Martin Jourdan
email: jourdan@uni-mainz.de

Fabian Grosse-Schulte

Michaela Hahn

Gerd Schönhense

Fabian Große-Schulte

Spectroscopy of the electronic states of the Heusler compounds Co 2 FeAl and Co 2 Cr 0.6 Fe 0.4 Al and the influence of oxidation

HAL is

Introduction

Band structure calculations, which predict half metallic properties for several Heusler compounds [1,2,3], initiated great experimental efforts concerning this class of materials. Such materials with a spin polarization at the Fermi energy close to 100% are highly desirable for spintronics applications. The validation of the calculated electronic properties remains difficult, though. These materials with medium strong electronic correlations provide a test for modern band structure calculation methods.

The preparation and investigation of planar magnetic tunneling junctions is one method which allows conclusions concerning the spin polarization and electronic density of states (DOS) near the Fermi energy (e. g. [4,5,6,7,8]). Another technique, which is in principle more powerful concerning the accessible energy range and directional sensitivity, is angle resolved photoemission spectroscopy (ARPES). However, ARPES is extremely surface sensitive and the preparation of Heusler compounds with clean and well ordered surface for such measurements poses a major challenge. Valence band spectroscopy is usually performed on bulk samples which are fractured or polished inside the spectroscopy chamber (e. g. [9,[START_REF] Miyazaki | Electron Spectrosc[END_REF]11]). Another approach is a sequence of sputter cleaning and annealing processes of a bulk or thin film sample (often capped with a protection layer) after ex-situ transfer into an analysis chamber [12,13]. If complex intermetallic alloys like the Heusler compounds are investigated, all these preparation methods result in more or less distorted surface order and stoichimetry. In order to deal with this problem, hard x-ray photoemission spectroscopy (HAXPES) is employed, which is able to detect photoemitted electrons from Heusler thin films capped by a thin MgO layer (e. g. [14,15]). Alternatively, uncapped Heusler thin films (grown by Pulsed Laser Deposition) were transported in a vacuum suitcase from the preparation chamber to a synchrotron facility for spin resolved photoemission spectroscopy [16].

Here, a different approach is presented: Uncapped Heusler thin films were prepared by a process optimized for tunneling junction preparation and were investigated in-situ by angle resolved ultraviolet photoemission spectroscopy (ARUPS). Thus it is possible to study undistorted atomically clean Heusler surfaces and the results can be related directly to our work on Heusler based tunneling junctions [5,6]. Our main intention is the identification of intrinsic band structure related features in the ARUPS data, which can be used to test the predictions of band structure calculations. By controlled oxidation of the samples, degradation related features in the spectra are distinguishable from intrinsic properties of the Heusler compounds.

The Heusler compounds Co 2 Cr 0.6 Fe 0.4 Al and Co 2 FeAl were investigated. The spin polarization of the former is predicted to be close to 100%, whereas in the case of the latter theory yields a reduced value [18,17].

Experiment

Epitaxial Heusler thin films were prepared by rf-magnetron sputtering from stoichiometric targets on MgO(100) substrates at room temperature. The sputtering chamber is part of an Omicron Molecular Beam Epitaxy (MBE)-/sputtering-/analysiscluster. The base pressure of the sputtering chamber was 2 × 10 -9 mbar. 6N purity Argon flowing through a cleaning cartridge (Linde Oxisorb), which is specified to reduce the amount of oxygen to <5 ppb, was used as sputtering gas at 0.1mbar. The thin films show epitaxial B2 order already as deposited. Subsequent annealing at 550 o C for 15 minutes in an MBE chamber improves the structural and morphological order (formation of terraces) of the thin film. The base pressure of the MBE annealing chamber amounts to 1×10 -10 mbar, this pressure increases up to 2×10 -9 mbar at maximum annealing temperature, the total time in the annealing chamber was 75 minutes. For additional details on the sample preparation and structural characterization (x-ray diffraction, electron diffraction, in-situ scanning tunneling microscopy), see [6,19]. The surface magnetic moments of the Co 2 Cr 0.6 Fe 0.4 Al and Co 2 FeAl thin films are identical with the bulk moments as determined by x-ray magnetic circular dichroism experiments (TEYand TM-XMCD) and amount to 3.5µ B /f. u. and 4.8µ B /f. u., respectively [20,21].

In-situ ARUPS is performed in a special analysis chamber (base pressure 5 × 10 -10 mbar) which is part of the UHV-cluster. We use a Leybold EA-10/100 hemispherical electron energy analyzer and a Helium gas discharge lamp, which produces unpolarized photons with energies of 21.2eV (He I) and 40.8eV (He II). The diameter of the visible spot amounts to 3mm. During operation of the gas discharge lamp the He partial pressure in the analysis chamber is 6 × 10 -8 mbar. The photon angle of incidence on the sample surface was 50 o . The sample can be tilted with a maximum angle of 20 o off-normal (the photon angle of incidence thereby changes to 70 o ), the angular resolution of the energy analyzer is 6 o . The set-up does not allow the comparison of absolute photoemission intensities of different samples or of samples transfered between measurements into other chambers. All ARUPS curves shown for comparison in one graph are scaled.

As concluded from the width of the Fermi edge of a silver thin film, the energy resolution of our ARUPS set-up amounts to 200meV.

In order to study the effect of oxidation on the Heusler thin films, the samples were exposed to small amounts of oxygen. In an additional chamber, which is part of the UHV-cluster, an oxygen flow resulting in a pressure of 1×10 -6 mbar (hot cathode gauge) was adjusted. The thin film to be oxidized was positioned directly in front a closed gate valve (size CF-100), which connects the UPS chamber (with running Ion-Getter-and Turbopumps) with the oxygen filled chamber. After opening the gate valve the pressure gauge showed immediately a value of 2 × 10 -7 mbar, which remained constant for the typical exposure times of the order of 10s. The oxidation was ended by closing the gate valve again. The combination of sputter deposition (typical method for Heusler thin film growth) and in-situ ARUPS has not been used yet. In order to demonstrate the feasibility of sufficient surface purity, first a most reactive Chromium (100) thin film prepared by exactly the same process as described above for the Heusler compounds, including the annealing step, was studied (dashed line in Fig. 1). The ARUPS spectrum is in good agreement with published data obtained on MBE grown Cr thin films [22,23] with one additional feature at -6.5eV, which can be attributed to weak oxidation with 1 Langmuir (L) O 2 [24]. Thus it was demonstrated that the combination of sputter deposition, annealing and ARUPS processes is able to produce the intrinsic features of highly reactive transition metals such as the Heusler compounds. As shown below, the Heusler samples prepared without controlled exposure to oxygen show at the utmost weak traces of oxidation. Those samples are called clean in the following.

The influence of oxidation

The influence of intended oxidation on the ARUPS spectra of the Heusler compound Co 2 Cr 0.6 Fe 0.4 Al was investigated. For both excitation energies hν=21.2eV (Fig. 1, solid lines) and hν=40.8eV (Fig. 2) increasing oxidation of the thin film produces a growing emission peak at E -E F -6eV and a decreasing relative intensity at higher energies. The broad -6eV signal is thus indicative for the oxidation of the Heusler compound. In both figures the data was scaled to show the same intensity at E -E F = 2.5eV.

Furthermore, some characteristic features in the spectra of clean samples (labeled A in the inset of Fig. 1 and E and H in Fig. 2), are already quenched by the exposure of the samples to only 1L of oxygen. Recently it was shown for Fe(001), that a surface coverage by a single layer of oxygen produces additional states at E F [25]. A similar mechanism could explain the quenching of the features A and E in the case of Co 2 Cr 0.6 Fe 0.4 Al.

This sensitivity of the Heusler compound to oxidation is highly relevant for the preparation of magnetic tunneling junctions. It explains the experimental observation that over-oxidation of the tunneling barrier or aging of the Heusler surface is a major problem if Co 2 Cr 0.6 Fe 0.4 Al or similar compounds are used as tunneling electrodes. In our previous experiments on Co 2 Cr 0.6 Fe 0.4 Al/AlO x /CoFe tunneling junctions we demonstrated that slightly aged Co 2 Cr 0.6 Fe 0.4 Al surfaces produce only half of the magnetoresistance of junctions prepared with optimized conditions [6].

In ARUPS, the high reactivity of Co 2 Cr 0.6 Fe 0.4 Al with oxygen is also demonstrated by the increase of the work function eΦ as determined from the lengths of the spectra (Fig. 1) [START_REF] Helander | [END_REF]. Whereas eΦ = 4.5eV for clean samples, it increases to 4.8eV, if the thin film was exposed to 1L of oxygen (5.1eV for 10L).

The 1L spectrum corresponds to a chemisorption phase of oxygen with a rather small but significant influence on the near Fermi region. The 10L oxygen spectrum reveals beginning bulk oxidation with loss of metallicity within the UPS probing depth as evident from the disappearance of the Fermi-cutoff in Figs. 1 and2. The relative emission intensity of the oxygen state (2p-like) referred to the Heusler states (3d-like) is reduced for hν = 40.8eV compared to hν = 21.2eV. This is due to the different energy dependence of the photoionisation cross sections [START_REF] Yeh | Atomic Data and Nuclear Data Tables[END_REF]. All features due to direct transition (see section V) are quenched by the exposure of the thin films to 10L oxygen for both excitation energies.

In ref. [6] we showed by in-situ scanning tunneling microscopy how aging the Co 2 Cr 0.6 Fe 0.4 Al thin films in moderate vacuum conditions results in a modified morphology of AlO x tunneling barriers deposited on top of the samples. The concluded change of the surface chemistry by aging of the Heusler thin films is consistent with the pronounced sensitivity to oxidation as observed by ARUPS. It should be considered that ARUPS is very surface sensitive (electron mean free path 0.4 -1nm [START_REF] Hüfner | Photoelectron Spectroscopy[END_REF]) and that in normal emission from the (001) oriented thin films, assuming an L2 1 structure, only the electronic states with momentum k parallel to the Γ-X direction contribute. However, only moderate DOS anisotropies are expected for cubic crystal structures. Within the usual three-step model of photoemission (ARUPS spectra dominated by direct optical transitions), the intensity is related to a joint density of an initial and a final electronic state [START_REF] Hüfner | Photoelectron Spectroscopy[END_REF](see section V). Nevertheless, emission peaks should always be related to an energy range with a high density of initial states.

ARUPS on clean samples and Density of States

The most obvious differences in the ARUPS spectra of Co 2 Cr 0.6 Fe 0.4 Al and Co 2 FeAl are observed near the Fermi edge, which is well developed for both compounds. In the case of Co 2 Cr 0.6 Fe 0.4 Al clear shoulder-like features (labeled A for hν = 21.2eV, Fig. 3, and E for hν = 40.8eV, Fig. 4) are observed on the rising edge of pronounced emission peaks (labeled B and F) at -1.1eV. In the case of Co 2 FeAl an emission peak (labeled B and F) without any shoulder on its edge is observed at -0.85eV.

This difference between the two compounds can be related to features of the theoretical (FLAPW-GGA) bulk total density of states (TDOS) as published by G. H. Fecher et al. [17]. Fig. 5 shows the TDOS of the two Heusler compounds with the L2 1 structure as derived from a digitized spin resolved graph of ref. [17]. Disorder on the Cr,Fe -Al sites (B2 structure) is in general expected to result in a smearing out of the DOS [18]. In the calculation there is a pronounced peak in the DOS of Co 2 Cr 0.6 Fe 0.4 Al (labeled A/E in Fig. 5), which does not exist for Co 2 FeAl. We account this peak to the origin of the feature A/E in the ARUPS data. Also the features B/F and C/G of the experimental ARUPS data can be identified in the theoretical TDOS. This good agreement between experiment and theory is consistent with our observation of a relatively large tunneling magnetoresistance of Co 2 Cr 0.6 Fe 0.4 Al/AlO x /CoFe junctions (101%), which is, again in agreement theory, evidence for a large spin polarization of this Heusler compound [6]. For Co 2 FeAl the position of the Fermi energy has to be shifted by -0.15eV in the theoretical data in order to be consistent with the ARUPS data . Considering that the exact determination of the position of the Fermi energy poses a challenge for band structure calculations, the agreement with the experimental results is relatively good. However, such a shift moves the Fermi energy toward the edge of the minority pseudo band gap in the calculation by G. H. Fecher et al. [17] and thus is reducing the spin polarization. This is in accordance with our observation of only a small tunneling magnetoresistance of Co 2 FeAl/AlO x /CoFe junctions (34%).

At hν = 21.2eV, a broad emission maximum (labeled D) is observed at -5.0eV for Co 2 Cr 0.6 Fe 0.4 Al and at -4.6eV for Co 2 FeAl. These maxima are at energies not identical with the energies at which the oxidation derived peaks discussed in section III are observed (Fig. 1 and Fig. 3). However, they are close and it cannot be excluded that feature D is related to beginning sample oxidation. At these energies, no peaks in the bulk densities of states were calculated for both compounds [18,17,9,30], but, as we will show in section V, these peaks could be generated by direct transitions into a free electron-like parabolic band.

Alternatively, a recent publication on the surface DOS (3-unit cell slab) obtained a peak in the relevant energy range for Co 2 FeAl [31]. The same publication shows that the other DOS features remain qualitatively unchanged in the surface region as compared to the bulk.

With a photon energy of hν = 40.8eV, another clear emission maximum (labeled H) [17]. Disorder on the Cr,Fe -Al sites (B2 structure) is in general expected to result in a smearing out of the density of states [18].

is seen at -3.3eV for Co 2 FeAl, which is also present but less distinct for Co 2 Cr 0.6 Fe 0.4 Al. Again, this feature can be related to a maximum in the calculated bulk TDOS of the two compounds (Fig. 5), which appears in theory at slightly lower energies. One more emission maximum is visible at -6eV (hν = 40.8eV, labeled I). This maximum may be related to sample oxidation, because its position is identical with the position of a huge peak which grows upon exposure of the thin films to oxygen. On the other hand, it could as well be related to the bulk band structure of the Heusler compounds as will be shown below.

Free electron final state model

The most simple assumption of the final state within the three-step model is that of a free electron (e. g. [32]), which was used successfully e. g. for the analysis of ARUPS data measured on Cr [22,23]. Within this model, the final state is described by a free electron-like parabolic band

E f (k) = h2 k 2 /2m + V 0 , (1)
where the only free parameter is V 0 , the bottom of a muffin tin potential referenced to the Fermi energy. We only apply this model to the compound Co 2 FeAl, for which more detailed band structure calculations are published.

In the calculated band structure of Co 2 FeAl [30], a free electron-like dispersion branch along the Γ-X direction is clearly visible (see Fig. 6). We take the bottom of this dispersion branch to determine V 0 = -9.8eV for the final state dispersion. In ref. [30] and Fig. 6 it is clearly visible that due to interactions with d-electron states the free electron-like dispersion is not simply folded back into the first Brillouin zone at energies close to E F . However, this is not relevant for our final state model because we intend to describe photoexcited electronic states well above the Fermi energy. Also due to their high energy those states are expected to show a substantial broadening of 2-3eV [32]. From the assumption of free electron-like final states follows a oneto-one correspondence between the energy of the emitted photoelectrons E kin and the momentum of the electronic states parallel to the Γ-X direction k Γ-X (eΦ: work function, E i : initial state energy):

(hk Γ-X ) 2 = 2m(E f -V 0 ) = 2m(E kin + eΦ -V 0 ) = 2m(E i + hν -V 0 )
In Fig. 6 the final state free electron parabola as described above is plotted in the first Brillouin zone of the L2 1 unit cell along the Γ-X direction (|Γ -X| = 2π/a = 1.01 Å-1 ). The initial state energies are determined from the positions of the ARUPS emission peaks of Co 2 FeAl. The direct transitions due to the absorption of photons are indicated by vertical arrows, the positions of the transitions on the momentum-axis were calculated by the formula given above. It is obvious that this simple parameter-free model for the description of the photoemission processes results in a consistent description of the experimental data.

The experimental data is compared with a band structure calculation by G. H. Fecher and C. Felser [30], in which they employ the same method as in [17], but show dispersion branches over a relatively wide energy region below E F . In Fig. 6 the digitized part of the spin resolved band structure for Co 2 FeAl along the Γ-X direction is taken from a graph of [30] and amended by a free electron-like dispersion branch as described above.

The initial states of the ARUPS emission maxima B/F and C/G appear just above very flat theoretical dispersion branches. The same applies to the emission maximum H. As already proposed above, the experimental data indicates that the position of the Fermi energy in the calculation should be shifted by ∆E -0.15eV.

The experimentally derived data from transition D is also positioned on a branch of the dispersion, but this branch shows a pronounced slope resulting in a small DOS. It is possible that the mechanism of a direct transition into the final state band produces this experimentally observed emission maximum as well. However, due to its distinctiveness beginning oxidation or a specific surface DOS [31] as discussed in section III seems a more likely origin. Also in section IV some evidence was given, that the emission maximum I may be associated with sample oxidation. From Fig. 6 it could be concluded that this emission maximum is generated by the flat band region close to the X-point, but this interpretation would not be consistent with the for all other transitions necessary shift of E F . Nevertheless, in this sense it could be associated with the upper edge of the local gap at X in the band structure.

Another band structure calculation from the same publication [30], derived with the LDA+U method, shows inferior agreement. [30] amended by a free electronlike final state dispersion branch (wide yellow lines). The thin black lines indicate the majority branches and the thin gray lines the minority branches. The experimentally observed ARUPS transitions are indicated by blue arrows. The labels provide the assignment to the emission peaks of Figs. 3 and4.

Additional data can be collected in principle by tilting the sample with respect to the analyzer (e. g. [32]). In our experiments no shift of the emission peak positions was observed by tilting by the maximum possible angle of 20 o . Such a behavior should be expected because this tilt changes k Γ-X only by 0.08|Γ -X| (for transition B), which, considering the flatness of the dispersion branches, does not result in a significant energy shift.

In the ARUPS data analysis the dipole selection rules for direct optical transitions (e. g. [START_REF] Hüfner | Photoelectron Spectroscopy[END_REF]) are not considered. Except for the lowest energy band (sp-like), all bands shown in Fig. 6 are mainly of d-character. For unpolarized photons with an angle of incidence of 50 o we thus expect that all transitions discussed above are allowed. However, the data analysis would be more reliable if a DOS calculation weighted by scattering cross sections was available.

Summary

By the combination of thin film deposition techniques optimized for tunneling junction preparation and in-situ ARUPS experiments intrinsic band structure properties of Heusler materials can be investigated. Employing two emission lines of a Helium gas discharge lamp, photoemission spectra of the Heusler compounds Co 2 Cr 0.6 Fe 0.4 Al and Co 2 FeAl were measured. These spectra show different characteristic features in agreement with the predictions of band structure calculations. The spectra show several energy dependent intensity maxima which can be analyzed within the three-step model of photoemission assuming a free electron-like final state. The experimental results for Co 2 FeAl were compared in detail with a theoretical bulk band structure. Good agreement was demonstrated taking into account a small shift of the Fermi energy.

Characteristic ARUPS features due to sample oxidation could be identified. The extreme chemical reactivity is obvious from spectra taken at different oxygen dosage. At only 1L the O 2 derived ARUPS spectrum close to E F changes significantly. At 10L the onset of oxidation leads to a quenching of the spectral features close to E F .

Figure 1 .

 1 Figure 1.Comparison of the ARUPS spectra (hν = 21.2eV) of a clean Co 2 Cr 0.6 Fe 0.4 Al thin film (black line), after exposure to 1L O 2 (blue line), to 5L O 2 (red line) and to 10L O 2 (green line). Additionally, the ARUPS spectrum of a clean Cr thin film is shown as a reference (dashed thin line). The inset shows the spectra near E F in more detail.

Figure 2 .

 2 Figure 2. Comparison of the ARUPS spectra (hν = 40.8eV) of a clean Co 2 Cr 0.6 Fe 0.4 Al thin film (black line), after exposure to 1L O 2 (blue line), to 5L O 2 (red line) and to 10L O 2 (green line).

Figure 3 .

 3 Figure 3. Comparison of the ARUPS spectra (hν =21.2eV) of a Co 2 Cr 0.6 Fe 0.4 Al thin film (red line) and a Co 2 FeAl thin film (blue line). The dashed curve shows the spectrum measured on the Co 2 FeAl sample after oxidation with 10L O 2 for comparison.

Figs. 3

 3 Figs. 3 and 4 compare the ARUPS spectra of the two compounds Co 2 Cr 0.6 Fe 0.4 Al and Co 2 FeAl excited by hν=21.2eV and hν=40.8eV photons.It should be considered that ARUPS is very surface sensitive (electron mean free path 0.4 -1nm[START_REF] Hüfner | Photoelectron Spectroscopy[END_REF]) and that in normal emission from the (001) oriented thin films, assuming an L2 1 structure, only the electronic states with momentum k parallel to the Γ-X direction contribute. However, only moderate DOS anisotropies are expected for cubic crystal structures. Within the usual three-step model of photoemission (ARUPS spectra dominated by direct optical transitions), the intensity is related to a joint density of an initial and a final electronic state[START_REF] Hüfner | Photoelectron Spectroscopy[END_REF](see section V). Nevertheless, emission peaks should always be related to an energy range with a high density of initial states.The most obvious differences in the ARUPS spectra of Co 2 Cr 0.6 Fe 0.4 Al and Co 2 FeAl are observed near the Fermi edge, which is well developed for both compounds. In the case of Co 2 Cr 0.6 Fe 0.4 Al clear shoulder-like features (labeled A for hν = 21.2eV, Fig.3, and E for hν = 40.8eV, Fig.4) are observed on the rising edge of pronounced emission peaks (labeled B and F) at -1.1eV. In the case of Co 2 FeAl an emission peak (labeled B and F) without any shoulder on its edge is observed at -0.85eV.This difference between the two compounds can be related to features of the theoretical (FLAPW-GGA) bulk total density of states (TDOS) as published by G. H. Fecher et al.[17]. Fig.5shows the TDOS of the two Heusler compounds with the L2 1 structure as derived from a digitized spin resolved graph of ref.[17]. Disorder on the Cr,Fe -Al sites (B2 structure) is in general expected to result in a smearing out of the DOS[18]. In the calculation there is a pronounced peak in the DOS of Co 2 Cr 0.6 Fe 0.4 Al (labeled A/E in Fig.5), which does not exist for Co 2 FeAl. We account this peak to the origin of the feature A/E in the ARUPS data. Also the features B/F and

Figure 4 .

 4 Figure 4. Comparison of the ARUPS spectra (hν =40.8eV) of a Co 2 Cr 0.6 Fe 0.4 Al thin film (red line) and a Co 2 FeAl thin film (blue line). The dashed curve shows the spectrum measured on the Co 2 FeAl sample after oxidation with 10L O 2 for comparison.

Figure 5 .

 5 Figure 5. Total density of states of Co 2 Cr 0.6 Fe 0.4 Al and Co 2 FeAl as derived from a digitized graph from G. H. Fecher et al.[17]. Disorder on the Cr,Fe -Al sites (B2 structure) is in general expected to result in a smearing out of the density of states[18].

Figure 6 .

 6 Figure 6. Calculated bulk dispersion branches of Co 2 FeAl parallel to the Γ-X direction digitized from a graph of G. H. Fecher and C. Felser [30] amended by a free electronlike final state dispersion branch (wide yellow lines). The thin black lines indicate the majority branches and the thin gray lines the minority branches. The experimentally observed ARUPS transitions are indicated by blue arrows. The labels provide the assignment to the emission peaks of Figs. 3 and 4.


MD Thomas Lehrnbecher
email: thomas.lehrnbecher@kgu.de

Ralf Schubert

Regina Allwinn

Kader Dogan

Ulrike Koehl

Hans-Peter Grüttner

Revaccination of Children after Completion of Standard Chemotherapy for Acute Lymphoblastic Leukaemia: a Pilot Study Comparing Different Schedules

Keywords:

Since a significant proportion of children with acute lymphoblastic leukaemia (ALL) lose immune protection to tetanus, diphtheria, and poliomyelitis, revaccination is indicated after chemotherapy. Our randomised pilot study comparing different revaccination schedules suggests that children with ALL might be revaccinated with non-live vaccines as early as 3 months after chemotherapy.

Introduction

Survival rates for children suffering from acute lymphoblastic leukaemia (ALL), the most common form of childhood malignancy, approach 80%, and therefore, long-term effects of therapy are of increasing importance. In this respect, various studies have demonstrated that chemotherapy may cause a loss of protective antibody levels to vaccine-preventable diseases in a significant proportion of children (van Tilburg, et al 2006). Unfortunately, data on responses to revaccination after chemotherapy with nonlive vaccines such as vaccines against tetanus or diphtheria are inconsistent (van Tilburg, et al 2006). It is therefore not surprising that considerable discrepancies exist between revaccination schedules, which are primarily based on theoretical considerations and expert opinion. Whereas early timing of revaccination (e.g., at 3 months after chemotherapy) may minimise the time leaving the vulnerable patient unprotected, others recommend revaccination not before 6 to 12 months after completion of chemotherapy to increase the immunogenicity of vaccination (Patel, et al 2007, Zignol, et al 2004). We performed a randomised pilot study to compare the efficacy of vaccines against tetanus,

Patients and Methods

The study was conducted from January 2001 through December 2003 in the Paediatric Haematology and Oncology, University of Frankfurt, Germany. Eligibility criteria included completed chemotherapy for de novo, non-high-risk ALL, no evidence of immunodeficiency before diagnosis of malignancy, and vaccination against diphtheria, tetanus, and poliovirus prior to diagnosis of ALL. The study was approved by the local Ethical Committee. Written informed consent was obtained from the patient and/or parents before patients were randomly assigned to one of the groups using a centralised computer system. According to randomisation, patients received booster DTaP-IPV and Hib conjugate vaccines [Infanrix ™ -IPV+Hib (GlaxoSmithKline)] either at 3, 6 or 9 months after completion of chemotherapy; a hepatitis B vaccine [Gen H-B-Vax ™ (Pasteur MSD/Chiron Behring)] was simultaneously given to some of the patients. Blood samples were obtained at the time of completion of chemotherapy, on the day of revaccination and 3 months after revaccination to assess lymphocyte subsets, immunoglobulin subclasses and serum antibody concentrations or titres. Lymphocyte subsets and immunoglobulin subclasses were analysed as previously reported [START_REF] Huenecke | Age-matched lymphocyte subpopulation reference values in childhood and adolescence: application of exponential regression analysis[END_REF], Lehrnbecher, et al 2009). Antibody concentrations specific to tetanus, diphtheria, and Hib were measured by standardised enzyme-linked immunosorbent assays (ELISA) following the manufacturer's instruction (PROGEN Biotechnik, Heidelberg, Germany). A neutralisation assay was used to detect neutralising antibodies against poliovirus -3 (Rabenau and Weber 1994). Titres against poliovirus ≥1:8 were considered to correlate with protection; antitetanus and antidiphtheria antibody concentrations of ≥0.01 and ≥0.1 IU/mL, and Hib antibody concentrations ≥0.15 and ≥1.0 µg/mL were correlated with short-term and long-term protection, respectively (Patel, et al 2007).

Statistical analysis

Lymphocyte subsets and immunoglobulin subclasses were compared to age-matched reference values (Lehrnbecher, et al 2009). For statistical analysis, antibody concentrations or titres that were below the limit of the assay´s sensitivity were assigned values equal to one-half of the lower detection limit. All antibody concentrations or titres were log-transformed to calculate geometric mean concentration or geometric mean, with 95% CI. Antibody levels before and after revaccinations were compared using Wilcoxon's test. Groups were compared using a multi-dimensional H-test (Kruskal-Wallis). A P value <0.05 (2-sided) was considered significant.

Results

A total of 24 patients [14 boys, 10 girls; median age 7.3 years (range, 4.1-20.7)] were included in the study. All patients were treated according to the clinical trials BFM-ALL 95 (n=15) or BFM-ALL 2000 (n=9). Seven patients received revaccination for tetanus, diphtheria, poliomyelitis, and Hib at 3 months after completion of chemotherapy, and 8 and 9 patients at 6 and 9 months, respectively. Lymphocyte subsets and immunoglobulin subclasses. When analysing the lymphocyte subsets at the time of vaccination, the median absolute number of CD4 + CD3 + cells was 766, 752 and 662 per µl for patients at 3, 6, and 9 months after completion of chemotherapy, respectively [range, 175-1230, 450-1574, and 326-1319, respectively; differences not significant (ns)]; similarly, the median number of CD8 + CD3 + cells was 541 (range, 85-1690), 628 (352-1575), and 558 (275-914) and for CD19 + cells 339 (84-1179), 526 (358-2336), and 413 (245-775) per µl for these patient groups (differences ns). In other words, at the time of vaccination, only one of 7 patients vaccinated at 3 months after completion of chemotherapy was below the age-matched 90% reference interval for CD4 + CD3 + and CD19 + cells, respectively, whereas this was seen for CD8 + CD3 + cells for one patient of each group. Similarly, there was no significant difference of the levels of immunoglobulin subclasses and the percentages of patients below the age-matched reference values assessed at the time of vaccination between the patients revaccinated at 3, 6, and 9 months after completion of chemotherapy, respectively (data not shown).

Tetanus and Diphtheria. At the time of completion of chemotherapy as well as at the time of revaccination, about two thirds of the patients evenly distributed in the different patient groups (e.g., revaccination at 3, 6, and 9 months after chemotherapy, respectively) had tetanus and diphtheria antibody concentrations <0.1 IU/mL, indicating that these patients required booster vaccination (Table ). For both tetanus and diphtheria antibody concentrations, there was a significant increase after vaccination (Table ). Importantly, all patients achieved protecting antibody concentration ≥0.1 IU/mL for tetanus and Poliovirus. Two third of the patients (n=12) evenly distributed in the different patient groups did not have protective levels of antibody titre to all 3 poliovirus serotypes at the time of completion of chemotherapy and at the time of revaccination, respectively (Table ). Specifically, at the time of revaccination, protective titres to serotypes 1, 2, and 3 were present in 81%, 81% and 48% of patients, respectively. For all groups, there was a significant increase of antibody titres after vaccination, and importantly, all 24 patients achieved protection to all serotypes (Table). In the three patients reassessed after 3 years of vaccination (one of each group revaccinated after 3, 6, and 9 after the end of chemotherapy, respectively), antibody titres remained unchanged.

Hib. All but three patients had received at least one dose of Hib conjugate vaccine prior to diagnosis of leukaemia. In contrast to tetanus, diphtheria, and polio, more than half of the patients demonstrated long-term immunity with antibody concentrations ≥1µg/mL at the time of completion of therapy (11/20; 55%) and at the time of revaccination (14/21; 67%), respectively (Table ). After revaccination, the antibody concentration significantly increased, and all patients achieved antibody concentration ≥1µg/mL indicating longterm protection (Table ). Notably, in 21/24 patients (87%), the antibody concentration rose to levels >5 µg/mL which correlates with protection against Hib colonization.

Adverse events. Only one patient revaccinated at 9 months after completion of therapy developed fever after vaccination. No other local or systemic adverse events were noted. The data of this pilot trial corroborate previous reports that a significant proportion of children undergoing therapy for non-high-risk ALL lose protection to tetanus, diphtheria, polio, and Hib (van Tilburg, et al 2006). Our data suggest that revaccination results in protective, long-lasting antibody concentration, independently whether revaccination is given 3, 6 or 9 months after completion of therapy, respectively. In line with the results of a previous study, in the majority of patients, lymphocyte subsets and immunoglobulin levels were already 3 months after the end of chemotherapy above the lower limit of the age-matched 90% reference intervals (Lehrnbecher, et al 2009). Notably, non-high-risk patients exhibit a faster immune recovery than high-risk patients [START_REF] Ek | Immune reconstitution after childhood acute lymphoblastic leukemia is most severely affected in the high risk group[END_REF], Lehrnbecher, et al 2009). A recent review demonstrated that protection rates after completion of chemotherapy vary widely depending on underlying malignancy and chemotherapeutic regimen (e.g., 20-98% for tetanus, 17-98% for diphtheria, 62-100% for poliomyelitis, and 35-100% for Hib) (van Tilburg, et al 2006), and most experts agree that revaccination in these children is indicated. Although in most studies, a significant increase of the levels of protective antibodies was seen after revaccination for tetanus, diphtheria, and poliomyelitis, even when administered already during maintenance chemotherapy for ALL, a comparison of these studies is difficult due to differences in the intensity of chemotherapy and study design [START_REF] Calaminus | Diphtheria (D) and tetanus (T) antibody values in children with acute lymphoblastic leukaemia (ALL) after treatment according to Co-ALL 05/92[END_REF][START_REF] Ercan | Antibody titers and immune response to diphtheria-tetanuspertussis and measles-mumps-rubella vaccination in children treated for acute lymphoblastic leukemia[END_REF], Patel, et al 2007, Zengin and Sarper 2009, Zignol, et al 2004). Unfortunately, no study to date compared the efficacy of different revaccination schedules after chemotherapy, which is always a compromise between reduced immunogenicity after immunosuppression and the risk to leave a patient unprotected for a period of time. Our data suggest that children with non-high-risk ALL might be revaccinated with vaccines to tetanus, diphtheria, and poliomyelitis as early as 3 months after completion of chemotherapy, since efficacy seems to be similar for the different schedules assessed (e.g., at 3, 6, or 9 months after chemotherapy). In contrast to tetanus, diphtheria, and poliomyelitis, the majority of patients demonstrated long-term protective antibody concentrations to Hib ≥1 µg/mL already at the time of completion of chemotherapy.

Discussion

However, revaccination increased antibody concentrations to ≥5 µg/mL in most of the patients, which correlates with protection against Hib colonization and therefore suggests a benefit of revaccination [START_REF] Fernandez | Prevention of Haemophilus influenzae type b colonization by vaccination: correlation with serum anti-capsular IgG concentration[END_REF]. Although we recognise the small study population, we believe that the unequivocal results of this trial may help to minimise the time of children being unprotected against vaccine-preventable diseases after chemotherapy for non-high-risk ALL, and this approach might be transferred to patients receiving less intensive chemotherapy, for example to patients with solid tumours.

  and Haemophilus influenzae type b (Hib) in children revaccinated at 3, 6 or 9 months after completion of non-high-risk ALL chemotherapy, respectively.

  vaccinated after 3, 6 or 9 months after completion of chemotherapy, respectively.

  Geometric mean concentrations (GMC) and titres (GMT) of antibodies to tetanus, diphtheria, poliomyelitis, and Haemophilus influenzae type b (Hib) in children vaccinated at 3 months (group I, n=7), 6 months (group II, n=8), or 9 months (group III, n=9) after completion of chemotherapy for non-high-risk acute lymphoblastic leukaemia. Antibody titres were assessed at the time of completion of therapy ("after therapy"), on the day of revaccination ("pre-vaccination"), and 3 months after revaccination ("post-vaccination"). %, percentage of protected patients * P<0.05 for prior to versus after revaccination ** P<0.005 for prior to versus after revaccination *** P<0.001 for prior to versus after revaccination # P<0.01 for comparison between group 3 and groups 1 and 2, respectively ., Schubert, R., Behl, M., Koenig, M., Rose, M.A., Koehl, U., Meisel, R.& Laws, H.J. (2009) Impaired pneumococcal immunity in children after treatment for acute lymphoblastic leukaemia. British Journal of Haematology, 147, 700-705. Patel, S.R., Ortin, M., Cohen, B.J., Borrow, R., Irving, D., Sheldon, J. & Heath, P.T. (2007) Revaccination of children after completion of standard chemotherapy for acute leukemia. Clinical Infectious Diseases, 44, 635-642. Rabenau, H. & Weber, B. (1994) Evaluation of a new automated microneutralization assay for the quantitative detection of neutralizing antibodies against enteroviruses. Zentralbl Bakteriol, 280, 534-539. van Tilburg, C.M., Sanders, E.A., Rovers, M.M., Wolfs, T.F. & Bierings, M.B. (2006) Loss of antibodies and response to (re-)vaccination in children after treatment for acute lymphocytic leukemia: a systematic review. Leukemia, 20, 1717-1722. Zengin, E. & Sarper, N. (2009) Humoral immunity to diphtheria, tetanus, measles, and hemophilus influenzae type b in children with acute lymphoblastic leukemia and response to re-vaccination. Pediatr Blood Cancer, 53, 967-972. Zignol, M., Peracchi, M., Tridello, G., Pillon, M., Fregonese, F., D'Elia, R., Zanesco, L. & Cesaro, S. (2004) Assessment of humoral immunity to poliomyelitis, tetanus, hepatitis B, measles, rubella, and mumps in children after chemotherapy. Cancer, 101, 635-641.

F

British Journal of Haematology


Yves Colin De Verdière
email: yves.colin-de-verdiere@ujf-grenoble.fr

Semi-classical trace formulas and heat expansions

published or not. The documents may come

Semi-classical trace formulas and heat expansions

Introduction

There is a strong similarity between the expansions of the heat kernel as worked out by people in Riemannian geometry in the seventies (starting with the famous "Can one hear the shape of a drum" by Mark Kac [Kac], the Berger paper [Berger] and the Mc-Kean-Singer paper [McK-Si]) and the so-called semi-classical trace formulas developed by people in semi-classical analysis (starting with Helffer-Robert [He-Ro]). In fact, this is not only a similarity, but, as we will prove, each of these expansions, even if they differ when expressed numerically for some example, can be deduced from the other one as formal expressions of the fields.

Let us look first at the heat expansion on a smooth closed Riemannian manifold of dimension d, (X, g), with the (negative) Laplacian ∆ g 1 . The heat kernel e(t, x, y), with t > 0 and x, y ∈ X, is the Schwartz kernel of exp(t∆ g ): the solution of the heat equation u t -∆ g u = 0 with initial datum u 0 is given by u(t, x) = X e(t, x, y)u 0 (y)|dy| g .

The function e(t, x, x) admits, as t → 0 + , the following asymptotic expansion:

e(t, x, x) ∼ (4πt) -d/2 1 + a 1 (x)t + • • • + a l (x)t l + • • • .
The a l 's are given explicitly in [START_REF] Gilkey | Asymptotic formulas in spectral geometry[END_REF], page 201, for l ≤ 3 and are known for l ≤ 5 [Avramidi, vdV]. See also the related works by Hitrik and Polterovich [START_REF] Hitrik | Existence of resonances in magnetic scattering. On the occasion of the 65th birthday of Professor Michael Eastham[END_REF][START_REF] Polterovich | Heat invariants of Riemannian manifolds[END_REF]. They are universal polynomials in the components of the curvature tensor and its co-variant derivatives. For example a 0 = 1, a 1 = τ g /6 where τ g is the scalar curvature.

1

The previous asymptotic expansion gives the expansion of the trace by integration over X and has been used as an important tool in spectral geometry:

trace(e t∆g ) = X e (t, x, x)

|dx| g = ∞ k=1 e λ k t , where -λ 1 = 0 ≤ -λ 2 ≤ • • • ≤ -λ k ≤ • • • is the sequence of eigenvalues of -∆ g with the usual convention about multiplicities. If d = 2, this gives trace(e t∆g ) = 1 4πt Area(X) + 2πχ(X) 6 t + 0(t 2 ) ,
where χ(X) is the Euler characteristic of X.

There is an extension of the previous expansion in the case of Laplace type operators on fiber bundles: the coefficients of the expansion are then polynomials in the co-variant derivatives of the curvature of the metric and of the connection on the fiber bundle. The heat expansion can be re-interpreted as an expansion of the Schwartz kernel of f (-2 ∆ g ) on the diagonal x = y in powers of with f (u) = exp(-u) and t = 2 . This is a particular case of the semi-classical trace.

Let us describe the semi-classical setting in the flat case: H is a self-adjoint -pseudo-differential operator with Weyl symbol H(x, ξ) in some open domain X in R d , or more generally on a Riemannian manifold. Let f ∈ S(R) and look at f ( H ). Under some suitable assumptions (ellipticity at infinity in ξ) on H, f ( H ) is a pseudo-differential operator whose Weyl symbol f ⋆ (H) is a formal power series in , given, using the Moyal product denoted by ⋆, by the following formula (see [Gracia] for explicit formulas and Section 4.2 for a proof, see also [Charles]) at the point z 0 ∈ T ⋆ X:

f ⋆ (H)(z 0 ) = (2π ) -d ∞ l=0 1 l! f (l) (H(z 0 )) (H -H(z 0 )) ⋆ l (z 0 ) . (1)
From the previous formula, we see that the symbol of f ( H ) at the point z depends only of the Taylor expansions of H at the point z and of f at the point H(z). In the paper [H-P], the authors study the case of the magnetic Schrödinger operator whose Weyl symbol is

H a,V (x, ξ) = d j=1 (ξ j -a j (x)) 2 + V (x)
and show that the Schwartz kernel of f ( H ,a,V ) at the point (x, x) admits an asymptotic expansion of the form

[f ( H ,a,V )](x, x) = (2π ) -d   ∞ j=0 2j   k j l=0 R d f (l) ( ξ 2 + V (x))Q a,V j,l (x, ξ)|dξ|    
where the Q a,V j,l (x, ξ)'s are polynomials in ξ calculated from the Taylor expansions of the magnetic field B = da and V at the point x. The proof in [H-P] uses a pseudo-differential calculus adapted to the magnetic field.

We will give a simplified version of the expansion replacing the (non unique) Q a,V j,l (x, ξ)'s by functions P B,V j,l (x)'s which are uniquely defined and are given by universal O(d)-invariant polynomials of the Taylor expansions of B and V at the point x. We present then two ways to compute the P B,V j,l 's: • we can first use Weyl's invariant theory (see Gilkey's book [Gilkey2]) in order to reduce the problem to the determination of a finite number of numerical coefficients; then simple examples, like harmonic oscillator and constant magnetic field, allow to determine (part of) these coefficients.

• The P B,V j,l 's are related in a very simple way to the coefficients of the heat expansion; it is possible to compute the P B,V j,l 's from the knowledge of the a l 's for j + 1 ≤ l ≤ 3j. This is enough to re-compute the coefficient of2 and also, in principle, the coefficients of 4 in the expansion, because the a l 's are known up to l = 6 in the case of a flat metric (see [vdV]).

In this note, we will first describe precisely the semi-classical expansion for Schrödinger operators (in the case of an Euclidian metric) and the properties of the functions P B,V j,l (x)'s. Then, we will show how to compute the P B,V j,l (x)'s using an adaptation of the method used for the heat kernel (Weyl's Theorem on invariants and explicit examples). Finally, we will explain how the a l 's are related to the P B,V j,l (x)'s. This gives us two proofs of the main formula given by Helffer and Purice in [H-P]; this paper was the initial motivation to this work.

Semi-classical trace for Schrödinger operators

In what follows, X is an open domain in R d , equipped with the canonical Euclidian metric, and Ω k (X) will denote the space of smooth exterior differential forms in X. Let us give a Schrödinger operator, with a smooth magnetic field B = 1≤i<j≤d b ij dx i ∧ dx j (a closed real 2-form) and a smooth electric potential V (a real valued smooth function) in X. We assume that V is bounded from below. We will assume also that the 2-form B is exact and can be written B = da and we introduce the Schrödinger operator defined by

H ,a,V = d j=1 i ∂ ∂x j -a j (x) The Weyl symbol of H ,a,V is H a,V (x, ξ) = ξ -a(x) 2 + V (x). We denote by H ,a,V a self-adjoint extension of H ,a,V in L 2 (X, |dx|). Let us give f ∈ S(R) and φ ∈ C ∞ o (X)
and consider the trace of φf ( H ,a,V ) as a distribution on X × R (the density of states):

Trace(φf ( H ,a,V )) = X Z ,a,V (g)(x)φ(x)|dx| , where Z ,a,V (g)(x) is the value at the point (x, x) of the Schwartz kernel of f ( H ,a,V ).
Theorem 1 We have the following asymptotic expansion in power of :

Z ,a,V (g)(x) ∼ (2π ) -d R d f ( ξ 2 + V (x))|dξ| + ∞ j=1 2j l=3j l=j+1 P B,V j,l (x) R d f (l) ( ξ 2 + V (x))|dξ|
We have the explicit formulas

P B,V 1,2 = - 1 6 ∆V + B 2 , P B,V 1,3 = - 1 12 ∇V 2 , P B,V 2,3 = - 1 180 8 ∇B 2 + d ⋆ B 2 + 12 ∆B|B + 3∆ 2 V .
Here

B 2 = 1≤i<j≤d b 2 ij , d ⋆ : Ω 2 (X) → Ω 1 (X)
is the formal adjoint of d used in the definition of the Hodge Laplacian on exterior forms. If d = 3, B is the Euclidean norm of the vector field associated to B.

The P B,V j,l (x) are polynomials of the derivatives of B and V at the point x. Moreover, if λ, µ, c are constants and we define λ ⋆ (f )(x) = f (λx), we have the following scaling properties:

1. P λ.λ ⋆ (B),λ ⋆ (V ) j,l (x) = λ 2j P B,V j,l (λx)
. This will be used with x = 0.

2. P µB,µ 2 V j,l (x) = µ 2(l-j) P B,V j,l (x) 3. P B,V +c j,l (x) = P B,V j,l (x) 4. P -B,V j,l (x) = P B,V j,l (x).
5. The P B,V j,l 's are invariant by the natural action of the orthogonal group O(d) on the Taylor expansions of B and V at the point x.

Remark 1 From the statement of the previous Theorem, we see that the expansion of the density of states is independent of the chosen self-adjoint extension.

As a consequence, we can get the following full trace expansion under some more assumptions:

Corollary 1 Let us assume that E 0 = inf V < E ∞ = lim inf x→∂X V (x) and that we have chosen the Dirichlet boundary conditions. Let f ∈ C ∞ o (] -∞, E ∞ [),
then the trace of f ( H ,a,V ) admits the following asymptotic expansion

Trace(f ( H ,a,V )) ∼ (2π ) -d X R d f ( ξ 2 + V (x))|dξ| + • • • • • • ∞ j=1 2j l=3j l=j+1 P B,V j,l (x) R d f (l) ( ξ 2 + V (x))|dξ| |dx| ,
The coefficient of 2 can be written as

- 1 12 X×R d f (2) ( ξ 2 + V (x)) ∆V (x) + 2 B(x) 2 |dxdξ| .
The expansion follows from [He-Ro]. An integration by part in x gives

X f (3) ( ξ 2 + V (x)) ∇V (x) 2 |dx| = - X f (2) ( ξ 2 + V (x))∆V (x)|dx| .
2 Existence of the -expansion of Z ,a,V Using Theorem 2 in the Appendix, we can work in R d with a and V compactly supported. The existence of the expansion is known in general from [He-Ro] and the calculus of the symbol of f ( H ,a,V ). We get

X Z ,a,V (f )(x)φ(x)|dx| = (2π ) -d   ∞ j=0 2j k j l=0 φ(x)f (l) (H a,V (x, ξ))Q j,l (x, ξ)|dxdξ|  
where the Q j,l (x, ξ)'s are polynomials in the Taylor expansion of H a,V at the point (x, ξ). The previous expansion is valid for any (admissible) pseudo-differential operator. In the case of Schrödinger operators we can make integrations by part in the integrals f (l) (H a,V (x, ξ))Q j,l (x, ξ)|dξ| which reduces to a similar formula where we can replace the Q j,l (x, ξ)'s by P j,l (x)'s. This is based on the expansion of Q j,l as a polynomial in ξ in powers of (ξ -a): odd powers give 0 and even powers can be reduced using

d ξ (ξ j -a j )f (l) (H a,V )ι(∂ ξ j )dξ = 2 ξ j -a j 2 f (l+1) (H a,V )dξ + f (l) (H a,V )dξ .
We have only to check that the powers of ξ in Q j,l (x, ξ) are less than l: this is based on Equation (1). The coefficients of the l-th Moyal power of H a,V (z) -H a,V (z 0 ) are homogeneous polynomials of degree l in the derivatives of H a,V (z). At the point z = z 0 only derivatives of order ≥ 1 are involved. They are all of degree ≤ 1 in ξ. Using Gauge invariance at the point x (Section 3), we can assume that a(x) = 0.

Gauge invariance

If S : X → R is a smooth function, we have Trace(φe -iS(x)/ f ( H ,a,V )e iS(x)/ ) = Trace(φf ( H ,a,V )) and e -iS(x)/ f ( H ,a,V )e iS(x)/ = f ( H a+dS,V ) .

Hence, we can chose any local gauge a in order to compute the expansion: using the synchronous gauge (see Section 4), we get the individual terms

f (l) (H 0,V )P B,V j,l (x)|dξ|
for the expansion, where the P B,V j,l (x) depend only of the Taylor expansions of B and V at the point x.

4

The synchronous gauge

The main idea is to find an appropriate gauge a adapted to the point x 0 where we want to make the symbolic computation. In a geometric language, we use the trivialization of the bundle by parallel transportation along the rays: the potential a vanishes on the radial vector field2 . Here, this is simply the fact that, for any closed 2-form B on R 2 , there exists an unique 1-form a = d j=1 a j dx j so that da = B and d j=1 x j a j = 0. We will do that for the Taylor expansions degree by degree. In what follows we will use a decomposition for 1-forms, but it works also for k-forms.

Let us denote by Ω k N the finite dimensional vector space of k-differential forms on R d whose coefficients are homogeneous polynomials of degree N and by W = d j=1 x j ∂ ∂x j the radial vector field. The exterior differential induces a linear map from Ω k N into Ω k+1 N -1 and the inner product ι(W ) a map from Ω k N into Ω k-1 N +1 . They define complexes which are exact except at k = N = 0. Moreover, we have a situation similar to Hodge theory:

Ω k N = dΩ k-1 N +1 ⊕ ι(W )Ω k+1 N -1
. This is due to Cartan's formula: the Lie derivative of a form ω ∈ Ω k N satisfies, from the direct calculation, L W ω = (k + N)ω, and, by Cartan's formula,

L W ω = d(ι(W )ω) + ι(W )dω. So ω = 1 k + N (d(ι(W )ω) + ι(W )dω) .
It remains to show that this is a direct sum: if ω = dα = ι(W )γ, we have ι(W )ω = 0 and dω = 0; from the previous decomposition, we see that ω = 0. Let us denote by J N ω, where ω is a differential form of degree k, the form in Ω k N which appears in the Taylor expansion of ω.

We get

Proposition 1 If P (J 0 a, J 1 a, • • • , J N a) is a polynomial in the Taylor expansion of the 1-form a at some order N which is invariant by a → a + dS, P is independent of J 0 a and

P J 1 a, • • • , J N a = P 1 2 J 1 ι(W )B, • • • , 1 N + 1 J N ι(W )B
is a polynomial of the Taylor expansion of B to the order N -1.

5 Properties of the P j,l 's 5.1 Range of l for j fixed

From the scaling properties, we deduce that, in a monomial

D α 1 B i 1 ,j 1 • • • D α k B i k ,j k D β 1 V • • • D βm V ,
belonging to P j,l , we have k + 2m = 2(l -j) and k

+ |α 1 | + • • • + |α k | + |β 1 | + • • • + |β m | = 2j. Moreover, for j ≥ 1, k + m ≥ 1 and |β p | ≥ 1. Hence j + 1 ≤ l ≤ 3j.
The previous bounds are sharp: take the monomials ∆ j V and ∇V 2j which give l = j + 1 and l = 3j.

Invariance properties

1. Let us assume that we look at the point x = 0 and consider the operator D µ (f )(x) = f (µx). We have

D µ • Ĥ ,A,V • D 1/µ = Ĥ /µ,A•Dµ,V •Dµ .
The same relation is true for any function f ( Ĥ ,A,V ) and then we have, looking at the Schwartz kernels and using the Jacobian µ d of D µ :

P B,V j,l (0) R d f (l) ( ξ 2 +V (0))|dξ| = µ -2j P µ.µ ⋆ B,µ ⋆ V j,l (0) R d f (l) ( ξ 2 +V (0))|dξ| .
2. We have

H ,µa,µ 2 V = µ 2 H µ ,a,V .
3. Changing V into V + c gives a translation by c in the function f but does not change the P B,V j,l 's.

4. Changing B into -B gives a complex conjugation in the computations. The final result is real valued.

5. Orthogonal invariance is clear: an orthogonal change of coordinates around the point x preserves the density of states.

The case d = 2

We deduce from the scaling properties and invariance by the orthogonal group, that there exists constants

a d , b d , c d so that P B,V 1,2 (x) = a d ∆V +b d B 2 , P 1,3 (x) = c d ∇V 2 .

Explicit examples

The calculation for the harmonic oscillators and the constant magnetic fields allows to determine the constants a d , b d , c d .

Harmonic oscillators

Let us consider Ω = -2 d 2 dx 2 + x 2 with d = 1. The kernel of P (t, x, y) of exp(-tΩ) is given by the Mehler formula:

P (t, x, y) = (2π sinh(2t )) -1 2 exp - 1 2 sinh(2t ) cosh(2t )(x 2 + y 2 ) -2xy . Hence P (t, x, x) ∼ (2π ) -1 e -tx 2 R e -tξ 2 dξ 1 -2 (t 2 -t 3 x 2 )/3 + 0( 4 ) . Hence P 1,2 (x) = -V ′′ (x)/6, P 1,3 (x) = -V ′ (x) 2 /12,.
Similarly, in dimension d > 1, we get P 1,2 (x) = -∆V (x)/6, P 1,3 (x) = -∇V 2 /12,.

Constant magnetic field

Let us consider the case of a constant magnetic field B in the plane and denote by Q(t, x, y) the kernel of exp(-tH B,0 ). We have (see [A-H-S])

Q(t, x, x) = B 4π sinh Bt .
Hence the asymptotic expansion

Q(t, x, x) = (2π ) -2 exp(-t ξ 2 )|dξ| 1 -t 2 2 B 2 /6 + 0( 4 ) , hence P 1,2 (x) = -B 2 /6, P 1,3 (x) = 0. Using the normal form B = b 12 dx 1 ∧ dx 2 + b 34 dx 3 ∧ dx 4 + • • • , we get in dimension d > 2, P 1,2 (x) = -B 2 /6, P 1,3 (x) = 0.
7 Heat expansion from the semi-classical expansions

We have t H 1,a,V = H √ t, √
ta,tV . Using the expansion of Theorem 1 with f (E) = e -E , we get easily the point-wise expansion of the heat kernel on the diagonal as

t → 0 + : [exp(-t H 1,a,V )](x, x) ∼ 1 (4πt) d/2 e -tV (x)   ∞ l=0   l/3≤j≤l-1 P B,V j,l (x)   (-t) l   .
In particular, a 1 (x) = -V (x) and the coefficient a 2 (x) is given by

a 2 (x) = 1 2 V (x) 2 - 1 6 ∆V (x) - 1 6 B(x) 2 .
This formula agrees with Equation (3) of Theorem 3.3.1 in [START_REF] Gilkey | Asymptotic formulas in spectral geometry[END_REF]. This gives another way to compute the P j,l 's: if, as power series in t,

∞ l=0 (-1) l b l (x)t l = e tV (x) ∞ l=0 a l (x)t l ,
we have l/3≤j≤l-1

P B,V j,l (x) = b l (x) . P B,V j,l
is the sum of monomials homogeneous of degree 2(l -j) in b l where B and its derivatives have weights 1 while V and its derivatives have weights 2.

The heat coefficients a l on flat spaces are known for l ≤ 6 from [vdV]. This is enough to check the term in 2 (uses a 2 and a 3 ) in [H-P] and to compute the term in 4 in the semi-classical expansion (uses the a l 's for 3 ≤ l ≤ 6).

We have also a mixed expansion writing t H ,a,V = H √ t , √ ta,tV , we get a power series expansion in powers of and t valid in the domain 2 t → 0 and 0 < t ≤ t 0 for the point-wise trace of exp(-t H ,a,V ):

Z t, (x) ∼ 1 (4πt) d/2 e -tV (x) 1 + j≥1, j+1≤l≤3j 2j (-t) l P B,V j,l (x) .
This shows that the integrals X V (x) k |dx| and X P B,V j,l (x)|dx| are recoverable from the semi-classical spectrum.

Appendix: functional calculus in domains and self-adjoint extensions (after Johannes Sjöstrand)

The content of this Appendix is due to Johannes Sjöstrand. I thank him very much for this contribution.

Let X ⊂ R d be an open set, we say that a linear operator A is a ΨDO in X, with Weyl symbol a, if, for any compact K ⊂ X, A acts on functions supported in K as a ΨDO of Weyl symbol a.

Theorem 2 Let H ,a,V be a Schrödinger operator with magnetic field given by

H ,a,V = d j=1 i ∂ ∂x j -a j (x) 2 + V (x) ,
defined in some open domain X ⊂ R d . We assume that a and V are smooth in X and that V is bounded from below, so that H ,a,V admits some self-adjoint extensions on the Hilbert space L 2 (X, |dx|). One of them will be denoted by H ,a,V . Then, for any f ∈ S(R), f ( H ,a,V ), given by the functional calculus, is a semiclassical ΨDO in X whose symbol is given by Equation (1) and is independent of the chosen extension.

The proof uses a multi-commutator method already used by Helffer and Sjöstrand. Proof.-We introduce, for s ∈ R, the semi-classical ( -dependent) Sobolev spaces

H s := {u ∈ S ′ (R d ) | Op (1 + ξ 2 ) s/2 u L 2 < ∞ } with the norm u s := Op (1 + ξ 2 ) s/2 u L 2 .
The ( -dependent) norm A s 1 ,s 2 is the norm of A as linear operator from H s 1 to H s 2 . A linear operator K is smoothing if, for all s 1 , s 2 , K s 1 ,s 2 = O( ∞ ) . This implies that the Schwartz kernel of K is smooth with all derivatives locally O( ∞ ). We have the Lemma 1 Let Y be an open set in R d . Let P j = P j ( ), j = 0, 1 be two self-adjoint operators on Hilbert spaces

H j = L 2 (X j , |dx|) with Y ⊂⊂ X 0 ⊆ X 1 ⊆ R d and with domains D j so that C ∞ o (Y ) ⊂ D j ⊂ H j . Let us assume that, on C ∞ o (Y ), P 0 = P 1 = H ,a,V (= P ). Then, for any f ∈ C ∞ o (R), f (P 0 ) -f (P 1
) is smoothing on Y . In particular, the densities of states [f (P j )](x, x), j = 0, 1, coïncide in Y modulo O( ∞ ).

Assuming Lemma 1, Theorem 2 follows by extending a and V smoothly outside Y so that they have compact support in R d . We take Y ⊂⊂ X = X 0 ⊂ R d = X 1 . It follows that P 1 is essentially self-adjoint and the functional calculus for P 1 follows then easily from [He-Ro]. The result is valid even for f ∈ S(R) because C ∞ o is dense in S and the result of [He-Ro] is valid for f ∈ S and the resulting formulas for the symbols are continuous w.r. to the topology of S.

Proof of Lemma 1.-If χ ∈ C ∞ o (Y ), then, for z / ∈ R and j, k ∈ {0, 1}, we have on L 2 (Y ): (P j -z) -1 • χ = χ • (P k -z) -1 -(P j -z) -1 [P, χ](P k -z) -1
(2)

Let χ 0 ≤ χ 1 ≤ • • • ≤ χ N with, for l = 0, • • • , N, χ l ∈ C ∞ o (Y )
and, for l = 0, • • • , N -1, χ l (1 -χ l+1 ) ≡ 0. By iterating Equation (2) and using χ l+1 [P, χ l ] = [P, χ l ], we find:

(P 1 -z) -1 • χ 0 = χ 1 • (P 0 -z) -1 χ 0 -χ 2 • (P 0 -z) -1 [P, χ 1 ](P 0 -z) -1 χ 0 + • • • ±χ N (P 0 -z) -1 [P, χ N -1 ](P 0 -z) -1 [P, χ N -2 ] • • • (P 0 -z) -1 χ 0 ∓(P 1 -z) -1 [P, χ N ](P 0 -z) -1 • • • (P 0 -z) -1 χ 0 Let us give now χ 0 , ψ ∈ C ∞ o (Y ) with disjoints supports. By choosing the χ ′ l s for l > 0 with supports disjoint from the support of ψ, we see, using Equation (2), that, for any N, ψ(P 1 -z) -1 χ 0 0,2 = O N |ℑz| -(N +1) .

The standard a priori elliptic estimates u s+2,Ω 1 ≤ C ( (P -z)u s,Ω 2 + u s,Ω 1 ) for z ∈ K ⊂⊂ C and Ω 1 ⊂⊂ Ω 2 ⊂⊂ R d , allow to prove that, for any N, s, there exists M(N, s) so that ψ(P 1 -z) -1 χ 0 s,s+N +2 = O( N |ℑz| -M (N,s) )

(3)

Let χ ∈ C ∞ o (Y ) so that χ ≡ 1 on the support of χ 0 . Let us apply a multiplication by χ 0 to the right and to the left to Equation (2) and choose ψ with support disjoint of χ 0 so that [P, χ]ψ = [P, χ]. Inserting ψ this way in Equation (2), we get, using Equation (3): χ 0 (P 1 -z) -1 χ 0 -χ 0 (P 0 -z) -1 χ 0 = K and, for any N, there exists M(N) so that K -N,N = O( N ℑz -M (N ) ). We now apply the formula (known to some people as the "Helffer-Sjöstrand formula", proved for example in the book [Di-Sj], p. 94-95), valid for f ∈ C ∞ o (R) and f an almost holomorphic extension of f :

f (P j ) = 1 π C ∂ z f (z)(P j -z) -1 dL(z) ,
where dL(z) is the canonical Lebesgue measure in the complex plane. From this, we see that f (P 0 ) -f (P 1 ) is smoothing in Y .

+ V (x) .

This gauge is sometimes called the Fock-Schwinger gauge; in [A-B-P], it is called the synchronous framing

Acknowledgments

I thank the referee for his careful reading of the paper which forced me to make the statements and proofs more precise. Many thanks also to Johannes Sjöstrand for his help in clarifying some points of semi-classical analysis and for allowing me to present them in an Appendix to this paper.


Guillaume Artignan
email: artignan@lirmm.fr

Mountaz Hascoët
email: mountaz@lirmm.fr

Visual Analysis of Clustering Algorithms A methodology and a case study

Clustering is probably one of the most frequently used approaches when facing a scaling problem in large collections of documents. In many situations, however, the choice of the most appropriate algorithm for clustering can turn into a real dilemma. Numerical criteria have been proposed to evaluate the quality of the results of clustering algorithms. However, so many different criteria have been proposed that the dilemma even worsens. Most criteria reveal different aspects of the quality of the results and hide others. The aim of this paper is to help with the understanding of clustering and to facilitate the comparison and the choice of clustering algorithm for a given purpose. Our proposal consists in studying both quality evaluation criteria and clustering algorithms. We start by discussing a selected set of representative criteria, and further conduct a case study on a large set of real data, measuring not only the quality of different representative clustering algorithms but also the impact of each criterion on the ranking of the algorithms. By providing empirical results on large scale corpus of either documents or lexical networks useful to digital library, we hope to clarify the field and facilitate designers' choices.

Introduction

In digital libraries, clustering can be used at various stages. Either as a post treatment for a search engine to cluster the results instead of displaying ranked lists, or as a tool to automatically extract thesauri from a set of documents. It can also be used to automatically organize a collection of documents into a catalog or a directory. In all cases, the designer eager to perform clustering has to choose amongst thousands of algorithms. As recalled by Jain in his recent review on that subject [START_REF] Jain | Data clustering: 50 years beyond K-means[END_REF], "There is no best clustering algorithm". Indeed, the choice of the appropriate algorithm for one purpose is multi-factorial. Amongst all possible important factors that would impact the choice, the quality or nature of the results is probably the most important, and at the same time, the most obscure and difficult to evaluate. Surprisingly, very few efforts have been made to help non clustering experts to understand what is at stake and how to compare the quality of various clustering algorithms. Our aim is to report the results of a case study conducted on large information collections relevant for most designers of information systems. Because our aim is to evaluate the quality of clustering, the data on which we have worked was chosen carefully. The first set of data was extracted from "Jeux de Mots", a lexical network of the French language. Jeux de Mots is one of the most accurate and complete publicly available lexical network for French [START_REF] Lafourcade | Making people play for lexical acquisition with the jeuxdemots prototype[END_REF]. The second set of data is extracted from papers published in the information retrieval field since 1980. Our approach to the multitude of models underlying clustering approaches is to use graph based representation as a common ground to simplify and unify notations. Most current data sets used in digital libraries can be conceptually represented by graphs and most current clustering algorithms can be simplified using graph based representations. Our methodology for analyzing the quality of various approaches is experimental and exploratory. For that purpose visual analysis was used to explore the results of our experiments. In this paper, we first describe the datasets used for the experiments. We further describe and discuss the quality criteria used to analytically evaluate the results. We then rapidly review the algorithms selected for the experiment. Finally, the two last sections report the results, and further discuss them.

Datasets

Four datasets are derived from "Jeux de Mots", JdmAll contains Jdm2000 which contains Jdm200 containing at his turn Jdm20 and that they respectively contains 111701 nodes, 2000 nodes, 200 nodes and 20 nodes. We built two datasets from a corpus of 635 research papers in the field of information retrieval and digital libraries. We compute similarities between each pair of documents using the TF-IDF measure [START_REF] Salton | Term-weighting approaches in automatic text retrieval[END_REF] and a Pearson's correlation. A complete graph is then obtained, where nodes are document and similarities are weighted links. We construct Sig1000 and Sig10000 by keeping respectively the 1000 and 10000 best similarity relations. For all these six datasets we studied the degree distribution. Jdm200, Jdm2000, JdmAll and Sig1000 have a degree distributions following a power-law tails. We note γ the exponent. The Tab. 2 describes the datasets by providing the name, the number of nodes N , the total number of edges E, the exponent γ, the graph diameter D, the averaged clustering coefficient C, an URL describing the different dataset and proposing a link for downloading them. To describe the six different criteria selected from the litterature and used in the experiment, we furtherintroduce the following notations:

n is the number of nodes in G, e is the number of edges in G, k is the number of clusters, n i is the number of nodes in the cluster C i , we i is the number of edges within the cluster C i , oe i is the number of edges outgoing from the cluster C i be ij is the number of edges between the two clusters C i and C j pe i is the number of possible edges between two different nodes in C i . For an undirected graph : pe i = n i (n i -1)/2. For a directed graph : pe i = n i (n i -1). me i is the number of missing edges in C i . Hence, me i = pe i -we i .

we, be,pe and me are the total number of respectively within cluster edges, between cluster edges possible edges and missing edges. -Therefore, we =

k i=1 we i , be = k i=1 be i , me = k i=1 me i .
Cut [START_REF] Boutin | Filtrage, partitionnement et visualisation multi-échelles de graphes d'interactions à partir d'un focus[END_REF] is computed as the number of between-edges (also called extra-edges) over the number of within-edges(also called intra-edges). Lowest values correspond to best clustering results.

Cut(G) = be we (1)
Perf [START_REF] Boutin | Filtrage, partitionnement et visualisation multi-échelles de graphes d'interactions à partir d'un focus[END_REF] takes into accounts for the number of links between clusters, as well as the number of missing within edges, e.g nodes grouped in the same cluster without edges relating each other.

P erf (G) = 1 - be + me pe (2)
Cond [START_REF] Boutin | Filtrage, partitionnement et visualisation multi-échelles de graphes d'interactions à partir d'un focus[END_REF] criterion is an average over the conductance of each pair of clusters. The conductance of a pair of cluster Ci and Cj is the proportion of edges between Ci to Cj divided by the minimum number of edges within Ci and Cj. Lowest values corresponds to best clustering results. We consider we i or we j equals to one in case of singleton clusters.

Cond(G) = k i<j beij min(wei,wej ) k(k -1)/2 (3)
Cov [START_REF] Boutin | Filtrage, partitionnement et visualisation multi-échelles de graphes d'interactions à partir d'un focus[END_REF] is the ratio of number of within-edges to number total edges in the graph.

Cov(G) = we e (4)
MQ [START_REF] Boutin | Filtrage, partitionnement et visualisation multi-échelles de graphes d'interactions à partir d'un focus[END_REF] MQ is a difference between the average within-cluster edge density and between-cluster edge density. Therefore it varies between -1 and +1, and highest values correspond to best clustering results. In case of singleton cluster, we i and pe i equal 0. In this case, we do not compute we i /pe i but use the value of 1 instead.

M Q(G) = k i=1 (we i /pe i ) k - k i<j (be ij /n i n j ) (k(k -1))/2 (5)
Mod [START_REF] Clauset | Finding community structure in very large networks[END_REF] can be considered as a measure of Cov defined above corrected by the Cov computed for a random clustering of the same graph and that we note rCov. Therefore, the highest values for Mod correspond to best clustering results according to Cov and values below 0 correspond to clustering worse than random according to the Cov criteria. However, the computation of rCov is still debatable and would deserve a discussion that would lead us beyond the scope of this paper.

M od(G) = Cov -rCov (6)

Clustering Algorithms

Clustering has a huge and multidisciplinary history since it has been used in many scientific fields including in information retrieval [START_REF] Wang | Regularized clustering for documents[END_REF], data visualization [START_REF] Abello | Ask-graphview: A large scale graph visualization system[END_REF], physics [START_REF] Clauset | Finding community structure in very large networks[END_REF], etc. Several surveys have reviewed partially this litterature [START_REF] Xu | Survey of clustering algorithms[END_REF][START_REF] Jain | Data clustering: 50 years beyond K-means[END_REF][START_REF] Schaeffer | Graph clustering[END_REF]. In order to choose the algorithms to be tested in our study we had three criteria in mind. First, source code for the proposed algorithm is provided by authors or the description of the algorithm is sufficiently clear, complete and precise to be implemented. Second, the algorithm is relevant to clustering data such as documents or keywords. The CNM algorithm [START_REF] Clauset | Finding community structure in very large networks[END_REF] has a bottom-up approach. Communities are made for each node and further merged iteratively merged with others to increase the criteria of modularity measure defined in equation Equ. [START_REF] Dhillon | Concept decompositions for large sparse text data using clustering[END_REF]. CNM results can be represented by a hierarchical clustered graph or a simple clustered graph depending on how merging is handled.

The BGLL algorithm [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF] approach is very similar to CNM, but the definition of modularity differs and it makes the hierarchical clustered graph explicit as well as the level at which the clusters are extracted from the hierarchical clustered graphs.

The CMJA algorithm [START_REF] Auber | Multiscale visualization of small world networks[END_REF] has a different approach from the two previous.

CMJA is proposed for detecting communities in small world networks by identifying weak edges. The algorithm operates in two steps. Firstly, it processes a score on each edge, this score is proportional with the number of 4-cycles and 3-cycles containing the edge. Secondly it removes the k edges with the lowest scores. Clusters are the resulting connected components.

The InfoMap approach [START_REF] Rosvall | Maps of random walks on complex networks reveal community structure[END_REF] treats the problem of finding community structures in networks as an information coding problem. The approach has three steps: (1) Infomap processes a random walk on the graph and generates the random path, (2) assigns a codeword to each node in the random pass using Huffman coding [START_REF] Huffman | A method for the construction of minimum-redundancy codes[END_REF], (3) searches a clustering minimizing the average number of bits useful to describe it.

The MCL Algorithm [START_REF] Van Dongen | Graph clustering by flow simulation[END_REF] detects communities using a Markov Matrix. The algorithm computes random walks by flow simulation. An operator named expansion computes n multiplications of the matrix with itself. An operator named inflation computes the Hadamard matrix [START_REF] Seberry | Hadamard matrices, sequences, and block designs[END_REF].

K-Means Algorithm [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF] is one of the most frequently used algorithm for clustering and many slightly different versions have been proposed. The main principle is to start with an arbitraty partition of the dataset and try to move each element to a better cluster as long as possible to improve the overall within clusters cohesion. It is one very efficient and very simple algorithm to implement. However, its based on centroid computation. This implies that its prerequisite is that computing centroids makes sense for the dataset to be clustered.

LinLog Algorithm [START_REF] Noack | An energy model for visual graph clustering[END_REF] is a layout algorithm based on an energy model that aims at geometrically exhibiting clusters. Its principle is to optimize the layout accounting mainly for attraction and repulsion forces between nodes.

The NCut Algorithm [START_REF] Shi | Normalized cuts and image segmentation[END_REF] comes from image segmentation domain but can be adapted to graphs. Its principle is to optimize a criteria named normalized cut, using a spectral technique.

The Cluto Toolkit [START_REF] Zhao | Criterion functions for document clustering: Experiments and analysis[END_REF] is a toolkit made of several clustering algorithms. Four approaches are tested in this paper: (1) The rb-based clustering approach proposed clustering computed by K-1 bisections, (2) the direct-based clustering approach, (3) an agglomerative approach, (4) the graph-based approach based on a similarity graph and a min-cut criterion.

The Spherical K-Means algorithm [START_REF] Dhillon | Concept decompositions for large sparse text data using clustering[END_REF] is an extension of the well-known Euclidian K-Means algorithm. This algorithm partitions the dimension using great hypercircle.

Results

In this paper, we combine different approaches to compare the results of clustering. First, we unified the analytical criteria extracted from the literature to facilitate the comparison of different criteria and discuss their interpretation. Second, we empirically measured these criteria on datasets clustered by the different clustering algorithms presented in previous sections. " To analyze the results we started by computing the ranking of each algorithm according to each criteria and for each dataset. Table 3 reports average ranking over different datasets of a subset of algorithms. It also reports the aggregated ranking over all criteria and computed as the average of all criteria. We further computed the Spearman correlation for each pair of criteria on each dataset and computed an average of all Spearman measure over all datasets. The results are reported in the diagram of Fig. 1 where average Spearman values between two criteria is written next to the edge connecting the two criteria on the diagram. For example, the average value found for Spearman correlation between cov and mod is 0.43. Overall, with the exception of 1/cut and cov, most criteria are not strictly correlated.

The variability of the results was an incentive to use a visual analysis approach to better understand the cause of variability. Therefore we used parallel coordinate diagrams [START_REF] Inselberg | Parallel coordinates: a tool for visualizing multidimensional geometry[END_REF] to interpret the results of the experiments. Figure 2 and3 show two such diagrams corresponding to two different datasets. In these diagrams several vertical axes are used to embody different dimensions of the data explored. In our case, the vertical axes were used to embody the different quality criteria. Each item, in our case each algorithm is further represented by polyline that joins the values corresponding to that same item for all axes. These diagrams offer well-known benefits: (1) covariance/contravariance of ranking between two adjacent criteria is visually obvious, (2) the distribution of the values for each criteria is also visually obvious for all criteria and (3) it is very easy to select an algorithm that performs best for one criteria and to see how it compares to other algorithms on the other criteria.

Discussion

General trends : important variations of ranking over criteria and datasets. Table 3 shows the average ranking for and algorithms used in the experiment. It illustrates (1) important variations over different criteria and (2) very average ranking for all algorithms when all ranking are combined. These results tend to corroborate the position of both Jain and Buxton. Indeed W. Buxton used to say: "everything that is best for something is worse for something else". And Jain, recently wrote: "While numerous clustering algorithms have been published and new ones continue to appear, there is no single clustering algorithm that has been shown to dominate other algorithms across all application domains [...] with the emergence of new applications, it has become increasingly clear that the task of seeking the best clustering principle might indeed be futile". However, that said, it is important to better understand the variations of the quality of algorithms measured by different criteria over varying datasets and how the three interact.

What are the dependencies between Cut, Cov, Cond and Mod ? Somehow all criteria used in the experiment try to capture how similar the elements inside clusters are and how dissimilar the clusters are one from another. Ratio of between-edges over within-edges are used in the definition of four out of the six criteria used in the experiment. However these ratio are not exactly computed the same way and small differences in their definition sometimes has huge impact on the results. Cut and 1/cov are strictly covariant. In fact, Cov can be considered as the normalized version of cut. This is confirmed by the empirical evidence as Spearman correlation between 1/cut and cov is 1, and parallel coordinates Both empirical and analytical results further suggests that mod can be considered as a measure of cov corrected by random. Therefore cov and mod are partially correlated. This is corroborated by both Spearman and parallel coordinate diagrams that clearly show that the two criteria are related but not strictly correlated. More work is needed to characterize the relationship more accurately.

Cut and cond use ratio of between-edges over within-edges. However cut has a global computation of the ratio, whereas cond not only computes the ratio at the cluster level but also consider only the minimum number of within-edges in each cluster. This observation is coherent with empirical evidence. Spearman average correlation between cond and cut is 0.20. Most parallel coordinate diagrams show that there are not too many crossings between cond and cut confirming a partial relation between the criteria. This implies that cond can discriminate among algorithms that provide clusters with highly variable numbers of within-edges than cut is expected to do. Note that cond and cut are the only two criteria that have to be minimized and not maximized. This is the reason why in the Spearman correlation we have computed 1/cut and 1/cond instead to facilitate their comparison with others. It is also the reason why we have reversed their axis in the parallel coordinate diagrams to make them visually coherent with the other where best values are on top, worst values at the bottom. with different numbers of clusters, MQ is very useful. Other criteria can exhibit severe bias. For example, in the extreme case where a clustering results in a single cluster, and is compared to a much better clustering that provides 10 clusters, no between-edge will be found in the first clustering and most criteria will compute a high quality measure despite the fact that the clustering is really doing a poor job compared to the second. The fact that MQ accounts for the number of clusters prevents it from that bias. Also, experiments showed no correlation at all with criteria such as cov or mod and these results suggest that using MQ in conjunction with cov can be useful to balance the number of cluster bias.

Perf is probably the most debatable criteria amongst those reported in this paper. It surely tries to capture a different aspect of the quality of the results than the others. Somehow Perf captures the number of errors compared to an ideal clustering that would ideally lead to a disconnected set of cliques. However, the fact that the computation of Perf computes a ratio of the number of errors (between edges and missing within edges) over the total number of possible edges can lead to very misleading interpretations in many real situations. For example, previous experiments showed that random clustering can get better Perf ratings than other clustering.

Conclusion

In order to compare clustering results, most existing numerical criteria found in the literature focus on evaluating the quality of the compromise between intra cluster cohesion and inter cluster differentiation. In this paper we report the a lot of different aspects of the quality of the results cannot be captured at all with existing criteria. This experiment suggests that a lot of work is needed to better understand the quality and characteristics of automatic clustering results for keywords or documents datasets.

Fig. 1 .

 1 Fig. 1. Spearman Rank Correlation on Quality Measures

Fig. 2 .

 2 Fig. 2. Usage of parallel coordinates on quality Criteria for JDM 200

Fig. 3 .

 3 Fig. 3. Usage of parallel coordinates on quality Criteria for JDM 2000

Table 2 .

 2 Algorithm's and Implementations used in this evaluation

	Algorithm Name Article Implementation	Author's Impl.
	CNM	[5] http://www.cs.unm.edu/~aaron/research/fastmodularity.htm	Yes
	SPK-MEANS	[6] http://www.cs.utexas.edu/users/dml/datamining/spkmeans.html	Yes
	Cluto	[21] http://glaros.dtc.umn.edu/gkhome/views/cluto	Yes
	LinLog	[13] http://www.informatik.tu-cottbus.de/~an/GD/	Yes
	InfoMap	[14] http://www.tp.umu.se/~rosvall/code.html	Yes
	CMJA	[2] our implementation (link after blind reviews)	No
	BGLL	[3] http://sites.google.com/site/findcommunities/	Yes
	Simple K-Means [12] our implementation (link after blind reviews)	No
	NCut Algorithm [18] http://www.cis.upenn.edu/~jshi/software/	Yes
	MCL	[7] http://www.arbylon.net/projects/	No

Third the set of algorithms tested should be representative of different types of approaches. The Tab. 3 summarizes the choices made in terms of algorithms and indicate the URL of the implementation used in the experiment.

Table 3 .

 3 Average of rankings