) n≥0 h p ≤ 1 = sup (x * n ) n≥0 ; (r n x n ) n≥0 ; (x n ) n≥0 h p ≤ 1 ≤ (x * n ) n≥0 (h p ) * .
So, for every r ∈ (0; 1) and for every (

x * n ) n≥0 ∈ (h p ) * , ∞ n=0 x * n r n a n ≤ M (x * n ) n≥0 (h p ) * .
Consequently, (r n a n ) n≥0 h p ≤ M for each r ∈ (0; 1), and thus (a n ) n≥0 is in h p . We have f (e iθ ) = a n e inθ ∈ H p (T) and a n = F n (ζ) -→ n→∞ 0. This contradiction completes the proof.

Proof of Corollary 4.7. We get from Lemma 4.9 and Theorem 4.2 that it is sufficient to prove that for any x ∈ X and x * ∈ X * we have

( F n (T )x; x * ) n≥0 ∈ h 1 .
From Lemma 2.9 we get that for any r > 1,

|φ(z)|=r | (zI -T ) -1 x; x * ||dz| = z∈rT | (ψ(z)I -T ) -1 x; x * ||ψ ′ (z)||dz| = z∈rT ∞ n=0 z -n-1 F n (T )x; x * |dz| = z∈r -1 T ∞ n=0 z n F n (T )x; x * r|dz|. This proves that z → ∞ n=0 z n F n (T )x; x * ∈ H 1 (D).
Remark 4.10. The space h 1 is not an admissible sequence space, thus Corollary 4.7 cannot be deduced from Theorem 3.5.

Explicit estimates

5.1. General case. We suppose in this section that Γ is an analytic Jordan curve. Let E be a Banach function space over N containing (r n ) n≥0 for any r ∈ (0; 1) and satisfying the condition

χ {0,...,n-1} E -→ n→∞ ∞.
Let T ∈ B(X ) be a weak type E operator.

As we have supposed Γ to be analytic, we get from Theorem 2.7 that the map ψ : D c -→ K c has an analytic extension to r 0 D c for some r 0 < 1.

Notation. For any r ∈ (r 0 ; 1), let Γ r be the analytic Jordan curve defined by Γ r = {ψ(z); |z| = r} and let K r be the simply connected compact delimited by Γ r .

We know from Theorem 3.5 that the spectrum of T is included in int(K). Having in mind possible applications in numerical analysis, it is the aim of this section to give an estimate of the "shrinking radius" r ∈ (r 0 ; 1) such σ(T ) ⊂ K r . In the case of the disc, this was done by N. Nikolski in [START_REF] Nikolski | Estimates of the spectral radius and the semigroup growth bound in terms of the resolvent and weak asymptotics[END_REF]. Our estimate of the "shrinking radius" is given in terms of the constant C(T, E) defined by

C(T, E) = sup F n (T )x; x * n≥0 E ; x * ≤ 1, x ≤ 1 .
Notation. For r ∈ (r 0 ; 1) we denote

p(r) = inf r≤|φ(ζ)|<1 (F n (ζ)) n≥0 E
and, for M > 0 p -1 (M) = r 0 if {r ∈ (r 0 ; 1); p(r) ≤ M} = ∅, sup{r ∈ (r 0 ; 1); p(r) ≤ M} otherwise. Theorem 5.1. Let T be a weak type E operator for the compact K and set r = p -1 (C(T, E)). Then σ(T ) ⊂ K r .

Proof. We argue by contradiction. Suppose that σ(T ) K r ; then there exists λ ∈ ∂σ(T )\K r . From Lemma 3.7 we get that for any ε > 0 and any N ∈ N, there exist x N ∈ X and This contradiction completes the proof.

Concrete examples.

In case of some concrete compacts, the previously described quantities can be explicitly estimated. We begin with the case of a compact K such that K r 0 is convex. The following proof has been kindly suggested to the author by a referee.

Corollary 5.2. Suppose r < 1 and 1 ≤ p < ∞. Let T be a weak type l p operator for a compact K delimited by an analytic Jordan curve. Suppose that K r 0 is convex and As F n -φ n is an analytic function on C\K 0 and vanishes at ∞, the maximum principle gives us that for any z / ∈ K r 0 ,

C(T, l p ) ≤ 1 1 -r p 1/p - 1 1 -r p
|F n (z) -φ(z) n | ≤ r n 0 . Let w ∈ C be such that |w| > r 0 . Then F n (ψ(w)) n≥0 l p = (w n + (F n (ψ(w)) -w n )) n≥0 l p ≥ (w n ) n≥0 l p -(r n 0 ) n≥0 l p . Thus, for every t ∈ (r 0 ; 1), we have

p(t) ≥ ∞ n=0 t np 1/p - ∞ n=0 (r 0 ) np 1/p ≥ 1 1 -t p 1/p - 1 1 -(r 0 ) p 1/p .
The conclusion follows now from Theorem 5.1.

We now turn to the more specific case of an ellipse. Let K be the compact delimited by the ellipse with the foci -1 and 1, and semi-axes

a = 1 2 R + 1 R and b = 1 2 R - 1 R ,
with R > 1. Then (cf. Section 2) the function ψ is given by

ψ(w) = 1 2 Rw + 1 Rw
and the sequence of Faber polynomials is given by

F n (z) = 2
R n C n (z), n ≥ 1, where (C n ) n∈N is the sequence of Chebychev polynomials. We also have F n (ψ(w)) = w n + 1 R 2n w n (n ≥ 1). Corollary 5.3. Let K be the elliptic compact set described above. Suppose r < 1 and 1 ≤ p < ∞. Let T ∈ B(X ) be a weak type l p operator for K. Suppose that

C(T, l p ) ≤ 1 1 -r p 1/p - 1 1 -1 rR 2 p 1/p . Then σ(T ) ⊂ K r .
Remark 5.4. This estimate is better than the one given in Corollary 5.2 as r 0 = 1/R > 1/rR 2 for r > r 0 .

Remark 3 . 10 .

 310 According to Remark 2.6, if ∂K is an Alper curve with (possible) angles, the conditions on the sequence (ω n ) n≥0 can be replaced by

n≥0ϕ(

  |a n |) < ∞, and satisfies χ {0,...,n-1} L Φ -→ n→∞ ∞.

Corollary 3 . 18 .

 318 We suppose that (|F n (ζ)|) n≥0 is eventually bounded below by a positive number for every ζ ∈ ∂K. Let T ∈ B(X ). Suppose there exists a positive function ϕ on [0; 1) such that √ 1 -r ϕ(r) -→ r→1 0 and such that 1

  x * N ∈ X * satisfying x N = 1, x * N = 1 and | x * N ; F k (T )x N | > |F k (λ)| -ε, k ∈ {0, 1, . . . N}. By definition of C(T, E), we have C(T, E) ≥ F n (T )x N ; x * N n≥0 E ≥ (F 0 (ζ), . . . , F N (ζ), 0, 0, . . . ) -εχ {0,...,N } E .Thus, for every N ∈ N,F 0 (λ), F 1 (λ), . . . , F N (λ), 0, 0, . . . E ≤ C(T, E). But F n (λ) ∼ φ(λ) n , n → ∞, and |φ(λ)| < 1. Therefore (F n (λ)) n≥0 ∈ E and F n (λ) n≥0 E ≤ C(T, E).Using λ / ∈ K r , and taking into account the definition of r as r = p -1 (C(T, E)), we have F n (λ) n≥0 E > C(T, E).

  ) ⊂ K r .Proof. One can get from the inequality of Kövari and Pommerenke[START_REF] Kövari | On Faber polynomials and Faber expansions[END_REF] (cf. Theorem 2.8) and a change of variable that for any z ∈ Γ r 0 and any n ∈ N, we have |F n (z) -φ(z) n | ≤ r n 0 .

  Definition 4.5 ([6]). For p ∈ [1; ∞), a function f analytic in C\K is said to be of class E p (C\K) if there exists a sequence of rectifiable Jordan curves C 1 , C 2 , . . . in C\K such that C n eventually surrounds each compact subset of C\K, and such that sup Suppose that lim n |F n (ζ)| = 0 for no ζ ∈ ∂K. Let T ∈ B(X ) be such that σ(T ) ⊂ K. If for any x ∈ X and x * ∈ X * we have z → (zI -T ) -1 x; x * ∈ E 1 (C\K), According to Remark 2.6, if ∂K is an Alper curve with (possible) angles, the condition on the sequence (|F n (ζ)|) n≥0 is automaticaly satisfied. Lemma 4.9. Suppose that lim n |F n (ζ)| = 0 for no ζ ∈ ∂K. Then the space h

	Remark 4.8.
	n	Cn	|f (z)| p |dz| < ∞.
	The following characterization ([6, Theorem 10.1]) of the functions in E p (C\K)
	is well adapted here as it uses the conformal representation φ.
	Theorem 4.6 ([6]). An analytic function f is of class E p (C\K) if and only if,
	sup r∈(1;2) |φ(z)|=r	|f (z)| p |dz| < ∞.
	We can now state the generalisation of Mlak's result [11] as a corollary to The-
	orem 4.2.
	Corollary 4.7. then σ(T ) ⊂ int(K).

Acknowledgement: I would like to express my sincere gratitude to my advisor Catalin Badea for proposing the topic of this paper and for his guidance and support.

I would also like to thank the two referees for their constructive remarks, critical reading and suggestions for improvements.

Proof. It is easy to verify that r 0 = 1/R. Let w ∈ C be such that |w| > 1/R. Then

Thus, for every t ∈ (1/R; 1),

The conclusion follows now from Theorem 5.1.


M Ali Mirzaei
email: ali.mirzaei@ensam.eu

Jean-R Émy Chardonnet
email: jean-remy.chardonnet@ensam.eu

Christian P Ère
email: christian.pere@ensam.eu

‡ Fr

Designing a 3D Navigation System Using Cognitive Factors

Keywords: H, 1, 2 [Models and Principles]: User/Machine Systems-Human factors; H, 5, 2 [Information Interfaces and Presentation]: User Interfaces-User-centered design

This paper focuses on the measurement and the mathematical definition of cognitive parameters of designing a navigation system based on these parameters. The nausea level due to different velocities of a 3D scene, the user head rotation around Yaw, Roll and Pitch axes, the delay between navigation device stimuli and the 3D display movement are measured. Appropriate mathematical functions are fitted to the measurements. A sickness level is defined as an accumulation of a nausea level due to the velocity and the delay. Assigning an analog control button on the navigation device will help the user to adjust the speed. The records of the test-bed and practical experiments prove the effectiveness of this kind of design. Moreover, due to the parametric design of the system, any maloperation can be readjusted with further inquiries over the specific applications. In addition, any amendment or modification performance can be compared with the parametric criteria.

INTRODUCTION

Navigation is an important part of any real or virtual dynamic system. A lot of navigation and interaction devices with different functionalities for different applications were developed during the last decades. However, the list of navigation and interaction devices is not limited only to what we literary know in robotics, haptics, remote control systems. These devices cover a wide spectrum of applications and devices, for instance a gear handle of our personal car is a kind of navigation device. Navigation and interaction devices of Virtual Reality (VR) systems or Virtual Environments (VE) attracted especial attention due to the rapid development of robotics and game consoles. Three-dimensional virtual environments (VEs) are used in fields as diverse as manufacturing [START_REF] Mujber | Virtual reality applications in manufacturing process simulation[END_REF], medicine [START_REF] John | The impact of web3d technologies on medical education and training[END_REF], construction [START_REF] Setareh | Development of a virtual reality structural analysis system[END_REF], psychotherapy [START_REF] Riva | Cybertherapy: Internet and virtual reality as assessment and rehabilitation tools for clinical psychology and neuroscience[END_REF], design [START_REF] Maher | An agent approach to supporting collaborative design in 3d virtual worlds[END_REF], and education [START_REF] Pan | Virtual reality and mixed reality for virtual learning environments[END_REF]. They also play an important role in the investigation of spatial processes, such as examining directional knowledge [START_REF] Waller | The walkabout: Using virtual environments to assess largescale spatial abilities[END_REF] or assessing spatial abilities [START_REF] Waller | The walkabout: Using virtual environments to assess largescale spatial abilities[END_REF] allowing researchers to design realistic experimental settings and a flexibly record user behavior [START_REF] Jansen-Osmann | Using desktop virtual environments to investigate the role of landmarks[END_REF].

Topics such as navigation devices, metaphors, menus, user's representations, etc., are just a short list of these research topics. The development of useful VE applications, however, needs an appropriate user interface in a real environment and requires optimization of the most basic interactions, in particular object manipulation, navigation, moving inside 3D; hence, users can concentrate on high-level tasks rather than on low level motor activities [START_REF] Stanney | Realizing the full potential of virtual reality: human factors issues that could stand in the way[END_REF]. However, recently, some cognitive issues have emerged in VE such as motion sickness, cyber sickness; the effect of different metaphors on the end-users turns this topic to a serious concern of researchers. It seems selection of interaction devices for navigation and manipulation considering cognitive issues is one of the most challenging topics of engineering researches.

Travel is the motor component of navigation. Navigation is the task of performing the actions that move us from our current location to a new target location or in the desired direction. In addition, watching a specific part of a 3D scene from different perspectives is demanding in almost all 3D applications [START_REF] Göbel | Industrial applications of ves[END_REF]. In physical environments, travel is often a no-brainer, or to be more precise, it involves unconscious cognition. Therefore, it is quite crucial that the 3D scene moves with the same (perceivable) speed as a human. There are many different reasons why a user might need to perform a 3D travel task. Understanding the various types of travel tasks is important because the usability of a particular technique often depends on the task for which it is used.

Although traveling is among the most important interactions in VEs [START_REF] Mine | Virtual environment interaction techniques[END_REF], we are not aware of any formal experimental studies which propose a design with the support of mathematics, and systematically evaluate and categorize traveling techniques for immersive traveling inside VEs. Prior research relates primarily to the assessment of user performance as a function of the properties of input and output devices [START_REF] Watson | Evaluation of the effects of frame time variation on vr task performance[END_REF]. In contrast, the focus of some study is on human factor aspects of mappings between the user input (captured by input devices) and the resulting actions in VEs [START_REF] Mine | Virtual environment interaction techniques[END_REF].

Tracking is one of the fundamental tasks for navigation in 3D environments.

Currently, there is little understanding of how navigation interfaces should be designed to maximize user performance in VE [START_REF] Mine | Virtual environment interaction techniques[END_REF] and provide sickness-free handling gadget. Research that systematically investigates human factors and design implications of immersive navigation tasks, devices, metaphors, remains sparse [START_REF] Herndon | The challenges of 3d interaction: a chi'94 workshop[END_REF]; consequently, VE designers have had to rely on their intuition and common sense rather than on the guidance of established theory and research results. However, as Brooks [START_REF] Brooks | Grasping reality through illusioninteractive graphics serving science[END_REF] has noted, the uninformed and untested intuition for metaphor design is almost always wrong.

Although the diversity of VE applications makes it necessary to design VE interfaces that support domain dependent needs [START_REF] Chen | Effectiveness of cloning techniques for architectural virtual environments[END_REF], some tasks are common to all VE applications and are essential, even when they are not the main objective of a user in a VE. Navigation can be defined as the process whereby people determine where they are, where everything else is and how to get to particular objects or places [START_REF] Jul | Navigation in electronic worlds: a chi 97 workshop[END_REF]. Navigation is the aggregate task of way finding and motion. Way finding is the cognitive element of navigation. It does not involve movement of any kind but only the tactical and strategic parts that guide movement [START_REF] Darken | Spatial orientation, wayfinding, and representation[END_REF]. The navigation behavior of users in VEs has been investigated to a large degree [START_REF] Darken | A toolset for navigation in virtual environments[END_REF]. To navigate successfully, people must plan their movements using spatial knowledge they have gained about the environment and which they store as a mental map. However, accurate spatial knowledge of VEs typically develops very slowly after long periods of navigation or study, and users may not always be willing to spend this time [START_REF] Ruddle | The effects of maps on navigation and search strategies in very-large-scale virtual environments[END_REF]. Thus, if the navigation support provided by user interfaces of VEs is insufficient, people become disoriented and get lost.

In fact the final objective of current researches is to answer these two questions: 1) how can we select a metaphor and the corresponding device for a specific task to have comfortable and more natural (close to human action in real life) navigation? 2) if a device is selected for interaction, how can the parameters of the device be adjusted to remove any cognitive problems? (or how can we have a sickness-free navigation with the selected device?) A lot of navigation metaphors and their corresponding devices have been proposed during the last decades [START_REF] Bowman | 3D user interfaces: theory and practice[END_REF]. However, the current literature will concentrate on tracking and traveling in a 3D scene, taking into account cognitive parameters. It means, first we introduce available navigation, we evaluate some cognitive aspects and introduce some cognitive parameters, then, based on these parameters, a mathematical model is proposed to be used in future designs or studies.

This paper is organized as follows: traveling and navigation metaphors will be described and implemented in section 2. In section 3, a test bench for measurement will be proposed and the apparatus of the real-test will be described. The measurements will be illustrated and interpreted based on mathematical definitions in section 4. In section 5, a mathematical model of navigation interfaces based on cognitive parameters will be proposed, before the conclusion.

IMPLEMENTATION

Knowing the sub-tasks definition for any kind of navigation in VEs will help to design a generic traveling and tracking metaphor (see Figure 1). Assume we are planning a travel from point A to E. Suppose points B, C and D are located between these two points. The very basic question is how to go from point A to E? If the basic function of traveling is defined as going from point A to B, then the entire travel is a combination of four small travels or applying four times this function. Moreover, this strategy could be employed everywhere with any kind of navigation tool, and it is quite a platform independent definition. That is why defining these detailed metaphors has great deal of importance. The travel function should have a starting point, a target point, a specification of velocity and the acceleration and deceleration rate. For this purpose, we use VR Jugglua. VR Jugglua scripting language manages the relation between virtual reality software platforms for a single and cluster support in one side, and those who are designing the efficient content for interaction and navigation metaphors through prototyping. VR JuggLua is a high-level virtual reality application framework based on combining Lua, as a dynamic and interpreted language, with VR Juggler and OpenScene-Graph. This framework allows fully-featured immersive applications to be written entirely in Lua, and also supports the embedding of the Lua engine in C++ applications. Jugglua has been successfully used in an immersive application implementing two different navigation techniques entirely in Lua, and a physically-based virtual assembly simulation, where C++ code handles physics computations while Lua code handles all display and configuration. The required function was entirely developed in this scripting language. Like native C++ VR Juggler applications, VR JuggLua-based applications can support a wide variety of systems ranging from a single desktop machine to a 49-node cluster successfully. Sometimes, it is necessary to have access to the child node of the 3D model. This facility is provided by adding some functions and classes. The osgLua (OpenSceneGraph Lua) introspection-based bindings facilitate scene-graph manipulation from Lua code. A thread-safe run buffer allows new Lua code to be passed to the interpreter during run time, supporting interactive creation of scene-graph structures. One of the best part of Lua language is a navigator test bench which facilitates the execution of files with lua extension ( * .lua). Figure 2 shows the test bench GUI. There is a part for scripting in the test bench that helps the user to test only a single line of command, exactly in the same way as done in the command line of MATLAB. After successful connection of the device, a 3D model can be easily uploaded and the user can navigate with a mouse or a keyboard on the laptop or desktop PC. For our visualization purpose, we used this interface to see the position of the input device. The position of the head can be seen as a ball in the middle of our 3D scene in Figure 2. The only way to see the coordinates of the head in Jugglua is to use the print function. As a model depicted in Figure 2, we used the digital mockup of our laboratory, the Institut Image in Chalon-sur-Saône. Not only the navigation function should be coded, but some subsidiary functions need to be written to provide enough feedback from the system. This requirement is achieved under the platform shown in Figure 3.

The test bench collects data from three essential components of the virtual environment: the user, the navigation device and the 3D system. Bio-feedback (e.g., EEG signal, blood pressure) will be recorded to show the situation of a user when operating with navigation tools inside a CAVE for example. EEG signal can provide the most complete feedback, however the feedback can be returned from simpler sensors like blood pressure, skin resistance sensor, heart rate sensor. Another signal is the image stream that comes from an external camera. This camera is responsible of showing the response of the system to the input command. Acceleration, velocity, the delay between input and output, etc., will be extracted from this signal.

Finally, VRPN interfacing will help us to record appropriate signals from the navigation tool and other input devices. Simultaneous analysis of these signals and feedback will help us to understand the system performance deeply. Since we collect data from different users, based on user characteristics, age, etc., we can select appropriate values for the setting units. Different criteria could be selected to verify the results of the tests and then be applied to the setting units. The highlighted region in blue color in Figure 3 is under the current research. The objective of the next step is to find an accurate mapping function with continuous measurement from the process, users and the navigation tool, to improve the quality of the navigation system.

EXPERIMENT 3.1 Apparatus

A virtual research set of a CAVE system with an ARTracker is used in this study. The indicators (a set of aligned balls) are mounted on 3D glasses. The system is promoted with an NVidia Quadruplex Graphic processing unit. Different navigation tools can be attached by VRPN or Gadgeteer interfacing. In this experiment, a fly-stick, a gamepad, a mouse and a keyboard are connected to the system as navigation tools by a wireless connection. The user wears 3D glasses in the CAVE to see a virtual 3D scene which is projected inside the CAVE (Figure 4).

The glasses are synchronized with the projectors so that each eye only sees the correct image. People using the CAVE can see objects and scenes apparently real, and can walk around them, getting a proper view of what they look in reality. This is possible with optical illusions and laser triangulation. The frame of the CAVE is made of non-magnetic stainless steel to interfere as little as possible with the electromagnetic sensors. A CAVE user's movements are tracked by a tracking system (like a head mounted tracker) and the 3D scene is updated based on the user's current location. The computers rapidly generate a pair of images, one for each of the user's eyes. The position of the head in the Cartesian coordinates (x, y, z) and roll, pitch, yaw coordinates of the head are registered by the interface software of the head tracker. The standard of different rotations is provided in Figure 5. This standard will be used later to record, analyse and demonstrate the data.

Navigation tasks

Two groups of tasks were examined in this experiment: 1) walking (forward, backward, up and down), 2) turning to the left and right. Complex movement is interpreted by these two groups of basic tasks. For example going from point A to B may contain a series of these two tasks. It can be formalized by rotation and translation matrices.

Data grabbing system

Different measurements are saved in a * .txt file with a time stamp for each sample and component. For example, forward, backward, rotation to the left and right, up, down movements, and their associated mechanical and software resolutions are recorded for a given navigation tool (for example a joystick). (x, y, z) coordinates and rotation around Pitch, Yaw and Roll axes are stored in the same * .txt file but in a second line. A third line is used to save measurements of the acceleration sensor at the first moment (the first time stamp). Then, other lines of the file are used to save the over mentioned variables of navigation, tracker and sensor for the second time (second time stamp) and so on. If our sampling time is one millisecond and the first index of the time stamp is 0.000s then the second index is 0.001s. Therefore, the second line of the * .txt file is "Tracker, 0.000, x 0 , y 0 , z 0 , θ x 0 , θ y 0 , θ z 0 ".

A couple of images are recorded by the virtual camera of OpenGL and an external camera for each location of the tracker, with the same time stamp as the tracker, acceleration sensor and navigation tool. For instance, the second couple of images gets 0.001s time stamp. Acceleration and velocity of the scene are calculated from these two image streams. The time stamp is used to calculate the time difference between the input and output of the system, as well as the delay between different processing units. Two sets of analysis tools were developed in MATLAB and C++. Currently, these tools are applied offline just to extract features from the measured data. However, with a fast platform, it can be used as a real-time solution. Figure 6 shows the interconnection of recording modules, processing and analysis units. As seen, one of the outputs of the tool is the delay of different processing units.

RESULTS

So far, the implementation of the system and the test bench for the data collection has been explained. Here, we try to interpret the information and take out some features from the measurement, to be able to parameterize the navigation system and have a full control over it in real-time performance. Figure 7 shows the measurements of the velocity around the Yaw and Roll axes. First, user movement and spatial shift of the scene were recorded, then the velocity was computed from this movement by a simple speed equation, which is the difference of the spatial shift over time (see Figure 7). The velocity is computed by using the images of a high-speed camera.

As seen in Figure 7, there are sharp variations of velocity in some points (at times: 3600ms, 3750ms, 3970ms, and 4025ms). Sharp variations in velocity mean an acceleration (positive variation) and a deceleration (negative variation). Deceleration in this context looks like making a sudden break in the real movement, like when we are driving a car on the road. Intuitively, a sudden break most of the time makes a motion sickness. Based on our experience, the same deceleration gives rise to cyber sickness. Acceleration of the user head tracker and the 3D scene can be directly related to the nausea and vomiting level of a user. But it is not still clear that this relation can be linear or non-linear via a multivariable function. Because speed can be seen as a relative parameter, thus either the user or the scene movement or both of them affects the visual perception, which in turn leads to nausea (in the worse case vomiting). Table 1 shows the acceleration and velocity around the Yaw and Roll axes. Here just the acceleration process is illustrated. The acceleration and velocity can be controled by the user via the navigation device. In fact when the user feels sick, he can pitch down or up the speed of movement by a button on the navigation device. The rate of these variations can be considered as another symptom of nausea. As a result, the speed rate signal of the navigation device plays a critical role in the sickness-free navigation design, because it can reflect human factors while it is recording navigation data. Figure 8 shows the simultaneous variation of the speed rate and the rotation variations along the vertical axis. As seen, when the step of movement is reduced and the velocity of movement gets faster and faster, rotations gets smoother. Between 3600ms and 3750ms, and 3750ms and 3900ms, the speed control was unchanged; it means that the velocity of movement was appropriate for the user, which in turns means less sickness. The duration that users navigate in the 3D system is very important, because it is another important factor leading to sickness, and, as the time of navigation increases, the level of sickness can increase. Figure 9 shows three different times of navigation: in the first case (#1), the time of navigation increases dramatically, so the level of nausea will increase in the low velocity phase faster than in case 2 or 3, and after 40 m.s -1 , navigation is stopped. The reduc- tion in the curve does not mean a dramatic reduction of the level of nausea in higher speed, in contrast, it means the user gets badly sick and leaves the system. This situation appears practically after 15 minutes of navigation with very low speed. The duration of experiment is less in the two other cases (#2, 3). Especially, in case 3, navigation is continued with higher speed because the level of nausea was less in the low speed phase, due to a short duration of this phase. However, in the second case, the level of nausea increases in the higher speed phase, because of a longer duration of a low speed movement. The average of five measurements under longer navigation, and one of the five cases are shown in Figure 9.

There is a curve fitting to the average curve which is mostly used in mathematical modeling. As seen, the behavior of the average curve is closer to the second category (#2) while the individual curve is closer to the third category (#3). The delay between navigation device stimuli and the scene movement also is important because delay leads to sickness too. To solve this issue, the delay of the system response is computed from the synchronous measurements, as explained in the test bench. Figure 10 shows the effect of rotations around three main axes on the nausea. As seen, when the delay increases, the level of nausea increases in all the cases but with different trends and curves.

The delay due to the stimuli of the rotation around Pitch and Roll axes increases the level of nausea with an exponential trend, as shown in trend 2 of Figure 10. The behavior of the nausea level versus the delay due to the stimuli of the rotation around Yaw axis is more similar to a sinh function.

DISCUSSION

A navigation system without parameters tuning for sure will induce sickness. The level of sickness is dependent on a lot of parameters like age, gender, race, profession, career, previous experiences, etc. That is why the level of sickness is different from a user to another one, and in the same situation, it might be perceived by a user while no symptoms appear in another user. As seen, velocity and delay have great effects on the level of nausea and total cyber sickness. These parameters will help us to propose an effective navigation system based on cognitive parameters. The proposed navigation system taking into account human cognitive issues with two factors of movement velocity and response delay is shown in Figure 11. One of the important aspects of this model is the parametric definition which makes it suitable for real-time application, and a comparison of the performance under different circumstances and situations. As explained before, the navigation parameters depend on different factors of the users. Therefore, the best solution is to provide a dynamic level of sickness and give a notification to the user before the sickness level goes beyond specific threshold.

CONCLUSION

Different steps of a navigation system design, the test-bed establishment and the data collection procedure have been explained. By analyzing the collected data, we came to conclusion that the relative velocity of the scene versus user movement and the delay between different processing units, especially the delay between inputs and outputs of the system, are two important cognitive factors. Using these factors, the navigation system were proposed. The proposed system is quite efficient because it provides a real-time solution for 3D applications and is not platform or user dependent. Moreover, it is quite easy to implement as a simple function. Besides, it is a quantitative representation that lets the designer adapt the solution and set it up for new circumstances, and the modification in the performance or the parameters of the system can be easily compared with previous situations, and any amendment can be easily verified.

Figure 1 :

 1 Figure 1: Generic metaphor definition for tracking and traveling inside VEs.

Figure 2 :

 2 Figure 2: 3D display of Jugglua GUI.

Figure 3 :

 3 Figure 3: Interconnection between feedback provider functions, main navigation functions, navigation devices, 3D display and the rendering system.

Figure 4 :

 4 Figure 4: Summary of CAVE system operation.

Figure 5 :

 5 Figure 5: Illustration of the viewer-central coordinates used in the head mounted tracker data recording.

Figure 6 :

 6 Figure 6: Interconnection between processing units and analysis tools.

Figure 7 :

 7 Figure 7: Velocity around the Yaw and Roll axes.

Figure 8 :

 8 Figure 8: Simultaneous variation of rotation angle variation and rotation acceleration/velocity index.

Figure 9 :

 9 Figure 9: Level of nausea versus speed of navigation and time.

Figure 10 :

 10 Figure 10: Relation between the level of nausea and the response delay.

Figure 11 :

 11 Figure 11: An efficient navigation controller based on cognitive factors.

Table 1 :

 1 Yaw, Roll acceleration and velocity variation during 8 steps

		1	2	3	4	5	6	7	8
	roll-velocity	2.04	1.76	2.75	15.61	21.2	25.51	30.02	33.1
	yaw-velocity	2.1	1.29	2.01	17.81	18.4	23.17	24.23	28.18
	roll-acceleration	1.01	1.09	2.08	37.2	38.5	40.31	45.13	45.89
	yaw-acceleration	1.41	6.77	5.1	22.71	30.1	32.84	34.2	34.15


Julie Wagner
email: wagner@lri.frmackay@lri.frhuot@lri.fr

Wendy E Mackay

Stéphane Huot

Left-over Windows Cause Window Clutter... But What Causes Left-over Windows?

Keywords: Left-over windows, Window clutter, Window logger. ACM Classification H.5.2 Information Interfaces and Presentation: User Interfaces-Windowing systems Experimentation, Human Factors

Sleep mode lets users go for days or weeks without rebooting, supporting work on multiple tasks that they can return to later. However, users also struggle with window clutter, facing an increasing number of 'left-over windows' that get in the way. Our goal is to understand how users create and cope with left-over windows. We conducted a two-week field study with ten notebook users. We found that they work in very short sessions, switching often between computerbased and external tasks. 34% of left-over windows remain untouched for a day or more, increasing in quantitity until they all disappear after a reboot. Some users reboot as a deliberate 'clean-up' strategy, whereas others lose left-over windows after an unexpected system crash. Users intentionally keep left-over windows as to-do lists, as reminders of upcoming tasks, and for facilitating future access; the rest are simply forgotten. Tools for visualizing and managing left-over windows should help users reduce window clutter, while maintaing the benefits of interruptible work sessions.

INTRODUCTION

Window managers have changed little over the past decade, despite major changes in their context of use. Laptops are now powerful enough for both work and home settings, enabling users to shift between office and personal activities in the course of a day. Travelling now involves taking along one's computer, increasing the scope and variety of tasks performed. Unlike early desktop computers, which forced users to close all open windows at the end of the day, modern laptops come equipped with a sleep mode that preserves the state and layout of their windows when the computer shuts down. Users can shift among multiple tasks, preserving context in the face of interruptions or when restarting the computer. Many users thus avoid rebooting for weeks or even months at a time.

There is, however, a drawback: leaving windows open can result in overwhelming 'window clutter'. Figure 1 shows an Exposé view of one user's screen, with over 50 open windows that have accumulated since the last reboot. These include left-over windows, i.e. duplicate or no-longer-needed windows that get in the way of finding windows needed for a particular task, reducing productivity and contributing to the user's sense of information overload.

This paper investigates the lifecycle of windows between reboots, with particular emphasis on 'left-over windows'. We conducted a field study that logged ten users' window activity for two weeks. In addition to semi-structured in situ interviews, we also introduced 'critical incident' questionnaires that asked about the status of a randomly chosen selection of left-over windows. We discuss the results and conclude with implications for design.

RELATED WORK

The first system to address window overload explicitly was Xerox PARC's Rooms [START_REF] Henderson | Rooms: the use of multiple virtual workspaces to reduce space contention in a window-based graphical user interface[END_REF], which let users organize groups of tasks into multiple virtual project rooms. Later window managers, e.g., Apple's Spaces, offer similar task-context organizational strategies, but do not explicitly address the problem of extraneous windows.

Although we know of no studies that investigate the specific causes of left-over windows, Barreau and Nardi's [START_REF] Barreau | Finding and reminding: file organization from the desktop[END_REF] early study of desktop management offers some clues. They identified three types of files, ephemeral, working and archived, noting that 'quick-and-dirty reports and memos [end up in a] no-man's land between ephemeral and working information'. Users must track but do not actively work on these files until after completing a follow-up activity.

Hutchings et al. [START_REF] Hutchings | Revisiting display space management: understanding current practice to inform next-generation design[END_REF] emphasized the fact that information workers are constantly interrupted, with an average of 50 tasks shifts over the course of a week. Resuming tasks requires effort, depending upon task complexity, duration, and length of absence, and they noted that some users leave windows open to remind them of previous tasks when they return after an interruption.

Salvucci [START_REF] Salvucci | On reconstruction of task context after interruption[END_REF] argues that reconstructing the task context after an interrruption is critical for successful resumption and depends upon sometimes subtle cues in the external environment. For laptop users, moving from one physical location to another not only affects the tasks performed, but may also disrupt the reconstruction process, potentially increasing the number of left-over windows.

This paper investigates the characteristic patterns and underlying causes of left-over windows. We hope this will lead to better window management tools that help users cope effectively with window clutter.

FIELD STUDY

We conducted a field study to help us understand the causes of window clutter, or why some users leave large numbers of windows open between work sessions. We asked the following research questions:

Q1: How do interruptions affect work-session duration?

Q2: What is the relationship between left-over windows and the number of work sessions between reboots?

Q3: Why don't users close left-over windows?

We define abandoned or left-over windows as those that remain open from one work session to another, without being modified by the user in any way. Some windows are all or partially visible, others may be minimized or hidden behind other windows. We define a work session as a continuous interaction between the user and the computer, bracketed by periods of inactivity. Rebooting or sleeping the computer always ends a work session, as does lack of input from the user for one hour or more.

Participants

We recruited ten laptop users (three women) aged 31 to 58: 1 CEO, 1 managing director, 3 professors, 2 associate professors, 1 researcher and 2 post-doctoral fellows. They engage in teaching (60%), engineering (60%), administration (70%) and research (80%).

WM-LISA Logger

We developed WM-LISA -Window Management Logger Investigating Session behavior and Abandoned windowsto log the opening and closing of windows, applications and system events. Similar to the wmtrace [START_REF] Chapuis | Gestion des fenêtres: enregistrement et visualisation de l'interaction[END_REF] logger for XWindows, WM-LISA runs as an unobtrusive background process on the Mac OS X operating system (10.5 and higher) that intercepts low-level window manager events and Apple Accessibility API information.

WM-LISA records IDs and times of all standard window events and uses Applescript to log information about tabs within Safari web browser windows, e.g. the number of tabs and their corresponding IDs. It uses low-level accessibility and device events to generate high-level window and tab manipulation events, e.g., window content changed and window viewport changed (scrolling). WM-LISA also logs when the computer goes to sleep, awakens, is rebooted or shut down. Temporary windows, e.g., dialog boxes, tooltips, menus, system menu bars and icons, are not logged.

Data collected

We recorded responses to semi-structured interviews and critical-incident style questionnaires. We also collected logs of every window, tab, sleep, wake-up and reboot event monitored by WM-LISA.

Procedure

We asked participants to avoid deleting windows prior to our initial interview, except any they considered confidential. We conducted each interview in the participants' offices and asked about their window management strategies. We installed WM-LISA and asked them to use the Safari web browser for the 14 days of the study. WM-LISA logged events silently for the first 10 days. For the final four days, whenever the participant started a new session, a questionnaire popped up based on five randomly chosen screenshots from their current set of left-over windows. Participants were encouraged to avoid speculation and offer specific, incontext reasons as to why they abandoned certain windows.

RESULTS

Seven users answered at least one critical incident questionnaire. We used ANOVA and Nominal Logistic analysis 1 to analyze 164,897 logged events.

Session duration

We identified 512 discrete sessions across all participants over the two-week study period, 50% of which were shorter than 65 minutes. Participants averaged 1-9 sessions per day (median = 4), divided into three periods: night (0am-6am), day (7am-5pm), and evening (6pm-11pm).

Most sessions occur during the day (F 2,12 = 30.73, p < 0.0001, mean = 27.1 day sessions and mean = 88.9 length). These sessions tend to be shorter than evening or night sessions, which do not differ significantly from each other. Several 1 We used the JMP 9 statistical package. participants commented that they are interrupted more frequently during office hours than in the evening or at night:

"Very often when I leave work, I say I have not worked at all today, because for me working is doing research ... and that only happens when I close my [office] door." (P2)

Reboot strategies

We asked participants to describe their rebooting strategies. Six avoid rebooting and only do so for system updates, performance issues or system crashes: "I never choose to reboot. ... I have to reboot because of updates. I do not choose to." (P10)

We classified these six participants as seldom rebooters (SR). They often become overwhelmed by their left-over windows, dealing with them by default when they are forced to reboot.

The remaining four are normal rebooters (NR). They control the number of left-over windows by setting aside a regular time to 'clean-up' their windows, at the end of the day or in conjunction with a scheduled reboot. Not surprisingly, SR participants rebooted less frequently than NR participants (p < 0.006, r 2 = 0.57): One participant (P1), a seldom rebooter, never rebooted during the two weeks of the study.

Figure 2 shows that, as the number of sessions between reboots increases, so does the overall number of open windows (p < 0.0001, r 2 = 0.03) and tabs (p < 0.0001, r 2 = 0.34). Seldom rebooters keep significantly more windows open simultaneously than normal rebooters (p < 0.0001, r 2 = 0.27): median = 17 vs. median = 5 windows. "It's a cycle: things get progressively more cluttered and one day I say, 'OK', I cannot do this anymore and so I clean up." (P1)

As the interval between reboots increases, the number of left-over windows tracks the overall increase in the number of windows (p < 0.01, r 2 = 0.5). However, the number of leftover tabs does not show a corresponding increase (p < 0.3, r 2 = 0.22), implying that tabs are more likely to be reused than left over.

Left-over window patterns

How long do windows stay open?

We define the window session lifetime as the number of sessions from when the window or tab was first opened and P1, who never rebooted, had the longest window session lifetime, 58 sessions for both windows and tabs. Even group C, with the shortest window session lifetime, left some windows open for up to 29 sessions and left some tabs open for up to 21 sessions.

How many windows are left over within a session?

The most striking finding was that, within a given session, all participants avoided interacting with approximately one third of their windows (mean = 36.6%), regardless of the overall number of open windows or reboot practices. However, if we examine participants' total numbers of left-over windows, we see three distinct groups: (A) P1, P3, P8, P4 keep a mean of 57.1% left-over windows, (B) P10, P9 keep a mean of 30.8% left-over windows and (C) P2 keep a mean of 4.7% left-over windows (F 9,9 = 144.6, p < 0.0001). The remaining three participants fit between groups (B) and (C).

Figure 2 shows the aggregate number of active (pale blue) and left-over (dark blue) windows per session: the numbers of each both increase as the number of sessions between reboots (inter-reboot interval) increases. Figure 3 compares active vs. left-over windows for three representative participants from each group: P1 (A), P10 (B), and P2 (C).

What causes long-term left-over windows?

Gonzalez et al. [START_REF] González | Constant, constant, multi-tasking craziness": managing multiple working spheres[END_REF] observed that information workers organize their daily activities into multiple high-level units of work, referred to as working spheres. They found that people switched to a new working sphere about one third of the time, without returning to it that day. This suggests that interruptions and task switching increase the number of work sessions in a day, while decreasing their duration.

We decided to examine the differences between short-term and long-term left-over windows. We categorize a window as 'short-term left-over' if it is opened in a session and, after a number of short, interrupt-driven sessions, dealt with later the same day. In opposition to such windows, which users return to relatively quickly, windows that are abandoned for days at a time are 'longer-term left-over'.

We found that the participants who seldom reboot (SR) leave a higher percentage of windows open and untouched over multiple days (34% vs. 22% for normal rebooters; F 1,1 = 5.3, p = 0.049). Although everyone had left-over windows, only seven had left-over tabs. Of these, only 16% were left over for more than a day, with no significant differences between SR and NR participants.

Why do users create left-over windows?

Although a number of left-over windows are simply forgotten, others persist for good reasons. Table 1 summarizes why participants create left-over windows, drawn from both the interviews and the critical-incident questionnaires.

DISCUSSION AND CONCLUSION

This paper describes a field study of how ten laptop users deal with the window clutter caused by left-over windows. We found that interruptions during the day not only result in shorter work sessions but also increase the total number of left-over windows, with a corresponding increase in window clutter (Q1). Users who seldom reboot are more affected than those who reboot often.

We also found that all participants accumulated left-over windows over time, with a corresponding increase in window clutter until the next reboot (Q2). Regardless of the overall number of open windows or reboot rate, approximately one third of each participant's windows remained untouched during the current work session.

Participants offered a number of positive reasons for keeping left-over windows (Q3), including reminders, to do lists and facilitating future access to specific windows. However, they also admitted that many were forgotten and caused undesireable window clutter.

Although our study was limited to a small group of Mac OS X users, it suggests that beneficial features such as using sleep mode to delay closing windows may have unintended side effects, in this case progressively increasing the user's window clutter. We argue that we need to consider this when designing window management tools, allowing users to quickly identify left-over windows and distinguish between those that serve a useful purpose and those that were simply forgotten and are now getting in the way.

Reason

Figure 1 .

 1 Figure 1. Example of a progressive clutter process over weeks: screenshot in Exposé mode of over 50 windows on a user's virtual desktop.

Figure 2 .

 2 Figure 2. Numbers of active (pale) and left-over (dark) windows both increase with greater inter-reboot intervals (all participants).

Figure 3 .

 3 Figure 3. Open vs. left-over windows: P1 (SR) never rebooted, has the most open and left-over windows; P10 (SR) rebooted 3 times, half as many open and left-over windows; P2 (NR) rebooted 7 times, few open and almost no left-over windows.

  Sample participant comment Intermittent Access This is documentation I program with ... I keep the window open because I still have [to work on it]. (P8) Permanent Access This is my email client which stays open when I'm in front of my computer. Sometimes I minimize it. (P6) Task Reminders [This] email window has probably been around since February (in a July interview.) I need to collect data and it isn't high enough on my priority list." (P1) Longterm Reminders I keep this window open as a bookmark but also to remind me to finish it. This is important as a mid-or long-term [task], but not for now." (P8) To do Lists I minimize [windows] ordered by priority. It is also a todo list. I look at the filename to see what I have to do and I order them accordingly. (P6) Awaiting Action This message arrived today that I want to remember to tell [name] about. I'll close it when I've talked to her about it (probably tomorrow). (P1) Completed Tasks This is a Finder window with the folder of an old presentation. I usually keep several Finder windows open, some are just forgotten. (P4) Forgotten Clean up I searched this PDF document for a presentation that I am preparing. I closed the window of the active presentation but forgot this window. (P8)

Table 1 .

 1 Reasons for keeping left-over windows.

ACKNOWLEDGMENTS

Our thanks to the study participants and to Olivier Chapuis for numerous discussions and feedback on this paper.