A Goldenschluger

A Nemirovski Fabienne Juditsky

Céline Comte

Valentine Duval

Genon-Catalot

Discussion on the paper "Hypotheses testing by convex optimization" by A

Keywords: subject classifications: Primary 62F03 Convex criterion, Hypothesis testing, Multiple hypothesis

Introduction.

We congratulate the authors for this very stimulating paper. Testing statistical composite hypotheses is a very difficult area of the mathematical statistics theory and optimal solutions are found in very seldom cases. It is precisely in this respect that the present paper brings a new insight and a powerful contribution. The optimality of solutions depends strongly on the criterion adopted for measuring the risk of a statistical procedure. In our opinion, the novelty here lies in the introduction of a new criterion different from the usual one (compare criterions (2.1) and (2.2) below). With this new criterion, a minimax optimal solution can be obtained for rather general classes of composite hypotheses and for a vast class of statistical models. This solution is nearly optimal with respect to the usual criterion. The more remarkable results are contained in Theorem 2.1 and Proposition 3. In what follows, we give some more precise details on the main results necessary to enlighten the strength and the limits of the new theory.

The main results.

Theorem 2.1

In this paper, the authors consider a parametric experiment (Ω, (P µ ) µ∈M ) where the parameter set M is a convex open subset of R m . From one observation ω, it is required to build a test deciding between two composite hypotheses H X : µ ∈ X, H Y : µ ∈ Y where X, Y are convex compact subsets of M. Assumptions on the subsets X, Y are thus quite general and the problem is taken as symmetric (no distinction is done between the hypotheses such as choosing H 0 versus H 1 ). We come back to this point later on.

A test φ is identified with a random variable φ : Ω → R and the decision rule for

H X versus H Y is φ < 0 ⇔ reject H X , φ ≥ 0 ⇔ accept H X so that -φ is a decision rule for H Y versus H X .
The theory does not consider randomized tests which are found to be optimal tests in the classical theory for discrete observations.

The usual risk of a test is based on the two following errors:

R(φ) := max{ X (φ) := sup x∈X P x (φ < 0), Y (φ) := sup y∈Y P y (φ ≥ 0)}. (2.1)
We use the notation X (φ), Y (φ) to stress on the dependence on φ. An optimal minimax test with respect to the classical criterion would be a test achieving min φ max (x,y)∈X×Y

(P x (φ < 0) + P y (φ ≥ 0)).
The authors introduce another risk r(φ), larger than R(φ) by the simple Markov inequality:

r(φ) := max{sup x∈X E x (e -φ ), sup y∈Y E y (e φ )}. (2.2)
The main result contained in Theorem 2.1 is that there exists an optimal minimax solution with respect to the augmented criterion: a triplet (φ * , x * , y * ) exists that achieves min Nevertheless, conditions on the parametric family are required and the optimal test is to be found in a specific class F of tests. The whole setting (conditions 1.-4.) must be "a good observation scheme" which roughly states:

• First, the parametric family must be an exponential family with its natural parameter set: ( denotes the transpose)

dP µ (ω) = p µ (ω) dP = exp (µ S(ω) -ψ(µ)) dP (ω), (2.4) M = interior of {µ ∈ R m , Ω exp (µ S) dP < +∞} S : ω ∈ Ω → S(ω) ∈ R m .
• The class F of possible tests is a finite-dimensional linear space of continuous functions, and contains constants and all Neyman-Pearson statistics log(p µ (.)/p ν (.)). This is not a surprising assumption but means that the class of tests among which an optimal solution is to be found is exactly

F = {a S + b, a ∈ R m , b ∈ R} (2.5)
• Condition 4. is more puzzling: the class F must be such that for all φ ∈ F,

µ → log Ω e φ p µ dP
is defined and concave in M. This means that: Because of (2.4) and (2.5), the computation of

µ → log Ω exp [(a + µ )S + b -ψ(µ)] dP
(φ, x, y) → E x (e -φ ) + E y (e φ )
is easy and the solution of (2.3) is obtained by solving a simple optimization problem. Theorem 2.1 provides the solution φ * = (1/2) log(p x * /p y * ) and a remarkable result is that:

ε * = ρ(x * , y * )
where ρ is the Hellinger affinity of P x * , P y * . An important point too concerns the translated detectors φ a * := φ * -a. Equation (4), p.4, shows that by a translation, the probabilities of wrong decision can be upper-bounded as follows:

X (φ a * ) ≤ e a ε * , Y (φ a * ) ≤ e -a ε * .
Therefore, by an appropriate choice of a, one can easily break the symmetry between hypotheses, choose H 0 and H 1 and reduce one error while increasing the other one. Examples (2.3.1, 2.3.2, 2.3.3) are particularly illuminating. All computations are easy and explicit. The extension of the theory to repeated observations is relatively straightforward due to the exponential structure of the parametric model and we will not discuss it.

Proposition 3.1

Another very powerful result concerns the case where one has to decide not only on a couple of hypotheses but on more than two hypotheses. The testing of unions is an impressive result. The problem is of deciding between:

H X : µ ∈ X = m i=1 X i , H Y : µ ∈ Y = n i=1 Y i
on the basis of one observation ω ∼ p µ . Here, X i , Y j are subsets of the parameter set. The authors consider tests φ ij available for the pair (H Xi , H Yj ) such that

E x (e -φij ) ≤ ij , ∀x ∈ X, E y (e φij ) ≤ ij , ∀y ∈ Y,
(for instance the optimal tests of Theorem 2.1, but any other test would suit) and define the matrices E = ( ij ) and

H = 0 E E 0 .
An explicit test φ for deciding between H X and H Y is built using an eigenvector of H and the risk r(φ) of this test is evaluated with accuracy: it is smaller than ||E|| 2 the spectral norm of E. A very interesting application, which is illustrated on numerous examples, is when H X = H 0 : µ ∈ X 0 is one hypothesis (not a union) and

H Y = n j=1 H j : µ ∈ n j=1
Y j is a union. One simply builds for j = 1, . . . , n the optimal tests φ * (H 0 , H j ) := φ 0j with errors bounded by 0j (obtained, for instance, by Theorem 2.1). The matrix

E is the (1, n) matrix E = [ 01 . . . 0n ]. As E E has rank 1, its only non nul eigenvalue is equal to T r(E E) = n j=1 2 0j = (||E|| 2 ) 2 . The eigenvector of H is given by [1 h 1 . . . h n ] with h i = 0i /( n j=1 2 0j ) 1/2
. Then, the optimal test for H 0 versus the union n j=1 H j is explicitly given by

φ = min 1≤j≤n φ 0j -log ( 0j /( n j=1 2 0j ) 1/2 ) . For all µ ∈ X 0 , E µ (e -φ ) ≤ n j=1 2 0j 1/2 and for all µ ∈ n j=1 Y j , E µ (e φ ) ≤ n j=1 2 0j 1/2 .
Then, given a value ε , if it is possible to tune the test φ 0j := φ 0j (ε) of H 0 versus H j to have a risk less than ε/ √ n, then the resulting test for the union φ(ε) has risk less than ε. This is remarkable: if we consider the test min 1≤j≤n {φ 0j } for H 0 versus the union n j=1 H j , we have

P 0 ( min 1≤j≤n {φ 0j }) ≤ n j=1 0j .
To get a risk bounded by ε, one would have to tune the test φ 0j of H 0 versus H j to have a risk less than ε/n.

Discussion.

• The theory is restricted to exponential families of distributions with natural parameter space. As noted by the authors in the introduction, this is the price to pay for having very general hypotheses. The problem of finding more general statistical experiments which would fit in the theory is open and worth investigating. The new risk criterion seems to be a more flexible one for finding new optimal solutions. • The fact that the sets X, Y must be compact sets is restrictive. We wonder if there are possibilities to weaken this constraint. • The combination of conditions 3. and 4. on the class of tests implies a reduction of the class of tests. Consider the case of exponential distributions, where

p µ (ω) = µ exp (-µ ω), ω ∈ (0, +∞), µ ∈ M = (0, +∞), S(ω) = -ω.
By condition 3., F contains p µ /p ν for all µ, ν > 0, hence F contains all tests φ(ω) = aω + b, a, b ∈ R.

For condition 4., to compute F (µ) the condition µ > a is required and

F (µ) = log [ +∞ 0 µ exp ((a -µ)ω + b)dω] = b + log [µ/(µ -a)].
As F (µ) = a(2µ -a)/(µ 2 (µ -a) 2 ), F is concave if and only if a ≤ 0. Therefore, condition 4. restricts the class of tests to F = {φ = aω + b, a ≤ 0, b ∈ R}. This raises a contradiction: condition 3) states that F must contain all log p µ /p ν , thus all φ = aω + b, a, b ∈ R. We wonder if condition 4. could be stated differently so as to avoid this contradiction.

Concluding remarks

To conclude, the criterion r(φ), defined in (2.2), provides a powerful new tool to build nearly optimal tests for multiple hypotheses. The large number of concrete and convincing examples is impressive. We believe that this paper offers a new insight on the theory of testing statistical hypotheses and will surely inspire new research and new developments.

  1 and are illustrated by numerous examples.

  (e -φ ) + E y (e φ )) := 2 log ε * (2.3) Such a triplet can be explicitly computed in classical examples (discrete, Gaussian, Poisson) together with the value ε * . Moreover, the usual error probabilities X (φ * ), Y (φ * ) can be computed too.

  is defined and concave in µ ∈ M. This may reduce the class F. On the three examples (discrete, Gaussian, Poisson) condition 4. is satisfied. However, on other examples (see the discussion below), condition 4. together with condition 3. requires a comment.

  • Generally, when testing statistical hypotheses, one chooses a hypothesis H 0 and builds a test of H 0 versus H 1 . One wishes to control the error of rejecting H 0 when it is true and the other error does not really matter. A discussion on this point is lacking in relation with Theorem 2.1, formula (4).• Randomized tests are not considered here. However, they are found as optimal solutions in the fundamental Neyman-Pearson lemma. In estimation theory, randomized estimators are of no use because, due to the convexity of loss functions, a non randomized estimator is better. In the setting of the paper, is there such a reason to eliminate randomized tests? • The criterion r(φ) is larger than the usual one, entailing a loss. The notion of "provably optimal test" introduced in this study is not commented in the text. So, it is difficult to understand or quantify it. Maybe more comments on Theorem 2.1 (ii) would help. At this point, let us notice that the notation X without dependence on the test statistic φ is a bit misleading. We find it would be useful to denote X (φ * ) instead of X (see Theorem 2.1, formula (4)). Another point is the comparison between X (φ * ) and ε * . Apart from the Markov inequality, would it be possible to quantify ε * -X (φ * )? Example 2.3.1 give both quantities without comment. Examples 2.3.2 and 2.3.3 only give ε * . • We looked especially at Section 4.2. The Poisson case is illuminating and helps understanding the application of Proposition 3.1. On the contrary, we had difficulties with Section 4.2.3 (Gaussian case) and do not understand why the optimal test of H 0 versus H j is not computed as in Section 2.3.1. Consequently, more details on the proof of Proposition 4.2 would have been useful. Also, in these examples, are the quantities 0j computed as ε * (φ * (H 0 , H j )) or as X0 (φ * (H 0 , H j ))? The numerical illustrations were also difficult to follow.


Charles Maragna

Xavier Rachez

Hydrothermal Modeling of Pile Heat Exchangers in the Unsaturated Zone

Hydrothermal Modeling of Pile Heat Exchangers in the Unsaturated Zone

Charles Maragna and Xavier Rachez BRGM, 3 Avenue Claude-Guillemin, BP 36009 -45060 Orléans cedex 2 -France c.maragna@brgm.fr

Abstract

Pile heat exchangers (PHE) are an attractive way to exchange thermal energy with shallow underground since they combine the structural role of geostructures with ground heat exchangers, making it possible to mutualize both investments. PHE are typically 5 to 20 m deep and can be partially located in the unsaturated (vadose) zone. Several cases were modelled for ground water table depths ranging from 0.5 m to 9.5 m (cf. Figure 1). The two mentioned bounds respectively correspond to situation where the PHE stands in almost saturated or unsaturated soil. The considered soil, a homogenous Fontainebleau sand, is subject to large water content variations which lead to large thermal conductivity variations: The estimated thermal conductivity is 3.01 W.K -1 .m -1 at saturation, i.e. matric potential ψ = 0 m, and 0.46 W.K -1 .m -1 for ψ = -10 m. Other soils ranging from clay to sandy loam exhibit a smaller influence of moisture content and matric potential upon thermal conductivity. To determine an effective thermal conductivity of the ground 𝜆 𝑚 the PHE was subject to a thermal response test (TRT)i.e. a constant power given to the heat-carrier fluid. 𝜆 𝑚 was estimated based on the evolution of fluid temperature, according to state-of-the-art procedures for TRT interpretations. The analytical model used for the interpretation considers that the pile behaves as an infinite line emitting a constant heat flow in a homogenous media. 𝜆 𝑚 is tuned so that the evolutions of inlet/outlet averaged fluid temperatures computed by both models (FE and analytical) match. As a result, 𝜆 𝑚 ranges from 3.01 ± 0.01 W.K -1 .m -1 for a groundwater level depth of 0.5 m to 1.33 ± 0.04 W.K -1 .m -1 for a groundwater level depth of 9.5 m. The effective thermal conductivity 𝜆 𝑚 is almost identical to the ground conductivity averaged over the PHE height. These preliminary results suggest that for a single PHE standing in an unsaturated soil the temperature evolution of the heat-carrier fluid can correctly be described by an analytical model considering a homogenous media whose thermal conductivity is equal to the averaged ground conductivity. Further development should focus on similar study for a group of PHE. page 1 / 2

Figure 1 :

 1 Figure 1 : Cross-sectional view of temperature field (°C) around a 10 m deep pile heat exchanger after 7.4 years with a constant exchanged power at 500 W. The water table is located at 5 m depth.

  Thermal properties of soil depend on water content, which is heterogeneous in the unsaturated zone. The presented study investigates the influence of groundwater table depth upon the amount of heat transferred by the PHE. It is a part of a research project ''GECKO'' (geo-structures and hybrid solar panel coupling for optimized energy storage) which is supported by a grant from the French National Research Agency (ANR). The authors focused on deriving an effective thermal conductivity of the ground, denoted 𝜆

𝑚 , as this value is essential to a consistent PHE sizing. A three-dimensional finite elements (FE) model developed in COMSOL-Multiphysics ® of a 10 m deep PHE has been set up. The geometry of the pipes, pile concrete and surrounding ground is discretized. Stationary distribution of matric potential is determined by solving the Richards equation while the resolution of the heat equation leads to the temperature evolution in the heat-carrier fluid, the concrete and the surrounding ground.


El Houssein

Chouaib Harik

François Guérin

Frédéric Guinand

Jean-François Brethé

Hervé Pelvillain

Decentralized Control Architecture for UAV-UGV Cooperation

We present a decentralized control architecture for an heterogeneous group of mobile robots made of one Unmanned Aerial Vehicles (UAV) and several Unmanned Ground Vehicles (UGVs) performing collaborative tasks (area inspection, object transportation, etc...). The UAV is used to help a human operator to supervise and guide a group of UGVs by providing an aerial coverage view of the navigation area. Our control scheme is based on minimalistic computation and communication requirements, as well as an architecture complexity kept at a simple level regardless of the deployed number of grounds robots. Simulation and experimentations are performed and show the efficiency of our proposed control architecture.

I. INTRODUCTION

The present research is partially funded by Le Havre town council (CODAH), CNRS and the Normandy Region and takes place in two projects conducted by several research laboratories of both University of Le Havre and University of Rouen. The main goal of the work is the design and implementation of a platform made up of ground and air mobile robots to carry out, in a coordinated way, a set of different tasks in an industrial area.

Unmanned Aerial and Ground Vehicles (UAVs, UGVs) cooperation attracts increasingly the attention of researchers, essentially for the complementary skills provided by each type to overcome the specific limitations of the others (payload, computing power, movement velocities, local vision, etc...), thus, deploying UAVs and UGVs in a single mission allows to envisage a wide range of applications.

In [?] the authors used a group of UAVs to coordinate a group of UGVs in urban environment.

An agricultural application of UGV-UAV coordination is presented in [?], where aerial images taken by a quadrotor are processed by a ground mobile robot to extract and to navigate towards a waypoint in order to perform a specific task (eg: apply pesticides/herbicides).

The authors in [?] proposed a reactive supervisory control strategy for a team of UAVs and UGVs used in search and rescue missions. Another application for UAV-UGV cooperation is target tracking, the authors in [?] used a Parrot drone to track an omnidirectional Mecanum Wheel UGV.

In [?] the authors fused information gathered from ground and aerial robots in order to track a moving target in an open environment.

1 LITIS, University of Le Havre. 25 Rue Philippe Lebon, 76600 Le Havre Cedex (France) -(el-houssein-chouaib.harik,frederic.guinand)@univlehavre.fr 2 GREAH, University of Le Havre. 75, Rue Bellot, 76058 Le Havre Cedex (France) -(francois.guerin,jean-francois.brethe)@univ-lehavre.fr 3 IUT GEII, University of Le Havre. Rue Boris Vian, 76610 Le Havre Cedex (France) -herve.pelvillain@univ-lehavre.fr For some applications the human must be part of the processing cycle. This observation motivates our work. For instance, the inspection of industrial sites cannot be done, legally, without a human for piloting and/or supervising the task. Thus, to fulfill the industrial expectations in this matter, we propose an adequate scheme: a pilot for operating the UAV, an expert for taking some decision about the supervision of the surveillance process and mobile robots (UAVs and/or UGVs). We made the choice of using an UAV for vision purpose only and one or several UGVs for both vision and action.

This paper is organized as follows: In section II we introduce the problem statement, define the objectives of our work and present the hardware configuration and the overall architecture. We discuss in section III the simulation and experimental results, and we conclude in section IV the present work and discuss future perspectives.

II. PROBLEM STATEMENT AND ARCHITECTURE DESIGN

For numerous and unavoidable reasons, monitoring, surveillance and inspection of industrial sites belonging to Seveso category (ranking highly critical sites) cannot be performed without a constant and careful attention of human experts. However, nothing forbids the use of additional elements for helping supervisors in their critical tasks, provided that the crucial decisions are still taken by men.

With these constraints in mind, we have proposed a global architecture including Unmanned Ground Vehicles (UGVs), Unmanned Aerial Vehicles (UAVs) and men. Figure 1 with a leader and followers (as illustrated in Figure 2. A UAV takes, thanks to a camera mounted on it, and continuously sends images of the monitored area where UGVs (mobile robots) are present. These images are sent to a ground station. A human expert gets these images and take some decision about the next direction of the UGV team. This decision is expressed through the choice of a waypoint (a simple clic in the window for the visualization of the video). These waypoints may also be chosen prior to the start of the mission as a path through the industrial site. In such a case the video taken by the UAV is a way of verifying the good progress of the mission with the possibility at any moment to change the direction of the team by a simple clic in the window displaying the video provided by the UAV.

The UGV team is made of a leader, that receives the waypoints from the expert, and of other UGVs called followers. As the leader navigates to each next waypoint, these followers take the leader as a target and follow it keeping a certain separation distance [?] in order to form a specific configuration that can be defined prior, or during the mission. In our study, we used two followers, but this number can be extended depending on the mission goals. On the top of the UGV leader two colored markers have been fixed. The orientation of the leader can be deduced from these markers. We have considered in this work three similar nonholonomic, unicycle-like, Wheeled Mobile Robots (WMR) and a quadcopter drone equipped with a color camera. The video flow is sent continuously to the ground station where it is processed to extract the position and the orientation of the leader relative to the next waypoint. These information are then sent to the leader in order for it to correct its trajectory in real time using a non-linear kinematic controller [?].

At a functional level our architecture (Figure 3) can be divided in two layers: Drone-Leader layer, and Leader-Followers layer. We will explain in the next section the functionality of each layer. The ground station processes the video flow sent continuously via a wireless communication from the color camera mounted on the drone, to extract the position and the orientation of the leader to a waypoint selected by a human operator (the expert). The ground station sends this information (distance and steering angle) to the leader via another wireless communication device.

The human operator can give waypoints w to the leader by selecting them in the image using the developed software.

III. RESULTS

We will present in this section simulations and experimental results. In both cases, we consider that the center of the image is the center of the drone. The altitude of the drone can be estimated using the distance between the colored markers on the leader, or using the drone's altimeter.

A. Simulation results

To check the efficiency of our architecture and controllers, we simulated a real-like scheme to navigate in an industrial area avoiding obstacles.

For each WMR (leader and followers), we have used the dynamic model of [?] and [?].

The chosen scenario consists in giving waypoints to navigate in a specific area. Figure 4 shows the trajectories of each WMR: The simulation gave promising results, it showed the efficiency of the developed architecture and the proposed controllers (not described in this paper but that can be found in [?]). The leader is able to follow the given waypoints and the followers are able to navigate considering the leader as a target while keeping a predefined separation distance (range) and angle (bearing).

B. Experimental results

For brevity, we only present the experimental results of the drone-leader layer. The leader-follower experiments (vision target tracking) were carried out successfully in [?]. Our unicycle-like WMR (Figure 3) has been designed in the GREAH laboratory of the University of Le Havre (France). We used as a UAV a quadcopter (Phantom 2 Vision) developed by DJI [?], with a sufficient payload to carry our open platform device (android phone) that is used to take a video flow and send it continuously via Wifi (IEEE 802.11) to the ground station in order to localize and correct the position of the leader regarding its next waypoint. During our experiments, the UAV was manually piloted, but the work remains valid for autonomous flights. We created a user interface (Figure 5) using the development environment Processing [?] running on the ground station, where we can supervise in real time our WMR through a video flow received continuously from the smartphone mounted beneath the drone. We can select waypoints in the image by a simple click, or define them prior to the mission. The program extracts and send the distance between the leader and the target and the orientation of the leader w.r.t. the target. The communication is performed via ZigBee module (IEEE 802.15.4) at a frequency of 50Hz (real time interrupt). With the leader in their vision field initially, the followers will have to track the leader keeping a predefined separation distance and bearing angle in order to form a specific configuration (triangle is our case). The experiments were carried out with the smartphone mounted under the quadcopter instead of the built-in camera that has a limited tilt angle (the camera has to face the ground). The drone was piloted by a professional (for safety and legal reasons), but autonomous flights can be envisaged, since the quadcopter we used can fulfill this task.

IV. PERSPECTIVES AND CONCLUSION

We introduced in this paper a hierarchical architecture for air-ground coordination. We presented our testbed and explained the features of the whole system. The efficiency of our architecture has been proved by both simulations and experimental results, where a UGV (leader) followed by two other UGVs (followers) in a certain configuration (triangle), were able to navigate through waypoints extracted from an image taken from a drone. The architecture is fully functional and we now envision new developments, in three main directions.

• At first we would like to design a new control algorithm for allowing our UGV team to collectively carry a given cumbersome load in an environment with obstacles. • In a second step we are planning to subject the UAV movement to the leader movement, based on image processing and using a specific communication protocol.

• Finally, as a mid-term goal, we plan to dramatically improve the autonomy of the UGV team allowing robots to behave as a swarm according to their environment.

V. ACKNOWLEDGMENTS

The authors would like to thank "Communauté De l'Agglomération Havraise" CODAH for their support under research grants, and "DroneXTR" for piloting the drone.

Fig. 1 .

 1 Fig. 1. Global architecture including UAV, a team of UGVs, an expert for moving decisions and a pilot.

Fig. 2 .

 2 Fig. 2. The leader and the followers moving among obstacles thanks to waypoints defined, before or during the mission, by experts.

Fig. 3 .

 3 Fig. 3. A tow-layer architecture.

Fig. 4 .

 4 Fig. 4. Waypoints tracking simulation

Fig. 5 .

 5 Fig. 5. Waypoint tracking experimentation. The four images represent the progression of the UGV toward the target. The robot changes its orientation and moves towards the target simultaneously

Fig. 6 .

 6 Fig.6. Experiments: The pilot (1) manipulates the drone (2) that takes a video of the navigation area and sends it to the ground station (3). The operator (4) selects new waypoints to guide the ground robot (5)


Erwan Rousseau
email: erwan.rousseau@univ-amu.fr

HYPERBOLICITY, AUTOMORPHIC FORMS AND SIEGEL MODULAR VARIETIES

We study the hyperbolicity of compactifications of quotients of bounded symmetric domains by arithmetic groups. We prove that, up to a finite étale cover, they are Kobayashi hyperbolic modulo the boundary. Applying our techniques to Siegel modular varieties, we improve some former results of Nadel on the non-existence of certain level structures on abelian varieties over complex function fields.

Introduction

Any complex space Y can be equipped with an intrinsic pseudo-distance d Y , the Kobayashi pseudo-distance [START_REF] Kobayashi | Hyperbolic complex spaces[END_REF]. It is the largest pseudo-distance such that every holomorphic map f : ∆ → Y from the unit disc, equipped with the Poincaré metric, is distance decreasing.

Recall [START_REF] Kobayashi | Hyperbolic complex spaces[END_REF] that a complex space Y is said to be (Kobayashi) hyperbolic modulo W ⊂ Y if for every pair of distinct points p, q of Y we have d Y (p, q) > 0 unless both are contained in W . In the case where Y is compact and W = ∅, Brody's lemma gives a criterion for hyperbolicity: Y is hyperbolic if and only if there are no curves f : C → Y .

In the case where W = ∅, we do not know an analogue of Brody's criterion for hyperbolicity modulo W . It is not known whether Y is necessarily hyperbolic modulo W if it is Brody hyperbolic modulo W i.e. every entire curve f : C → Y has image in W . The reason is that the classical Brody's lemma does not provide any information on the image of the entire curve.

In this article, we investigate hyperbolic properties of compactifications of quotients of bounded symmetric domains by arithmetic groups. It is well known that such quotients may be far from being hyperbolic, as rational Hilbert modular surfaces show.

Let X := B/Γ be a quotient of a bounded symmetric domain by an arithmetic subgroup Γ ⊂ Aut(B). By passing to a subgroup of finite index if necessary, we will always suppose that Γ is neat [START_REF] Borel | Introduction aux groupes arithmétiques[END_REF]. In particular, X admits a smooth toroidal compactification X where D = X \ X is a normal crossings divisor [START_REF] Ash | Smooth compactification of locally symmetric varieties[END_REF]. Moreover, X may be chosen to be projective [START_REF] Ash | Smooth compactification of locally symmetric varieties[END_REF].

2010 Mathematics Subject Classification. Primary: 32Q45, 32M15; Secondary: 11G99, 14K15. Key words and phrases. entire curves, Kobayashi hyperbolicity, bounded symmetric domains, automorphic forms.

Partially supported by the ANR project "POSITIVE", ANR-2010-BLAN-0119-01. This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government programme managed by the French National Research Agency (ANR).

We obtain the following result Theorem 1.1. Let X := B/Γ be a quotient of a bounded symmetric domain by a neat arithmetic subgroup Γ ⊂ Aut(B). Then there exists a finite étale cover

X 1 = B/Γ 1 determined by a subgroup Γ 1 ⊂ Γ such that X 1 , a smooth projective compactification of X 1 , is Kobayashi hyperbolic modulo D 1 := X 1 \ X 1 .
As a corollary, we obtain a new simple proof of the main result of [START_REF] Nadel | The nonexistence of certain level structures on abelian varieties over complex function fields[END_REF], where it is proved that X 1 is Brody hyperbolic modulo D 1 : the image f (C) of any non-constant holomorphic map

f : C → X 1 is contained in D 1 .
While Nadel's proof [START_REF] Nadel | The nonexistence of certain level structures on abelian varieties over complex function fields[END_REF] was based on Nevanlinna theory, our proof consists in using the fundamental distance decreasing property of the Kobayashi pseudo-metric mentioned above. To prove Theorem 1.1, we will construct pseudo-distances on X 1 for which holomorphic maps from the unit disc are distance decreasing.

As an application, we study Siegel modular varieties from this point of view. Their geometry has attracted a lot of attention (see [START_REF] Hulek | The geometry of Siegel modular varieties[END_REF] for a survey). In particular, the birational geometry of A g and A g (n), the moduli spaces of principally polarized abelian varieties with level structures, has been extensively investigated. The Kodaira dimension of compactifications A g (n) has been studied by Tai, Freitag, Mumford, Hulek and others, proving the following result. Therefore it is very natural to study hyperbolicity of these spaces in the light of the Green-Griffiths-Lang conjecture Conjecture 1.3 (Green-Griffiths, Lang). Let X be a projective variety of general type. Then there exists a proper algebraic subvariety Y ⊂ X such that X is hyperbolic modulo Y .

In this setting, using the theory of automorphic forms, we can apply the same strategy as above and obtain

Theorem 1.4. A g (n) is hyperbolic modulo D := A g (n) \ A g (n) if n > 6g.
As a corollary we have Theorem 1.5. If A is a principally polarized abelian variety of dimension g ≥ 1 which is defined and non-constant over k, a complex function field of genus ≤ 1, then A does not admit a level-n structure for n > 6g.

In particular, this improves the previous bound, n ≥ max( 12 g(g + 1), 28), obtained by Nadel in [START_REF] Nadel | The nonexistence of certain level structures on abelian varieties over complex function fields[END_REF].

Acknowledgments. The author is grateful to Carlo Gasbarri and Xavier Roulleau for several interesting and fruitful discussions and remarks. He also thanks CNRS for the opportunity to spend a semester in Montreal at the UMI CNRS-CRM and the hospitality of UQAM-Cirget where part of this work was done.

Distance decreasing pseudo-distances

Let X := B/Γ be a quotient of a bounded symmetric domain by an arithmetic subgroup Γ ⊂ Aut(B). We first give a criterion to ensure the existence of a pseudo-distance g on X non-degenerate at a given x ∈ X and satisfying the distance decreasing property: every holomorphic map f : (∆, g P ) → (X, g) from the unit disc, equipped with the Poincaré metric g P , is distance decreasing i.e.

f * g ≤ g P . Let g denote the Bergman metric on B satisfying the Kähler-Einstein property Ric(g) = -g. Let γ > 0 be a rational number such that g has holomorphic sectional curvature ≤ -γ. We want to construct a continuous function ψ : X → R + such that ψ.g defines a pseudo-metric on X.

Theorem 2.1. Let x ∈ X and suppose there is a section s ∈ H 0 (X, l(K X + D)) such that

• s(x) = 0 • s vanishes on D with multiplicity m > l γ . Then there exists a pseudo-metric g on X non-degenerate at x satisfying the distance decreasing property. In particular, the Kobayashi pseudo-metric on X is non-degenerate at x.

We will prove this theorem in 3 steps. From [START_REF] Mumford | Hirzebruch's proportionality theorem in the noncompact case[END_REF], we know that the Bergman metric induces a good singular metric h := (det g) -1 on K X + D. Let x ∈ X and s be as given above. Let 0 < ǫ be such that

γ -ǫ > l m .
We define

ψ := ||s|| 2(γ-ǫ) l h
. Despite the singularities of h, we have Proposition 2.2. ψ is a continuous function on X vanishing on D.

Proof. We know [START_REF] Mumford | Hirzebruch's proportionality theorem in the noncompact case[END_REF] that K X + D is a good extension of K X with the metric h. This implies that near D, which locally is given by {z 1 . . . z k = 0}, we have (2.1)

||s|| h = O(|z 1 . . . z k | m log 2N |z 1 . . . z k |),
which suffices to obtain the conclusion.

The second step is Proposition 2.3. ψ • g is a continuous pseudo-metric on X vanishing on D.

Proof. We know that g has Poincaré growth near D (Lemma 2.1 [START_REF] Nadel | The nonexistence of certain level structures on abelian varieties over complex function fields[END_REF]) i.e.

g = O(g P ),
where

g P := k i=1 dz i ⊗ dz i |z i | 2 log 2 |z i | 2 + n i=k+1 dz i ⊗ dz i
is the metric obtained by taking the Poincaré metric on the punctured disc. Therefore if we combine this with (2.1), we have

||s|| 2(γ-ǫ) l h • g ij = O(|z 1 . . . z k | 2m(γ-ǫ) l -2 log α |z 1 . . . z k |).
This estimate clearly implies the proposition.

Finally as a third step, we are in position to construct a distance decreasing pseudo-distance which finishes the proof of Theorem 2.1.

Proposition 2.4.

There exists a constant β > 0 such that g := β •ψ•g is a distance decreasing pseudo-distance: for any holomorphic map f : ∆ → X from the unit disc equipped with the Poincaré metric g P , we have

f * g ≤ g P . Proof. Let u(z) := (1 -|z| 2 ) 2 • f * g.
Restricting to a smaller disc, we may assume that u vanishes on the boundary of ∆. Take a point z 0 at which u is maximum. z 0 ∈ ∆ \ f -1 ({ψ = 0}) otherwise u would vanish identically. Therefore at z 0 we have i∂∂ log u(z) z=z 0 ≤ 0. We have

i∂∂ log u(z) = i∂∂ log f * ψ + i∂∂ log f * g + i∂∂ log(1 -|z| 2 ) 2 .
On X, we have i∂∂ log ψ = (γǫ) Ric(g) = -(γǫ)g by the Kähler-Einstein property.

g has holomorphic sectional curvature ≤ -γ and therefore

i∂∂ log f * g ≥ γf * g. Moreover i∂∂ log(1 -|z| 2 ) 2 = -2 (1 -|z| 2 ) 2 .
Combining the above inequalities, we obtain

-(γ -ǫ)f * g(z 0 ) + γf * g - 2 (1 -|z 0 | 2 ) 2 ≤ 0. So ǫ/2.f * g(z 0 ) • (1 -|z 0 | 2 ) 2 ≤ 1.
Finally, we have

u(z) ≤ u(z 0 ) = (1 -|z 0 | 2 ) 2 • βψ(f (z 0 )) • f * g(z 0 ) ≤ 2βψ(f (z 0 )) ǫ .
Therefore if we take

β = ǫ/2 sup ψ ,
we obtain the distance decreasing property.

Proof of Theorem 1.1

To prove Theorem 1.1 it is now sufficient to show the existence of an étale cover X 1 = B/Γ 1 satisfying the hypothesis of Theorem 2.1 for every x ∈ X 1 .

First we recall the following result Proposition 3.1 ([6], 4.2). For any point x ∈ X, there exists a section

s ∈ H 0 (X, ( n + 1 2 + 2)(K X + D))
such that s(x) = 0 and s| D ≡ 0, where D := X \ X and n = dim X.

Let l := n+1 2 + 2 and take a positive integer m such that m > l γ . Now, we have Proposition 3.2. There exists a covering X 1 of X such that for any x ∈ X 1 , there is a section s ∈ H 0 (X 1 , l(K X 1 + D 1 )) vanishing on D 1 := X 1 \ X 1 with multiplicity at least m such that s(x) = 0.

Proof. By [START_REF] Mumford | Hirzebruch's proportionality theorem in the noncompact case[END_REF], one can find a covering X 1 such that π : X 1 → X is ramified to order at least m along D 1 . Let x ∈ X 1 . From Proposition 3.1, there is a section s 0 ∈ H 0 (X, l(K X + D)) such that s 0 (π(x)) = 0 and s 0 | D ≡ 0. We just have to take

s := π * (s 0 ) ∈ H 0 (X 1 , l(K X 1 + D 1 ) -mD 1 ).
As a consequence, we obtain a proof of Theorem 1.1.

Proof. Proposition 3.2 and Theorem 2.1 imply that for any x ∈ X 1 , there is a pseudo-distance ρ non-zero at x such that

d X 1 ≥ ρ.
This implies that X 1 is Kobayashi hyperbolic modulo D 1 .

Siegel modular varieties

We will apply Theorem 2.1 to the particular case of quotients of the Siegel upper half space by principal congruence subgroups.

Here B = H g , the Siegel upper half space of rank g. The arithmetic group Γ is Sp(2g, Z) and Γ(n) is the kernel of Sp(2g, Z) → Sp(2g, Z/n).

The space A g (n) := H g /Γ(n) is the moduli space of principally polarized abelian varieties with a level-n structure. Recall that if A is an abelian variety of dimension g over a field k of characteristic 0, a level-n structure is a 2g-tuple (x 1 , . . . , x 2g ) of points in A(k) which generate the subgroup of all n-torsion points in A(k) and form a symplectic basis with respect to the Weil pairing.

It is known that the Bergman metric on H g has holomorphic sectional curvature ≤ -2 g(g+1) . We can therefore take γ = 2 g(g + 1)

.

We are going to see that in this situation, we can use the theory of automorphic forms to obtain the section needed to apply Theorem 2.1.

First, recall (see for example [START_REF] Hulek | The geometry of Siegel modular varieties[END_REF]) that the automorphy factor of modular forms on H g defines a line bundle L on the quotient X := H g /Γ(n), the (Q-) line bundle of modular forms of weight 1. If the spaces X = A g (n) are smooth (for example Γ(n) is neat when n ≥ 3), we have K X = (g + 1)L -D, where D is the boundary divisor.

Let F be a modular form with respect to the full modular group Sp(2g, Z). Then following [START_REF] Hulek | Nef Divisors on Moduli Spaces of Abelian Varieties[END_REF], we define the order o(F ) of F as the quotient of the vanishing order of F at the boundary divided by the weight of F .

Particularly useful for us is the following result of Weissauer on the existence of cusp forms of high order which do not vanish at a given point of the Siegel space. Theorem 4.1 ([10]). For every point τ ∈ H g and every ǫ > 0 there exists a modular form F of order o(F ) ≥ 1 12+ǫ which does not vanish at τ. We obtain as a corollary Proposition 4.2. Let X := A g (n), x ∈ X and suppose n > 6g. Then there is a section s ∈ H 0 (X, l(K X + D)) such that

• s(x) = 0 • s vanishes on D with multiplicity m > l γ . Proof. We consider the natural map

π : A g (n) → A g .
This map is ramified to order at least n over the divisor at infinity [START_REF] Nadel | The nonexistence of certain level structures on abelian varieties over complex function fields[END_REF]. We choose some ǫ > 0 such that ǫ < 2 g . Now, we take a modular form given by Weissauer's theorem non-vanishing at π(x). In terms of line bundles, it gives a section s ∈ H 0 (A g , kL -mD) such that m k ≥ 1 12+ǫ non-vanishing at π(x). We take s := π * ( s). It gives a section in H 0 (X, kL -mnD) nonvanishing at x, where we make the slight abuse of denoting the line bundles corresponding to the modular forms and the boundary again by L and D. Therefore, using the equality K X = (g + 1)L -D, we have obtained a section in H 0 (X, l(K X + D)) with l = k g+1 vanishing on D with multiplicity mn. Finally, we observe that mn ≥ (6g + 1)k 12 + ǫ > kg 2 = l γ .

Theorem 1.4 is an immediate corollary of Proposition 4.2 and Theorem 2.1.

Theorem 1.5 is a consequence of Theorem 1.4, since a non-constant principally polarized abelian variety over k, a complex function field of genus ≤ 1, which admits a level-n structure induces a non-constant holomorphic map C → A g (n) whose image does not lie in D.

Theorem 1 . 2 (

 12 Tai, Freitag, Mumford, Hulek). A g (n) is of general type for the following values of g and n ≥ n 0 :


Delphine Le Roux

Brian E Root

Jeffrey A Hickey

Orion N Scott

Anchi Tsuei

Jingyi Li

David J Saul

Luc Chassagne

James P Landers

Philippe De Mazancourt

An integrated sample-in-answer-out microfluidic chip for rapid human identification by STR analysis

Keywords: microfluidic chip fabrication, sample collection, analytical processes inside the microfluidic chip, conventional analysis, data analysis

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Over the past decades, extensive research has gone into the development of micro-total analysis systems (µ-TAS) in genetic analysis and diagnostics [1][2][3][4][5][6] . Microfluidics offer a large number of advantages for rapidly analyzing samples because several steps commonly performed in different benchtop instruments can be integrated into a single microfluidic device -commonly called labon-a-chip devices 7 . The entire sample processing often encompasses (1) sample collection and treatment to release the target of interest (e.g., bacteria, DNA, proteins, cell), (2) amplification of the target and (3) detection and interpretation of this amplification product to get the result of the assay. Many of these benchtop processes have been independently demonstrated on microfluidic chips [8][9][10][11] . When performed in a conventional laboratory, these steps require trained technicians to transfer the samples to the instruments performing each of the different steps and are frequently performed in batches that lead to a long turnaround time. Additionally, current benchtop analytical processes are often long, and contamination can occur because samples are transferred from one tube to another between steps. Some efforts have been made to try to automate the bentchop processes 12 , but microfluidic devices may offer a greater possibility of faster sub-assay times and automated integration of multiple processes. However, realizing this vision of integrating multiple steps has been challenging, especially when it comes to integrating the sample preparation 13,14 .

Demonstrations of integrated microfluidic analysis systems have varied widely in terms of the target and in the analytical detection method (e.g., qPCR, electrophoresis, colorimetric, etc). [15][16][17][18][19][20][21][22][23][24] Some genetic assays currently require an electrophoretic separation as the assay output. One such assay requiring electrophoretic separation is forensic short-tandem repeat (STR) analysis for human identification. [START_REF] Butler | Fundamentals of Forensic DNA Typing[END_REF] This assay is particularly challenging for integrated microfluidics as it has very stringent requirements and requires at least three analytical steps: (1) DNA preparation from the sample, (2) DNA amplification, and (3) DNA separation with multicolour fluorescence detection. [START_REF] Horsman | [END_REF] Often, the reagents used for one of these steps is incompatible or detrimental to downstream processes which further increases the complexity of the microfluidic architecture in order to isolate any incompatible reagents [START_REF] Easley | Proc. Natl. Acad. Sci[END_REF] .

The first demonstration of a sample-in-answer-out microfluidic chip for genetic analysis by Easley et al. [START_REF] Easley | Proc. Natl. Acad. Sci[END_REF] detected B. anthracis from blood by solid phase extraction, PCR, and electrophoretic separation. However, several limitations of this system, such as single-color detection and separation resolution, make it non-applicable for STR analysis. Liu et al. [START_REF] Liu | [END_REF] expanded on this work to demonstrate an integrated microchip for STR analysis. While the level of assay integration is significant, the complex microchip is manufactured from glass and requires pre-treatment of the channel (e.g., polymer coating the separation channel), which make this approach prohibitively expensive for widespread use.

Other systems have been demonstrated in recent years, each with advantages and limitations.

Two groups working toward point-of-collection STR analysis demonstrated end-to-end analysis, but with the electrophoretic separation performed either on a separate separation microchip, or a capillary 29,30 . This arrangement results in multiple consumable components and begins to move away from the goal of developing true µTAS systems. Another demonstration of an integrated chip required a relatively large footprint (296 mm x 166 mm) with the device being a multicomponent structure connected together 31 . The result is a bulky consumable that requires complex and expensive manufacturing. While each of these systems represents a significant advance for STR analysis, taken together, their limitations demonstrate the challenge of creating a compact, low-cost, fully integrated disposable microchip for this type of assay.

Here, we report on the development of an integrated microfluidic chip for "sample in-answerout" human identification that is a single injection-molded piece approximately the size of a 96 well plate with simple, low-cost fabrication. The compact size of the device is achieved by a direct DNA extraction without any washing steps by using the protease from Bacillus sp. Erebus Antarctica 1 (EA1) 32,33 , a small PCR chamber heated by an IR laser 34 and a 7 cm microchannel capillary for DNA separation permitted by the use of a proprietary polymer. Moreover, the microfluidic chip is completely contained: no liquids leave the microchip during the assay and the instrument is not touched by liquid so contamination risks are minimized. The extracted DNA and amplified products are easily recoverable for conventional post-assay analysis if desired. The total assay time is reduced to almost 2 hours compared to the conventional processes in a forensic laboratory of up to 10 hours.

The instrument associated with the microfluidic chip contains the associated hardware to perform fluid flow control and each of the molecular assays that occur within the microchip. The chip contains no active, functional features. In this paper, we report the development and demonstration of each sub-assay inside the microfluidic chip (Liquid Extraction (LE), Powerplex ® 18 Fast System PCR and Microchip Electrophoresis (ME)) as well as the integration of two of the sub-assays together (for buccal swabs and FTA paper DNA inputs) to finally achieve the full integration of the process. The full functionality of the integrated microchip and instrument is demonstrated showing for the first time that a low cost and manufacturablecomplete micro-total analysis system is able to generate a full genetic profile in about two hours.

Methods and experimental section

Instrument design

Fig. 1a shows the instrument that drives the analysis within the microfluidic chip. The instrument has been designed in parallel with the microfluidic chip (shown in Fig. 1b), thereby ensuring effective control of the microfluidic movements and the different molecular assays for STR analysis. The main features interacting with the microfluidic chip are located within a portion of the instrument called the Chip Interface Module (CIM). The CIM is designed to receive the microfluidic chip with the sample acceptors attached (Fig. 1b) with a precision of 100 µm to ensure proper alignment of the CIM features with the microchip element. An important component of this module is an air supply system that meters different volumes with controllable flow rates to seven different ports per channel that link to different chambers and channels to control the fluidic movements. The interaction between the CIM and the microchip is described below in microfluidic chip design.

Other instrument modules include: heating modules for liquid extraction and electrophoretic separation, detection laser and CCD assembly module, power and controller modules, IR laser modules to perform the IR laser mediated PCR, and high voltage modules to perform the electrophoretic separation. Each of these is designed to meet the specifications determined by either the microfluidic fluid flow or the molecular biological assays. Data analysis of the STR profiles is described in supplementary information.

Microfluidic chip design

Previous work has shown that PCR and separation can be integrated for forensic STR analysis on a plastic microfluidic chip with the dimensions of a 96-well plate 35 . The plastic microfluidic chip demonstrated here provides the fully integrated process of DNA liquid extraction, aliquoting a volume of the DNA extract, mixing with PCR reagents, PCR and electrophoretic separation with embedded electrodes -all within a completely enclosed disposable chip. Two critical dimensions from prior work 35 , the PCR chamber volume and effective separation length, were carried over to this design. The additional assay processes as well as reagent metering and mixing significantly increase the complexity of the microfluidic chip design. Therefore, the injection molded chip has features on both sides (top and bottom), creating two layers to achieve this additional functionality while retaining a minimal footprint. These two layers are a fluidic layer, containing many reagent storage reservoirs and transport channels, and an analytical layer, containing the PCR chamber and separation channel. The analytical layer also contains some transport channels by necessity of design. The design allows a simple manufacturing process capable of high volume production. This is described in more detail in the supplementary information (S.1 microfluidic chip fabrication). Fig. 1c and Fig. 1d shows an overview of the chip design. Development of individual components of the design is discussed in the results section with respect to verifying functionality such as reagent mixing and DNA extract volume. Fig. 1c shows the top of the integrated microfluidic chip. Box 1 is a Luer lock connector that is injection molded as part of the chip for connecting the sample acceptor to it. Two heating plates, one on the inside CIM wall and one on the CIM door, contact the sample acceptors when the chip is inserted. Boxes 2 show electrode pads that pogo pins contact when the CIM top engages to physically contact the chip.

There are four electrode pads per channel to transfer the voltage from the CIM to the electrophoresis reservoirs. Box 3 is the pneumatic interface region that allows fluids to move.

The CIM top includes a silicon gasket that mates with this interface region. The pneumatic interface of the device is covered with a hydrophobic membrane to prevent liquid from contacting the CIM. The pneumatic system can deliver positive and negative pressures for fluidic movement. Box 4 is the detection region for one of the four channels where optics are aligned for sample detection. The detection optics are mounted on actuators that allow two-axes of motion for micrometer alignment precision. There are two pneumatic pistons per channel that can contact the chip for additional fluidic control by microfluidic valves (not highlighted). Fig. 1d shows the integrated chip filled with dye to highlight each of the reagent reservoirs. The reservoirs and their volumes are given in Table 1. The loading volumes are different because the filling lines from where the liquids are introduced into the chip are different for each of the reservoirs (the liquids are introduced from the filling ports highlighted by a red box in Fig. 1d).

Four sample collectors were attached to this disposable consumable (Fig. 1b) and inserted in to the CIM (Fig. 1b). The instrument door is closed locking the chip inside and a main piston brings the CIM down in to contact with the chip.

Microfluidic chip loading

The reagent reservoirs are filled by pipetting into the filling port of each reservoir (see the red box in Fig. 1d, 2 filling ports per reservoir, 8 in total per channel). Once the reagents are loading, the filling area is sealed with a PSA strip (Adhesives Research, Glen Rock, PA, USA). The separation polymer is loaded to the polymer reservoir to the right of the detection zone in Fig. 1d and is sealed with a pressure sensitive adhesive.

Microfluidic chip loading for a complete integrated run comprised the following steps:

Process optimization

For each of the sub-assays (LE, PCR and ME), fluidic movement from one step to the next and sub-integration of LE-PCR and PCR-ME were tested to verify and optimize functionality.

Fluidic movement tests were initially evaluated based on colorimetric results such as mixing and then verified with the assay result. For these sub-assay and sub-integration tests, only a portion of the microchip is loaded -the optimization approach here refers to the chip loading step numbers above. Sample collection details can be found in supplementary information. Different sample sizes of FTA were tested for the optimization step (25 mm², 50 mm², 75 mm², 100 mm² and 125 mm² out of 490 mm² for the entire FTA mini card circle). Four repeats were performed. These pieces were introduced into the 1 mL syringe attached to the chip and pushed toward the lower part (to be in contact with the LE heater).

For LE-PCR, (1) and (2) were loaded. PCR test chips filling is based on the PCR mix described for PCR-ME with an appropriate volume of DNA (extracted from benchtop extraction) corresponding to the aliquot portion within the mix. To verify the PCR-ME process, the same PCR mix was introduced into the PCR chamber. Reagents (3) and (4) are filled the same way than previously described. PCR and ME protocols are described in supplementary information.

In the system described here, the polymer is pneumatically loaded in to the microchannel by the instrument. As the polymer is loaded through the separation channel on the bottom layer of the chip and comes up through the vias, the liquid contacts the electrodes almost immediately upon entering the buffer and sample waste reservoirs resulting in connectivity between the electrodes.

Additional polymer is pushed into these reservoirs to provide buffering capacity for electrophoresis. This optimization was done by observing initial connectivity over several chips and applying an additional time for polymer loading beyond this.

To assist with fluidic movement to the sample reservoir, the current between the sample electrode and the buffer electrode was monitored. As the sample electrode is located in the second half of the sample reservoir from where the sample enters, an increase in current between the sample and buffer electrodes provides an indicator for fluid flow. This is used as an active feedback sensor in the instrument control software for fluid flow to this reservoir. When adequate sample enters the sample reservoir and the increase in current is detected between the electrodes, the software stops the fluidic movement. When an inadequate volume of fluid is in the sample reservoir and there is no change in current, an additional fluidic movement is prompted by the software.

Results

Extraction optimization

The microfluidic chip contains the extraction reagents which become fluidically connected to the sample acceptor when the pneumatic module pushes the ZyGEM reagent from one port into the sample collector as shown in Fig. 2a (blue dyed reagents into the collector for all four channels).

The instrument door and the CIM contain contact heaters (red in Mixing after the denaturation step was tested to determine if a higher DNA yield will be obtained when extract is pumped in and out of the sample acceptor. Doing so increased the yield by The fluid flow rate is the critical factor and therefore is the factor that is the primary interest for development.

FTA paper optimization began with trying different punch sizes of FTA mini cards (as described the FTA paper was found to be lower than the buccal swabs for some of the data points (Fig. 2c).

However inhibition in the qPCR assay was observed for some of these samples, which may have resulted in this reduction. Moreover, the sample recovery and sample collection process difference can lead to variability in the results. The graph also shows that the DNA yield from the FTA paper can be as high as that obtained by buccal swabs (for example points around 10 µL/ng) when only 100mm² of the FTA paper is put into the system compared to a buccal swab.

More importantly, despite the difference in DNA yield, all samples from both the FTA paper and buccal swabs samples gave full profiles using the PowerPlex ® 18 Fast System on a conventional both methods were comparable since the peak heights fall within the same range for all the samples. The Powerplex 18 PCR did not appear to suffer the inhibition seen in some qPCR results. Moreover, the peak height ratio is greater than 50% for almost all of the samples (Fig.

S.3e

) although buccal swabs peak height values are a little lower for some of the samples. The LE extracts from both sample types were able to provide extracted DNA samples yielding balanced multiplexed PCR for human identification and 100% full profiles.

PCR and LE-PCR optimization

Microchip PCR has been previously demonstrated on a simpler microchip that required the preparation of ZyGEM DNA and mixing of reagents outside the chip 35 . Therefore the first step to achieve LE-PCR is to demonstrate that a DNA aliquot, with the target yield, can be microfluidically mixed with PCR reagents and flowed in to the PCR chamber. An aliquot is taken by drawing the extract from the ZyGEM reservoir toward the hydrophobic membrane as

shown by the red arrows in Fig. 3a. The aliquot volume is defined by the microchannel dimensions between the hydrophobic membrane port (blue dot formed by the liquid stopped at the membrane) and the active valve (oval with two vias in the middle). The active valve is then actuated after the aliquot is taken. To prevent PCR reagents being drawn toward the hydrophobic membrane, surface tension valves are placed on either side of the PCR reservoirs.

The aliquot is then pushed off the hydrophobic membrane toward the PCR reagent mixing chamber through the PCR reagents chambers (Fig. 3b). The DNA aliquot begins mixing with the PCR reagents during the movement to the mixing chamber. The mixing chamber spans the depth of the injection molded piece (1.5 mm) so it can be connected to both layers: the reagents enter the mixing chamber on the bottom layer and a vent line on the top layer on the opposite side of the chamber allows air to vent. Air is pushed into the chamber after the PCR reagents and DNA sample and bubbles through the liquid to the top of the chamber and vents out of the microchip through a pneumatic port. Colorimetric tests indicate good mixing within this chamber (Fig. 3b).

During the fluidic movement to the mixing chamber, the active valve downstream of the PCR chamber was closed. This valve is then allowed to open and PCR mix is pushed from the mixing chamber to the PCR chamber (Fig. 3c). The channel exiting the mixing chamber toward the PCR chamber is on the bottom layer to reduce the probability of bubbles being pushed in to this channel. Once the PCR chamber is filled, the active valve downstream of the PCR chamber is closed. The PCR chamber is then pressurized to 5 PSI against the active valve to prevent bubble formation and liquid movement during PCR.

In preliminary studies, the impact of DNA mass in the PCR chamber on PCR peak heights was investigated. It was found that when DNA mass falls below 1.2 ng in the PCR chamber, peak heights begin to decrease below the target threshold of 1000 RFU on a conventional CE instrument. With an average DNA yield of 9.0 ng/µl, and a lower standard deviation bound of 6 ng/µl, the aliquot microchannel volume was designed to hold 4 µl. This volume provides an average DNA mass of 2.4 ng with a lower standard deviation of 1.6 ng.

The microfluidic movement and mixing efficiency during LE-PCR were verified by comparing integrated results with a microchip amplification where the DNA extraction and reagent mixing steps were performed off-chip. Fig. 3d and 3e show LE-PCR from a buccal swab and FTA paper, respectively. Each method produced a full profile (18-loci) with similar peak balances and similar peak heights (with variations due to sample to sample variation, amount of unreacted master mix pulled from the chip with the sample…) to each other and to profiles obtained with PCR on chip (off-chip mixing). These LE-PCR tests demonstrated that the microchip fluidic control is capable of generating sufficient DNA, drawing an aliquot of the extracted DNA, mixing with the PCR reagents, and filling the PCR chamber without bubbles.

Automated Alignment

The separation channels have a width of 50 µm and therefore the detection sensitivity is expected to have a low tolerance to variations in chip-to-chip alignment. shows that to maintain 80% of the maximum signal, the allowable deviation in the Y-axis and Z-axis is approximately 10 µm and 38 µm, respectively. The lines in Fig. 4a represent this allowable deviation. While the Z-axis conforms to this threshold for three out of four channels, the Y-axis is well above maximum allowable deviation. Therefore, an automated alignment method for every chip is required to detect the fluorescent DNA. To achieve this, a high wavelength dye, outside of the dye spectrum used for the DNA, was added to the separation buffer. This dye is excited and collected by the same laser induced fluorescence system as for the STR DNA fragments. The optics system rasters the focusing and collecting lenses in the Y-axis with step changes in the Z-axis (focusing axis) between lateral scans. The high wavelength dye signal is maximized to find the separation channel. Fig. 4b shows a representative trace of the high wavelength dye signal during the alignment procedure with the fitting curve to identify to optimal alignment point.

ME and PCR-ME Optimization

Prior to ME, the polymer reservoirs are pressurized to 30 PSI (see Fig. 5a for the air movement with the red arrows) to push the polymer through the separation channels to each of the buffer, sample and sample waste electrodes. This loading pressure was applied for 30 minutes in addition to connectivity time to allow excess polymer to provide buffing capacity the buffer and sample waste electrodes for stable electrophoresis current during the entire separation. This polymer loading step was performed during the PCR in the PCR-ME and the full integrated runs.

ME in a plastic microchip were demonstrated on a simpler device were the polymer was loaded manually 35 . To verify that ME performed similarly to that previous simpler design, an allelic ladder was separated after fluid flow from separation reservoir. Voltages were adjusted to keep similar field strength to those used in the previous study. In order to validate the ME, the allelic ladder and separation reagents were mixed and introduced directly into the sample reservoir. The ME is able to resolve all of the peaks in the ladder to generate a binning matrix (Fig 5b). TH01 9.3 and 10 are resolved that demonstrate that the system is able to discriminate the single base pair required by the forensic community for that multiplex kit (Fig. 5b left bottom corner).

Integration of the PCR and ME was tested to verify mixing of the PCR product with the separation reagents. Following PCR thermocycling pressurized PCR chamber is purged slowly and the active valves are opened. Separation buffer is pushed through the PCR chamber into the separation sample reservoir with current assist as described in Section 2.4. The arrows in Fig. 5c show the fluidic movement. At this point, the stage is heated to the separation temperature for ME. Fig. 5d shows a typical integrated PCR-ME test for the 18-plex PCR amplification. Both the PCR products and ILS ladder are detected verifying sufficient mixing of the two liquids and good injection, separation and detection of all fragments, the good mixing shown here was consistent with colorimetric tests. The allele calls in the profiles were compared with the conventional analysis performed for the same donor to verify correct allele calling (always found to be concordant).

Integrated LE-PCR-ME

Following the successful demonstration of LE-PCR and PCR-ME partial integration, microchip LE-PCR-ME is tested. Buccal swabs were loaded into the sample acceptors and attached to a chip loaded into the instrument. The complete full end-to-end integrated process is then run on the device leading to a genetic profile analyzed by GeneMarker™ and compared with the conventional reference profiles for concordance.

LE-PCR-ME analyses of buccal swabs are shown in Fig. 6. Three different donors were tested for this preliminary demonstration of the fully integrated microfluidic chip for "sample in-answer out" human identification (Fig. 6 a-c). The profiles show each of the 18 loci were successfully amplified and separated with the internal lane standard. The alleles from the sample were analyzed using GeneMarker™ to call the allele numbers using a binning palette created from an allelic ladder separation run on the system under the same condition. The allele calling was compared against the conventional method and found to be 100% correct for each of the sample.

Discussion

The advantages of a fully integrated µTAS for forensic STR analysis are clear: (i) a fully enclosed microfluidic system reduces the opportunity for contamination from conventional liquid transfer steps, (ii) a simple work flow allows less experienced users to perform the analysis at the point of sample collection, and (iii) the analysis time is reduced substantially compared with the current process. While the sample-to-answer analytical time is reduced, the greater impact of these systems may be the time-savings of eliminating the need to send the samples to a central laboratory for testing.

Other integrated DNA systems have been previously reported [START_REF] Liu | [END_REF]29,31 , it is important to note several critical features that distinguish the system described here. First, the main component of the microchip is a single injection molded piece of Cyclo Olefin Polymer (COP). This significantly reduces the cost of a fabrication compared with devices made from glass [START_REF] Liu | [END_REF] from multiple components 29,30. The final microchip assembly processes of sputtering electrodes, solvent bonding and lamination of the capping layers on to the flat chip are well-established and can be easily scaled to larger batches of microchips. This straightforward fabrication process, along with no requirement for microchip surface pretreatment, enables a simple and low-cost fabrication process.

Second, this microchip has the footprint roughly the size of a 96-well plate and is capable of analyzing four samples. This is a stark contrast to other reports for assays that achieve high resolution separation by using a much larger microchip, or that require an interface to a capillary for the separation. The reduced size is enabled by two different technologies First, the unique hydrophobically-modified polyacrylamide polymer enable the high resolution DNA separation to be achieved in an effective length of 7 cm compared with other reports for similar assays with lengths of 22.5 cm in a plastic substrate 31 , 14 cm in a glass substrate [START_REF] Liu | [END_REF] , or in a one the shelf capillary 29 . Second, the ZyGEM enzyme preparation method allows footprint compression through reduction in the number of reagents, fewer fluidic transport channels, and no reservoir for liquid waste (e.g., adsorption and wash buffers for solid phase extraction).

The microchip design also allows easy recovery of both the amplified product and the extracted DNA if additional post-assay processing if desired. The ZyGEM extraction method leaves the excess DNA in an easily recoverable solution rather than in a waste reservoir. The Luer lock connector design allows different DNA sources to be input in to the system for analysis increasing the flexibility of the design for different assays. While this study focused of forensic applications, the DNA preparation -PCR amplification -electrophoretic separation process is used in other diagnostics including, clinical, environmental and food testing.

Conclusions

A fully integrated sample-in-answer-out microchip for human identification base on STR analysis was demonstrated. A combination of enzymatic DNA preparation, microliter PCR, and a unique separation polymer enabled a compact footprint, plastic microchip substrate, and eliminated all surface pretreatment. The enzymatic DNA preparation with appropriate microfluidic movements lead to full genetic profiles from buccal brush swabs and buccal cells on FTA paper that show the system can be used for at least two different DNA inputs. An aliquot of this DNA sample allowed an 18-plex STR amplification that provides similar profiles to the conventional method. The PCR product was separated in an effective length of 7 cm using a polymer that is demonstrated to resolve each of the peaks of the allelic ladder and achieved concordant and accurate allele calling for three different individuals. This proof-of-concept micro-total analysis system is a critical step towards achieving a low cost microchip for a pointof-collection human identification system. A top laminate was an aluminum-acrylic laminate composite and was laser cut to the dimensions of the chip and with several critical features. Two perforated lines were created that align both sides of the pneumatic interface region (Fig. 1c). This allowed the top film to be laminated as a single piece yet allowing an easy peel action just prior to reveal the protected hydrophobic membrane (from debris, scratch etc). Cutouts in this layer were created for the detection zone and around the sample waste electrode. A track outlining the sample reservoir was also created to further isolate that electrode as the acrylate layer may not provide a perfect

Sample collection.

Buccal cells were collected after a 30 seconds mouth rinsing in both cases, by turning the buccal brush swabs against the cheek for 30 seconds and using a DNA collector for FTA ™ (sterile foamtipped applicator, Puritan Medical Products Company LLC, Guilford, ME, USA) after putting it S-4

against the tongue for 30 seconds to moisturize it and scrub each of the cheeks with each of the side of the collector. The buccal swab was directly put into the swab acceptor attached to the chip. The DNA collector was applied onto one circle of the FTA mini card to get the maximum of cells.

Analytical processes inside the microfluidic chip.

Microchip PCR is accomplished by IR-heating and non-contact temperature sensing as described in Le Roux et al 2 . Based on these results, the thermocycling conditions were an initial denature of 94 °C for one minute, 29 cycles of 94 °C for 10 seconds and 59 °C for 40 seconds, and a final extension of 59 °C for 2 minutes. Compared with the previous report 2 , these conditions increase the number of cycles from 27 to 29 and slightly increase the length of the anneal and denature times to increase peak heights. The increased in peak heights with longer anneal and denature times may be a function of differences in thermal mass between the two systems causing a difference in heating and cooling ramp rates.

Microchip electrophoresis for STR analysis was previously demonstrated in a 7 cm effective separation length following a standard cross-T injection method. The microchip used the same effective separation length but with different channel lengths from the cross-T node to the electrodes. The voltage drop along each arm was kept between the two designs requiring different applied voltages. The sample was electrokinetically injected by applying 300 V at the sample waste with the sample electrode grounded for 360 sec. A 6 second gated injection was used with 1500 V at the outlet and the sample electrode again grounded. The separation was performed with 1500 V on the outlet electrode, 250 V and 300 V on the sample and sample S-5

waste electrode, respectively, and the buffer electrode grounded. The stage heater temperature that controlled the separation polymer temperature within the microchip channel was set to 43°C.

To test automated alignment of the system, the separation channel was loaded in the same manner as for an end-to-end sample analysis. The sample reservoir was loaded with a fluorescein solution. The channel was aligned as normal by maximizing the signal of the alignment dye. To determine the change in signal with alignment position, the fluorescence intensity is measured when no fluorescein is in the channel and when fluorescein is electrophoresed to the detection zone. The signal is taken as the difference in intensity and the noise is taken as the standard deviation of the fluorescence intensity when no fluorescein is present in the channel. The alignment actuators are then moved in the Y-or Z-direction away from the optimal alignment position and the measurement is taken again.

Conventional analysis.

Liquid Extraction/PCR and Capillary Electrophoresis are performed to compare the results of some sub-assays on the microfluidic chip following the manufacturer's instructions and using a GeneAmp® PCR System 9700 thermal cycler (Life Technologies, Carlsbad, CA, USA), a ABI PRISM® 310 Genetic Analyzer (Life Technologies) and a ABI PRISM® 3130-Avant Genetic Analyzer (Life Technologies).

Data analysis.

Data analysis from both microfluidic and conventional processes was performed with SoftGenetics GeneMarker® HID v2.4.0 software (SoftGenetics, LLC, State College, PA, USA).

S-6

The microfluidic platform performed the data processing of the assay using proprietary software (Lockheed Martin Corp., Bethesda, MD, USA) that interfaced with GeneMarker®.

6. Supplementary figures. The actuators were then moved set distances in the X-and Z-directions. The signal was calculated from the difference in pixel intensity of when the fluorescein was present in the

( 1 )System) 36 . ( 3 )

 1363 adding a volume greater than 200 µL to ensure complete filling of the reservoir and filling channels of the ZyGEM extraction reagents. A total of 900 µL is prepared for 4 channels to allow extra volume for pipette errors: 846 µL extraction buffer and 54 µL EA1 enzyme based on prepGEM™ Saliva kit (ZyGEM corp., Hamilton, New Zealand). (2) 13 µL of PCR reagents is used to fill both PCR reservoirs. PCR reagents are prepared with PowerPlex® 18 Fast System Promega Corp., Madison, WI, USA), 69 µL Promega PowerPlex® 18 Fast System primers, 35 µL Promega PowerPlex® 18 Fast System reaction mix and 6 µL water. The Promega primer and reaction mix were modified by Promega for this specific PCR (based on the PowerPlex® 18D The separation reservoir is filled with 25 µL of a separation reagent mix of 18 µL Internal Lane Standard (ILS) 500 (Promega) and 90 µL dilution buffer (Hi-DI TM Formamide, Life Technologies, Carlsbad, CA, USA) prepared for four channels. Finally, (4) 20 µL hydrophobically-modified polyacrylamide polymer 35 (Microlab Diagnostics, Inc, VA, USA) containing DY-680 far red excitation dye (Dyomics GmbH, Jena, Germany) are introduced directly to the polymer reservoir. For an allelic ladder separation, both PCR reservoirs are filled with water and 25 µL separation reagents/allelic ladder mix is introduced into the separation reagent reservoir (90 µL Hi-DI TM Formamide, 18 µL Internal Lane Standard 500 and 25 µL PowerPlex® 18D allelic ladder).

For DNA extraction tests, only chip loading step ( 1 ) 9 -

 19 was introduced into the chip. This assay modifies the published protocol 32 by reducing the enzyme activation time at 75 °C and deactivation time at 95 °C to 2 minutes each, the extraction is reduced to these unique two steps allowing PCR ready extract. Parameters such as enzyme activation time and mixing could then be varied compared to the recommended protocol for the prepGEM Saliva kit. Two DNA sources were tested: buccal swabs (MasterAmp™ Buccal Swab Brush, Epicentre, Madison, WI, USA) and FTA TM mini card (Indicating FTA TM Mini Card, Whatman Inc., Clifton, NJ, USA).

  Fig 2a) that clamp the sample collector to raise the sample to the ZyGEM enzyme activation and denaturation temperatures (in this picture, the door is open so that chip and sample collectors can be seen). As shown in the image, the ZyGEM reagent covers approximately two-thirds of the swab so that an additional benchtop extraction can be performed after microchip analysis if desired. Following the LE, the solution is drawn back into the ZyGEM reservoir. DNA was extracted from two different sources -a brush buccal swab and FTA paper. The extraction protocol was first optimized for a buccal swab. Initial extractions pumped the liquid extraction reagents into the sample acceptor, heated the solution to the target activation and denaturation temperatures, and drew the solution out of the sample acceptor. This "Simple Flow" protocol resulted in an average DNA yield of 2.5 ng/µl (Fig. 2b), which is below the yield that will produce full STR profiles in microchip PCR amplifications. Longer extraction times were tested but did not increase the yield (Fig. S.2).

  fivefold and the yield was flow rate dependent with slower mixing (~3.5 µl/sec) yielding 6.0 ng/µl and faster mixing (approximately 15 µl/sec in and 30 µl/sec out) yielding 9.0 ng/µl. In the final protocol, the syringe compression rate was set to 50 µl/sec which yielded a fluid flow rate of approximately 20 µl/sec. The difference in syringe rate and fluid flow rate is due to compressibility of air and air flow restrictions in the instrument between the syringe and the chip.

in 2 . 4 )

 24 inside the sample acceptor and performing the same LE protocol as buccal swab with mixing. The PCR-ready extracts from the two smallest FTA punches did not give full conventional profiles indicating insufficient DNA yield. The largest FTA punches gave full conventional profiles but the FTA paper absorbed too much liquid to be able to draw back a sufficient amount to have liquid proximal to the aliquot line and this would have resulted to a failure for LE-PCR on chip. The 100 mm² FTA paper pieces were able to give full conventional profiles and the amount of liquid absorbed was acceptable for the aliquot step, with a small modification of the fluid flow script taking the absorbed liquid into account. Confirmatory tests were done with 20 samples of 100 mm² FTA paper punches in parallel with 20 samples of buccal brush swab to allow direct comparison (4 donors repeated 5 times). The extracted DNA was then quantified using Quantifiler® Duo DNA Quantification kit (Life Technologies). The yield from

  PCR system (See representative full profiles inFig. S.3a and b). Analysis of these full profiles shows that the PCR quality from both methods is comparable. Assuming that the height of each peak reflects the number of copies of the microsatellite present in the DNA (i.e. DNA concentration for PCR), Fig. S.3c and S.3d show that the DNA concentration used for PCR from

  Fig. 4a (bars) shows the standard deviation of the alignment position for 19 chips. This variation in microchannel position from one chip to the next is then compared with the change in detection sensitivity to alignment position. Fig. S.4

Fig. 6 c

 6 Fig. 6 c-d shows the same donor analyzed three times by the end-to-end microchip

Figures:Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 . 1 . 1

 1234511 Figures:

S- 3 Fig. S. 1 .

 31 Fig. S.1. Microfluidic chip manufacture process. (a) Picture of the bottom molding part of the

Fig. S. 2 .Fig. S. 3 .

 23 Fig. S.2. DNA yield from ZyGEM extractions on chip with increasing amounts of extraction

Fig. S. 4 .

 4 Fig. S.4. Tolerance of sensitivity to misalignment in the X-and Z-directions. The optics were

Table 1 .

 1 Reagent volumes inside the microchip reservoirs and dye colors corresponding to each of them for identification in the

Fig. 1d Reagent Reservoir Color Volume (µl)

	ZyGEM extraction reagent Blue	200
	PCR Mix #1	Red	5.5
	PCR Mix #2	Orange	5.5
	Separation reagents	Green	17
	Polymer	Purple	20

Acknowledgements

The authors are grateful for the technical support of Michael E. Egan, Darren Albert, Douglas J.

South from Lockheed Martin Corp, Bethesda, MD. We also thank Doug Storts from Promega Corporation, for reagents improvements and supporting the development and Morpho for financial support. Finally, the authors are especially grateful to the collaboration with Minifab, Victoria, Australia for microfluidic chip design development and fabrication.

the separation reagents are then ready to be electrophoretically separated (cross T injection and separation microchannel showed in orange). (d) A PCR-ME performed inside the microfluidic chip. The Internal Lane Standard is well separated aswell as the DNA peaks.

Lab on a Chip

Supplementary Information

Integrated sample-in-answer-out microfluidic chip for Rapid Human Identification by STR analysis

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