J Marteau
email: marteau@ipnl.in2p3.fr

The OPERA experiment: a direct search of the ν µ -→ ν τ oscillations

Keywords: Neutrino oscillations, long baseline experiment, hybrid detectors, nuclear emulsions. PACS: 14.60.Pq

The aim of the OPERA experiment is to search for the appearance of the tau neutrino in the quasi pure muon neutrino beam produced at CERN (CNGS). The detector, installed in the Gran Sasso underground laboratory 730 km away from CERN, consists of a lead/emulsion target complemented with electronic detectors. After a short pilot run in 2007, a first physics run took place from June to November 2008. The second physics run started in June 2009. At present a total (2008+2009) of 4.2 10 19 protons on target were delivered by the CNGS, producing more than 25,000 events in time coincidence in the OPERA detector. Among them 4000 events occured in the target of the detector. In this paper the detector and the analysis strategy are described and the status of the analysis of the 2008 and 2009 runs is discussed.

Introduction

In the last decades solar and atmospheric neutrino experiments observed deficits in the measured fluxes which are all well reproduced in a neutrino oscillations model, implying non vanishing, not degenerate neutrino masses and neutrino mixing. Within such hypothesis weak interactions eigenstates differ from the mass eigenstates. The mixing can be parametrized in an unitary matrix whose parameters (3 angles and 1 or 3 phases depending on the Dirac or Majorana nature of neutrinos) associated to the square masses differences ∆m 2 drive the amplitude of the disappearance (P (ν α → ν β )) or survival (P (ν α → ν α )) probabilities.

The major experimental results for solar, 1-7 atmospheric, 4-9 reactor 10,11 or accelerator 12,13 neutrinos were obtained by observing the "disappearance" of neutrinos wrt to a close position measurement or a predicted flux.

OPERA 14 has been designed to perform an unique appearance observation of the oscillation products to confirm (or infirm) the neutrino oscillation hypothesis in the atmospheric sector through the ν µ → ν τ channel and also to set limits on the θ 13 angle through the ν µ → ν e channel.

The OPERA experiment

The CNGS beam

The CNGS [START_REF]CNGS project[END_REF] programme of neutrino beam from CERN to Gran Sasso has been approved in 1999. The beam has been optimized to maximize the number of τ events in the detector (convolution of the neutrino flux, the appearance probability and the detection efficiency). The neutrino average energy is 17 GeV. Measured in the number of interactions in the detector, the νµ contamination is ∼ 2%, the ν e ( νe ) is < 1% and the number of prompt ν τ is negligible. The beam configuration starts from the SPS protons directed to the target chamber where mainly pions and kaons are produced, which decay then into a 1km long decay tunnel followed by hadron stops and muon detectors. From CERN to Gran Sasso the escaping neutrinos travel for 2.44 ms and their mean direction w.r.t. the horizontal in Gran Sasso forms a 3 • angle due to the earth curvature. There are two fast extractions separated by 50 ms in each CNGS cycle (6s long). These cycles are repeated 3 or 4 times during each SPS supercycle. The nomimal beam intensity is 4.5 • 10 19 pot /year. The so far achieved performance in 2008 and 2009 is respectively 1.78 19 and 2.4 19 pot (Fig. 1).

The detection technique

The challenge of the experiment is to measure the appearance of ν τ from ν µ oscillations through CC τ interactions. The events induced by the shortlived τ have a characteristic topology (with a "kink" due to the presence of undetected neutrinos in the τ decay) but extends over ∼ mm 3 typical volumes. The detector should therefore match a large mass for statistics, a high spatial resolution and high rejection power to limit background contamination. These requirements are satisfied using the proven ECC (Emulsion Cloud Chamber) technique which already worked successfully in the DONUT experiment. [START_REF] Kodama | [END_REF] The passive target consists of lead plates. Particles are tracked in nuclear emulsions films with a sub-micrometric intrinsic resolution. 57 emulsions films are assembled and interspaced with 56 lead plates 1 mm wide in a detector basic cell called "brick". An additional doublet of emulsion film (Changeable Sheets, CS) is attached on the downstream face of each brick to guide the tracks predictions inside the brick itself. A brick is a 12.7 × 10.2 cm 2 object with a thickness along the beam direction of 7.5 cm (about 10 radiation lengths). Bricks are assembled in 31 walls (52 × 64 bricks) separated by electronic detectors planes to trigger the event and identify the brick with the interaction vertex. The total OPERA target contains 150000 bricks with a total mass of 1.25 ktons. The ECC bricks were assembled underground at an average rate of 700 per day by a dedicated fully automated Brick Assembly Machine (BAM); the OPERA target was filled using two automated manipulator systems (BMS). The passage of a m.i.p. in an emulsion film results, after development, in a set of aligned grains, 35 grains /100µm.

The off-line emulsions scanning

After development emulsions are scanned by automatic microscopes whose nominal speed is higher than ∼ 20 cm 2 /h per emulsion layer (44 µm thick). There are two different approaches developed by the OPERA collaboration, in Europe (ESS, 20 based on software image reconstruction) and in Japan (S-UTS 21 based on hard-coded algorithms). The scanning sequence proceeds with the division of the emulsion thickness into ∼ 16 tomographic images by focal plane adjustment, images digitization and track finding algorithms. Track grains are identified and separated from "fog" grains and associated into "micro-tracks".

The event validation chain

Every time a trigger in the electronic detectors (see later) is compatible with an interaction inside the target and on time with the beam arrival, the brick with the highest probability to contain the neutrino interaction vertex is extracted from the apparatus and exposed to X-rays for film-tofilm alignment between the CS and the last brick film. CS films are scanned looking for tracks in an area of about 50 cm 2 around the TT prediction. The CS-to-TT connection is made with an accuracy of 1 cm in position and 20 mrad in angle. If tracks from the CS scanning are compatible with those reconstructed in the electronic detectors the brick is brought to the surface laboratory and exposed to cosmic rays in a dedicated pit in order to provide straight tracks for a finer (sub-micrometric) film-to-film alignment. Brick emulsion films are then developed and dispatched to the scanning laboratories to be analysed. All tracks reconstructed in the CS are sought in the most downstream films of the brick with an accuracy of about 70 µm in position and 10 mrad in angle. Confirmed tracks are followed back along the brick. A stopping point is a signature either of a primary or a secondary vertex. Vertex confirmation is made by scanning a volume of 1 cm 2 transverse size for 11 films located upstream and downstream of the stopping point. All tracks originating inside the volume are input for a vertex reconstruction algorithm which is tuned to find also decay topologies. The tracks attached to the primary vertex are followed downstream to check whether they show any decay topology and to measure their momentum from their Coulomb multiple scattering.

Expected performances

At the expected nominal beam intensity and for five years data taking a total of 25000 charged and neutral current interactions is expected in the nominal mass target of OPERA. Among these about 100 ν τ CC interactions are expected for oscillation parameter values ∆m 2 23 =2.4 × 10 -3 eV 2 and sin 2 2θ 23 =1. The overall detection efficiencies have been estimated in Monte Carlo simulations and are currently under re-evaluation by exploiting the data themselves. A total of 10 identified τ events is expected (in all physics channels) for less than 1 background event. In the sub-dominant ν µ → ν e channel OPERA should, in the same run conditions, set a limit of sin 2 2θ 13 < 0.06 (90% C.L.), 17 the main militation being the beam contamination in -ν e .

The OPERA electronic detectors

OPERA is a hybrid detector made of two identical Super Modules (SM1 and SM2) consisting of a target section followed by a muon spectrometer (see Fig. 3). The target section is made of the already mentionned 31 brick walls each one being followed by a highly segmented scintillator tracker plane. A large VETO plane is placed in front of the detector to further discriminate beam events from horizontal cosmics and beam neutrino interactions in the rock. The construction of the experiment started in Spring 2003. The two instrumented magnets were completed in May 2004 and beginning of 2005 respectively. In Spring 2006 all scintillator planes were installed.

The target tracker covers a total area of 7000 m 2 and is built of 32000 scintillator strips, each 7 m long and of 25 mm×15 mm cross section. Along the strip, a wavelength shifting fiber of 1 mm diameter transmits the light signals to both ends, where the opto-electonics readout chain includes the Hamamatsu 64 channels MaPMT, a dedicated auto-triggerable front-end ASIC 18 allowing gain correction and an Ethernet capable microprocessor digital board. [START_REF] Marteau | The OPERA global readout and GPS distribution system[END_REF] The muon spectrometer consists of a large 8 × 8 m 2 dipolar magnet delivering a magnetic field of 1.55 T and instrumented with RPC's and drift tubes. Each magnet arm consists of twelve 5 cm thick iron slabs, alternating with RPC planes. This sandwich structure allows the tracking in the magnetic field to identify the muons and to determine their momentum and sign. In addition the precision tracker 19 measures the muon track coordinates in the horizontal plane. It is made of 8 m long drift tubes with an outer diameter of 38 mm. The charge misidentification is expected to be 0.1 % -0.3 % in the relevant momentum range which is efficient enough to minimize the background originating from the charmed particles produced in ν µ interactions. With the muon spectrometer a momentum resolution of ∆p/p ≤ 0.25 for all muon momenta p up to a maximum of p = 25 GeV/c can be achieved.

Status of the analysis

This section gives an overview of the present status of the data collection and analysis in the electronic detectors and in the emulsions during the 2008 and 2009 physics run. Currently a total of 4.2 10 19 pot have been delivered by CERN, resulting in 20 (8 + 12) millions of raw events (L2 trigger), 25,000 (10,000 + 15,000) events tagged by the OPERA detector in time coincidence with the CNGS. The event timestamp in OPERA is given with a 10ns accuracy through the common clock distribution system locked on the GPS. [START_REF] Marteau | The OPERA global readout and GPS distribution system[END_REF] This large statistics is used to confront MC with data results to re-evaluate the real performance of the experiment. [START_REF] Anokhina | [END_REF]24

Electronic detectors analysis

Recent progresses in the reconstruction of kinematics distributions in the electronic detectors allow to improve the identification and the brick finding efficiencies (Figure 4 displays for example the good data/MC agreement for muon momentum and the visible energy in the Target Tracker).

A lot of progress has been recorded as well in the event identification with a special emphasis on the low multiplicity events (mostly soft NC events) which intervene as background for τ physics channel with low energy deposit (τ → e quasi-elastic for instance).

Vertex location

The validation of events by the CS scanning is displayed on Fig. 5 (left) for the 2008 run. Also shown is the vertex distribution projected on the section of the brick (same Figure, right panel). The analysis is continuously on-going but also upgraded to improve the location efficiency.

Emulsions data analysis

Important steps are under careful and systematic (re)evaluation for the validation of the data analysis. Among them the distributions of the impact parameters for the tracks wrt the primary vertex are important. Fig. 6 (left) displays an example of IP measurements for gammas attached to primary or secondary vertices. Another crucial point is the decay search and the charm production analysis. First of all because charm decays exhibit the same topology as τ decays, therefore measuring the charm-like event reconstruction efficiency provides an important cross-check of the τ event reconstruction capability. Furthermore charm events are a potential source of background, in particular if the muon at the primary vertex is not identified. Therefore, searching for charm-decays in events with the primary muon correctly identified provides a direct measurement of this background. Charm decay topologies are searched for in the sample of located neutrino interactions (Fig. 6, right panel). At present a set of 15 charm candidates has been identified and is deeply studied. A new systematic decay search procedure has been recently endorsed and is being applied to increase the understanding of that important data sample. A constant effort is brought on test beam programmes to validate the kinematics measurement by Multiple Coulomb Scattering (MCS). Tests with pion beams of different energies gave a good data/MC agreement showing the validity of the procedure (Fig. 7)

Complementary studies

On top of the oscillations physics, OPERA is also measuring the charged ratios (µ + /µ -) of cosmic muons, exploiting the muon spectrometer magnetic field. An independant study of atmospheric neutrinos detection, using a time-of-flight analysis, is also performed and the up/down analysis has been shown to be efficient although the OPERA configuration is optimized for horizontal events topologies. Finally a measurement is also done in parallel of the neutrino velocity after a detailed intercalibration of the LNGS and CERN timing systems. Publications of those results are in preparation.

Conclusions

The analysis of the events collected in 2008-2009 is quite advanced. The results obtained prove that OPERA is able to accomplish the task of selecting decay topologies in the emulsions from a large number of interactions triggered by the electronic detectors and that the scene has been set for the discovery of ν τ appearance. The large data set recorded and under analysis allows a constant and systematic improvement of the detection procedures and efficiencies. Important data samples such as the charm decays are carefully studied since they provide crucial information in the τ appearance hunting. At the end of the 2009 run around 2 τ neutrinos events are expected...

Fig. 1 .

 1 Fig. 1. Integrated potdelivery versus time during 2008 run (left). Protons per extraction during 2009 run (right).

Fig. 2 .

 2 Fig. 2. Pictures of one of the ESS microscopes (left) and of the S-UTS (right).

Fig. 3 .

 3 Fig. 3. Side view of the OPERA detector. The two target modules, 31 bricks walls spaced by black covered scintillator biplanes, are separated by the first spectrometer, 6 stations of drift tubes upstream, between and downstream the two arms of a dipolar magnet instrumented with RPC.

Fig. 4 .

 4 Fig. 4. TT analysis: muon momentum reconstruction (left) and visible energy (right).

Fig. 5 .

 5 Fig. 5. 2008 events scanning progress: number of vertex found (left) and vertex spatial distribution (right).

Fig. 6 .

 6 Fig. 6. IP measurements for γ reconstruction in the emulsions (left). Charm decay candidate (right).

Fig. 7 .

 7 Fig. 7. Pions energy reconstruction with MCS.


Luciana Benotti
email: luciana.benotti@loria.fr

Frolog: an accommodating text-adventure game

Frolog is a text-adventure game whose goal is to serve as a laboratory for testing pragmatic theories of accommodation. To this end, rather than implementing ad-hoc mechanisms for each task that is necessary in such a conversational agent, Frolog integrates recently developed tools from computational linguistics, theorem proving and artificial intelligence planning.

Introduction

If we take a dialogue perspective on Lewis ' (1979) notion of accommodation and assume that the state of a dialogue is changed by the acts performed by the dialogue participants, it is natural to interpret Lewis' broad notion of accommodation as tacit (or implicit) dialogue acts. This is the approach adopted by [START_REF] Kreutel | Context-dependent interpretation and implicit dialogue acts[END_REF] who formalize the treatment of tacit dialogue acts in the information state update framework. According to them, accommodation is ruled by the following principle:

Context Accommodation (CA): For any move m that ocurrs in a given scenario sc i : if assignment of a context-dependent interpretation to m in sc i fails, try to accommodate sc i to a new context sc i+1 in an appropriate way by assuming implicit dialogue acts performed in m, and start interpretation of m again in sc i+1 .

The authors concentrate on the treatment of implicit acceptance acts but suggest that the CA principle can be seen as a general means of contextdependent interpretation. This principle opens up the question of how to find the appropriate tacit dialogue acts. Finding them is an inference problem that is addressed using special-purpose algorithms in [START_REF] Thomason | Enlightened update: A computational architecture for presupposition and other pragmatic phenomena[END_REF], where the authors present a unified architecture for both contextdependent interpretation and context-dependent generation. In Frolog, we investigate how this inference process can be implemented using recent tools from artificial intelligence planning.

The resulting framework naturally lends itself to studying the pressing problem for current theories of accommodation called missing accommodation [START_REF] Beaver | Accommodation[END_REF]. These theories can neither explain why accommodation is sometimes easier and sometimes much more difficult, nor how cases of missing accommodation relate to clarification subdialogues in conversation. We review what Frolog has to offer to the understanding of accommodation in general and missing accommodation in particular in Section 3. But first, we have to introduce Frolog and describe its components, and we do so in Section 2.

The text-adventure game

Text-adventures are computer games that simulate a physical environment which can be manipulated by means of natural language requests. The game provides feedback in the form of natural language descriptions of the game world and of the results of the players' actions.

Frolog is based on a previous text-adventure called FrOz [START_REF] Koller | Put my galakmid coin into the dispenser and kick it: Computational linguistics and theorem proving in a computer game[END_REF]) and its design is depicted in Figure 1. The architecture is organized in three natural language understanding (NLU) modules and three natural language generation (NLG) modules, and the state of the interaction is represented in two knowledge bases (KBs). The two KBs codify, in Description Logic [START_REF] Baader | The Description Logic Handbook: Theory, Implementation, and Applications[END_REF], assertions and concepts relevant for a given game scenario. The game KB represents the true state of the game world, while the player KB keeps track of the player's beliefs about the game world. Frolog's modules are scenario-independent; the player can play different game scenarios by plugging in the different information resources that constitute the scenario.

Frolog uses generic external tools for the most heavy-loaded tasks (depicted in grey in Figure 1); namely, a generic parser and a generic realizer for parsing and realization, an automated theorem prover for knowledge base management, and artificial intelligence planners for implementing its accommodating capabilities. The rest of the modules (depicted in white) were implemented by us in Prolog and Java. Frolog's interface shows the interaction with the player, the input/output of each module and the content of the KBs.

We now present Frolog's modules in pairs of an NLU module and its NLG counterpart; each pair uses a particular kind of information resource and has analogous input/output.

Parsing and Realization

The parsing and the realization modules use the same linguistic resources, namely a reversible grammar, a lemma lexicon and a morphological lexicon represented in the XMG grammatical formalism [START_REF] Crabbé | Metagrammar redux[END_REF]. The XMG grammar used specifies a Tree Adjoining Grammar (TAG) of around 500 trees and integrates a semantic dimension à la [START_REF] Gardent | Integrating a unification-based semantics in a large scale lexicalised tree adjoininig grammar for french[END_REF]). An example of the semantics associated with the player input "open the chest" is depicted in Figure 2. The parsing module performs the syntactic analysis of a command issued by the player, and constructs its semantic representation using the TAG parser Tulipa [START_REF] Kallmeyer | TuLiPA: Towards a multi-formalism parsing environment for grammar engineering[END_REF] (illustrated in the Figure 2 by ⇓). The realization module works in the opposite direction, verbalizing the results of the execution of the command from the semantic representation using the TAG surface realizer GenI [START_REF] Gardent | A symbolic approach to near-deterministic surface realisation using tree adjoining grammar[END_REF] (illustrated in the Figure 2 by ⇑).

Reference Resolution and Reference Generation

The reference resolution (RR) module is responsible for mapping the semantic representations of definite and indefinite noun phrases and pronouns to individuals in the knowledge bases (illustrated in Figure 3 by ⇓). The reference generation (RG) module performs the inverse task, that is it generates the semantic representation of a noun phrase that uniquely identifies an individual in the knowledge bases (illustrated in the Figure 3 by ⇑). The algorithms used for RR and RG are described in [START_REF] Koller | Put my galakmid coin into the dispenser and kick it: Computational linguistics and theorem proving in a computer game[END_REF]. [START_REF] Haarslev | RACER system description[END_REF] to query the KBs and perform RR and RG. In order to manage the ambiguity of referring expressions two levels of saliency are considered. The player KB is queried (instead of the game KB) naturally capturing the fact that the player will not refer to individuals he doesn't know about (even if they exist in the game KB). Among the objects that the player already knows, a second level of saliency is modelled employing a simple stack of discourse referents which keeps track of the most recently referred individuals. A new individual gets into the player KB when the player explores the world.

Action Execution and Content Determination

These two last modules share the last information resource that constitute an scenario, namely, the action database. The action database includes the definitions of the actions that can be executed by the player (such as take or open). Each action is specified as a STRIPS-like operator [START_REF] Fikes | Learning and executing generalized robot plans[END_REF] detailing its arguments, preconditions and effects as illustrated below. The arguments show the thematic roles of the verb (for instance, the verb open requires a patient and an agent), the preconditions indicate the conditions that the game world must satisfy so that the action can be executed (for instance, in order to open the chest, it has to be accessible, unlocked and closed); the effects determine how the action changes the game world when it is executed (after opening the chest, it will be open).

action: open(E) agent(E,A) patient(E,P) preconditions: accessible(P), not(locked(P)), closed(P) effects: opened(P)

Executing a player's command amounts to verifying whether the preconditions of the actions involved by the command hold in the game world and, if they do, changing the game KB according to the effects. After the command is executed, the content determination module constructs the semantic representation of the effects that were applied, updates the player KB with it and passes it to the next module for its verbalization (so that the player knows what changed in the world). For our running example the following modules will verbalize "the chest is open" closing a complete cycle of the system as illustrated in Figure 1.

If a precondition of an action does not hold then Frolog tries to accommodate as we will explain in following section.

Accommodation in Frolog

In the previous section we presented the execution of the system when everything "goes well", that is (to come back to the terminology used in Section 1) when the assignment of a contextdependent interpretation to the player's move succeeds. However, during the interaction with Frolog, it often happens that the player issues a command that cannot be directly executed in the current state of the game but needs accommodation or clarification. This is the topic of the next two subsections.

Tacit acts are inferable and executable: accommodation succeeds

Suppose that the player has just locked the little chest and left its key on the table when she realizes that she forgot to take the sword from it, so she utters "open the chest". If Frolog is in its nonaccommodating mode then it answers "the chest is locked" because the precondition not(locked(P)) does not hold in the game world. In this mode, the interactions with the game can get quite long and repetitive as illustrated below. In its accommodating mode, Frolog tries to accommodate the current state sc i of the game to a new state sc i+1 in which the precondition hold, by assuming tacit dialogue acts performed, and starts the interpretation of the command again in sc i+1 . That is, the game assumes that "take the key and unlock the chest with it" are tacit acts that are performed when the player says "open the chest".

The inference of such tacit dialogue acts is done using artificial intelligence planners. The planning problems are generated on the fly during a game each time a precondition does not hold; the initial state being the player KB, the goal being the precondition that failed, and the action schemas those actions available in the action database. The size of the plans can be configured, when the length is zero we say that Frolog is in its nonaccommodating mode. For detailed discussion of the subtleties involved in the kind of information that has to be used to infer the tacit acts see [START_REF] Benotti | Incomplete knowledge and tacit action: Enlightened update in a dialogue game[END_REF].

Two planners have been integrated in Frolog (the player can decide which one to use): Blackbox [START_REF] Kautz | Unifying SAT-based and graph-based planning[END_REF] which is fast and deterministic and PKS [START_REF] Petrick | Extending the knowledge-based approach to planning with incomplete information and sensing[END_REF] which can reason over non-deterministic actions. For detailed discussion and examples including non-deterministic actions see [START_REF] Benotti | Accommodation through tacit sensing[END_REF].

Accommodation fails: clarification starts

Tacit acts are inferred using the information available to the player (the player KB) but their execution is verified with respect to the accurate and complete state of the world (the game KB). So Frolog distinguishes three ways in which accommodation can fail: there is no plan, there is more than one plan, or there is a plan which is not executable in the game world. For reasons of space we will only illustrate the last case here.

Suppose that the golden key, which was lying on the table, was taken by a thief without the player knowing. As a consequence, the key is on the table in the player KB, but in the game KB the thief has it. In this situation, the player issues the command "Open the chest" and the sequence of tacit acts inferred (given the player beliefs) is "take the key from the table and unlock the chest with it". When trying to execute the tacit acts, the game finds the precondition that does not hold and verbalizes it with "the key is not on the table, you don't know where it is". Such answer can be seen as a clarification request (CR), it has the effect of assigning to the player the responsability of finding the key before trying to open the chest. The same responsability that would be assigned by more commonly used CR that can happen in this scenario, namely "Where is the key?".

In the game, such clarifications vary according to the knowledge that is currently available to the player. If the player knows that the dragon has the key and she can only take it while the dragon is asleep an answer such as "the dragon is not sleeping" is generated in the same fashion.

Conclusion and future work

In this paper we have presented a text-adventure game which is an interesting test-bed for experimenting with accommodation. The text-adventure framework makes evident the strong relation between accommodation and clarification (which is not commonly studied), highlighting the importance of investigating accommodation in dialogue and not in isolation.

Our work is in its early stages and can be advanced in many directions. We are particularly interested in modifying the architecture of the system in order to model reference as another action instead of preprocessing references with specialpurpose algorithms. In this way we would not only obtain a more elegant architecture, but also be able to investigate the interactions between reference and other kinds of actions, which occur in every-day conversations.

Figure 1 :

 1 Figure 1: Architecture of Frolog

  Figure 2: Parsing/realization for "open the chest"

  Figure 3: RR/RG for "the little chest on the table"

		⇑	⇓
	chest little	has-location		table
	big chest	h a s -l o c a ti o n		little chest
	Frolog	uses	the	theorem	prover
	RACER

det(C), chest(C), little(C), has-location(C,T), table(T)


Holger Schwenk
email: schwenk@lium.univ-lemans.fr

Sadaf Abdul-Rauf

Loïc Barrault
email: barrault@lium.univ-lemans.fr

Jean Senellart
email: senellart@systran.fr

SMT and SPE Machine Translation Systems for WMT'09

This paper describes the development of several machine translation systems for the 2009 WMT shared task evaluation. We only consider the translation between French and English. We describe a statistical system based on the Moses decoder and a statistical post-editing system using SYSTRAN's rule-based system. We also investigated techniques to automatically extract additional bilingual texts from comparable corpora.

Introduction

This paper describes the machine translation systems developed by the Computer Science laboratory at the University of Le Mans (LIUM) for the 2009 WMT shared task evaluation. This work was performed in cooperation with the company SYSTRAN. We only consider the translation between French and English (in both directions). The main differences to the previous year's system [START_REF] Schwenk | First steps towards a general purpose French/English statistical machine translation system[END_REF] are as follows: better usage of SYSTRAN's bilingual dictionary in the statistical system, less bilingual training data, additional language model training data (news-train08 as distributed by the organizers), usage of comparable corpora to improve the translation model, and development of a statistical post-editing system (SPE). These different components are described in the following.

Used Resources

In the frame work of the 2009 WMT shared translation task many resources were made available. The following sections describe how they were used to train the translation and language models of the systems.

Bilingual data

The latest version of the French/English Europarl and news-commentary corpus were used. We realized that the first corpus contains parts with foreign languages. About 1200 such lines were excluded.1 Additional bilingual corpora were available, namely the Canadian Hansard corpus (about 68M English words) and an UN corpus (about 198M English words). In several initial experiments, we found no evidence that adding this data improves the overall system and they were not used in the final system, in order to keep the phrase-table small. We also performed experiments with the provided so-called bilingual French/English Gigaword corpus (575M English words in release 3). Again, we were not able to achieve any improvement by adding this data to the training material of the translation model. These findings are somehow surprising since it was eventually believed by the community that adding large amounts of bitexts should improve the translation model, as it is usually observed for the language model [START_REF] Brants | Large language models in machine translation[END_REF].

In addition to these human generated bitexts, we also integrated a high quality bilingual dictionary from SYSTRAN. The entries of the dictionary were directly added to the bitexts. This technique has the potential advantage that the dictionary words could improve the alignments of these words when they also appear in the other bitexts. However, it is not guaranteed that multi-word expressions will be correctly aligned by GIZA++ and that only meaningful translations will actually appear in the phrase-table. A typical example is fire engine -camion de pompiers, for which the individual constituent words are not good translations of each other. The use of a dictionary to improve an SMT system was also investigated by ( [START_REF] Peter | But dictionaries are data too[END_REF].

In comparison to our previous work [START_REF] Schwenk | First steps towards a general purpose French/English statistical machine translation system[END_REF], we also included all verbs in the French subjonctif and passé simple tense. In fact, those tenses seem to be frequently used in news material. In total about 10,000 verbs, 1,500 adjectives/adverbs and more than 100,000 noun forms were added.

Use of Comparable corpora

Available human translated bitexts such as the UN and the Hansard corpus seem to be out-of domain for this task, as mentioned above. Therefore, we investigated a new method to automatically extract and align parallel sentences from comparable indomain corpora. In this work we used the AFP news texts since there are available in the French and English LDC Gigaword corpora.

The general architecture of our parallel sentence extraction system is shown in figure 1. We first translate 174M words from French into English using an SMT system. These English sentences are then used to search for translations in the English AFP texts of the Gigaword corpus using information retrieval techniques. The Lemur toolkit [START_REF] Ogilvie | Experiments using the Lemur toolkit[END_REF] was used for this purpose. Search was limited to a window of ±5 days of the date of the French news text. The retrieved candidate sentences were then filtered using the word error rate with respect to the automatic translations. In this study, sentences with an error rate below 32% were kept. Sentences with a large length difference (French versus English) or containing a large fraction of numbers were also discarded. By these means, about 9M words of additional bitexts were obtained. An improved version of this algorithm using TER instead of the word error rate is described in detail in [START_REF] Abdul | On the use of comparable corpora to improve SMT performance[END_REF].

Monolingual data

The French and English target language models were trained on all provided monolingual data. We realized that the news-train08 corpora contained some foreign texts, in particular in German. We tried to filter those lines using simple regular expressions. We also discarded lines with a large fraction of numerical expressions. In addition, LDC's Gigaword collection, the Hansard corpus and the UN corpus were used for both languages. Finally, about 30M words crawled from the WEB were used for the French LM. All this data predated the evaluation period.

Development data

All development was done on news-dev2009a and news-dev2009b was used as internal test set. The default Moses tokenization was used. All our models are case sensitive and include punctuation. The BLEU scores reported in this paper were calculated with the NIST tool and are case sensitive.

Language Modeling

Language modeling plays an important role in SMT systems. 4-gram back-off language models (LM) were used in all our systems. The word list contains all the words of the bitext used to train the translation model and all words that appear at least ten times in the news-train08 corpus. Separate LMs were build on each data source with the SRI LM toolkit [START_REF] Stolcke | SRILM -an extensible language modeling toolkit[END_REF]

Architecture of the SMT system

The goal of statistical machine translation (SMT) is to produce a target sentence e from a source sentence f . It is today common practice to use phrases as translation units [START_REF] Koehn | Statistical phrased-based machine translation[END_REF][START_REF] Franz | A systematic comparison of various statistical alignement models[END_REF] and a log linear framework in order to introduce several models explaining the translation process:

e * = arg max p(e|f ) = arg max e {exp( i λ i h i (e, f ))} (1)
The feature functions h i are the system models and the λ i weights are typically optimized to maximize a scoring function on a development set [START_REF] Franz | Discriminative training and maximum entropy models for statistical machine translation[END_REF]. In our system fourteen features functions were used, namely phrase and lexical translation probabilities in both directions, seven features for the lexicalized distortion model, a word and a phrase penalty and a target language model (LM).

The system is based on the Moses SMT toolkit [START_REF] Koehn | Moses: Open source toolkit for statistical machine translation[END_REF] and constructed as follows. First, word alignments in both directions are calculated. We used a multi-threaded version of the GIZA++ tool [START_REF] Gao | Parallel implementations of word alignment tool[END_REF].2 This speeds up the process and corrects an error of GIZA++ that can appear with rare words. This previously caused problems when adding the entries of the bilingual dictionary to the bitexts.

Phrases and lexical reorderings are extracted using the default settings of the Moses toolkit. The parameters of Moses are tuned on news-dev2009a, using the cmert tool. The basic architecture of the system is identical to the one used in the 2008 WMT evaluation [START_REF] Schwenk | First steps towards a general purpose French/English statistical machine translation system[END_REF]), but we did not use two pass decoding and n-best list rescoring with a continuous space language model.

The results of the SMT systems are summarized in the upper part of Table 1 and3. The dictionary and the additional automatically produced AFP bitexts achieved small improvements when translating from French to English. In the opposite translation direction, the systems that include the additional AFP texts exhibit a bad generalisation behavior. We provide also the performance of the different systems on the official test set, calculated after the evaluation. In most of the cases, the observed improvements carry over on the test set.

Architecture of the SPE system

During the last years statistical post-editing systems have shown to achieve very competitive performance [START_REF] Simard | Rule-based translation with statistical phrase-based post-editing[END_REF][START_REF] Dugast | Statistical post-editing on SYSTRAN's rule-based translation system[END_REF] Table 3: Case sensitive NIST BLEU scores for the English-French systems. "NC" denotes the newscommentary bitexts, "dict" denotes SYSTRAN's bilingual dictionary and "AFP" the automatically aligned news texts ( * =primary, # =contrastive system)

an SMT system to correct the errors of a rulebased translation system. In this work, SYSTRAN server version 6, followed by an SMT system based on Moses were used. The post-editing systems uses exactly the same language models than the above described stand-alone SMT systems.

The translation model was trained on the Europarl, the news-commentary and the extracted AFP bitexts. The results of these SPE systems are summarized in the lower part of Table 1 and3. SYS-TRAN's rule-based system alone already achieves remarkable BLEU scores although it was not optimized or adapted to this task. This could be significantly improved using statistical post-editing. The additional AFP texts were not useful when translating form French to English, but helped to improve the generalisation behavior for the English/French systems. When translating from English to French (Table 3), the SPE system is clearly better than the carefully optimized SMT system. Consequently, it was submitted as primary system and the SMT system as contrastive one.

Conclusion and discussion

We described the development of two complementary machine translation systems for the 2009 WMT shared translation task: an SMT and an SPE system. The last one is based on SYSTRAN's rule-based system. Interesting findings of this research include the fact that the SPE system outperforms the SMT system when translating into French. This system has also obtained the best scores in the human evaluation.

With respect to the SMT system, we were not able to improve the translation model by adding large amounts of bitexts, although different sources were available (Canadian Hansard, UN or WEB data). Eventually these corpora are too noisy or out-of-domain. On the other hand, the integration of a high quality bilingual dictionary was helpful, as well as the automatic alignment of news texts from comparable corpora.

Future work will concentrate on the integration of previously successful techniques, in particular continuous space language models and lightlysupervised training [START_REF] Schwenk | Investigations on largescale lightly-supervised training for statistical machine translation[END_REF]. We also believe that the tokenization could be improved, in particular for the French sources texts. Numbers, dates and other numerical expressions could be translated by a rule-based system.

System combination has recently shown to provide important improvements of translation quality. We are currently working on a combination of the SMT and SPE system. It may be also interesting to add a third (hierarchical) MT system.

Figure 1 :

 1 Figure 1: Architecture of the parallel sentence extraction system (Rauf and Schwenk, 2009).

Table 1 :

 1 and then linearly interpolated, optimizing the coefficients with an EM procedure. The perplexities of these LMs Case sensitive NIST BLEU scores for the French-English systems. "NC" denotes the newscommentary bitexts, "dict" SYSTRAN's bilingual dictionary and "AFP" the automatically aligned news texts ( * =primary, # =contrastive system) are given in Table2. Adding the new news-train08 monolingual data had an important impact on the quality of the LM, even when the Gigaword data is already included.

	Corpus		# Fr words Dev09a Dev09b Test09
	SMT system
	Eparl+NC		46.5M	22.44	22.38	25.60
	Eparl+NC+dict		48.5M	22.60	22.55	26.01
	Eparl+NC+dict+AFP	57.8M	22.82	22.63 *	26.18
	SPE system
	SYSTRAN		-	17.76	18.13	19.98
	Eparl+NC		45.5M	22.84	22.59 #	25.59
	Eparl+NC+AFP		54.4M	22.72	21.96	25.40
	Data French English
	Vocabulary size	407k	299k
	Eparl+news 248.8	416.7
	+ LDC Gigaword 142.2	194.9
	+ Hansard and UN 137.5	187.5
	news-train08 alone 165.0	245.9
	all 120.6	174.8

Table 2 :

 2 Perplexities on the development data of various language models.

  . The main idea of this techniques is to use

	Corpus	# En words Dev09a Dev09b Test09
	SMT system
	Eparl+NC	41.6M	21.89	21.78	23.80
	Eparl+NC+dict	44.0M	22.28	22.35 #	24.13
	Eparl+NC+dict+AFP	51.7M	22.21	21.43	23.88
	SPE system
	SYSTRAN	-	18.68	18.84	20.29
	Eparl+NC	44.2M	23.03	23.15	24.36
	Eparl+NC+AFP	53.3M	22.95	23.15 *	24.62

Lines 580934-581316 and 599839-600662.

The source is available at http://www.cs.cmu. edu/ ˜qing/

Acknowledgments

This work has been partially funded by the French Government under the project INSTAR (ANR JCJC06 143038) and the by the Higher Education Commission, Pakistan through the HEC Overseas Scholarship 2005.


N Mellouli
email: mellouli@math.univ-lyon1.fr

Conformally equivariant second-order differential operators in dimension 1|2: Quantization and symbol calculus

This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle S 1|2 equipped with the standard contact structure. The conformal Lie superalgebra K(2) of contact vector fields on S 1|2 contains the Lie superalgebra osp(2|2). We study the spaces of linear differential operators on the spaces of weighted densities as modules over osp(2|2). We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as osp(2|2)-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.

Introduction and the main results

The concept of equivariant quantization first appeared in [START_REF] Lecomte | Projectively invariant symbol calculus[END_REF] and [START_REF] Duval | Conformally equivariant quantization: existence and uniqueness[END_REF]. The general idea is to identify, in a canonical way, the space of linear differential operators on a manifold acting on weighted densities with the corresponding space of symbols. Such an identification is called a quantization (or symbol) map. It turns out that for an arbitrary projectively/conformally flat manifold, there exists a unique quantization map commuting with the action of the group of projective/conformal transformations.

Equivariant quantization on supermanifolds was initiated by [START_REF] Cohen | Automorphic pseudodifferential operators, Algebraic aspects of integrable systems[END_REF] and further investigated in [START_REF] Gargoubi | Differential operators on supercircle: conformally equivariant quantization and symbol calculus[END_REF]. In these works, the authors considered supermanifolds of dimension 1|1. Already in this simplest case, the calculations are much more difficult than in the case of This is in part due to the fact that, in the super cases considered, one has to take into account a non-integrable distribution, namely the contact structure, see [START_REF] Leites | Supermanifold theory[END_REF][START_REF] Grozman | Lie superalgebras of string theories[END_REF], dubbed "SUSY" structure in [START_REF] Cohen | Automorphic pseudodifferential operators, Algebraic aspects of integrable systems[END_REF].

In this paper, we consider the space of linear differential operators on the supercircle S 1|2 acting from the space of λ-densities to the space of µ-densities, where λ and µ are (real or complex) numbers. This space of operators is naturally a module over the Lie superalgebra of contact vector fields (see Section 3 below) also known as the stringy superalgebra K(2), see [START_REF] Grozman | Lie superalgebras of string theories[END_REF]. We denote these modules by D λ,µ S 1|2 .

Our main result concerns the spaces containing second-order differential operators, D λµ S 1|2 is contained in D 2 λµ S 1|2 , so we will interest to the space D 2 λµ S 1|2 . This space is not isomorphic to the corresponding space of symbols, as a K(2)-module. The obstructions to the existence of such an isomorphism are given by (the infinitesimal version of) the Schwarzian derivative, see [START_REF] Michel | On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative[END_REF] and references therein. We thus restrict the module structure on D 2 λµ S 1|2 to the orthosymplectic Lie superalgebra osp (2|2) naturally embedded to K(2).

The main result of this paper is as follows.

Theorem 1.1. (i) The space D 2 λµ S 1|2 and the corresponding space of symbols are isomorphic as an osp (2|2)-modules, provided µ -λ = 0, 1 2 , 1, 3 2 , 2. (ii) The above isomorphism is unique.

The particular values of λ and µ such that µ -λ ∈ {0, 1 2 , 1, 3 2 , 2} are called resonant. We do not study here the corresponding "resonant modules" of differential operators. Note that these modules are of particular interest and deserve further study.

We think that a similar result holds for the space of differential operators of arbitrary order, but such a result is out of reach so far. We would like to mention however, that most of the known interesting examples of differential operators in geometry and mathematical physics are of order 2. This allows one to expect concrete applications of the above theorem.

Geometry of the supercircle S 1|2

The supercircle S 1|2 is a supermanifold of dimension 1|2 which generalizes the circle S 1 . To fix the notation, let us give the basic definitions of geometrical objects on S 1|2 , see [START_REF] Leites | Supermanifold theory[END_REF][START_REF] Grozman | Lie superalgebras of string theories[END_REF] for more details.

We define the supercircle S 1|2 by describing its graded commutative algebra of (complex valued) functions that we note by C ∞ S 1|2 , consisting of the elements

f (x, ξ 1 , ξ 2 ) = f 0 (x) + ξ 1 f 1 (x) + ξ 2 f 2 (x) + ξ 1 ξ 2 f 12 (x), (2.1)
where x the Fourier image of the angle parameter on S 1 and ξ 1 , ξ 2 are odd Grassmann coordinates, i.e., ξ 2 i = 0 and where f 0 , f 12 , f 1 , f 2 ∈ C ∞ (S) are functions with complex values. We define the parity function p by setting p (x) = 0 and p (ξ i ) = 1.

Vector fields and differential forms

Any vector field on S 1|2 is a derivation of the algebra C ∞ S 1|2 , it can be expressed as

X = f ∂ x + g 1 ∂ ξ 1 + g 2 ∂ ξ 2 (2.2) with f, g i ∈ C ∞ S 1|2 , ∂ x = ∂ ∂x and ∂ ξ i = ∂ ∂ξ i , for i = 1, 2.
The space of vector fields on S 1|2 is a Lie superalgebra which we note by Vect S 1|2 .

Any differential form is a skew-symmetric multi-linear map (over C ∞ S 1|2 ) from Vect S 1|2 to C ∞ S 1|2 . To fix the notation, we set ∂ u i , du j = δ ij , for u = (x, ξ 1 , ξ 2 ). The space of differential forms Ω 1 S 1|2 is a right C ∞ S 1|2 -module and a left Vect S 1|2module, the action being given by the Lie derivative, i.e., X, L Y α := [X, Y ] , α for any Y, X ∈ Vect S 1|2 , α ∈ Ω 1 S 1|2 .

The Lie superalgebra of contact vector fields

The standard contact structure on S 1|2 is defined by the data of a linear distribution D 1 , D 2 on S 1|2 generated by the odd vector fields

D 1 = ∂ ξ 1 -ξ 1 ∂ x , D 2 = ∂ ξ 2 -ξ 2 ∂ x . (2.3)
This contact structure can also be defined as the kernel of the differential 1-form:

α = dx + ξ 1 dξ 1 + ξ 2 dξ 2 .
(2.4)

We refer to [START_REF] Shchepochkina | How to realize Lie algebras by vector fields[END_REF] for more details.

A vector field X on S 1|2 is called a contact vector field if it preserves the contact distribution, that is, satisfies the condition:

X, D 1 = ψ 1 X D 1 + ψ 2 X D 2 , X, D 2 = φ 1 X D 1 + φ 2 X D 2 (2.5)
where

ψ 1 X , ψ 2 X , φ 1 X , φ 2 X ∈ C ∞ S 1|2
are functions depending on X . The space of the contact vector fields is a Lie superalgebra which we note by K (2).

The following fact is well-known.

Lemma 2.1. Every contact vector field can be expressed, for some function f ∈ C ∞ S 1|2 , by

X f = f ∂ x -(-1) p(f ) 1 2 D 1 (f ) D 1 + D 2 (f ) D 2 (2.6)
where

D i = ∂ ξ i + ξ i ∂ x ; i = 1, 2.
The function f is said to be a contact Hamiltonian of the field X f . The space C ∞ S 1|2 is therefore identified with the Lie superalgebra K (2) and equipped with the structure of Lie superalgebra with respect to the contact bracket:

{f, g} = f g ′ -f ′ g -(-1) p(f ) 1 2 D 1 (f ) D 1 (g) + D 2 (f ) D 2 (g) , (2.7)
where

f ′ = ∂ x (f ).

Conformal symmetry: The orthosymplectic algebra

The conformal (or projective) structure on the supercircle S 1|2 (see [START_REF] Ovsienko | Classification of contact-projective structures on the supercircle[END_REF]) is defined by the action of the 4|4-dimensional Lie superalgebra osp (2|2). This action is spanned by the contact vector fields with the contact Hamiltonians:

1, x, x 2 , ξ 1 ξ 2 ; ξ 1 , ξ 2 , xξ 1 , xξ 2 .
The embedding of osp (2|2) into K (2) is then given by (2.6).

The subalgebra Aff (2|2) of osp (2|2) spanned by the contact vector fields with the contact Hamiltonians {1, x, ξ 1 ξ 2 ; ξ 1 , ξ 2 } will be called the affine Lie superalgebra.

Modules of weighted densities

We introduce a family of K (2)-modules with a parameter. For any contact vector field, we define a family of differential operators of order one on

C ∞ S 1|2 L λ X f := X f + λf ′ , (2.8)
where the parameter λ is an arbitrary (complex) number and the function is considered as a 0-order differential operator of left multiplication by this function. The map

X f → L λ X f
is a homomorphism of Lie superalgebras. We thus obtain a family of K (2)-modules on C ∞ S 1|2 that we note by F λ S 1|2 and that we call spaces of weighted densities of weight λ.

Viewed as vector spaces, but not as K (2)-modules, the spaces

F λ S 1|2 are isomorphic to C ∞ S 1|2 .
The space of weighted densities possesses a Poisson superalgebra structure with respect to the contact bracket {., .} :

F λ S 1|2 ⊗ F µ S 1|2 → F λ+µ+1 S 1|2 given explicitly by {f, g} = µ f ′ g -λ f g ′ -(-1) p(f ) 1 2 D 1 (f ) D 1 (g) + D 2 (f ) D 2 (g) (2.9) for all f ∈ F λ S 1|2 , g ∈ F µ S 1|2 . Note that if f ∈ F -1 S 1|2 , then {f, g} = L µ X f (g). The space F -1 S 1|2 is a subalgebra isomorphic to K (2), see formula (2.7).

Differential operators on the spaces of weighted densities

In this section we introduce the space of differential operators acting on the spaces of weighted densities and the corresponding space of symbols on S 1|2 . We refer to [START_REF] Leites | New invariant differential operators on supermanifolds and pseudo-(co)homology[END_REF][START_REF] Grozman | Invariant operators on supermanifolds and standard models[END_REF][START_REF] Gargoubi | Differential operators on supercircle: conformally equivariant quantization and symbol calculus[END_REF][START_REF] Conley | Conformal symbols and the action of contact vector fields over the superline[END_REF] for further details. This space is naturally a module over the Lie superalgebra K(2). We also define a K(2)-invariant "finer filtration" on the modules of differential operators that plays the key role in this paper. The graded K(2)-mudule associated to the finer filtration is called the module of symbols.

Definition of the modules

D (k) λµ S 1|2
Let D λµ S 1|2 be the space of linear differential operators A :

F λ S 1|2 → F µ S 1|2 .
This space is naturally filtered:

D (0) λµ S 1|2 ⊂ D (1) λµ S 1|2 ⊂ • • • ⊂ D (k-1) λµ S 1|2 ⊂ D (k) λµ S 1|2 ⊂ • • • where D (k)
λµ S 1|2 is the space of linear differential operators of order k. The space D λµ S 1|2 and every subspace D (k) λµ S 1|2 is naturally a module over the Lie superalgebra of contact vector fields K (2). The above filtration is of course K (2)invariant. Note that, in the case λ = µ = 0, the space of differential operators is a module over the full Lie superalgebra Vect S 1|2 and the above filtration is Vect S 1|2 -invariant.

The finer filtration: Modules

D k λµ S 1|2
It turns out that there is another, finer filtration:

D 0 λµ S 1|2 ⊂ D 1 2 λµ S 1|2 ⊂ D 1 λµ S 1|2 ⊂ D 3 2 λµ S 1|2 ⊂ D 2 λµ S 1|2 ⊂ • • • (3.1)
on the space of differential operators on S 1|2 . This finer filtration is invariant with respect to the action of K (2) (but it cannot be invariant with respect to the action of the full algebra of vector fields).

Proposition 3.1. Every differential operator can be expressed in the form

A = ℓ,m,n a ℓ,m,n (∂ x ) ℓ D m 1 D n 2 , (3.2)
where a ℓ,m,n ∈ C ∞ S 1|2 , the index ℓ is arbitrary while m, n ≤ 1, and where only finitely many terms are non-zero.

Proof.

If A ∈ D k λµ S 1|2 , then A = a ℓ,m,n (∂ x ) ℓ ∂ m ξ 1 ∂ n ξ 2
and since one has:

∂ ξ 1 = D 1 + ξ 1 ∂ x , ∂ ξ 2 = D 2 + ξ 2 ∂ x , D 2 i = -∂ x ,
we have the form desired.

For every (half)-integer k, we denote by D k λµ S 1|2 the space of differential operators of the form

A = ℓ+2m+2n≤k a ℓ,m,n (∂ x ) ℓ D m 1 D n 2 (3.3)
where a ℓ,m,n ∈ C ∞ S 1|2 . Furthermore, since

∂ x = -D 2 1 = -D 2 2
, we can assume m, n ≤ 1.

Proposition 3.2. The form (3.3) is stable with respect to the action of K (2).

Proof. Let X f be a contact vector field, see formula (2.3). The action of X f on the space D λµ S 1|2 is given by

L X f (A) = L µ X f • A -(-1) p(f )p(A) A • L λ X f , (3.4)
where L λ X f is the Lie derivative (2.9). The invariance of the form (3.3) is subject to a straightforward calculation. λµ S 1|2 ⊂ D k λµ S 1|2 but these modules do not coincide. Indeed, the module D k λµ S 1|2 contains terms proportional to ∂ k x D 1 D 2 which are, of course, of order k + 1. An element of D k λµ S 1|2 will be called k-differential operator, it does not have to be of order k, it can be of order k + 1.

2) A similar finer filtration exists for an arbitrary contact manifold, cf. [START_REF] Ovsienko | Vector fields in the presence of a contact structure[END_REF][START_REF] Fregier | Decomposition of symmetric tensor fields in the presence of a flat contact projective structure[END_REF]. In the 1|1-dimensional case this finer filtration was used in [START_REF] Gargoubi | Differential operators on supercircle: conformally equivariant quantization and symbol calculus[END_REF].

Example 3.4. The module D 2 λµ S 1|2 will be particularly interesting for us. Every differential operator A ∈ D 2 λµ S 1|2 can be expressed in the form A = a 0,0,0

+ a 0,1,0 D 1 + a 0,0,1 D 2 + a 1,0,0 ∂ x + a 0,1,1 D 1 D 2 + a 1,1,0 ∂ x D 1 + a 1,0,1 ∂ x D 2 + a 2,0,0 ∂ 2 x + a 1,1,1 ∂ x D 1 D 2 .
In this formula, the terms are collected order-wise.

Space of symbols of differential operators

We consider the graded K (2)-module associated to the fine filtration (3.1):

grD λµ S 1|2 = ∞ i=0 gr i 2 D λµ S 1|2 , (3.5)
where

gr k D λµ S 1|2 = D k λµ S 1|2 /D k-1 2 λµ S 1|2
for every (half)integer k. This module is called the space of symbols of differential operators.

The image of a differential operator A under the natural projection

σ pr : D k λµ S 1|2 → gr k D λµ S 1|2
defined by the filtration (3.1) is called the principal symbol.

We need to know the action of the Lie superalgebra K (2) on the space of symbols.

Proposition 3.5. If k is an integer, then

gr k D λµ S 1|2 = F µ-λ-k F µ-λ-k (3.6)
Proof. By definition (see formula (3.3)), a given operator

A ∈ D k λµ S 1|2 with integer k is of the form A = F 1 ∂ k x + F 2 ∂ k-1 x D 1 D 2 + • • •
, where • • • stand for lower order terms. The principal symbol of A is then encoded by the pair (F 1 , F 2 ). From (3.4), one can easily calculate the K(2)-action on the principal symbol:

L X f (F 1 , F 2 ) = L µ-λ-k X f (F 1 ) , L µ-λ-k X f (F 2 ) .
In other words, both F 1 and F 2 transform as (µ -λ -k)-densities.

Surprisingly enough, the situation is more complicated in the case of half-integer k.

Proposition 3.6. If k is a half-integer, then the K (2)-action is as follows:

L X f (F 1 , F 2 ) = (L µ-λ-k X f (F 1 ) -2 D 1 D 2 (f ) F 2 , L µ-λ-k X f (F 2 ) + 2D 1 D 2 (f ) F 1 ). (3.7)
Proof. Directly from (3.4).

This means that the space of symbols of half-integer order is not isomorphic to the space of weighted densities. It would be nice to understand the geometric nature of the action (3.7).

Corollary 3.7. The module grD λµ S 1|2 depends only on µ -λ.

Following [START_REF] Lecomte | Projectively invariant symbol calculus[END_REF][START_REF] Duval | Conformally equivariant quantization: existence and uniqueness[END_REF][START_REF] Gargoubi | Differential operators on supercircle: conformally equivariant quantization and symbol calculus[END_REF] and to simplify the notation, we will denote by S µ-λ S 1|2 he full space of symbols grD λµ S 1|2 and S k µ-λ S 1|2 the space of symbols of order k.

Corollary 3.8. The space D k λµ S 1|2 is isomorphic to S µ-λ S 1|2 as a module over the affine Lie superalgebra Aff (2|2).

Proof. To define an Aff (2|2)-equivariant quantization map, it suffice to consider the inverse of the principal symbol:

Q = σ -1 pr .
A linear map Q : S µ-λ S 1|2 → D λµ S 1|2 is called a quantization map if it is bijective and preserves the principal symbol of every differential operator, i.e., σ pr • Q = Id. The inverse map σ = Q -1 is called a symbol map.

Conformally equivariant quantization on S 1|2

In this section we prove the main results of this paper -Theorem 1.1 -on the existence and uniqueness of the conformally equivariant quantization map on the space D 2 λµ S 1|2 . We calculate this quantization map explicitly.

We already proved that the space D 2 λµ S 1|2 is isomorphic to the corresponding space of symbols as a module over the affine Lie superalgebra Aff (2|2). We will now show how to extend this isomorphism to that of the osp (2|2)-modules.

Equivariant quantization map in the case of 1 2 -differential operators

Let us first consider the quantization of symbols of 1 2 -differential operators. By linearity, we can assume that the symbols of differential operators are homogeneous (purely even or purely odd). Since for any symbol (F 1 , F 2 ) ∈ S 1 2 µ-λ S 1|2 , we have p(F 1 ) = p(F 2 ), we can define parity of the symbol (F 1 , F 2 ) as p(F ) := p(F 1 ) = p(F 2 ). µ-λ S 1|2 provided µ = λ:

Q (F 1 , F 2 ) = F 1 D 1 + F 2 D 2 + (-1) p(F ) λ λ -µ D 1 (F 1 ) + D 2 (F 2 ) . (4.1)
Proof. First, one easily checks that this quantization map is, indeed, osp (2|2)-equivariant. Let us prove the uniqueness.

Consider first an arbitrary differentiable linear map Q : S

1 2 µ-λ S 1|2 → D 1 2
λµ S 1|2 preserving the principal symbol. Such a map is of the form:

Q (F 1 , F 2 ) = F 1 D 1 + F 2 D 2 + Q 1 (F 1 ) + Q 2 (F 2 )
where Q 1 and Q 2 are differential operators with coefficients in F µ-λ , cf. formula (3.2).

One then easily checks the following: a) This map commutes with the action of the vector fields D 1 , D 2 ∈ osp (1|2) if and only if the differential operators Q 1 and Q 2 are with constant coefficients.

b) This map commutes with the linear vector fields X x , X ξ 1 , X ξ 2 if and only if the differential operators Q 1 and Q 2 are of order 1 and moreover have the form:

Q 1 (F 1 , F 2 ) = c 1 D 1 (F 1 ) Q 2 (F 1 , F 2 ) = c 2 D 2 (F 2 )
where c 1 , c 2 are arbitrary constants. We thus determined the general form of a quantization map commuting with the action of the affine subalgebra Aff (2|2).

In order to satisfy the full condition of osp (1|2)-equivariance, it remains to impose the equivariance with respect to the vector field X x 2 .

c) The above quantization map commutes with the action of X x 2 if and only if c 1 , c 2 satisfy the following condition:

2 µ -λ -1 2 c 1 + c 2 = (-1) p(F )+1 2λ c 1 + 2 µ -λ -1 2 c 2 = (-1) p(F )+1 2λ
If µ-λ = 0, this system can be easily solved and the solution is c 1 = c 2 = (-1) p(F ) λ λ-µ .

Equivariant quantization map in the case of 1-differential operators

Let us consider the next case. All the calculations are similar (yet more involved) to the above calculations so we omit the details.

Proposition 4.2. The unique osp (2|2)-equivariant quantization map associates the following differential operator to a symbol

(F 1 , F 2 ) ∈ S 1 µ-λ S 1|1 : Q (F 1 , F 2 ) = F 1 ∂ x + F 2 D 1 D 2 +(-1) p(F ) 1 2(µ-λ-1) D 1 (F 1 ) D 1 + D 2 (F 1 ) D 2 +(-1) p(F ) λ+ 1 2 µ-λ-1 D 2 (F 2 ) D 1 -D 1 (F 2 ) D 2 -λ µ-λ-1 ∂ x (F 1 ) -1+2λ 2(µ-λ)-1 D 1 D 2 (F 2 ) (4.2) provided µ -λ = 1 2 , 1.
Proof. First, we check by a straightforward calculation that an arbitrary Aff (2|2)-equivariant quantization map is given by

Q (F 1 , F 2 ) = F 1 ∂ x + F 2 D 1 D 2 + c 1 11 D 1 (F 1 ) + c 1 12 D 2 (F 2 ) D 1 + c 1 21 D 2 (F 1 ) + c 1 22 D 1 (F 2 ) D 2 +c 2 1 ∂ x (F 1 ) + c 2 2 D 1 D 2 (F 2 ) ,
where c 1 ij and c 2 i are arbitrary constants depending on λ and µ. The above quantization map commutes with the action of X x 2 if and only if the coefficients c 1 ij and c 2 i satisfy a certain system of linear equations. Solving this system, one obtains formula (4.2).

The next case corresponds to the space of differential operators D The proof of this statement is similar to that of Proposition 4.2.

Q (F 1 , F 2 ) = F 1 ∂ x D 1 + F 2 ∂ x D 2 +(-1) p(F )
The last case we consider is the space of differential operators D 2 λµ S 1|2 .

Case of 2-order differential operators

Proposition 4.4. The unique osp (2|2)-equivariant quantization map associates the following differential operator to a symbol (F 1 , F 2 ) ∈ S

Q (F 1 , F 2 ) = F 1 ∂ 2 x + F 2 ∂ x D 1 D 2 +(-1) p(F ) 1 µ-λ-2 D 1 (F 1 ) ∂ x D 1 + D 2 (F 1 ) ∂ x D 2 +(-1) p(F ) λ+1 µ-λ-2 D 2 (F 2 ) ∂ x D 1 -D 1 (F 2 ) ∂ x D 2 + 2λ+1 λ-µ+2 ∂ x (F 1 ) ∂ x + λ+1 λ-µ+2 ∂ x (F 2 ) D 1 D 2 - 1 (λ-µ+2)(2λ-2µ+3) D 1 D 2 (F 1 ) D 1 D 2 + (2λ+1) (λ+1) (λ-µ+2)(2λ-2µ+3) D 1 D 2 (F 2 ) ∂ x -(-1) p(F ) 2λ+1 (λ-µ+2)(2λ-2µ+3) ∂ x D 1 (F 1 ) D 1 + ∂ x D 2 (F 1 ) D 2 -(-1) p(F ) (2λ+1) (λ+1) (λ-µ+2)(2λ-2µ+3) ∂ x D 2 (F 2 ) D 1 -∂ x D 1 (F 2 ) D 2 + λ (2λ+1) (λ-µ+2)(2λ-2µ+3) ∂ 2 x (F 1 ) + λ (2λ+1)(λ+1) (λ-µ+2)(2λ-2µ+3)(λ-µ+1) ∂ x D 1 D 2 (F 2 ) (4.4)
provided µ -λ = 1, 3 2 , 2. The proof of this statement is again similar to that of Proposition 4.2. However, the computations are quite involved.

3 2 λµ

 32 S 1|2 and D 2 λµ S 1|2 . The space D 3 2

Remark 3 . 3 . 1 )

 331 It is worth noticing that for k integer, one has D (k)

Proposition 4 . 1 .

 41 The unique osp (2|2)-equivariant quantization map associates the following1 2 -differential operator to a symbol (F 1 , F 2 ) ∈ S 1 2

3 2 λµ S 1|2 . 4 . 3 2 µ

 32432 Equivariant quantization map in the case of 3 2 -differential operators Proposition 4.3. The unique osp (2|2)-equivariant quantization map associates the following differential operator to a symbol (F 1 , F 2 ) ∈ S 3 -λ S 1|1 :

λ+ 1 2 λ 2 ∂ 2 - λ+ 1 2 2

 2222 -µ+1 D 1 (F 1 ) + D 2 (F 2 ) ∂ x +(-1) p(F ) 1 2 (λ-µ+1) D 2 (F 1 ) -D 1 (F 2 ) D 1 D 2 + x (F 1 ) D 1 + ∂ x (F 2 ) D (λ-µ+1) 2 D 1 D 2 (F 1 ) D 2 -D 1 D 2 (F 2 ) D 1 +(-1) p(F ) λ (λ+ 1 2 ) (λ-µ+1) 2 ∂ x D 1 (F 1 ) + ∂ x D 2 (F 2 ) (4.3)

2µ-λ S 1|1 :

Acknowledgements. I am grateful to H. Gargoubi and V. Ovsienko for the statement of the problem and constant help. I am also pleased to thank D. Leites for critical reading of this paper and a number of helpful suggestions.


Ameur Dhahri
email: adhahri@puc.cl

On the quadratic Fock functor *

published or not. The documents may come

Introduction

The renormalized square of white noise (RSWN) was firstly introduced by Accardi-Lu-Volovich in [AcLuVo]. Later, Sniady studied the connection between the RSWN and the free white noise (cf [Sn]). Subsequently, its relation with the Lévy processes on real Lie algebras was established in [AcFrSk].

Recently, in [START_REF] Accardi | Quadratic exponential vectors[END_REF]-[AcDh2], the authors obtained the Fock representation of the RSWN. They started defining the quadratic Fock space and the quadratic second quantization. After doing that, they characterized the operators on the one-particle Hilbert algebra whose quadratic second quantization is isometric (resp. unitary). A sufficient condition for the contractivity of the quadratic second quantization was derived too.

It is well known that the first order second quantization Γ 1 (p) of an operator p, defined on the usual Fock space, is an orthogonal projection if and only if p is an orthogonal projection (cf [Par]). Within this paper, it is shown that the set of orthogonal projections p, for which its quadratic second quantization Γ 2 (p) is an orthogonal projection, is quite reduced. More precisely, we prove that Γ 2 (p) is an orthogonal projection if and only if p is a multiplication operator by a characteristic function χ

I , I ⊂ R d .
This paper is organized as follows. In section 2, we recall some main properties of the quadratic Fock space and the quadratic second quantization. The main result is proved in section 3.

Quadratic Fock functor

The algebra of the renormalized square of white noise (RSWN) with test function Hilbert algebra

A := L 2 (R) ∩ L ∞ (R) is the * -Lie-algebra, with central element denoted 1, generators B + f , B h , N g , f, g, h ∈ L 2 (R d ) ∩ L ∞ (R d )}, involution (B + f ) * = B f , N * f = N f and commutation relations [B f , B + g ] = 2c f, g + 4N f g , [N a , B + f ] = 2B + af (1) [B + f , B + g ] = [B f , B g ] = [N a , N a ′ ] = 0, for all a, a ′ , f , g ∈ L 2 (R d ) ∩ L ∞ (R d ).
The Fock representation of the RSWN is characterized by a cyclic vector Φ satisfying [AcAmFr], [AcFrSk]).

B f Φ = N g Φ = 0 for all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ) (cf

Quadratic Fock space

In this subsection, we recall some basic definitions and properties of the quadratic exponential vectors and the quadratic Fock space. We refer the interested reader to

[AcDh1]-[AcDh2] for more details. The quadratic Fock space Γ 2 (L 2 (R d ) ∩ L ∞ (R d )) is the closed linear span of {B +n f Φ, n ∈ N, f ∈ L 2 (R d ) ∩ L ∞ (R d )}, where B +0 f Φ = Φ, for all f ∈ L 2 (R d ) ∩ L ∞ (R d ). From [AcDh2] it follows that Γ 2 (L 2 (R d ) ∩ L ∞ (R d )
) is an interacting Fock space. Moreover, the scalar product between two n-particle vectors is given by the following (cf [START_REF] Accardi | Quadratic exponential vectors[END_REF]).

Proposition 1 For all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ), one has B +n f Φ, B +n g Φ = c n-1 k=0 2 2k+1 n!(n -1)! ((n -k -1)!) 2 f k+1 , g k+1 B +(n-k-1) f Φ, B +(n-k-1) g Φ .
The quadratic exponential vector of an element

f ∈ L 2 (R d ) ∩ L ∞ (R d ), if it exists, is given by Ψ(f ) = n≥0 B +n f Φ n!
where by definition

Ψ(0) = B +0 f Φ = Φ. ( 2
)
It is proved in [START_REF] Accardi | Quadratic exponential vectors[END_REF] that the quadratic exponential vector Ψ(f ) exists if and only if f ∞ < 1 2 . Furthermore, the scalar product between two exponential vectors, Ψ(f ) and Ψ(g), is given by

Ψ(f ), Ψ(g) = e -c 2 R R d ln(1-4 f (s)g(s))ds .
(3)

Now, we refer to [START_REF] Accardi | Quadratic exponential vectors[END_REF] for the proof of the following theorem.

Theorem 1 The quadratic exponential vectors are linearly independents. Moreover, the set of quadratic exponential vectors is a total set in

Γ 2 (L 2 (R d )∩ L ∞ (R d )).

Quadratic second quantization

For all linear operator T on

L 2 (R d ) ∩ L ∞ (R d ), we define its quadratic second quantization, if it is well defined, by Γ 2 (T )Ψ(f ) = Ψ(T f ) for all f ∈ L 2 (R d ) ∩ L ∞ (R d ). Note that in [AcDh2], the authors have proved that Γ 2 (T ) is well defined if and only if T is a contraction on L 2 (R d )∩L ∞ (R d ) with respect to the norm ||.|| ∞ . Moreover, they have given a characterization of operators T on L 2 (R d ) ∩ L ∞ (R d )
whose quadratic second quantization is isometric (resp. unitary). The contractivity of Γ 2 (T ) was also investigated.

Main result

Given a contraction p on L 2 (R d ) ∩ L ∞ (R d ) with respect to . ∞ , the aim of this section is to prove under which condition Γ 2 (p) is an orthogonal projec- tion on Γ 2 (L 2 (R d ) ∩ L ∞ (R d )). Lemma 1 For all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ), one has B +n f Φ, B +n g Φ = n! d n dt n t=0 Ψ( √ tf ), Ψ( √ tg) . Proof. Let f, g ∈ L 2 (R d ) ∩ L ∞ (R d ) such that f ∞ > 0 and g ∞ > 0. Consider 0 ≤ t ≤ δ, where δ < 1 4 inf 1 f 2 ∞ , 1 g 2 ∞ .
It is clear that

√ tf ∞ < 1 2 and √ tg ∞ < 1 2 . Moreover, one has Ψ( √ tf ), Ψ( √ tg) = m≥0 t m (m!) 2 B +m f Φ, B +m g Φ .
Note that for all m ≥ n, one has

d n dt n t m (m!) 2 B +m f Φ, B +m g Φ = m!t m-n (m!) 2 (m -n)! B +m f Φ, B +m g Φ = t m-n m!(m -n)! B +m f Φ, B +m g Φ . Put K m = δ m-n m!(m -n)! B +m f Φ B +m g Φ .
Then, from Proposition 1, it follows that

||B +m f Φ|| 2 = c m-1 k=0 2 2k+1 m!(m -1)! ((m -k -1)!) 2 | f k+1 2 2 B +(m-k-1) f Φ 2 = c m-1 k=1 2 2k+1 m!(m -1)! ((m -k -1)!) 2 | f k+1 2 2 B +(m-k-1) f Φ 2 4 +2mc f 2 2 B +(m-1) f Φ 2 = c m-2 k=0 2 2k+3 m!(m -1)! (((m -1) -k -1)!) 2 | f k+2 2 2 B +((m-1)-k-1) f Φ 2 +2mc f 2 2 B +(m-1) f Φ 2 ≤ 4m(m -1) f 2 ∞ c m-2 k=0 2 2k+1 (m -1)!(m -2)! (((m -1) -k -1)!) 2 f k+1 2 2 B +((m-1)-k-1) f Φ 2 .
But, one has

||B +(m-1) f Φ|| 2 = c m-2 k=0 2 2k+1 (m -1)!(m -2)! ((m -1) -k -1)!) 2 | f k+1 2 2 B +((m-1)-k-1) f Φ 2 .
Therefore, one gets

||B +m f Φ|| 2 ≤ 4m(m -1) f 2 ∞ + 2m f 2 2 B +(m-1) f Φ 2 .
This proves that

K m K m-1 ≤ 4m(m -1) f 2 ∞ + 2m f 2 2 4m(m -1) g 2 ∞ + 2m g 2 2 m(m -n) δ.

It follows that lim

m→∞ K m K m-1 ≤ 4 f ∞ g ∞ δ < 1.
Hence, the series m K m converges. Finally, we have proved that

d n dt n Ψ( √ tf ), Ψ( √ tg) = m≥n t m-n m!(m -n)! B +m f Φ, B +m g Φ . (4)
Thus, by taking t = 0 in the right hand side of (4), the result of the above lemma holds. As a consequence of the above lemma, we prove the following.

Lemma 2 Let p be a contraction on

L 2 (R d ) ∩ L ∞ (R d ) with respect to the norm . ∞ . If Γ 2 (p) is an orthogonal projection on Γ 2 (L 2 (R d ) ∩ L ∞ (R d ))
, then one has

B +n p(f ) Φ, B +n p(g) Φ = B +n p(f ) Φ, B +n g Φ = B +n f Φ, B +n p(g) Φ for all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ) and all n ≥ 1. Proof. It Γ 2 (p) is an orthogonal projection on Γ 2 (L 2 (R d ) ∩ L ∞ (R d )), then Ψ( √ tp(f )), Ψ( √ tp(g)) = Ψ( √ tp(f )), Ψ( √ tg) = Ψ( √ tf ), Ψ( √ tp(g)) (5) for all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ) (with f ∞ > 0 and g ∞ > 0) and all 0 ≤ t ≤ δ such that δ < 1 4 inf 1 f 2 ∞ , 1 g 2 ∞ .
Therefore, the result of the above lemma follows from Lemma 1 and identity (5). Lemma 2 ensures that the following result holds true.

Lemma 3 Let p be a contraction on

L 2 (R d ) ∩ L ∞ (R d ) with respect to the norm . ∞ . If Γ 2 (p) is an orthogonal projection on Γ 2 (L 2 (R d ) ∩ L ∞ (R d )), then one has (p(f )) n , (p(g)) n = f n , (p(g)) n = (p(f )) n , g n (6) for all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ) and all n ≥ 1. Proof. Suppose that Γ 2 (p) is an orthogonal projection on Γ 2 (L 2 (R d ) ∩ L ∞ (R d ))
. Then, in order to prove the above lemma we have to use induction.

-For n = 1: Lemma 2 implies that

B + p(f ) Φ, B + p(g) Φ = B + p(f ) Φ, B + g Φ = B + f Φ, B + p(g) Φ for all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ). Using the fact that B f Φ = 0 for all f ∈ L 2 (R d ) ∩ L ∞ (R d
) and the commutation relations in (1) to get

B + p(f ) Φ, B + p(g) Φ = Φ, B p(f ) B + p(g) Φ = 2c p(f ), p(g) B + p(f ) Φ, B + g Φ = Φ, B p(f ) B + g Φ = 2c p(f ), g B + f Φ, B + p(g) Φ = Φ, B f B + p(g) Φ = 2c f, p(g) .
This proves that identity (6) holds true for n = 1.

-Let n ≥ 1 and suppose that identity ( 6) is satisfied. Then, from Lemma 2, it follows that for all f, g ∈

L 2 (R d ) ∩ L ∞ (R d ) one has B +(n+1) p(f ) Φ, B +(n+1) p(g) Φ = B +(n+1) p(f ) Φ, B +(n+1) g Φ = B +(n+1) f Φ, B +(n+1) p(g) Φ . (7)
Identity ( 7) and Proposition 1 imply that

B +(n+1) p(f ) Φ, B +(n+1) p(g) Φ = 2 2n+3 cn!(n + 1)! (p(f )) n+1 , (p(g)) n+1 +c n-1 k=0 2 2k+1 n!(n + 1)! ((n -k)!) 2 (p(f )) k+1 , (p(g)) k+1 B +(n-k) p(f ) Φ, B +(n-k) p(g) Φ = 2 2n+3 cn!(n + 1)! f n+1 , (p(g)) n+1 +c n-1 k=0 2 2k+1 n!(n + 1)! ((n -k)!) 2 f k+1 , (p(g)) k+1 (8) B +(n-k) f Φ, B +(n-k) p(g) Φ = 2 2n+3 cn!(n + 1)! (p(f )) n+1 , g n+1 +c n-1 k=0 2 2k+1 n!(n + 1)! ((n -k)!) 2 (p(f )) k+1 , g k+1 B +(n-k) p(f ) Φ, B +(n-k) g Φ .
Note that by induction assumption, one has

(p(f )) k+1 , (p(g)) k+1 = f k+1 , (p(g)) k+1 = (p(f )) k+1 , g k+1 (9)
for all k = 0, . . . , n -1. Therefore, from Lemma 2 and identity (9), one gets

c n-1 k=0 2 2k+1 n!(n + 1)! ((n -k)!) 2 (p(f )) k+1 , (p(g)) k+1 B +(n-k) p(f ) Φ, B +(n-k) p(g) Φ = c n-1 k=0 2 2k+1 n!(n + 1)! ((n -k)!) 2 f k+1 , (p(g)) k+1 B +(n-k) f Φ, B +(n-k) p(g) Φ = c n-1 k=0 2 2k+1 n!(n + 1)! ((n -k)!) 2 (p(f )) k+1 , g k+1 B +(n-k) p(f ) Φ, B +(n-k) g Φ .
Finally, from (8) one can conclude. Note that the set of contractions

p on L 2 (R d ) ∩ L ∞ (R d ) with respect to . ∞ , such that Γ 2 (p) is an orthogonal projection on Γ 2 (L 2 (R d ) ∩ L ∞ (R d )), is reduced to the following. Lemma 4 Let p be a contraction on L 2 (R d ) ∩ L ∞ (R d ) with respect to the norm . ∞ . If Γ 2 (p) is an orthogonal projection on Γ 2 (L 2 (R d ) ∩ L ∞ (R d )), then p( f ) = p(f ). for all f ∈ L 2 (R d ) ∩ L ∞ (R d ).
Proof. Let p be a contraction on

L 2 (R d ) ∩ L ∞ (R d ) with respect to the norm . ∞ such that Γ 2 (p) is an orthogonal projection on Γ 2 (L 2 (R d ) ∩ L ∞ (R d )). Then, from Lemma 3 it is clear that p = p * = p 2 (taking n = 1 in (6)). Moreover, for all f 1 , f 2 , g 1 , g 2 ∈ L 2 (R d ) ∩ L ∞ (R d ), one has (p(f 1 + f 2 )) 2 , (g 1 + g 2 ) 2 = (p(f 1 + f 2 )) 2 , (p(g 1 + g 2 )) 2 . It follows that (p(f 1 )) 2 , g 2 1 + g 2 2 + (p(f 2 )) 2 , g 2 1 + g 2 2 + 4 p(f 1 )p(f 2 ), g 1 g 2 = (p(f 1 )) 2 , (p(g 1 )) 2 + (p(g 2 )) 2 + (p(f 2 )) 2 , (p(g 1 )) 2 + (p(g 2 )) 2 (10) +4 p(f 1 )p(f 2 ), p(g 1 )p(g 2 ) .
Then, using ( 6) and ( 10) to obtain p(f 1 )p(f 2 ), g 1 g 2 = p(f 1 )p(f 2 ), p(g 1 )p(g 2 ) (11)

for all f 1 , f 2 , g 1 , g 2 ∈ L 2 (R d )∩L ∞ (R d
). Now, denote by M a the multiplication operator by the function a

∈ L 2 (R d ) ∩ L ∞ (R d ). Then, identity (11) implies that M p(f 2 ) ḡ2 p(f 1 ), g 1 = M p(f 2 )p(g 2 ) p(f 1 ), p(g 1 ) for all f 1 , f 2 , g 1 , g 2 ∈ L 2 (R d ) ∩ L ∞ (R d ). This gives that M p(f 2 ) ḡ2 p = pM p(f 2 )p(g 2 ) p (12) for all f 2 , g 2 ∈ L 2 (R d ) ∩ L ∞ (R d ).
Taking the adjoint in ( 12), one gets

pM p(f 2 ) g 2 = pM p(f 2 )p(g 2 ) p. ( 13
) Note that, for all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ), identity (12) implies that M p(f )ḡ p = pM p(f ) p(g) p. (14)
Moreover, from ( 13), one has

pM p(f ) p(g) p = pM f p(g) . (15)
Therefore, identities ( 14) and ( 15) yield

M p(f )ḡ p = pM f p(g) (16) for all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ). Hence, for all f, g, h ∈ L 2 (R d ) ∩ L ∞ (R d ), identity (16) gives p(f p(g)h) = p(f )ḡp(h). ( 17
)
Taking f = g = p(h) in ( 16) to get

M |p(h)| 2 p = pM |p(h)| 2 . ( 18
)
Then, if we put f = h = p(g) in ( 17), one has

p(p(g) p(g) 2 ) = p(|p(g)| 2 p(g)) = (p(g)) 2 ḡ.
But, from ( 18), one has

p(|p(g)| 2 p(g)) = (pM |p(g)| 2 )(p(g)) = M |p(g)| 2 p(p(g)) = |p(g)| 2 p(g). Hence, one obtains |p(g)| 2 p(g) = (p(g)) 2 ḡ (19) for all g ∈ L 2 (R d )∩L ∞ (R d ). Now, let g be a real function in L 2 (R d )∩L ∞ (R d ).
So, the polar decomposition of p(g) is given by p(g) = |p(g)|e iθ p(g) .

Thus, identity (19) implies that

|p(g)| 3 e -iθ p(g) = |p(g)| 2 g. This proves that for all x ∈ R d , θ p(g) (x) = k x π, k x ∈ Z. Therefore, p(g) is a real function. Now, taking f = f 1 + if 2 ∈ L 2 (R d ) ∩ L ∞ (R d )
, where f 1 , f 2 are real functions on R d . It is clear that

p( f ) = p(f 1 -if 2 ) = p(f 1 ) + ip(f 2 ) = p(f ).
This completes the proof of the above lemma.

As a consequence of Lemmas 3 and 4 we prove the following theorem.

for all f, g ∈ L 2 (R d ) ∩ L ∞ (R d ). Hence, one obtains

M (p(g)) 2 = pM g 2 p. (22)
In particular, for g = χ I where I ⊂ R d , one has

M (p(χ I )) 2 = pM χ I p. ( 23
)
If I tends to R d , the operator M χ I converges to id (identity of L 2 (R d ) ∩ L ∞ (R d )) for the strong topology. From ( 23), it follows that p(f ) = p 2 (f ) = lim

I↑R d M (p(χ I )) 2 f for all f ∈ L 2 (R d ) ∩ L ∞ (R d ).
But, the set of multiplication operators is a closed set for the strong topology. This proves that p = M a , where a ∈ L 2 (R d ) ∩ L ∞ (R d ). Note that p = p 2 is a positive operator. This implies that a is a positive function. Moreover, one has p n = p for all n ∈ N * . This gives M a n = M a for all n ∈ N * . It follows that a n = a for all n ∈ N * . Therefore, the operator a is necessarily a characteristic function on R d .

Acknowledgments

The author gratefully acknowledges stimulating discussions with Eric Ricard.

* Research partially supported by PBCT-ADI 13 grant "Laboratorio de Análisis y Estocástico", Chile

Theorem 2 Let p be a contraction on L 2 (R d ) ∩ L ∞ (R d ) with respect to the norm . ∞ . Then, Γ(p) is an orthogonal projection on

Using identity (20) and Lemma 3 to get

This yields

Then, identity (21) implies that f 1 , M (p(g)) 2 f 2 = f 1 , (pM g 2 p)f 2