Tamer Badran Alhassan Sayed

Student Member, IEEE Marie-Minerve Louërat

Member, IEEE Hassan Aboushady

.5-to-3.0GHz Tunable RF Σ∆ ADC with a Fixed Set of Coefficients and a Programmable Loop Delay

Keywords: Tunable Bandpass Sigma-Delta, LC, RF, Loop Delay

In this paper, we present a tunable bandpass RF Sigma-Delta modulator with a fixed set of feedback DAC coefficients. The sampling frequency, fs, is tuned with the center frequency, f0, in order to maintain a fixed normalized center frequency, f0/fs. Any variation in the loop delay is compensated to maintain a fixed normalized loop-delay t d /Ts. The bandpass Sigma-Delta, reported in this paper, is based on an LC resonator with tunable center frequency from 1.5 to 3.0 GHz and a corresponding sampling frequency from 6.0 to 12.0 GHz. For an oversampling ratio of 64, the ADC achieves the same SNDR of 37 dB and the same Dynamic Range of 45 dB over the complete tuning range. This performance is achieved for a bandwith of 47 MHz at 1.5GHz and a Bandwidth of 93 MHz at 3 GHz. The ADC, fabricated in a 65 nm CMOS process, consumes only 13 mW from a 1.2 V supply. In order to compare this circuit with the state of the art, we use not only the conventional ADC Figure of Merit but we also use a Figure of Merit dedicated to RF circuits. In this case, the measured chip center frequency, noise figure and non-linearity are also taken into account in the comparison.

I. INTRODUCTION

B andpass (BP) Σ∆ modulation is a very power-efficient technique to realize RF Analog-to-Digital Converters (ADCs) [START_REF] Beilleau | A 1.3V 26mW 3.2GS/s Undersampled LC Bandpass Sigma-Delta ADC for a SDR ISM-band Receiver in 130nm CMOS[END_REF]- [START_REF] Jang | A 16-Element 4-Beam 1 GHz IF 100 MHz Bandwidth Interleaved Bit Stream Digital Beamformer in 40 nm CMOS[END_REF]. In order to cover a wide frequency range, BP Σ∆ ADCs with tunable center frequencies have been proposed [START_REF] Shibata | A DC-to-1 GHz Tunable RF ∆Σ ADC Achieving DR=74 dB and BW=150 MHz atf 0 =450 MHz Using 550 mW[END_REF]- [START_REF] Chae | A 69 dB SNDR, 25 MHz BW, 800 MS/s Continuous-Time Bandpass ∆Σ Modulator Using a Duty-Cycle-Controlled DAC for Low Power and Reconfigurability[END_REF]. One important challenge in designing a tunable BP Σ∆ ADC, shown in Fig. 1, relies in varying the modulator's feedback DAC coefficients in order to maintain the same Signal Transfer Function (STF) and Noise Transfer Function (NTF) for each center frequency, f 0 , in the complete tuning range, ∆f 0 , of the ADC. The fact that the BP Σ∆ feedback DAC coefficients have to be tuned for each center frequency has considerably complicated the modulator architecture. This resulted in a very large power consumption of 750 mW for a 1 GHz tuning range in [START_REF] Shibata | A DC-to-1 GHz Tunable RF ∆Σ ADC Achieving DR=74 dB and BW=150 MHz atf 0 =450 MHz Using 550 mW[END_REF], a rather low SNR of 40 dB with a bandwidth of 1 MHz for a 1.2 GHz tuning range in [START_REF] Gupta | A 0.8-2 GHz Fully-Integrated QPLL-Timed Direct-RF-Sampling Bandpass Σ∆ ADC in 0.13 µm CMOS[END_REF] or a very limited tuning range of 40 MHz in [START_REF] Chae | A 69 dB SNDR, 25 MHz BW, 800 MS/s Continuous-Time Bandpass ∆Σ Modulator Using a Duty-Cycle-Controlled DAC for Low Power and Reconfigurability[END_REF]. In the BP Σ∆ architecture, illustrated in Fig. 1, the sampling frequency, f s , is fixed, so any variation in the center frequency, f 0 , leads to a variation in the normalized center frequency, f 0 /f s , and subsequently in the overall STF and NTF of the Σ∆ modulator. In this case, all the Σ∆ feedback DAC coefficients have to be tuned to compensate for this f 0 /f s variation and Alhassan Sayed, Tamer Badran, Marie-Minerve Louërat and Hassan Aboushady are with Sorbonne Université, CNRS, LIP6 Laboratory, 75005 Paris, France (e-mail: alhassan.sayed@lip6.fr). Alhassan Sayed is also with Faculty of Engineering, Minia University, Egypt. Tamer Badran is on leave from Faculty of Engineering, AASTMT, Alexandria, Egypt. In this paper, we propose the tunable BP Σ∆ architecture, illustrated in Fig. 2, where the normalized center frequency, f 0 /f s , is fixed. In this case, any variation in the center frequency, f 0 , is tied to a variation in the sampling frequency, f s , in order to keep the normalized center frequency, f 0 /f s , equal to 1/4. Ideally in this case, all the Σ∆ feedback DAC coefficients can be fixed because f 0 /f s is fixed. Practically, varying the sampling frequency, f s = 1/T s , changes the normalized loop delay, t d /T s , which leads to a modification of the modulator's NTF. As illustrated in Fig. 2, in order to maintain the same NTF without tuning the feedback DAC coefficients, we propose to modify the loop delay, t d , in order to compensate for any variation in the normalized loop delay t d /T s [START_REF] Aboushady | Tunable Bandpass Sigma-Delta Modulator[END_REF]. The proposed architecture is validated through the implementation of a 2 nd order LC BP Σ∆ modulator in a 65 nm CMOS process. For an OSR of 64, the modulator achieves an SNDR of 37 dB over the entire tuning range, ∆f 0 = 1.5 GHz. This SNDR is achieved for a bandwidth BW = 47 MHz at f 0 = 1.5 GHz and for a bandwidth BW = 93 MHz at f 0 = 3.0 GHz. The fabricated chip consumes 13 mW and achieves the highest reported tuning range, ∆f 0 , and the highest reported center frequency, f 0 , for a CMOS BP Σ∆ ADC.

II. PROPOSED TUNABLE Σ∆ ADC

The proposed 2 nd order tunable BP Σ∆ modulator is illustrated in Fig. 3. The modulator is composed of an LC resonator with a tunable center frequency, f 0 , a single-bit comparator with a tunable sampling frequency, f s , a tunable loop-delay, t d , and two Non-Return-to-Zero (NRZ) feedback Finite Impulse Response Digital-to-Analog Converters (FIRDACs) with a fixed set of coefficients.

A. System-Level Design

The system level design of the proposed BP Σ∆ modulator, shown in Fig. 3, is based on the Σ∆ Discrete-Time (DT) to Continuous-Time (CT) transformation described in [START_REF] Ashry | A 4th Order 3.6GS/s RF Sigma-Delta ADC with a FoM of 1pJ/bit[END_REF]. This transformation method is used the calculate the feedback FIR-DAC coefficients. As shown in Fig. 4, the FIRDAC coefficients depend on the value of the Normalized Loop Delay, t d /T s . In order to simplify this design, we choose, t d /T s = 1.5. In this case, the number of FIRDAC coefficients is reduced to a single coefficient out of four (only m 0 has a nonzero value). The loop delay, t d , is the time difference between the risingedge of the sampling clock and the feedback DAC output. In practice, t d represents the quantizer response-time and the DAC nonzero switching time [START_REF] Cherry | Excess Loop Delay in Continuous-Time Delta-Sigma Modulators[END_REF]. In the proposed tunable BP Σ∆ modulator both the center frequency, f 0 , and the sampling frequency, f s = 1/T s , are tuned together in order to maintain a fixed normalized center frequency, f 0 /f s , equal to 1/4. Having f 0 /f s = 1/4, greatly simplifies the digital signal processing at the modulator's output [START_REF] Haghighitalab | A 2.4 GHz ISM-band Highly Digitized Receiver based on a Variable Gain LNA and a Subsampled Sigma-Delta ADC[END_REF]. For a specific circuit, it is expected that the loop delay, t d , is constant, but since the sampling period, T s , is tunable then the normalized loop delay, t d /T s will be variable. In Fig. 5, it can be seen that, a 2 nd order CT BP Σ∆ modulator designed for a Normalized loop delay of 1.5, does not suffer from SNR degradation if t d /T s is kept in the range from 1.2 to 1.7. Within this range, the SNR degradation is less than 3 dB. In the next subsection, we propose a circuit technique to control the normalized loop-delay and to maintain t d /T s within this acceptable range.

B. Circuit Design

The circuit details of the proposed BP RF Σ∆ ADC are shown in Fig. 6. The modulator is based on an LC loop filter, a single-bit quantizer and a NRZ feedback DAC. The LC filter is composed of an input transconductance, G m , and an LC tank. The input G m converts the input voltage to current enabling current summation at the feedback node. The LC tank is composed of a differential inductor, two capacitor banks and a negative transconductance, G mq . The coarse and fine capacitor bank are used to obtain a wide and high resolution tuning range.

The coarse 63-capacitor bank, C c , realized using a 440 fF Metal-Insulator-Metal (MIM) unit-capacitor, is used to obtain a wide tuning range. Another 63-capacitor bank, C f , realized using an 8 fF Metal-Oxide-Metal (MOM) unit-capacitor, is used for fine-tuning and high resolution trimming. G mq , has a programmable negative transconductance to compensate for the losses introduced by the integrated inductor and the capacitor bank. The NRZ feedback signal is realized using a current steering DAC. The comparator block is composed of a pre-amplifier followed by two Source-Coupled-Logic (SCL) latches (latch-1 and latch-2) having inverted clock signals. latches are inverted.In this case, latch-3 increases t d /T s by 0.5 and latch-4 adds another 0.5 to t d /T s . In these cases, the loop delay is defined as: t d = (N -1)T s /2 + t par , where N is the number of toggled latches and t par is the total parasitic delay due to the latch regeneration time, the inverter propagation delay and the DAC switching time. The first part of the loop delay, (N -1)T s /2, is welldefined and is equal to integer multiples of half the sampling period, T s /2. On the other hand, the second part of the loop delay t par depends on the circuit characteristics and is estimated by post-layout simulation. The normalized loop delay, t d /T s , is estimated for the complete tuning range of the sampling frequency from 6.0-to-12.0 GHz. The estimated t d /T s for each case described above is listed in Table I. The tunable delay block can be used to increase the normalized loop delay by steps of 0.5. As shown in Fig. 5, there is very little degradation in the Σ∆ performance when the normalized loop delay, t d /T s , is between 1.2 and 1.7. According to table I, these values can be achieved using the configuration for "Case 1" and "Case 2". Although "Case 3" does not seem necessary, it was implemented to allow for more flexibility and to account for any process variation. In the proposed method for loop delay tuning, only a coarse tuning by, T s /2, is possible. Although, more precise techniques may be used [START_REF] Maymandi-Nejad | A digitally programmable delay element: design and analysis[END_REF], the proposed technique has the advantage of requiring only 2 bits of configuration to cover the complete sampling frequency range from 6.0-to-12.0 GHz.

III. MEASUREMENT RESULTS

The chip was fabricated in a 65 nm CMOS technology and packaged in an 28-lead quad flat no-leads package. The chip micrograph is shown in Fig. 7. The active area is 0.3 mm 2 . Most of the active area is occupied by the inductors, and the four capacitor banks. For measurements, we used two synchronized signal generators: one generator was used for the input signal and the second generator was used as a clock for the ADC. The measured and the simulated Power Spectral Density (PSD) of the BP Σ∆ single-bit output are shown in Fig. 8. In this figure, we show the measurements and simulations for two different center frequencies, at f 0 = 1.5 GHz and at f 0 = 2.9 GHz, to validate the fact that the proposed BP Σ∆ has the same performance over the entire tuning range. The Signal-to-Noise and Distortion Ratio (SNDR) versus input power is also measured at the same two center frequencies and the results are shown in Fig. 9. As expected from simulation results, the ADC achieves an SNDR of 37 dB and a DR of 45 dB for an OSR of 64 over the complete tuning range, ∆f 0 . Note that in these measurements and simulations the normalized center frequency, f 0 /f s is always fixed to 1/4. The variation of the SNDR and DR with respect to the OSR has also been studied. For OSR values higher than 64, the Fig. 8: Measured PSD at the output of the BP Σ∆ ADC. Fig. 9: Measured SNDR and DR at center frequencies of 1.5 GHz and 2.9 GHz for OSRs of 64 and 1024. thermal-noise is dominant and the noise-floor is relatively flat within the band-of-interest. In this case, each doubling of the OSR results in a 3 dB increase in SNDR. On the other hand, for OSRs lower than 64, the noise-shaped quantization-noise is dominant and dividing the OSR by two results in a 9 dB degradation of the SNDR. Fig. 9 also shows that the ADC achieves an SNDR of 51 dB and a DR of 59 dB for an OSR of 1024 over the complete tuning range. In order to evaluate the linearity of the BP Σ∆, an IQ vector signal generator was used to generate a two-tone input signal with a power of -30 dBm around 1.5 GHz with a frequency spacing, ∆f in , of 1 MHz. In this case, the Σ∆ output PSD is shown in Fig. 10(a). The normalized third intermodulation distortion, IM 3 , is -66 dBc. In Fig. 10(b), it is shown that the ADC achieves an IIP 3 of +5 dBm and a Spurious-Free-Dynamic-Range (SFDR) of 45 dB in a bandwidth of 47 MHz. Due to the lack of the appropriate measurement equipment, it was not possible to perform a 2-tone test at 2.9 GHz. It is worth noting however, that the measured IM 3 at 1.5 GHz is only 2 dB below postlayout simulations. We can then assume that we have the same linearity performance over the complete tuning range as predicted by post-layout simulations. The measurement results presented in Fig. 8, Fig. 9 and Fig. 10 have been obtained using the following configurations: -Case 2 of Table I for f 0 = 1.5 GHz and f s = 6.0 GHz.

-Case 1 of Table I for f 0 = 2.9 GHz and f s = 11.6 GHz. In Table II, the performance of the proposed BP Σ∆ ADC is compared to recently published tunable and non-tunable RF BP Σ∆ ADCs. We notice that the proposed circuit achieves the highest center frequency, f 0 = 3.0 GHz, and the widest tuning range, ∆f 0 . These performances are achieved with the lowest power consumption of 13 mW. In order to compare the different implementations we used the widely used Figure of Merit, F oM ADC , for high speed ADCs [START_REF] Ali | A 16-bit 250-MS/s IF Sampling Pipelined ADC With Background Calibration[END_REF]:

F oM ADC (dB) = SN DR + 10log BW P (1)
where, BW , is the ADC bandwidth and, P , its power consumption. Unfortunately, F oM ADC , does not take into account neither the center frequency, f 0 , nor the frequency tuning range, ∆ f , of the ADC. In fact, the SNDR, in equation (1), does not really evaluate the circuit's linearity since, in the case of RF BP Σ∆ ADCs, all the harmonics of the input signal are located outside the band of interest. Furthermore, F oM ADC does not consider many important aspects in RF circuits like The Gain for all ADCs is assumed to be equal to one.

F oM RF (kHz) = Gain 1 IIP 3 f 0 (kHz) (F -
sensitivity and Noise Figure.

In order to give credit to RF BP Σ∆ ADCs achieving a high center frequency and a high linearity, we propose to use a Figure of Merit used to evaluate RF circuits [START_REF] Brederlow | A Mixed-Signal Design Roadmap[END_REF]:

F oM RF (kHz) = Gain IIP 3 f 0 (kHz) (F -1) P (2)
where we assume that the Gain in ADCs is equal to one, IIP 3 is the third order intercept point in mW, F is the noise factor and P the power consumption in mW. In case, the center frequency, f 0 , is given in kHz, F oM RF will also be in kHz. Note that we do not propose to substitute F oM ADC by F oM RF for RF BP Σ∆ ADCs but rather to consider the two FoMs in order to give credit to f 0 and IIP 3 along with the BW . As reported in table II, the proposed tunable RF bandpass Σ∆ ADC achieves the best F oM RF thanks to its very high center frequency and its high IIP 3 compared to the state of the art. Note also that the proposed ADC achieves the widest tuning range, ∆f 0 = 1.5 GHz.

IV. CONCLUSION

In this paper, a tunable BP LC Σ∆ ADC is presented. The 1-bit Σ∆ modulator has a fixed set of coefficients on the complete tuning range thanks to fixing the normalized center frequency, f s /f 0 , to 1/4. The normalized loop delay, t d /T s is also kept constant on the entire tuning range by adding a tunable delay in the Σ∆ loop. The circuit, fabricated in a 65 nm CMOS, achieves 37 dB of SNDR and +5 dBm of IIP 3 for an OSR of 64. The same performance is measured on the complete tuning range. In addition to the conventional F oM ADC , the performance of the measured chip is compared with F oM RF , a Figure of Merit, dedicated to RF circuits which takes into account f 0 and IIP 3 . Compared to the state of the art, the proposed BP Σ∆ ADC achieves the highest center frequency of 3.0 GHz and the widest tuning range of 1.5 GHz while consuming only 13 mW.

Fig. 1 :Fig. 2 :

 12 Fig.1:A tunable bandpass Σ∆ ADC having a variable normalized center-frequency, f 0 /f s . All Feedback DAC coefficients have to be tuned with f 0 .

Fig. 3 :

 3 Fig. 3: Proposed 2 nd order tunable BP Σ∆ ADC.

Fig. 4 :

 4 Fig. 4: The FIRDAC coefficients (m 0 , m 1 , c 0 and c 1 ) in function of the Normalized Loop Delay, t d /T s .

Fig. 5 :

 5 Fig. 5: Post-layout simulation results of maximum SNR versus Normalized Loop Delay, t d /T s .

Fig. 6 :

 6 Fig. 6: Circuit details of the proposed tunable BP Σ∆ ADC.

Fig. 7 :

 7 Fig. 7: Chip micrograph.

Fig. 10 :

 10 Fig. 10: (a) Measured output spectrum for a 2-tone test signal with ∆f 0 = 1 MHz. (b) IIP 3 and SFDR measured using 2-tone test signals with different input power.

TABLE I :

 I Estimated Normalized Loop Delay, t d /T s , at each sampling frequency and for each configuration case of the tunable delay. t d /T s values which do not cause SNR degradation are indicated in bold.

		Normalized loop delay, t d /Ts
	Sampling Frequency, fs	Case 1 Case 2	Case 3
	6.0 GHz	1.1	1.6	2.1
	7.0 GHz	1.2	1.7	2.2
	8.0 GHz	1.3	1.8	2.3
	9.0 GHz	1.4	1.9	2.4
	10.0 GHz	1.5	2.0	2.5
	11.0 GHz	1.6	2.1	2.6
	12.0 GHz	1.7	2.2	2.7

ACKNOWLEDGMENT

This work was partially funded by the French National Research Agency: TOLTECA Project ANR-16-CE04-001301.

This

Martens [START_REF] Martens | RF-to-Baseband Digitization in 40nm CMOS With RF Bandpass ∆Σ Modulator and Polyphase Decimation Filter[END_REF] Shibata [START_REF] Shibata | A DC-to-1 GHz Tunable RF ∆Σ ADC Achieving DR=74 dB and BW=150 MHz atf 0 =450 MHz Using 550 mW[END_REF] Jang [START_REF] Jang | A 16-Element 4-Beam 1 GHz IF 100 MHz Bandwidth Interleaved Bit Stream Digital Beamformer in 40 nm CMOS[END_REF] Ashry [START_REF] Ashry | A 4th Order 3.6GS/s RF Sigma-Delta ADC with a FoM of 1pJ/bit[END_REF] Gupta [


Guillaume Monneret
email: guillaume.monneret@chu-lyon.fr

Fabienne Venet

Christian Meisel

Joerg C Schefold

Assessment of monocytic HLA-DR expression in ICU patients: analytical issues for multicentric flow cytometry studies

published or not. The documents may come   L'archive ouverte pluridisciplinaire

We read with interest the recent report by Gogos and colleagues [START_REF] Gogos | Early alterations of the innate and adaptive immune statuses in sepsis according to the type of underlying infection[END_REF]. While the rationale for their study is excellent, we would like to comment on technical issues that may have infl uenced the results.

As stated, a time limit of 8 hours between sample drawing and staining at a central laboratory was specifi ed [START_REF] Gogos | Early alterations of the innate and adaptive immune statuses in sepsis according to the type of underlying infection[END_REF]. Unfortunately, information regarding transport conditions is missing (average time, temperature). Th is seems important since monocytic HLA-DR expression (mHLA-DR) increases artifi cially over time [START_REF] Monneret | Analytical requirements for measuring monocytic human lymphocyte antigen DR by fl ow cytometry: application to the monitoring of patients with septic shock[END_REF][START_REF] Döcke | Monitoring temporary immunodepression by fl ow cytometric measurement of monocytic HLA-DR expression: a multicenter standardized study[END_REF]. Conse quently, recommendations suggest that sample staining for mHLA-DR should occur within 4 hours [START_REF] Monneret | Analytical requirements for measuring monocytic human lymphocyte antigen DR by fl ow cytometry: application to the monitoring of patients with septic shock[END_REF][START_REF] Döcke | Monitoring temporary immunodepression by fl ow cytometric measurement of monocytic HLA-DR expression: a multicenter standardized study[END_REF]. Although the authors aimed to address the eff ect of transportation, they inappropriately used samples present ing with already near-maximal mHLA-DR values (>90%) before storage. We therefore assume that mHLA-DR results may be falsely elevated due to prolonged transportation times. Furthermore, mHLA-DR modulation during sepsis takes days and consecutive measurements are required [START_REF] Monneret | Persisting low monocyte human leukocyte antigen-DR expression predicts mortality in septic shock[END_REF]. Assessment of one early sample (within the fi rst 24 hours) is probably inappropriate to investigate the impact of infection on mHLA-DR. Similarly, apoptosis staining should not be performed after 8 hours and experts' recommendations highlight the need for dedicated protocols on fresh cells [START_REF] Ayala | Apoptosis in sepsis: mechanisms, clinical impact and potential therapeutic targets[END_REF].

We are convinced that successful future trials in sepsis will rely on our capacity to accurately assess immune responses. In that sense, fl ow cytometry multicentric clinical studies are essential. Such trials should be performed in standardized environments in accordance with specifi c (pre)analytical requirements. Otherwise, results might be misinterpreted and may impede promising new avenues in future care of septic patients.

Abbreviations

ICU, intensive care unit; mHLA-DR, monocytic human leukocyte antigen DR-1.

Author details

Competing interests

The authors declare that they have no competing interests.

Published: 27 July 2010


Antoine Mhanna

TANGO-QUADRILATERALS

Keywords: Mathematics Subject Classification. 51N20, 33B10, 26B99. Key words and phrases. Tangential and extangential quadrilaterals, Trigonometric Identities. 1

In this note we represent and characterize a tango-quadrilateral as a quadrilateral having a circle that is tangent to the lines supporting its sides.

Introduction

The problem of tangential resp. extangential quadrilaterals and their characterizations is a well known problem, see [START_REF] Josefsson | More characterizations of Tangential Quadrilaterals[END_REF] resp. [START_REF] Josefsson | New Characterizations of Extangential Quadrilaterals[END_REF]. By definition an extangential quadrilateral is a convex quadrilateral with an external circle tangent to the extensions of all four sides. Similarly a tangential quadrilateral is a convex quadrilateral with an incircle, that is a circle tangent at the inside of the quadrilateral to all four sides.

Our purpose here is to give a unified presentation for both tangential and extangential quadrilaterals. Throughout this paper we assume quadrilaterals without three aligned vertices. Definition 1. A tango-quadrilateral is any quadrilateral having a circle tangent to the lines supporting its four sides, (see Figure 1).

Convex Concave Crossed

We start by some classic properties of tangential and extangential quadrilaterals:

Lemma 1 (Pitot's theorem). [START_REF] Josefsson | More characterizations of Tangential Quadrilaterals[END_REF] A convex quadrilateral is tangential if and only if the sum of opposite sides are equal.

Lemma 2. [2]

A convex quadrilateral is extangential if and only if the sum of two adjacent sides is equal to the sum of the other two.

The reader can see [START_REF] Josefsson | More characterizations of Tangential Quadrilaterals[END_REF][START_REF] Josefsson | New Characterizations of Extangential Quadrilaterals[END_REF] and the references therein for further equivalent properties of such quadrilaterals.

Lemma 3 (Newton's line). [START_REF] Josefsson | New Characterizations of Extangential Quadrilaterals[END_REF][START_REF] Bogomolny | Newton's Theorem and Leon Anne's Theorem[END_REF] Suppose ABCD is a tangential or extangential quadrilateral. If M and N are the midpoints of its diagonals and O the center of the tangent circle then M ,N and O are collinear.

See also [START_REF] Ayme | La ponctuelle de Newton d'un quadrilatère circonscriptible[END_REF] for a proof of Lemma 3 with a historical view. We reprove previous lemmas analytically in a slightly more general setting together with a trigonometric characterization for tango-quadrilaterals.

Given a quadrilateral ABCD with corners A, B, C, D and sides [AB], [BC], [CD] and [DA], we call an inside (internal) bisector of a corner angle, the bisector of the angle formed by the segments of adjacent sides. The other bisector is called the outside (external) bisector (the one of the supplementary angle). The following is a basic formulation of tango-quadrilaterals: Proposition 1. A quadrilateral ABCD is a tango-quadrilateral if and only if there exist four (equivalently 3) angle bisectors one from each corner such that they intersect at a single point denoted by O.

Main Results

Before stating the main results we consider which angle bisectors are like to intersect.

Proposition 2. For a tango-quadrilateral ABCD, if 4 corner bisectors intersect at O then no consecutive two are both outside bisectors.

Proof. This is straightforward, for suppose not the two outside consecutive bisectors intersect at point O outside of ABCD and it is easy to see that the fourth side (line) can not be tangent to that circle (using that from a point T on circle (C) there is a single line tangent to (C) and from a point T outside of (C) there are only two tangents to (C)).

Similarly we have: Proposition 3. For a crossed tango-quadrilateral ABCD, if 4 corner bisectors intersect at O, no consecutive two are both inside bisectors.

Figure 2

2.1. A slope formula. We shall prove a slope formula that may be used in a Newton's line characterization of tango-quadrilaterals. We also give a direct application of the formula as an exercise.

Given an orthonormal axis (O; -→ i ; -→ j ), the equation of a line (l) is y = αx + β for some reals α and β, α is called the slope of (l). In the sequel we take α (without loss of generality) to be finite and well defined, the case of a vertical line is taken as a tending limit for α → ±∞ and α = y -y A x -x A for some point A ∈ (l).

(CM ) = st -r(ta + sb) sa + bt -r . Proof. Set f = b(y B -y C ), i = b(x B -x C ), h = a(y A = y C ), k = a(x A -x C ).

Consequently we have

t = h k and s = f i , slope (CM ) = ay A + by B -y C ax A + bx B -x C = h + f k + i . For x = k i , it can be verified that r = hb -f a kb -ia = txb -sa xb -a and after simplification x = (r -s)a (r -t)b
. We also have slope (CM ) = txi + si xi + i which is the formula given after simplifying i.

As an application consider the points O(0; 0), E(-a; Proof. We take here an orthonormal axis centered at A, up to a scalar multiple let d(O; (AB)) = 1, so the circle is of radius one. We find the slopes of (OC), (OA) and (AC) in order to apply Proposition 4 on the median (OJ). The slope of (OI) can be deduced by replacing a ↔ b, d ↔ x and adding a minus sign to the corresponding slopes (vertical reflection). Call t = tan(a) the slope of (OA) and s = -tan(2b + d) that of (OC).

M (x M ; y M ) = M ( ax A + bx B a + b ; ay A + by B a + b ).
From the angle definitions we have A(0; 0), O( cos A similar characterization holds if we consider two angle bisectors of two opposite corners for the quadrilateral ABCD.

Lemma 5. In an orthonormal system (O; -→ i ; -→ j ), let A(r cos(θ); r sin(θ)), C(c; 0) for some positives r, c and θ ∈ R. If a point B(t cos(α); t sin(α)) is taken such that (OB) is an angle bisector of the lines (BA) and (BC) at B then t = rc sin(2α -θ) c sin(α) -r sin(θ -α) .

Proof. We use the characterization of an angle bisector so:

d(O; (BA)) k = d(O; (BC)) h .
By the law of cosines:

4Area(OCB) 2 =h 2 (t 2 + c 2 -2tc cos(α)) = t 2 c 2 sin(α) 2 . 4Area(OAB) 2 =k 2 (t 2 + r 2 -2tr cos(θ -α)) = t 2 r 2 sin(θ -α) 2 .
The equation h 2 = k 2 gives the quadratic (t(c sin(α) + r sin(θ -α)) -rc sin(θ))(t(r sin(α -θ) + c sin(α)) -rc sin(2α -θ)) = 0.

The roots are t 1 = rc sin(θ) c sin(α) + r sin(θ -α) and t 2 = rc sin(2α -θ) c sin(α) -r sin(θ -α)

. The first root gives A, B and C aligned which we have already excluded.

(OC). Assume that (B

1 A) ∩ (B 2 C) = D. It can be verified that (B 2 C) is the line y = y B x c -x B + y B c x B -c and (B 1 A) is y = x B sin(2θ) -y B cos(2θ) -r sin(θ) x B cos(2θ) + y B sin(2θ) -r cos(θ) x + ry B cos(θ) -rx B sin(θ) x B cos(2θ) + y B sin(2θ) -r cos(θ)
.

This gives point D(x D ; y D ) where

x D = rx B sin(θ)(c -x B ) + y B (rx B cos(θ) -cx B cos(2θ) -cy B sin(2θ)) (cx B -t 2 ) sin(2θ) + y B (r cos(θ) -c cos(2θ)) + r sin(θ)(x B -c) . Simplify y D + y B x D + x B to get slope (OI) = sin(α) t -r cos(α) cos(θ) cos(α) t - c sin(2θ -2α) + r sin(2α) cos(θ) sin(2θ) cos(α) , (ID = IB), slope (OJ) = r sin(θ) r cos(θ) + c = m, (JA = JC).
Equivalently slope

(OI) = 2 sin(θ) sin(α)(r cos(α) -t cos(θ)) c sin(2(θ -α)) + r sin(2α) cos(θ) -t sin(2θ) cos(α) = n.
Solving n = m gives sin(θ) cos(θ)(t(c sin(α) + r sin(α -θ)) + rc sin(θ -2α)) = 0, this is true for θ ≡ 0 (mod 90 • ) and for t = rc sin(2α -θ) c sin(α) + r sin(α -θ)

. The case θ ≡ 0 (mod 180 • ) gives ABCD a kite, otherwise the given solution is the one for ABCD a tango-quadrilateral by Lemma 5. Now assume θ ≡ 0 (mod 90 • ) the particular case when

r cos(α) cos(θ) = c sin(2θ -2α) + r sin(2α) cos(θ) sin(2θ) cos(α)
gives c = r cos(α) cos(θ -α) = c 0 . For c = c 0 , from t = r cos(α) cos(θ) , (B 1 A) ≡ (B 2 C) (excluded) and from slope (OI) = sin(α) cos(α) = m we get c = r sin(θ-α) sin(α)

= c 1 . The equation c 0 = c 1 is equivalent to sin(2α) = sin(2θ -2α) so α ≡ θ 2 (mod 90 • ), c = r and ABCD is a kite, or θ = 90 • . This completes the proof.

A particular case of Theorem 1 is proved in [START_REF] Josefsson | New Characterizations of Extangential Quadrilaterals[END_REF] (Theorem 5.2). A rhombus is a tangential quadrilateral and if only one diagonal is the perpendicular bisector of the other one then such quadrilateral (a kite) has an incircle and an external circle tangent to the lines of its sides. Proof. The case y J = 0 is already discussed so assume y J = 0 and so is y I , we need to prove y I y J < 0, equivalently • Say 90

• < b + d ⇔ 2b + d > 180 • -d ⇔ b + d -a > 180 • -b -d -a. Since b + d <

Proposition 4 .

 4 Take three non collinear points in the plan A(x A ; y A ), B(x B ; y B ), C(x C ; y C ) and a point M ∈ (AB) with M (ax A + bx B ; ay A + by B ) for a, b ∈ R * : a + b = 1. If slope (CB) = s, slope (CA) = t and slope (BA) = r, then slope

  1), F (b; 1), E = F for a, b ∈ R. A line (l) passes through O of equation y = ex, E / ∈ (l) and F / ∈ (l); (l) ∩ (EF ) = K and the perpendiculars at E resp. F to (OE) resp. (OF ) intersect at C.

Finally

  from H ∈ (l) let a perpendicular line to (l) intersect (CF ) at A and (CE) at B. If (KC) ∩ (HA) = M , show that

Figure 3 2. 2 .

 32 Figure 3

Figure 4 Remark 1 .Lemma 4 .

 414 Figure 4

  (a) sin(a) ; 1), B( cos(a) sin(a) + cos(b) sin(b) ; 0), C( cos(a) sin(a) -cos(2b+d) sin(d) ; 1 + sin(2b+d) sin(d) ). Letting r denote the slope of (AC), r = sin(a) sin(d) + sin(2b + d) sin(a) sin(d) cos(a) -sin(a) cos(2b + d) . If m is the slope of (OJ), m = 2ts -r(t + s) s + t -2r , it can be verified that m = sin(a)(sin(d) -sin(2b + d)) sin(a) cos(2b + d) + sin(d) cos(a) consequently the slope n of (OI) is n = -sin(b)(sin(x) -sin(2a + x)) sin(b) cos(2a + x) + sin(x) cos(b) . Solving n = m in x gives: sin(a)(sin(d + b -b) -sin(b + b + d)) sin(a) cos(b + b + d) + sin(d + b -b) cos(a) = -sin(b)(sin(x + a -a) -sin(a + a + x)) sin(b) cos(a + a + x) + sin(x + a -a) cos(b) , after expanding and simplifying we get: 2 sin(a) sin(b) sin(b -a) + tan(a + x) cos(a + b) = 2 sin(a) sin(b) sin(b -a) -tan(b + d) cos(a + b) , equivalently (sin(a) sin(b) = 0) tan(a + x) cos(a + b) = tan(-b -d) cos(a + b) which implies x + a + b + d ≡ 0 (mod 180 • ) as cos(a + b) = 0.

Theorem 1 .

 1 Given a quadrilateral ABCD with no three aligned vertices; let the angle bisectors of two opposite corners intersect at O with an angle θ. If θ = 90 • , the midpoints of the diagonals (I and J) are collinear with O if and only if ABCD is a tango-quadrilateral. Proof. Take as in Lemma 5 the points O(0; 0), A(r cos(θ); r sin(θ)), C(c; 0) and B(t cos(α); t sin(α)) where r and c are positives reals with θ ∈ [0; 180 • ]. We start by the following figure, let B 1 resp. B 2 be the symmetric of B with respect to (OA) resp.

  Now consider an orthonormal system centered at O with d(O; (AB)) = 1, we have A(-cos(a) sin(a) ; -1), B( cos(b) sin(b) ; -1), C(-cos(2b+d) sin(d) ; sin(2b+d) sin(d) ). The midpoints I and J are: and y J = 0 if and only if sin(2b + d) = sin(d) thus b + d ≡ 90 • (mod 180 • ). Equivalently b + d = 90 • for b acute and b + d = 450 • for b obtuse (d verifies d < 180 • -b or d > 360 • -b), this gives J(-cos(a) 2 sin(a) + sin(b) 2 cos(b) ; 0). When a + x = 90 • or a + x = 450 • we get I( cos(b) 2 sin(b) -sin(a) 2 cos(a) ; 0) and it is easy to verify that x I x J > 0 resp. x I x J < 0 for a or b obtuse resp. a and b acute with x I = x J = 0 if and only if a + b = 90 • (ABCD is a rhombus). With the same previous notations: Proposition 5. For x + a + b + d = 180 • , -→ OJ = α -→ IO with α ∈ R + , α is nul if and only if ABCD is a rhombus.

(2. 1 )

 1 (sin(2b + d) -sin(d))(sin(b + d -a) -sin(b + d + a)) < 0.

  180 • and a acute this implies that sin(2b + d) < sin(d) and sin(b + d -a) > sin(b + d + a). • When 2b + d < 180 • -d so 0 < b + d < 90 • ; sin(2b + d) > sin(d) and sin(b + d -a) < sin(b + d + a).Proposition 6. For x + a + b + d = 540 • , -→ OJ = α -→ IO with α ∈ R * -.Proof. We may assume y I and y J are not zero, by Remark 1 we know that sin(x) sin(d) < 0. Let without loss of generality d < 180• -b < 180 • so 180 • < b + d + a < 360 • and (2.1) is proved as follows:• If 180 • < b + d + a < 270 • , for b + d > 90 • , sin(b + d -a) -sin(b + d + a) > 0 and similarly to the last proof sin(2b + d) < sin(d). When b + d -a < 180 • -a -b -d, b + d < 90 • , a > 90 • with sin(b + d -a) -sin(b + d + a) < 0 and sin(2b + d) > sin(d). • If 270 • ≤ b + d + a < 360 • , from 180 • > b + d > 90 • , a > 90• and b < 90 • : sin(b + d -a) -sin(b + d + a) > 0 and sin(2b + d) < sin(d). Proposition 7. Under the previous notations: ) sin(d) cos(2b + 2a + d + x) if and only if x + a + b + d ≡ 0 (mod 180 • ). Proof. Expanding and simplifying gives cos(x + d) = cos(2a + 2b + d + x) and thus x + a + b + d ≡ 0 (mod 180 • ). It can be verified that: sin(d) cos(2b + 2a + d + x). Notice that ( cos(x) sin(x) + cos(d) sin(d) ) has the same sign as sin(x+d) sin(x) sin(d) . From Remark 1, given a tango-quadrilateral sin(x + d) > 0. For x + a + b + d = 180 • : sin(x) sin(d) > 0 and for x + a + b + d = 540 • : sin(x) sin(d) < 0. For d < 180 • (resp. x < 180 • ), BC = cos(b) sin(b) + cos(d) sin(d) with b + d < 180 • (resp. AD = cos(x) sin(x) + cos(a) sin(a) with a + x < 180 • ). For d > 180 • (resp. x > 180 • ), BC = cos(180 • -b) sin(180 • -b) + cos(360 • -d) sin(360 • -d) = -( cos(b) sin(b) + cos(d) sin(d) ) with b + d > 360 • (resp. AD = -( cos(a) sin(a) + cos(x) sin(x) ) with a + x > 360 • . Consequently we have the following: • If x < 180 • and d < 180 • , then ABCD is a tango-quadrilateral with an incircle if and only if AB + DC = AD + BC. • If x > 180 • and d < 180 • , then ABCD is a tango-quadrilateral with an excircle if and only if AD + AB = DC + BC. • If x < 180 • and d > 180 • , then ABCD is a tango-quadrilateral with an excircle if and only if AB + BC = DC + AD. • If x > 180 • and d > 180 • , ABCD is not a tango-quadrilateral.

Kfardebian, Lebanon E-mail address: tmhanat@yahoo.com

Acknowledgments

Some computations are assisted through Maple software and on https://www. dcode.fr.


Ritamaria Bucciarelli

Francesco Saverio Tortoriello

Andrea Rodrigo

Colomba Javier Julian Enriquez

La Ragione

Ilaria Veronesi

Smart technologies and digital intelligence

Keywords: topological quantum computation, R4 exotic, finite geometry fundamental group

la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

With the advent of descriptive grammars arises the concept of semantic operations by categories and linguistics mediates in production techniques for the transmission of knowledge from quantum physics and beyond. The idea of describing a formal process with the mediation of a quantum paradigm is born to generate a continuum of the relationship mathematics -linguisticsquantum physics, but above all, to produce a link between the different areas of knowledge, which scientifically convinces the community concerned. Among the goals set, the research team aims to undertake research on the different textual typologies, but with absolute priority the literary communication in the poetic tradition of the ancient and modern world to find the points of contact that over the centuries have transmitted semantics intact. Nowadays, between utopia and reality, the wonderful phrase# I love you # becomes N0 V N1 and again $NO, $V and $N1. See [START_REF] Silberztein | Les outils informatiques au service des linguistes: présentation[END_REF]. We witness these formal processes before our eyes every day, but often even specialists, technicians, and teachers do not know how to explain the evolution of these processes. The present work is part of the research project Literary Communication in Digital and includes quantum physics, studies ranging from group theory to graph theory. The research team is led by M. Planat, Institut FEMTO-ST Besançon, France, Department of Micro Nano Sciences and Systems (MN2S); Skills and expertise, Number Theory, Quantum Theory, Graph Theory. For the formal process we comply with the methodologies Lexicon-grammar A. Elia (EMDA) and the Nooj M. Silberztein linguistic environments for graph construction and paraphrase generation. Likewise, we will deal with implementations in blended domains using BuViTeGMS techniques with the D.I. Word Tool software for editing and visualization of chosen traits.

FROM GROUP THEORY TO GRAPH THEORY

The aim of these studies is to find a point of contact between scientific and humanistic disciplines, in short, between all disciplines. If we need to proceed with a feedback for our research, it must be said that we started with the desire to give a quantum explanation to the literary text. In this case for the detection of the emotional triplet of the Canto XXXIII of Paradise and Canto III of Inferno and to identify the points of contact that explain the emotional capture. The triangle in the Fano plane gave us a scientific answer because the projective plane includes 7 points (each of which is contained in three lines) and 7 lines (each of which contains three points). It was easy to see in the said points (nodes) and in the straight lines the interested directions (see Bucciarelli et al. 2021 1 ). It remained to clarify the relationship node-lexical sequence, was not at all easy but the studies conducted have shown us otherwise such as: G. De Bueriis and A. M. Langella in which 2 ... analyze some equivalences of parts of sentences, syntagms or parts of syntagms, resulting from local transformations, using notions and tools of graph theory........ that a sentence can be represented using graph theory: in it, the nodes represent the lexical units that constitute it, and the arcs represent the relationships that exist between those same units; a sentence, therefore, as a network consisting of elements (lexical) and relationships (syntactic). Studies continue by relating the model Planat et al. (2021) Full-text request: Quantum Computation and Measurements from an Exotic Space-Time R-4 a wonderful exploration that confirmed our hypothesis about topological computation for a small exotic (space-time) R4 with an example, has two remarkable properties of which the first shows the presence of standard contextual geometries such as the Fano plane:

The authors have previously found a model of universal quantum computation making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a 'magic' |ψ⟩ state in d -dimensional Hilbert space that encodes for an informational complete minimal quantum measure (or MIC), which could contain a finite 'contextual' geometry. In the present work, we choose G as the fundamental group π1(V) of an exotic 4-variety V, more precisely a 'small exotic' (space-time) R4 (which is homeomorphic and isometric, but not diffeomorphic to Euclidean R4). Our selected example, due to S.

Akbulut and RE Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), the Mermin pentagram (at index 10), the two -image switching qubits GQ(2,2) (at index 15), and the Grassmannian combinatorics Gr (2,8) (at index 28); and (b) it allows the interpretation of MIC measurements as arising from such exotic (space-time) R4 S. Our new framework concerning a topological quantum computation and an exotic space-time is also intended to 1 Topological S-T R quantum computation and contextual geometries standard NLG fixed sentences.

2 A large number of texts, both advanced and introductory, can be consulted on graph theory: among the latter, we note at least Chartrand & Zhang (2012), Voloshin (2009), and Wilson (1972[START_REF] Silvestri | Analisi linguistica della poesia[END_REF]. become an approach to "quantum gravity". From topological computation to elementary computation and contextually numerical group and language group was easy 0. 3 :. For M. Planat they are conjugation classes of groups and subgroups inserted in sequences and the secondary structures of the lines of a poem are obtained from an encoding of word types that translated into Lexicon-grammar are groups of phrases or syntagms grouped into nuclear and nonnuclear, thereby defining the theory of groups in a contextualized setting. Like this manner, we can say that:

We define a group: Let G be a non-empty set. To obtain a language group defined as such we filter through the composition of the group belonging to classes and holding a function.

A binary operation defined on G is a function

: G × G G (1.1)
Operation 1.1 is said to satisfy the: associative property, if for every a,b,c, ∈G there is identity (ab) c = a(bc)

When we define a group G, it is said to be such if all the membership conditions are met

: G × G G (1.1)
..... A sentence hides a great quantity of facts (data and relations) expressible with numerical values; in essence, a sentence -an essentially "literary" sentence -is paradoxically "full of numbers." The objective is then -paraphrasing the title of Devlin's book (1998) The Language of Mathematics: Making the Invisible Visible -"to make the invisible visible", not only transforming into numbers everything that can be expressed through them, but showing at the same time as "In general, language is presented to us in the form of strings of sounds (or letters)."Yet, linguists of persuasion have argued that there is evidence for more structure than the eye meets » [START_REF] Kracht | The emergence of syntactic structure[END_REF]. Thus, it is about a 'structural order' -underlying a 'linear order' -determined by dependency relations between the elements that constitute the utterances4 .

HHIIIIICCCCCCCCCCCCCCCCCCTTTTTCCCCCCCCCHHHHHHHHHHH... CCCTTTTTCCCCCTTTTTCCCC44444EEEEEECC 5

Consequently, if the associative property is satisfied it is possible to define recursively in an unambiguous way the product of three or more elements: Therefore, with a syntactic experiment of transformation and distribution we will operate on the validity of the commutative property applied to a semantic unit of the sentence: N0V N1

.

Displacement

An element of the sentence is dislocated with a pause (#), with a pronominal form of the type Ppv

From what has been said briefly, it can be deduced that fixed languages are groups and there is a context between the numerical group and the linguistic group.

PHASE A: MORPHO-SYNTACTIC PROSODIC ANALYSIS A CALCULATION AND AN AVERAGE

The team, then, focuses on samples from the Iliad and Odyssey and traits for textual prediction of fixed forms, with particular attention to prosodic analysis and phonetic phenomena from quantum metrics, and takes its time in some aspects of phonic recursion or non-causal, presence of vowels and consonants in a poetic corpus, in order to identify the DPH, which is the tool that allows identifying the rhythm of the tone, l'Humanitas that emerges and detects l'Ego, as well as semantics of traits in the recitation. The phonic recursion is that phenomenon known as alliteration that the poet pursues in the choice of words with a specific phonic body underpinned by a linguistic or phonetic-stylistic operation and that consists of the non-causal repetition of sounds in sequential figures such as iteration 6 , parallelism 7 , and sequence manipulation 8 , consistent with [START_REF] Silvestri | Analisi linguistica della poesia[END_REF].

Homer -Odyssey -Beginning of the poem (Hom. Od. I 1-2) 9

Ἄˉνδρα˘ μοι˘ ἔˉννε˘πε˘, Μοῦˉσα˘, πο˘λύˉτρο˘πο˘ν, ὃˉς μά˘λα˘ ποˉλλὰ 5 Repeat PLANAT 6 Iteration occurs when two or more homophonic vowels follow each other in contiguous syllables

7 Parallelism occurs when two or more vowels are repeated in the same order with at least one interrupting vowel 8 Finally, there is sequence manipulation when two or more vowels correspond to an inversion of such a sequence 9 http://www.poesialatina.it/_ns/Greek/tt2/Omero/Odissea0001-021.html

Recursion phenomenon: πο˘λύˉτρο˘πο˘ν, ὃˉς μά˘λα˘ ποˉλλὰ If this denotes a sequential contiguity of a vowel voice or even a combination of false vowels, simple iteration, as it is based on vowel monophony and in relation to the number of false involved vowels, we will distinguish simple bimembri iterations (or Oo), trimembers ( AI OI OI) and so on; in the second case we will deal with the syntagmatic iteration, (Oi IO SS), since it is based on vowel polyphonies, which can be biphonic, homophonic, and diaphonic.

LINGUISTIC MECHANISMS AND LEXICO-GRAMMATICAL MANIPULATIONS

Omero-Iliade I 1-32 (Hom. Il. I 1-32).

PHASE A: FIXED FORM ANALYSIS

In this second part of phase A, the team tries to identify fixed phrases by which the forms predisposed to the formulaic style are defined, such as the super lexical units (Dardano, 1978), complex lexeme [START_REF] Voghera | Promemoria per una teoria del linguaggio[END_REF]De Mauro, Voghera, 1996 ), syntagmatic word (Masini, 2009) and complex word [START_REF] Simone | I verbi sintagmatici come costruzione e come categoria[END_REF]. Among the most frequent Homeric forms and in the classical world we find: -the patronymics are the expressions delegated to indicate a link such as Peliades, it is the shortest form and is close to our NLG encoded languages; --The Homeric epithet not only has the function of characterizing the hero, but it does provide the already prepared portions of verse aedo, so that he can compose the poem to suit his song. Thus, "the divine and resistant Ulysses" (in ancient Greek): πολύτλας δῖος Ὀδυσσεύς) it is a hemistich ready to combine with an invented half verse or even another hemistich ready to compose the dactyl hemistich in ancient Greek: ὣς φάτο, γήθησεν δὲ / πολύτλας δῖος Ὀδυσσεύς («With these words, he was filled with joy / the divine resistant Ulysses », Od., XIII, 250). Milman Parry, in his thesis contained in the traditional epithet in Homer, essay on a problem of Homeric style (1930), He was the first to show the role of the Homeric epithet in the mechanics of the "formulaic style." This research, which points to the existence of a system of adjectives in Homer, is at the origin of the theory of oral style. 10 We still remember some features of formal Epithets like fixed form "worms" that are included in higher lexical units such as:

The expressions (idiomatic), or rather the higher lexical units that can be found very frequently in the Homeric text, are compounds that make use of a metaphor (idiomatic compounds). The objective of this essay is precisely the description of the syntagmatic condition of signification, that is, the analysis of those morphemic units that can be historically assigned to one or more instances of designation based on the conditions of signification and a phrase in the languages to use; those superior lexical units that designate the significance of values and anthropologies of traits. The notion of higher lexical units is confirmed by the stability of their frequency in languages. They are characterized by the fact that the succession of the different elements cannot be modified or interrupted. These types of expressions affect various areas of the composition:

a) Compounds with structure of the type: N+Agg;Agg+N; prep+ N; N+Prep+N ecc… b) Verbal locutions and support verbs: V+N; V+Art+N ecc. c) Compounds that use a metaphor (or idiomatic compounds): In the parameters of textual analysis, it should also be considered the essentially syntagmatic condition of signification. Thus, the ways in which we can historically assign one or more instances of designation to a specific morphemic unit from different syntagmatic conditions of signification must be taken into account. Hence, we come to consider a particular type of higher lexical units, that of idiomatic expressions. When the meaning of a word or an expression is transferred from its proper sense to another figurative one that has a similar relationship with the first one, the rhetorical figure of the metaphor is constituted. The metaphor is then the basis of the semantic change that occurs due to the similarity of meanings, but In this simile, the substitution mechanism is apparently quite simple: the poet substitutes the words "the shepherd kissed Telemachus excited and happy" by those introduced by the "how" (ὡϛ δὲ) and concluded with "thus ... then" (ὣϛ τότε). Emotion and happiness are made known to us, instead of with two generic words, through a phrase that tells us how a father receives his son after a long time and after suffering a lot of pain11 .

NOOJ LANGUAGE ENVIRONMENT: DATA ANALYSIS OF FIXED FORMS

We enter to validate or refute the choices made in a highly calculated linguistic environment and we analyze the fixed forms such as: the simple sentence and the metaphor It is an environment in which lexical-grammatical data are validated with transformational manipulation tests and fed back into PECO tables and graphs. In this high-computing environment, the team tries to reproduce fixed sentences, that is, phrases and idiomatic expressions from the classical world and sentences transformed into fixed sentences during a formal process with the methodologies and models proposed by M. Planat and Max Silberztein before reconverting and implementing them. In Digital Intelligence WT, the automatic analysis of fixed sentences is carried out with NooJ, a program for Natural Language Processing, developed by Max [START_REF] Koeva | Formaliser les langues avec l'ordinateur[END_REF][START_REF] Silberztein | Disambiguation tools for NooJ[END_REF]. Another important computer tool is NooJ and the Intex [START_REF] Silberztein | Dictionnaires électroniques et analyse automatique de textes: le système INTEX[END_REF][START_REF] Silberztein | Text indexation with INTEX[END_REF], in which this author worked in the decade 1992-2002 under the direction of Maurice Gross at LADL. NooJ allows the construction of electronic dictionaries and grammars and their application to large corpus (www.nooj4nlp.net). Nooj is a language engine that includes computational devices used both for the formal description of linguistic phenomena and for the analysis of written text phenomena and to analyze written texts: automata and finite state transducers, recursive networks, regular expressions and free grammars of context. In this sense, we point out that NooJ model, according to M. Silberztein, allows linguists to formalize various types of linguistic description:

Orthography and spelling, lexicons for simple words, multiword units and frozen expressions, inflectional and derivational morphology, local, structural, and transformational syntax. One important characteristic of NooJ is that the linguistic descriptions are reversible, i.e. they can be used both by a parser (to recognize sentences) as well as a generator (to produce sentences). (Silberztein, 2011[START_REF] Silberztein | Formalizing natural languages: The NooJ approach[END_REF] shows how, by combining a parser and a generator and applying them to a syntactic grammar, we can build a system that takes one sentence as its input, and produce the sentences that share the same lexical material with the original sentence 12 . Homeric tract similarity is analyzed 12 I am using the term transformation as in (Harris 1968): an operator that links sentences that share common semantic material, as opposed to (Chomsky, 1957) whose transformations link deep and surface structures.

A lexical-grammatical data table is constructed and formalized in a dictionary or electro-dictionary or lexicon, in turn associated with an electronic grammar. The purpose of the electronic dictionary is to identify the lexical item (or sequence of lexical items) for formal understanding. We tried a conditional electronic grammar model (PECO.dic) associated with a complex grammar (PECO.nog) that in turn contains three Homeric similarity graphs: the first graph allows the annotation of sentences whose defining structure N 0 G n C 13

DIGITAL INTELLIGENCE IMPLEMENTATIONS

In the implementation phase, we construct the taxonomies of the Homeric text into acronyms of the fixed forms. To this end, we draw on the database (DB) of Homeric texts in Greek. The DB is backed up by a spreadsheet that reproduces sounds and signs in text or in parts of sentences. The writing of the text is a mixed system, in other words, the operator can transfer the acronyms and insert them into the text and at the same time transcribe the sentence on the axis of the language, thus obtaining an intact and reformulated writing from time to time. In this way, Digital Intelligence W.T (BuViTeGMS© ( 2019) is a redactor text a reformulator, a translator, a reconvertor, composed of a multilingual database and a writing sheet-. 1) It is a scientific model (Nooj mind), which analyzes and identifies with phonological calculations the rhetoric of the speaker, which is combined with NooJ descriptive linguistic techniques (lexicon-grammar). 2) It scientifically describes the transformational and substitutive manipulations of a sentence. 3) It produces probabilistic calculation of a sentence on the NooJ model in linguistic environments. The digital worksheet produces in real time a text composed of fixed sentences, with the DB, free sentences and, on command, translates and reconverts the text, including iconic languages such as: Arabic, Chinese and sign languages; it also produces verisimilar diachronic texts. If we want to describe the WT software, we can say that it is composed of: -a database of fixed sentences transformed into acronyms, synthesized with deletion and then reduction techniques, in which it focuses the nuclear of the sentence. It is built and prepared with precise syntagmatic manipulative techniques of nuclear and extra nuclear positional elements. Finally, it determines, from the deictic positional ones, the degree and emotional level scientifically validated by studies of linguistic engineers, which explain the facts in quantum physics (Planat, 2016). Of course, the DB is reconvertible and transforms even iconic languages into acronyms. To support the database, WT uses Adalta Wolfram Cloud techniques, combines a state-of-the-art notebook interface with the scalable programming language to huge programs, with immediate access to a wide range of algorithms and embedded knowledge.

CONCLUSION

Mood digital thinking implies a professional management of the communicative message, the possession of multi-skills in disciplinary fields, new cultural horizons, new formal codes, and thus new grammars that explain to us the functions and processes of digital communication. The idea arises from the intuition that perhaps LSP languages and the style of classical poetics have common processes. With the advent of descriptive grammars arises the concept of semantic operations by categories and linguistics mediates in production techniques for the transmission of knowledge from quantum physics and beyond. We witness these formal processes before our eyes every day, but often even specialists, technicians, and teachers do not know how to explain the evolution of these processes. Throughout this paper, among the goals set, the research team aimed to undertake research on the different textual typologies, but with absolute priority the literary communication in the poetic tradition of the ancient and modern world to find the points of contact that over the centuries have transmitted semantics intact. The team, then, focuses on samples from the Iliad and Odyssey and traits for textual prediction of fixed forms, with particular attention to prosodic analysis and phonetic phenomena, in keeping with M. Plant regarding Quantum Physics .For the formal process, we complied with the methodologies Lexicon-grammar, the Nooj M. Silberztein linguistic environments for graph construction and paraphrase generation. Like this manner, we dealt with implementations in blended domains by using BuViTeGMS techniques with the D.I. Word Tool software for editing and visualization of chosen traits, with access to a wide range of algorithms and embedded knowledge.

BIBLIOGRAPHY



  Associative property: = (a,b)= a.b for each a,b ∈ G N 0 V N 1The operation (1.1) is said to satisfy the associative property, if for each a, b, c G is true the identity (ab)c = a(bc) = e.g. Planat p. 5 where the structure of a group is a sequence that we will call semantic unit rel(H,E,C,G,I, .; Aschheim, R.; Amaral, M. M; Fang, F.; Irwin, K. Quantum information in the protein codes, 3manifolds and the Kummer surface,Symmetry 2020Symmetry , 13, 1146. .

  abc=(ab) c, abcd= (abc)d , ecc. (Us+ (Us+ Us) In particular, for n = 1,2,3, ... c a ∈ G we set A n = a....a, n factors An element u ∈ G is said to be a neutral element for operation (1.1) if for each a ∈ G we apply identity: au=ua= a  Commutative property, if for each a, b ∈ G holds the identity ab = ba

  many and strong lives to Hades he threw ἡ ˉρώ ˉωˉν , αὐ ˉτοὺ ˉς δὲ ˘ ἑ ˘λώ ˉρι ˘α˘ τεῦ ˉχε ˘ κύ ˘νε ˉσσι ˉν of their heroes and surrendered prey by afflicts Homer-Odyssey-Beginning of the poem (Hom. Od. I 1-21) Ἄ ˉνδρα ˘ μοι ˘ ἔ ˉννε ˘πε ˘, Μοῦ ˉσα ˘, πο ˘λύ ˉτρο ˘πο ˘ν, ὃ ˉς μά ˘λα ˘ πο ˉλλὰ Man inspires me, O Muse, with many ways Ppt μοι ˘ πλά ˉγχθη ˘, ἐ ˘πεὶ ˉ Τροί ˉηˉς ἱ ˘ε˘ρ ὸ ˉν πτο ˘λί ˘εˉθ ρο ˘ν ἔ ˘πε ˉρσε ˘· He erred when the sacred citadel of Troy destroyed Dislc.

Fig 1 .

 1 Fig 1. An elementary calculation

  let Homer speak about this rhetorical figure and what he has transmitted to us in: Odissea, XVI, 14-21 […]κιρνὰς αἴθοπα οἶνον. ὁ δ' ἀντίος ἦλθεν ἄνακτος, κύσσε δέ μιν κεφαλήν τε καὶ ἄμφω φάεα καλὰ χεῖράς τ' ἀμφοτέρας· θαλερὸν δέ οἱ ἔκπεσε δάκρυ. ὡς δὲ πατὴρ ὃν παῖδα φίλα φρονέων ἀγαπάζῃ ἐλθόντ' ἐξ ἀπίης γαίης δεκάτῳ ἐνιαυτῷ, μοῦνον τηλύγετον, τῷ ἐπ' ἄλγεα πολλὰ μογήσῃ, ὣς τότε Τηλέμαχον θεοειδέα δῖος ὑφορβὸς πάντα κύσεν περιφύς, ὡς ἐκ θανάτοιο φυγόντα·

  Basic sentence: Μῆνιν ἄειδε θεὰ Πηληϊάδεω Ἀχιλῆος = Simple sentence: Μῆνιν ἄειδ ….. Ppt V N 1 Fixed form: Μῆνιν ἄειδε C…. Πηληϊάδεω Ἀχιλῆος

Fig

  Fig 2. Reversible descriptions

  Fig 3. Sentence annotation The graph, fig 3, allows us to recognize those comparative structures in which verbs such as look, stay, remain can occur, etc. N 0 PN as C

˘λῆ ˉοˉς Μῆ ˉνι ˘ν ἄ ˘ει ˉδε ˘ θε ˘ὰˉ Πη ˉλη ˉϊ˘ά˘δ εω ˉ Ἀ ˘χι ˘λῆ ˉοˉς

  Μῆ ˉνι ˘ν ἄ ˘ει ˉδε ˘ θε ˘ὰˉ Πη ˉλη ˉϊ˘ά˘δ εω ˉ Ἀ ˘χι

	Ira, sing Oh goddess of Pelis Achilles	Nominal dislocation

οὐ ˉλο ˘μέ ˘νη ˉν, ἣ ˉ μυ ˉρί ˘' Ἀ ˘χαι ˉοῖ ˉς ἄ ˉλγε ˘' ἔ ˘θη ˉκε ˘, fatal that the infinite pains caused the Achaeans verbal dislocation πο ˉλλὰ ˉς δ' ἰ ˉφθί ˉμου ˉς ψυ ˉχὰ ˉς Ἄ ˘ϊ˘δ ι ˉ προ ˘ΐ˘αˉψ ε ˉν

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 De Mauro, Tullio, and Miriam Voghera. Scala mobile. Un punto di vista sui lessemi complessi. NA, 1996.

G. De Bueriis. A. M.Langella I numeri nelle parole (in corso di stampa, volume in onore di A.Elia 2021)

https://it.wikipedia.org/wiki/Epiteto_omerico

https://www.micheletortorici.it/omero-la-straordinaria-invenzione-della-metafora-specchio/

http://www.poesialatina.it/_ns/Greek/tt2/Omero/Iliade06-119-149.html

Authors : Ritamaria Bucciarelli, Francesco Saverio Tortoriello, Andrea Rodrigo, Javier Julian Enriquez, Colomba La Ragione, Ilaria Veronesi


Mohamed Hamdouche

Pierre Henry-Labordere

Huyên Pham

Generative modeling for time series via Schrödinger bridge

Keywords: generative models, time series, Schrödinger bridge, kernel estimation, deep hedging

We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting the temporal dynamics of the time series distribution. We can estimate the drift function from data samples either by kernel regression methods or with LSTM neural networks, and the simulation of the SB diffusion yields new synthetic data samples of the time series.

The performance of our generative model is evaluated through a series of numerical experiments. First, we test with a toy autoregressive model, a GARCH Model, and the example of fractional Brownian motion, and measure the accuracy of our algorithm with marginal and temporal dependencies metrics. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. Finally, we illustrate the SB approach for generating sequence of images.

Introduction

Sequential data appear widely in our society like in video and audio data, and simulation of time series models are used in various industrial applications including clinical predictions [START_REF] Lyu | Improving clinical predictions through unsupervised time series representation learning[END_REF], and weather forecasts [START_REF] Chen | Model-free renewable scenario generation using generative adversarial networks[END_REF]. In the financial industry, simulations of dynamical scenarios are considered in market stress tests, risk measurement, and risk management, e.g. in deep hedging [START_REF] Buehler | Deep hedging[END_REF]. The design of time series model is a delicate issue, requiring expensive calibration task, and subject to error misspecification and model risk. Therefore, the generation for synthetic samples of time series has gained an increasing attention over the last years, and opens the door in the financial sector for a pure data driven approach in risk management.

Generative modeling (GM) has become over the last years a successful machine learning task for data synthesis notably in (static) image processing. Several competing methods have been developed and state-of-the-art includes Likelihood-based models like energy-based models (EBM) [START_REF] Lecun | A tutorial on energybased modeling[END_REF], variational auto-encoders (VAE) [START_REF] Kingma | Auto-encoding variational Bayes[END_REF], Implicit generative models with the prominent works on generative adversarial network (GAN) [START_REF] Goodfellow | Generative adversarial nets[END_REF] and its extensions [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF], and recently the new generation of score-based models using Langevin dynamics, [START_REF] Song | Generative modeling by estimating gradients of the distribution[END_REF], [START_REF] Song | Scorebased generative modeling through stochastic differential equations[END_REF], [START_REF] Guth | Wawelet score-based generative modeling[END_REF], and diffusion models via Schrödinger bridge, see [START_REF] Henry-Labordere | From (martingale) Schrodinger bridges to a new class of stochastic volatility model[END_REF] for the application to a class of stochastic volatility models, and [START_REF] Wang | Deep generative learning via schrödinger bridge[END_REF], [START_REF] Bortoli | Diffusion Schrödinger bridge with applications to score-based generative modeling[END_REF]. Generative methods for time series raises challenging issues for learning efficiently the temporal dependencies. Indeed, in order to capture the potentially complex dynamics of variables across time, it is not sufficient to learn the time marginals or even the joint distribution without exploiting the sequential structure. An increasing attention has been paid to these methods in the literature and state-of-the-art generative methods for time series are: Time series GAN [START_REF] Yoon | Time-series generative adversarial networks[END_REF] which combines an unsupervised adversarial loss on real/synthetic data and supervised loss for generating sequential data, Quant GAN [START_REF] Wiese | Quant gans: deep generation of financial time series[END_REF] with an adversarial generator using temporal convolutional networks, Causal optimal transport COT-GAN [START_REF] Xu | COT-GAN: Generative Sequential Data via Causal Optimal Transport[END_REF] with adversarial generator using the adapted Wasserstein distance for processes, Conditional loss Euler generator [START_REF] Remlinger | Conditional versus adversarial Euler-based generators for time series[END_REF] starting from a diffusion representation time series and minimizing the conditional distance between transition probabilities of real/synthetic samples, Signature embedding of time series [START_REF] Fermanian | Embedding and learning with signatures[END_REF], [START_REF] Ni | Conditional Sig-Wasserstein GANs for time series generation[END_REF], [START_REF] Buehler | A data-driven market simulator for small data environments[END_REF], and Functional data analysis with neural SDEs [START_REF] Choudhary | FuNVol: A Multi-Asset Implied Volatility Market Simulator using Functional Principal Components and Neural SDEs[END_REF].

In this paper, we develop a novel generative model based on Schrödinger bridge approach that captures the temporal dynamics of the time series. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a pathdependent drift function, called Schrödinger bridge time series (SBTS) diffusion, and the simulation of the SBTS diffusion yields new synthetic data samples of the time series.

Our SB approach differs from related works that have been recently designed for learning marginal (or static) distributions. In [START_REF] Wang | Deep generative learning via schrödinger bridge[END_REF], the authors perform generative models by solving two SB problems. The paper [START_REF] Bortoli | Diffusion Schrödinger bridge with applications to score-based generative modeling[END_REF] formulates generative modeling by computing the SB problem between the data and prior distribution. The very recent work [START_REF] Chen | Deep momentum multi-marginal Schrödinger bridge[END_REF] proposes momentum SB by considering an additional velocity variable for learning multi marginal distributions. Let us mention also the recent paper [START_REF] Somnath | Aligned diffusion Schrödinger bridges[END_REF] that combines SB with h-transform in order to respect aligned data. Instead, our SBTS diffusion interpolates the joint time series distribution starting from an initial deterministic value. Moreover, we propose an alternate method for the estimation of the drift function, which is path-dependent in our case. While [START_REF] Wang | Deep generative learning via schrödinger bridge[END_REF] uses a logistic regression for estimating the density ratio and then the drift function, which requires additional samples from Gaussian noises, and [START_REF] Bortoli | Diffusion Schrödinger bridge with applications to score-based generative modeling[END_REF] performs an extension of the Sinkhorn algorithm, we propose a kernel regression method relying solely on data samples, and this turns out to be quite simple, efficient and low-cost computationally. Compared to GAN type methods, the simulation of synthetic samples from SBTS is much faster as it does not require the training of neural networks.

We validate our methodology with several numerical experiments. We first test on some time series models like autoregressive, GARCH models, and also for the fractional Brownian motion with rough paths. The accuracy is measured by some metrics aiming to capture the temporal dynamics and the correlation structure. We also provide operational metrics of interest for the financial industry by implementing our results on real datasets, and applying to the deep hedging of call options. Finally, we show some numerical illustrations of our SB method in high dimension for the generation of sequential images.

Problem formulation

Let µ be the distribution of a time series representing the evolution of some R d -valued process of interest (e.g. asset price, claim process, audio/video data, etc), and suppose that one can observe samples of this process at given fixed times of a discrete time grid T = {t i , i = 1, . . . , N } on (0, ∞). We set T = t N as the terminal observation horizon. Our goal is to construct a model that generates time series samples according to the unknown target distribution µ ∈ P((R d ) N ) the set of probability measures on (R d ) N . For that objective, we propose a dynamic modification of the Schrödinger bridge as follows. Let Ω = C([0, T ]; R d ) be the space of R d -valued continuous functions on [0, T ], X = (X t ) t the canonical process with initial value X 0 = 0, and F = (F t ) t the canonical filtration. Denoting by P(Ω) as the space of probability measures on Ω, we search for P (representing the theoretical generative model) in P(Ω), close to the Wiener measure W in the sense of Kullback-Leibler (or relative entropy), and consistent with the observation samples. In other words, we look for a probability measure P * ∈ P(Ω) solution to:

P * ∈ arg min P∈P µ T (Ω) H(P|W), (2.1)
where P µ T (Ω) is the set of probability measures P on Ω with joint distribution µ at (t 1 , . . . , t N ), i.e., P • (X t 1 , . . . , X t N ) -1 = µ, and H(.|.) is the relative entropy between two probability measures defined by

H(P|W) = ln dP dW dP, if P W ∞,
otherwise .

Denoting by E P and E W the expectation under P and W, we see that H(P|W) = E P [ln dP dW ] = E W [ dP dW ln dP dW ] when P W. Compared to the classical Schrödinger bridge (SB) (see [START_REF] Leonard | A survey of the schrodinger problem and some of its connections with optimal transport[END_REF], and the application to generative modeling in [START_REF] Wang | Deep generative learning via schrödinger bridge[END_REF]), which looks for a probability measure P that interpolates between an initial probability measure and a target probability distribution at terminal time T , here, we take into account via the constraint in P µ T the temporal dependence of the process observed at sequential times t 1 < . . . < t N , and look for an entropic interpolation of the time series distribution. We call (2.1) the Schrödinger bridge for time series (SBTS) problem.

Let us now formulate (SBTS) as a stochastic control problem following the well-known connection established for classical (SB) in [START_REF] Pra | A stochastic control approach to reciprocal diffusion process[END_REF]. Given P ∈ P(Ω) with finite relative entropy H(P|W) < ∞, it is known by Girsanov's theorem that one can associate to P an F-adapted R d -valued process α = (α t ) with finite energy:

E P [ T 0 |α t | 2 dt] < ∞ such that ln dP dW = T 0 α t .dX t - 1 2 T 0 |α t | 2 dt, (2.2)
and X t -t 0 α s ds, 0 ≤ t ≤ T , is a Brownian motion under P. We then have

H(P|W) = E P 1 2 T 0 |α t | 2 dt .
Therefore, (SBTS) is reformulated equivalently in the language of stochastic control as:

   Minimize over α ∈ A, J(α) = E P 1 2 T 0 |α t | 2 dt subject to dX t = α t dt + dW t , X 0 = 0, (X t 1 , . . . , X t N ) P ∼ µ,
where W is a Brownian motion under P, A is the set of R d -valued F-adapted processes s.t.

E P [ T 0 |α t | 2 dt
] < ∞, and (X t 1 , . . . , X t N ) P ∼ µ is the usual notation for P • (X t 1 , . . . , X t N ) -1 = µ. In the sequel, when there is no ambiguity, we omit the reference on P in E = E P and ∼ = P ∼. We denote by V SBT S the infimum of this stochastic control problem under joint distribution constraint:

V SBT S := inf α∈A µ T J(α),
where

A µ T is the set of controls α in A satisfying (X t 1 , . . . , X t N ) ∼ µ with X t = t 0 α s ds+W t .
Our goal is to prove the existence of an optimal control α * that can be explicitly derived, and then used to generate samples of the time series distribution µ via the probability measure P * on Ω, i.e., the simulation of the optimal diffusion process X controlled by the drift α * .

Solution to Schrödinger bridge for time series

Similarly as for the classical Schrödinger bridge problem, we assume that the target distribution µ admits a density with respect to the Lebesgue measure on (R d ) N , and by misuse of notation, we denote by µ(x 1 , . . . , x N ) this density function. Denote by µ W T the distribution of the Brownian motion on T , i.e. of (W t 1 , . . . , W t N ), which admits a density given by (by abuse of language, we use the same notation for the measure and its density)

µ W T (x 1 , . . . , x N ) = N -1 i=0 1 2π(t i+1 -t i ) exp - |x i+1 -x i | 2 2(t i+1 -t i ) , (3.1)
for (x 1 , . . . , x N ) ∈ (R d ) N (with the convention that t 0 = 0, x 0 = 0). The measure µ is absolutely continuous with respect to µ W T , and we shall assume that its relative entropy is finite, i.e.,

H(µ|µ

W T ) = ln µ µ W T dµ < ∞.
The solution to the (SBTS) problem is provided in the following theorem.

Theorem 3.1. The diffusion process X t = t 0 α * s ds + W t , 0 ≤ t ≤ T , with α * defined as

α * t = a * (t, X t ; (X t i ) t i ≤t ), 0 ≤ t < T, with a * (t, x; x i ), for t ∈ [t i , t i+1 ), x i = (x 1 , . . . , x i ) ∈ (R d ) i , x ∈ R d , given by a * (t, x; x i ) = ∇ x ln E W µ µ W T (X t 1 , . . . , X t N ) X t i = x i , X t = x ,
where we set X t i = (X t 1 , . . . , X t i ), induces a probability measure

P * = µ µ W T (X t 1 , .
. . , X t N )W, which solves the Schrödinger bridge time series problem. Moreover, we have

V SBT S = H(P * |W) = H(µ|µ W T ). Proof. First, observe that E W [ µ µ W T (X t 1 , . . . , X t N )] = 1
, and thus one can define a probability measure P * W with density process

Z t = E W dP * dW F t = E W µ µ W T (X t 1 , . . . , X t N ) F t , 0 ≤ t ≤ T.
Notice from the Markov and Gaussian properties of the Brownian motion that for t ∈

[t i , t i+1 ), i = 0, . . . , N -1, we have Z t = h i (t, X t ; X t i )
, where for a path

x i = (x 1 , . . . , x i ) ∈ (R d ) i , h i (; x i ) is defined on [t i , t i+1 ) × R d by h i (t, x; x i ) = E Y ∼N (0,I d ) µ µ W T (x i , x + t i+1 -tY, . . . , x + √ t N -tY ) for t ∈ [t i , t i+1 ), x ∈ R d , and E Y ∼N (0,I d ) [.]
is the expectation when Y is distributed according to the Gaussian law N (0, I d ). Moreover, by the law of conditional expectations, we have

h i (t, x; x i ) = E W h i+1 (t i+1 , X t i+1 ; x i , X t i+1 ) X t = x , with the convention that h N (t N , x; x N -1 , x) = µ µ W T (x 1 , . . . , x N -1 , x). Therefore, for i = 0, . . . , N -1, and x ∈ (R d ) i , the function (t, x) → h i (t, x; x i ) is a strictly positive C 1,2 ([t i , t i+1 ) × R d ) ∩ C 0 ([t i , t i+1 ] × R d ) classical solution to the heat equation ∂h i (.; x i ) ∂t + 1 2 ∆ x h i (.; x i ) = 0, on [t i , t i+1 ) × R d ,
with the terminal condition: h i (t i+1 , x; x i ) = h i+1 (t i+1 , x; x i , x) (here ∆ x is the Laplacian operator). By applying Itô's formula to the martingale density process Z of P * under the Wiener measure W, we derive

dZ t = ∇ x h i (t, X t ; X t i )dX t , = Z t ∇ x ln h i (t, X t ; X t i )dX t , t i ≤ t < t i+1 ,
for i = 0, . . . , N -1. Thus, by defining the process α * by α *

t = ∇ x ln h i (t, X t ; X t i ), for t ∈ [t i , t i+1 ), i = 0, . . . , N -1, we have dP * dW = exp T 0 α * t dX t - 1 2 T 0 |α * t | 2 dt ,
and by Girsanov's theorem, X t -t 0 α * s ds is a Brownian motion under P * . On the other hand, by definition of P * , and Bayes formula, we have for any bounded measurable function ϕ on (R d ) N :

E P * ϕ(X t 1 , . . . , X t N ) = E W µ µ W T (X t 1 , . . . , X t N )ϕ(X t 1 , . . . , X t N ) = µ µ W T (x 1 , . . . , x N )ϕ(x 1 , . . . , X N )µ W T (x 1 , . . . , x N )dx 1 . . . dx N = ϕ(x 1 , . . . , X N )µ(x 1 , . . . , x N )dx 1 . . . dx N ,
which shows that (X t 1 , . . . , X t N ) P * ∼ µ. Moreover, by noting that

J(α * ) = E P * T 0 1 2 |α * t | 2 dt = E P * ln dP * dW = H(P * |W) = H(µ|µ W T ) < ∞,
where we used in the last inequality the fact that (X t 1 , . . . , X t N )

P *
∼ µ, this shows in particular that α * ∈ A µ T . It remains to show that for any α ∈ A µ T associated to a probability measure P W with density given by (2.2), i.e. J(α) = H(P|W), we have

J(α) ≥ H(µ|µ W T ). (3.2)
For this, we write from Bayes formula and since W t = X t -t 0 α s ds is a Brownian motion under P by Girsanov theorem:

1 = E W µ µ W T (X t 1 , . . . , X t N ) = E P exp ln µ µ W T (X t 1 , . . . , X t N ) - T 0 α t dW t - 1 2 T 0 |α t | 2 dt ≥ exp E P ln µ µ W T (X t 1 , . . . , X t N ) - T 0 α t dW t - 1 2 T 0 |α t | 2 dt = exp H(µ|µ W T ) -J(α) ,
where we use Jensen's inequality, and the fact that (X t 1 , . . . , X t N ) P ∼ µ in the last equality. This proves the required inequality (3.2), and ends the proof.

Remark 3.2. The optimal drift of the Schrödinger bridge time series diffusion is in general path-dependent: it depends at given time t not only on its current state X t , but also on the past values X η(t) = (X t 1 , . . . , X η(t) ), where η(t) = max{t i : t i ≤ t}, and we have:

dX t = a * (t, X t ; X η(t) )dt + dW t , 0 ≤ t ≤ T, X 0 = 0.
Moreover, the proof of the above theorem shows that this drift function is explicitly given by a * (t, x;

x i ) = ∇ x h i (t, x; x i ) h i (t, x; x i ) , t ∈ [t i , t i+1 ), x i ∈ (R d ) i , x ∈ R d , (3.3)
for i = 0, . . . , N -1, where

h i (t, x; x i ) = E Y ∼N (0,I d ) ρ(x i , x + t i+1 -tY, . . . , x + √ t N -tY ) , with ρ := µ µ W
T the density ratio.

The following result states an alternate representation of the drift function that will be useful in the next section for estimation.

Proposition 3.3. For i = 0, . . . , N -1, t ∈ [t i , t i+1 ), x i = (x 1 , . . . , x i ) ∈ (R d ) i , x ∈ R d , we have a * (t, x; x i ) = 1 t i+1 -t E µ (X t i+1 -x)F i (t, x i , x, X t i+1 ) X t i = x i E µ F i (t, x i , x, X t i+1 ) X t i = x i , (3.4)
where

F i (t, x i , x, x i+1 ) = exp - |x i+1 -x| 2 2(t i+1 -t) + |x i+1 -x i | 2 2(t i+1 -t i ) ,
and E µ [•] denotes the expectation under µ.

Proof. Fix i ∈ 0, N -1 , and t ∈ [t i , t i+1 ). From the expression of µ W T in (3.1), we have

E W µ µ W T (X t 1 , • • • , X t N ) (X t 1 , • • • , X t i ) = (x 1 , • • • , x i ), X t = x (3.5) = C µ µ W T (x 1 , • • • , x N ) exp - |x i+1 -x| 2 2(t i+1 -t) N -1 j=i+1 exp - |x j+1 -x j | 2 2∆t j dx i+1 • • • dx N = C F i (t, x i , x, x i+1 ) µ(x 1 , . . . , x N ) µ i (x 1 , . . . , x i ) dx i+1 • • • dx N = CE µ F i (t, x i , x, X t i+1 ) X t i = x i ,
where C is a constant varying from line to line and depending only on t and x i = (x 1 , • • • , x i ), but not on x, and µ i is the density of (X t 1 , • • • , X t i ) under µ, i.e.,

µ i (x 1 , • • • , x i ) = µ(x 1 , • • • , x N )dx i+1 • • • dx N .
By plugging the new expression (3.5) into a * and differentiating with respect to x, we then get a * (t, x;

x i ) = 1 t i+1 -t E µ (X t i+1 -x)F i (t, x i , x, X t i+1 ) X t i = x i E µ F i (t, x i , x, X t i+1 ) X t i = x i .
Remark 3.4. In the case where µ is the distribution arising from a Markov chain, i.e., in the form µ(dx 1 , . . . , x N ) = N -1 i=1 ν i (x i , dx i+1 ), for some transition kernels ν i on R d , then the conditional expectations in (3.4) will depend on the past values X t i = (X t 1 , . . . , X t i ) only via the last value X t i .

Generative learning

From Theorem 3.1, we can run an Euler scheme for simulating the Schrödinger bridge diffusion, and then samples of the target distribution µ. For that purpose, we need an accurate estimation of the drift terms, i.e., of the functions a * i , for i = 0, . . . , N -1. We propose several estimation methods. In the sequel, for a probability measure ν on (R d ) N , we denote by E ν [•] the expectation under the distribution ν.

Drift estimation

Estimation of the density ratio. This method follows the idea in [START_REF] Wang | Deep generative learning via schrödinger bridge[END_REF]. Denote by ρ = µ µ W T the density ratio, and observe that the log-density ratio ln ρ minimizes over functions r on (R d ) N the logistic regression function

L logistic (r) = E µ ln 1 + exp(-r(X) + E µ W T ln 1 + exp(r(X) .
Therefore, given data samples X (m) = (X (m) t 1 , . . . , X (m t N ), m = 1, . . . , M from µ, and using samples Y (m) from µ W T , we estimate the density ratio ρ by ρ = exp(r θ)

where r θ is the neural network that minimizes the empirical logistic loss function:

θ → 1 M M m=1 ln 1 + exp(-r(X (m) ) + ln 1 + exp(r(Y (m) ) .
By writing from (3.3) the Schrödinger drift a * as a * (t, x;

x i ) = E Y ∼N (0,I d ) ρ∇ x ρ(x i , x + √ t i+1 -tY, . . . , x + √ t N -tY ) E Y ∼N (0,I d ) ρ(x i , x + √ t i+1 -tY, . . . , x + √ t N -tY ) , (4.1)
for t ∈ [t i , t i+1 ), we obtain an estimator of a * by plugging into (4.1) the estimate ρ, and r θ of ρ, and ln ρ, and then computing the expectation with Monte-Carlo approximations from samples in N (0, I d ). Notice that this method is very costly as it requires in addition to the training of the neural networks for estimating the density ratio, another Monte-Carlo sampling for estimating finally the drift.

Kernel estimation of the drift. In order to overcome the computational issue of the above estimation method, we propose an alternative approach that relies on the representation of the drift term in Proposition 3.3. Indeed, the key feature of the formula (3.4) is that it involves (conditional) expectations under the target distribution µ, which is amenable to direct estimation using data samples. For the approximation of the conditional expectation, we can then use the classical kernel methods.

From data samples X (m) = (X (m)

t 1 , . . . , X (m)
t N ), m = 1, . . . , M from µ, the Nadaraya-Watson estimator of the drift function is given by â(t, x;

x i ) = 1 t i+1 -t M m=1 (X (m) t i+1 -x)F i (t, X (m) t i , x, X (m) t i+1 ) i j=1 K h (x j -X (m) t j ) M m=1 F i (t, X (m) t i , x, X (m) t i+1 ) i j=1 K h (x j -X (m) t j ) , (4.2) for t ∈ [t i , t i+1 ), x i ∈ (R d ) i , x ∈ R d , i = 0, . . . , N -1,
where K h is a kernel, i.e., a non-negative real-valued integrable and symmetric function on R d , with bandwith h > 0. Common kernel function is the Gaussian density function, but we shall use here for lower time complexity reason, the quartic kernel

K h (x) = 1 h K( x h ) with K(x) = (1 -|x| 2 )1 |x|≤1 .
LSTM network approximation of the path-dependent drift. The conditional expectations in the numerator and denominator of the drift terms can alternately be approximated by neural networks. In order to achieve this, we need a neural network architecture that fits well with the path-dependency of the drift term, i.e., the a priori non Markov feature of the data time series distribution µ. We shall then consider a combination of feed-forward and LSTM (Long Short Term Memory) neural network.

Figure 1: Architecture of the neural network For i = 0, . . . , N -1, the conditional expectation function in the numerator:

(t, x i , x) ∈ [t i , t i+1 ) × (R d ) i × R d -→ E µ (X t i+1 -X t )F (t, X t i , X t , X t i+1 ) (X t i , X t ) = (x i , x) ,
is approximated by Ψ θ f (t, p t i , x), where Ψ θ f ∈ N N d+k+1,d is a feed-forward neural network with input dimension 1 + k + d, and output dimension d, and p t i is an output vector of dimension k from an LSTM network, i.e., p t i = ψ θr (x i ) with ψ θr ∈ LSTM i,d,k at time t i . This neural network is trained by minimizing over the parameter θ = (θ f , θ r ) the quadratic loss function

L(θ) = N -1 i=0 Ê (X t i+1 -X)F (τ, X t i , X, X t i+1 ) -Ψ θ f (τ, ψ θr (X t i ), X) 2 .
Here Ê is the empirical loss expectation where (X t 1 , . . . , X t N ) are sampled from the data distribution µ, τ is sampled according to an uniform law on [t i , t i+1 ), and X is sampled e.g. from a Gaussian law with mean X t i for i = 0, . . . , N -1. The conditional expectation function in the denominator is similarly approximated. The output of this neural network training yields an approximation:

t ∈ [0, T ), x ∈ (R d ) η(t) , x ∈ R d -→ â(t, x; x),
of the drift term function, which is then used for generating samples (X t 1 , . . . , X t N ) of µ from the simulation of the diffusion:

dX t = â(t, X t ; X η(t) )dt + dW t , X 0 = 0. (4.3)

Schrödinger bridge time series algorithm

From the estimator â of the drift path-dependent function, we can now simulate the SB SDE (4.3) by an Euler scheme. Let N π be the number of uniform time steps between two consecutive observation dates t i and t i+1 , for i = 0, . . . , N -1, and t π k,i = t i + k N π , k = 0, . . . , N π -1, the associated time grid. The pseudo-code of the Schrödinger bridge time series (SBTS) algorithm is described in Algorithm 1.

Algorithm 1: SBTS Simulation

Input: data samples of time series (X (m)

t 1 , • • • , X (m)
t N ), m = 1, . . . , M , and N π . Initialization: initial state x 0 = 0; for i = 0, . . . , N -1 do Initialize state y 0 = x i ; for k = 0, . . . , N π -1 do Compute â(t π k,i , y k ; x i ) e.g. by kernel estimator (4.2); Sample ε k ∈ N (0, 1) and compute

y k+1 = y k + 1 N π â(t π k,i , y k ; x i ) + 1 √ N π ε k , end Set x i+1 = y N π . end Return: x 1 , • • • , x N

Numerical experiments

In this section, we demonstrate the effectiveness of our SBTS algorithm on several examples of time series models, as well as on real data sets for an application to deep hedging. The algorithms are performed on a computer with the following characteristics: Intel(R) i7-7500U CPU @ 2.7GHz, 2 Core(s).

Evaluation metrics

In addition to visual plot of data vs generator samples path, we use some metrics to evaluate the accuracy of our generators:

• Marginal metrics for quantifying how well are the marginal distributions from the generated samples compared to the data ones. These include -Classical statistics like mean, 95% and 5% percentiles -Kolmogorov-Smirnov test: we compute the p-valued, and when p > α (usually 5%), we do not reject the null-hypothesis (generator came from data of reference distribution)

• Temporal dynamics metrics for quantifying the ability of the generator to capture the time structure of the time series data. We compute the empirical distribution of the quadratic variation

: i |X t i+1 -X t i | 2 .
• Correlation structure for evaluating the ability of the generator to capture the multidimensional structure of the time series. We shall compare the empirical covariance or correlation matrix induced by the generator SBTS and the ones from the data samples.

Toy autoregressive model of time series

We consider the following toy autoregressive (AR) model:

     X t 1 = b + ε 1 , X t 2 = β 1 X t 1 + ε 2 , X t 3 = β 2 X t 2 + |X t 1 | + ε 3 ,
where the noises ε i ∼ N (0, σ 2 i ), i = 1, . . . , 3 are mutually independent. The model parameters are b = 0.7, σ 1 = 0.1, σ 2 = σ 3 = 0.05 and

β 1 = β 2 = -1.
We use samples of size M = 1000 for simulated data of the AR model. The drift of the SBTS diffusion is estimated with a kernel of bandwith h = 0.05, and simulated from euler scheme with N π = 100. The runtime for generating 500 paths of SBTS is 8 seconds.

In Figure 2, we plot the empirical distribution of each pair (X t i , X t j ) from the AR model, and from the generated SBTS. We also show the marginal empirical distributions. Table 1 presents the marginal metrics for the AR model and generator (p-value and percentiles at level 5% and 95%). In Table 2, we give the difference between the empirical correlation from generated samples and AR data samples.

GARCH Model

We consider a GARCH model:

X t i+1 = σ t i+1 ε t i+1 σ 2 t i+1 = α 0 + α 1 X 2 t i + α 2 X 2 t i-1 , i = 1, . . . , N,
with α 0 = 5, α 1 = 0.4, α 2 = 0.1, and the noises ε t i ∼ N (0, 0.1), i = 1, . . . , N , are i.i.d. The size of the time series is N = 60. The hyperparameters for the training and generation of SBTS are M = 1000, N π = 100, and a bandwith for the kernel estimation h = 0.2 (larger than for the AR model since by nature the GARCH process is more "volatile"). The runtime for generating 1000 paths is 120 seconds.

In Figure 3, we plot four sample paths of the GARCH process to be compared with four sample paths of the SBTS. Figure 4 represents samples plot of the joint distribution between (X t 1 and the terminal value X t N ) of the time series. Figure 5 provides some metrics to evaluate the performance of SBTS. On the left, we represent the p-value for each of the marginals of the generated SBTS. On the right, we compute for each marginal index i = 1, . . . , N = 60, the difference between ρ i and ρi where ρ i (resp. ρi ) is the sum over j of the empirical correlation between X i and X t j from GARCH (resp. generated SBTS), and plot its mean and standard deviation.

Fractional Brownian Motion

We consider a fractional Brownian motion (FBM) with Hurst index H that measures the roughness of this Gaussian process. We plot in Figure 6 Finally, we provide estimate of the Hurst index from our generated SBTS with the standard estimator (see e.g. [START_REF] Gairing | Hurst index estimation in stochastic differential equations driven by fractional brownian motion[END_REF]) given by: Ĥ

= 1 2 1 - log N -1 i=0 |X t i+1 -X t i | 2
log N .

For N = 60, we get: Ĥ = 0.2016, Std = 0.004.

Application to deep hedging on real-data sets

In this paragraph, we use generated time series for applications to risk management, and notably the pricing of derivatives and the computation of associated hedging strategies via deep hedging approach. We use samples of historical data for generating by SBTS new synthetic time series samples. We then compute deep hedging strategies that are trained from these synthetic samples, and we compare with the PnL and the replication error based on historical dataset. The general backtest procedure is illustrated in Figure 9. We consider stock price S from the company Apple with data (ticker is AAPL) from january 1, 2010 to january 30, 2020, and produce M = 2500 samples of N = 60 successive days, with a sliding window. The hyperparameters for the generation of SBTS synthetic samples are N π = 100, bandwith h = 0.05.

We plot in Figure 10 four sample paths of the SBTS diffusion to be compared with the real ones from Apple. We illustrate the excess of kurtosis of the real data by plotting in Figure 11 the tail distribution for the return

R t i = St i+1 St i -1: x in log-scale → P[|R| ≥ x],
and found that the excess of kurtosis of real data is 1.96, to be compared with the one from SBTS, and equal to 2.34. Figure 12 represents the empirical distribution of the Apple time series data vs SBTS, while Figure 13 shows their covariance matrices. ∆(t i , S t i )(S t i+1 -S t i ) -g(S T ).

We then compare with the deep hedging on historical data by looking at the PnL and replication errors. The historical data set of Apple is split in the chronological order, namely training data set from 01/01/2007 to 31/12/2017, validation data set from 01/01/2018 to 31/12/2028, and test set from 01/01/2019 to 30/01/2020. As pointed out in [START_REF] Wang | A note on spurious model selection[END_REF], it is important not to break the time structure as it may lead to an overestimation of the model performance.

In Figure 14, we plot the empirical distribution of the PnL with deep hedging obtained from real data vs SBTS, and backtested on the validation and test sets. It appears that the PnL from SBTS has a smaller variance (hence smaller replication error), and yields less extreme values, i.e. outside the zero value, than the PnL from real data. This is also quantified in Table 3 where we note that the premium obtained from SBST is higher than the one from real data, which means that one is more conservative with SBTS by charging a higher premium.

Further tests in high dimension

In this section, we illustrate how our SB approach can be used for generating samples in very high dimension.

We use a data set of images from MNIST with training size M = 10000. We first start with handwritten digital numbers, and plot in Figure 15 the static images from real MNIST data set and the ones generated by SBTS. The number of pixels is 28 × 28, and the runtime for generating 16 iamges is equal to 120 seconds. Next, in Figures 16 and17, we plot sequential images sampled from real data set, and compare with the sequence generated by SBTS algorithm. The number of pixels is 14×14, and the runtime for generating 100 paths of sequential images is equal to 9 minutes.

Figure 2 :

 2 Figure 2: Comparison between the true and generated distribution for each couple (Xt i , Xt j ) with i, j ∈ 1, 3 with i = j

Figure 3 :Figure 4 :

 34 Figure 3: Samples path of reference GARCH (left) and generator SBTS (right)

Figure 5 :

 5 Figure 5: Left: p-value for the marginals Xt i . Right: Difference between the term-by-term empirical correlation from generated samples and reference samples.

Figure 8 :

 8 Figure 8: Quadratic variation distribution for N = 30 (left), N = 60 (right) and T = tN = 1

Figure 9 :

 9 Figure 9: Procedure of backtest for deep hedging

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: Four paths generated by Schrodinger bridge(Right) vs real ones (Left)

Figure 13 :E

 13 Figure 13: Covariance matrix for real-data and generative SBTS

Figure 14 :

 14 Figure 14: Deep hedging PnL distribution with backtest from validation set (left) and test set (right).

Figure 15 :

 15 Figure 15: Left: MNIST samples. Right: Generated SBTS samples.

Figure 16 :

 16 Figure 16: Top: a time series sampled from real distribution. Bottom: generated time series via SBTS

Figure 17 :

 17 Figure 17: Top: a time series sampled from real distribution. Bottom: generated time series via SBTS

Table 1 :

 1 Marginal metrics for AR model and generator (q for percentile)

	value	q 5	q5	q 95	q95
	X t 1 0.98	0.535 0.528 0.855 0.861
	X t 2 0.74	-0.873 -0.861 -0.516 -0.514
	X t 3 0.90	1.243 1.251 1.808 1.793
		X t 1	X t 2	X t 3
	X t 1 0	0.014 -0.01
	X t 2 0.014 0	0.013
	X t 3 -0.01 0.013 0

Table 2 :

 2 Difference between empirical correlation from generated samples and reference samples

Table 3 :

 3 Mean of PnL and its Std (replication error).

			Training Set	Validation Set	Test Set
		Premium Mean Std	Mean	Std	Mean	Std
	Data	0.0415	0.0008 0.0098 -0.0154 0.0371 0.003	0.012
	SBTS 0.0471	0.0004 0.0109 -0.0075 0.0164 -0.0024 0.0076

This author is supported by the BNP-PAR Chair "Futures of Quantitative Finance", and by FiME, Laboratoire de Finance des Marchés de l'Energie, and the "Finance and Sustainable Development" EDF -CACIB Chair

1000 sample paths, and the hyperparameters used for the simulation are N π = 100, with bandwith h = 0.05 for the kernel estimation of the Schrödinger drift. The runtime for 1000 paths is 120 seconds. Figure 7 represents the covariance matrix of (X t 1 , . . . , X t N ) for N = 60 of the FBM and of the generated SBTS, while we plot in Figure 8 the empirical distribution of the quadratic variation N -1 i=0 |X t i+1 -X t i | 2 for the FBM and the SBTS.