R Gay

V Della Marca

Hassen Aziza

M Mantelli

F Trenteseaux

F La Rosa

A Regnier

S Niel

A Marzaki

A Novel Trench-Based Triple Gate Transistor With Enhanced Driving Capability

Keywords: Multigate transistor, MOS devices, Trench transistor, Driving capability, Flash memory

This paper addresses the design, implementation, and characterization of a novel highdensity Triple Gate Transistor in a 40 nm embedded Non-Volatile Memory technology. Deep trenches are used to integrate two vertical transistors connected in parallel with the main planar transistor. Thanks to the built-in trenches, the proposed manufacturing process increases the transistor width without impacting its footprint. The voltage/current characteristics of a planar MOS structure are compared with the features of the new Triple Gate Transistor. The new architecture provides an improved driving capability, with an on-state drain current twice as high as its equivalent standard MOS, combined with a lower threshold voltage, suitable for low-voltage applications. Finally, the gate oxide and junction reliability are validated over the operating voltage range.

I. INTRODUCTION

HE continuous development of consumer electronics underlines the high demand for low-cost and reliable microelectronic devices [START_REF] Lin | A High-Performance Polysilicon Thin-Film Transistor Built on a Trenched Body[END_REF], [START_REF] Xue | Single-Side Fabricated p+Si/Al Thermopile-Based Gas Flow Sensor for IC-Foundry-Compatible, High-Yield, and Low-Cost Volume Manufacturing[END_REF]. The need for more integration has accelerated the current design trend for more scalability in almost every device parameter, such as channel length, gate dielectric thickness, supply voltage, and device leakage [START_REF] Bohr | CMOS Scaling Trends and Beyond[END_REF]. As most of these parameters are now reaching fundamental limits, new alternatives to the existing devices have to be identified in order to meet higher performance levels using new innovative architectures. In recent years, several devices, using multi-gate or gate-all-around transistors as an alternative to classical MOS, have emerged enabling enhancements in circuit performance [START_REF] Ferain | Multigate transistors as the future of classical metal-oxide-semiconductor field-effect transistors[END_REF]- [START_REF] Colinge | Multiple-gate SOI MOSFETs[END_REF]. Alternatively, and mostly for integrated passive devices, deep trenches are used in various ways, including as insulators, inductors, or DRAM capacitors to increase integration density [START_REF] Sunami | Development of three-dimensional MOS structures from trench-capacitor DRAM cell to pillar-type transistor[END_REF]. The aim of this work is to build an area-efficient transistor with a high driving capability. Starting from a classical MOS transistor denoted Standard Gate Transistor (SGT), two lateral conduction channels controlled by vertical trenches enhance the driving capability. This combination results in the proposed Triple Gate Transistor (TGT) [START_REF] Patent | Transistor MOS à triple grille et procédé de fabrication d'un tel transistor[END_REF]. Note that vertical deep trenches are currently used in embedded NOR Flash fabrication processes [START_REF] Rosa | 40nm embedded Select in Trench Memory (eSTM™) Technology Overview[END_REF]. The next paragraph presents the TGT device. In Section III, experimental results are presented to demonstrate the functioning of the device. Finally, Section IV concludes this paper.

II. DEVICE DESCRIPTION AND MANUFACTURING

The TGT is integrated in a Non-Volatile Memory (NVM) environment and designed following the process flow of an embedded NOR Flash technology. No additional steps or masks are required to build the device. Fig. 1 shows (a) a 3D schematic model of the TGT and (b) (c) TEM versus schematic crossdetailed views. The TGT is created as follows: active definition and NVM implantations are first performed. Deep trenches are then etched and an oxide is grown along sidewalls of the trenches. The latter are filled with an in-situ doped polycrystalline silicon to create two vertical gates. Finally, the planar gate patterning is processed. It includes two polysilicon layers (poly1 and poly2), an oxide/nitride/oxide (ONO) interpoly dielectric and a tunnel oxide. The planar gate contact is realized on poly1 through the poly2/ONO stack while two vertical trenches contacts are shifted with respect to the gate stack and connected to trenches polycrystalline silicon. Source and drain contacts are formed with n+ implantation. 3 (a)-(c), we observe that the TGT provides an Ion driving current (463 µA for VD = 2 V) three times higher than the SGT (143 µA for VD = 2 V), because the TGT benefits from a triple conduction (i.e., trenches conduction in addition to the classical MOS channel conduction). Regarding the leakage current (Ioff) extracted at VG = 0 V, the TGT and SGT values are comparable (less than 1 pA). Quantitative results are provided in Fig. 3 (d) MOS Gate Ion (102 µA) represents around 25% of the TGT Ion current and MOS Trenches Ion (341 µA) represents 75% of the TGT Ion current. Moreover, we can see a higher threshold voltage for the SGT (Vt = 0.98 V) compared to the TGT (Vt = 0.56 V). This difference is explained by a thinner trench oxide thickness and by the MT geometry. In Fig. 3 (b) we can note at the beginning of I-V characteristics (around ID = 10E -11 A and VG = 0.2 V) a hump effect due to the parasitic transistors formed by the active rounding [START_REF] Schwantes | Characterization of a new hump-free device structure for smart power and embedded memory technologies[END_REF], [START_REF] Locati | AC stress reliability study on a novel vertical MOS transistor for non-volatile memory technology[END_REF]. This effect is reflected in the TGT characteristics (Fig. 3 (b)) even though its impact is negligeable.

In Fig. 4 (a), the test results of gate voltage stress are provided for the TGT and the SGT. We note that the tunnel-oxide breakdown is reached at 15 V for both devices. However, the MG gate current start increasing before the gate current of the SGT does. This early leakage current is caused by an oxide weakness at trench-gate corners induced by the manufacturing process (as already mentioned before during Vt extraction). Regarding the MT transistor, it starts to leak at 3 V with a breakdown voltage of 10 V. This lower value is explained by the irregular thickness of the trench oxide caused by a nonuniform oxide growth along sidewalls. In Fig. 4 (b) are featured the test results of Drain/Bulk junction voltage stress for the TGT and the SGT in off-state (VG = 0 V). Comparable Drain/Bulk junction breakdown characteristics reported for TGT and SGT. These last stress tests demonstrate the robustness of the TGT: indeed, the maximum gate and drain voltages reported for the addressed technology are respectively VG = 4 V and VD = 3 V, far from the breakdown threshold (around 7.7 V).

Fig. 5 reports the evolution of the driving current (Ion) versus different device footprints with constant L. The blue linear curve which is related to the SGT is compared with the characteristics of the proposed TGT. For equivalent Ion current, the trench transistors integration allows to divide the device area by a factor two. Therefore, for an equivalent area, Ion current is doubled if the TGT architecture is considered.

IV. CONCLUSION

In this paper we propose a new transistor architecture denoted by Triple Gate Transistor that uses deep polysilicon trenches to build side transistor. The TGT manufacturing process and its different conduction modes are first introduced. The experimental results demonstrate the functionality and reliability of the TGT based on I-V characteristics and stress tests results. The driving capability of the TGT device is two times higher than its equivalent standard MOS counterpart. The Ioff current is maintained constant and a Vt decrease is reported. The new proposed architecture can provide significant chiplevel area savings as only half the silicon area is needed to reach an equivalent driving capability. For future work, we plan to use the TGT device with decorrelated gates (i.e., two different control signals for the trench transistors) to explore new opportunities for low-voltage and analog applications.

A

  Trench-Based Triple Gate Transistor With Enhanced Driving Capability R. Gay, V. Della Marca, H. Aziza, M. Mantelli, F. Trenteseaux, F. La Rosa, A. Regnier, S. Niel and A. Marzaki T R. Gay, A. Marzaki, F. La Rosa, A. Regnier and M. Mantelli are with STMicroelectronics Rousset, 13106 Rousset, France (e-mail: romeric.gay@st.com). S. Niel and F. Trenteseaux are with STMicroelectronics Crolles, 38926 Crolles, France. H. Aziza and V. Della Marca are with the Institut Matériaux

Fig. 1 .

 1 Fig. 1. (a) Triple Gate Transistor 3D view and TEM/schematic crosssections (b) in length (L) direction, (c) in width (W) direction

Fig. 2

 2 Fig.2maps the electrical schematic model of the TGT. The TGT is modeled by three MOS transistors connected in parallel, and therefore has three conduction channels. The first channel, which is located between the Drain and the Source, is controlled by the stacked gate (Poly1+tunnel oxide) and belongs to the device denoted by MOS Gate (MG) transistor. The second and third channels, which are also located between the Drain and the Source, are controlled by the trenches (shorted) and belong to the devices denoted by MOS Trenches (MT) transistors.III. EXPERIMENTAL RESULTSFig. 3 shows I-V characteristics in linear mode with drain voltage VD = 0.1 V and in saturated mode with VD = 2 V for: (a) MG and SGT, (b) MT and (c) TGT. The TGT characteristics include the contribution of three transistors: the MG and two trench transistors. A summary of the most relevant parameters in saturated mode is shown in Table I of Fig. 3 (d). The driving current parameter (Ion) is extracted for gate voltage VG = 4 V and the threshold voltage (Vt) is extracted for the drain current ID = 0.1 µA. Moreover, we highlight the Ion/Vt ratio that can be used as a design parameter figure of merit. In Fig. 3 (a) the standard SGT MOS is compared with the MG transistor of triple-gate device. The conduction currents are equivalent, while the MG threshold voltage slightly decreases due to the presence of vertical transistors. The latter impact the oxide thickness at the edges of the active substrate. Based on Fig. 3 (a)-(c), we observe that the TGT provides an Ion driving current (463 µA for VD = 2 V) three times higher than the SGT (143 µA for VD = 2 V), because the TGT benefits from a triple conduction (i.e., trenches conduction in addition to the classical MOS channel conduction). Regarding the leakage current (Ioff) extracted at VG = 0 V, the TGT and SGT values are comparable

Fig. 3 .

 3 Fig. 3. I-V characteristics in linear and saturated modes for: (a) MOS Gate and SGT, (b) MOS Trenches and (c) Triple Gate Transistor. (d) Comparison metrics in saturated mode.

Fig. 2 .Fig. 4 .Fig. 5 .

 245 Fig. 2. Triple Gate Transistor electrical schematic

  Marzaki, F. La Rosa, A. Regnier and M. Mantelli are with STMicroelectronics Rousset, 13106 Rousset, France (e-mail: romeric.gay@st.com).

S. Niel and F. Trenteseaux are with STMicroelectronics Crolles, 38926 Crolles, France. H. Aziza and V. Della Marca are with the Institut Matériaux Microélectronique Nanosciences de Provence, UMR CNRS 7334, University of Aix-Marseille, 13451 Marseille, France.


Hussein Bazzi

Jeremy Postel-Pellerin

Hassen Aziza

Mathieu Moreau

Adnan Harb

Resistive RAM SET and RESET Switching Voltage Evaluation as an Entropy Source for Random Number Generation

Keywords: Resistive memories, Resistive RAM (RRAM), Random Number Generator (RNG), Stochastic switching, Memory array. I

The intrinsic variability of the switching parameters in resistive memories has been a major wall that limits their adoption as the next generation memories. In contrast, this natural stochasticity can be beneficial for other applications such as Random Number Generators (RNGs). This paper presents two RNG approaches based on a 130nm HfO2-based Resistive RAM (RRAM) memory array. The memory array is programmed with a voltage close to the median value of the SET (resp. RESET) voltage distribution to benefit from the SET (resp. RESET) voltage variability. In both cases, only a subset of the memory array is programmed, resulting in a stochastic distribution of cell resistance values. Resistance values are next converted into a bit stream and confronted to National Institute of Standards and Technology (NIST) test benchmarks.

INTRODUCTION

A True Random Number Generator (TRNG) is a device that generates random numbers from a physical process, rather than by means of an algorithm. These processes are, in theory, stochastic and completely unpredictable. Conventional TRNG have been demonstrated based on Random Telegraph Noise (RTN) that occurs in semiconductors and ultra-thin gate oxide films [START_REF] Brederlow | A low-power true random number generator using random telegraph noise of single oxide-traps[END_REF][START_REF] Lelmini | Resistance-Dependent Amplitude of Random Telegraph-Signal Noise in Resistive Switching Memories[END_REF]. In the last decade, TRNG-based resistive-switching devices have been widely studied by several research groups [START_REF] Wang | A novel true random number generator design leveraging emerging memristor technology[END_REF][START_REF] Sahay | Recent trend in hardware security exploiting hybrid CMOS-resistive memory circuits[END_REF]. RRAMs have been utilized in several TRNG applications due to their appealing characteristics, including intrinsic stochastic nature [START_REF] Van Der Leest | Hardware intrinsic security to protect value in the mobile market[END_REF], immunity to radiation [START_REF] Wang | Highly Stable Radiation-Hardened Resistive-Switching Memory[END_REF], high endurance, CMOS compatibility [START_REF] Ielmini | Physical modeling of voltage-driven resistive switching in oxide RRAM[END_REF] and low power consumption [START_REF] Cagli | Study of the Energy Consumption Optimization on RRAM Memory Array for SCM Applications[END_REF]. It is worth noticing that in literature, the generic term RRAM is often used to generically refer to both Oxide Random Access Memory (OxRAM) and Conductive-Bridging Random Access Memory (CBRAM) [START_REF] Wang | Overview of Resistive Random Access Memory (RRAM): Materials, Filament Mechanisms, Performance Optimization, and Prospects[END_REF]. In the rest of the paper, OxRAM technology will be considered.

OxRAM devices suffer from a significant variability that can be divided into two categories, Device to Device (D2D) and Cycle to Cycle (C2C) variabilities [START_REF] Fantini | Intrinsic switching variability in HfO2 RRAM[END_REF]. This downside is analyzed and explored in this paper at a memory array level to assess the potential to use OxRAMs as TRNGs. Two sources of entropy are considered, the first one is based on the SET switching voltage variability and the second one is based on the RESET switching voltage variability. In both cases, programming voltages close to the SET and RESET voltage median distributions are applied. In these conditions, half of the cells in the array are expected to be in Low Resistive State (LRS) and the other half in High Resistive State (HRS). Thus, a stochastic distribution of resistance values across the array is anticipated. These resistance values are converted into logical values to generate a random bit stream with a perfect balance between 0s and 1s during the OxRAM cell READ operation by carefully trimming the reading voltages/currents. As the proposed TRNG combines C2C and D2D variabilities within a memory array, it overcomes limitations of state of the art OxRAM-based TRNGs which are mainly based on a single memory cell [START_REF] Balatti | True random number generation by variability of resistive switching in oxide-based devices[END_REF][START_REF] Sahay | OxRAM RNG Circuits Exploiting Multiple Undesirable Nanoscale Phenomena[END_REF][START_REF] Jiang | A novel true random number generator based on a stochastic diffusive memristor[END_REF]. Moreover, using a memory array instead of a single cell turns the random number generation process more reliable.

In section II, experimental results obtained from a memory array based on an OxRAM technology are presented. In section III, the random number generation process is detailed and random numbers are generated after both SET and RESET operations and converted into a bit stream. In Section IV, the bit stream is confronted to the National Institute of Standards and Technology (NIST) test bench. Finally, section V provides some concluding remarks.

II. OXRAM TECHNOLOGY

A. 1T1R memory cell

Devices used in our TRNG circuit are bipolar OxRAMs where bipolar voltage sweeps are required to switch the resistive element. An OxRAM consists of two metallic electrodes that sandwich a thin dielectric layer serving as a storage medium. This structure can be easily integrated in the Back-End-Of-Line (BEOL) in combination with advanced CMOS technologies. As reported in [START_REF] Molas | Resistive Memories (RRAM) Variability: Challenges and Solutions[END_REF], the MIM structure is integrated on top of Metal 4 copper layer (Cu). A TiN Bottom Electrode (BE) is first deposited. Then, a 10nm-HfO2/10nm-Ti/TiN stack is added. In this configuration one MOS transistor is serially connected to an OxRAM cell. The select transistor acts as current limiting device. It controls the amount of current flowing through the cell according to its gate voltage value in order to prevent memory cell damage during programming operations [START_REF] Grossi | Experimental Investigation of 4-kb RRAM Arrays Programming Conditions Suitable for TCAM[END_REF]. Fig. 1 (a) and (b) present typical 1T1R OxRAM I-V characteristics in linear scale. Based on these I-V curves, the memory cell behavior can be seen as follows: after an initial electro-FORMING step (not presented here), the memory element can be reversibly switched between a High Resistance State (HRS) and a Low Resistance State (LRS). Resistive switching in an OxRAM corresponds to an abrupt change between HRS and LRS. This resistance change is achieved by applying specific biases across the 1T1R cell (i.e. VSET, to switch from HRS to LRS and VRESET, to switch from LRS to HRS). Fig. 1 The proposed random number generation methodology exploits the HfO2-based OxRAM variability (C2C and D2D) to generate a sequence of random bits. Indeed, HfO2-based OxRAM technology suffers from stochastic switching. These statistical fluctuations are generally attributed to the formation/dissolution of a Conductive Filament (CF) between metallic electrodes [START_REF] Larentis | Resistive switching by voltage-driven ion migration in bipolar RRAM -Part II: Modeling[END_REF][START_REF] Miranda | Compact model for the major and minor hysteretic I-V loops in nonlinear memristive devices[END_REF][START_REF] Larcher | Microscopic understanding and modeling of HfO2 RRAM device physics[END_REF]. The number of defects and the distance between traps within the CF in both SET and RESET states are considered as the main causes of variability [START_REF] Puglisi | Bipolar Resistive RAM Based on HfO2: Physics, Compact Modeling, and Variability Control[END_REF]. To extract the RESET distribution, the memory array is first SET. Then, memory cells are RESET one by one to extract the RESET voltage value referred to as VRESET. Regarding VSET distribution, the memory array is first RESET. Then, memory cells are SET one by one to extract the VSET threshold. The process is repeated 1000 times for both VRESET and VSET. Fig. 3 (resp. Fig. 4) presents the resulting VRESET (resp. VSET) distribution. From these distributions, the median value of the RESET distribution referred to as VRESET_M is extracted, as well as the median value of the SET distribution referred to as VSET_M. Results are summarized in Table II.

B. 1T1R memory array

.5 V -0.86 V 1.8 V SET 0.5 V 0.84 V 1.2 V
III. TRNG METHODOLOGY Note that VRESET_M and VSET_M extraction is a key point in the presented random number generation methodology as these values will be used as the conventional SET and RESET voltages for the next programming operations.

A. RESET switching voltage-based RNG

This RN generation process consists in setting the whole memory array and resetting each cell individually with VRESET_M = -0.86 V ("weak" RESET). As a result, some cells are switched to HRS, while others remain in LRS, resulting in a random distribution of resistance values. 100 runs are applied to the memory array to obtain a sufficiently large number of data (49x100). Fig. 5 presents an example of two 2D bitmaps obtained after two consecutive runs. Resistance values are represented by a greyscale from white (low values) to black (high values). Note that no spatial pattern reproduction has been detected after consecutive runs. Fig. 6 presents the resistance distribution obtained after the weak RESET whose median value is equal to 32 kΩ. This median is used in order to convert resistance values into a stream of bits presenting a perfect balance between '0' and '1'. Despite external (temperature, etc.) or internal variations (degradation, etc.) the 50% ratio of '1' cells (and thus of '0' cells) has to be ensured. To this aim, real time voltage tracking techniques such as the one presented in [START_REF] Choi | A magnetic tunnel junction based true random number generator with conditional perturb and real-time output probability tracking[END_REF] can be used. Indeed, NIST tests require an equal parity between '0' and '1'. To convert resistance values into '0' and '1', all values lower than RM = 32 kΩ (dashed line in Fig. 6) are associated with a '1' logical state while all higher values are associated with a '0' logical state.

The reading circuitry which basically consists in a simple biasing circuit with its output connected to a comparator is in charge of the distribution split [START_REF] Aziza | A built-in self-test structure (BIST) for resistive RAMs characterization: Application to bipolar OxRRAM[END_REF]. Thus, the analog bitmaps presented in Fig. 5 are turned into logical bitmaps presented in Fig. 7. At run N we obtain 29 bits at '1' level (and thus 20 bits at '0' level) and at run N+1 we obtain 23 bits at '1' level (and thus 26 bits at '0' level). Moreover, between the two presented runs 21 cells out of the 49 have changed their state and no systematic binary pattern has been observed between two consecutive runs.

B. SET switching voltage-based RNG

In this configuration, the RN generation process consists in resetting the whole memory array and setting each cell individually with VSET_M = 0.84 V ("weak" SET). This study was reported in a previous work [START_REF] Postel-Pellerin | True Random Number Generation Exploiting SET Voltage Variability in Resistive RAM Memory Arrays[END_REF] where the SET voltage variability was exploited for RNG. Here also, some cells are switched to LRS state, while others remain in HRS state, resulting in a random distribution of resistance values. However, the resistance dispersion is larger. Fig. 8 presents an example of two 2D bitmaps obtained after two consecutive runs. Fig. 9 presents the resistance distribution whose median value is equal to 145 kΩ with values ranging from 10 kΩ to 1 MΩ. To convert resistance values into '0' and '1', all values lower than RM = 145 kΩ (dashed line of Fig. 9) are associated with a '1' logical state while all higher values are associated with a '0' logical state. Logical bitmaps are presented in Fig. 10. One can notice that at run N we obtain 28 bits at '1' level (and thus 21 bits at '0' level) and at run N+1 we obtain 28 bits at "1" level (and thus 21 bits at "0" level). Moreover, between the two presented consecutive runs, 13 cells out of the 49 have changed their state. Also, no systematic pattern is visible between the two runs. NIST TEST SUITE RESULTS

The random numbers (4.9 kbits) generated after 100 runs and extracted from distributions presented in Fig. 6 (obtained after weak RESET) and Fig. 9 (obtained after weak SET) are verified against a statistical test suite. The statistical calculation is performed based on the NIST test suite for random and pseudorandom number generators for cryptographic applications [START_REF] Rukhin | A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications[END_REF]. For the distribution obtain after a weak RESET, the bit stream passes 9 NIST tests over 15 (see Table III). For the distribution obtained after a weak SET, the bit stream passes 11 NIST tests over 15 (see Table IV). It should be noted that these results are obtained without any post-processing step.

Tests 05-Binary Matrix Rank, 08-Overlapping Template Matching, 09-Maurer's Universal Statistical and 10-Linear Complexity are failed for both RESET and SET distributions. Indeed, these tests need a larger number of bits as recommended in [START_REF] Rukhin | A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications[END_REF]. Test 12-Approximate Entropy checks the frequency of all possible overlapping m-bit patterns across the entire sequence. The purpose of the test is to compare the frequency of overlapping blocks of two consecutive/adjacent lengths (m and m+1) against the expected result for a random sequence, ensuring the quality of the random number sequence. Only the SET switching voltage-based bit stream passes this test (p-value > 0.01). Same goes for test 6-Discrete Fourier Transform or DFT used to detect periodic features (i.e. repetitive patterns that are near each other).

A. Comparison and Discussion

Table V shows a comparison between the two proposed methods. The comparison is based on five parameters including NIST test result, resistance median distribution, expected energy/cell, expected bit-rate and the use of a post-processing step. The resistance distribution median value for RESET (32 kΩ) is lower than the SET one (145 kΩ). This difference has a straight impact on the energy consumption during the read operation preceding the bit stream generation. Thus, assuming that the RNG is properly configured (weak SET and weak RESET operation achieved and threshold resistances properly extracted), the energy/cell during RNG use is evaluated to 0.125 fJ and 0.027 fJ after weak RESET and weak SET respectively (values computed for a 100ns read pulse at 0.2V considering the average post weak RESET and post weak SET resistances). Regarding the expected bit-rate, it has been demonstrated that read time in the 10ns order can be achieved for hafnium oxide devices resulting in bit rate in the order of 100 Mb/s [START_REF] Traore | Investigation of HfO2-based resistive RAM cells by electrical characterization and atomistic simulations[END_REF]. Finally, the proposed RNGs do not include postprocessing circuits such as XOR gates [START_REF] Kaya | Random Number Generators Based on Irregular Sampling and Fibonacci-Galois Ring Oscillators[END_REF] and thus, can be improved in terms of randomness quality. From a physical point of view, generally, the HRS is more affected by switching variability, because the applied negative voltage results in a partial depletion of defects from the CF. Any variation in the conduction path and in the energy barrier impacts exponentially the resistance, therefore the statistical variations of HRS resistance are relatively large. Moreover, the resistance cumulative distributions slightly increases with VRESET, as previously demonstrated by experiments and simulations of HfOx-based RRAM [START_REF] Ambrogio | Statistical fluctuations in HfOx resistive-switching memory (RRAM): Part I -Set/Reset variability[END_REF]. On the other hand, resistance fluctuations in the LRS are generally due to geometrical variations in the CF cross section, which affect LRS resistance only linearly resulting in tighter resistance distributions. Regarding randomness, no correlation with variability is observed as the weak SET programming approach produce better randomness than the weak RESET programming approach.

V. CONCLUSION

The proposed TRNG is implemented based on an elementary array of 1T1R OxRAM cells. Based on two programming method exploiting weak RESET and weak SET resistance distributions, it is possible to obtain random numbers. However, it has been demonstrated that the capability to produce randomness is higher for weak SET resistance distributions compared to weak RESET resistance distributions. These results have been confirmed after NIST tests. Indeed, the obtained bit streams have passed 9 and 11 NIST tests over 15 without any post-processing step for RESET and SET respectively.

Fig. 1 .

 1 Fig. 1. I-V characteristics highlighting (a) D2D variability and (b) C2C variability. VSET and VRESET variabilities are highlighted in both cases.

Fig. 2 (Fig. 2 .

 22 Fig. 2. (a) 8x8 OxRAM memory array and (b) corresponding layout view.

Fig. 3 .

 3 Fig. 3. RESET switching voltage (VRESET) distribution obtained after 1000 SET/RESET cycles applied to the elementary memory array.

Fig. 4 .

 4 Fig. 4. SET switching voltage (VSET) distribution obtained after 1000 RESET/SET cycles applied to the elementary memory array.

Fig. 5 .

 5 Fig. 5. Analog 2D bitmap (RESET) after Run N (left) and Run N+1 (right)

Fig. 6 .

 6 Fig. 6. Resistance distribution obtained after 100 consecutive memory array runs with a RESET value equals to VRESET_M (-0.86 V).

Fig. 7 .

 7 Fig. 7. Digital 2D bitmap (RESET) after Run N (left) and Run N+1 (right)

Fig. 8 .

 8 Fig. 8. Analog 2D bitmap (SET) after Run N (left) and Run N+1 (right)

TABLE I .

 I STANDARD OPERATING VOLTAGES (CELL LEVEL)

		FORMING RESET	SET	READ
	WL	2V	4.5V	2V	4V
	BL	2V	0V	1.2V	0.2V
	SL	0V	1.8V	0V	0V

TABLE II .

 II VRESET/VSET DISTRIBUTION PARAMETERS

		Min	Median	Max
	RESET	0

TABLE III .

 III NIST TEST SUITE RESULTS AFTER WEAK RESET

	Test	p-value	Result
	01-Frequency	0.8639	Pass
	02-Frequency within a block	0.0872	Pass
	03-Runs	0.7314	Pass
	04-Longest Run of ones in a block	0.2693	Pass
	05-Binary Matrix Rank	-	Fail
	06-Discrete Fourier Transform	-	Fail
	07-Non-Overlapping Template	0.5209	Pass
	08-Overlapping Template Matching	-	Fail
	09-Maurer's Universal Statistical	-	Fail
	10-Linear Complexity	-	Fail
	11-Serial	0.4781	Pass
	12-Approximate Entropy	-	Fail
	13-Cumulative Sums	0.1627	Pass
	14-Random Excursions	0.3540	Pass
	15-Random Excursions Variant	0.0726	Pass
	Global NIST Score	9/15

TABLE IV .

 IV NIST TEST SUITE RESULTS AFTER WEAK SET

	Test	p-value	Result
	01-Frequency	0.1736	Pass
	02-Frequency within a block	0.6711	Pass
	03-Runs	0.8157	Pass
	04-Longest Run of ones in a block	0.9077	Pass
	05-Binary Matrix Rank	-	Fail
	06-Discrete Fourier Transform	0.8623	Pass
	07-Non-Overlapping Template	0.9970	Pass
	08-Overlapping Template Matching	-	Fail
	09-Maurer's Universal Statistical	-	Fail
	10-Linear Complexity	-	Fail
	11-Serial	0.4921	Pass
	12-Approximate Entropy	0.9999	Pass
	13-Cumulative Sums	0.3248	Pass
	14-Random Excursions	0.4072	Pass
	15-Random Excursions Variant	0.5929

Pass Global NIST Score 11/15

TABLE V .

 V COMPARISON BETWEEN THE TWO PROPOSED METHODS

		NIST	Median	Energy/cell	Bit-rate	Post-
		Results	Resistance	(fJ)	(Mb/s)	processing
	RESET	9	32 kΩ	0.125	100	No
	SET	11	145 kΩ	0.027	100	No

ACKNOWLEDGMENT

The authors wish to acknowledge the support from Aix-Marseille University, and Lebanese International University.


Matthias Renoirt

Frédéric Angelier

Marion Cheron

Paco Bustamante

Yves Cherel

Francois Brischoux
email: francois.brischoux@gmail.com

François Brischoux

Stable isotopes of a terrestrial amphibian illustrate fertilizer-related nitrogen enrichment of food webs in agricultural habitats

Keywords: Bufo spinosus, agriculture, amphibians, habitat use, forest, δ 15 N, δ 13 C

à 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

Anthropogenic activities are considered as the main factors responsible for the current loss of biodiversity [START_REF] Chapin | Consequences of changing biodiversity[END_REF][START_REF] Myers | The biotic crisis and the future of evolution[END_REF][START_REF] Brooks | Habitat loss and extinction in the hotspots of biodiversity[END_REF]. Among these anthropogenic changes, modern agricultural practices have been shown to negatively influence biodiversity both directly and indirectly. The direct negative impacts of agriculture on flora and fauna are mainly linked to the destruction and simplification of habitat structures [START_REF] Fahrig | Effects of habitat fragmentation on biodiversity[END_REF]. Indirect effects are mainly mediated by the increasing reliance on chemical inputs that aim at improving crop productivity [START_REF] K¨ohler | Wildlife ecotoxicology of pesticides: can we track effects to the population level and beyond?[END_REF]. For instance, the use of large amounts of fertilizers can eventually led to disruptions of ecosystem functioning [START_REF] Huang | Nitrogen and phosphorus losses and eutrophication potential associated with fertilizer application to cropland in China[END_REF]. In addition, the toxic effects of pesticides on non-target components have attracted considerable interest (Köhler and Triebskorn, 2013). Clearly, both direct and indirect effects are expected to affect the species inhabiting agricultural landscapes [START_REF] Mclaughlin | The impact of agricultural practices on biodiversity[END_REF]Köhler and Triebskorn, 2013).

In some cases, it is relatively straightforward to assess the consequences of these habitat modifications on the ecology of animal species (e.g., when nesting trees or shrubs are lacking for birds, [START_REF] Mohring | Vineyards, but not cities, are associated with lower presence of a generalist bird, the Common Blackbird (Turdus merula), in Western France[END_REF]. Yet, in most cases, agricultural landscapes will be intersected with remnants of other types of habitats (e.g., small woods, hedgerows) that should allow the persistence of populations. In these cases, assessing the actual use of agricultural habitats versus remnant of native habitats is logistically complicated, especially when the species under focus is relatively mobile (i.e., most animal species). Nonetheless, in order to comprehensively assess the consequences of agricultural practices on biodiversity, it is necessary to evaluate to which extent such habitat is actually used by a given species [START_REF] Street | Habitat functional response mitigates reduced foraging opportunity: implications for animal fitness and space use[END_REF].

Stable isotopes can provide insights in this respect [START_REF] Robinson | δ15N as an integrator of the nitrogen cycle[END_REF][START_REF] Rubenstein | From birds to butterflies: animal movement patterns and stable isotopes[END_REF][START_REF] Perkins | Application of nitrogen and carbon stable isotopes (δ(15)N and δ(13)C) to quantify food chain length and trophic structure[END_REF][START_REF] Newton | Stable Isotopes as Tools in Ecological Research[END_REF]. The concept of the isotopic niche is based on the fact that an animal's chemical composition is influenced by what it consumes [START_REF] Fry | Stable Isotope Ecology[END_REF]. Stable nitrogen isotope values (δ 15 N) are mostly used as a proxy of trophic position, but can be also a relevant proxy of consumers' foraging habitat [START_REF] Kelly | Stable isotopes of carbon and nitrogen in the study of avian and mammalian trophic ecology[END_REF]. In our context, because fertilizers widely used in agriculture show relatively high δ 15 N (e.g., manure and compost [δ 15 N up to 16.2 ‰], ammonium sulphate [δ 15 N up to 6.6 ‰] and ammonium nitrate [δ 15 N up to 2.2 ‰], [START_REF] Bateman | Fertilizer nitrogen isotope signatures[END_REF], it is expected that trophic chains influenced by fertilization will be enriched in δ 15 N [START_REF] Anderson | δ 15 N in riverine food webs: effects of N inputs from agricultural watersheds[END_REF][START_REF] Bateman | Fertilizer nitrogen isotope signatures[END_REF]. As a consequence, individuals relying on agricultural areas to forage should display higher δ 15 N values than individuals using other types of habitats. In contrast, stable carbon values (δ 13 C) vary little along the food chain, and often depend on foraging habitats (δ 13 C source). Typically, δ 13 C varies among specific primary producers [START_REF] Farquhar | Carbon isotope discrimination and photosynthesis[END_REF], and we can use this parameter to examine differences in trophic support and thus presumably habitats.

In this study, we tested whether blood δ 15 N and δ 13 C values can help to discriminate the foraging habitats of a wild vertebrate, the spined toad (Bufo spinosus). This species is particularly well suited to test these hypotheses for several reasons. First, this widespread amphibian can live in a variety of habitats and persist even in highly modified agricultural areas [START_REF] Guillot | Landscape influences the morphology of male common toads (Bufo bufo)[END_REF], see also [START_REF] Salazar | Mapping the relative probability of common toad occurrence in terrestrial lowland farm habitat in the United Kingdom[END_REF][START_REF] Leeb | Potential pesticide exposure during the postbreeding migration of the common toad (Bufo bufo) in a vineyard dominated landscape[END_REF]. Second, spined toads forage for invertebrates and its prey spectrum has been shown to be highly conserved between habitats (Zamora-Camacho and Comas, 2017).

Third, the terrestrial part of the life cycle occurs within 1 km from the breeding (sampling) sites, which allow a straightforward classification of the surrounding landscapes potentially used in the day-to-day life of individuals [START_REF] Janin | Beyond occurrence: body condition and stress hormone as integrative indicators of habitat availability and fragmentation in the common toad[END_REF][START_REF] Guillot | Landscape influences the morphology of male common toads (Bufo bufo)[END_REF]. Finally, the remarkably long lifetime of erythrocytes of amphibians [START_REF] Altland | Red cell life span in the turtle and toad[END_REF] indicates that stable isotopes from red blood cells actually reflect habitat use during the terrestrial life of our study species prior to breeding (see also [START_REF] Cloyed | Trophic discrimination factors and incorporation rates of carbon-and nitrogen-stable isotopes in adult green frogs, Lithobates clamitans[END_REF]. In order to test whether isotopic values of spined toads can discriminate their foraging habitats, we assessed δ 15 N and δ 13 C of red blood cells (herein blood) from 114 individuals from seven sites ranging from forested areas to highly agricultural sites.

Material and methods

Study species

In Western Europe, Spined toad (Bufo spinosus) is one of the most common species of amphibians. As most anuran species, spined toads have a biphasic life-cycle with an extensive use of terrestrial habitats during most of the year, and a short breeding season (~1 month) in ponds. During breeding, male toads massively migrate towards ponds where they wait for females [START_REF] Brischoux | Osmotic cost of reproduction in breeding male toads[END_REF] and a large number of males can be easily sampled at each pond.

Study sites and sampling

Sampling took place in February 2020 on seven breeding ponds situated in the south of the "Département des Deux-Sèvres" nearby the laboratory (46°8′48.64"N; 0°25'30.86"W). Two sites were located in highly forested areas, three sites in agricultural landscapes, and the remaining two sites at the interface between forested areas and agricultural areas (Appendix 1). Distances between different sites within a habitat type (e.g., ~18 km between the two forested sites; at least ~12 km between agricultural sites) were large enough to minimize spatial autocorrelation [START_REF] Janin | Beyond occurrence: body condition and stress hormone as integrative indicators of habitat availability and fragmentation in the common toad[END_REF][START_REF] Guillot | Landscape influences the morphology of male common toads (Bufo bufo)[END_REF]. Such site selection allowed making simple habitat classifications (Table 1). Using QGIS.org 2.18.2, 2016 and satellite images (Google Earth), we drew a buffer around each pond (1000 m radius spanning the spatial scale traveled by toads during the breeding migration, [START_REF] Janin | Beyond occurrence: body condition and stress hormone as integrative indicators of habitat availability and fragmentation in the common toad[END_REF][START_REF] Guillot | Landscape influences the morphology of male common toads (Bufo bufo)[END_REF] and we extracted the surface area of agricultural fields (Table 1).

We focused our sampling on the first breeding males arriving at each breeding sites.

Captures occurred at night using a headlamp to locate individuals. Upon sighting, each toad (total N = 114, 3-21 different individuals per sampling sites, Table 1) was captured with a net, and a blood sample was collected (approx. 100 μl) via cardiocentesis using a 1 ml syringe and a 30-G heparinized needle [START_REF] Brischoux | Influence of temperature, size and confinement on testosterone and corticosterone levels in breeding male spined toads (Bufo spinosus)[END_REF]. All individuals were released at their location of capture after blood collection.

Stable isotope analyses

Whole blood was centrifuged, and red blood cells subsequently stored at -20 °C until analysis. Isotopic analyses were carried out on freeze-dried red blood cells at the LIENSs (La Rochelle, France). Aliquots of ~0.3 mg dry mass were analyzed with a continuous flow mass spectrometer (Thermo Scientific Delta V Advantage) coupled to an elemental analyzer (Thermo Scientific Flash EA 1112). Results are in δ notation relative to Vienna PeeDee Belemnite and atmospheric N2 for δ 13 C and δ 15 N, respectively. Internal laboratory standards (acetanilide) were used to check accuracy.

Measurement errors were <0.15‰ for both δ 13 C and δ 15 N values.

Statistical analyses

All data were tested for homogeneity of variance, residues independence and normality with the Bartlett test, Dubin-Watson test and Shapiro-Wilks test, respectively. We also checked the residues normality using diagnostic plots. All statistical analyses were carried out with R. Studio v 1.2.1335 (R Core Team, 2019). We fitted linear mixed models (LMER, package lmerTest, [START_REF] Kuznetsova | Package "lmertest[END_REF] to assess differences in blood δ 15 N and δ 13 C values across agricultural surface area, with "sites" as a random factor. We analyzed these models with variances analysis (ANOVA). We also analyzed differences in blood δ 15 N and δ 13 C values between categories (see habitat classification). We followed these analyses by post-hoc tests to performed pair-wise comparisons between categories using Tukey-Kramer tests for unbalanced sample sizes (implemented in the Tukey HSD function).

A site was characterized by very low sample size (N = 3, site #6, Table 1, Appendix 1).

Excluding this site from our analyses yielded similar results and we present results with the whole dataset below.

Results

Blood δ 15 N values were positively related to agricultural surface area (F1,107 = 16.022, p < 0.001). Blood δ 15 N values were significantly different between habitat types (F2,107 = 6.967, p < 0.001, Fig. 1) with post-hoc tests showing that δ 15 N of toads from forest habitats were significantly lower than δ 15 N of toads from agricultural habitats (p < 0.001, Fig. 1), while individuals from mixed habitats were not different from the two other categories (both p > 0.089, Fig. 1).

Blood δ 13 C values did not vary according to the agricultural surface area (F1,107 = 0.120, p = 0.730). Accordingly, δ 13 C values were similar between habitat types (F2,107 = 0.201, p = 0.81).

Discussion

Overall, we found that habitat type influences isotopic values of toads. Blood δ 15 N values were positively related to agricultural surface area. These results were corroborated by diverging toad δ 15 N values between habitat categories with lower δ 15 N values found in individuals from forest habitats and higher δ 15 N values found in toads from agricultural areas. Conversely, blood δ 13 C values of toads did not vary according to the habitat.

Toads that breed in ponds surrounded by agricultural environments, and thus likely living in such environments [START_REF] Guillot | Landscape influences the morphology of male common toads (Bufo bufo)[END_REF][START_REF] Salazar | Mapping the relative probability of common toad occurrence in terrestrial lowland farm habitat in the United Kingdom[END_REF][START_REF] Leeb | Potential pesticide exposure during the postbreeding migration of the common toad (Bufo bufo) in a vineyard dominated landscape[END_REF], were characterized by higher blood δ 15 N values than individuals breeding in ponds surrounded by forest. Two different hypotheses could explain such results. First, toads from agricultural landscapes may forage on different food items (higher in the trophic web) than individuals living in forest. This hypothesis seems unlikely as toad diet remains similar amongst habitats (Zamora-Camacho and Comas, 2017). In addition, the remarkably large variation of δ 15 N values between habitat types (-0.9 to 7.9 ‰, Fig. 1), spans approximately 2.1-2.7 theoretical trophic levels [START_REF] Deniro | Influence of diet on the distribution of nitrogen isotopes in animals[END_REF].

Although proportional abundances of invertebrates of different trophic levels may differ between habitat types (but see Zamora-Camacho and Comas, 2017), it can hardly explain the large variation of δ 15 N -hence "trophic levels" -we found. Detailed analyses of the diet of toads and the δ 15 N values of prey items in each habitat (see [START_REF] Perkins | Application of nitrogen and carbon stable isotopes (δ(15)N and δ(13)C) to quantify food chain length and trophic structure[END_REF] for an example of food web stable isotopic table) are required to test this hypothesis. More likely, agricultural fertilization leads to very high amounts of 15 N-enriched fertilizers (e.g., manure, compost, ammonium sulphate and to a lesser extent ammonium nitrate, [START_REF] Bateman | Fertilizer nitrogen isotope signatures[END_REF] deposited on arable land [START_REF] Tamm | Nitrogen in Terrestrial Ecosystems: Questions of Productivity, Vegetational Changes, and Ecosystem Stability[END_REF]. Relatively high δ 15 N values of these fertilizers increases δ 15 N baselines that propagate through the trophic webs in agricultural landscapes [START_REF] Anderson | δ 15 N in riverine food webs: effects of N inputs from agricultural watersheds[END_REF]. Such process has already been highlighted in tadpoles developing in waters with high nitrate concentrations [START_REF] Trakimas | Ontogenetic dietary shifts in European common frog (Rana temporaria) revealed by stable isotopes[END_REF].

It is important to emphasize the large variations in δ 15 N values between individuals from the same habitat type or study site (Table 1, Fig. 1). Such result seems to suggest different individual strategies of (micro-) habitat use to forage within a similar landscape [START_REF] Miaud | Terrestrial habitat preferences of the natterjack toad during and after the breeding season in a landscape of intensive agricultural activity[END_REF][START_REF] Indermaur | Differentia resource selection within shared habitat types across spatial scales in sympatric toads[END_REF]. This indicates that the influence of nitrogen fertilization on toad δ 15 N values is spatially restricted. Such result suggests that, at least in our setting, δ 15 N values can help to understand the use of micro-habitats, movement and/or dispersal and individual strategies in agricultural landscapes [START_REF] Rickers | Stable isotope analyses document intraguild predation in wolf spiders (Araneae: Lycosidae) and underline beneficial effects of alternative prey and microhabitat structure on intraguild prey survival[END_REF][START_REF] Dammhahn | Trophic niche differentiation and microhabitat utilization revealed by stable isotope analyses in a dry-forest bat assemblage at Ankarana, Northern Madagascar[END_REF]. Future studies using both individual tracking (i.e, radio-tracking) and isotope analyses will be critical to test for this hypothesis.

In contrast to δ 15 N values, our results showed that blood δ 13 C values did not differ between toads from agricultural habitats and individuals living in forest. It is likely that the surface area occupied by corn fields (the main C4 plant expected to influence δ 13 C values in agricultural habitats, [START_REF] Schwertl | Carbon and nitrogen stable isotope composition of cattle hair: ecological fingerprints of production systems?[END_REF] is too low to significantly influence δ 13 C values of individual toads using agricultural habitats: In our study area, corn crop represents only ~4.5% of agricultural surface area [START_REF] Agreste | Agriculture des Deux-Sèvres[END_REF]. Future studies are required to investigate both the presence of toads and their potential prey in different crop types. Additionally the prey of toads could be sampled to assess their trophic level in each habitat to provide a better understanding of the habitat's nutrient flow and its possible influence on toads' use of various habitats [START_REF] Perkins | Application of nitrogen and carbon stable isotopes (δ(15)N and δ(13)C) to quantify food chain length and trophic structure[END_REF].

Finally, future studies could incorporate other isotope markers such as sulphur or oxygen to further determine the nutrients in each habitat's food web [START_REF] Penna | Water sources for root water uptake: using stable isotopes of hydrogen and oxygen as a research tool in agricultural and agroforestry systems[END_REF]. Complementary approaches such as compound-specific isotopic analyses of amino acids may well prove useful to assess both fertilizer-related (e.g., phenylalanine) and trophic-related (e.g., glutamic acid) effects on δ 15 N [START_REF] Mcmahon | Embracing variability in amino acid δ 15 N fractionation: mechanisms, implications, and applications for trophic ecology[END_REF].

To conclude, we suggest that δ 15 N values could be a powerful tool to assess habitat use in a terrestrial meso-predator such as the spined toad. Whether such approach could be used on other taxa living in agricultural landscape deserve further investigations. Interestingly, δ 15 N values, as an index of agricultural habitat use, may also help to reveal individual susceptibility to disturbances linked to modern agricultural practices (i.e., pesticides). Futures studies should usefully investigate whether individual δ 15 N values can be used as a fingerprint of concentrations of contaminants in agricultural landscapes.

Figure 1 .

 1 Figure 1. Blood δ 15 N values of toads from forest, mixed and agricultural habitats. The

Table 1 .

 1 Summary of the sampling design (habitat categories, number of individuals sampled) and corresponding toad blood mean (±standard error) of δ 15 N and δ 13 C values.

				Agricultural surface
	Sites	Categories	n		δ 15 N (mean ± sd ) δ 13 C (mean ± sd )
				area (ha)
	1	Forest	16	0	0.49 ± 0.54	-24.79 ± 0.26
	2	Forest	21	15	1.33 ± 1.61	-23.93 ± 0.53
	3	Mixed	16	54	2.52 ± 1.83	-23.62 ± 0.38
	4	Mixed	20	104	2.72 ± 1.70	-24.41 ± 0.59
	5	Agricultural	19	260	3.43 ± 1.64	-24.35 ± 0.47
	6	Agricultural	3	261	7.28 ± 0.20	-23.10 ± 0.76
	7	Agricultural	19	286	4.77 ± 1.86	-24.45 ± 0.63

Acknowledgments

The authors are grateful to G. Guillou from the Plateforme Analyses Isotopiques of LIENSs for running the stable isotope analysis. Three anonymous reviewers provided insightful comments on previous versions of this manuscript. Thanks to the CPER (Contrat de Projet Etat-Région) and the FEDER (Fonds Européen de Développement Régional) for funding the IRMS of LIENSs laboratory. The IUF (Institut Universitaire de France) is acknowledged for its support to PB as a Senior Member. Funding was provided by the CNRS (Centre Nationale de la Recherche Scientifique), the Agence de l′Eau Loire-Bretagne, the Agence de l′Eau Adour-Garonne, the Région Nouvelle-Aquitaine (Multistress 2017-1R20110, Aquastress 2018-1R20214, Amphitox 2019-1R20216), the ANSES (Agence nationale de sécurité sanitaire de l'alimentation, de l'environnement et du travail) (BiodiTox 2019/1/031), the CPER ECONAT, and the Conseil Départemental des Deux-Sèvres.

Appendix

Appendix 1. Satellite images (Google Earth) of each study site centred on the breeding (sampling) pond and depicting the buffer (1 km radius) drawn to characterize habitat types (agricultural fields and forest surface areas). For our analyses using categories, we classified sites 1 & 2 as "forest", sites 3 & 4 as "mixed" and sites 5 to 7 as "agricultural". The numbers in the pictures relate to site numbers in Table 1.


Alexandre Goldsztejn

MANIFOLD APPROXIMATION OF SET-VALUED FUNCTIONS

Notations

Sequences are denoted by x (k) -although C ∞ functions are involved in the sequel, their derivative are not explicitly used so this notation for sequences will not interfere with the usual one for derivatives. A function f : R 0 -→ P(R m ) is a subset of R m . Notations and theorems related to manifolds are taken from Hirsch [START_REF] Hirsch | Differential Topology[END_REF]. Points belonging to manifolds will be denoted with tildes. The closed n-dimensional disk is denoted by D n and the n-dimensional sphere by S n , so ∂D n = S n-1 .

Definition of ma-continuity for set-valued maps

Informally, a set-valued map is ma-continuous if it can be approximated by smooth manifolds. Up to now, the definition of ma-continuity is restricted to functions defined inside boxes.

Definition 2.1. Consider a set-valued function F : x ⊆ R n -→ P(R m ). If n > 0, F is ma-continuous if and only if there exists • a closed disk d ⊆ R n such that x ⊆ int d; • M (k) a sequence of C ∞ compact n-manifolds such that ∂M (k) = S n-1 ; • g (k) : M (k) -→ R n+m a sequence of C ∞ maps such that g (k) | ∂M (k) is a C ∞ diffeomorphism between ∂M (k) and (∂d) × {0}.
such that any sequence x(k) ∈ M (k) , satisfies

• the sequence g (k) (x (k) ) is bounded; • if furthermore g (k) (x (k) ) ∈ x (k) × R m for all k ∈ N, then any accumulation point (x * , y * ) T of the sequence g (k) (x (k) ) satisfies y * ∈ F (x * ); If n = 0, f is ma-continuous if and only if F = ∅.
The condition satisfied by the boundary of the manifolds was somewhat difficult to find and can certainly be improved. It has the advantage of simplifying the proof of the JOINT property although it may turn out to be difficult to prove the PROD property -because Cartesian products of closed disks are not easily related to disks, are they ?

JOINT

The following theorem is slightly more restrictive than the JOINT property as it asks for a strict inclusion instead of a simple inclusion. However, the JOINT property is certainly an easy consequence of this weaker property.

Date: February 21st, 2005. Theorem 3.1. Let F : x ⊆ R n -→ P(R m ), m > 0, n > 0, be a ma-continuous set-valued function. Given i ∈ [1..m] and j ∈ [1..n], suppose that there exists an interval x ⊆ int x j such that ∀x ∈ x , F i (x) ⊆ x (1)
where F i (x) stands for the projection of f (x) onto the i-axis. Then the function G :

x =j ⊂ R n-1 -→ P(R m ) defined by G(x =j ) = {y ∈ F (x) | y i = x j }
is also ma-continuous.

Sketch of the proof. First, suppose that n > 1. Consider the C ∞ function h : R -→ x j defined like illustrated in the figure 1 -it can be formally constructed using the function λ defined in Hirsch [START_REF] Hirsch | Differential Topology[END_REF] page 42. As we can see, h satisfies both

y ∈ x =⇒ h(y) = y (2) and x = h(y) + ǫ ∧ |ǫ| ≤ α =⇒ x ∈ x j (3)
where α = 1 2 min{|x j -x|, |x j -x|} -we have α > 0 because by hypothesis x ⊆ int x j . Now define the following functions.

h : R n+m -→ R ; h(x, y) = h(y i ) -x j and h (k) : M (k) -→ R ; h (k) (x) = h(g (k) (x))
They are both C ∞ because they are both composed of C ∞ functions. The Morse-Sard theorem -theorem 1.3 page 69 of Hirsch [START_REF] Hirsch | Differential Topology[END_REF] -proves that critical values of h (k) and the critical values of h (k) | ∂M (k) have measure zero. So, we can pickup a sequence ǫ (k) ∈ R, 0 ≤ |ǫ (k) | < α, which converges to zero such that ǫ (k) is a regular value of both h (k) and h (k) | ∂M (k) . Now define the sequence N (k) ⊆ M (k) by

N (k) = {x ∈ M (k) |h (k) (x) = ǫ (k) }
By the regular value theorem 4.1 page 31 of Hirsch

[1] 1 , N (k) is a compact (n -1)- submanifold of M (k) which satisfies ∂N (k) = N (k) ∩ ∂M (k) . We define g ′(k) : N (k) -→ R n-1 × R m by g ′(k) (x) = (x =j , y)
T where (x, y) T = g (k) (x). We now prove that the sequences (N (k) ) k∈N and (g ′(k) ) k∈N satisfies the definition of macontinuity for the function G. First, notice that g ′(k) is a C ∞ diffeomorphism between ∂N (k) and ∂d ′ × {0} where d ′ ⊆ R n-1 is a disk which contains x =j -somehow because x ⊆ int d and by definition of h, we have h(x, 0) = x jc = 0 with c ∈ x j , so the (n -1)-disk d ′ is the intersection of the n-disk d and the hyperplane x jc = 0. Second, consider any sequence x(k) ∈ N (k) . We denote

g (k) (x (k) ) by (x (k) , y (k) ) T so g ′(k) (x (k) ) = (x (k) =j , y (k) ) T
. By definition of ma-continuous functions we know that the sequence g (k) (x (k) ) is bounded so the sequence g ′(k) (x (k) ) is also bounded. On the other hand, any accumulation point (x * =j , y * ) of the sequence (x (k) =j , y (k) ) T corresponds to at least one accumulation point (x * , y * ) of the sequence (x (k) , y (k) ) T . 1 The dimension and compactness of the resulting manifolds are not explicitly written in the regular value theorem 4.1. However, they are easily proved to hold. Consider any accumulation point (x * =j , y * ) of (x

(k) =j , y (k)
) T and one corresponding accumulation point (x * , y * ) of the sequence (x (k) , y (k) ) T . We suppose that g ′(k) (x (k) ) ∈ x =j × R m for all k ∈ N and we have to prove that y * ∈ G(x * =j ). We have

x (k) =j ∈ x =j . Now, by definition of N (k) , we have h (k) (x (k) ) = ǫ (k) , that is h(x (k) , y (k) ) = ǫ (k) , that is h(y (k) i ) = x (k) j + ǫ k . This entails ∀k ∈ N, x (k) j
∈ x j because ǫ k < α and thanks to the equation (3). Therefore, we have x (k) ∈ x and therefore ∀k ∈ N,

g (k) (x (k) ) ∈ x × R m
So, by definition of ma-continuity, the accumulation point (x * , y * ) of the sequence (x (k) , y (k) ) T satisfies x * ∈ x because x is closed, y * ∈ F (x * ) by definition of macontinuity, and y * i ∈ x by [START_REF] Hirsch | Differential Topology[END_REF]. Furthermore, because h is continuous and because h(x (k) , y (k) ) = ǫ (k) for all k ∈ N, the accumulation point (x * , y * ) also satisfies h(x * , y * ) = 0, that is y * i = x * j because y * i ∈ x. Therefore, we have finally y * ∈ G(x * =j ). The case n = 1 is different. It is a consequence of the fact that a compact 1-manifold with boundary S 0 , i.e. two points, is homeomorph to a segment. Then the proof should be achived thank to the intermediate value theorem.

Continuous functions

Finally, the next proposition proves that continuous functions are ma-continuous. Therefore, f can be approximated by C ∞ functions f (k) : d -→ R m which satisfy f (k) (∂d) = 0 -check Hirsch [START_REF] Hirsch | Differential Topology[END_REF] section 2 for approximation theorems. Then M (k) = D n and g (k) = f (k) • i -where i : D n -→ d × {0} is the canonical C ∞ embedding of D n into R n × {0} -satisfy the hypothesis of the definition of ma-continuity.

Proposition 4.1. Let f : x ⊆ R n -→ R m be a continuous function. Then, F : x ⊆ R n -→ R m defined by F (x) = {f (x)} is ma-continuous.

Conclusion

The property COMP seems to be impossible to achieved in a direct way. However, PROD and PROJ seems to be possible to achieve, and the COMP would then be a consequence of these latter.

Figure 1 .

 1 Figure 1.

  Sketch of the proof. Consider a disk d ⊆ R n such that x ⊆ int d. Define the continuous function f : d -→ R m in such a way that f | x = f and f (∂d) = 0 -such a continuous function certainly exists. As f | ∂d is constant equal to zero, it is C ∞ .