Amir Dembo
email: adembo@stanford.edu

Milton Jara

AND Stefano Olla
email: olla@ceremade.dauphine.fr

Amir Dembo ⋆

Ceremade

THE INFINITE ATLAS PROCESS: CONVERGENCE TO EQUILIBRIUM

Keywords: September 9, 2017. 2010 Mathematics Subject Classification. 60K35, 82C22, 60F17, 60J60 Interacting particles, reflecting Brownian motions, non-equilibrium hydrodynamics, infinite Atlas process

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Introduction

Systems of competing Brownian particles interacting through their rank dependent drift and diffusion coefficient vectors have received much recent attention (for example, in stochastic portfolio theory, where they appear under the name first-order market model, see [START_REF] Fernholz | Stochastic portfolio theory: an overview[END_REF] and the references therein). For a fixed number of particles m ∈ N, such system corresponds to the unique weak solution of

dY i (t) = j≥1 γ j 1 {Y i (t)=Y (j) (t)} dt + j≥1 σ j 1 {Y i (t)=Y (j) (t)} dW i (t) , (1.1)
for i = 1, . . . , m, where γ = (γ 1 , . . . , γ m ) and σ = (σ 1 , . . . , σ m ) are some constant drift and diffusion coefficient vectors and (W i (t), t ≥ 0), i ≥ 1 are independent standard Brownian motions. Here Y (1) (t) ≤ Y (2) (t) ≤ . . . ≤ Y (m) (t) are the ranked particles at time t and the R m-1 + -valued spacings process Z(t) = (Z 1 (t), Z 2 (t), . . . , Z m-1 (t)), t ≥ 0, is given by

Z k (t) := X k+1 (t) -X k (t) := Y (k+1) (t) -Y (k) (t) , k ≥ 1 . (1.2)
The variables X k (•) and Z k (•) correspond to the k-th ranked particle and kth spacing, respectively, with X 1 = min i Y i denoting the leftmost particle. In particular, existence and uniqueness of such weak solution to (1.1) has been shown already in [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF]. The corresponding ranked process X(t) solves the system dX j (t) = γ j dt + σ j dB j (t) + dL j-1 (t) -dL j (t) , j = 1, . . . , m

for independent standard Brownian motions (B j (t), t ≥ 0), where L j (t) denotes the local time at zero of the non-negative semi-martingale Z j (•), during [0, t], with L 0 ≡ 0 and L m ≡ 0. The spacing process Z(t) is thus a reflected Brownian motions (rbm) in a polyhedral domain. That is, the solution in R m-1 + of dZ j = (γ j+1γ j )dt + σ j+1 dB j+1σ j dB j + 2dL j -dL j+1 -dL j-1 .

(1.4)

The general theory of such rbm (due to [START_REF] Harrison | Multidimensional reflected Brownian motions having exponential stationary sistributions[END_REF][START_REF] Williams | Reflected Brownian motion with skew symmetric data in a polyhedral domain[END_REF], c.f. the survey [START_REF] Williams | Semimartingale reflecting Brownian motions in the orthant. Stochastic networks[END_REF]), characterizes those (γ, σ) for which the stationary distribution of Z(t) is a product of exponential random variables. Further utilizing this theory, [START_REF] Sarantsev | Comparison techniques for competing Brownian particles[END_REF] deduces verious stochastic comparison results, whereas [START_REF] Ichiba | Convergence rates for rank-based models with applications to portfolio theory[END_REF] and the references therein, estimate the rate t m , m ≫ 1 of convergence in distribution of the spacing process Z(t).

In particular, the Atlas model of m particles, denoted by atlas m (γ) (or atlas m when γ = 1), corresponds to σ j ≡ 1 and γ j = γ1 j=1 . For atlas m (γ) it is shown in [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF]Corollary 10] that the spacing process Z(t) has the unique invariant measure

µ (m,2γ) ⋆ := m-1 k=1 Exp 2γ(1 -k/m)) , m ∈ N , (1.5)
out of which [15, Theorem 1] deduces that

µ (∞,2γ) ⋆ := ∞ k=1 Exp(2γ), (1.6)
is an invariant measure for the spacings process of atlas ∞ (γ) (see also [START_REF] Rost | Hydrodynamics of a one dimensional nearest neighbor model[END_REF] for invariant measures of spacings when the particles follow linear Brownian motions which are repelled by their nearest neighbors through a potential). By time-space scaling we hereafter set γ = 1 wlog and recall in passing that to rigorously construct the atlas ∞ we call y = (y i ) i≥1 ∈ R ∞ rankable if there exists a bijective mapping to the ranked terms y (1) ≤ y (2) ≤ y (3) ≤ . . . of y. The solution of (1.1) starting at a fixed y ∈ R ∞ is then well defined if a.s. the resulting process Y = (Y 1 (t), Y 2 (t), . . .) is rankable at all t (under some measurable ranking permutation). The atlas ∞ process is unique in law and well defined, when

P i≥1 e -αY i (0) 2 < ∞ = 1 , for any α > 0 (1.7)
(see [START_REF] Shkolnikov | Competing particle systems evolving by interacting Lévy processes[END_REF]Prop. 3.1]), and [14, Theorem 2] further constructs a strong solution of (1.1) in this setting (more generally, whenever σ 2 0 = 0, j → σ 2 j+1 is concave and eventually, both σ 2 j = 1 and γ j = 0). We focus here on atlas ∞ , where wlog all atlas m , m ∈ N ∪ {∞}, evolutions considered, start at a ranked configuration Y (0) = X(0) having leftmost particle at zero (i.e. X 1 (0) = 0), and which satisfies (1.7). For example, this applies when Z(0) is drawn from the product measure

µ (∞,λ,a) ⋆ := ∞ k=1 Exp(λ + ka) , λ > 0, a ≥ 0 . (1.8)
The natural conjecture made in [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF] that µ

(∞,2) ⋆
is the only invariant measure for spacing of atlas ∞ , has been refuted by [START_REF] Sarantsev | Stationary gap distributions for infinite systems of competing Brownian particles[END_REF] showing that {µ (∞,2,a) ⋆ , a ≥ 0}, forms an infinite family of such invariant measures (similar invariant spacings measures appeared earlier, in the non-interacting discrete model studied by [START_REF] Ruzmaikina | Characterization of invariant measures at the leading edge for competing particle systems[END_REF][START_REF] Shkolnikov | Competing particle systems evolving by i.i.d. increments[END_REF]).

As for the behavior of the leading particle of atlas ∞ , [START_REF] Dembo | Equilibrium fluctuation of the Atlas model[END_REF] verifies [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF]Conj. 3], that starting with spacing at the translation invariant equilibrium law µ

(∞,2) ⋆ results with t -1/4 X 1 (t) d → N (0, c) , when t → ∞ , some c ∈ (0, ∞) .
(1.9)

Similar asymptotic fluctuations at equilibrium were established in [START_REF] Dürr | Asymptotics of particle trajectories in infinite one-dimensional systems with collisions[END_REF][START_REF] Harris | Diffusion with "collisions" between particles[END_REF] for a tagged particle in the doubly-infinite Harris system (the non-interacting model with γ j ≡ 0, σ j ≡ 1), for the symmetric exclusion process associated with the srw on Z (starting with [START_REF] Arratia | The motion of tagged particle in the simple exclusion system in[END_REF]), and for a discrete version of the Atlas model (see [START_REF] Hernández | Equilibrium fluctuations for a discrete Atlas model[END_REF]).

In contrast, initial spacing of law µ (∞,2,a) ⋆ for a > 0, induces a negative ballistic motion of the leading particle. Specifically, [START_REF] Tsai | Stationary distributions of the Atlas model[END_REF] and [START_REF] Sarantsev | Stationary gap distributions for infinite systems of competing Brownian particles[END_REF] show respectively, that {X 1 (t) + at} is then a tight collection, of zero-mean variables.

Little is known about the challenging out of equilibrium behavior of atlas ∞ . From [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF]Theorem 1] we learn that at critical spacing density λ = 2 the unit drift to the leftmost particle compensates the spreading of bulk particles to the left, thereby keeping the gaps at equilibrium. Such interplay between spacing density and drift is re-affirmed by [START_REF] Cabezas | Brownian particles with rank-dependent drifts: out-of-equilibrium behavior[END_REF], which shows that initial spacing law µ (∞,λ) ⋆ induces the a.s. convergence t -1/2 X 1 (t) → κ with sign(κ) = sign(2λ). From [19, Theorem 4.7] it follows that if atlas ∞ starts at spacing law ν 0 which stochastically dominates µ Theorem 1.1. Suppose the atlas ∞ process starts at Z(0) = (z j ) j≥1 such that for eventually non-decreasing θ(m) with inf m {θ(m -1)/θ(m)} > 0 and β ∈ [1, 2), lim sup

m→∞ 1 m β θ(m) m j=1 z j < ∞ , (1.10) lim sup m→∞ 1 m β θ(m) m j=1 (log z j ) -< ∞ , (1.11) lim inf m→∞ 1 m β ′ θ(m) m j=1 z j = ∞ , β ′ := β 2 /(1 + β), (1.12)
further assuming in case β = 1 that θ(m) ≥ log m. Then, the fdd of Z(t) for the atlas ∞ converge as t → ∞ to those of µ (∞,2) ⋆

.

For example, if λ j ∈ [c -1 , c] and λ j z j are i.i.d. of finite mean such that E(log z 1 ) -< ∞, then taking θ(m) ≡ 1 and β > 1 small enough for β ′ < 1, it follows by the slln that (1.10)-(1.12) hold a.s. Namely, when ν 0 is any such product measure, Z(t) converges in fdd to µ (∞,2) ⋆ . For independent z j ∼ Exp(λ j ) this applies even when λ j ↑ ∞ slow enough so m j=1 λ -1 j /( √ m log m) diverges (and hence (1.12) holds a.s.), or when λ j ↓ 0 slow enough to have m -β m j=1 λ -1 j bounded (for some β < 2, so (1.10) holds).

Remark 1.2. As matter of comparison, note that if z j decays to zero slower than j -1/2 log j, then {z j } satisfies the hypothesis of Theorem 1.1 (for β = 1), while for measures of the form (1.8), generically z j decays like j -1 .

The key to proving Theorem 1.1 is a novel control of the atlas m particle spacing distance from equilibrium at time t, in terms of the relative entropy distance of its initial law from equilibrium.

(∂ z j-1 -∂ z j ) √ g t 2 m-1 j=1 2e -2z j dz j ≤ 1 2t H(ν (m) 0 |µ (m,2) ⋆ ) + 1 m . (1.13)
Combining Proposition 1.3 with Lyapunov functions for finite atlas systems (constructed for example in [START_REF] Dupuis | Lyapunov functions for semimartingale reflecting Brownian motions[END_REF][START_REF] Ichiba | Convergence rates for rank-based models with applications to portfolio theory[END_REF]), yields the following information on convergence of the atlas m particle spacing fdd at times t m → ∞ fast enough.

Corollary 1.4. Starting the atlas m system at initial spacing law ν (m) 0 of finite second moment, for any fixed k ≥ 1, the joint density of (Z 1 (t m ), . . . , Z k (t m )) with respect to the corresponding marginal of µ

(∞,2) ⋆ , converges to one, provided t m is large enough so both t -1 m H ν (m) 0 |µ (m,2) ⋆ → 0, and t -1 m k j=1 Z j (0) → 0 (in ν (m) 0 -probability), as m → ∞. Remark 1.5.
For concreteness we focused on the atlas ∞ process, but a similar proof applies for systems of competing Brownian particles where σ 2 j ≡ 1, γ 1 > 0 and j → γ j is non-increasing and eventually zero. We further expect this to extend to some of the two-sided infinite systems considered in [START_REF] Sarantsev | Two-sided infinite systems of competing Brownian particles[END_REF]Sec. 3], and that such an approach may help in proving the attractivity of µ

(∞,2,a) ⋆ in the atlas ∞ system.
In Section 2 we prove Proposition 1.3 and Corollary 1.4, whereas in Section 3, we deduce Theorem 1.1 from Corollary 1.4 by a suitable coupling of the atlas m system and the left-most k particles of atlas ∞ up to time t m .

Entropy control for atlas

m : Proposition 1.3 and Corollary 1.4 Recall (1.3), which for atlas m is X j (t) = X j (0) + 1 {j=1} t + B j (t) + L j-1 (t) -L j (t) j = 1, . . . , m , (2.1)
where L j (t) denotes the local time on {s ∈ [0, t] :

Z j (s) = 0} for 1 ≤ j < m, with L 0 (t) = L m (t) ≡ 0. Let X m := {x : x 1 ≤ x 2 . . . ≤ x m } ⊂ R m . The generator of the X m -valued Markov process X(t) is then ( L m g)(x) := 1 2 m j=1 ∂ 2 x j + ∂ x 1 (2.2)
defined on the core of smooth bounded functions g(•) on X m satisfying the Neumann boundary conditions

∂ x j -∂ x j+1 g x j =x j+1 = 0, j = 1, . . . , m -1 . Specializing (1.4) the corresponding R m-1 + -valued spacings Z j (t) = X j+1 (t)-X j (t) are then such that for 1 ≤ j ≤ m -1, Z j (t) = Z j (0) -1 {j=1} t + B j+1 (t) -B j (t) + 2L j (t) -L j+1 (j) -L j-1 (t) . (2.3)
Let ∆ (m) denote the discrete Laplacian with Dirichlet boundary conditions at 0 and m. Hence, using hereafter the convention of

∂ z 0 = ∂ zm ≡ 0, ∆ (m) ∂ z j := ∂ z j-1 -2∂ z j + ∂ z j+1 , j = 1, . . . , m -1 . (2.4)
Following this convention, in combination with the rule

∂ x j = ∂ z j-1 -∂ z j , j = 1, . . . , m , the generator of the R m-1 + -valued Markov process Z(t) is thus L m = 1 2 m j=1 ∂ z j-1 -∂ z j 2 -∂ z 1 = - 1 2 m-1 j=1 ∂ z j (∆ (m) ∂ z j ) -∂ z 1 (2.5) defined on the core C m of local, smooth functions h(z) such that (∆ (m) ∂ z j )h z j =0 = 0, j = 1, . . . , m -1 . (2.6) Recall that µ (m,2) ⋆ (•)
is the (unique) stationary law of Z(t) for atlas m . In fact, for the density on R m-1

+ of µ (m,2) ⋆ (•), p m (z) := m-1 j=1 α j e -α j z j , α j := 2(1 -j/m) , (2.7)
a direct calculation shows that

1 2 m-1 j=1 α j ∆ (m) ∂ z j = -∂ z 1 . (2.8)
Combining the lhs of (2.5) with (2.8) yields the symmetric form of the generator

L m = - 1 2p m m-1 j=1 ∂ z j p m ∆ (m) ∂ z j (2.9)
Using (2.9) and integration by parts, we have for bounded, smooth g, h satisfying (2.6)

g(-L m h)dµ (m,2) ⋆ = h(-L m g)dµ (m,2) ⋆ = 1 2 m j=1 [(∂ z j-1 -∂ z j )g][(∂ z j-1 -∂ z j )h] dµ (m,2) ⋆ := D m (g, h) .
(2.10)

We see that dµ

D m (h) = 1 2 m j=1 ∂ z j-1 -∂ z j h 2 p m (z)dz , (2.11)
to the Sobolev space W 1,2 (µ

(2,m) ⋆ ) of functions on R m-1 + with L 2 (µ (2,m) ⋆ )-derivatives.
We also consider the Markov dynamics on R m-1 + for the spacing process of an atlas m whose m-th particle X m (s) = X m (0) is frozen. Under the same convention as before, the generator for this, right-anchored dynamics, is

L m = 1 2 m-1 j=1 ∂ z j-1 -∂ z j 2 -∂ z 1 (2.12)
for the core C m of local, smooth functions h(z) such that

( ∆ (m) ∂ z j )h z j =0 = 0, j = 1, . . . , m -1 , (2.13)
where ∆ (m) is the discrete Laplacian with mixed boundary conditions. Specifically,

∆ (m) ∂ z j = ∂ z j-1 -2∂ z j + ∂ z j+1 , 1 ≤ j ≤ m -2 , ∆ (m) ∂ z m-1 = ∂ z m-2 -∂ z m-1
(2.14) For the remainder of this section we identify µ (∞,2) ⋆ with its marginal on z = (z 1 , . . . , z m-1 ), whose density on R m-1

+ is q m (z) := m-1 j=1 2e -2z j .
(2.15)

Analogously to (2.9) we find that

L m = - 1 2q m m-1 j=1 ∂ z j q m ∆ (m) ∂ z j .
(2.16)

This (marginal of) µ (∞,2) ⋆
is thus reversible (stationary and ergodic) for the rightanchored dynamics, and similarly to (2.10)-(2.11) for bounded, smooth h satisfying (2.13), the associated Dirichlet form is given (on W 1,2 (q m dz)) by

D m (h) = 1 2 m-1 j=1 (∂ z j-1 -∂ z j )h 2 q m (z)dz . (2.17)
Indeed, this reversible measure corresponds to starting the right-anchored dynamics with X 1 (0) = 0 and a Gamma(2, m -1) law for the frozen X m , with the remainder m -2 initial particle positions chosen independently and uniformly on [0, X m ].

Proof of Proposition 1.3. Fixing m ≥ 2, we start the finite particle dynamics of generator L m of (2.9), with initial law ν

(m) 0 on R m-1 +
whose density

f 0 := dν (m) 0 dµ (m,2) ⋆ , (2.18)
has the finite entropy

H(ν (m) 0 |µ (m,2) ⋆ ) = [f 0 log f 0 ](z)p m (z)dz =: H m (f 0 ) . (2.19)
Recall [START_REF] Savaré | Gradient flows of probability measures[END_REF] that a Wasserstein solution of the Fokker-Planck equation

∂ t f t = L m f t , (2.20)
starting at f 0 , is a continuous (in the topology of weak convergence) collection of probability measures t → f t µ

(2,m) ⋆

such that for any s, the derivatives (∂

z j-1 - ∂ z j ) √ f s exist a.e. in R m-1 + , with t 0 D m ( f s )ds < ∞ ∀t < ∞ (2.21)
and moreover for any fixed compactly supported smooth function

ζ(t, z) on R + × R m-1 + , ∞ 0 D m (f t , ζ(t, •)) -∂ t ζ(t, z)f t (z)p m (z)dz dt = 0 . (2.22)
By [1, Theorem 6.6], the law ν (m) t that corresponds to a starting measure ν (m) 0 of finite entropy and finite second moment, is a Wasserstein solution ν

(m) t = f t µ (m,2) ⋆
of (2.20). 1 From [1, Theorem 6.6] we further have that then

√ f t ∈ W 1,2 (µ (2,m) ⋆ ) and D m f t < ∞ ∀t > 0 , (2.23)
with t → D m √ f t non-increasing and

H m (f t ) -H m (f 0 ) = -4 t 0 D m f s ds . (2.24)
Consequently, for any t ≥ 0,

4t D m f t ≤ 4 t 0 D m f s ds = H m (f 0 ) -H m (f t ) ≤ H m (f 0 ) . (2.25)
Next, comparing (2.7) with (2.15), notice that q m = p m h m for the strictly positive

h m (z) := m-1 j=1 2 α j e -2j m z j , such that (∂ z j-1 -∂ z j ) h m = 1 m -1 {j=m} h m . (2.26)
Hence, for f t = h m g t , using that m j=1 ∂ z j-1 -∂ z j = 0 and g t q m dz = 1, we arrive at where the supremum is also over all initial configurations.

2D m ( f t ) = m j=1 ∂ z j-1 -∂ z j √ g t + 1 m -1 {j=m} √ g t 2 q m (z)dz =2 D m ( √ g t ) - 1 m + √ g t -∂ z m-1 √ g t 2 q m (z)dz . ( 2
Proof. Building on the construction in [START_REF] Dupuis | Lyapunov functions for semimartingale reflecting Brownian motions[END_REF]Sec. 3] of Lyapunov functions for rbm in polyhedral domains, while proving [13, Thm. 3] the authors show that for the atlas k+1 (and more generally, for the spacing associated with (1.1), whenever j → γ j is non-increasing and j → σ 2 j forms an arithmetic progression), one has

E[V (Z(t))] ≤ e -t [V (Z(0))] + c 2 , ∀t ≥ 0 , (2.30)
where V (z) = e v,z for some strictly positive v and c 2 < ∞ (see [13, inequality (51)]). Noting that v, Z(t) /D(t) ∈ [c -1 1 , c 1 ] (with c 1 := max j {v j ∨ v -1 j }), we get upon combining (2.30) with Markov's inequality that for any initial configuration Z(0),

P(D(t) ≥ x) ≤ P( v, Z(t) ≥ x/c 1 ) ≤ e -x/c 1 [e -t e c 1 D(0) + c 2 ] .
(2.31)

For t ≥ c 1 D(0) the rhs of (2.31) is at most e -x/c 1 (1 + c 2 ), yielding (2.29).

Proof of Corollary 1.4. Fix probability densities h

0 = h 1 wrt the law q m dz on R m-1 + , such that √ h 0 , √ h 1 ∈ W 1,2
(q m dz). Both properties then apply for h λ := λh 1 + (1λ)h 0 , any λ ∈ (0, 1), and it is not hard to verify that

d 2 D m ( √ h λ ) d 2 λ = m-1 j=1 √ α 0 (∂ z j-1 -∂ z j ) h 1 - √ α 1 (∂ z j-1 -∂ z j ) h 0 2 α 0 α 1 q m dz ,
where the non-negative α 0 := h 0 /h λ , α 1 := h 1 /h λ are uniformly bounded (per λ).

Consequently, the map h → D m ( √ h) is convex on the set of probability densities h with respect to the product law q m dz on R m-1 + . The marginal density on (z 1 , . . . , z k ) (wrt the k-th marginal of µ

(∞,2) ⋆ ), is given for ν (m) t (dz) = g t q m dz and 1 ≤ k < m, by g t,k (z 1 , . . . , z k ) := g t (z) m-1 j=k+1 2e -2z j dz j .
(2.32) Thus, by the convexity of D m ( √ •) and the formula (2.17), we have that

D m ( √ g t ) = D m ( √ g t ) m-1 j=k+1 2e -2z j dz j ≥ D m ( √ g t,k ) ≥ D k+1 ( √ g t,k ) . (2.33)
In particular, fixing k ≥ 1 and choosing t m → ∞ as in the statement of the corollary, we deduce from (1.13) and (2.33) that lim m→∞ D k+1 √ g tm,k = 0 .

(2.34)

For r ≥ 2 and the Markov generator L r of (2.12) consider the functional on the collection of probability measures ν on R r-1 + defined by

I r (ν) := sup h≫0 h -1 (-L r h) dν , (2.35)
where the supremum is taken over all bounded away from zero, twice continuously differentiable functions having the boundary conditions (2.13) at m = r. With h -1 (-L r h) then continuous and bounded, clearly I r (•) is l.s.c. in the weak topology on probability measures in R r-1 + . Further, recall from [6, Thm. 5] that I r (ν) = ∞ unless ν = gq r dz for a probability density g such that √ g ∈ W 1,2 (q r dz),

or equivalently D r ( √ g) < ∞, in which case I r (ν) = D r ( √ g).
Hence, (2.34)

amounts to I k+1 (ν (m,k) tm
) → 0 for the joint law ν (m,k) tm of (Z 1 (t m ), . . . , Z k (t m )) and any weak limit point of these laws must have a density g wrt q k+1 dz such that D k+1 ( √ g) = 0. From (2.17) it is thus necessarily that throughout

R k + , ∂ z 1 √ g = 0, ∂ z j-1 -∂ z j √ g = 0, j = 2, . . . , k ,
so as claimed, the only possible limit point is g ≡ 1. By Prohorov's theorem, it remains to verify that {ν (m,k) tm } are uniformly tight, namely, to provide a uniform in m tail-decay for k j=1 Z j (t m ) in the corresponding atlas m system. To this end, recall [START_REF] Sarantsev | Infinite systems of competing Brownian particles[END_REF]Cor. 3.10(ii)] that under the same driving Brownian motions {B j (s)} and initial configuration, the first k spacings increase when all particles to the right of X k+1 (0) are removed. Consequently, it suffices to provide a uniform in m tail decay for the diameter D(t m ) of an atlas k+1 system of initial spacing distribution ν (m,k) 0

. Fixing ǫ > 0 we have from (2.29) the existence of finite c 1 = c 1 (k) and x = x(ǫ) such that for a given initial configuration, if t ≥ c 1 D(0) then

P(D(t) ≥ x) ≤ ǫ. By our assumption that t -1 m ρ m → 0 in ν (m,k) 0
-probability, for the (random) initial diameter ρ m := X k+1 (0) -X 1 (0), we have that P(c 1 ρ m ≥ t m ) ≤ ǫ for all m ≥ m 0 (ǫ), in which case

P(D(t m ) ≥ x) ≤ P(D(t m ) ≥ x, t m ≥ c 1 ρ m ) + P(c 1 ρ m ≥ t m ) ≤ 2ǫ .
With ǫ > 0 arbitrarily small and x = x(ǫ) independent of m, we have thus established the required uniform tightness.

Remark 2.2. The proof of Proposition 1.3 is easily adapted to deal with the right-anchored dynamic (of the generator L m given in (2.12)). It yields for the latter dynamic the bound of (1.13), now with (2t) -1 H(ν

(m) 0 | ⊗ m-1 k=1 Exp(2)
) in the rhs. The proof of Lemma 2.1 also adapts to right-anchored dynamics, hence the conclusion of Corollary 1.4 applies for sequences of right-anchored dynamics when the latter expression decays to zero at t = t m → ∞ such that t -1 m k j=1 Z j (0) → 0.

Coupling to atlas

∞ : Proof of Theorem 1.1 Let G(a) = (2π) -1/2 ∞
a e -x 2 /2 dx and consider the atlas ∞ process Y (t) = {Y i (t)}, denoting by X(t) = {X j (t)} the corresponding ranked configuration. We first provide three elementary bounds for this process that are key to the proof of Theorem 1.1. Lemma 3.1. For any initial condition X(0), ℓ ≥ 1 and t, Γ > 0,

P sup s∈[0,t] {X 1 (s)} ≥ Γ ≤ 2G ℓ Γ -t -ℓ j=1 X j (0) √ ℓt . (3.1)
Proof. Starting wlog at Y (0) = X(0), we have that for any s ≥ 0,

X 1 (s) ≤ 1 ℓ ℓ i=1 Y i (s) ≤ s ℓ + 1 ℓ ℓ j=1 X j (0) + 1 √ ℓ W (s) ,
where W (s) := ℓ -1/2 ℓ i=1 W i (s) is a standard Brownian motion. Thus, by the reflection principle,

P sup s∈[0,t] {X 1 (s)} ≥ Γ ≤ P sup s∈[0,t] { 1 √ ℓ W (s)} ≥ Γ - t ℓ - 1 ℓ ℓ j=1 X j (0) = 2P W (t) ≥ √ ℓ Γ - t √ ℓ - 1 √ ℓ ℓ j=1 X j (0) ,
which upon Brownian scaling yields the stated bound of (3.1).

Lemma 3.2. For X 1 (0) ≥ 0, Γ and k ≥ 2 such that Γ (k) := (Γ -X k (0))/3 > 0, any ℓ ≥ 1 and t > 0, we have that

P sup s∈[0,t] {X k (s)} ≥ Γ ≤ 2G ℓ Γ (k) -t -ℓ j=1 X j (0) √ ℓt + 4kG Γ (k) √ t . (3.2)
Proof. Recall [START_REF] Sarantsev | Infinite systems of competing Brownian particles[END_REF]Cor. 3.12(ii)] that keeping the same driving Brownian motions {B j (s)} and initial configuration X(0), the spacing vector Z(t) is pointwise decreasing in γ. Further, by [START_REF] Sarantsev | Infinite systems of competing Brownian particles[END_REF]Cor. 3.10(ii)], the first k -1 spacings increase when all particles to the right of X k (0) are removed. Consequently, that value of X k (t)-X 1 (t) at the drift γ = 1 of atlas ∞ is bounded by its value for a k-particle Harris system (of γ = 0), starting at same positions as the original atlas ∞ process left-most k particles. In the latter case, letting V k (s) := max k j=1 {B j (s)} and the identically distributed V ′ k (s) := max k j=1 {-B j (s)}, our assumption that X 1 (0) ≥ 0 results with

X k (s) -X 1 (s) ≤ X k (0) + V k (s) + V ′ k (s) .
Thus, with Γ (k) = (Γ -X k (0))/3 and { B(s)} denoting a standard Brownian motion, we get by the union bound that

P sup s∈[0,t] {X k (s)} ≥ Γ ≤ P sup s∈[0,t] {X 1 (s) + V k (s) + V ′ k (s)} ≥ Γ -X k (0) ≤ P sup s∈[0,t] {X 1 (s)} ≥ Γ (k) + 2kP sup s∈[0,t] { B(s)} ≥ Γ (k) .
Consequently, by (3.1) and the reflection principle,

P sup s∈[0,t] {X k (s)} ≥ Γ ≤ 2G ℓ Γ (k) -t -ℓ j=1 X j (0) √ ℓt + 4kG Γ (k) √ t ,
as claimed.

Lemma 3.3. For any m ≥ 0, t, Γ > 0 and initial configuration Y (0) = X(0),

P inf s∈[0,t] inf i>m {Y i (s)} ≤ Γ ≤ 2 i>m G X i (0) -Γ √ t . (3.3)
Proof. Removing the drift in the atlas model decreases all coordinates of Y (s) and correspondingly increases the lhs of (3.3). Thus,

P inf i>m inf s∈[0,t] {Y i (s)} ≤ Γ ≤ i>m P inf s∈[0,t] {W i (s)} ≤ Γ -X i (0) = 2 i>m P W (1) ≤ Γ -X i (0) √ t = 2 i>m X i (0) -Γ √ t ,
as claimed.

Proof of Theorem 1.1. Given initial spacing configuration z that satisfies (1.10) and (1.11), consider the following two sequences of initial distributions for the finite increment vectors Z m (0) := (Z 1 (0), . . . , Z m-1 (0)) of the atlas m process, m ≥ 2. Starting at the measure ν (m,-) 0

(•) = µ (m,2) ⋆ (•|Z m (0) ≤ z m )
for the given z m = (z 1 , . . . , z m-1 ) yields for same driving Brownian motions an atlas m spacing process Z - m (s) which is dominated at all times s ≥ 0 by the corresponding process that started at spacing z m , whereas ν (m,+) 0

(•) = µ (m,2) ⋆ (•|Z m (0) ≥ z m )
similarly yields a spacing process Z + m (s) that dominates the spacing for the original process which started at z m . The corresponding relative entropies are then ). For ψ(m) ↑ ∞, with inf m {ψ(m -1)/ψ(m)} > 0, let

H + m := H(ν (m,+) 0 |µ (m,2) ⋆ ) = -log µ (m,2) ⋆ ({ m-1 j=1 [z j , ∞)}) = m-1 j=1 α j z j ≤ 2 m-1 j=1 z j (3.4) H - m := H(ν (m,-) 0 |µ (m,2) ⋆ ) = -log µ (m,2) ⋆ ({ m-1 j=1 [0, z j ]}) = m-1 j=1 -log(1 -e -α j z j ) ≤ m-1 j=1 1 + (log α j z j ) -≤ 2m log m + m-1 j=1 (log z j ) -, (3.5
t m := 2m β θ(m)ψ(m) . (3.6)
Setting m n = m n (s) := inf{m ≥ 2 : t m ≥ s n }, our constraints on θ(•) and ψ(•) yields for t m of (3.6) and any s n ↑ ∞,

inf n≥1 s n t mn ≥ inf m≥2 t m-1 t m > 0 .
Thus, by the preceding, upon applying Corollary 1.4 to the atlas m model initialized at ν (m,±) 0

we have that the joint law of the first k coordinates of Z ± mn (s n ), converges as n → ∞ to the corresponding marginal of µ (∞,2) ⋆ . The same limit in distribution then applies for the spacing (Z 1 (s n ), . . . , Z k (s n )) of atlas mn started at z mn (which is sandwiched between the corresponding marginals of Z - mn (s n ) and Z + mn (s n )). Assuming further that sup m {ψ(m)/m} ≤ 1, we claim that for X 1 (0) = 0 and the given initial spacing Z(0) = z, the rhs of (3.2) is summable over m, at t = t m of (3.6) and and recalling that θ(ℓ m ) ≥ 1 2 log m when β = 1, it is easy to verify that the preceding bound is summable in m. Next, utilizing (1.12), we can further make sure that ψ(m) ↑ ∞ slowly enough so that for any fixed κ < ∞, X m (0) ≥ (κ + 1)Γ m , ∀m ≥ m κ (3.9) so that for m ≥ m κ the rhs of (3. occur for all m large enough. Note that A m implies that throughout [0, t m ] the k left-most particles of the atlas ∞ process are from among the initially left-most m particles. From this it follows that under A mn the spacing (Z 1 (s n ), . . . , Z k-1 (s n )) for the atlas mn coincide with those for the atlas ∞ , when using the same initial configuration z and driving Brownian motions {W i (s)}. Having proved already the convergence in distribution when n → ∞, of the first k -1 spacing for atlas mn at time s n and that the events A m occur for all m large enough, we conclude that the fdd of spacing for atlas ∞ converge to those of µ (∞,2) ⋆

, along any (non-random) sequence s n ↑ ∞, as claimed.

Proposition 1 . 3 .

 13 Start the atlas m system at initial spacing law ν , for any t > 0 the spacing law ν (m) t is absolutely continuous with respect to the marginal of µ (∞,2) ⋆ and the Radon-Nikodym derivative g t satisfies m-1 j=1

(m, 2

 2 ) ⋆ = p m (z)dz is reversible for this dynamic, and the corresponding Dirichlet form D m (h) := D m (h, h), extends from C m , now only as

Lemma 2 . 1 .

 21 For atlas k+1 , some c 1 = c 1 (k) finite and D(t) := k j=1 Z j (t), lim x→∞ sup t≥c 1 D(0) { P(D(t) ≥ x) } = 0 , (2.29)

Fixing k ≥ 2 ,

 2 ) sincelog(1e -u ) ≤ 1 + (log u) -for all u ≥ 0, while α j ≥ 2/m (see (2.7)), hence log(e/α j ) ≤ 2 log m. By (1.10) and(3.4),lim sup m→∞ H(ν (m,+) 0 |µ (m,2) ⋆ ) m β θ(m) < ∞ .With θ(m) ≥ log m in case β = 1, we similarly deduce from (1.11) and (3the uniform over m ≥ 2k first moment bound, any t m → ∞. Further, the second moment of ν (m,-) 0 is finite (being at most z 2 ), as is the second moment of ν (m,+) 0 (being at most the product of e H + m and the finite second moment of µ (m,2) ⋆

Γ

  m := 36m β ′ θ(m)ψ(m) β/(1+β) , ℓ m := [m β/(1+β) ψ(m) 1/(1+β ] . (3.7)Indeed, since θ(•) is eventually non-decreasing, we have then that1 12 Γ m ℓ m ≥ t m ≥ ℓ 1+β m θ(ℓ m ) , ∀m ≥ m ⋆ (3.8)Further, with k fixed and Γ m ↑ ∞, necessarily X k (0) ≤ Γ m /8 for all m ≥ m ⋆ large enough, in which case from (3.8), the rhs of (3.2) is bounded above for such m, t m , Γ m and ℓ m , by 2G ℓ β m θ(ℓ m ) + 4kG 3 ℓ β-1 m θ(ℓ m )

3 ) 2 ∞ j=1 G 2 ∞ j=1 G

 32j=12j=1 is bounded above, at t = t m and Γ = Γ m , by κΓ m+j / √ t m . (3.10) Note that β ′ ≥ 1/2 and δ := β/(1 + β) -1/2 ≥ 0 is strictly positive when β > 1. Thus, β ′β/2 = δβ ≥ 0 and setting κ ′ := (18κ) 2 , we deduce from (3.6) and (3.7) that κ Γ m √ t m Γ m+j Γ m ≥ m βδ θ(m) 1/2 2κ ′ (1 + j/m) .Increasing m κ if needed, we have that m 2βδ θ(m) ≥ log m for all m ≥ m κ , in which case for b m := m -1 log m, the expression (3.10) is further bounded by 2κ′ (1 + j/m) log m ≤ 2m -κ ′ ∞ j=1 e -κ ′ jbm ≤ 2 κ ′ b m m -κ ′(recall the elementary bound G(x) ≤ e -x 2 /2 for x ≥ 0). Thus, for any κ ′ > 2, such choices of t m and Γ m guarantee that the rhs of (3.3) is also summable over m. Combining Lemma 3.2, Lemma 3.3 and the Borel-Cantelli lemma we deduce that almost surely, the events A m := sup s∈[0,tm] {X k (s)} < Γ m ≤ inf s∈[0,tm],i>m {Y i (s)} ,

(∞,2) ⋆ (e.g. when ν 0 = µ (∞,λ) ⋆ any λ ≤ 2), then the finite dimensional distributions (fdd) of Z(t) converge to those of µ (∞,2) ⋆ as t → ∞. However, nothing else is known about the domain of attraction of µ (∞,2) ⋆ (or about those of {µ (∞,2,a) ⋆ , a > 0}). For example, what happens when ν 0 = µ (∞,λ) ⋆ with λ > 2? Our main result, stated next, answers this question by drastically increasing the established domain of attraction of µ (∞,2) ⋆ spacing for atlas ∞ .

Though we could not find a reference for it, we expect f t to be also a strong solution of (2.20) which satisfies the boundary conditions of (2.6).

⋆ Research partially supported by NSF grant #DMS-1613091. †Research partially supported by the ERC Horizon 2020 grant #715734. ‡Research partially supported by ANR-15-CE40-0020-01 grant LSD. .


Xun Wang
email: xunwang@ust.hk

Mohamed S Ghidaoui
email: ghidaoui@ust.hk

MATCHED-FIELD PROCESSING METHOD FOR LEAK DETECTION IN A PIPE

Keywords: defect detection, leak localization, fluid transient, matched-field processing 1

This paper proposes a transient leak detection method in a pipeline system which uses head measurements in the frequency domain. A matched-field processing (MFP) approach is used to separately estimate location and size of a leak. The leak location is determined by matching the model with the measurement, more exactly by maximizing the inner product between a unit weighting vector and the measurement. This maximum occurs in the case that the weighting vector and the measurement have the same direction implying that the parameters behind are identical. Then, the leak size is estimated via least square. This MFP approach is able to use not only the resonant frequencies but also all the frequencies being invoked. Besides, the proposed method is a maximum likelihood estimation and equivalent to a matched-filter approach which maximizes the signal-to-noise ratio (SNR). Experiments on simulated data for a single pipe system illustrate the estimation efficiency of the MFP method. The simulation results justify that the proposed leak detection method is able to localize not only a single leak but also multiple leaks.

INTRODUCTION

Leakage detection in a pipeline system is an important issue in water supply systems because leaks result in financial losses, wastage of water, energy and other resources used to treat the water, and pose health risk issues since leaks are potential entry points for contaminants [START_REF] Colombo | A selective literature review of transient-based leak detection methods[END_REF]). In Hong Kong, for example, leak water annually costs more than one billion HK dollar due to the aging water infrastructure. Therefore, a fast, accurate and low-cost leak localization method is under searching.

Fluid transients-based defect detection methodology, where a hydraulic pressure wave is introduced into the pipe system and the pressure response is measured at specified location(s), is a promising general approach for the detection of leaks and other defects. The basic premise of this approach to defect identification and characterization is that a measured wave signal in the fluid in the conduit is modified by its interaction with the physical system as it propagates and reflects throughout the system. Accordingly, it contains information of the conduit's properties and state. Examples of such approach are the (1) Transient Reflection based Method (TRM), such as [START_REF] Brunone | Transient test-based technique for leak detection in outfall pipes[END_REF], [START_REF] Brunone | Detecting leaks in pressurised pipes by means of transients[END_REF], [START_REF] Covas | Application of hydraulic transients for leak detection in water supply systems[END_REF]); (2) Transient Damping based Method (TDM) by [START_REF] Wang | Leak detection in pipelines using the damping of fluid transients[END_REF]; (3) Frequency Response based Method (FRM) by [START_REF] Liou | Pipeline leak detection by impulse response extraction[END_REF], [START_REF] Mpesha | Leak detection in pipes by frequency response method[END_REF], [START_REF] Ferrante | Pipe system diagnosis and leak detection by unsteady-state tests harmonic analysis[END_REF], (Covas et al. 2005a), [START_REF] Lee | Frequency response coding for the location of leaks in single pipeline systems[END_REF], (Lee et al. 2005a), [START_REF] Lee | Experimental verification of the frequency response method for pipeline leak detection[END_REF], [START_REF] Sattar | Leak detection in pipelines by frequency response method[END_REF] and [START_REF] Taghvaei | Leak detection in pipes using induced water hammer pulses and cepstrum analysis[END_REF]; and (4) Inverse Transient based Method (ITM) studied in [START_REF] Liggett | Inverse transient analysis in pipe networks[END_REF], [START_REF] Vítkovsky | Leak detection and calibration using transients and genetic algorithms[END_REF], [START_REF] Stephens | Transient response analysis for fault detection and pipeline wall condition assessment in field water transmission and distribution pipelines and networks[END_REF], and [START_REF] Covas | Case studies of leak detection and location in water pipe systems by inverse transient analysis[END_REF].

Matched-field processing (MFP) is originally an array processing technique for the source localization problem, which generalizes the plane wave beamforming method [START_REF] Krim | Two decades of array signal processing research[END_REF]. It estimates the model parameter by matching the computational sound field to the measurements and is proved to be a maximum likelihood estimation (MLE) [START_REF] Krim | Two decades of array signal processing research[END_REF], (Wang et al. 2016a), (Wang et al. 2016b). The MFP approach has been applied to the sound source reconstruction in ocean [START_REF] Baggeroer | An overview of matched field methods in ocean acoustics[END_REF]), [START_REF] Kuperman | Shallow-water acoustics[END_REF]. Recently, [START_REF] Tolstoy | Detecting pipe changes via acoustic matched field processing[END_REF]) and [START_REF] Tolstoy | Waveguide monitoring (such as sewer pipes or ocean zones) via matched field processing[END_REF]) made an initial attempt of applying MFP to blockage detection in a sewer pipe. Note that in the last two works the acoustic field in pipe is not modeled such that some of the concerned unknown parameters cannot be estimated. In the present article, the leakage detection problem is solved using the MFP approach. The leak position and size are parametrized and explicitly estimated. This method is proved to be a matched-filter approach, which maximizes the signal-to-noise ratio (SNR) and thus is robust with respect to noises. The noise can be assumed as either white or non-white noise; in the latter case a whitening filter is applied to meet the model assumption.

This paper begins with a model description. Then, the MFP method is introduced: the leak position and size estimates are explicitly given. Numerical simulation results justify the estimation efficiencies. The performance of the proposed method for multiple leaks is also discussed.

Model

This section presents the pipeline model considered in the paper. Consider a single pipeline bounded by an upstream and a downstream node, whose coordinates are  0 U xx and  D xx, respectively. A sensor is assumed to be set near the downstream node whose coordinate is denoted by M

x . Here, the model of single leak is considered, the leak location is

 () L L M
x x x , L z denotes the elevation of the pipe at the leak, 0 L q and 0 L h stand for the steady-state discharge and head at the leak, respectively. The leak size is represented by the lumped leak parameter

 L d L s C A , where d C is the discharge coefficient of

the leak and L

A is the flow area of the leak orifice. The steady-state discharge of leak is related to the lumped leak parameter by  00 2 ( )

L L L L
q s g h z in which g is the gravitational acceleration. The discharge and head oscillations due to a fluid transient are represented by q and h. Given the discharge () U qx and head () U hx , the quantities at M x can be computed in the following way [START_REF] Chaudhry | Applied Hydraulic Transients[END_REF]:

                        0 0 ( ) ( ) 1 ( ) ( ) 2( ) ( ) ( ) 01 L MU NL M L NL L LL MU q q x q x M x x M x hz h x h x (1) in which           1 cosh( ) sinh( ) () sinh( ) cosh( ) NL xx Mx Z Z x x
is the field matrix, (2) Note that since the head in the frequency domain is obtained by a discrete Fourier transform (DFT) which is the summation of a large amount of time domain signal, according to the Central Limit Theorem the noise in the frequency domain can be reasonably assumed to follow a (zero-mean) Gaussian distribution. In the next section, the leak detection problem is solved under the assumption of Gaussian white noise. For non-white noise, the measurement signal can be whitened before using the proposed leak detection method, such that the noise distribution assumption holds.

Leak detection using matched-field processing

The leakage localization problem is solved using the matched-field processing (MFP) method.

The noise vector is assumed to follow a zero-mean Gaussian distribution. The idea is to adjust a unit vector  1 ( ,..., ) T J w w w

(  1 w
) to have the same direction (in the J-dimensional complex vector space) with the measurement. Here, the output function is defined by the inner product between the weighting vector w and the measured head difference h : in which the superscript H stands for the conjugate transpose. When w has the same direction as h , i.e., 2 || B reaches maximum, so that the concerned parameters behind w and h are the same. Therefore, the optimal weight is obtained by maximizing By inserting Eq. (2) into the above equation and maximizing its expectation with respect to w , the optimal weight is obtained: in which "arg max" represents the argument of the maximum. Then, the leakage location can be estimated by substituting ŵ back to 2 || B :

It is remarkable that it is also a maximum likelihood estimator (MLE) of the leak position [START_REF] Krim | Two decades of array signal processing research[END_REF], (Wang et al. 2016a), (Wang et al. 2016b). Then, by solving a least square, the leak size can be estimated by

According to the property of MLE, the above two equations are consistent estimators, i.e., they converge in probability to the actual values. Furthermore, the proposed method is equivalent to a time reversal approach [START_REF] Khazaie | Localization of random acoustic sources in an inhomogeneous medium[END_REF]) and a matched-filter processing method which maximizes the signal-to-noise ratio (SNR). The latter implies that the proposed method minimizes the influence of noise. (Lee et al. 2005b) indicates that the shape and the magnitude of the frequency response diagram (FRD) independently decide the location and the size of the leak. In the proposed approach, the estimates of location and size are separated: they reflect information of shape and amplitude of FRD, respectively. Furthermore, the proposed method can use not only the peak frequencies but also the frequencies in between, which is not the case of the frequency response approach in (Lee et al. 2005a) and the peak-sequencing method in (Lee et al. 2005b) .

Numerical simulation

In this section, numerical examples are introduced to test the MFP method. The cases of single leak and multiple leaks are respectively considered.

The case of single leak

First, the case of single leak is considered. The leakage problem in a single pipeline is numerically simulated, the setup is shown in Figure 1. Two reservoirs are connected by a pipe in a horizontal plane; the heads of the upstream and downstream reservoirs are H1=25 m and H2=20 m, respectively. The pipe length is l=2000 m and the diameter is D=0.5 m. The Darcy-Weisbach friction factor of the pipe is F=0.02 and the steady-state discharge is  3 0 0.0153 / q m s . The wave speed is 1200 m/s. A valve is located at the downstream end of the pipe and a sensor is just upstream of the valve. It is assumed that an impulse wave is generated by rapidly closing and opening the valve, thus the boundary conditions

 ( ) 0 U hx and  ( ) 1 U qx
are applied. The wave propagation simulation in the frequency domain is realized by the transfer matrix method. The discharge at the upstream is also assumed to be obtained. Gaussian white noises with 0-mean are added to both the head at M

x and the discharge at U

x . The signal-to-noise ratio SNR=10 dB, which is defined by

  10 20log ( / ) SNR h
, where h is the average head difference and  is the standard deviation of noise. Gx is symmetric with respect to M

x , which implies that while the objective function reaches maximum at both L x and  ML xx . Obviously, the latter is a wrong estimate which must be excluded. The problems can be solved by using another measurement station with a different location. Figure2 (b) shows the power output for measurement station at 1800. It is clear that when the hydrophone location is changed, the peak corresponding to the actual leak stays at the same position, while the symmetric peak moves, which can therefore be excluded. Furthermore, the leak size L s can also be estimated, which is m . The wave propagation and measurement are simulated and the leaks are localized using MFP based on the single-leak model. As previously, two different sensor locations are considered, the corresponding results are displayed in Figure 3. Similar to the single-leak case, a single sensor is able to localize both leaks but two symmetric maxima appear. Again, using another sensor is able to exclude the two false estimates: the two peaks corresponding to the actual leaks remains while the other two moves.

CONCLUSIONS

This paper addresses the problem of leak detection in a pipeline using a fluid transient. The leak location and size are estimated via the matched-field processing method. The model is based on the assumption of single leak but multiple leaks can still be localized. The proposed method is not limited to resonant frequencies but is able to use the head at all the frequencies.

The results obtained on simulated data illustrate the interest of the proposed method. The accuracy of the leak localization and the ability to estimate two leaks not very close to each other are demonstrated.

  the propagation function, a is the speed of sound,  is the angular frequency, A is the area of pipeline, and R is the frictional resistance term. Note that  for turbulent flows where F is the Darcy-Weisbach friction factor, 0 q is the steady-state discharge and D is the pipe diameter. If the pipe is frictionless (F=0),   ik where   / ka is the wavenumber. The transfer matrix on the right hand side of Eq. (1) can be simplified as in which is a matrix related to the location of the leak, but independent of the leakage size. By combining the above equations, the head at the measurement station for a given angular frequency is wherein and Let M j h denote the measured head at the frequency  j . The measurement is assumed to be contaminated by a noise j n , i.e.The purpose of this work is to use the measured head at J frequencies, denoted as to estimate the leak position and the leak size. Here, let us define the head difference due to leakage and and denote the error vector by then the head difference is represented by

Figure 1 .

 1 Figure 1. Setup of numerical simulation

Figure 2 .

 2 Figure 2. Localization of single leak (at 300 m) using MFP. The dash line and the cross stand for the leak and hydrophone locations, respectively.

Figure 3 .

 3 Figure 3. Localization of double leak (at 200 m and 700 m) using MFP. The dash line and the cross stand for the leak and hydrophone locations, respectively.

ACKNOWLEDGEMENTS

This work has been supported by research grants from the Research Grant Council of the Hong Kong SAR, China (Project No. T21-602/15R). The authors would like to thank Duncan A. McInnis, Bryan Karney, Fedi Zouari, Alireza Keramat, Moez Louati, and Jingrong Lin for their helpful comments and discussions.


Mihai Dascalu
email: mihai.dascalu@cs.pub.ro

Gabriel Gutu
email: gabriel.gutu@cs.pub.ro

Stefan Ruseti
email: stefan.ruseti@cs.pub.ro

Ionut Cristian Paraschiv
email: ionut.paraschiv@cs.pub.ro

Philippe Dessus
email: philippe.dessus@univ-grenoble-alpes.fr

Danielle S Mcnamara
email: dsmcnama@asu.edu

Scott A Crossley
email: scrossley@gsu.edu

Stefan Trausan-Matu
email: stefan.trausan@cs.pub.ro

ReaderBench: A Multi-Lingual Framework for Analyzing Text Complexity

Keywords: Multi-Lingual Text Analysis, Textual Complexity, Comprehension Prediction, Natural Language Processing, Textual Cohesion, Writing Style

Assessing textual complexity is a difficult, but important endeavor, especially for adapting learning materials to students' and readers' levels of understanding. With the continuous growth of information technologies spanning through various research fields, automated assessment tools have become reliable solutions to automatically assessing textual complexity. ReaderBench is a text processing framework relying on advanced Natural Language Processing techniques that encompass a wide range of text analysis modules available in a variety of languages, including English, French, Romanian, and Dutch. To our knowledge, ReaderBench is the only open-source multilingual textual analysis solution that provides unified access to more than 200 textual complexity indices including: surface, syntactic, morphological, semantic, and discourse specific factors, alongside cohesion metrics derived from specific lexicalized ontologies and semantic models.

Introduction

Two important and cumbersome tasks, which often face many teachers, are selecting reading materials suitable for their students' levels of understanding, and assessing their written productions (e.g., essays, summaries). In order to support both tasks, ReaderBench [START_REF] Dascalu | Analyzing discourse and text complexity for learning and collaborating[END_REF], a multilingual, open-source framework centered on discourse analysis, was developed. From an architectural perspective, as shown in Figure 1, our framework comprises three layers: a) linguistic resources that provide solid language background knowledge and can be used to train the semantic models and compute various measures; b) linguistic services used to process and append semantic metainformation to text resources, and c) linguistic applications that rely on machine learning and data mining techniques, and are designed for various educational experiments and visualizations. ReaderBench implements various metrics and categories of textual complexity indices that can be used to leverage the automated classification of datasets in multiple languages, such as English [START_REF] Allen | Modeling Individual Differences among Writers Using ReaderBench[END_REF], French [START_REF] Dascalu | Reflecting Comprehension through French Textual Complexity Factors[END_REF], Romanian [START_REF] Gifu | Time Evolution of Writing Styles in Romanian Language[END_REF] and Dutch [START_REF] Dascalu | ReaderBench Learns Dutch: Building a Comprehensive Automated Essay Scoring System for Dutch[END_REF]. Surface indices. These are the simplest measures that consider only the form of the text. This category includes indices such as sentence length, word length, the number of unique words used, and word entropy. All these indices rely on the assumption that more complex texts contain more information and, inherently, more diverse concepts.

Word complexity indices. This category of indices focuses on the complexity of words, but goes way beyond their form. Thus, the complexity of a word is estimated by the number of syllables and how different the flectional form is from its lemma or stem, considering that adding suffixes and prefixes increases the difficulty of using a given word. Moreover, a word's complexity is measured by considering the number of potential meanings derived from the word's senses available in WordNet, as well as a word's specificity reflected in its depth within the lexicalized ontology.

Syntactic and morphologic indices. These indices are computed at the sentence level. The words' corresponding parts of speech and the types of dependencies that appear in each sentence can be used as relevant measures, reflective of a text's complexity. In addition, named entity-based features are tightly correlated with the amount of cognitive resources required to understand the given text.

Semantic cohesion indices. Cohesion plays an important role in text comprehension and our framework makes extensive usage of Cohesion Network

Analysis. ReaderBench estimates both local and global cohesion by considering lexical chains, different semantic models (semantic distances in different multilingual WordNets, LSA -Latent Semantic Analysis, LDA -Latent Dirichlet Allocation, and Word2Vec), as well as co-reference chains.

Discourse structure indices. Specific discourse connectives and metrics derived from polyphonic model of discourse [START_REF] Dascalu | Analyzing discourse and text complexity for learning and collaborating[END_REF], which considers the evolution of expressed points of view, provide additional valuable insights in terms of the text's degree of elaboration. Word features and vectors from the integrated linguistic resources are also used to reflect specific discourse traits.

Validation Experiments

Multiple experiments have been performed to validate ReaderBench as a multilingual text analysis software framework. This section focuses on the latest and most representative experiments conducted in English, French, Romanian, and Dutch languages. The first experiment [START_REF] Allen | Modeling Individual Differences among Writers Using ReaderBench[END_REF] was performed on a set of 108 argumentative essays written in English and timed to 25 minutes. For the analysis, only essays that contained three or more paragraphs were considered in order to use global cohesion measures reflective of inter-paragraph relations. Individual difference measures such as vocabulary knowledge and reading comprehension scores were assessed. The results showed that writers with stronger vocabulary knowledge used longer words with multiple senses and higher entropy, but also created more cohesive essays. Also, students with higher reading comprehension scores created more cohesive and more lexically sophisticated essays, using longer words, and with higher entropy.

The second experiment [START_REF] Dascalu | Reflecting Comprehension through French Textual Complexity Factors[END_REF] relied on a set of 200 documents collected from primary school French manuals. The documents were pre-classified into five complexity classes mapped onto the first five primary grade levels of the French national education system. A Support Vector Machine (SVM) was used to classify the documents. The pre-trained model was used to determine the complexity for an additional set of 16 documents that were manually classified into three primary grades. Students belonging to the three classes had to read the texts and answer a posttest. Correlations between the textual complexity factors' scores and the students' average scores were computed. This allowed the computation of the impact for each factor in calculating the reliability of prediction of the textual complexity score for a given document.

The third experiment [START_REF] Gifu | Time Evolution of Writing Styles in Romanian Language[END_REF] was conducted on a set of 137 documents written in Romanian language. The documents were collected from two time periods, 1941-1991 and 1992-present, and two regions, Bessarabia and Romania. The first period altered the Romanian language spoken in the country because of the implementation of the Russian language into the education system of Bessarabia. The aim of the experiment was to determine whether differences between the two regions and the two time periods could be observed in relation to the complexity of written texts. The analysis showed that more elaborated texts were created in the second period for both Bessarabia and Romania, while more unique words have been used in the second period for Bessarabia, but remained the same for Romania. The semantic cohesion of the texts increased over time, but no significant differences were observed between the two regions.

The fourth experiment [START_REF] Dascalu | ReaderBench Learns Dutch: Building a Comprehensive Automated Essay Scoring System for Dutch[END_REF] was run on a set of 173 technical reports written in Dutch language belonging to high or low performance students. Due to the length of the documents, a multi-level hierarchical structure was automatically generated based on the section headings. The experiment showed that students who received higher scores had longer reports, but also greater word entropy. They used more pronouns, discourse connectors and unique words, but also had lower inner cohesion scores per paragraph which is indicative of more sophisticated paragraphs.

Conclusion

Many pedagogical scenarios can fully integrate the use of ReaderBench, thanks to its versatility. The wide range of textual assessment features can support both teachers' assessment and learners' writing self-regulation. Moreover, multiple learning contexts take advantage from ReaderBench's support: either individual textual production and reflection, or collaborative knowledge building.

The presented experiments support the ReaderBench framework for determining the textual complexity of texts written in English, French, Romanian, and Dutch languages. Other languages, such as Spanish, Italian, and Latin are also partially supported. To our knowledge, ReaderBench is a unique multilingual system that provides access to a wide range of textual complexity indices and to various textual cohesion analyses.

Fig. 1 .

 1 Fig. 1. ReaderBench processing architecture.

Acknowledgments. This research was partially supported by the FP7 2008-212578 LTfLL project, by the 644187 EC H2020 RAGE project, by the ANR-10-blan-1907-01 DEVCOMP project, as well as by University Politehnica of Bucharest through the "Excellence Research Grants" Program UPB-GEX 12/26.09.2016.


Ḥimyar

JÉRÉMIE SCHIETTECATTE

Ḥimyar was the geographic area stretching over the southern highlands of Yemen and also the name of the tribal confederacy settled here. After having seceded from the kingdom of Qatabān, this tribal confederacy assumed its independence: Ḥimyar adopted its own era (Raydānite era), starting in 110 BCE; it was then headed by the kings of dhū-Raydān, after the name of their lineage. The first known one, Shammar dhū-Raydān, is mentioned in south-Arabic inscriptions in the mid-first century BCE. According to classical sources, Ẓafār was the capital of the Ḥimyarites (Lat.: Homeritae) (Plin. HN 6.26.104;Peripl . Mar . Erythr . 22), a town spread over 110 ha where the royal palace Raydān was built (Figure 1). The other major cities of the Ḥimyarite confederation were significant tribal centres: Samiʿan (today Maṣnaʿat Mariya) (tribe of Muhaqra); Hakir (tribe of Maytam); Baynūn (tribe of Shaddad); Yaklaʾ (now Nakhlat al-Ḥamrāʾ) (tribe of Dhamārī); Jabal al-ʿAwd (tribe of Dhubhan); Sawā (tribe of Maʿafir); Waʿlan (now al-Miʿsāl) (tribe of Radmān); Hidū (today al-Hadīm) (tribe of Khawlān); Ḥaṣī (the Madhà). The Ḥimyarite economy depended on the aromatics trade through the ports of Makhawān (today al-Makhāʾ) and Aden, and on efficient agriculture resulting from the building of ambitious hydraulic structures including monumental dams. Ḥimyar, unlike the other south-Arabian kingdoms, was not characterized by a tribal pantheon dominated by a tutelary deity. Its people worshiped most of the south-Arabian deities (Wadd, ʿAthtar, ʿAmm, Almaqah, etc.). During his reign, Abikarib Asʿad (before 384 to after 433 CE) embraced a monotheistic creed influenced by Judaism. This was also adopted by the tribal aristocracy behind him and by other people, shown by the rapid abandonment of the pagan temples. From the first century BCE, Ḥimyar appeared as a competing power to the weakened kingdom of Sabaʾ. In the early Christian era, the rulers claimed their legitimacy over the throne of Sabaʾ by adopting the title of "king of Sabaʾ and dhu-Raydān," and Sabaic as the official language, and by making offerings to the Sabean tutelary deity Almaqah, in the Awwam temple in Marib. When the Sabean aristocracy broke free from the Ḥimyarites and restored the Sabean kingdom in the early second century CE, both powers entered a 200-year-long period of rivalry and conflict which ended with the final annexation of Sabaʾ. After having unified the whole of South Arabia, Ḥimyar started to stretch its dominion over the Hijaz and the Najd under the rule of Abikarib Asʿad, subjugating the main tribe in Central Arabia, Maʿadd (Procop. HW 1.19.14), and granting its government to its vassals, the princes of the tribe of Kinda. In the early sixth century, the two Ḥimyarite kings Marthadʾilan Yanuf (ca. 500-515) and Maʿdikarib Yaʿfur (ca. 515-522) became subjects of the Aksumite Negus and adopted Christianity. This belief spread in particular to the outskirts of the kingdom (Najrān, Tihāma, Socotra). The accession of Yusuf Asʾar to the throne in 522 CE put an end to this subjugation. Communities of Christian converts supported by Aksum came into conflict with the Ḥimyarite ruler, resulting in the slaughter of Christians and an Ethiopian military intervention (Cosmas Christian Topogr . 2.140-1), with Aksum seizing control of Ḥimyar for the next fifty years. The Ḥimyarite crown was assumed by the Negus officer, Abraha (ca. 530-560), whose great deeds have left their imprint on both pre-Islamic and Islamic sources: the restoration of the great dam of Marib; the building of the church of Sanaʾa; the submission of all the tribes of Central and North Arabia, Maʿadd, Hagar-and-Khatt, Tayyiʾ, Yathrib, and Guzam. But above all, it was the Expedition of the Elephant carried out by Abraha against Mecca that made the deepest impression on his contemporaries and which is possibly echoed in the Qurʾān (sura of the Elephant). In around 575 CE, Ḥimyar was conquered by the Persians under the rule of king Khusraw, and its importance diminished.

REFERENCES AND SUGGESTED READINGS

  GAJDA, I. (2009) Le royaume de Ḥimyar à l'époque monothéiste. Paris. ROBIN, C. J. (2004) "Ḥimyar et Israël." Compte rendus des séances de l'Académie des Inscriptions et Belles-Lettres 148, 2: 831-908. YULE, P. (2007) Ḥimyar. Spätantike im Yemen. Late Antique Yemen. Aichwald. YULE, P. (2013) Late antique Arabia: Ẓafār, capital of Ḥimyar. Wiesbaden. Bas-relief of a crowned figure (z607) from the Himyarite capital Ẓafār, ca. fifth century CE. High-resolution 3D scan (Credit: J. Lindenbeck, Heidelberg University Expedition, 2010). CITATION The Encyclopedia of Ancient History. Edited by Roger S. Bagnall, Kai Brodersen, Craige B. Champion, Andrew Erskine, and Sabine R. Huebner. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd. DOI: 10.1002/9781444338386.wbeah30219

	CAPTION
	Figure 1


Najrān

JÉRÉMIE SCHIETTECATTE

Najrān (Sab. Nagrān ; Gr. Nεγράνων /Nαγαρα ; Lat. Negrana /Nagara): 40-km-long oasis located along the wādī Najrān in southwest Saudi Arabia. Najrān is also the name of the main city, and that of the tribe which was settled there in the early Christian era. The luxuriance of the oasis and its unique location on the caravan roads fostered the regional development from the first millennium BCE onward. The richness of the oasis is echoed in a third century Sabaic inscription (Ja 577) which mentions sixty-eight settlements, sixty thousand fields, and ninety-seven wells. Ammianus Marcellinus counted Najrān as one of the seven chief cities in Arabia Felix (Amm. Marc. Res Gestae,23.6.47). When leaving Arabia Felix, Najrān was the last fertile valley before entering the barren areas of Arabia Deserta, at the crossroads for caravaneers heading toward the Mediterranean Sea, along the Hijaz range, or toward the Persian Gulf, through the Wādī Dawāsir. From the eighth to the sixth century BCE, the valley of Najrān was the heart of the kingdom of Muhaʾmir, with its capital city at Ragmat (Hebr. Raʿmah -Gen 10:7,7; Ezek 27:22). At around the fifth century BCE, the kingdom of Amīr succeeded that of Muhaʾmir; Ragmat was abandoned for the benefit of a new capital city, and Zirbān, later called Najrān, was occupied until the very early Islamic era. This town is identified with the modern ruins of al-Ukhdūd, on the right bank of the wādī Najrān. Archaeological excavations revealed a planned city, being a regular square bisected by a main street into two equal parts (Figure 1). Surrounded by an extra muros settlement, the town extended over 50 ha. The main deity dhu-Samāwī was worshiped in the temple Kaʾbatan. Owing to the strategic location of Najrān, neighboring kingdoms took a keen interest in its control. Coveted by the kingdom of Sabaʾ, the city of Ragmat was burned down during amilitary expedition. Thereafter, Najrān joined the trade coalition headed by the south Arabian kingdom of Maʿin (fourth to second centuries BCE); in 25 BCE, the town was seized by Aelius Gallus' expedition (Strab. Geogr. 16.4.24; Plin. HN 6.32.160); a few decades later, Sabaʾ took control of the oasis, a rule which was disputed by Aksum for a short period (ca. 210-240 CE). In 328 CE, in his epitaph found at al-Namara in Syria (Louvre 205), Maraʾ al-Qays, son of ʿAmraw king of all ʿArab, said he fought with the tribe of Madhhij " until he struck with his spear the gates of Najrān, the city of Shammar [Yuharʾish, the king of Himyar]." The fate of the inhabitants of Najrān was then intrinsically tied to Himyarite policy. When the Himyarite kings chose Judaism (ca. 380 CE) and rejected pagan cults, it is likely that the inhabitants of the oasis -under their control -did likewise. At around 450 CE, groups of Najrānites converted to Christianity through contact with the Monophysite milieu of Syria and Nestorians from al-Hīra in Iraq [START_REF] Robin | Joseph, dernier roi de Himyar[END_REF]. The proselytism of the Monophysite community and their connections to Byzantium led to the first wave of persecutions by the Himyarite king Shurihbiʾīl Yakkuf: the beheading of Azqīr, priest of Najrān, reported ca. 475 in the Martyrdom of Azqir (Conti Rossini 1910); the stoning of Paul, bishop of Najrān, ca. 500, mentioned in the Letter of Simeon of Beth-Arsham (Shahid 1971). The second wave of persecutions was conducted by the Jewish Himyarite king Yūsuf Asʾar in 523 CE : the Book of the Himyarites describes a five-month siege followed by the slaughter of many Christians [START_REF] Moberg | The book of the Himyarites: fragments of a hitherto unknown Syriac work[END_REF]. The massacre was fraught with consequences: an expedition carried out by the Aksumite king Kaleb led to the elimination of Yūsuf and the subjugation of HIMYAR to the Aksumite princes for half a century. Under the authority of the descendants of the leader of the Christian community, al-Hārith bin Kaʿb (Gr. Arethas), executed during the persecution, the tribal group of the inhabitants of Najrān took the name of its ruling dynasty: the banū al-Hārith bin Kaʿb. Najrān became again an active trade centre, halfway between Mecca and Yemen, headed by a triumvirate described in the delegation of the Christians of Najrān coming to Muhammad at Medina ca. , which included a governor, a tribal chief, and a bishop. During the caliphate of ʿUmar, the Monophysites were expelled, and they settled at Najrāniyya, near al-Kufa (Iraq) [START_REF] Robin | Joseph, dernier roi de Himyar[END_REF]. By the eleventh century, the ancient city of Najrān, today al-Ukhdūd, had fallen into ruins. The main settlement shifted to the left bank of the wādī, where the modern city of Najrān is located. Jewish and Christian communities were settled there at least until the early thirteenth century (Ibn al-Mujāwir, Taʾrīkh al-Mustabsir : 209).

FIGURE 1

 1 FIGURE 1 Plan of the intra muros area at al-Ukhdūd (Schiettecatte, J. 2011: figure 143). Reproduced with permission from Schiettecatte, J. 2011. Figure originally based on material from Zarins et al., 1983.