David Borges

Paulo Montezuma

Rui Dinis

Pedro Viegas

Energy Efficient Massive MIMO Point-to-Point Communications with Physical Layer Security: BPSK vs QPSK Decomposition 

Keywords: massive MIMO systems, efficient power amplification, constellation shaping, low complexity detection

Massive multiple-input multiple-output systems (mMIMO) are the most prevalent candidates for the next generation of wireless communication. Yet even with mMIMO systems the joint optimization of spectral and energy efficiencies can be only attained by combining high order signal constellations and efficient power amplification. In order to push this limitation, the transmitter can spread the information into several amplification branches, which are the result of the decomposition of multilevel constellation symbols into quasi constant envelope signals. Nevertheless, the high number of antennas involved in this type of communication leads to an increase of the channel matrix's size and therefore the complexity of the equalization process can create drawbacks for the power consumption and latency. In this paper we will study the combination of a multi-layer transmitter with a low complexity receivers based on an iterative block decision feedback equalizer (IB-DFE). These receivers avoid the matrix inversion operation in the equalizer the feed-forward by replacing it with an equal gain combiner (EGC) or a maximum ratio combiner (MRC) module. Results show that can be used without penalties on performance provided that the number of antennas involved is high.

Introduction

Massive multiple-input multiple-output (mMIMO) systems allow many improvements in the communication performance bounds [START_REF] Larsson | Massive MIMO for next generation wireless systems[END_REF]. Additionally, to maximize spectral efficiency, high order multilevel constellations can be employed. However, high order multilevel constellations have high peak-to-average power ratio (PAPR) which restricts the efficiency of the amplification [START_REF] Dinis | Performance trade-offs with quasi-linearly amplified OFDM through a two-branch combining technique[END_REF]. In order to improve the power amplification efficiency, the symbols with high envelope fluctuations can be decomposed into low order modulation symbols, which benefit of low envelope fluctuations [START_REF] Astucia | On the use of Multiple grossly Nonlinear amplifiers for Higly Efficient Linear amplification of multilevel constellations[END_REF]. As example, we can decompose one 16 Quadrature Amplitude Modulation (QAM) symbol into four Binary Phase Shift Keying (BPSK) symbols or two Quadrature Phase Shift Keying (QPSK) symbols. By doing this, each component can be amplified separately without distortion by NonLinear (NL) power amplifiers, since the PAPR of a BPSK or a QPSK is much lower than the PAPR of the original constellation. Using not oversized amplifiers and working in their optimal operation zone, energy efficiency of each amplifier can be highly improved. In this architecture, each power amplifier is directly connected to a single antenna. This means that each antenna will transmit a single component from the decomposed higher order symbol, i.e. a BPSK or QPSK symbol. At the channel level, the original symbol can be obtained through the combination of all the components. In other words, at the receiver side, each antenna will receive the signal from all transmitting antennas which will add up to form the original high order constellation symbol. Moreover, a multi-layer system could be created using this approach. The first layer is composed by the multi-branch amplification structure, the second layer is composed by the antenna array connected to each amplifier's output and the third layer associated to spatial multiplexing is composed by several layers 1 and 2 in parallel.

However, the high number of antennas employed in the multi layered transmitter and receiver communication process will imply an increasing size of the channel matrix, which means a higher complexity in the equalization step due to matrix inversion operation. In order to overcome this problem, a low complexity receiver, combining an Iterative Block Decision Feedback Equalization (IB-DFE) and a Maximum Ratio Combiner (MRC) or an Equal Gain Combiner (EGC) is implemented for systems based on these multi layered transmitters and analyzed in this paper. These simplified receivers avoid the usual matrix inversion inherent to mMIMO conventional detection scheme, but as we shall see the conditions for such approach must be fulfilled to avoid any impact on system performance.

The rest of this paper is organized as follows: Section 2 characterizes the system's architecture; Section 3 contains the description of the system; Simulation results and performance analysis are presented in section 4; Section 5 resumes the paper.

System Architecture and Characterization

It was already shown that hig order constellations can be decomposed in polar components with the constellation symbols expressed as a function of the corresponding bits [START_REF] Dinis | Iterative Frequency-Domain Equalization for General Constellations[END_REF]. Let 𝒮 = {𝑠 0 , 𝑠 1 , ..., 𝑠 (N-1) } , a constellation with 𝑀 points (i.e., #𝒮 = 𝑀), where 𝑠 𝑛 ∈ ℂ. To each constellation point 𝑠 𝑛 we associate a set of 𝜇 = log 2 (𝑀) bits in polar form ℬ = {𝑏 𝑛 0 , 𝑏 𝑛 1 , ..., 𝑏 𝑛 (𝜇-1) }, with 𝑏 𝑛 (𝑖) = ±1 = 2𝛽 𝑛 (𝑖) -1 and 𝛽 𝑛 (𝑖) = 0 or 1. The set of 𝜇 bits can be decomposed in 𝑀 = 2 𝜇 different subsets

ℬ 𝑚 , 𝑚 = 0,1, . . . , 𝑀 -1.
Having 𝑀 constellation points in 𝒮 and 𝑀 different subsets of ℬ, ℬ 0 , ℬ 1 , ..., ℬ 𝑀-1 , we can write

s n = ∑ M-1 m=0 g m ∏ b n (i) ∈ℬ m b n (i) , n = 0,1, . . . , M -1, (1)
which corresponds to a system of 𝑀 equations (one for each 𝑠 𝑛 and 𝑀 unknown variables 𝑔 𝑚 ). Moreover, by doing 𝑚 = (𝛾 (𝜇-1,𝑚) , 𝛾 (𝜇-2,𝑚) , . . . , 𝛾 (1,𝑚) , 𝛾 (0,𝑚) ) we can associate 𝑚 to its corresponding binary representation with 𝜇 bits, which turns possible to define ℬ 𝑚 as the set of bits where the bit 𝑏 𝑛 (𝑖) is included if and only 𝛾 (𝑖,𝑚) is 1. Based on that we may write

𝑠 𝑛 = ∑ 𝑀-1 𝑚=0 𝑔 𝑚 ∏ 𝜇-1 𝑖=0 (𝑏 𝑛 (𝑖) ) 𝛾 (𝑖,𝑚) . ( 2
)
The transmitter layered structure of Fig. 1 takes advantage of this decomposition in polar components, and uses a mMIMO scheme with 𝑁 𝑣 × 𝑁 𝑚 × 𝑁 𝑏 antenna elements, arranged in 𝑁 𝑣 sets of 𝑁 𝑚 × 𝑁 𝑏 antennas. Conventional beamforming schemes could be implemented by a layer 2 with 𝑁 𝑏 antenna elements connected to each one of 𝑁 𝑚 amplification branches. For spatial multiplexing a layer 3 with a set of 𝑁 𝑣 × 𝑁 𝑚 antennas is needed, where the 𝑁 𝑚 antennas of layer 1 are associated to the signal components of the constellation symbol and 𝑁 𝑣 sets of 𝑁 𝑚 antennas are used to transmit simultaneously 𝑁 𝑣 different constellation symbols (an example of a transmitter based on a layer 1 and layer 3 combination can be seen in Fig. 2). It follows that power efficiency comes improved due the lower PAPR of the component signals and the possible use of NL amplifiers in layer 1 [START_REF] Astucia | On the use of Multiple grossly Nonlinear amplifiers for Higly Efficient Linear amplification of multilevel constellations[END_REF][START_REF] Montezuma | Design of TC-OQAM Schemes Using a Generalised Nonlinear OQPSK-type Format[END_REF]. Another important aspect lies on uncorrelated signals associated to each RF chain, which means an omnidirectional radiation pattern for each set of 𝑁 𝑚 antennas. But it is important to mention that each set of 𝑁 𝑚 antennas implements directivity at information level, since the constellation points of the transmitted signal maintain their positions at the desired direction Θ but are scrambled for other directions. In Fig. 3 it can be seen that the transmitted components will arrive to the receiver with different delays, related with the spacing 𝑑 between antennas and the desired transmission angle Θ. However, these delays are compensated by introducing phase shifts between antennas related to the spacing between the amplification branches in the transmitter. This means that in direction Θ the receiver sees a symbol constellation without any distortion due to phase rotations of signal components. In other directions, phase shifts between antennas do not compensate the phase shifts due the different propagation delays and the received symbol suffers distortions since it results from the combination of signal components with the wrong phases. Obviously antenna spacing can be adjusted to assure the desired configuration of information directivity for a specific direction. Therefore, any user can demodulate the received signal if is in the desired direction while other users in other directions receive a scrambled constellation. Hence, to decode the information, each 𝑗 user needs to be in the right direction Θ 𝑗 .

System Description

Consider now the MIMO scenario shown in fig. 4 characterized by a point-to-point communication link between a transmitter with 𝑇 = 𝑁 𝑣 × 𝑁 𝑚 antennas and a receiver with 𝑅 ≥ 𝑇 receive antennas. The transmitter's configuration has the layered structure shown in fig. 2, but were Layer 3 is composed by a set of 𝑁 𝑣 transmitters each one composed by 𝑁 𝑚 amplification branches in parallel (for a 16-quadrature amplitude modulation (16-QAM) decomposition into QPSK or BPSK components we have 𝑁 𝑚 = 2 and 𝑁 𝑚 = 4 , respectively). This means a third layer with 𝑇 = 𝑁 𝑣 × 𝑁 𝑚 antennas that transmits simultaneously 𝑁 𝑣 symbol constellations (more exactly the components of 𝑁 𝑣 symbols from a 16-QAM constellation). The transmitter configuration is known and the receiver is assumed to be in the right direction Θ (in which the constellation is optimized).

Coupling effects among antennas are avoided by a horizontal spacing of 𝜆/2 between antennas of layer 1 and a vertical spacing of 𝜆 between each set of 𝑁 𝑚 antennas. It is assumed that all antennas at transmitter are assigned to an authorized receiver with 𝑅 antennas.

It is assumed that we have frequency slective channels linking transmit and receive antennas and a SC-FDE block transmission technique. To compensate the channel's frequency selectivity it can be adopted an IB-DFE receiver with the structure shown in fig. 3 [START_REF] Benvenuto | Block Iterative DFE for Single Carrier Modulation[END_REF][START_REF] Dinis | Iterative Layered Space-Time Receivers for Single-Carrier Transmission over Severe Time-Dispersive Channels[END_REF]. At transmitter the 𝑡 𝑡ℎ antenna sends the block of 𝑁 data symbols {𝑥 𝑛 (𝑡) ; 𝑛 = 0,1, . . . , 𝑁 -1} being {𝑦 𝑛 (𝑟) ; 𝑘 = 0,1, . . . , 𝑁 -1} the received block at the 𝑟 th receiver's antenna. As usual, a cyclic prefix with duration higher than the overall channel delay is appended to each transmitted block and removed at the receiver. Under these conditions, the corresponding frequency-domain received block {𝑌 𝑘 (𝑟) ; 𝑘 = 0,1, . . . , 𝑁 -1} is given by 1) . .. 𝑌 𝑘 (𝑅) ]

𝐘 𝑘 = [𝑌 𝑘 (
𝑇 = 𝐇 𝑘 𝐗 𝑘 + 𝐍 𝑘 , (3)
where 𝐇 𝑘 denotes the 𝑅 × 𝑇 channel matrix for the 𝑘th frequency, with (𝑟, 𝑡)th element 𝐻 𝑘 (𝑟,𝑡) , 𝐗 𝑘 = [𝑋 𝑘 (1) . .. 𝑋 𝑘 (𝑇) ] 𝑇 and 𝐍 𝑘 represents the channel noise.

In an iterative Minimum Mean Squared Error (MMSE) receiver the data symbols in each iteration are obtained through the inverse discrete Fourier transform (IDFT) of the block {𝑋 ̃𝑘 (𝑡) ; 𝑘 = 0,1, . . . , 𝑁 -1}, and are given by

𝐗 ̃𝑘 = [𝑋 ̃𝑘 1 . . . 𝑋 ̃𝑘 (𝑅) ] 𝑇 = 𝐅 𝑘 𝐘 𝑘 -𝐁 𝑘 𝐗 𝑘 , (4)
(see details in [START_REF] Silva | Frequency-domain multiuser detection for CDMA systems[END_REF], e.g.). Interference cancelation is done using 𝐗 𝑘 = [𝑋 0 . . . 𝑋 𝑁-1 ], with 𝑋 𝑘 denoting the frequency-domain average values conditioned to the FDE output for the previous iteration, which can be computed as described in [START_REF] Silva | Estimation of the feedback reliability for IB-DFE receivers[END_REF][START_REF] Silva | Frequency-domain multiuser detection for CDMA systems[END_REF].

Coefficients 𝐹 𝑘 , 𝐵 𝑘 and the correlation coefficient 𝜌 can be computed as described in [START_REF] Dinis | Iterative Frequency-Domain Equalization for General Constellations[END_REF][START_REF] Dinis | Iterative Layered Space-Time Receivers for Single-Carrier Transmission over Severe Time-Dispersive Channels[END_REF][START_REF] Silva | Frequency-domain multiuser detection for CDMA systems[END_REF]. In first iteration no information is available about the transmitted symbols and 𝐗 𝑘 = 𝟎 , which means that this receiver is equivalent to the linear frequency-domain MMSE receiver. Subsequent iterations use the average values conditioned to the receiver's output from previous iteration to remove the residual intersymbol interference (ISI). Due to matrix inversions the IB-DFE receiver has a computational complexity of 𝑂(𝑁 𝑅 3 ). To avoid this problem other two low-complexity iterative frequency-domain receivers, denoted as maximum ratio detection (MRD) and equal gain detection (EGD) are also considered [START_REF] Borges | Low complexity MRC and EGC based receivers for SC-FDE modulations with massive MIMO schemes[END_REF]. Similarly to [START_REF] Borges | Low complexity MRC and EGC based receivers for SC-FDE modulations with massive MIMO schemes[END_REF] the ratios 𝑅/𝑇 between receiving and transmitting antennas considered here are at least equal or higher than 4.

As mentioned in [START_REF] Borges | Low complexity MRC and EGC based receivers for SC-FDE modulations with massive MIMO schemes[END_REF], the MRD receiver (see Fig. 3 B) is characterized by

𝐗 ̃𝑘 = 𝚿𝐇 𝑘 𝐻 𝐘 𝑘 -𝐁 𝑘 𝐗 𝑘 , (5)
and

𝐁 𝑘 = 𝚿𝐇 𝑘 𝐻 𝐇 𝑘 -𝐈, (6)
where 𝚿 denotes a diagonal matrix in which the (𝑡, 𝑡) th element is given by

(∑ 𝑁-1 𝑘=0 ∑ 𝑅 𝑟=1 |𝐻 𝑘 (𝑟,𝑡) | 2 ) -1
, takes advantage of the fact that

𝐇 𝑘 𝐻 𝐇 𝑘 ≈ 𝑅𝐈, (7)
which is accurate when 𝑅 >> 1 and the correlation between different channels have small correlation (i.e., different transmit and receive antennas). On the other hand, the EGD characterized by

𝐁 𝑘 = 𝚿𝐀 𝑘 𝐻 𝐇 𝑘 -𝐈, (8)
where 𝚿 denotes a diagonal matrix whose (𝑡, 𝑡) th element is given by

(∑ 𝑁-1 𝑘=0 ∑ 𝑅 𝑟=1 |𝐻 𝑘 (𝑟,𝑡) |) -1
, takes advantage of the fact that for mMIMO systems with 𝑅 ≫ 1 and small correlation between the channels associated to different transmit and receive antennas, the elements outside the main diagonal of 𝐀 𝑘 𝐻 𝐇 𝑘 [START_REF] Silva | Estimation of the feedback reliability for IB-DFE receivers[END_REF] are much lower than the ones at its diagonal, where (𝑖, 𝑖′)𝑡ℎ element of the matrix 𝐀 is [𝐀] 𝑖,𝑙 = exp(𝑗arg([𝐀] 𝑖,𝑙 )).

Results

Bit error rate (BER) results are obtained through Monte Carlo experiments (at least 150 error events are considered for each value of BER). A 16-QAM constellation is adopted in Layer 1, where the decomposition applied can be based on QPSK or BPSK components. This means that in first case we have 𝑁 𝑚 = 2 branches with QPSK components and in second case we have 𝑁 𝑚 = 4 BPSK components. In both cases the transmitted symbols 𝑠 𝑛 are selected with equal probability from the original constellation. It is assumed linear power amplification at the transmitter and perfect time and frequency synchronization conditions at the receiver. The results are expressed as function of

𝐸 𝑏 𝑁 0
, where 𝑁 0 /2 denotes the noise variance and 𝐸 𝑏 represents the energy of the transmitted bits. The number of users is always settled at 𝑇 = 5. For EGD, MRD and IB-DFE receivers the number of iterations is 4 (previous study and showed than there is no advantage to consider higher number of iterations, since there are no significant improvements in performance) [START_REF] Borges | Low complexity MRC and EGC based receivers for SC-FDE modulations with massive MIMO schemes[END_REF]. Results shown in Figs. 5 and 6 refer a layer 1 with both 𝑁 𝑚 = 2 and 𝑁 𝑚 = 4 and angular separation to the direction Θ of ΔΘ = 0 𝑜 (i. e. the user is placed in the correct direction where the components add in phase). Values of 𝑅/𝑇 between 4 and 10 are also adopted. From the comparison of the results, it is clear that when the ratio 𝑅/𝑇 grows, the performance improvement in ZF receiver can be considered as negligible. However for the other receivers only values of 𝑅/𝑇 higher than 6 turn performance improvements to be considered as negligible (this means that the validity conditions for the approach done to avoid the matrix inversion stands for 𝑅/𝑇 equal or higher than 6). As expected ZF and EGD perform worst that IB-DFE and MRD receivers, with the two later ones showing similar performances. Another important conclusion is the fact there is no performance disparity between both cases adopted for layer 1, since performances for the decomposition into BPSK and QPSK components are the same. It is important to mention, that the results for ΔΘ = 0 𝑜 are the same obtained by a conventional spatial multiplexing scheme, which means that combination of layer 1 and layer 3 can be employed without compromising system's performance.

To analyze the impact of errors in the alignment of the users with the direction Θ, we consider a fixed ratio of 𝑅/𝑇 = 6 and two values of ΔΘ (0º and 5 𝑜 ). The results are presented in Figs. 7, 8 and 9 for different receivers compared with ZF. From the comparison of figures, it becomes clear that when the ratio 𝑅/𝑇 grows, the gap between the MRD and the EGD receiver vanishes (as we can see in Figs. 7 and8 performance curves of receivers overlap which is not the case of Fig. 5). It is also clear that despite the performance improvement achieved by the higher value of 𝑅/𝑇, the performance impact of constellation shaping due to the decomposition of the constellation into into BPSK components is higher than QPSK case. Indeed, when at layer 1, it is adopted a decomposition in BPSK components the data link becomes compromised for the ZF receiver, due to a BER of 0.5. Even the other receivers only have reasonable values of BERs for the decomposition into QPSK components. It is also observed that the IB-DFE exhibits more robustness to the constellation shaping, even for the angular error of ΔΘ = 5 𝑜 . On the other hand, both MRD and EGD are severely affected by the angular error. Also, as expected the IB-DFE and the MRD receivers perform better than the EGD one. From the results of Figs. 7, 8 and 9 it is clear that both IB-DFE and MRD receivers have similar performances . Also, as expected, these two receivers perform better than the EGD one. On the other hand, better security is assured. This can be seen in the results regarding the ZF and EGD receivers where it is obvious the higher impact on system performance of any angular error for 𝑁 𝑚 = 4. For ΔΘ = 5 𝑜 the BER near 0.5 means that users can't decode any useful information. Regarding the security level assured by this transmitter, it is evident that the decomposition into BPSK components applied in layer 1 will be more effective than the decomposition into QPSK components. Performance results also show that security due to constellation shaping is achieved independently the type of adopted receiver, since the performance of all receivers is severely affected when ΔΘ ≥ 5 𝑜 for 𝑁 𝑚 = 4. More interestingly, good physical security can be assured in point-to-point communication links with only 4 antennas (for 𝑁 𝑚 = 2 previous results showed good tolerance against ΔΘ ≤ 5 𝑜 , which can compromise the security). Thus, angular shaping of the transmitted symbol performed in layer 1 can be used to separate user streams and achieve security between them (clearly the angular separation between users should be equal or higher than 5 𝑜 or the the number of components could be increased). Security can be even increased if beamforming is adopted at layer 2, which will be analyzed in further works.

It should be mentioned that in modern communication systems such as long term evolution (LTE) or further 5G system, various kinds of channel coding and error correction techniques are employed in different layers (mainly in physical and logical layers) to correct residual errors associated to a specific degree of BER. Thus, the BER level of 10 -4 the bit error rate attainable by the proposed tranmission schemes is more than sufficient to achieve the desired quaility of service, since when combined with an error correction we may see much lower error rate than the without error correction (shown in previous figures). Block error rate (BLER) of frame error rate (FER) are commonly used to assess the system performance, since they result from the sum of contributions of the several error correction techniques employed in the layers. This also means that the previous BER results are within the perfomance ranges employed in physical layer of modern communication systems.

Conclusions

In this paper it was shown that a double layer structure with information directivity and spatial multiplexing could be employed at the transmitter to improve power amplification efficiency and to assure physical security. For an aligned receiver (𝑖. 𝑒. ΔΘ = 0 𝑜 ) , since BER results achieved by both decompositions options are similar to those of classical spatial MIMO multiplexing, we may say that both higher power amplification efficiency and physical layer security come without sacrificing system's performance. Despite of using more antennas than the traditional scheme, in further millimetric wave communication systems this factor will be not a restriction.

Fig. 1 .

 1 Fig.1. Transmitter layered structure.

Fig. 2 .

 2 Fig. 2. Layered transmitter structure with layer 1 and layer 3.

Fig. 3 .

 3 Fig. 3. Constellation rotation due to delay.

Fig. 4 .

 4 Fig. 4. Massive MIMO system architecture for SC-FDE schemes (A) and detail of the low complexity massive MIMO receiver and equalization (B).

Fig. 5 .

 5 Fig. 5. BER performance for MRD and EGD with ΔΘ = 0 𝑜 and variant 𝑅/𝑇.

Fig. 6 .

 6 Fig. 6. BER performance for IB-DFE and ZF with ΔΘ = 0 𝑜 and variant 𝑅/𝑇.

Fig. 7 .

 7 Fig. 7. BER performance comparing ZF and EGD with diferrent ΔΘ .

Fig. 8 .

 8 Fig. 8. BER performance for ZF and MRD with ΔΘ = 5 𝑜 and variant 𝑅/𝑇.

Fig. 9 .

 9 Fig. 9. BER performance for ZF and IB-DFE with ΔΘ = 5 𝑜 and variant 𝑅/𝑇.

 This work was supported in part by FCT -Portuguese Foundation for Science and Technology through the PhD scholarship SFRH/BD/131093/2017, IT UID/EEA/50008/2013 (plurianual founding and project GLANCES), EnAcoMIMOCo EXPL/EEI-TEL/2408/2013 and UID/EEA/50008/2013 -MM5G.


Jeferson C Dias

Jônatas C Dias

Edinei Legaspe

Rodrigo Lima Stoeterau

Fabrício Junqueira

Newton Maruyama

Lucas Antônio Moscato

Paulo E Miyagi

Diolino Filho

Jeferson C Dias

Jônatas C Dias

Edinei Legaspe¹

Rodrigo Lima Stoeterau¹

Fabrício Junqueira¹

Newton Maruyama¹

Lucas Antônio Moscato¹

Diolino J Santos Filho

Treatment of Ventricular Assist Device Test Bench Data for Prediction of Failures and Improved Intrinsic Reliability

Keywords: Cardiovascular, Ventricular Assist Device (VAD), In Vitro, Big Data Analytics, Industry 4.0, Test Bench

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

Introduction

Among cardiovascular diseases (CVDs), heart failure (HF) has a remarkable prevalence and a great impact on morbidity and mortality worldwide [START_REF] Magalhães | Tratado de Cardiologia SOCESP[END_REF]. HF is considered to be a syndrome that renders the heart unable to supply oxygen at an adequate rate to the tissues, or at the expense of increasing the filling pressure (preload). Generally, HF results from structural or functional dysfunction of the heart, which compromises the ability to fill with blood or to eject it [START_REF] Martins | Doenças Cardiovasculares, Doenças Respiratórias, Emergências e Terapia Intensiva[END_REF].

For refractory HF cases and when no further treatment is available, cardiac transplantation (CT) is considered the most effective therapeutic modality to significantly prolong the patient's life [START_REF] Magalhães | Tratado de Cardiologia SOCESP[END_REF] [3] [START_REF] Azeka | I DIRETRIZ DE INSUFICIÊNCIA CARDÍACA (IC) E TRANSPLANTE CARDÍACO[END_REF]. However, there is a fundamental problem in the practice of organ transplantation, according to the "Thirty-fourth Adult Heart Transplantation Report-2017" [START_REF] Chambers | The Registry of the International Society for Heart and Lung Transplantation: Thirty-fourth Adult Lung And Heart-Lung Transplantation Report-2017; Focus Theme: Allograft ischemic time[END_REF], the number of cardiac surgeries done for transplants in Europe, for example, does not grow significantly. For the rest of the world, just as in the US inclusive, there is a growing trend, but it is not enough to balance supply and demand. Drug therapy alone for refractory HF continues to have disappointing results. Because of this, VADs have emerged as a therapy for refractory HF and are used, for example, as a long-term use strategy, to provide conditions so that the patient can receive the heart transplant at the expected time, a practice called by "bridge to the transplant", or if the patient is not able to perform the surgery because of a high probability of rejecting the transplant, VAD is recommended as a permanent practice known as "destination therapy" [START_REF] Magalhães | Tratado de Cardiologia SOCESP[END_REF].

Although VADs have become an advantageous opportunity in refractory HF therapy, this alternative has opportunities for improvement in the face of adverse events caused by the use of these devices. For example, some of the main adverse events are: bleeding, infection, stroke and malfunction, the latter event being directly related to the failure of the device functionality, according to the Interagency Registry for Mechanically Assisted Circulatory Support1 in its eighth annual INTERMACS report [START_REF] Kirklin | Eighth annual INTERMACS report: Special focus on framing the impact of adverse events[END_REF]. This leads us to reflect on the need to record reliability data and improve the reliability of these devices.

Relation with Industrial and Service Systems

The technological evolution observed in recent years; mainly related to: a) the digitalization that transforms the way in which we collect data to transform into useful information for decision making, b) the creation of increasingly smaller sensors from microelectronics with innovations in microchips and c) the technologies of communication networks wireless; has provided the development of products and processes in various market segments, from consumer goods to industrial automation systems [START_REF] Faceli | Inteligência Artificial -Uma Abordagem de Aprendizado de Máquina[END_REF].

This prospective article proposes the use of this technological evolution in the improvement of mechatronic devices, as is the case of VAD. The technological evolution based on multidisciplinary concepts that created the mechatronic devices, when applied in a project, and in this case for VADs, with resources that allow the communication through Internet (IoT), robotics, software and manufacturing accompanied by an integrated digital system, it was possible to provide the creation of smarter devices [START_REF] Atoche | Nuevas Tendencias en Sistemas Mecatrónicos[END_REF].

This paper highlights how this technological innovation can be used to improvement of VADs of continuous flow and with the mechanism of operation of the pump of the centrifugal type. The importance of technological application to improve these devices is justified by two reasons: a) To predict and avoid adverse events attributed to malfunction and b) The increasing use of VADs as destination therapy.

Objective

To propose and test a method of improving the intrinsic reliability of ventricular assist devices (VADs) using: i) an automated test bench and ii) a knowledge base, allowing data-based decision making using the technologies present in the industry 4.0 such as: Cloud computing, Internet of Things (IoT) and technologies linked to Artificial Intelligence (AI) present in tools of Big Data Analytics and Data Mining.

The experimental performance, according to the proposed method, will result in: a) a knowledge base, coming from the data of failure of the behavior of a VAD throughout the tests, acquired knowledge, will allow intelligent control to predict future consequences and b) a knowledge analysis obtained will result in a set of improvements, which will be applied to the design of new devices, making these devices more durable.

Materials and Methods

Intrinsic reliability is not simple to achiev in time-varying devices, such as a VAD. For this reason, a five-stage cyclic process was developed, as proposed in Fig. 1. "4. VAD Test" is described in detail in the following sections. A VAD project, step 1, is drawn from demand requirements, in which it is evaluated in step "2. Risk Analysis" in such a way as to enable it to provide sufficient reliability for rapid prototyping production, provided in step "3.Production (3D-PETG)", of a bench test device in step "4. VAD Test on Bench ". In these steps 1 through 4, the risk analysis evaluates how reliable the project, production, or testing is to then allow the project to be validated in step "5. Project Validation". This last stage guarantees the intrinsic reliability of the device, as well as its respective design, allowing a future verification to verify the reduction of adverse events due to the malfunction of these devices. A "Knowledge Base" in the cloud collects information for adjusting control parameters of the actuators in the test bench, so that the "Bench Test Procedure" described in the section below can be controlled [START_REF] Dias | In Vitro Test Bench with Intelligent Behavior to Ventricular Assist Devices[END_REF].

VADs Test Bench

The test bench used in this work consists of the reservoirs 1 and 2 which store the liquid used is transported from tank T1 to tank T2, performed from a pump (VAD), which imposes energy on the system, according to Fig. 2. Sensors perform the collection of fluid pressure at the output of reservoirs 1, rotation, vibration and electrical current of the motor of the VAD, fluid pressure, flow and temperature at the output of the VAD, position and the electric current of the Vvalve motor 2 and the pressure of the liquid at the inlet of the reservoir 2. The test bench has its own motor to ensure control of the rotation of the blood pump and the motor 2 that contours the valve V, so it is possible to control the flow of the liquid into the system.

Collection and Processing of Data

To collect information from the "context" [START_REF] De Andrade | Jogos Digitais, Cidade e (Trans)mídia: A Próxima Fase[END_REF] of the assisted device, emitted by the sensors in the "Test Bench", Fig. 2, and impose the new references to the actuators, a communication manager module is required for distribution and targeting of the collected data to the cloud knowledge base. These data from the sensors are processed for real-time error checking (failures), at which time the adjustment decisions are made. This knowledge base is handled by a standards analyzer for knowledge discovery using "Big Data Analytics". In this article "context" is any information that can be used to characterize the situation of an entity (object of interest), it is also known as sentient computing [START_REF] De Andrade | Jogos Digitais, Cidade e (Trans)mídia: A Próxima Fase[END_REF].

Knowledge Base Data Analysis

Big Data Analytics approaches to pattern discovery are based on: i) DESCRIPTIVE ANALYSIS: It is based on real-time understanding of events to make immediate decisions. Descriptive analysis works with historical data, cross-information to generate a clear and precise understanding of the relevant topics for the present moment, it is not necessary to relate past or future patterns [START_REF] Marquesone | Big Data: Técnicas e tecnologias para extração de valor dos dados[END_REF]. ii) DIAGNOSTIC ANALYSIS: The objective is to understand the cause and effect relationship over time. The diagnostic analysis works based on the collection of data related to a certain subject and crosses information to understand which factors influenced the current outcome [START_REF] Marquesone | Big Data: Técnicas e tecnologias para extração de valor dos dados[END_REF]. iii) PREDICTIVE ANALYSIS: It uses data mining as: statistical and historical data to know future trends [START_REF] Marquesone | Big Data: Técnicas e tecnologias para extração de valor dos dados[END_REF]. iv) PRESCRIPTION ANALYSIS: Prescriptive analysis commonly confused with predictive analysis, because it works with the same logic. However, for different purposes, while the predictive analysis identifies future trends, the prescriptive delineates the possible consequences of each action [START_REF] Marquesone | Big Data: Técnicas e tecnologias para extração de valor dos dados[END_REF].

Inspection Test Procedure

The inspection test, procedure 4.1 of Fig. 1, required for continuity in the following tests, is performed by means of roughness recognition through a top image. The method uses top images of calibrated roughness patterns. From these images computational information is obtained that allows the characterization of the surface and recognition of the roughness, according to technique of surface profilometry with and without contact.

Test Bench Procedure

The procedure "4.2 Test Bench", Fig. 1, occurs in three phases:

Standard Curve Survey

The VAD is fixed in the test bench to perform data collection of independent variables (velocity and rotor flow) that are manipulated with progressive values adjustment, impacting the behavior of the dependent variables (pressure, vibration, temperature and energy consumption). The data collection cycle (dependent variable values) begins with the fixed rotation speed for all flow variations. Thus, for each variation of the flow, the respective values of the dependent variables are collected.

After having performed all the variations of values referring to the flow variable; with the respective collection (reading) of the dependent variables (already mentioned); a new configuration is performed on the rotor speed variable and a new cycle is started. A pre-programming is established in the system that controls the execution of tests on the devices; for example: configuration speed ranges and flow variations. At the end of the test, the device is able to perform the next tests.

Gathering information during this step is useful for improving the default curve for this type of device.

Once the various standard curve tests have been performed considering the same type of device, the knowledge base for the behavior of dependent and independent variables will be used to observe deviations in the following test phases.

Short Term Testing

The short duration test consists of continuous repeatability of the standard curve test until the first failure is obtained, at which time the failure occurrence is recorded in the database. Thereafter, the pump is disassembled, so that each pump part goes through the surface inspection protocols. The defective part is replaced, just as the identified fault is recorded in the database. After identifying the causes of failure and risk analysis, the pump returns to the bench test until at least one failure is recorded for each pump part (VAD).

Long-Term Testing

Once the standard curve of the same type of device is defined, within its variables any observed deviations will be recorded and plotted on the graph. For the operating limits of the independent variables, only the speed limits are defined. Thus, tissue perfusion values simulated by speed limits will be appropriate for the human. Throughout the test deviations of the behavior of the variables can occur in relation to those registered in the standard curve.

Before the simulated tissue perfusion (independent variables) in the test bench, during the long-term tests become impracticable, through self-adjustments of the dependent variables, or a critical failure occurs, an AI manager module interrupts the device. The purpose of the long-term test is to obtain as much fault data as possible within the operating range of dependent and independent variables for a simulated human perfusion.

Results and Discussion

At the moment of the implantation of the method proposed in this work, according to Fig. 1, a VAD project was already underway, and a VAD made of aluminum with subtractive technology had already undergone a bench test; with the occurrence of failure. This is because the magnetic rotor had scraped on the upper casing, damaging the VAD. For this reason, the validation process of VADs was put into practice in the process "2. Risk Analysis" in which he submitted the damaged VAD parts to the "4.1 Inspection Test process".

The analytical procedure presents in the process "2. Risk Analysis", identified a fault in the bearing of the lower housing. A callosity at the center of the bearing was identified as the fault that caused the magnetic rotor to misalign, damaging the upper housing of the tested device.

An initial report was provided with improvement items, as Table 1, indicating the main points for increasing intrinsic reliability. The improvement items also caused a change in the design of the VAD, process "1. VAD design "resulting in the construction of the base of the mobile cradle for adjusting the lower bearing clearance, with tightening torque connection, as Fig. 3.

Conclusions

This prospective project, still in progress of its finalization, has already been able to demonstrate its positive effects, according to results section. The next steps include the implementation of intelligent control of the test bench with the use of the knowledge base for the prescription and prediction of events for the decisions of adjustment of the variables of the bench to the device in test. Thus, during every test step, the collected data is stored for an evaluation of the failures. The data collected, when analyzed using Big Data Anlytics, allow the prediction of failures for analyzed sample. Thus, when performing the evaluation in the new devices tested for fault predictivity, based on the information acquired, it is possible to make a decision before the failures appear. This assists in the longevity of the device being tested, in a similar human perfusion condition.

With regard to the gap presented in the INTERMACS report on adverse events related to device malfunction, this work seeks to improve reliability by reducing failures that may cause malfunction of these devices. Thus, the tangible contributions are first divided into a test bench with an intelligent control system, improving from the collected data, and assisting in a geographical expansion in the treatment of failures of other distributed devices. And the second contribution is a verification and validation of devices, from the knowledge stored reliability data.

Fig. 1 .

 1 Fig. 1. VADs Validation Process. Source: Dias, J. C. 2019 (Author)

Fig. 2 .

 2 Fig. 2. Scheme of the "Test Bench" existing in the Department of Mechatronics of USP. Source: Dias, J. C. 2019 (Author)

Fig. 3 .

 3 Fig. 3. Result of project improvements. Source: Dias, J. C. 2019 (Author)

Table 1 .

 1 Improvement items proposed during phase "2. Risk analysis"

	PROBLEM	EFFECTS	CAUSE	SOLUTION
				R1.1 Ensure the production according to
				the project.
		Engine rotation error	1. Bearing inclination	R1.2 Ensure compliance with the assembly procedure. R1.3 Overall geometric and dimensional
	Bearing	Change of pump flow		surface analysis with test protocols. R2.1 Make the cradle base mobile for adjusting the lower bearing clearance
	failure	Motor torque friction	2. Clearance between bearing and cradle	with wrench connection for clearance adjustment. R2.2 Ensure compliance with the
		Rotor Locking	3. Deformity in bearing cradle	assembly procedure.
			(manufacturing
			flha)

R3.1 Ensure manufacturing according to design. R3.2 Perform surface, geometric and dimensional analysis with test protocols. Source: Dias, J. C. 2019 (Author)

INTERMACS -Interagency Registry for Mechanically Assisted Circulatory Support contemplates a database with a total of

22,866 patients who received a mechanical circulatory support device approved by the FDA. With registration from 06/23/2006 to 12/31/2016 of 185 participating hospitals.


Frank O Wagner
email: wagner@math.univ-lyon1.fr

BOUNDED MORPHISMS

Keywords: Mathematics Subject Classification. 03C45, 12H05, 12H10. Key words and phrases. bounded automorphism, Pfaffian family, bounded endomorphism. Partially

A bounded automorphism of a field or of a group with trivial almost-centre is definable. In an expansion of a field by a Pfaffian family F of additive endomorphisms such that the algebraic closure of any F-substructure in the expansion coincides with relative field-theoretic algebraic closure, a bounded ring endomorphism, possibly composed with a power of the Frobenius, is a composition of ring endomorphisms canonically associated to F.

Introduction

In [START_REF] Lascar | Les automorphismes d'un ensemble fortement minimal[END_REF] Lascar calls an automorphism σ of a structure M bounded if there is a small set A of parameters such that for all m ∈ M we have σ(m) ∈ acl(A, m). He shows that for a strongly minimal set M the subgroup aut f (M) of strong automorphisms (which fix acl eq (∅)) is simple modulo the normal subgroup of bounded automorphisms. This result was generalized in [START_REF] Evans | Simplicity of the automorphism groups of some Hrushovski constructions[END_REF] to structures with an integer dimension compatible with a notion of stationary independence.

More generally, given an invariant family Σ of partial types, an automorphism σ of a field K with simple theory is called Σ-bounded if there is a small set A of parameters such that for all m ∈ M we have σ(m) ∈ Σcl(A, m), where the Σ-closure Σcl(B) of B is the set of all elements b such that tp(b/B) is Σ-analysable. If Σ is the class of the algebraic types, then the Σ-closure is the algebraic closure, and Σ-bounded coincides with bounded. In [START_REF] Blossier | Sur les automorphismes bornés de corps munis d'opérateurs[END_REF] Blossier, Hardouin and Martín Pizarro show that for a field with of operators, any Σ-bounded automorphism is definable and obtained by composition of a power of the Frobenius with automorphisms among the operators or their inverses, where Σ is taken as the family of all partial types co-foreign to the generic types of the field. In particular, this yields a uniform proof of results of Lascar [START_REF] Lascar | Les automorphismes d'un ensemble fortement minimal[END_REF], Ziegler and Konnerth [START_REF] Konnerth | Automorphism groups of differentially closed fields[END_REF].

Of course, the definition of a bounded automorphism is meaningful for any permutation σ of a set X; it need not be induced by an automorphism of the ambient structure, nor preserve all the induced structure on X. Similarly, the definition of Σ-bounded is valid for any hyperdefinable set X in a simple theory; moreover we can freely choose Σ, but we should take Σ small with respect to X: for instance, any permutation of X is X-bounded. As a priori a Σ-bounded automorphism of a hyperdefinable set X M in a structure M need not extend to a Σ-bounded automorphism of X N in an elementary extension N (and even less so with the same small set A of parameters), one should suppose that M is at least |T (A)| + -saturated, in order not to have to change the model one is working in.

In this paper we shall generalize the main theorem of [START_REF] Blossier | Sur les automorphismes bornés de corps munis d'opérateurs[END_REF] in two directions. On the one hand, we consider group-theoretic Σ-bounded automorphisms of a hyperdefinable group G in a simple theory, where Σ is still the class of partial types (with parameters) co-foreign to the generic types of G. We show that if any non-trivial element has a centralizer of unbounded index (that is, if the almost centre Z(G) of G is trivial), then any Σ-bounded automorphism is definable.

On the other hand, we will look at bounded endomorphisms in an expansion of a field by a family F = {f i : i < α} of additive endomorphisms, such that (1) The family F is Pfaffian, that is f i (xy) is a linear combination of products f j (x)f k (y) with j, k < i, for all i < α.

(2) The algebraic closure of an F-substructure A in the sense of the expansion is equal to the relative field-theoretic algebraic closure of A. These conditions are satisfied in particular by

• differentially closed fields in characteristic zero with several commuting derivations [START_REF] Mcgrail | The model theory of differential fields with finitely many commuting derivations[END_REF]; • generic difference fields in any characteristic [START_REF] Chatzidakis | Model Theory of difference fields[END_REF];

• separably closed fields of finite imperfection degree with a p-base named [START_REF] Delon | Idéaux et types sur les corps séparablement clos[END_REF][START_REF] Wood | Notes on the stability of separably closed fields[END_REF]; • differential generic difference fields in characteristic zero, where the automorphism commutes with the derivation [START_REF] Bustamante Medina | Differentially closed fields of characteristic zero with a generic automorphism[END_REF][START_REF] Bustamante Medina | Rank and dimension in difference-differential fields[END_REF]; • fields with free operators in characteristic zero [START_REF] Moosa | Model theory of fields with free operators in characteristic zero[END_REF]. (For the Pfaffian condition, use [START_REF] Blossier | Sur les automorphismes bornés de corps munis d'opérateurs[END_REF]Proposition 1.4].) We shall show that any bounded endomorphism of K is the composition of endomorphisms canonically associated with functions in F, which moreover are linear combinations, followed by a power of the inverse of Frobenius. In the cases mentioned above, the endomorphisms (except for the Frobenius for the separably closed fields) will be surjective, and the associated automorphisms will be the automorphisms among the functions in F, and possibly the Frobenius.

Preliminaries

In this section we will recall some less known concepts and results for simple groups and theories. For more details, we refer to [START_REF] Wagner | Simple Theories[END_REF]. As usual, we will work in a monster model of a simple theory, and we fix an ∅-invariant family Σ of partial types with parameters.

Definition 1.1. The Σ-closure Σcl(A) of A is the collection of all hyperimaginaries a such that tp(a/A) is Σ-analysable.

We always have bdd(A) ⊆ Σcl(A), with equality if and only if Σ only contains algebraic types. In general one will choose a set Σ of partial types that are small with respect to X (in particular X ⊂ Σcl(∅)). Fact 1.2. [14, Lemma 3.5.3] The following are equivalent:

(1) tp(a/A) is foreign to Σ.

(

) a | ⌣A Σcl(A). (3) a | ⌣A dcl(aA) ∩ Σcl(A). (4) dcl(aA) ∩ Σcl(A) ⊆ bdd(A). 2
Except in case of equality with the bounded closure, the Σ-closure has the size of the monster model. The equivalence (2) ⇔ (3) allows us to use only a small part. More properties of the Σ-closure can be found in [START_REF] Wagner | Some remarks on one-basedness[END_REF][START_REF] Palacín | Ample thoughts[END_REF], as well as a refinement into different levels. Now let G be a group hyperdefinable over ∅, and recall that the ambient theory is simple. Definition 1.4.

( We also recall the notion of the almost centre.

Definition 1.6. The almost centre of a hyperdefinable group G, denoted Z(G), is the characteristic subgroup of elements whose centralizer is of finite index in G.

In a simple theory, the approximate centre of G is hyperdefinable over the same parameters as G itself [START_REF] Wagner | Simple Theories[END_REF]Proposition 4.4.10].

Finally, we need Ziegler's lemma ([17, Theorem 1] for the stable abelian case, and [2, Lemma 1.2 and Remark 1.3] for the general case).

Fact 1.7. Let g and g ′ in G be such that g, g ′ and gg ′ are pairwise independent. Then the left stabilizers of g and gg ′ are equal, and commensurable with gStab(g ′ )g -1 , and all three are ∅-connected. Moreover, g is generic in the right coset Stab(g)g, which is bdd(∅)-hyperdefinable, and similarly for g ′ and gg ′ .

Σ-bounded group automorphisms

In this section G will still be a group hyperdefinable over ∅ in a simple theory; we shall asusme that it is sufficiently saturated. We let Σ be the class of all partial types co-foreign to generic types of G. (Since generic types have non-forking extensions that are translates of one another, co-foreign to one is equivalent to co-foreign to all.) In particular, if G has a regular generic type, Σ contains of all non-generic partial types.

Theorem 2.1. If Z(G) is trivial, then any Σ-bounded automorphism σ of G is hyperdefinable.
Proof. Let A be a set such that σ(g) ∈ Σcl(A, g) for all g in G, and recall that we shall need the ambient model to be |T (A)| + -saturated. Consider first g ∈ G generic over A, and g ′ ∈ G generic over A, g, σ -1 (g). Put A ′ = Σcl(A) ∩ bdd(A, g ′ , σ(g ′ )). According to Fact 1.3 we have g ′ , σ(g ′ ) |

⌣A ′ σ -1 (g), g. Since generics of G are foreign to Σ, the elements g and g ′ remain generic over A ′ . So g 0 = g ′ σ -1 (g) and σ(g 0 ) = σ(g ′ )g are both generic over A ′ , whence over A. Now choose g 2 generic in G over A, g 0 , and put g 1 = g -1 0 g 2 . Then g 0 , g 1 and g 2 = g 0 g 1 are pairwise independent over A. By Fact 1.3 their Σ-closures Σcl(A, g 0 ), Σcl(A, g 1 ) and Σcl(A, g 2 ) are pairwise independent over Σcl(A). In particular, (g 0 , σ(g 0 )), (g 1 , σ(g 1 )) and (g 2 , σ(g 2 )) are pairwise independent over Σcl(A), whence by Fact 1.3 also over

B = bdd(B) = Σcl(A) ∩ bdd(A, g 0 , σ(g 0 ), g 1 , σ(g 1 ), g 2 , σ(g 2 )).
Moreover, since the generic types of G are foreign to Σ, the elements g 0 , σ(g 0 ), g 1 and g 2 each remain generic over B.

For i ≤ 2 let S i = Stab(g i , σ(g i )/B) be the (left) stabilizer in G × G. So S i is connected over B by Fact 1.7, the right coset S • (g i , σ(g i )) is hyperdefinable over B, and the pair (g i , σ(g i )) is generic in that coset. Furthermore, S 0 = S 2 ; let S = S loc 0 be its locally connected component.

Since

g i , σ(g i ) | ⌣ bdd(A,gi,σ(gi))∩Σcl(A) B, S i is commensurable with Stab(g i , σ(g i )/bdd(A, g i , σ(g i )) ∩ Σcl(A)) for i = 0, 2,
whose locally connected component is also S. It follows that S and the coset S • (g i , σ(g i )) are hyperdefinable over bdd(A, g i , σ(g i )) ∩ Σcl(A) for i = 0, 2. Now g 2 was any generic over A, g 0 . So S is hyperdefinable over Σcl(A) ∩ bdd(A, a, σ(a)) for any generic a, and hence over Ā = Σcl(A) ∩ a generic bdd(A, a, σ(a)).

Note that since g 0 and σ(g 0 ) are generic over B, the projections of S 0 , and therefore S, to the first and to the second coordinate are generic in G, and thus subgroups of bounded index. Since σ(g 0 ) ∈ Σcl(A, g 0 ) and all generic types of S are translates of tp(g 0 , σ(g 0 )/B) over independent parameters, we have g ′ ∈ Σcl(B, g) for any generic (g,

g ′ ) of S over B. If h ∈ coker(S) = {g ∈ G : (1, g) ∈ S},
and (g, g ′ ) ∈ S is generic over B, h, then (g, g ′ h) is still generic over B, h, and h ∈ Σcl(B, g). Since h | ⌣B g we have h ∈ Σcl(B) = Σcl(A), and coker(S) ⊆ Σcl(A). Now let g ′′ ∈ G be arbitrary, and g ∈ G generic over Ā, g ′′ , σ(g ′′ ). We put g ′ = g -g ′′ and B ′ = Σcl(A) ∩ bdd(A, g, g ′ , σ(g), σ(g ′ )). So g and g ′ are generic over B ′ , and

(g, σ(g)) ∈ (S • (g ′ , σ(g ′ )) (g ′′ ,σ(g ′′ )) ∩ (S • (g, σ(g)).
As the two cosets S • (g, σ(g)) and S • (g ′ , σ(g ′ )) are hyperdefinable over B ′ ,

g, σ(g) | ⌣ B ′ g ′′ , σ(g ′′ )
by Fact 1.3, and (g, σ(g)) is generic in S • (g, σ(g)) over B ′ , the two groups S and S (λ,σ(λ)) are commensurable, and therefore equal by local connectivity. It follows that S is normalized by the subgroup {g, σ(g) : g ∈ G} ≤ G × G. In particular, if h ∈ coker(S), then h σ(G) ∈ coker(S) ⊂ Σcl(A). Now σ is an automorphism, therefore surjective, and h G ⊆ Σcl(A). So for g ∈ G generic over A, h we have g | ⌣A,h h g , since tp(g/A, h) is foreign to Σ. So C G (h) has bounded index in G, and h ∈ Z(G) = {1}. Thus coker(S) is trivial.

By Fact 1.5 the normalizer N G (S) is hyperdefinable. Since (g, σ(g)) normalizes S for all g ∈ G, the projection to the first coordinate of N G (S) is the whole of G. But for (h, h ′ ) ∈ N (S) and (g, g ′ ) ∈ S with g ′ ∈ G generic over h, σ(h), h ′ we have (g h , g ′h ′ ) ∈ S and (g h , g ′σ(h) ) ∈ S (h,σ(h)) = S.

Since coker(S) is trivial, we get g ′h ′ = g ′σ(h) and σ(h

)h ′-1 ∈ C G (g ′ ). Thus g ′ ∈ C G (σ(h)h ′-1 ). But g ′ is generic over σ(h), h ′ , so σ(h)h ′-1 ∈ Z(G) = {1} and h ′ = σ(h). Thus N G (S) hyperdefines σ.
Corollary 2.2. Let σ be a Σ-bounded automorphism of a field K hyperdefinable in a simple theory, where Σ is the class of partial types co-foreign to the generic types of K. Then σ is hyperdefinable.

Proof. Let G = K + ⋊ K × . So G is hyperdefinable, and Σ is also the set of partial types co-foreign to generics of G (which are pairs of two independent generics of K). Furthermore, Z(G) = {1}, and σ induces an automorphism τ : (a, b) → (σ(a), σ(b)) of G. According to the previous theorem τ is hyperdefinable, and so is σ.

Bounded endomorphisms of a field with operators

In this section we shall consider a hyperdefinable field K in a simple theory, together with a well-ordered family F = {f i : i < α} of additive endomorphisms. We assume that (1) The family F is K-free: if i λ i f i = 0 for λ i ∈ K, then λ i = 0 for all i < α.

(2) The family is K-Pfaffian, i.e. for all i < α there are a i j,k ∈ K with j, k ≤ i, almost all 0, such that

f i (xy) = j,k≤i a i j,k f j (x)f k (y).
Remark 3.1. If we start with a K-Pfaffian family that is not K-free, we can extract a K-free and K-Pfaffian subfamily. In particular, in positive characteristic we can assume that the Frobenius is part of F.

Lemma 3.2. If ῑ a ῑf i0 (x 0 ) • • • f in (x n ) = ῑ b ῑf i0 (x 0 ) • • • f in (x n ) for a ῑ, b ῑ ∈ K, then a ῑ = b ῑ for all ῑ = (i 0 , . . . , i n ).
Proof. By induction on n. If n =0, this is just K-freeness of F. We therefore assume the assertion true for n -1, and

ῑ a ῑf i0 (x 0 ) • • • f in (x n ) = ῑ b ῑf i0 (x 0 ) • • • f in (x n ). So 0 = in i0,...,in-1 (a ῑ -b ῑ)f i0 (x 0 ) • • • f in-1 (x n-1 ) f in (x n ).
If there is some k with ak = bk, then by induction hypothesis

g in (x 0 , . . . , x n-1 ) = i0,...,in-1 (a ῑ -b ῑ)f i0 (x 0 ) • • • f in-1 (x n-1 )
is not identically 0 for i n = k n , and there is ᾱ ∈ K n with g kn (ᾱ) = 0. But then in g in (ᾱ)f in (x n ) = 0, which again contradicts K-freeness of F. Corollary 3.3. The coefficients a i j,k are uniquely determined. Proof. Immediate. Remark 3.4. a i j,k = a i k,j for all j, k ≤ i, and j≤i a i j,k f j (1) = δ ki . Proof. The first equation follows from f i (xy) = f i (yx), and the second from f i (1y) = f i (y).

Corollary 3.5. For all i < α the function

σ i (x) = j≤i a i j,i f j (x)
is a ring endomorphism of K. Moreover, we can assume σ i ∈ F.

Proof. We have

f i (xy) = j,k≤i a i j,k f j (x)f k (y) = j≤i a i j,i f j (x)f i (y) + j≤i, k<i a i j,k f j (x)f k (y) = σ i (x)f i (y) + R(x, y), where R(x, y) = j≤i, k<i a i j,k f j (x)f k (y). Thus f i (xx ′ y) = σ i (xx ′ )f i (y) + R(xx ′ , y) = σ i (xx ′ )f i (y) + j≤i, k<i a i j,k ℓ,m≤j a j ℓ,m f ℓ (x)f m (x ′ ) f k (y) = σ i (xx ′ )f i (y) + j≤i, k<i ℓ,m≤j a i j,k a j ℓ,m f ℓ (x)f m (x ′ )f k (y) and f i (xx ′ y) = σ i (x)f i (x ′ y) + R(x, x ′ y) = σ i (x)σ i (x ′ )f i (y) + σ i (x)R(x ′ , y) + R(x, x ′ y) = σ i (x)σ i (x ′ )f i (y) + j≤i a i j,i f j (x) ℓ≤i, m<i a i ℓ,m f ℓ (x ′ )f m (y) + j≤i, k<i a i j,k f j (x) ℓ,m≤k a k ℓ,m f ℓ (x ′ )f m (y) = σ i (x)σ i (x ′ )f i (y) + j≤i ℓ≤i, m<i a i j,i a i ℓ,m f j (x)f ℓ (x ′ )f m (y) + j≤i, k<i ℓ,m≤k a i j,k a k ℓ,m f j (x)f ℓ (x ′ )f m (y).
Comparing the coefficient of f i (y), Lemma 3.2 yields

σ i (xx ′ ) = σ i (x)σ i (x ′ )
and σ i is a ring endomorphism of K. Note that σ i (1) = 1, so σ i is non-zero. The K-linear combinations of functions in F form a K-vector space with basis F. Since every family of (non-zero) ring endomorphisms of K is linearly independent, we can start another basis with (σ i : i < α) (eliminating any repetitions which might occur), followed by the functions of F, eliminating those which are already in the span of the previous ones together with (σ i : i < α). It is clear that this family is still Pfaffian, and K-free. Definition 3.6. We call σ i (x) = j≤i a i j,i f j (x) the endomorphism associated to f i . Note that the endomorphism associated with a ring endomorphism σ is σ itself, and the endomorphism associated with a derivation is the identity. Remark 3.7. [1, Corollary 1.5] If we well-order compositions of additive endomorphisms in F first by the length of the composition and then by lexicographic order of the indices of the functions of F used, the endomorphism associated to such a composition will be the composition of the associated endomorphisms of the functions in F.

The following lemma will not be used for the proof of the main theorem of this section.

Lemma 3.8. Any finite subfamily {f i : i ∈ I 0 } of F is contained in a finite Pfaffian subfamily {f i : i ∈ I}, with max I = max I 0 .

Proof. We put I -1 = ∅, and for n > 0 we build inductively a string of finite families

I 0 ⊆ I 1 ⊆ I 2 ⊆ • • • by posing I n+1 = I n ∪ {j < α : ∃ i ∈ I n ∃k a i j,k = 0}. Since the sequence max(I n \ I n-1
) is strictly decreasing, the sequence of the I n becomes stationary, and I = n<ω I n gives a Pfaffian family {f i : i ∈ I}. Theorem 3.9. Let K be a field, and F a Pfaffian family of additive endomorphisms of K. For A ⊆ K let A be the closure of A under all functions f ∈ F. We assume that acl(A) = A alg ∩ K, where acl is the algebraic closure in the sense of the structure (K, +, •, f : f ∈ F), and A alg is the field-theoretic algebraic closure. Then any bounded endomorphism σ of the field K satisfies an identity of the form

Frob j • σ = σ i ,
where σ i is a composition of associated endomorphisms of F, and the Frobenius.

Proof. We can assume that F is K-free, and that in positive characteristic the Frobenius is in F. Let B = acl(B) be parameters over which σ is bounded.

For all a ∈ K the image σ(a) is in

acl(Ba) = Ba alg ∩ K = (B ∪ a ) alg ∩ K.
By compactness, there is a finite number of polynomials (P i (x i , y) : i < ℓ) over B, and for all i < ℓ a sequence (f θi,j : j < |x i |) of compositions of functions in F such that for all a ∈ K there is i < ℓ with P i ( fθi,j (a), y) non-trivial in y, and satisfied by σ(a). Let K = K alg be the algebraic closure of K, and (f θi : i < n) the sequence of functions f θi,j , with f 0 = id. We consider the following additive subgroup of Kn+1 :

G = {((f θi (a) : i < n), σ(a)) : a ∈ K}.
Let Γ be its Zariski closure, and Γ 0 the connected component of Γ. Then Γ 0 is the Zariski closure of G 0 = Γ 0 ∩ G, and the latter is of finite index in G.

Since every element (x, y) of G satisfies an equation P i (x, y) = 0 (with a suitable enumeration of x), non-trivial in y, for some i < ℓ, the disjunction of these equations is generically satisfied in Γ 0 and non-trivial in y by [START_REF] Wagner | Subgroups of stable groups[END_REF]Remark 27].

However, the only proper additive algebraic subgroups of Kn+1 are given by additive polynomials. So there is a polynomial P ∈ K[X 0 , . . . , X n ] of the form P (x, x n ) = i≤n, j<ω λ i,j x p j i such that P x, x n ) = 0 is satisfied on Γ 0 , and such that P ( 0, x n ) is non-trivial.

Since G and G 0 are Gal(K)-invariant, so are Γ and Γ 0 , and we can assume that P has coefficients in the purely inseparable closure of K. By composing with a power of Frobenius, we can even suppose that P ∈ K[X 0 , . . . , X n ].

The function x → P ((f θi (x)) i<n , σ(x)) is an additive endomorphism of K whose image F is a finite additive subgroup. In characteristic zero F is trivial; in positive characteristic there is an additive nontrivial polynomial Q ∈ K[X] which vanishes on F . So Q(P ((f θi (x)) i<n , σ(x)) = 0 for all x ∈ K. Since F contains Frobenius in positive characteristic, we obtain a non-trivial equation of the form

i<n λ i f θi (x) = j<ω µ j σ(x) p j
satisfied on K (possibly with n and θ i different). We choose such an equation with

θ m = max{θ i : λ i = 0}
minimal. Moreover, we can assume that there is only one non-zero µ, since for j 0 maximal with µ j0 = 0 and a ∈ K transcendental, This proof is inspired by that of Blossier, Hardouin and Martin Pizarro [1, Théorème 3.1], but it uses the generic properties of the definable envelope (which here is equal to the Zariski closure) of [START_REF] Wagner | Subgroups of stable groups[END_REF] instead of the stabilizer. We need for our Theorem 3.9 only part of hypothesis 4 of [START_REF] Blossier | Sur les automorphismes bornés de corps munis d'opérateurs[END_REF] (the characterization of the algebraic closure) and we make no assumptions about the simplicity of the expansion; on the other hand, we assume from the beginning the Pfaffian property ([1, Proposition 1.4], which follows from their hypothesis 1). Note that we are not considering the inverses of automorphisms in F: we can always add them to F and preserve the Pfaffian property; and in the applications this will be necessary to obtain the characterization of the algebraic closure.

In order to obtain [1, Théorème 3.1] (note that bounded there means Σ-bounded in our sense), it suffices to see that any Σ-bounded automorphism of a simple field must be definable by Corollary 2.2, whence bounded, and then to apply Theorem 3.9.

Fact 1 . 3 .

 13 [START_REF] Wagner | Simple Theories[END_REF] Lemma 3.5.5] If A | ⌣B C then Σcl(A) | ⌣Σcl(B) Σcl(C). More precisely, for all A 0 ⊂ Σcl(A) we have A 0 | ⌣B 0 Σcl(C), where B 0 = dcl(A 0 B)∩Σcl(B).

i<n λ i f

  θi (ax)σ(a) p j 0 i<n λ i f θi (x) = j<j0 µ j [σ(a) p jσ(a) p j 0 ] σ(x) p j allows us to reduce the number of non-trivial µ j while preserving θ m by Corollary 3.7. Then we can take µ = 1, and i<n λ i f θi (x) = σ(x) p j for all x ∈ K.For all a ∈ K we have from Remark 3.7 thati<n λ i f θi (ax) = λ m σ θm (a)f θm (x) + θ<θm α θ f θ (x) = σ(ax) p j = σ(a) p j σ(x) p j for some coefficients α θ ∈ K that depend on a. If σ θm (a) = σ(a) p j , then [σ(a) p jσ θm (a)] σ(x) p j = λ m σ θm (a)f θm (x) + θ<θm α θ f θ (x)σ θm (a) i<n λ i f θi (x) = θ<θm α θ f θ (x) -i<n, i =m σ θm (a)λ i f θi (x)gives a equation with a smaller maximal θ, a contradiction. The theorem follows.

  1) Two hyperdefinable subgroups H and K are commensurable if their intersection is of bounded index in either one. (2) A hyperdefinable subgroup H is locally connected if it is equal to any commensurable group-theoretical or automorphic conjugate. Fact 1.5. [14, Corollary 4.5.16, Lemmas 4.5.18 and 4.5.19]

(1) A locally connected hyperdefinable subgroup of G has a canonical parameter. (2) The normalizer of a locally connected hyperdefinable group is again hyperdefinabe. (3) Any hyperdefinable subgroup H of G has a locally connected component H loc , which is the smallest hyperdefinable locally connected subgoup commensurable with H. Its canonical parameter is definable over the parameters used to hyperdefine H.

-CE40-0026 Agrume.