Ivan Beschastnyi
email: i.beschastnyi@gmail.com

Ugo Boscain
email: ugo.boscain@upmc.fr

Mario Sigalotti
email: mario.sigalotti@inria.fr

An obstruction to small-time controllability of the bilinear Schrödinger equation *

In this article we discuss which controllability properties of classical Hamiltonian systems are preserved after quantization. We discuss some necessary and some sufficient conditions for small-time controllability of classical systems and quantum systems using the WKB method. In particular, we investigate the conjecture that if the classical system is not small-time controllable, then the corresponding quantum system is not small-time controllable either.

Introduction

The relation between classical and quantum dynamics has been an important question for physicists and mathematicians since the very beginning of quantum mechanics. The set of tools which allow to study a quantum system using the properties of the corresponding classical system is known as semi-classical analysis. Such methods try to capture the general idea that, when the Planck constant goes to zero, quantum dynamics approaches classical dynamics. In this way many interesting properties of a classical system can be observed in the behaviour of the corresponding quantum system (see for instance [START_REF] Nonnenmacher | Anatomy of quantum chaotic eigenstates[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF] and the references therein).

The goal of this article is to explore which properties of the controlled Schrödinger equation

i∂ t ψ = - ∆ψ 2 + V (x)ψ + u(t)W (x)ψ, ψ ∈ L 2 (M ), (1)
on a complete Riemannian manifold M can be deduced from the behaviour of the corresponding controlled classical system λ = H(t, λ), λ ∈ T * M,

with Hamiltonian

H(t, λ) = p 2 2 + V (x) + u(t)W (x), (3)
where λ = (p, x) ∈ T * x M and • denotes the usual norm on the cotangent bundle induced by the Riemannian metric on M . The manifold setting enables to include relevant physical systems such as rotating molecules [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF][START_REF] Koch | Quantum control of molecular rotation[END_REF].

The first idea that comes to mind is that maybe both systems share some similar controllability properties. Let L 2 (M ) be the space of square integrable functions with respect to the standard Riemannian volume. Since the Schrödinger equation preserves the L 2 -norm, we denote by S(M ) = {ψ ∈ L 2 (M ) : ψ L 2 (M ) = 1} the L 2 -sphere and consider quantum and classical reachable sets in time less than T ∈ R + from states ψ 0 ∈ S(M ) and λ 0 ∈ T * M :

A q
≤T (ψ 0 ) = {φ ∈ S(M ) : ∃0 ≤ s ≤ T, u ∈ U, ψ(•) solution of (1) s.t. ψ(0) = ψ 0 , ψ(s) = φ} , A c ≤T (λ 0 ) = {µ ∈ T * M : ∃0 ≤ s ≤ T, u ∈ U, λ(•) solution of (2) s.t. λ(0) = λ 0 , λ(s) = µ} .

Here U denotes the set of admissible controls. Under reasonable assumptions on V and W , one can take U to be the space of piecewise constant functions. In particular we will make the following general assumption:

Assumption 1. We assume that V, W are C ∞ functions such that the operators

- ∆ 2 + V (x) + uW (x)
are essentially self-adjoint on L 2 (M ) for every u ∈ R.

Assumption 1 holds, for instance, if M is a compact manifold or if M = R n , V is bounded from below and W ∈ L ∞ (M ) ∩ C ∞ (M ) [START_REF] Reed | Methods of modern mathematical physics. II. Fourier analysis, self-adjointness[END_REF]Theorem X.28].

It is well-known that one cannot generally expect full controllability of equation ( 1) from a given state ψ 0 [START_REF] Ball | Controllability for distributed bilinear systems[END_REF], i.e., in general

T ≥0 A q ≤T (ψ 0 ) S(M ). (4)
Nevertheless, it has been proved that, under suitable generic conditions, the left-hand side of (4) is dense in S(M ) [START_REF] Mason | Generic controllability properties for the bilinear Schrödinger equation[END_REF]. If

T ≥0
A q ≤T (ψ 0 ) = S(M ) for every state ψ 0 ∈ S(M ), then we say that the Schrödinger equation ( 1) is approximately controllable. The full and approximate classical controllabilities could be defined in the same way, but in view of the relation with quantum mechanics it makes sense to define approximate controllability of the system (2) as the property that there exists a dense set Υ ∈ T * M such that for every initial state λ 0 ∈ Υ

T ≥0 A c ≤T (λ 0 ) = T * M. (5)
For brevity we write "QA" for "quantum approximate" and "CA" for "classical approximate". Coming back to our question, the first naive guess could be that CA controllability should imply QA controllability. However simple examples show that this is not the case in general. For example, it is known that the harmonic oscillator in a controlled electric field

i∂ t ψ = - 1 2 ∂ 2 x ψ + x 2 ψ + u(t)xψ, ψ ∈ L 2 (R),
is not QA controllable [START_REF] Coron | Mathematical Surveys and Monographs[END_REF][START_REF] Mirrahimi | Controllability of quantum harmonic oscillators[END_REF], even though the corresponding classical system

ẋ = p, ṗ = -x -u,
is controllable as it follows easily from the Kalman condition (for, instance, [START_REF] Sontag | Mathematical control theory[END_REF]). On the other hand, if a classical system is not CA controllable, it does not necessarily follow that the corresponding quantum system is not approximately controllable either, since quantum effects such as tunneling allow a particle to move into classically forbidden regions. In the appendix we discuss an example of this kind. In the following we go more deeply inside the relation between classical and quantum controllability. We define the controllability diameter of (1) as

T q = sup ψ 0 ∈S(M ) inf T ≥ 0 : A q ≤T (ψ 0 ) = S(M ) ,
i.e., T q is the smallest time such that any state in S(M ) can be transferred to an arbitrarily small neighborhood of any other state in time less than T q . We define analogously the controllability diameter of the classical system (2) as

T c = inf T ≥ 0 : ∃Υ ⊂ T * M dense s.t. ∀λ 0 ∈ Υ, A c ≤T (λ 0 ) = T * M .
The reason for considering, in the definition of the controllability diameter T c , only a dense set of initial conditions is that we want to take into account situations in which a common critical point x for V and W prevents the usual approximate controllability property to hold true in T * M , even if it holds in the punctured manifold T * M {(x, 0)}. In this situation, we expect that such slightly reduced controllability property of the classical system is reasonably informative about the controllability of the corresponding quantum system. It is quite natural to expect that there is a connection between the two quantities T q and T c . Indeed, semiclassical analysis allows to construct approximate solutions to the Schrödinger equation using the solutions of the corresponding Hamiltonian system. Notice however that if a system is not CA or QA controllable, then the corresponding controllability diameter is equal to infinity. Thus the controllability discussion above shows that we should not expect nice estimates such as cT c ≤ T q ≤ CT c for some positive constants c, C. Instead, something that is much more reasonable is the following implication that we give in the form of a conjecture. Conjecture 1. Let M be a connected complete Riemannian manifold. For any Hamiltonian of the form (3) the following implication is true

T c > 0 ⇒ T q > 0. ( 6
)
This implication basically says that all the quantum effects which are due to non-locality of the Schrödinger equation have a limited effect at small times. Understanding if a system can or cannot be controlled in arbitrarily small time is important for applications, in particular when the system is subject to decoherence and hence it must be controlled in a time much smaller than the decoherence time [START_REF] Glaser | Training Schrödinger's cat: quantum optimal control[END_REF].

When T q = 0 (respectively, T c = 0), we say that the quantum system (1) (respectively, the classical system (2)) is small-time controllable. Most of the questions related to small-time controllability are completely open. In the classical case T c can indeed be equal to zero, since the classical harmonic oscillator in an electric field is a linear system, and all controllable linear systems have T c = 0 (see, for instance, [START_REF] Seidman | How violent are fast controls? II[END_REF][START_REF] Sontag | Mathematical control theory[END_REF]). But in general the question is open even locally for polynomial systems [START_REF] Agrachev | Is it possible to recognize local controllability in a finite number of differentiations?[END_REF].

In the quantum case it is known that for systems evolving in a finite-dimensional Hilbert space (e.g., spin systems) i ψ = (A + u(t)B)ψ, where ψ ∈ C n and A, B are Hermitian matrices, we necessarily have T q > 0 for n > 1. Indeed, if we denote by e an eigenvector of B, one can check that the corresponding occupation probability cannot change too quickly, since it satisfies the estimate

| ψ(t), e | ≤ | ψ(0), e | + A t. (7)
(See [START_REF] Agrachev | A Note on Time-Zero Controllability and Density of Orbits for Quantum Systems[END_REF]). Some upper and lower bounds on the controllability diameter in this case were studied in the articles [START_REF] Agrachev | An estimation of the controllability time for singleinput systems on compact Lie groups[END_REF][START_REF] Khaneja | Time optimal control in spin systems[END_REF]. For a system of the form (1) evolving in an infinite-dimensional space, A is not a bounded operator anymore. Hence the estimate [START_REF] Beauchard | Minimal time for the bilinear control of Schrödinger equations[END_REF] is not an obstacle for small-time controllability. This gives hope that in some infinite-dimensional cases small-time controllability can be achieved. An example of such a system was given in [START_REF] Boussaïd | Small time reachable set of bilinear quantum systems[END_REF], where the authors proved that

i∂ t ψ = -|∂ 2 x | α ψ + u(t) cos xψ, ψ 0 (-π) = ψ 0 (π) = 0,
is small-time controllable for α > 5/2. The physically interesting case α = 1 is however open. In contrast to this positive result, negative results were presented in [START_REF] Beauchard | Minimal time for the bilinear control of Schrödinger equations[END_REF][START_REF]Minimal time for the approximate bilinear control of Schrödinger equations[END_REF], where the authors proved that for a particle in an electric field we always have T q > 0. All the small-time uncontrollability results that we cite in the present paper are based on proving that some properties of the considered system are uniform with respect to the control. For example, in the finite-dimensional case we have estimates on the occupation probabilities [START_REF] Beauchard | Minimal time for the bilinear control of Schrödinger equations[END_REF] that satisfy this property. In [START_REF] Beauchard | Minimal time for the bilinear control of Schrödinger equations[END_REF] the authors exploit instead the fact that, for an electric potential, Gaussian wave-packets remain localized along a given classical trajectory until a time (called the Ehrenfest time) that is independent of the control u(t). Thus a Gaussian state has to remain almost Gaussian for small times and hence T q > 0.

In this article we study some obstructions to small-time controllability that are related to the behaviour of projections of classical trajectories to the base manifold M . The paper is divided into two sections. In the first section we give some sufficient conditions to have T c = 0 or T c > 0. In the second section we see how in specific examples one can deduce that T q > 0 from T c > 0.

In particular, in Section 2 we show along which directions in the classical phase space it is possible to move arbitrarily fast and we prove the following theorem.

→ R such that d 1 V (x, y) T * x N 1 ≤ c(x) for all (x, y) ∈ M , where d 1 V (x, y) denotes the differential at x of V (•, y) : N 1 → R. Under these assumptions T c > 0.
In Section 3 for the corresponding quantum case we prove the following theorem.

Theorem 1.2. Assume that conditions of Theorem 1.1 are satisfied. Then T q > 0.

Let us discuss a couple of practical situations where Theorems 1.1 and 1.2 can be applied. Assume, for example, that N 2 is just a point. In this case M = N 1 and W is constant on a nonempty open subset Ω of M and T c > 0 because in Ω the classical dynamics is completely independent of the control u. This means that until a classical particle exists Ω it is unaffected by the control and thus the positivity of the exit time from Ω is an obstruction for having T c = 0.

We note the fact that T q > 0 is not immediately obvious from the previous discussion, since the Schrödinger equation is not local. So even if initially the support of ψ 0 is concentrated in Ω, a certain amount of the occupation probability exits Ω instantaneously. This effect can be enough to ensure QA controllability of a system whose classical counterpart is not CA controllable as in the example presented in the Appendix. Nevertheless the rationale of Theorem 1.2 is that the tunneling effect is not strong enough to transfer occupation probability outside Ω in arbitrarily small time. Thus the state remains essentially localized for sufficiently small times.

Another situation when Theorem 1.2 applies is when

M = R n = R n 1 × R n 2 endowed
with a flat product metric. If we denote by (x, y) the coordinates on R n 1 ×R n 2 , then the first condition of the theorem simply says that W (x, y) = W (y) for all x ∈ Ω. In this case the gradient lines of W preserve the fibre bundle Ω × R n 2 , and what one can show is that a classical particle again cannot leave Ω × R n 2 in arbitrarily small time, because the second condition ensures that the speed along the base Ω is uniformly bounded on each fibre {x} × R n 2 . Similarly to the previous case one can show that, if the wave function at the initial time is concentrated in Ω × R n 2 , it will stay essentially localized in this set for some positive time independent of the control u.

The proof of the theorem is based on the WKB method. We will discuss its geometric meaning, and we will see how one can extract information about the controllability diameter. After that we will prove the theorem above by a modification of the WKB ansatz. A possible variant of the proof not relying on semiclassical estimates are suggested in Remark 3.4.

Classical small-time controllability 2.1 Small-time controllability in phase space

We start by looking at which states can be reached in an arbitrarily small time from a given state λ 0 ∈ T * M . Let λ 0 = (p 0 , x 0 ) and as usual by π : T * M → M we denote the projection to the base manifold M . Definition 2.1. Let λ 0 , λ 1 be in T * M . We say that the Hamiltonian system (2) can be steered in small time from λ 0 arbitrarily close to λ 1 if for any T > 0 and any neighborhood Ω ⊂ T * M of λ 1 , there exists an admissible control function u : [0, T ] → R such that T ≤ T and the corresponding solution λ(•) of ( 2) with initial condition λ(0) = λ 0 satisfies λ(T ) ∈ Ω. Lemma 2.2. Consider a Hamiltonian of the form (3). Given an initial state λ 0 one can steer (2) in small time from λ 0 arbitrarily close to 1. the state λ 0 + kdW (x 0 ), ∀k ∈ R; 2. a state λ whose projection π(λ) is arbitrarily close to any chosen point on the geodesic issued from x 0 with initial covector dW (x 0 ).

Proof. Let us write

H g (λ) = p 2 2 , H V (λ) = V (x), H W (λ) = W (x).
Then the classical system (2) can be written as

λ = H g (λ) + H V (λ) + u(t) H W (λ) (8)
or, in local coordinates and using the Einstein summation convention, as

ẋi = g ij p j , ṗi = - ∂g jk ∂x i p j p k - ∂V ∂x i -u(t) ∂W ∂x i . (9)
For every ε > 0 we consider the constant control of the form u(t) = -k/ε defined on the interval [0, ε]. We rescale the time variable by taking t → t/ε. In this way we map the interval [0, ε] to [0, 1] and the endpoint λ(ε) is mapped to the endpoint λ(1) of the Cauchy problem

λ = ε H g (λ) + ε H V (λ) -k H W (λ), λ(0) = λ 0 . ( 10
)
We obtain a system that depends smoothly on the parameter ε. Thus, by the continuous dependence of solutions of ODEs on parameters, the solution of ( 10) converges uniformly to the solution corresponding to ε = 0, which is exactly t → λ 0 + tkd x 0 W . The second part is proved similarly. We take u(t) ≡ 0 and for every ε > 0 we consider the solution of (10) on the interval [0, ε] with initial condition λ(0) = λ 0 + (k/ε)d x 0 W . It is well known that the joint rescaling p → pε, t → t/ε of the fibre variables and time is a symmetry of the geodesic equations. Indeed, from the coordinate expressions [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] it is easy to see that the

solution of λ = H g (λ) + H V (λ) with λ(0) = λ 0 + (k/ε)d x 0 W on [0, ε] is mapped to the solution of λ = H g (λ) + ε 2 H V (λ)
on [0, 1] with λ(0) = ελ 0 + kd x 0 W . Once again by continuous dependence of solutions of ODEs on the parameter, we can conclude that π(λ(1)) converges, as ε → 0, to the endpoint at time 1 of the geodesic issued from x 0 with initial covector kdW (x 0 ). Note that at this point we still need to justify why we can take λ(0) = λ 0 + (k/ε)d x 0 W as the initial condition, because using the first statement we can only steer (2) from λ 0 to a neighborhood Ω of λ 0 + (k/ε)d x 0 W . But since Ω can be made arbitrarily small, the result follows from the continuous dependence of solutions on the initial value.

This result gives us a number of interesting corollaries.

Corollary 2.3. Given an initial state λ 0 such that dW (π(λ 0 )) = 0, let γ be the gradient curve of ∇W passing through π(λ 0 ) and let q 1 be a point on γ. Then (2) can be steered in small time from λ 0 to a state λ 1 such that π(λ 1 ) is arbitrarily close to q 1 . Proof. We note that the geodesic starting from x 0 with initial covector dW (x 0 ) has the same tangent line as the gradient curve γ at x 0 , as can be easily seen from [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF]. Thus γ can be approximated with geodesic arcs and we simply apply Lemma 2.2 several times using the continuous dependence of solutions of ODEs on initial values.

Corollary 2.4. Consider a classical system (2) with Hamiltonian

H(t, λ) = p 2 2 + V (x) + n i=1 u i (t)W i (x), (11)
where n = dim M , all potentials are C 2 and u i ∈ R. If the set

O = {x ∈ M : dW 1 ∧ • • • ∧ dW n (x) = 0}
has empty interior, then (2) is small-time CA-controllable. If dim M = 1 then the converse is true. Namely if (2) is small-time CA-controllable, then O has an empty interior.

Proof. Even though the statement is about systems with several controls, the proof can be done by using Lemma 2.2 and Corollary 2.3. Let us consider the set F = π -1 (M O) M where the last M should be thought as the zero section of T * M . By construction F does not contain fixed points of (2), for which the dynamics is independent of the control. If λ 0 and λ 1 are two states in F , then π(λ 0 ) can be connected to π(λ 1 ) via a geodesic, whose lift to the cotangent bundle is given by [0, 1] t → µ(t). But since π(µ(0)) = π(λ 0 ) / ∈ O, we can decompose µ(0) in the basis of dW 1 , . . . , dW n , i.e., we can write

µ(0) = n i=1 a i dW i (π(λ 0 ))
for some constants a i ∈ R. We can then apply Corollary 2.3 to the control system governed by the Hamiltonian

H a (t, λ) = p 2 2 + V (x) + v(t) n i=1 a i W i (x),
where v(t) is the new control, i.e., we take u i (t) = a i v(t) as controls in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF]. It follows that we can steer (2) in small time from λ 0 arbitrarily close to some λ with π(λ) arbitrarily close to π(λ 1 ). In particular, we can assume that π(λ) is in O and, using the same trick as above, we can choose a 1 , . . . , a n such that

n i=1 a i dW i (π(λ 1 )) = λ 1 -λ.
By applying the first statement of Lemma 2.2, we conclude the proof that ( 2) is small-time CA-controllable.

In dimension one the emptiness of the interior of O is also a necessary condition for smalltime controllability, since on an open set where W (x) ≡ 0 the Hamiltonian system (8) is independent of the control and thus T c > 0.

Concerning Conjecture 1, Corollary 2.4 shows that the set of one-dimensional systems with a nowhere vanishing potential W is probably the best place to start looking for concrete examples of small-time QA controllable systems, since there are no other possible semi-classical obstructions than the vanishing of W on an open set. In the next subsection we show that even though being rare, there can be other obstructions in higher dimensions.

In order to find them, we define the controllability diameter in the configuration space as

T cs c = inf T ≥ 0 : ∃Υ ⊂ T * M dense s.t. ∀λ 0 ∈ Υ, π(A c ≤T (λ 0 )) = M .
In this definition we essentially ignore information about the momenta and focus on the position of a particle in the configuration space. From the definition it clearly follows that T cs c ≤ T c . And hence in order to prove Theorem 1.1 we prove that T cs c > 0. Proof of Theorem 1.1. By possibly taking a smaller subset in Ω, we can assume that T * Ω is a trivial bundle and we can introduce local coordinates (x, p x ). Similarly by (y, p y ) we denote a point in T * N 2 . Using the product structure of M , due to the assumptions of the theorem, in Ω × N 2 the Hamiltonian system (8) can be written as

ẋi = g ij x p x j , ṗx i = - ∂g jk x ∂x i p x j p x k - ∂V (x, y) ∂x i , ẏi = g ij y p y j , ṗy i = - ∂g jk y ∂y i p y j p y k - ∂V (x, y) ∂y i -u(t) ∂W (x, y) ∂y i . (12)
We note that

-K(x)c(x) ≤ - ∂V ∂x i (x, y) ≤ - ∂V (x, y) ∂x i ≤ ∂V ∂x i (x, y) ≤ K(x)c(x),
where

K : Ω → [0, +∞) is a locally Lipschitz function such that |∂ x i V (x, y)| ≤ K(x) d 1 V (x, y) TxN 1 for every (x, y) ∈ Ω × N 2 .
Its existence follows from the equivalence of norms in finitedimensional spaces.

The result now follows from Chaplygin's lemma that states that if z(•) and z(•) are solutions of the differential equations

ż = f (z), ż = f (z), z, z ∈ R n
with f, f locally Lipschitz, f (z) ≤ f (z) coordinate-wise for all z ∈ R n , and z(0) ≤ z(0), then z(t) ≤ z(t) for all t ≥ 0 for which both solutions exist.

In our case we apply the Chaplgin lemma to compare the solutions of ( 12) with those of the two control systems ẋi = g ij x p x j ,

ṗx i = - ∂g jk x ∂x i p x j p x k ± K(x)c(x), ẏi = g ij y p y j , ṗy i = - ∂g jk y ∂y i p y j p y k - ∂V (x, y) ∂y i -u(t) ∂W (x, y) ∂y i , (13)
one for each choice of the sign in front of the term K(x)c(x). The right-hand side of the equations for the variables (x, p x ) in ( 13) is now autonomous, locally Lipschitz, and independent of the control. Thus for a given initial state λ 0 ∈ T * Ω × T * N 2 the corresponding trajectory will remain in T * Ω × T * N 2 for small times independent of the control function u(•). So we can apply the Chaplygin lemma for each interval on which u(•) is constant, and the projection onto N 1 of the solution of the Hamiltonian system [START_REF] Coron | Mathematical Surveys and Monographs[END_REF] with the same initial condition λ 0 will also remain in Ω for all times in a small interval independent of the control function u(•). Moreover, the bound on the exit time from T * Ω × T * N 2 can be made uniform with respect to all initial conditions in a neighborhood of λ 0 . Hence T cs c > 0 and T c > 0.

3 Obstructions to small-time controllability of quantum systems

WKB method

The goal of this section is to prove Theorem 1.2. We start by recalling the WKB method which is going to be our main tool. Let us recall quickly some basic definitions that we need in the geometric presentation of the WKB method (for more details see [START_REF] Anantharaman | Eigenfunctions of the Laplacian on negatively curved manifolds: a semiclassical approach[END_REF] or [START_REF] Bates | Lectures on the geometry of quantization[END_REF]). Let M be a Riemannian manifold and let us denote by •, • the scalar product on T M . Recall that the gradient of a function f is characterized by the identity

∇f, Y = Y [f ], ∀Y ∈ Vec(M ),
Using a local orthonormal frame X 1 , . . . , X n of vector fields we can equivalently write

∇f = n i=1 X i [f ]X i .
The volume Vol allows to define the divergence of a vector field X ∈ Vec(M ) as div(X) Vol = L X Vol, where L X is the Lie derivative in the direction X. Thus we can define the Laplace operator in the usual way as ∆f = div ∇f . In a local orthonormal frame the operator ∆ has the form

∆ = n i=1 X 2 i + div(X i )X i .
From here it is easy to verify that the formula for the Laplacian of a product of two functions a, b is the same as in the Euclidean case: ∆(ab) = a∆b + 2 ∇a, ∇b + b∆a.

Let us now consider a Schrödinger equation on M of the form

i ∂ t ψ + 2 ∆ψ 2 -V (t, x)ψ = 0, ( 14
)
where V is a smooth time-dependent potential. Here we introduce the parameter since we are going to study the formal expansion of ( 14) with respect to . We start with the usual WKB ansatz ψ(t, x) = a(t, x)e i S(t,x) .

By plugging this ansatz into ( 14), we obtain the expression

i ∂ t a + i a∂ t S + 2 2 ∆a + 2i ∇a, ∇S + i a∆S - 1 2 a ∇S 2 -V (t, x)a = 0.
We now collect terms of different order in . For 0 we obtain the Hamilton-Jacobi equation

∂ t S + ∇S 2 2 + V (t, x) = 0 ( 15
)
and for the order 1 the transport equation

∂ t a + ∇a, ∇S + ∆S 2 a = 0. ( 16
)
If we can solve the last two equations, then we are also able to construct a WKB approximation ψ, which satisfies the following Schrödinger equation

i ∂ t ψ + 2 ∆ ψ 2 -V (t, x) ψ = 2 ∆a 2 . ( 17
)
Thus if ψ(t, x) is a solution of ( 14) with initial condition ψ(0, x) = ψ(0, x), by Duhamel's formula we have

ψ(t, •) -ψ(t, •) = 2 2 t 0 (U (s, t)∆a)(s, •)ds, (18)
where U (s, t) is the propagator from time s to time t of ( 14).

Let us now comment on the solutions of ( 15) and ( 16). We begin with the Hamilton-Jacobi equation [START_REF] Khaneja | Time optimal control in spin systems[END_REF]. With each smooth function S 0 : M → R we can associate a differential 1-form dS 0 , which can be identified with its graph L = {(x, d x S 0 ) : x ∈ M }. One can easily check that this graph is a Lagrangian submanifold, i.e., it has dimension dim M and the symplectic form of T * M vanishes on L.

In order to find a solution of the Hamilton-Jacobi equation [START_REF] Khaneja | Time optimal control in spin systems[END_REF] we can try to construct it from a curve of Lagrangian submanifolds L t . We denote by Φ t : T * M → T * M the flow from time 0 to time t of the classical Hamiltonian system associated with the Hamiltonian

H(τ, p, x) = p 2 2 + V (τ, x). ( 19
)
Assuming that L 0 = {(x, d x S 0 ) : x ∈ M }, where S 0 : M → R is smooth, we define L t = Φ t (L 0 ). If there exists a smooth function S : [0, ε]×M → R such that S(0, •) = S 0 and each L t , t ∈ [0, ε], is the graph of the differential of S(t, •), then S is a solution of ( 15) as follows from the method of characteristics [START_REF] Anantharaman | Eigenfunctions of the Laplacian on negatively curved manifolds: a semiclassical approach[END_REF]. Of course this is not possible, in general, for all t larger than ε, because the restriction π| Lt can stop being one-to-one. We will explain how to avoid this issue in the next section.

Let us for now assume that we were able to construct a solution of the Hamilton-Jacobi equation [START_REF] Khaneja | Time optimal control in spin systems[END_REF]. Then we can solve the transport equation ( 16) as well, via the method of characteristics. Since ∇a, ∇S = ∇S[a], the equation for characteristics can be written as

ṫ = 1, ( 20
) ẋ = ∇S, ( 21
) ȧ = - ∆S 2 a. (22)
The first equation simply says that we can parametrize characteristic curves using the original time. In the second one we note that by construction ∇S = d x S, • , which is the projection of the Hamiltonian vector field H to M . Therefore solutions of ( 21) with initial value x(0) = x 0 are given by x(t, x 0 ) = πΦ t (x 0 , d x 0 S 0 ). Let us denote the flow of (21) from time 0 to time t by G t . Then G t takes a point x 0 in M , lifts it to the cotangent bundle as (x 0 , d x 0 S 0 ), transfers it along the Hamiltonian flow Φ t , and projects it back to M . Then the characteristics method gives us a solution of ( 22) of the form

a(x, t) = a 0 ((G t ) -1 x) exp - 1 2 t 0 ∆S(τ, x(τ ))dτ . ( 23
)
We can further simplify the solution by giving a geometric interpretation of the exponential term. We claim that

J t (x) = exp t 0 ∆S(τ, G τ x)dτ
is equal to the Jacobian of the flow G t . Here the Jacobian is defined intrinsically via the identity

(det dG t ) Vol = (G t ) * Vol .
Differentiating this equality with respect to t, we obtain that

d dt det dG t Vol = d dt (G t ) * Vol = L ∇S Vol = (∆S) Vol .
Solving this differential equation, we obtain the equality

J t (x) = det dG t (x).

Relation to the standard calculus of variations

From formulas [START_REF] Mirrahimi | Controllability of quantum harmonic oscillators[END_REF] and ( 23) it follows that the L 2 norm of the WKB approximation blows up if J t (x) is zero for some t and x. The set of zeros of J t is called the caustic and, if it is nonempty, the WKB approximation breaks down. If we multiply the WKB ansatz by a smooth function whose support has empty intersection with the caustic for sufficiently small times, then this new approximation is going to be a well-defined smooth function. The goal of this section is to recall some results that will help us to prove that caustics in certain regions of the configuration space cannot develop for sufficiently small times.

In order to prove it, we use a link between the caustics and the minimality of extremal curves of the action functional

S[x(•)] = S 0 (x 0 ) + t 0 ẋ(s) 2 2 -V (s, x(s)) ds.
It is known that each extremal curve must satisfy a Hamiltonian system with Hamiltonian given by ( 19) and with initial momentum p(0) = d x(0) S 0 [START_REF] Giaquinta | Calculus of variations[END_REF]. Thus, the wavefront at time t, given by the endpoints at time t of all such extremal curves in T * M , coincides with the Lagrangian surface L t from which we constructed solutions of the Hamilton-Jacobi equation in the previous section.

Caustics play an important role in calculus of variations: in the case of the functional S they indicate whether or not there are infinitesimal variations of the extremal curve that decrease the value of S with fixed final point [START_REF] Giaquinta | Calculus of variations[END_REF]. An intersection point between a given extremal trajectory and the caustic is called a conjugate point and the corresponding time a conjugate time. Techniques from calculus of variations allow to estimate the time t at which L t starts developing caustics. In particular, we will rely on the following result.

Theorem 3.1 ([13]

). Sufficiently small arcs of extremal curves of S do not contain conjugate points. This is a consequence of the Legendre-Clebsch condition, which is verified for the functional S. It is a sufficient condition for local minimality of small arcs of extremal curves.

Corollary 3.2. The first conjugate time is a lower semi-continuous function from M to (0, +∞] of the initial point x ∈ M . In particular, the restriction of the first conjugate time to a compact subset of M has a positive lower bound.

The positivity of the first conjugate time follows from Theorem 3.1. Its lower-semicontinuity follows from the definition of the caustic as the set of zeroes of the smooth function det dG t (x).

In terms of the functions a and J t introduced in the previous section, we can rephrase the second part of the previous corollary as follows.

Corollary 3.3. Let Ω be compactly contained in M . The function |J t | restricted to Ω has a uniform positive lower bound for sufficiently small times. In particular, given a 0 : Ω → R smooth and Ω 0 compactly contained in Ω, there exists ε > 0 such that the system of equations ( 21)-( 22) admits a smooth solution (S, a) on [0, ε] × Ω 0 that depends only on a 0 , S 0 | Ω , and V | [0,ε]×Ω .

Proof of Theorem 1.2 in the case when N 2 is a point

In the case when N 2 is just a point, we have that M = N 1 , the second condition of Theorem 1.2 is trivially satisfied and the first condition is simply given by W | Ω = c.

Let us fix a nonempty subset Ω compactly contained in Ω. In the open set Ω the classical dynamics are independent of the control, so it will take some time for each trajectory starting from Ω to exit from Ω, uniformly with respect to the control function. We wish to exploit this fact to prove that not only T c , but also T q is positive.

Consider a smooth cut-off function χ whose support is equal to Ω . Fix two smooth functions a 0 , S 0 : Ω → R such that χa 0 L 2 (Ω) = 1.

(24) Following Corollary 3.3, there exists a time ε > 0 such that, for every admissible control u : [0, ε] → R, the system of equations ( 21)-( 22) admits a smooth solution (S, a) on [0, ε] × Ω depending only on a 0 , S 0 , and V | Ω . Then, for every admissible control u : [0, ε] → R, define

ϕ(t, x) = χ(x) ψ(t, x) exp -ic t 0 u(s)ds , t ∈ [0, ε], x ∈ M, (25)
where ψ(t, x) = a(t, x)e iS(t,x) (with the convention that χ(x) ψ(t, x) = 0 for x / ∈ supp χ, even if ψ(t, x) is not defined). Notice that ψ(0, •) is in S(M ) thanks to (24). If we plug the ansatz above in the Schrödinger equation we find

exp ic t 0 u(s)ds i∂ t + ∆ 2 -V -u(t)W ϕ = i∂ t + ∆ 2 -V (χ ψ) = =χ i∂ t + ∆ 2 -V ψ + ∇χ, ∇ ψ + ∆χ 2 ψ = χ ∆a 2 + ∇χ, ∇ ψ + ∆χ 2 ψ,
where the first equality is a consequence of the identity χ(W -c) ≡ 0 and the last one follows from [START_REF] Mason | Generic controllability properties for the bilinear Schrödinger equation[END_REF]. Define r :

[0, ε] × M → C by r(t, x) = exp -ic t 0 u(s)ds χ(x) ∆a(t, x) 2 + ∇χ(x), ∇ ψ(t, x) + ∆χ(x) 2 ψ(t, x) , so that i∂ t + ∆ 2 -V -u(t)W ϕ = r on [0, ε] × M.
Notice that r is smooth with respect to x and that it depends only on a 0 , S 0 and V | Ω , and not on the control function u.

By the Duhamel formula we have

ϕ(t, •) -ψ(t, •) = t 0 U (s, t)r(s, •)ds,
where U (s, t) is the propagator from time s to time t of (1) and ψ is the solution of (1) with initial condition ψ(0, x) = ϕ(0, x), that is, ψ(t, •) = U (0, t)ϕ(0, •). Using the fact that U (s, t) is unitary we obtain

ϕ(t, •) -ψ(t, •) L 2 (M ) ≤ t 0 U (s, t)r(s, •) L 2 (M ) ds ≤ t 0 r(s, •) L 2 (M ) ds. (26)
Up to eventually reducing ε, assume that

δ := ε 0 r(s, •) L 2 (M ) ds < 1. ( 27
)
Take now ψ 1 ∈ S(M ) supported in M Ω, so that ψ 1 -ϕ(t, •) L 2 (M ) = 1 + ϕ(t, •) L 2 (M ) ≥ 1 for every t ∈ [0, ε] and every control law u. Then, by triangular inequality and because of ( 26) and ( 27), for every t ∈ [0, ε] and every u(•),

ψ 1 -ψ(t, •) L 2 (M ) ≥ ψ 1 -ϕ(t, •) L 2 (M ) -ϕ(t, •) -ψ(t, •) L 2 (M ) ≥ 1 -δ > 0,
concluding the proof that T q > 0.

3.4 Proof of Theorem 1.2

In order to complete the proof of Theorem 1.2, we wish to exploit the product structure of the manifold M . The Schrödinger equation takes the form

i∂ t ψ = - ∆ 1 ψ + ∆ 2 ψ 2 + V ψ + uW ψ, (28)
where ∆ 1 , ∆ 2 are the Laplace-Beltrami operators on N 1 , N 2 . Similarly to the case when N 2 is a point, we fix Ω compactly contained in Ω and we look for an approximate solution of the form ϕ(t, x, y) = χ(x)ψ 1 (t, x)ψ 2 (t, y)

where x ∈ N 1 , y ∈ N 2 , and χ is a smooth cut-off function whose support is Ω . We can assume without any loss of generality that Ω is a coordinate neighborhood centered at some x 0 ∈ Ω . The first condition of the theorem tells us that W (x, y) = W (y) for some function W ∈ C 2 (N 2 ) and all x ∈ Ω, y ∈ N 2 . So let us assume that ψ 2 is a solution of

i∂ t ψ 2 (t, y) = - ∆ 2 ψ 2 (t, y) 2 + V (0, y)ψ 2 (t, y) + u(t) W (y)ψ 2 (t, y).
Plugging the ansatz (29) into the Schrödinger equation (28) above we find that

i∂ t + ∆ 2 -V (x, y) -u(t)W (x, y) ϕ(t, x, y) = ψ 2 (t, y) χ(x) i∂ t + ∆ 1 2 ψ 1 (t, x)+ + ∇χ(x), ∇ψ 1 (t, x) + ∆ 1 χ(x) 2 ψ 1 (t, x) + χ(x)(V (0, y) -V (x, y))ψ 1 (t, x)+ + u(t)χ(x)(W (x, y) -W (y))ψ 1 (t, x) =: r(t, x, y).
Notice that, by construction, χ(x)(W (x, y) -W (y)) ≡ 0.

As in the previous section, let a 0 , S 0 : Ω → R be smooth and, thanks to Corollary 3.3, fix ε > 0 such that there exist a, S :

[0, ε] × Ω → R smooth such that ψ 1 (t, x) = a(t, x)e iS(t,x) is a solution of i∂ t ψ 1 + ∆ 1 ψ 1 2 = ∆ 1 a 2 on [0, ε] × Ω . Then r(t, x, y) = ψ 2 (t, y) χ(x) ∆ 1 a 2 + ∇χ(x), ∇ψ 1 (t, x) + ∆ 1 χ(x) 2 ψ 1 (t, x) +χ(x)(V (0, y) -V (x, y))ψ 1 (t, x) . The L 2 norm of M (x, y) → ψ 2 (t, y) χ(x) ∆ 1 a 2 + ∇χ(x), ∇ψ 1 (t, x) + ∆ 1 χ(x) 2 ψ 1 (t, x)
is uniformly bounded with respect to t ∈ [0, ε].

Let us now focus on the term of r involving V . In local coordinates on Ω we have

χ(x)(V (0, y) -V (x, y)) = χ(x) 1 0 ∇ 1 V (tx, y), x E dt ≤ K(x)χ(x) x E 1 0 c(tx)dt
where •, • E , • E are the Euclidean scalar product and norm and K(x) is a function that comes from equivalence of the Euclidean norm and the norm coming from the Riemannian scalar product. We then deduce that the L 2 norm of

M (x, y) → ψ 2 (t, y)χ(x)(V (0, y) -V (x, y))ψ 1 (t, x)
is uniformly bounded with respect to t ∈ [0, ε].

Summarizing everything we find that, as in the previous section, ϕ is an approximate solution of (28) concentrated in Ω × N 2 with a uniformly bounded error term t 0 r(s, •) L 2 (M ) ds which tends to 0 as t → 0+ uniformly with respect to u. Thus the occupation probability cannot be transferred outside Ω × N 2 in arbitrarily small time.

Remark 3.4. An alternative construction of the ansatz ϕ in (25) and (29) could be obtained by replacing ψ and ψ 1 by local regular (bounded in H 1 ) solutions of the Schrödinger equation. We have chosen to rely on the WKB approximation, since it permits a better parallel with the controllability of the corresponding classical system.

Appendix: An example of a quantum, but not classically controllable system

In [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF] it was proven that the harmonic oscillator with Gaussian control potential i∂ t ψ(t, x) = -∂ 2

x + x 2 + u(t)e ax 2 +bx+c ψ(t, x)

is controllable for a generic choice of real numbers a < 0, b, c. In particular, the Gaussian must be non-centered in order to avoid reflection symmetry. It was an application of the following theorem proven in the same article.

Theorem 3.5. Consider a controlled evolution equation on a Hilbert space H of the form

i∂ t ψ = (A + v(t)B) ψ, (31)
where v(t) takes values in an interval (0, δ) and A, B are self-adjoint operators on H. Suppose that A has discrete spectrum {λ i } i∈N , that the corresponding eigenvectors {φ i } i∈N form an orthonormal basis of H and belong to D(B). Suppose, moreover, that A+vB : span{φ i | i ∈ N} → H is essentially self-adjoint for every v ∈ (0, δ). Denote by b ij = Bφ i , φ j the components of B in this basis and by B (k) the first principal minor of order k. If -the elements of {λ i+1 -λ i } i∈N are Q-linearly independent, -for any j ∈ N there exists k ≥ j, such that B (k) is connected, then the Schrödinger equation (30) is approximately controllable.

Recall that with each Hermitian matrix C we can associate a graph with n vertices in such a way that the vertices i and j are connected by an edge if and only if c ij = 0. Then the matrix C is said to be connected if the corresponding graph is connected.

Note that Theorem 3.5 does not apply immediately to the harmonic oscillator since λ i+1λ i = 1 for all i ∈ N. However, it is proven in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF] that for a suitable choice of constants a, b, c and almost every µ ∈ R the operator A µ = -∂ 2

x + x 2 + µe ax 2 +bx+c has Q-linearly independent differences λ i+1 -λ i . This result is proven using some analyticity arguments, which reduce the proof of the statement to the proof of Q-linear independence of the elements b ii . Thus one can prove controllability of the harmonic oscillator with a Gaussian potential by considering an equivalent system of the form (31) with A = A µ , B = e ax 2 +bx+c , v(t) = u(t) -µ.

Let us fix a, b, c such that system (30) is controllable. We claim that there exists ε > 0 such that i∂ t ψ(t, x) = -∂ 2 x + x 2 + u(t)χ {x>ε} e ax 2 +bx+c ψ(t, x)

is also approximately controllable. Indeed, let us denote B = χ {x>ε} e ax 2 +bx+c . Then using the arguments of [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF] it is sufficient to prove that bii are all Q-linearly independent and that B(k) is connected for every k.

We know from the explicit formula for eigenfunctions of the harmonic oscillator that bij = b ij -C ij ε -ε e (a-1)x 2 +bx+c H i (x)H j (x)dx, where C ij are constants and H i are Hermite polynomials. We note that under the integral we have analytic functions, since H i are polynomials. Hence the integrals

f ij (ε) = C ij ε -ε
e (a-1)x 2 +bx+c H i (x)H j (x)dx depend analytically on ε and converge to 0 as ε → 0. Hence, if b ij is different from 0 then the set f -1 ij (b ij ) has zero measure, which means that bij = 0 for almost every ε > 0. Therefore, since the finite union of zero measure sets has measure zero, if the matrix B (k) is connected then B(k) is also connected for almost every ε > 0. Finally, for almost every ε > 0 the following holds true: for any j ∈ N there exists k ≥ j, such that B(k) is connected.

Similarly we can prove that the set of ε for which the elements of { bii } i∈N are Q-linearly dependent has measure zero. Indeed if the elements of { bii } i∈N are linearly dependent, then there exists a finite number of λ 1 , . . . , λ k ∈ Q, such that

0 = k i=1 λ i bii = k i=1 λ i b ii - k i=1 λ i f ij (ε).
Reasoning as above,

k i=1 λ i f ij -1 k i=1 λ i b ii
has measure zero for each fixed set λ 1 , . . . , λ k ∈ Q. Thus from countability of Q the claim follows.

On the other hand, the classical system of the form (2) with Hamiltonian H(t, p, x) = p 2 + x 2 + u(t)χ {x>ε} e ax 2 +bx+c

is not approximately controllable, because discs centered at the origin in phase space and of radius smaller than ε are invariant under the Hamiltonian flow.

Theorem 1 . 1 .

 11 Assume that M = N 1 ×N 2 is the product of two connected complete Riemannian manifolds N 1 , N 2 . Consider a Schrödinger equation of the form (1) on M and assume that the potentials V, W : M → R satisfy Assumption 1. Suppose that there exists a nonempty open set Ω ∈ N 1 such that 1. the restrictions W | Ω×{y} are constant for all y ∈ N 2 , 2. there exists a Lipschitz continuous function c : Ω

Acknowledgements. The authors would like to thank Patrick Gerard, Camille Laurent and Holger Teismann for helpful discussions and useful remarks.

* This work was supported by the ANR project SRGI ANR-15-CE40-0018 and by the ANR project Quaco ANR-17-CE40-0007-01.


Extensions, Intensities and the Convergent Advertisement

Convergence media disrupts how advertising is constructed, disseminated and used by consumers. This paper asks whether assemblage thinking can be mobilized to apprehend this new advertising/consumer landscape. If viewed as complex assemblages, rather than as texts or persuasive devices, a more reflexive theoretical frame for convergent advertising may evolve.

INTRODUCTION

The current era of technological innovation has enabled the development of new media forms [START_REF] Jenkins | Convergence Culture: Where Old and New Media Collide[END_REF], new consumer identities (Shau and Gilly 2003) and access to ever-evolving visual-affective worlds. The impact on advertising has been profound, suggesting "a disappearance of the advertisement in its traditional form… and its intensification and hybridization in oblique ways and through new media" (Campbell 2013,142). In effect, this transformation from static persuasive mechanism to integrated entertainment form (Campbell 2013) fundamentally challenges how advertising is constructed, disseminated and ultimately used by consumers (de Burgh-Woodman and Payne 2013). However, as different media channels continue to shape advertising consumption, equally traditional approaches to understanding both the dissemination and uses of advertising content come under increasing scrutiny as their capacity to adequately interpret and capitalize upon new consumer engagements with advertising is questioned [START_REF] Taylor | The Future of International Advertising Research: Suggestions for Moving the Discipline Forward[END_REF].

The essential question posed in this conceptual paper is whether assemblage thinking, as imagined by [START_REF] Deleuze | Mille plateaux Paris: Minuit[END_REF], can be fruitfully mobilized to better apprehend this new advertising/consumer landscape? While consumer engagement with the marketplace has radically changed through the advent of digital media, the theoretical evolution to understand these changes has lagged. As researchers grapple with the reality of the digital world, they are confronted with the need for theoretical resources for interpreting this landscape. In response to repeated calls for more nuanced theoretical perspectives that can deal with a rapidly transforming media landscape [START_REF] Duguet | la mémoire au poing[END_REF][START_REF] Kuntzel | Video about Video[END_REF][START_REF] Schroeder | Critical Visual Analysis[END_REF][START_REF] Speilman | The Reflexive Medium[END_REF][START_REF] Taylor | The Future of International Advertising Research: Suggestions for Moving the Discipline Forward[END_REF], the present paper aims to show how assemblage thinking [START_REF] Deleuze | Mille plateaux Paris: Minuit[END_REF] as perhaps distinct from assemblage theory [START_REF] Landa | Assemblage Theory[END_REF], 2006) enables a more reflexive perspective on how both the construction, dissemination and uses of advertising can be understood. To date, assemblage thinking has been applied in various contexts such as brands (Lury 2009), consumer communities (Parmentier andFischer 2014) and the public sphere (Campbell 2013). However, assemblage thinking has not been applied extensively to advertising. In this paper, we focus specifically on how the advertisements by noted film directors blur cinema/art/advertising lines to form multi-faceted assemblages composed of filmic, artistic and persuasive components.

CONVERGENCE AND ASSEMBLAGE

Convergence

Based on [START_REF] Levy | Cyberculture[END_REF], the term convergence was coined by [START_REF] Jenkins | Convergence Culture: Where Old and New Media Collide[END_REF] to explain the flow of content across multiple media platforms, the cooperation between multiple media industries, the search for new structures of media financing that fall at the interstices between old and new media, and the migratory behavior of media audiences" (p.3). The emergence of convergence catalyzes the shift from static advertising forms to the "re-appearance" (Campbell 2013) of advertising as a sophisticated entertainment form akin to a film, documentary or video game. The effect of convergence is that advertising becomes constructed for multiple media platforms, engages disparate audiences and utilizes creative modalities such as film, documentary and photographic production to generate content [START_REF] Dena | Current State of Cross Media Storytelling: Preliminary Observations for Future Design[END_REF]). Convergence extends advertising aesthetics (Brown, and Patterson 2001;[START_REF] Joy | Speaking of Art as Embodied Imaginations: A Multisensory Approach to Understanding Aesthetic Experience[END_REF][START_REF] Schroeder | Critical Visual Analysis[END_REF]Venkatesh and Meamber 2006) and provides a platform for diverse, enriched new forms of advertising production. These forms give rise to consumers engaging in 'spreading' [START_REF] Jenkins | Spreadable Media: Creating Value and Meaning in a Networked Culture[END_REF] activities, whereby ads become shared cultural resources. Equally, the advent of embedded marketing "reflects the merging interests and priorities of the entertainment, communication, advertising and brand marketing industries" [START_REF] Hackley | Advertising and Promotion: Communicating Brands[END_REF]. This evolution has given rise to the "mini-film" [START_REF] Copping | David Lynch's New Film for Christian Dior[END_REF][START_REF] Chen | The Persuasive Effectiveness of Mini-films: Narrative Transportation and Fantasy Proneness[END_REF] ad genre that consciously hybridizes advertising and traditional cinematic forms. The spreadable nature of convergence media potentially prompts the same self-extension, symbolic meaning and personal reflexivity we attribute to the object-possession [START_REF] Belk | Possessions and the Extended Self[END_REF]) transposed into a digital context [START_REF] Belk | Extended Self in a Digital World[END_REF]. It becomes important, then, to view advertising potentially as part of these kinds of culturally mediated collections.

Assemblage

Assemblage theory has attracted attention from consumer researchers seeking to understand diverse contexts such as individual consumption experiences [START_REF] Canniford | Purifying Practices: How Consumers Assemble Romantic Experiences of Nature[END_REF], consumer communities [START_REF] Scaraboto | Triggers, Tensions and Trajectories: Towards an Understanding of the Dynamics of Consumer Enrolment in Uneasily Intersecting Assemblages[END_REF]Thomas, Price, and Schau 2013) and families [START_REF] Epp | The Role of Brands and Mediating Technologies in Assembling Long-distance Family Practices[END_REF][START_REF] Epp | Outsourcing parenthood? How Families Manage Care Assemblages using Paid Commercial Services[END_REF]. As mentioned, assemblage theory has also been used to understand brands (Enwistle andSlater 2012 2013;[START_REF] Lury | Brands as Assemblage[END_REF]. Assemblage theory allows us to excavate how ideas are composed to create new interactions between artistic, market and media platforms. Assemblage thinking was originally flagged by [START_REF] Deleuze | Mille plateaux Paris: Minuit[END_REF] and elaborated as theory by DeLanda (2006DeLanda ( , 2015)). DeLanda describes assemblages as "agentic systems of diverse components that interact with one another in ways that can either stabilize or destabilize an assemblage's identity" (DeLanda 2006, 12). Assemblages "draw attention to the material and expressive "capacities" of components: capacities are defined as what components have the potential to do when they interact with other entities" (Parmentier and Fischer 2015 p.1229). Material components have a function or presence, others have an expressive or symbolic function that organize the assemblage. Equally, assemblages are characterized by extensions (the evolving inclusion of new components via expansion) and intensities (the greater or lesser power of individual components within the assemblage) (DeLanda 2015). Finally, assemblages are not fixed in time or space. They are able to adapt and shift over time (DeLanda 2006). In the same way that research has suggested brands can hold together or dissipate as assemblages [START_REF] Lury | Brands as Assemblage[END_REF]Parmentier and Fischer 2015), advertisements can also be regarded as assemblages in that they collate diverse components that interact to produce a tangible, stabilized (yet open to destabilization or deterriorialization) entity -the ad or campaign itself. Embedded within the ad or ads are an amalgam of components that acquire expressive capacities as they interact with other entities.

METHOD

In this research, a two-step method was utilized. The first step was to undertake an analysis of the assemblage's intensities. To achieve this, critical visual analysis (Schroeder 2002;[START_REF] Schroeder | Marketing Images of Gender: A Visual Analysis[END_REF] was used. Following the operations of description, interpretation, contextualization and finally comparison, Schroeder's visual analysis method was adapted to take in diverse elements of the moving image such as its complex intertextual and technological qualities [START_REF] Conomos | Mutant Media: Essays on Cinema, Video Art and New Media[END_REF], spatio-temporal structures, perceptual and affective dimensions [START_REF] Deleuze | Cinema 2: The Time Image[END_REF]) and audio-visual relationships [START_REF] Chion | Film, A Sound Art[END_REF]. The data used included 69 campaigns (sets of ads -some of which comprised multiple/numerous ads for the one campaign but captured here as one campaign count) directed by Wes Anderson, Darren Aronofksy, Michael Bay, Sofia Coppola, David Fincher, Jean-Pierre Jeunet, Spike Jonz, Ang Lee, Baz Luhrmann, David Lynch, Errol Morris, Martin Scorsese, Ridley Scott, and Joe Wright.

The second step was to map a classificatory typology of certain advertising forms. This classificatory typology reflects assemblage extension. Assemblages can be understood as accumulative to the extent that an assemblage can exist microcosmically (i.e. an individual ad), extend to a larger level (i.e. a suite of connected ads that extend the same set of structures and can be read as one large assemblage) or sit within an overarching assemblage (i.e. form a component of an entire oeuvre of inter-related transmedia communications for a particular brand). Therefore, the first step involved isolating a specific data set of ads made by film directors (described below), classifying said data set into one of three types i.e. 1. Singular 2. Suite or 3. Oeuvre. Once the data had been organized into one of these three types, the nature of the typology was elaborated, tracing the accumulation of visual, rhetorical and symbolic force. [START_REF] Deleuze | Mille plateaux Paris: Minuit[END_REF] emphasize the rhizomic nature of assemblages as entities with 'multiple roots' that spring up across time and space. This inherent quality is reflected in convergence media itself and, by extension, convergent advertising as it moves across multiple platforms, springing 'new roots'. Advertising created by film directors further augments this rhizomic nature as the ads can be viewed as rhizomic 'lines of flight' [START_REF] Deleuze | Mille plateaux Paris: Minuit[END_REF] connecting to greater cinematic, narrational, authorial and stylistic multiplicities that render our view of rhizomic advertising as 'always in the middle, between things, inter-being, intermezzo…the rhizome is alliance, uniquely alliance" (pg. 26). Viewing advertising in this way also requires us to see its extensions and intensities as perpetually engaged, shifting and changing the nature of the assemblage. The following figure (Figure 1) captures some of the dominant components at work in an advertising assemblage, thus representing the core of the typology evolved from the classification of the data along the axes of extension and intensity. The figure presented here is not exhaustive but rather designed to demonstrate some of the core relationships between components and how the nature of the assemblage either alters or preserves certain core components depending on whether it is a standalone ad, a suite or an oeuvre. The relationship between extensions and intensities is symbiotic as the affective, narrative, stylistic and referent intensities of the ad enable both de-territorialization in the form of larger, emergent assemblages and re-territorialization of key assemblage components. Like certain forms of cinema itself [START_REF] Brown | Deterritorialization and Schizoanalysis in David Fincher's Fight Club[END_REF], the simultaneous capacity for de-territorialization and re-territorialization through extension and intensification of affective, stylistic, narrative and referent intensities is shown to characterize the data presented here. As the valence of different intensities morph and re-align within the assemblage, so too does the assemblage itself. Thus, we should bear in mind these intensities are not static but always in an iterative dialogue of becoming with one another. In the case of convergent advertising, seemingly the most powerful intensity that affects all others within the assemblage is narrative. In its shift from persuasive spectacle to entertainment [START_REF] Gurevitch | The Cinema of Transactions: The Exchangeable Currency of the Digital Attraction[END_REF], convergent advertising is marked by a return to strong narratives (i.e. Lee's mini-film for BMW, Anderson's set of ads for Prada) that galvanize the other assemblage components. Chen (2015) also notes that narrative facilitates consumer transportation and fantasy pronesses, both of which appear as necessary intensities in the convergent assemblage as they drive consumer affect (in the Deleuzian sense) and the desire to seek extensions across platforms.

DISCUSSION

Drawing from narrative, particularly in the context of film directors who make ads, visual-narrative affect (or the synergy between the storyline, its visual realization and the emotional affect on the consumer) and the stylistic visual effects intensify the narrative through aesthetic and audio cues [START_REF] Chion | Film, A Sound Art[END_REF]. Linked to stylistic character, the style and effect of the ad is often a replication of the film director's cinematic signature (Wes Anderson for H&M, David Lynch for Calvin Klein), intensified by similar visual styling, use of the same cast (Lara Flynn Boyle for David Lynch's Calvin Klein Obsession, Adrien Brody as train conductor in a similar role to his Grand Budapest character for Wes Anderson for H&M, Nicole Kidman referencing her Moulin Rouge 'Satine' character for Baz Luhrmann for Chanel) and a replication of similar cinematic techniques and strategies to create a rich storyworld that harkens to other similarly rich storyworlds (Jean-Pierre Jeunet for Chanel referencing Amélie). Further intensities include the authorial presence or stamp of the director [START_REF] Gerstner | Authorship and Film[END_REF] whose distinctive stylistics and filmic techniques inhabit and frame the viewing experience. Finally, convergent ads embody an inherent reflexivity, moving somewhere between ad and art, low and high culture, disposable snippet and cinematic experience. These multiple intensities thus work within the ad itself but also affect consumer perception, engagement and use.

Extensions

The convergent nature of these ads, many of which are part of a set (in some cases like Errol Morris' Miller ads, the set runs to 80 ads and each extends the 'world' of the Miller Man as he moves from home, to diners, to lakes and open roads. Here one territorialized space spills into the next, i.e. de-territorializes, only to be recuperated or re-territorialized by the constant narrative thread), opens up what [START_REF] Deleuze | Mille plateaux Paris: Minuit[END_REF] call 'lines of flight', reflecting, again, a rhizomic multiplicity that can 'turn up' at any time. Extensions in the convergent ad assemblage enable core intensities such as narrative, stylistics, spatio-temporal parameters and authorial stamp to spill over either into other ads to compose a suite i.e. a larger assemblage that at once extends and intensifies core components. David Lynch's Calvin Klein Obsession ads represent this suite approach where each one of the four ads draws from a great writer (D.H. Lawrence, F. Scott Fitzgerald, Ernest Hemingway, Gustave Flaubert), utilize black and white photography and evoke a similar brooding mood with tell-tale Lynchian close-ups and abstractions of bodily forms. While each ad is complete (or territorialized) in its own right, there is nonetheless a larger narrative, stylistic and interpretative becoming that spills across the four ads. Where these components are further extended into larger assemblages again, an oeuvre is composed. Unlike a suite, which essentially replicates the intensities of the singular ad across a further two or three ads, an oeuvre further intensifies its original components but embeds within a much larger brand, marketing and cultural assemblage where the original assemblage potentially faces deterritorialization as its components moves across a greater diversity of platforms, confronts additional, more complex branding/marketing/ cultural components and offers maximum opportunity for consumer spreadabilitywhich, in turn, replicates further de-territorialization as components find new, possibility de-commodified, assemblages with which to interact. One such example of an oeuvre is David Lynch's four ads for Georgia Coffee. The four ads make no sense unless they are viewed in sequence. However, the entire aesthetic, narrative and symbolic composition of the ads rely on the consumer's thorough knowledge of the Twin Peaks series. The ads' mise en scènes such as the Red Room in the Black Lodge, Ghostwood Forest and the local diner harken back to the symbolic spaces of Twin Peaks with the return of Agent Cooper and his love of coffee introducing the customary comedic relief. More broadly, the ad provokes troubling questions in the context of oeuvre as the Twin Peaks season 2 cliffhanger (1990) left Agent Cooper imprisoned in the Red Room with his body/soul taken over by the omnipotent evil of "Bob". While the fate of Agent Cooper was resolved in the series 3 return (2017), these ads' airing between seasons posed questions around where and who Agent Cooper was. Was he still trapped in the Red Room with his imposter doppelganger promoting Georgia Coffee? Read against the overwhelming cult following, the copious textual extensions (such as Laura Palmer's Diary and the Guide to Twin Peaks) and the larger cinematic opus of Lynch himself, these ads take on monumental symbolic and narrative intensity within a highly extended assemblage.

CONCLUSION

The question posed in this paper is whether assemblage thinking can be mobilized to apprehend a rapidly evolving advertising/consumer landscape. The analysis of 69 ad campaigns by noted film directors demonstrates how assemblage thinking enables us to re-conceptualize convergent advertising as a rhizomic intersection of intensities (within the assemblage) and extensions that implicate diverse components from media, filmic, cultural and persuasive influences. From the consumer perspective, new terrains of engagement, sharing and spreadability (to borrow Jenkins' term) are made possible if advertising as static 'genre' is exploded, challenged by a conceptualization of advertising as a liminal entity, working at once within and across its territories or planes. This paper is an early step towards considering how assemblage thinking may account for the fluid nature of convergent advertising as it shifts and morphs into new forms. As such, further research is needed to see how this thinking might work as a sustainable framework. However, assemblage thinking animates the necessity to consider the intersectional, embedded nature of convergent advertisements, seeing them not as snippets of media 'content' or grabs of persuasive rhetoric but rather emergent forces, driven by the authorial, narrational, stylistic and cultural components that propel them across platforms, linking rich terrains of cultural referents, filmic influences and fantastical possibilities. At once de-territorialized so as to surge across platforms and re-territorialized back into the storyworld at the heart of the assemblage, this way of conceptualizing convergent advertising in the contemporary media environment opens up new possibilities for consumers and scholars alike.

Figure 1 -

 1 Figure 1 -Key Extensions and Intensities


Boris Jung

Aurelien Daurat

Audrey De Jong

Gerald Chanques

Martin Mahul

Marion Monnin

Nicolas Molinari

Samir Jaber

Rapid response team and hospital mortality in hospitalized patients

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Themis Palpanas
email: themis@mi.parisdescartes.fr

Volker Beckmann
email: beckmann@in2p3.fr

Report on the First and Second Interdisciplinary Time Series Analysis Workshop (ITISA)

The analysis of time-series data associated with modernday industrial operations and scientific experiments is now pushing both computational power and resources to their limits. In order to analyze the existing and (more importantly) future very large time series collections, new technologies and the development of more efficient and smarter algorithms are required. The two editions of the Interdisciplinary Time Series Analysis Workshop brought together data analysts from the fields of computer science, astrophysics, neuroscience, engineering, electricity networks, and music. The focus of these workshops was on the requirements of different applications in the various domains, and also on the advances in both academia and industry, in the areas of time-series management and analysis. In this paper, we summarize the experiences presented in and the results obtained from the two workshops, highlighting the relevant state-ofthe-art-techniques and open research problems.

INTRODUCTION

Time series 1 have gathered the attention of the data management community for more than two decades [START_REF] Agrawal | Efficient similarity search in sequence databases[END_REF][START_REF] Keogh | Locally adaptive dimensionality reduction for indexing large time series databases[END_REF][START_REF] Zoumpatianos | ADS: the adaptive data series index[END_REF]. They are one of the most common data types, present in virtually every scientific and social domain: they appear as audio sequences [START_REF] Kashino | Time-series active search for quick retrieval of audio and video[END_REF], shape and image data [START_REF] Ye | Time series shapelets: a new primitive for data mining[END_REF], financial [START_REF] Shasha | Tuning time series queries in finance: Case studies and recommendations[END_REF], environmental monitoring [START_REF] Raza | Practical data prediction for real-world wireless sensor networks[END_REF] and scientific data [START_REF] Huijse | Computational intelligence challenges and applications on large-scale astronomical time series databases[END_REF], and they have many diverse applications, such as in health care, astronomy, biology, economics, etc.

Recent advances in sensing, networking, data processing and storage technologies have significantly eased the process of generating and collecting data series. It is not unusual for applications to involve numbers of sequences in the order of hundreds of millions to billions [START_REF] Palpanas | Data series management: The road to big sequence analytics[END_REF]. These data have to be analyzed to identify patterns, gain insights, detect anomalies, and extract new knowledge.

A key observation is that analysts need to process a sequence (or subsequence) of values as a single object, rather than the individual points independently, which is what makes the management and analysis of data sequences a hard problem. Note that even though a sequence can be regarded as a point in n-dimensional space, traditional multidimensional approaches fail in this case, mainly due to the combination of the following two reasons: (a) the dimensionality is typically very high, i.e., in the order of several hundreds to several thousands, and (b) dimensions are strictly ordered (imposed by the sequence) and neighboring values are correlated.

Current time series analysis solutions require custom code, which implies huge investments in time and effort, and duplication of effort across different teams. Existing systems (e.g., based on DBMSs, Column Stores, or Array Databases) do not provide a viable solution, since they have not been designed for managing and processing sequence data [START_REF] Palpanas | Data series management: The road to big sequence analytics[END_REF]. Therefore, they do not offer a suitable declarative query language, storage model, auxiliary data structures, and optimization mechanism that can support a variety of sequence query workloads in an efficient manner [START_REF] Echihabi | The lernaean hydra of data series similarity search: An experimental evaluation of the state of the art[END_REF][START_REF] Zoumpatianos | Data series management: Fulfilling the need for big sequence analytics[END_REF]. (In Section 4, we discuss more reasons why existing solutions are inadequate.)

The Interdisciplinary Time Series Analysis Workshop provided a unique forum for researchers and practitioners that approach time series from different angles, ranging from data management and processing, to analysis, mining and machine learning. The core research issues considered in the workshop include: management and indexing, interactive visualization, machine learning, privacy preserving analytics, handling uncertainty and missing values, as well as applications of the above in astrophysics, neuroscience, engineering, electricity networks, and music.

The program of the two editions of the workshop included 2 keynote talks, 2 hands-on sessions, and a panel discussion. In this report, we summarize the ideas that were presented and discussed in the workshop, which had over 80 participants in total. The detailed program and the slides for the talks are available at the ITISA web pages. 1st edition: https://indico.in2p3.fr/event/13186/ 2nd edition: https://indico.in2p3.fr/event/13934/

KEYNOTE TALKS

In this section, we summarize the keynote talks.

Computer Science

Prof. Anthony Bagnall (University of East Anglia) focused on Time series classification problems (TSC). He described the recent advances in time series classification, and presented a taxonomy of algorithms based on the nature of discriminatory features used to classify. Finally, he presented an experimental comparison of over 20 algorithms on 85 of the UCR-UEA datasets. The results showed that the collective of transformation-based ensembles (COTE) was significantly more accurate than all other approaches, because it could utilize features from each of the five promising representations identified in the algorithm taxonomy.

Prof. Abdullah Mueen (University of New Mexico) talked about algorithms and applications of three primitive temporal patterns, namely, motifs, shapelets, and discords. Motifs are repeating segments in seemingly random time series data; Shapelets are small segments of long time series characterizing their sources; and Discords are anomalous waveforms in long time series that do not repeat anywhere else. He discussed efficient algorithms to discover these patterns, and presented corresponding use cases. Applications included activity classification using accelerometer and brain activity data, correlated clusters in social media data, and anomaly detection in online review data.

Prof. Themis Palpanas (Paris Descartes University) presented techniques for time series indexing. He described recent efforts in designing techniques for indexing and mining massive collections of time series, and showed that the main bottleneck is the time taken to build the index. He presented the state of the art techniques that adaptively create time series indexes, allowing users to correctly answer queries before the indexing task is finished.

Prof. Anastasia Bezerianos (University Paris-Sud), Dr. Theophanis Tsandilas (Inria), and Ms. Anna Gogolou (Inria) talked about interactive visual exploration of large time series collections. They provided an overview of existing interaction and visualization techniques for time series exploration and analysis, and noted the absence of focus on their scalability to multi-terabyte time series collections. To this end, they described work directions for achieving both visual scalability (how can we visualize billions of data series) and response-time scalability (how can we get answers quickly in interactive response times), including approximate and progressive result mechanisms.

Astrophysics

Dr. Dimitrios Emmanoulopoulos (University of Southampton) talked about astrophysical time series, and in particular AGN light curves2 , whose variability allow astrophysicists to study the physical conditions around a black hole. In order to test any theoretical model though, it is crucial to attribute a precise statistical significance to any timing property. Dr. Emmanolopoulos presented a new statistical method that can produce random light curves that contain all the genuine statistical and variability properties of the observed ones, i.e., the same flux distribution (quantified by the probability density function) and same power spectral density.

Dr. Jerome Rodriguez (CEA) discussed methods for diagnosing and analyzing the fast time variability in X-ray binaries 3 . He introduced the Fourier analysis and generic techniques used in high energy astrophysics, and explained how these tools help understand the properties of fast variabilities in X-ray binaries time series.

Dr. Gabriele Ponti (Max Planck Institute) also focused on the variabilities of X-ray binaries time series. He showed that the characteristic time-scales of such variations depend linearly on the mass of the black hole, and that by studying the correlations between the emission at various energy bands (through cross spectra and lag frequency spectra), it is possible to determine delays between radiation produced by different components of the system. From this, it is possible to draw conclusions about the geometry of the regions around black holes.

Dr. Vivien Raymond (Cardiff University) focused on Gravitational Wave (GW) 4 detection using the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO). He stressed that at the core of this new observational medium for GW-astronomy is the analysis of the time series of space-time deformations recorded by the GW detectors. He presented the time series analysis techniques currently used in signal detection, detector noise analyses, and source properties inference.

Dr. Eric Chassande-Mottin (CNRS) talked about the detection of GW in space. It is believed that the space-based LISA GW detector data stream will contain approximately 60 million simultaneous sources. He emphasized that in analyzing these time series data, there are two important goals: the first is to separate the various sources from each other, and the second is to estimate the astrophysical parameters of each source. In this task, matched filtering is the main tool of GW astronomy, a method that requires the use of accurate theoretical templates for each source type. development of sophisticated Bayesian algorithms.

Neuroscience

Dr. Katia Lehongre (ICM Institute for Brain and Spinal Cord) talked about times series analysis in neuroscience, and elaborated on the electrophysiology of epilepsy. Patients with epilepsy present abnormal brain activity, like epileptic spikes and seizures that can be recorded with electroencephalography (EEG). In order to localize the region of the brain that produces this abnormal activity, EEG from the patients is recorded continuously for 2 to 3 weeks. Usual clinical practice involves a neurologist reviewing visually the signal in order to determine the spatial localization and the temporal dynamics of the epileptic activity. As Dr. Lehongre pointed out, several studies tried to develop an automatic and reliable detection / characterization of the epileptic events in time and space, however, no fully non-supervised methods are commonly used by the neurologists, because they are not accurate enough. An efficient time series analysis could be of great interest to speed up the signal analysis, and in turn to increase the number of patients handled.

Prof. Uri Hasson (University of Trento) discussed time series in relation to the brain. He began with a brief overview of the sorts of time series that modern cognitive neuroscience can obtain from human participants and the principles of the instrumentation used to obtain those. The second part of the talk focused on approaches for analyzing these times series. These include quantification of correlations between different brain regions and network partitioning strategies. It was noted that more recent work is focusing on fast, non-oscillatory signatures in brain dynamics that also contain important information. These signatures are either driven by an input or internally generated. Several methods based on the analysis of peaks and pits in neural time series were discussed, as well as methods for decomposing spatiotemporal data into series of micro-states and motifs. The last part summarized these technologies from the perspective of temporal search engines, highlighting the importance of approximate searches on multivariate time series, and in the context of real-time analysis.

Engineering and Electricity Networks

Dr. Dohy Hong (Safran) shared his experience on multivariate time series analysis in aeronautics, and aircraft engines in particular. He pointed out that one of the main future challenges in aeronautics is the use of available data in order to enable the optimal management of maintenance processes (e.g., a per engine-based individual management strategy), and later its integration in the design process. The overall data processing along the engine cycle includes several technical challenges: weak signal detection in continuous multivariate usage time series (hidden layer/rules learning/extraction), management of heterogeneous data (maintenance repair and operation data, test bench or inspection data, configuration data, etc.), integrating and consolidating the existing expert knowledge (from design model to residual life time estimate practice), and others. All these challenges have to be addressed under the constraint of certifiability, which implies interpretability and robustness.

Dr. Georges Hebrail (EDF) made the case for privacy-preserving use of individual smart meter data for customer services. The advent of on-body/athome sensors connected to personal devices leads to the generation of fine grain highly sensitive personal data at an unprecedented rate. However, despite the promises of large scale analytics there are obvious privacy concerns that prevent individuals to share their personal data. Dr. Hebrail presented Chiaroscuro, a solution for clustering personal time series, with strong privacy guarantees. The execution sequence produced by Chiaroscuro is massively distributed on personal devices, coping with arbitrary connections/disconnections. Chiaroscuro builds on a novel data structure, which allows the participating devices to collaborate privately by combining encryption with differential privacy.

Music

Prof. Philippe Esling (IRCAM) talked about musical time series. Music inherently conveys several open and interesting scientific questions, which all embed the notion of time. Specifically, musical orchestration is the subtle art of writing musical pieces for orchestra, by combining the spectral properties specific to each instrument in order to achieve a particular sonic goal. Prof. Esling described novel learning and mining algorithms on multivariate time series that can cope with the various time scales that are inherent in musical perception, and can be used for orchestration. His current research is focused on automatic inference through deep representational learning to allow the automatic deciphering of these dimensions in order to provide optimal features for orchestration, by targeting correlations existing in the work of notorious composers.

HANDS-ON SESSIONS

The workshop also offered two hands-on sessions. The first hands-on session was organized by Dr. Vivien Raymond, and walked participants through the process of analyzing real time series signals collected at LIGO, and visualize a GW that was buried in the signal. This session aimed at teaching participants all the preprocessing steps necessary for cleaning the time series, and for amplifying the true signal, i.e., the GW. This process involved filtering and noise removal and downsampling steps5 , which had to be performed either in the time-or frequency-domain. The main take-away message of this session was that the time series processing workflow that analysts apply (often times) requires many preprocessing and data transformation steps.

The second hands-on session was led by Ms. Anna Gogolou, and dealt with the challenges in interactive visual exploration of large time series. The participants were asked to reply to a questionnaire that was divided in three parts: background information, scenarios, and detailed examples of their questions and problems at hand. Participants came mainly from two different domains: neuroscience and astrophysics. Analyzing their answers, led to the conclusion that their goal on multi-dimensional time series data is to find: similar patterns, abnormal patterns, time length of events, specific times of variability in data (periodicity), and correlation. Moreover, some are interested in working with quick, but rough results (e.g., approximations or incomplete answers), while they wait for the complete and exact answer.

DISCUSSION SESSIONS

Finally, we report on the discussion sessions of the workshop, which greatly helped in putting all previous information in perspective, and in identifying the research directions that are useful and promising. Below, we summarize the main points of the discussion, and relevant open research problems.

(1) Preprocessing: In most cases, time series must be preprocessed before being analyzed: this involves selecting the series and intervals of interest, and applying techniques from signal processing (e.g., band filters, denoising), several of which operate in the frequency domain. Thus, time series management systems [START_REF] Jensen | Time series management systems: A survey[END_REF] and analysis workflows should allow analysts to easily extract subsets of interest, and embed their data in different spaces (e.g., time, frequency), suitable for the various analysis techniques.

(2) Analysis Operations: The analysis task itself encompasses several different operations, including similarity search, correlation, clustering, classification, anomaly (or discord) detection, motif (or frequent patterns) discovery, and causal modeling 6 . Similarity search is a key operation that is expensive per se, and if performed fast then it can help speedup several of the other analysis operations, as well. We note that in some cases, the analysis operations need to be performed in a way that takes into account spatial information 7 , pointing to the need for the development of corresponding query languages and index structures.

(3) Versatility: Even though there exists a sizable number of studies on time series analysis techniques in the literature (and some of them have found their way into real systems), these techniques are usually not versatile enough for use in the real world. Real applications require scalable techniques that can serve ad-hoc queries and analysis workflows, have the ability to select and operate on sets of sequences selected using complex conditions8 , operate on both entire sequences and subsequences, support the analysis of variable-length subsequences 9 , treat value uncertainty 10 [4] as a first class citizen and be able to carry confidence and significance values along the analysis workflow, as well as allow for privacy-preserving analytics. (4) Interaction with Users: Despite the significant effort of the visualization community in this area [START_REF] Aigner | Visualization of Time-Oriented Data[END_REF], most of the available systems are far from scaling to the sizes of time series collections that are used in practice. Recent advances in time series indexing can be of help here, though, novel approaches are required in order to really address the current and future needs. Interactive visualizations are important for users, and when the datasets grow very large some of the most promising ways to achieve interactive response times is through the use of fast approximate and/or progressive answers [START_REF] Turkay | Progressive data science: Potential and challenges[END_REF], with the support of appropriate visualizations.

[Summary] Overall, we observed a slight disconnect between the needs of scientists and practitioners that process and analyze time series, and Computer Science (CS) researchers that work on time series. The problems that CS researchers have been studying are for the most part simplified, clean, and sanitized versions of the real problems and analysis workflows that practitioners have to address in the real world. Despite the remarkable progress in this area by the CS community during the recent years [START_REF] Echihabi | The lernaean hydra of data series similarity search: An experimental evaluation of the state of the art[END_REF][START_REF] Faloutsos | Forecasting big time series: Old and new[END_REF][START_REF] Jensen | Time series management systems: A survey[END_REF][START_REF] Keogh | Indexing and mining time series data[END_REF][START_REF] Large | Can automated smoothing significantly improve benchmark time series classification algorithms?[END_REF][START_REF] Palpanas | Big sequence management: A glimpse of the past, the present, and the future[END_REF], there are still many challenging open problems.

Efficiently supporting similarity search [START_REF] Chatzigeorgakidis | Local pair and bundle discovery over co-evolving time series[END_REF][START_REF] Kondylakis | Coconut palm: Static and streaming data series exploration now in your palm[END_REF][START_REF] Peng | Paris: The next destination for fast data series indexing and query answering[END_REF][START_REF] Yagoubi | Massively distributed time series indexing and querying[END_REF] is still challenging for large (out of memory) data series collections [START_REF] Echihabi | The lernaean hydra of data series similarity search: An experimental evaluation of the state of the art[END_REF]. Only very recently attention has been given to solutions that can support variable-length queries [START_REF] Linardi | Scalable, variable-length similarity search in data series: The ULISSE approach[END_REF][START_REF] Linardi | Matrix profile X: VALMOD -scalable discovery of variable-length motifs in data series[END_REF], and there are still a lot to be done in terms of supporting uncertain series [START_REF] Dallachiesa | Top-k nearest neighbor search in uncertain data series[END_REF]. Scalable visualization solutions are direly needed, especially in support of progressive analytics [START_REF] Gogolou | Progressive similarity search on time series data[END_REF][START_REF] Turkay | Progressive data science: Potential and challenges[END_REF]. At the same time, even some basic problems, such as the interplay between visual perception and similarity measures [START_REF] Gogolou | Comparing similarity perception in time series visualizations[END_REF], deserves to be studied in more detail. Evidently, in order to be used in practice, all the above components should be combined in general, easy-to-use by non-experts, time series management systems [START_REF] Echihabi | The lernaean hydra of data series similarity search: An experimental evaluation of the state of the art[END_REF][START_REF] Jensen | Time series management systems: A survey[END_REF], a task that is by itself a challenge.

WORKSHOP CONCLUSIONS

Even though time series are a very common data type, no available system can inherently accommodate and support the dataset sizes and complex analytics required by users. Our discussions showed that applications across different domains share common requirements: fulfilling them is a challenging goal, involving many interesting research problems.

Time series, or data series, or sequences are values measured and ordered over a dimension (usually time, but could also be mass in mass spectroscopy, angle in radial chemical profiles, or position in genome sequences).

An Active Galactic Nucleus (AGN) is a compact region at the center of a galaxy that has an unusually high luminosity. A light curve is an astrophysical time series that measures the amount of light as a function of time.

X-ray binaries (a.k.a. microquasars) are a class of binary stars that host the most compact objects (neutron stars and black holes), and are luminous in X-rays.

[START_REF] Dallachiesa | Uncertain time-series similarity: Return to the basics[END_REF] Gravitational waves are ripples in space-time caused by violent and energetic processes in the universe (e.g., the merge of two black holes). Albert Einstein predicted the existence of gravitational waves in 1916 in his general theory of relativity, and they were first detected in 2016.

https://www.gw-openscience.org/tutorials/

For example, Granger causality.

Neuroscientists for instance are interested in correlations among signals recorded by sensors that are spatially close to one

another.[START_REF] Gogolou | Comparing similarity perception in time series visualizations[END_REF] Based on metadata, time intervals, value

thresholds.[START_REF] Gogolou | Progressive similarity search on time series data[END_REF] Consider for example that most of the current time series indexes only support fixed-

length queries.[START_REF] Huijse | Computational intelligence challenges and applications on large-scale astronomical time series databases[END_REF] Note that in several cases this uncertainty comes from the actual measurement instrument, and is therefore inherent in the time series.

Acknowledgements

The organization of the workshops was supported by the CNRS Mastodons TimeClean project.