Hui Li

Alain Fourmigue

Sébastien Le Beux
email: sebastien.le-beux@ec-lyon.fr

Xavier Letartre

Ian O'connor

Gabriela Nicolescu

Thermal Aware Design Method for VCSEL-based On-Chip Optical Interconnect

Keywords: ONoC, design methodology, thermal simulation. I

Optical Network-on-Chip (ONoC) is an emerging technology considered as one of the key solutions for future generation on-chip interconnects. However, silicon photonic devices in ONoC are highly sensitive to temperature variation, which leads to a lower efficiency of Vertical-Cavity Surface-Emitting Lasers (VCSELs), a resonant wavelength shift of Microring Resonators (MR), and results in a lower Signal to Noise Ratio (SNR). In this paper, we propose a methodology enabling thermal-aware design for optical interconnects relying on CMOS-compatible VCSEL. Thermal simulations allow designing ONoC interfaces with low gradient temperature and analytical models allow evaluating the SNR.

INTRODUCTION

Technology scaling down to the ultra-deep submicron domain provides for billions of transistors on chip, enabling the integration of hundreds of cores. Many-core designs, integrating interconnect that can support low latency and high data bandwidth, are increasingly required in modern embedded systems to address the increasingly stringent power and performance constraints of embedded applications. Designing such systems using traditional electrical interconnect presents a significant challenge: due to capacitive and inductive coupling [START_REF] Ho | The Future of Wires[END_REF], both interconnect noise and propagation delay of global interconnect increase. The increase in propagation delay requires global interconnect to be clocked at a low rate, which limits the achievable bandwidth and system performance.

In this context, Optical Network-on-Chip (ONoC) is an emerging technology considered as one of the key solutions for future generations of on-chip interconnects. It relies on optical waveguides to carry optical signals, so as to replace electrical interconnect, and provide the low latency and high bandwidth characteristic of optical interconnect. Among the proposed ONoCs, the wavelength-routing based interconnect solutions are of considerable interest among the major players in the field, since they do not require any arbitration [START_REF] O'connor | Reduction Methods for Adapting Optical Network on Chip Topologies to Specific Routing Applications[END_REF][2] [START_REF] Ramini | Contrasting wavelength-routed optical NoC topologies for power-efficient 3D-stacked multicore processors using physical-layer analysis[END_REF] to propagate the optical signals. They rely on passive Microring Resonators (MRs) that filter the optical signals based on their resonant wavelengths. However, silicon photonic devices are highly sensitive to temperature variation, which leads to a drift of the MRs resonant wavelength, and consequently a lower Signal to Noise Ratio (SNR). Furthermore, the power efficiency of integrated laser sources decreases at higher temperatures, which further decreases the interconnect power efficiency. Among the available laser sources candidates, Vertical-Cavity Surface-Emitting Laser (VCSEL) compatible to CMOS are of high of interest despite the use of less mature technologies (they usually require the inclusion of III-V semiconductors). Indeed CMOS-compatible VCSELs allow direct modulation of the optical signals and can thus be dedicated to a communication channel. Their size is of the same order of magnitude as the size of a MR used to modulate continuous waves emitted by shared lasers. VCSELs are thus sufficiently compact to be implemented in a large number and at any position, which leads to the following key advantages. First, the integration is easier since layout constraints are relaxed. Second, power saving is expected since waveguide lengths are reduced and waveguide crossings are avoided. Third, higher scalability is obtained since the laser sources are fully distributed.

For the first time, this paper proposes a thermal aware methodology to design ONoC relying on fully distributed CMOS-compatible VCSELs. The methodology relies on i) steady-state thermal simulations and ii) SNR analyses taking into consideration the temperature of the VCSEL and the MRs. Design space exploration on MR heating power and laser current modulation allow reducing ONoC gradient temperature while keeping acceptable the VCSEL temperature. SNR analyses allows estimating the ONoC reliability and power efficiency under a given chip activity.

The paper is structured as follows. Section II presents the related work. Then, the considered architecture models and laser sources are described in Section III. Section IV details the design methodology and Section V presents the case study and gives results. Section VI concludes the paper.

II. RELATED WORK

Reducing the thermal sensitivity of optical communication has been addressed at both device and system levels. At device level, solutions relying on athermal devices [START_REF] Djordjevic | CMOS-compatible, athermal silicon ring modulators clad with titanium dioxide[END_REF], voltage tuning [START_REF] Manipatruni | Wide temperature range operation of micrometer-scale silicon electro-optic modulators[END_REF], local heating [START_REF] Biberman | Thermally Active 4x4 Non-Blocking Switch for Networkson-Chip[END_REF], and feed-back control schemes [START_REF] Padmaraju | Thermal stabilization of a microring modulator using feedback control[END_REF] have been explored to limit the thermal impact on or control the resonant wavelength of MRs. In [START_REF] Ye | System-Level Modeling and Analysis of Thermal Effects in Optical Networks-on-Chip[END_REF], system level analyses allow evaluating the influence of temperature variation on the optical signal power received by the photodetectors. In order to counter-balance the temperature variation, communication channels can be remapped through ONoC reconfiguration [START_REF] Zhang | Power-Efficient Calibration and Reconfiguration for Optical Network-on-Chip[END_REF] and DVFS and workload migration techniques can be applied [START_REF] Li | Reliability Modeling and Management of Nanophotonic On-Chip Networks[END_REF]. These techniques depend on redundant resources to re-map the wavelength channel or detect run-time temperature variation.

In [START_REF] Zhang | Thermal management of Manycore Systems with Silicon-Photonic Networks[END_REF], the authors proposed a thermally-aware job allocation policy to minimize the temperature gradients among MRs. This work relies on device characteristics such as the Free Spectral Range (FSR), the number of waveguides and wavelengths. In our work, we evaluate the influence of the temperature variation on the SNR, taking into account the actual efficiency of the VCSEL under a given chip activity. In [START_REF] Chen | Sharing and Placement of On-chip Laser Sources in Silicon-Photonic NoCs[END_REF], the authors explore the placement of shared on-chip lasers on a layer located on top of the optical interconnect. In our work, we focus on CMOS-compatible VCSEL (allowing dedicated communication channel assignment) distributed on a layer dedicated to optical interconnects.

III. 3D ARCHITECTURE

This section presents the architecture including ONoC. The challenges addressed in this paper are then introduced.

A. Architecture Overview

Figure 1-a illustrates the MPSoC architecture we consider. It is composed of i) an electrical layer implementing processors (in tiles) and memories and ii) an optical layer dedicated to the implementation of a ring-based ONoC. The activity of the processors leads to local and global communications that are implemented with electrical interconnect and ONoC respectively. The communication hierarchy is defined at design time and depends mainly on the number of processors and the ONoC complexity and bandwidth. The silicon photonic fabrication process is compatible with the CMOS one allows integrating VCSEL, waveguides, MRs and photodetectors. These devices are gathered into so-called Optical Network Interface (ONI), which are responsible for emitting the light, modulating optical signals with the data to be transmitted, transporting the modulated signals and receiving them on the destination side (Figure 1-c). VCSEL and photodetectors are respectively connected to CMOS drivers and CMOS receiver through TSVs (Figure 1-c). We choose ORNoC [START_REF] Beux | Layout Guidelines for 3D Architectures including Optical Ring Network-on-Chip (ORNoC)[END_REF] for the implementation of the optical communications. ORNoC is a ring-based network allowing point-to-point communications between source and destination, with passive MRs. As reported in [START_REF] Beux | Optical Crossbars on Chip, A Comparative Study based on Worst-Case Losses[END_REF], ORNoC demonstrates reduced worst-case and average insertion losses compared with related optical crossbars including Matrix [START_REF] Bianco | Optical Interconnection Networks based on Microring Resonators[END_REF], -router [START_REF] O'connor | Reduction Methods for Adapting Optical Network on Chip Topologies to Specific Routing Applications[END_REF] and Snake [START_REF] Ramini | Contrasting wavelength-routed optical NoC topologies for power-efficient 3D-stacked multicore processors using physical-layer analysis[END_REF] (e.g., on average, 42.5% reduction for worst-case and 38% for average in 4x4 scale), which is a significant advantage to reduce the laser power consumption.

B. ONoC Interface and Thermal Sensitivity

The ONIs are responsible for emitting, transmitting and receiving the optical signals on the optical layer, as illustrated in Figure 1-c. For this purpose, the signals are injected into waveguides and modulated with VCSELs. The signals then propagate on the waveguide, crossing the intermediate ONIs.

Once they reach their destination, the signals are dropped from the waveguide with passive MRs toward large band photodetectors. For proper filtering operation, it is mandatory that resonant wavelengths of destination MRs are well aligned with the wavelength of the emitted signals. However, silicon based optical devices are sensitive to temperature variation (0.1nm/°C typically), which leads to a reduced SNR due to reduced coupling of signals into MRs (i.e. lower signal power and higher crosstalk noise on photodetector).

Device-level calibration processes [START_REF] Li | Reliability Modeling and Management of Nanophotonic On-Chip Networks[END_REF] help improving the SNR by aligning the resonant wavelength. However, such technique suffers from significant power consumption overhead: voltage tuning and heat tuning of MRs (that allow blue shift and red shift of the resonant wavelength respectively) lead to 130uW/nm and 190uW/nm respectively, as reported in [START_REF] Ahn | Devices and architectures for photonic chip-scale integration[END_REF]. For large-scale ONoCs (e.g. Corona [START_REF] Ahn | Devices and architectures for photonic chip-scale integration[END_REF] including approximately 1.1×10 6 MRs), the power dedicated to the calibration process represents more than 50% of the total network power. Since the calibration comes with performances overhead due to algorithm execution and heating latency, they are generally coupled to MRs clustering techniques. Indeed, clustering the MRs helps reducing the algorithm complexity by assuming a same local temperature among MRs close enough. However, this technique requires careful design of the ONIs to ensure a homogeneous temperature under different processing activities.

Maintaining low gradient temperature within an ONI including on-chip lasers is a challenging task since they dissipate a relatively high power. Heating MRs to reduce the gradient temperature is thus mandatory, which can be obtained by using a resistance on top of each MR. In addition, alternatively placing VCSEL and MRs contributes to reduce MRs heating power through a better initial distribution of the heat generated by VCSELs. This assumption leads to the chessboard-like layout illustrated in Figure 1

C. CMOS-Compatible VCSEL

VCSEL-based lasers [START_REF] Sciancalepore | Modal, and Polarization Features of Double Photonic Crystal Vertical-Cavity Surface-Emitting Lasers[END_REF][8] offer a direct emission of data through current modulation. While the fabrication processes of CMOS-compatible VCSEL is less mature than those of microdisk lasers [START_REF] Van Campenhout | Electrically pumped InP-based microdisk lasers integrated with a nanophotonic silicon-on-insulator waveguide circuit[END_REF], they offer significant advantage in terms of scalability (higher laser output power is achievable) and spectral density due to their small 3dB bandwidth (typically 0.1nm). The drawback of on-chip lasers over off-chip counterpart is their intrinsically lower efficiency and their higher sensitivity to the chip activity variation since they are located above the processing layer. More precisely, each VCSEL is located above a CMOS driver that converts an input electrical data coming from an IP core (represented as a binary voltage) into a current, as illustrated in Figure 2-a. The current propagates through a TSV and directly modulates the VCSEL. An optical signal is vertically emitted and is redirected to a horizontal waveguide through a taper. The power of the optical signal injected into the network (OP net ) thus depends on i) the intensity of the modulation current I VCSEL , ii) the laser efficiency (η laser ) and the taper coupling efficiency (η coupling , assumed to be 70%). The VCSEL efficiency is highly sensitive to its temperature: it can drop from 15% at 40°C to 4% at 60°C. This rather low efficiency leads to a high dissipated power (P VCSEL ) which, together with the power dissipated by the CMOS driver (P driver ) and chip (P chip ), influences the onchip laser temperature. Hence, for a given modulation current, the power of the emitted optical signal (OP VCSEL ) depends on the laser temperature, which is influenced by P chip , P VCSEL and P driver , as illustrated in Figure 2-b. Furthermore, for an increasing activity of the processing layer (which is expected to result in additional communications), either the optical interconnect bandwidth will decrease assuming a same modulation current (the SNR being lower, data will be reemitted) or the optical interconnect power consumption will increase (a higher modulation current is required to compensate the reduced efficiency). The modulation current thus must be carefully selected since i) a too small value will lead to low SNR and ii) a too high value will lead to a power hungry solution.

D. Contribution

The gradient temperature and average temperature in ONI are critical to design VCSEL-based optical interconnect: i) low gradient temperature within an ONI eases the run-time calibration process and reduces the design complexity; ii) low ONI average temperature helps maintaining a reasonable power efficiency of the VCSEL.

For the first time, we propose a temperature aware design methodology allowing efficient use of CMOS-compatible VCSEL. Temperature evaluation and system level analyses allow estimating the ONoC SNR.

IV. PROPOSED DESIGN METHODOLOGY

In this section, we describe the methodology allowing to design thermal-aware optical interconnect.

A. Design Methodology Overview

Figure 3 illustrates the design methodology. System level inputs include information on the packaging (heat sink base and fins, fan, etc.) and the considered architecture (die size, location of the ONIs, distance between the optical layer and the electrical layer, material used for each layer, etc.). The ONIs are more accurately described since it is possible to specify the number and the type of each device (VCSELs, MRs/heater, TSVs and photodetectors), their size and their relative location. Key parameters such as the I VCSEL and the MR heater power (P heater ) are specified by the user. The VCSEL electrical characteristics are fetched from a library and different activities can be considered to simulate the power dissipated by the processing layer (e.g. uniform and diagonal). Steady-state thermal simulations are achieved under a given chip activity. The result is a thermal map giving the gradient temperature and the average temperatures of each ONI. Reduction of the gradient temperature is obtained by exploring P heater values. SNR analyses are achieved to evaluate the optical interconnect reliability by considering the actual temperature of each ONI. The design space can be further explored by considering different I VCSEL values.

B. System Specification

In order to perform thermal evaluations, our architecture model is based on the real physical structure of the system. The different components of the system (i.e. package, die, heat sources, and optical devices) are represented as rectangular blocks, defined by their dimension, their position, and a constitutive material. The blocks can be assigned to power values, which allow modeling the heat sources of the system. The Back-End-Of-Line (BEOL) is modeled as a thin layer (10µm) and the heat sources (cores, cache, router, etc.) are represented as rectangular blocks with power values, situated in the BEOL layer. IcTherm 1 [START_REF] Fourmigue | Efficient Transient Thermal Simulation of 3D ICs with Liquid-Cooling and Through Silicon Vias[END_REF]is a thermal simulator for electronic devices which accurately models their complex structure and provides 3D full-chip temperature maps. IcTherm solves the physical equations that govern the temperature in the chip, using the Finite Volume Method [START_REF] Chapra | Numerical Methods for Engineers[END_REF], a numerical method for solving partial differential equations. IcTherm was validated against the commercial simulator COMSOL [START_REF]COMSOL[END_REF]: its maximal error was found to be less than 1% [START_REF] Fourmigue | Efficient Transient Thermal Simulation of 3D ICs with Liquid-Cooling and Through Silicon Vias[END_REF]. The structure of the system is discretized into small cubic cells that match the distribution 1 IcTherm website : http://www.ictherm.com/

MPSoC activity

Uniform, diagonal, random, benchmark of the materials and the heat sources. Figure 4 illustrates the discretization of a section of the system. Because the interfaces contain micro-scale components (e.g. TSVs, VCSELs and CMOS drivers), we use a fine-grain resolution with a cell size of 5 µm x 5 µm for meshing the region containing the interfaces. For the rest of the system, we use a coarser resolution with a cell size of 100 µm x 100 µm for the heat sources and 500 µm x 500 µm for the package.

C. SNR Analysis

For a given activity scenario, the thermal map gives the temperature of the lasers and the MRs, from which the gradient temperature of each ONI is extracted. We assume that the gradient temperature within an ONI must remain below 1°C (i.e. since we consider MRs with 1.55nm 3dB bandwidth, 0.1nm drift of their resonant wavelength corresponds to 6.5% transmission loss). Design space on the MRs heater power can be explored in order to satisfy the 1°C gradient temperature constraint inside each ONI. However, the temperature gradient among ONIs influences the SNR, as detailed in the following.

Figure 5-a illustrates the transmission of an optical signal OP in at wavelength λ signal into an MR with a resonant wavelength λ MR . The 3dB bandwidth of the signal is assumed to be small compared to MR's one (0.1nm and 1.55nm respectively). The level of the signal power at the through port (OP through ) and drop port (OP drop ) depends on the alignment between λ signal and λ MR , (Figure 5-b). A maximum transmission to the drop port is obtained when λ signal =λ MR . In case both wavelengths are significantly different (above 1.5nm), most of the input signal power continue propagating along the waveguide to the through port. The misalignment of the wavelengths may be a required from proper ONoC operation (for wavelength routed routing) or it can be a side effect related to a temperature difference between the ONIs. By assuming λ signal equals to λ MR for a same laser source and MR temperature, 50% of the signal will be (wrongly) dropped from the waveguide for a 7.7°C temperature difference (0.77nm misalignment). This will lead to crosstalk and signal attenuation, which is evaluated with an analytical model. The losses encountered by the signal C s  d depend on the waveguide length and the crossed MRs (in R i  j ) in the intermediate ONI J . In case signal wavelength λ of C s  d (denoted λS s  d ) is far enough from the MR resonant wavelength λR i  j (in R i  j ), the losses are expected to be small. This optimistic scenario occurs if the temperature of the intermediate ONI (ONI J in the figure) is strictly equal to the temperature of the source ONI. A temperature difference among different ONIs will lead to a misalignment of the wavelengths: part of the signal power will be ejected from the waveguide and will reach a wrong photodetector. This leads to additional crosstalk and reduced signal power on intermediate and destination photodetectors respectively (e.g. X s-d [j  d] and OP s-d [s  d] in the figure). The losses are formulated as follows:

X i  j [s  d]=OP in,i  j [s  d]×∆λ i  j [s  d] Where X i  j [s  d] is the signal power dropped to R i  j , OP in,i  j [s  d] is the power of the signal at the R i  j , ∆λ i  j [s  d] is the signal ratio dropped by MR (in R i  j , ∆λ k  0 [s  d]=∆λ k  N [s  d])
. N is the number of ONIs in the optical interconnect. L k is the propagation loss (in %) along a communication path (e.g. L 1 for C 1  2 and L 2 for C 2  3 ), l k is the corresponding length of the waveguide and L propagation is the propagation loss (in % per cm).

The crosstalk noise received by each photodetector is estimated by considering the total signal power dropped due to wavelengths misalignment and is formulated as follows:

X s  d [j  d]=OP in,s  d [j  d]× ∆λ s-d [j  d]
T sd R ds T sj R js T si R is ONI J ONI D T jd R dj T ji R ij T js R sj R sd T ds R id T di R jd T dj T sd : transmitter for ONIS ONID communication (Csd) R sd : receiver for ONIS  ONID communication (Csd) OP in,i-j [sd] X i-j [sd] OP s-d [sd] X s-d [jd]

OP in,s-d [jd]

]) [

L d s OP d s OP N m k j m if N m k N j m if N m k i k s j if N j s j if j s m N k s j if N j s j if j s k net j i in                                k l n propagatio k L L  1 ( ] [ ] [ mod , mod , , mod , 1 1 , , 1 mod , 1 , 1 , d s
given chip activity. This crucial information allows the exploration of the design space and particularly the driver power consumption. Indeed, P driver is directly related to the laser modulation current and, therefore, it impacts the laser efficiency and the optical signal power. Such exploration is illustrated in the following section.

V. RESULTS

We first detail the considered architecture and VCSEL model. Then, MR heater power is explored through thermal simulations in order to reduce the gradient temperature. Finally, ONoC solutions are compared according to the SNR.

A. Case Study

The targeted system used in our experiments is based on Intel's Single-Chip Cloud Computer (SCC) [START_REF] Howard | A 48-core IA-32 processor in 45 nm CMOS using on-die messagepassing and DVFS for performance and power scaling[END_REF], a 24-tile, 48core IA-32 45nm processor with a maximum power dissipation of 125W. Given the large number of cores, the SCC is a good candidate for silicon-photonic communications. We model the same package as the one used by Intel [START_REF] Howard | A 48-core IA-32 processor in 45 nm CMOS using on-die messagepassing and DVFS for performance and power scaling[END_REF]. Figure 7 shows the assembly view of the targeted system, which contains the following components: steel back-plate, motherboard, socket, SCC chip with silicon-photonic links and on-chip laser sources, copper lid and heat sink. We assume ORNoC relying on 4 waveguides and 4 lasers per waveguide per ONI. The CMOS drivers and receivers are placed on an empty area of the electrical router of SCC tile. We consider a VCSEL with a 15x30µm² footprint size [7][8]. It relies on mirrors redirecting the vertically generated light into to horizontal waveguide, which allow reducing the thickness of the laser below 4µm (i.e. thus making the VCSEL compatible to CMOS). The direct modulation bandwidth is 12GHz and the 3dB bandwidth is about 0.1nm. Figure 8-a represents a 3D view of the laser extracted from [START_REF] Sciancalepore | Modal, and Polarization Features of Double Photonic Crystal Vertical-Cavity Surface-Emitting Lasers[END_REF]: it is composed of 3 layers of III-V material (0.6µm, 0.45µm and 0.4µm thickness respectively). The laser effect is generated in the central active layer (in red color) and the power is dissipated in the adjacent layers (in grey color). Two contacts allow driving the current from the CMOS layer through 5µm-diameter TSVs. The active layers are surrounded by Si/SiO 2 line constituting the mirror structure, which allows coupling the light into the horizontal waveguide (optical signal shown in blue arrow) using a tapper with an estimated 70% efficiency. Figure 8-b gives the laser efficiency according to its temperature and I VCSEL . Figure 8-c gives the influence of P VCSEL and its temperature on the actual emitted light OP VCSEL .

B. Reduction of the ONI Gradient Temperature

We first evaluate the influence of P chip and P VCSEL on the ONI average and gradient temperature. We run thermal simulations under homogeneous 12.5W, 18.75W, 25W and 31.25W chip activities and we explored P VCSEL ranging from 0 to 6mW (we assumed P VCSEL =P driver , which corresponds to the worst case scenario since the total energy received by the VCSEL is dissipated as heat). Figure 9-a illustrates the average temperature results: a 6W increase of the total chip activity roughly leads to +3.3°C on the average temperature while a 6mW increase of P VCSEL leads to +11°C. This demonstrates how important the calibration of the laser modulation current according to the requirements is: a slightly over-sized current will lead to significant power consumption overhead. Results also show a significant impact of P VCSEL on the gradient temperature between lasers and MRs (1.7°C/mW). Such gradient temperature does not realistically allow using clustering technique for run-time calibration process. We thus explore MR heating power values (P heater ) to reduce this gradient at design time. As illustrated in Figure 9-b, the smallest gradient is obtained for P heater = 0.3xP VCSEL . In Figure 10, we compare the temperature results with and without MR heater: for P VCSEL =1mW, solutions with and without heater lead to 0.3°C and 1°C gradient temperature respectively. Significant improvement of the gradient temperature is obtained for higher P VCSEL values: for 6mW, the gradient temperature drops from 5.8°C to 1.3°C (i.e. -4.5°C), which is significant compared to the reasonable 0.8°C increase of the average laser temperature. As illustrated in Figure 11, we consider 3 scenarios leading to 18mm (in red), 32.4mm (in blue) and 46.8mm (in green) waveguide lengths. P VCSEL and P heater are set to 3.6mW and 1.08mW respectively (simulations validated that the gradient temperature within each ONI remains below 1°C). OP VCSEL is estimated from the ONI average temperature and Figure 8-c. Table 1 summarizes the technological parameters we have considered and Figure 12 gives the worst-case SNR results.

Under a uniform activity, the asymmetric structure of the SCC chip leads to a 3°C difference among the ONIs for 46.8mm length. The crosstalk is relatively small and the SNR thus only depends on the signal power, which depends on the propagation losses. The SNR drops from 38dB for 18mm length to 13dB for 46.8mm. Under the diagonal activity, the upper-right and bottom-left parts of the chip dissipate each 4W while the upper-left and bottom-right parts dissipate 8W each. This leads to heterogeneous temperature of the ONIs (54.6255.92°C for case 1, 54.3356.92°C for case 2, 56.1660.85°C for case 3 respectively). Compared to the uniform activity, the diagonal activity exhibits lower average temperature since upper-right and bottom-left parts have lower power. This leads to higher laser efficiency, further resulting in higher OP VCSEL . However, the temperature gradient among the ONIs being higher, additional crosstalk occurs, which results in a lower SNR compared to the uniform activity. A random activity leads to intermediate SNR results. This analysis validates that the ONoC matches with the receiver sensitivity and SNR requirements. Further explorations of the design space allow optimizing the ONoC. For instance, in case a lower SNR is acceptable, P VCSEL and P heater can be reduced for energy saving.

VI. CONCLUSION

In this paper, we proposed a thermal aware design methodology for CMOS-compatible VCSEL-based ONoC. Thermal simulations allow exploring design parameters such as the laser modulation current and heater power in order to reduce the gradient temperature within an ONoC interface. A heater dissipating 30% of the VCSEL power leads to an optimal solution for the considered architecture. SNR analyses allow comparing ONoC under different chip activities.

Figure 1 :

 1 Figure 1: Considered 3D architecture: a) MPSoC with a stacked ONoC, b) ONI layout and c) implementation of an ONI and its operations.

  -b: 4 waveguides propagating signals in clockwise and counter-clockwise rotation are alternatively placed and, for each waveguide, 4 receivers and 4 transmitters are alternatively placed.

  Photodetector (1.5µm×15µm)

Figure 2 :

 2 Figure 2: The efficiency of a VCSEL depends on its temperature, which is influenced by the CMOS driver and the chip activity.

Figure 3 :

 3 Figure 3: Proposed methodology.

Figure 4 :

 4 Figure 4: IcTherm computes the heat transfers between the cells and outputs the temperature value of each cell. This thermal map allows computing the gradient temperature between any points of the system.

Figure 5 MR

 5 Figure 5 MR transmission model: a) MR structure, b) signal transmission according to the alignment between λsignal and λMR.

Figure 6

 6 Figure6illustrates a waveguide propagating optical signals injected and ejected by the ONIs. Since we consider ORNoC[START_REF] Beux | Layout Guidelines for 3D Architectures including Optical Ring Network-on-Chip (ORNoC)[END_REF], no arbitration is needed and passive MRs are used (i.e. the MRs resonant wavelength is defined at design time but it shifts with the temperature variation). A communication between a source interface ONI S and a destination interface ONI D (i.e. denoted as C s  d and represented by the red line) implies that an optical signal is 1) generated and modulated by a VCSEL (T s  d ) and 2) dropped to a photodetector by a MR (R s  d ). The quality of the communication C s  d depends upon the insertion loss along the path and the crosstalk induced by the other communications C i  j , where i≠s and j≠d (e.g. the blue line). The worst-case SNR for C s  d is formalized as follow:

Figure 6 :

 6 Figure 6: Signal propagation in a waveguide. The quality of a communication depends on the insertion losses and the crosstalk.

  SNR analysis allow estimating the ONoC reliability under a 0 λ MR -λ signal (nm)

Figure 7 :

 7 Figure 7: Packaging of the SCC chip and the optical interconnect.

Figure 8 :

 8 VCSEL: a) 3D view extracted from [7], b) efficiency and c) laser output power according to the temperature.

Figure 9 :

 9 a) Influence of PVCSEL and Pchip on the average temperature and b) influence of PVCSEL and Pheater on the gradient temperature.

Figure 10 :

 10 Figure 10: Average and gradient temperatures w/ and w/o MR heater.

Figure 11 :

 11 Figure 11: The location of the ONI leads to the following waveguide lengths: a) 18mm, b) 32.4m and c) 46.8mm.

Figure 12 :

 12 Figure 12: SNR results under uniform, diagonal and random activities

Table 1 : Technological parameters

 1

	Parameters	Value
	Wavelength range	1550nm
	BW3-dB	1.55 nm
	Phtodetector sensitivity	-20dBm (0.01mW)
	Thermal sensitivity	0.1nm/°C
	Lpropagation	0.5 dB/cm [3]

ACKNOWLEDGMENT

Hui LI is supported by China Scholarship Council (CSC).


Corentin Léna
email: clena@unito.it

Courant-sharp eigenvalues of a two-dimensional torus Valeurs propres Courant-strictes d'un tore bidimensionnel

Keywords: Nodal domains, Courant theorem, Pleijel theorem, torus. MSC classification. 35P05, 35P20, 58J50

published or not. The documents may come

Introduction

Let us first recall two classical results on the eigenvalues and eigenfunctions of the Dirichlet-Laplacian on a bounded domain Ω in the plane. According to a well-known result by R. Courant (in [START_REF] Courant | Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke[END_REF]), an eigenfunction (real valued) associated with the k-th eigenvalue λ k (Ω) of this operator has at most k nodal domains. In [START_REF] Pleijel | Remarks on Courant's nodal theorem[END_REF], Å. Pleijel sharpened this result by showing that, for a given domain, an eigenfunction associated with λ k (Ω) has less than k nodal domains, except for a finite number of indices k . This was generalized in [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF] by P. Bérard and D. Meyer to the case of a compact Riemannian manifold, with or without boundary, in any dimension. It has been shown by I. Polterovich in [START_REF] Polterovich | Pleijel's nodal theorem for free membranes[END_REF], using estimates from [START_REF] Toth | Counting nodal lines that touch the boundary of an analytic domain[END_REF], that the analogous result also holds for the Neumann-Laplacian on a planar domain with a piecewise-analytic boundary.

These results leave open the question of determining, for a specific domain or manifold, the cases of equality. It is stated in [START_REF] Pleijel | Remarks on Courant's nodal theorem[END_REF] that when Ω is a square, equality can only occur for eigenfunctions having one, two or four nodal domains, associated with the first, the second (which has multiplicity two), and the fourth eigenvalue respectively. The proof in [START_REF] Pleijel | Remarks on Courant's nodal theorem[END_REF] is however incomplete and was corrected by P. Bérard and B. Helffer in [START_REF] Bérard | Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses[END_REF]. The case of the sphere is treated in [START_REF] Helffer | On spectral minimal partitions: the case of the sphere. In Around the research of Vladimir Maz'ya. III[END_REF], see also [START_REF] Leydold | Knotenlinien und Knotengebiete von Eigenfunktionen[END_REF][START_REF] Leydold | On the number of nodal domains of spherical harmonics[END_REF] for a related study. The cases of an equilateral torus and an equilateral triangle are investigated in [START_REF] Bérard | Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle[END_REF], and the case of the Neumann-Laplacian in a square is treated in [START_REF] Helffer | Nodal domains in the square-the Neumann case[END_REF]. In this note, we will show that for the flat torus (R/Z) 2 , equality holds only for eigenfunctions having one or two nodal domains, respectively associated with the first and the second eigenvalue (this last eigenvalue has multiplicity four). This complements the result [9, Theorem 7.1], which determines the cases of equality for a flat torus (R/Z) × (R/bZ) with b 2 irrational.

Let us give a more precise statement of the above result, and fix some notation that will be used in the following. In the rest of this paper, T 2 stands for the two-dimensional torus (R/Z) 2 equipped with the standard flat metric, and -∆ T 2 stands for the Laplace-Beltrami operator on T 2 . If Ω is an open set in T 2 with a sufficiently regular boundary, we write (λ k (Ω)) k≥1 for the eigenvalues of -∆ T 2 in Ω with a Dirichlet boundary condition on ∂Ω , arranged in non-decreasing order and counted with multiplicity. In particular, λ k (T 2 ) is the k-th eigenvalue of -∆ T 2 (in that case the boundary is empty). If u is an eigenfunction of -∆ T 2 , we call nodal domains of u the connected components of T 2 \ u -1 ({0}) , and we denote by µ(u) the cardinal of the set of nodal domains. With any eigenvalue λ of -∆ T 2 , we associate the integer

ν(λ) = min{k ∈ N * : λ k (T 2 ) = λ} .
We say that an eigenvalue λ of -∆ T 2 is Courant-sharp if there exists an associated eigenfunction u such that µ(u) = ν(λ) . Following [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF], we also use the adjective Courant-sharp for such an eigenfunction u . We will prove the following result.

Theorem 1. The only Courant-sharp eigenvalues of -∆ T 2 are λ k (T 2 ) with k ∈ {1, 2, 3, 4, 5} .
The proof follows the approach used by Å. Pleijel in [START_REF] Pleijel | Remarks on Courant's nodal theorem[END_REF] and in the case of a compact manifold by P. Bérard and D. Meyer in [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF] (see also [START_REF] Bérard | Inégalités isopérimétriques et applications. Domaines nodaux des fonctions propres[END_REF][START_REF] Bérard | Volume des ensembles nodaux des fonctions propres du Laplacien[END_REF]). We first establish a Faber-Krahn type inequality for domains in T 2 whose area is sufficiently small. We deduce it from an isoperimetric inequality proved in [START_REF] Howards | The isoperimetric problem on surfaces[END_REF][START_REF] Chavel | Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives[END_REF]. We then combine this information with an explicit lower bound of the counting function (similar to Weyl's law) to show that large eigenvalues cannot be Courant-sharp.

Let us point out that interest in Courant-sharp eigenvalues has grown recently thanks to their connection to minimal partition problems. This appears clearly in the paper [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF], where the authors consider the following problem: given a two-dimensional domain Ω and an integer k , find a k-partition of Ω , that is to say a family (D i ) 1≤i≤k of k open, connected, and disjoint subsets of Ω , that minimizes the energy

max 1≤i≤k λ 1 (D i ) .
Such a k-partition is said to be minimal. Existence and regularity for minimal partitions are proved in [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF]. Following [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF], let us say that a k-partition is nodal if it is the family of the nodal domains for some eigenfunction of the Dirichlet-Laplacian on Ω . It is shown in [START_REF] Helffer | Nodal domains and spectral minimal partitions[END_REF] that a nodal partition is minimal if, and only if, the corresponding eigenfunction is Courant-sharp. Finding nodal minimal partitions is therefore equivalent to finding Courant-sharp eigenfunctions. In particular, Theorem 1 has the following consequence.

Corollary 2. If k ≥ 3 , minimal k-partitions of T 2 are not nodal.
The problem of finding minimal k-partitions of T 2 , with k ≥ 3 , is studied in [START_REF] Léna | Spectral minimal partitions for a family of tori[END_REF]. In this paper, a numerical method, based on [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF], is used to produce candidates to be minimal partitions for k ∈ {3, 4, 5}. They seem to be tillings of T 2 by hexagons or squares. Proposition 3. If Ω is an open set in T 2 such that |Ω| ≤ 1 π , we have the inequality

H 1 (∂Ω) 2 ≥ 4π|Ω| . (1)
In this proposition, |Ω| stands for the usual two-dimensional area measure of Ω and H 1 (Ω) for the one-dimensional Hausdorff measure of ∂Ω . This inequality is also proved in [START_REF] Léna | Courant-sharp eigenvalues of a two-dimensional torus[END_REF], by a more elementary method than in [START_REF] Howards | The isoperimetric problem on surfaces[END_REF].

Proposition 4. If Ω is an open set in T 2 such that |Ω| ≤ 1 π , then λ 1 (Ω)|Ω| ≥ πj 2 0,1 . (2)
The constant j 0,1 in Equation ( 2) is the first positive zero of the Bessel function of the first kind J 0 . Let us note that πj 2 0,1 is the value of the product λ 1 (D)|D| , when D is a disk in the euclidean plane R 2 . As in the planar case, the proof uses Schwarz symmetrization of the level sets Ω t = {x : u(x) > t} , where u is a positive eigenfunction associated with λ 1 (Ω) and t > 0 . We go through the same steps as in [5, I.9], or [START_REF] Chavel | Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives[END_REF]III.3]. Note that since |Ω t | ≤ |Ω| ≤ 1 π for all t > 0 , we can use Inequality (1).

Weyl's law with explicit bounds

For λ ≥ 0 , we define the counting function by N (λ) = ♯{k : λ k (T 2 ) ≤ λ}.

Proposition 5. We have, for all λ ≥ 0 ,

λ 4π - 2 √ λ π -3 ≤ N (λ) . (3)
Proof. The proof consists in counting lattice points contained in a planar region. Indeed, the eigenvalues of -∆ T 2 are of the form

λ m,n = 4π 2 (m 2 + n 2 ) , with (m, n) ∈ N 2 .
With each pair of integers (m, n) we associate a finite dimensional space E m,n of eigenfunctions such that

L 2 (T 2 ) = (m,n) E m,n .
The vector space E m,n is generated by products of trigonometric functions, see for instance the proof of [9, Theorem 2.2] for details. The dimension of E m,n is 1 if (m, n) = (0, 0) , 2 if either m or n, but not both, is 0 , and 4 if m > 0 and n > 0 . Let us denote by n(λ) the number of points with positive and integer coordinates contained in the disk of center 0 and radius √ λ 2π . Taking the dimension of the spaces E m,n into account, we have the following exact formula for the counting function:

N (λ) = 4n(λ) -4 √ λ 2π -3 , (4)
where ⌊x⌋ denotes the largest integer not greater than x . By covering the upper right-hand quarter of the disk of center 0 and radius √ λ 2π with squares of side 1 , we see that n(λ) ≥ λ 16π , and we obtain the desired lower bound.

Courant-sharp eigenvalues of the torus

We now turn to the proof of Theorem 1. We will use the following lemmas. Lemma 6. If λ is an eigenvalue of -∆ T 2 that has an associated eigenfunction u with k nodal domains, and if k ≥ 4 , πj 2 0,1 k ≤ λ .

λ = λ 1 (D) ≥ πj 2 0,1 |D| ≥ πj 2 0,1 k . Corollary 7. If λ is a Courant-sharp eigenvalue of -∆ T 2 with ν(λ) ≥ 4 , πj 2 0,1 ν(λ) ≤ λ . (5)
Lemma 8. For all all k ∈ N ,

λ k (T 2 ) ≤ 4 + 2 4 + π(k + 3) 2 . ( 6
)
Proof. The proof is immediate from the following remark: if λ is a non-negative number such that N (λ) ≥ k , then λ k (T 2 ) ≤ λ . The lower bound for N (λ) given in Inequality (3) then implies the desired upper bound.

Comparing the lower bound (5) with the upper bound (6), we can easily show that if λ is an eigenvalue of -∆ T 2 with ν(λ) ≥ 50 , it is not Courant-sharp. Table 1 gives the first fifty-seven eigenvalues of -∆ T 2 . It shows that we have to test Inequality (5) for λ = λ k (T 2 ) with k ∈ {6, 10, 14, 22, 26, 30, 38, 46} . Table 2 displays the ratio λ k (T 2 ) 4kπ 2 , which should be greater than j 2 0,1 4π ≃ 0.4602 in case λ k (T 2 ) is Courant-sharp. This does not happen in the cases considered, and therefore Theorem 1 is proved.

Table 1 :

 1 The first 57 eigenvalues (Les 57 premières valeurs propres)

			λ 4π 2 0	indices (0, 0)	multiplicity ν(λ) 1 1
			1	(1, 0), (0, 1)	4		2
			2	(1, 1)	4		6
			4	(2, 0), (0, 2)	4		10
			5	(2, 1), (1, 2)	8		14
			8	(2, 2)	4		22
			9	(3, 0), (0, 3)	4		26
			10 (3, 1), (1, 3)	8		30
			13 (3, 2), (2, 3)	8		38
			16 (4, 0), (0, 4)	4		46
			17 (4, 1), (1, 4)	8		50
	k	6	10	14	22	26	30	38	46
	λ k (T 2 ) 4kπ 2	0.3333 0.4000 0.3571 0.3636 0.3462 0.3333 0.3421 0.3478

Table 2 :

 2 Table of ratios (Tableau des rapports) Proof. Since |T 2 | = 1 , one of the nodal domains of u has an area no larger than 1 k . Let us denote this nodal domain by D . Since k ≥ 4 , |D| ≤ 1 k < 1 π . According to Proposition 4,

Acknowledgements I thank Bernard Helffer for introducing me to this problem and for numerous discussions and advices. I also thank Pierre Bérard for explaining to me the approach used in [5], pointing out the reference [13], and suggesting several improvements. I thank Virginie Bonnaillie-Noël for her corrections and Thibault Paolantini for his help with topological questions. I also thank the anonymous referee for their constructive comments. This work was partially supported by the ANR (Agence Nationale de la Recherche), project OPTIFORM n • ANR-12-BS01-0007-02.


Jean-Bernard Brissaud
email: jb.brissaud@gmail.com

MaxEnt Mechanics

Keywords: Maxent, Least-action principle, Lagrangian Mechanics. MSC 2000 codes: 94A17 (Measures of information, entropy) 70H03 (Lagrange's equations) PACS codes: Lagrangian and Hamiltonian mechanics, 45.20.Jj statistical mehanics, 05.20.y

Introduction

Physics is fundamentally divided in two parts. Mechanics, which includes classical, relativistic and quantum mechanics, is ruled by the least-action principle (LAP) and characterized by laws invariant by time reversal. Thermodynamics, on the other hand, can be entirely deduced from the maximum entropy principle (Maxent), and introduces a so-called "arrow of time". LAP is equivalent to Newton's law in classical mechanics, and can be used in relativistic and quantum mechanics with an adequate reformulation of the Lagrangian. It has been historically applied, from Fermat to Feynmann, with such success that today it is a fundamental principle of physics. It has never been, to our knowledge, deduced from more fundamental principles. On the other hand, Maxent is justified by the Bayesian rules of probability, and simply gives the most probable probability distribution according to given constraints. For instance, in equilibrium thermodynamics, Maxent proves that the average values of the extensive quantities used to describe the system (energy, volume, ...) are appropriate constraints. For an excellent summary of Maxent properties and their applications to thermodynamics, see [START_REF] Rc Dewar | Maximum entropy production and the fluctuation theorem[END_REF]. Since Maxent is a principle of logic, it can and has been successfully applied in many different fields, for example biology, economy or ecology [START_REF] Garbaczewski | INFORMATION -Is There Anything Where Nobody Looks[END_REF][3]. A mechanical system is defined by the values of a set of generalized coordinates at a given time. We make the hypothesis that these values, including time, are only known in average. Applying Maxent, we will logically deduce that the system will follow the Lagrangian laws of motion. LAP is a consequence of Maxent with appropriate constraints.

In the first section, a least information principle will be deduced from Maxent. This section does not involve physics and is entirely mathematical. In the second section, the least information principle will be applied to a mechanical system whose space and time coordinates are only known in average. The least information principle will appear to be, up to a multiplicative constant with dimension Action, the least action principle. A third section will be devoted to possible physical interpretations of this result, and to new questions that then naturally emerge.

The least information principle

In this section, a least information principle, similar in form to the least action principle, will be deduced from Maxent applied with linear constraints. Provided that time is one of the constrained quantities, the rate of information will satisfy the equations of Cauchy-Riemann.

The proof

Notations: we write a sequence (s i ) instead of (s 1 , s 2 , ..., s imax ). Two sequences using the same name of indice should have the same length. We write i instead of imax i=1 . A sequence using two indices is written (s i,j ) instead of (s 1,1 , ..., s 1,jmax , ..., s i,1 , ..., s i,jmax , ..., s imax,1 , ..., s imax,jmax ). Some numbers (A k,i ) are given once and for all, and will always be considered fixed. Some other numbers (A k ) are given and we wish to find the probability law (p i ) satisfying the constraints:

   i p i A k,i = A k f or all k i p i = 1 (1)
Following Jaynes, we will choose the probability law which maximizes the information:

I = - i p i ln p i
Using Lagrange multipliers (α k ) (see demonstration in Annex 1), we obtain the standard results:

dI = k α k dA k (2)
∂α k ∂A l = ∂α l ∂A k f or all k, l (3)
Adding another constraint on a new variable t that the reader can conveniently consider as time, the system (1) then becomes:

         i p i A k,i = A k f or all k i p i t i = t i p i = 1 (4)
Calling β the Lagrange multiplier of t, the results ( 2) and (3) become:

dI = k α k dA k + βdt ( 5
)
∂α k ∂A l = ∂α l ∂A k f or all k, l ∂α k ∂t = ∂β ∂A k (6)
The notation .

A k is an abbreviation for the exact differential dA k dt . Equation ( 5) can be written

dI = k α k . A k + β dt Let L((A k ), ( . A k ), t) = k α k . A k + β (7)
We have:

dI = Ldt (8)
We now prove our main result, which is that L satisfies the equations of Cauchy-Riemann. Since

∂ . A l ∂A k = 0 (but note that .
dA l dA k may be = 0), we have from equation ( 7)

∂L ∂A k = l ∂α l ∂A k . A l + ∂β ∂A k (9)
The (α k ) are only functions of the (A k ) and t, ∂α l

∂ . A k = ∂β ∂ . A k
= 0 for all k, l. So we have:

∂L ∂ . A k = α k (10)
and

d dt ∂L ∂ . A k = dα k dt = l ∂α k ∂A l . A l + ∂α k ∂t (11)
As a result of (6),

d dt ∂L ∂ . A k = l ∂α l ∂A k . A l + ∂β ∂A k
and, using (9), L satisfies the equations of Cauchy-Riemann:

d dt ∂L ∂ . A k = ∂L ∂A k f or all k (12)
As a consequence, the information

I = t 1 t 0 L(t)dt (13)
is stationary. For any variations δA k (t) such that δA k (t 0 ) = δA k (t 1 ) = 0 for all k, we have δI = 0.

By analogy with LAP, this consequence will be subsequently referred as the "least information principle".

Conservation laws

From equations ( 11) and (12), we obtain

dα k dt = ∂L ∂A k f or all k (14)
One can also prove that (see demonstration in Annex 2)

dβ dt = ∂L ∂t (15)
If L does not depend explicitly on a given A k , then the conjugate quantity α k does not vary with t. If L does not depend explicitly on time t, then the quantity β does not vary with t.

Degree of validity

Note that in this problem, there are no assumptions about the nature of the (A k ). In particular: -The (A k ) do not have to be frequency averages of the values (A k,i ) in an experiment. Neither do we need a notion of ensemble. We want to find a probability law which reflects our state of knowledge, not a property of some system.

-The (A k ) do not have to be extensive quantities (a thermodynamical concept not necessary for Maxent), neither do they need to scale together or have any other relationship.

Physical application

Let us consider a mechanical system whose state is defined by a set of independent coordinates (q k ). The motion of such a system can be described by a parameterized curve (q k (λ), t(λ)) in the ((q k ), t) space. We call such a curve a path. The system can potentially take many different paths from a given starting position (q k (0), t(0)). Let us denote by (i) the set of all these paths, and adopt, without loss of generality, a common parameter λ to describe all these paths. A given path i is then described by the k + 1 functions ((q k,i (λ)), t i (λ)).

We make the hypothesis that the observed path ((q k (λ)), t(λ)) is the average path of all the paths i, each one occurring with a probability p i . Mathematically:

         i p i q k,i (λ) = q k (λ) f or all k i p i t i (λ) = t(λ) i p i = 1
f or all λ

We now fix the parameter λ, and no longer write the dependence on λ. The Maxent distribution which satisfies the preceding constraints is the solution of system (4), with:

q k,i = A k,i q k = A k
Equations ( 7), ( 8), ( 14) and (15) become:

L((q k ), ( . q k ), t) = k α k . q k + β dI = k α k dq k + βdt = Ldt dα k dt = ∂L ∂q k f or all k dβ dt = ∂L ∂t (16)
We recognize the equations of Lagrangian mechanics: L being the Lagrangian, the (α k ) the generalized momentum, β the opposite of the Hamiltonian and I the action. But this can not be correct because, for instance, the dimension of action is Action = Energy × T ime while I is dimensionless. However, K being an appropriate constant of dimension Action, we recover all lagrangian mechanics with the following identifications:

K I = S action K α k = p k generalized momentum K β = -H -Hamiltonian K L = L Lagrangian which give L((q k ), ( . q k ), t) = k p k . q k -H dS = k p k dq k -Hdt = Ldt dp k dt = ∂L ∂q k f or all k d(-H) dt = ∂L ∂t (17)
The least information principle becomes the least action principle. The action

S = t 1 t 0 L(t)dt
is stationary. For any variations δA k (t) such that δA k (t 0 ) = δA k (t 1 ) = 0 for all k, we have δS = 0.

Equations ( 14) and (15) become:

dp k dt = ∂L ∂q k f or all k d(-H) dt = ∂L ∂t
which is Noether's theorem [START_REF]Wikipedia: Noether's theorem[END_REF]. The generalized momentum is conserved if L does not explicitly depend on the associated generalized coordinate. Energy is conserved if L does not explicitly depend on time.

The relation between the Lagrangian L and the Hamiltonian H appears naturally in the Maxent formalism. However, while one is the Legendre transform of the other, they do not play roles similar to I and ln(Z). In fact, the partition function Z does not play any particular role in our description (the properties of ln(Z) mirror the properties of I [START_REF] Rc Dewar | Maximum entropy production and the fluctuation theorem[END_REF]), and it is the presence of time which induces the relation between L and H.

One can also note that, fundamentally, L satisfies the equations of Cauchy-Riemann because dI is an exact differential.

In the case of conservative forces, LAP is equivalent to Newton's law. This means that Laplace's equations could have emerge without any hint of Newton, if the work of Jaynes (1922-1998) had been known. Maxent as a fundamental physical principle certainly has epistemological implications.

For a given physical path, there is a constant K with dimension Action such that

K I = S action K α k = p k generalized momentum K β = -H -Hamiltonian K L = L Lagrangian
But nothing a priori prevents the value of K to be different in different experiments. One can not calculate the values of I and of the (α k ) for a given physical path, since the (q k,i ) are a priori unknown. However, the (q k,i ) could eventually be known using quantum mechanics (QM). To simplify, let us state that there is one single coordinate x. In QM, a system is defined by its wave function, which is a function of space and time ϕ(x, t). We can identify a path i of probability p i with the set of all (x, t) such that |ϕ(x, t)| 2 = p i (of course, to be rigorous, the discrete set of paths (i) first has to be replaced by a continuous set). Since the paths i are known, the (x i (λ)) and (t i (λ)) are also known if we can parameterize all these paths with a common parameter λ. Deducing the value of K rests an open question.

Feynmann's path integral formulation of QM [START_REF]Wikipedia: Path integral formulation[END_REF] offers similarities with our description. However, the two theories also present fundamental differences. In particular, Feynmann assigns equal probabilities to all paths, and does not average the action S, but the quantity e iS/h . A possible link has to be investigated.

An analogy between mechanics and thermodynamics has already been found [START_REF] Gaies | Analogy between mechanics and thermodynamics[END_REF]. It uses the formalism of differential forms, but the main results can be obtain using Maxent. This analogy comes fundamentally from the fact that Lagrangian mechanics and equilibrium thermodynamics can both be described by a set of linear constraints as has been shown in this paper.

The least action principle (LAP) is a consequence of Maxent, provided that the constraints concern the average coordinates and the average time of a mechanical system. The simplicity of the demonstration and the high degree of generality of Maxent explain why LAP is so general in mechanics. This demonstration of LAP sheds a new light on the relationship between thermodynamics and mechanics. It offers an opportunity to unify these two branches of physics, with Maxent as a common basis.

Annex 1

This annex demonstrates classical results about Maxent distributions [START_REF] Jaynes | Probability Theory: The Logic of Science[END_REF]. Some numbers (A k,i ) are given once and for all, and will always be considered fixed. Some other numbers (A k ) are given and we wish to find the probability law (p i ) satisfying the constraints:

   i p i A k,i = A k f or all k i p i = 1 (18)
The Maxent principle consists of choosing the distribution (p i ) which maximizes the information

I = - i p i ln(p i )
Using the method of Lagrange multipliers, let

£ = I - k α k i p i A k,i -γ i p i
where (α k ) and γ are new variables called the Lagrange multipliers. The distribution (p i ) should satisfy

∂£ ∂p i = 0 = -ln(p i ) -1 - k α k A k,i -γ for all i Calling Z i = e - k α k A k,i Z = i Z i (19)
we obtain

p i = Z i Z and -ln(p i ) = k α k A k,i + ln(Z) Therefore I = i p i (-ln(p i )) = k α k A k + ln(Z) (20)
Differentiating (19), we obtain:

∂Z i ∂α k = -A k,i Z i = -A k,i p i Z ∂Z ∂α k = i ∂Z i ∂α k = -A k Z
that we write:

∂ ln(Z) ∂α k = -A k for all k (21)
The (α k ) can be found by solving the k max equations of this last system. ln(Z) is an exact differential and can be written:

d(ln(Z)) = - k A k dα k
Using equation (20), we can now find an expression for dI. Since the quantities (A k ) are independent, ∂A k ∂A l = δ k,l (δ k,l is Kronecker symbol) and:

∂I ∂A l = k ∂α k ∂A l A k + α l + ∂ ln Z ∂A l
Since Z is a function of the α k ,

∂ ln Z ∂A l = k ∂ ln Z ∂α k ∂α k ∂A l
and using (21):

∂ ln Z ∂A l = k -A k ∂α k ∂A l
We finally obtain:

∂I ∂A k = α k for all k
I is a function of the (A k ):

dI = k α k dA k
dI is an exact differential:

∂ 2 S ∂A k ∂A l = ∂ 2 S ∂A l ∂A k therefore: ∂α k ∂A l = ∂α l ∂A k f or all k, l Annex 2
We have, by definition of L (equation ( 7)):

β = L -
dβ dt = ∂L ∂t

  into account (14) and (10), we obtain (15):


Christos Papageorgakis

Sebastian Hitziger

Théodore Papadopoulo

Dictionary learning for M/EEG multidimensional data

BaCI Introduction

Signals obtained from magneto-or electroencephalography (M/EEG) are very noisy and inherently multidimensional, i.e. provide a vector of measurements at each single time instant. To cope with noise, researchers traditionally acquire measurements over multiple repetitions (trials) and average them to classify various patterns of activity. This is not optimal because of trial-to-trial variability (waveform variation, jitters). The jitter-adaptive dictionary learning method (JADL [START_REF] Hitziger | Jitter-adaptive dictionary learning-application to multi-trial neuroelectric signals[END_REF]) has been developed to better handle for this variability (with a particular emphasis on jitters). JADL is a data-driven method that learns a dictionary (prototype pieces) from a set of signals, but is currently limited to a single channel, which restricts its capacity to work with very noisy data such as M/EEG. We propose an extension to the jitter-adaptive dictionary learning method, that is able to handle multidimensional measurements such as M/EEG.

Jitter-adaptive dictionary learning model (JADL)

JADL is a dictionary learning framework:

Compensate for small variations in latency and phase of atoms d i .

Atoms learned by JADL are defined on the entire signal domain. Hypotheses:

The set of signals of interest {x j } M j=1 can be generated by a dictionary. Atoms present in a signal can suffer from unknown time delays (jitter).

x j = K i=1 a ij δ ij (d i ) + K : #atoms, M : #trials a ij ∈ R : coefficient δ ij ∈ ∆ : shift operator ∆ : finite set of allowed shifts d i : atom D = {d i } K i=1 : dictionary : Gaussian noise min d i ,a ij ,δ ij M j=1 1 2 x j - K i=1 a ij δ ij (d i ) 2 2 + λ a j 1 , s.t. d i 2 = 1.
The algorithm solving the JADL problem, is based on an implementation in [START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF] for common dictionary learning, which iteratively alternates between:

(i) Sparse coding: finding the coefficients {a ij } and the jitters {δ ij } Let an "unrolled" version of the dictionary D be a dictionary D S containing all allowed shifts (S = |∆|) of all its atoms:

The sparse coding problem is solved using a modification of least angle regression (LARS) [START_REF] Efron | Least angle regression The[END_REF] by rewriting the problem as follows:

Once an atom is chosen all its shifts are forbidden.

D S = {δ(d ) : d ∈ D, δ ∈ ∆}, a matrix of dimension N × KS. a S j ← argmin 1 2 x j -D S a S j 2 2 +λ a S j 1 , s.t. a S,i j 0 ≤ 1, i = 1, . . . , K .
(ii) Dictionary update: finding the shapes {d i }.

Block coordinate descent is used to iteratively solve the constrained minimization problem for each atom:

d k = argmin d k M j=1 1 2 x j - K i=1 a ij δ ij (d i ) 2 2 s.t. d k 2 = 1

Our modified JADL model

We propose an extension to the jitter-adaptive dictionary learning method, that: Is able to handle multidimensional measurements such as M/EEG. Learns a dictionary over M/EEG recordings that have the same waveform and jitter over all the channels in a single trial. Is still able to account for different jitters across trials.

Significant modifications are applied to the original JADL framework, especially in:

The solution of sparse coding problem (i) by least angle regression algorithm (LARS):

1. Atom Selection: The best shifted versions of the atoms contained in the extended dictionary D S are selected, over all the channels, leading to a compressed dictionary D S .

d S j = argmax d S j ∈D S C c=1 s c • d S j ,
where C is the number of channels of the EEG data, s c is the signal of channel c and d S j is the j-th atom of the extended dictionary D S .

2. Standard LARS sparse coding over the channels for the current atom set: During this step the multidimensional coefficients a ijc are computed using the compressed dictionary D S selected by the previous step and the multi-channel signals for the given trial.

The dictionary update problem (ii) is also slightly modified to treat the measurements corresponding to the different channels as additional trials.

Synthetic data generation

Create a dictionary of K = 3 synthetic atoms. Generate an extended dictionary of 9 signals:

Introducing random jitters (from the set ∆ of S = 103 contiguous allowed shifts) to the dictionary's atoms . Select 3 source groups, each of them containing 3 neighboring sources.

Each source group is associated to shifted versions of the same atom. Combine the generated signals with a lead field matrix G computed from real EEG measurements [START_REF] Henson | Effective Connectivity between OFA and FFA during face perception: DCM of evoked MEG, EEG and fMRI[END_REF].

M = G S ,
where M ∈ R C ×N is the measurement matrix either MEG or EEG, G ∈ R C ×Q is the lead field matrix, S ∈ R Q×N is the sources matrix. C , Q and N are the numbers of channels, sources and time samples respectively. Perform the above procedure for M trials:

Introducing new random jitters to the dictionary of K = 3 synthetic atoms. ⇒ Generated clean M/EEG measurements of C = 6 channels, M = 200 trials and N = 515 time samples.

Results on lead field synthetic data

A comparison between the original and our multi-dimensional JADL model

Both algorithms are executed with the same signals, initial random dictionary and latency parameters. The multi-channel algorithm is executed using all the channels from the input data, while the single-channel algorithm is executed several times, each time using a different channel.

The results of our multi-channel algorithm show: A very good fit of the learned dictionary to the generated one.

A good fit also in the case where the signals were contaminated by noise. The comparison of our multi-channel approach with the single-channel algorithm showed: Similar results when the best channel is used by the single-channel algorithm.

Worse results for the single-channel algorithm when a medium or the worse channel is used.

Cases where the single-channel algorithm is unable to recover correctly all the atoms of the dictionary used to generate the signals.

A small but superior performance for the multi-channel approach based on the coefficients vectors obtained by the goodness of fit metric: 0.995, 0.996 and 0.995 instead of 0.992, 0.977 and 0.964 for the singlechannel approach using the best channel and 0.939, 0.512, 0.512 using the worst channel.

Goodness of fit metric:

cor i = max j,τ |a i a j τ | ,
where a i is a generated atom, a j is a learned atom and

a

Results on real data

The multi-dimensional approach is tested using real MEG and EEG data: Input parameters: S = 103 contiguous allowed shifts, K = 3 atoms. Figure 3 : The single-channel (left) and the multi-channel method (right).

Conclusions

The method shows superior performance and less noisy estimated waveforms compared to the original single-channel JADL framework, both on synthetic and real data.

It is more robust to various levels of noise.

Using the JADL framework allows one to deal with signal variabilities such as jitters which is difficult to do with standard methods such as PCA or ICA. Not having to select a "best" channel (as with the JADL method) is both a user simplification and allows the exploitation of all the available information for M/EEG trial by trial signal decomposition. It thus provides better estimations of waveforms in the dictionary.

http://team.inria.fr/athena/ christos.papageorgakis@inria.fr

Figure 1 :

 1 Figure 1 : The generated (blue color) and learned (red color) dictonary using our model (left). The learned dictionary on contaminated signals by noise of SNR : 0.021 (right).

Figure 2 :

 2 Figure 2 : The learned dictionaries by the single-channel method: using the best (left) and the worst (right) channel. Wrong recovered components are marked by the red ellipses.

C

  = 200 channels, M = 63 trials, N = 541 time samples, contaminated by ambient noise.


Carlos Flores

Vicente Milanés

Joshué Pérez

Fawzi Nashashibi

An Energy-Saving Speed Profile Algorithm for Cybernetic Transport Systems

Cybernetic Transport Systems (CTS) are now available on some cities worldwide covering the so-called lastmile for people transportation. These electric vehicles are expecting to operate during several hours without interruptions. It implies that energy consumption plays a key role when it comes to extend its autonomy. This paper presents a new algorithm for reference speed generation of CTS with energy consumption considerations. It consists on a two layer controller: the first one is in charge of generating the speed reference based on an energetic model, and the second one is a high-level fuzzy system to improve some low-level undesirable behaviors. The designed system was compared with three other algorithms in a real platform, showing promising results in terms of energy saving.

I. INTRODUCTION

The development of new automotive systems aid drivers not only on the safety-related driving aspects but also on improving users' quality of life. Efforts on this direction are mainly focused on the development of zero-emission vehicles. Autonomy is one of the key aspects to increase the market penetration of electric vehicles compared with gas-propelled ones, leading the change toward eco-friendly cities.

Among the different environments where vehicles drive, urban areas represent the most complex ones. This is because the constant interaction with other vehicles either at intersections, roundabouts or merging zones. One of the most promising urban areas for introducing fully automated vehicles is the so-called last-mile transportation sector [START_REF] Chong | Autonomous personal vehicle for the first-and last-mile transportation services[END_REF]. It refers to the vehicles in charge of moving people from a transport hub-i.e. bus or train station-to their final destination. Cybernetic Transportation Systems (CTS) [START_REF] Parent | Cybercars: Review of first projects[END_REF] have rose as the adequate solution to satisfy the last-mile sector requirements. CTSs are urban mobility platforms formed by electric-powered car-like robots. They are equipped with a set of sensors providing the ability of autonomously drive in urban areas [START_REF] Boisse | Cybernetic transportation systems design and development: Simulation software[END_REF].

CTSs have been actively investigated on recent years. There are currently some different solutions that have been developed to improve transportation network. Some of them are already working on public areas. An example of such systems is implemented over Heathrow London Airport, which consists on several driverless vehicles called Heathrow pods, which cover specified routes over one of the airport Authors are with the Robotics and Intelligent Transportation Systems (RITS) Team, Inria Paris-Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay, France {carlos-eduardo.flores-pino, vicente.milanes, joshue.perez rastelli, fawzi.nashashibi} @inria.fr terminal depending on the passenger desired destination. Additionally, there are several European projects dealing with the development of these transportation systems as Cybercars I and II, CitynetMobil or CityMobil I and II, among others. These projects carried out multiple demonstrations showing the CTS ability for moving people in urban areas.

CityMobil-2 project have demonstrated a fleet of CTS vehicles for covering last-mile routes in some European cities. Some of these demonstrations have been successfully carried out at Oristano (Italy) and at Leon (Spain). During these demonstrations, a shuttle service with several stops was installed in the cities, integrating new technologies to improve urban mobility and to determinate how people accepts these transportation systems.

The main objective of this solution is to provide a transportation service for people, being able to cover a pre-defined route constantly. Among the advantages of CTSs, they are able to travel over any unstructured environment without requiring a special infrastructure modification. However, one of the current limitations of CTS is coming from the fact of being electric cars, limiting the autonomy. Even though remarkable demonstrations have been carried out-i.e. vehicle fleet that covered urban routes as buses would, one of its main limitations is coming from the battery life. It is required that the service works with an uninterrupted schedule such as an urban bus does. The route used has been previously pre-defined, being always the same. It allows to optimize and reduce the vehicle energy consumption for each route.

This article aims to find a solution to the CTS vehicles autonomy problem by using an energy optimization algorithm. Considering a pre-defined route, the vehicle identifies the different stretches to be covered, modifying accordingly its speed with energy-saving purposes. One of the key factors is that the vehicle has to cover these routes in a fixed time that cannot be modified. This is used as input parameter to cover the pre-defined route, but with a monitoring on the energy consumption.

II. UNDERSTANDING CYBERCARS

CTSs or Cybercars are a new concept of fully automated electric vehicles with the purpose of creating an efficient transportation service on urban environments. They are composed of two main intelligence blocks: one that carries out the environment understanding and another for planning, decision and control. These blocks follow predefined trajectories without any driver intervention.

Cybercars have been designed aiming to create a lowcost transportation service for last-mile sector requirements,

A. CTS Constraints

As a consequence of being low-cost vehicles, they present some hardware constraints that can significantly affect their performance in comparison with regular vehicles. Below, the different limitations associated to the prototype cybercar used in this work-a Cybus-are detailed:

1) It is not equipped with a regenerative braking system. This causes that the embedded motor driver is not able to generate energy when driving downgrade.

2) The vehicle is propelled by a DC motor directly coupled to the wheel axis. It means that even in downgrades, the cybercar can consume energy. 3) When the vehicle brakes, a reverse torque is demanded to reduce the speed which is also causing an energy consumption. 4) On-board sensors-i.e. embedded encoders-are also low-cost devices with low resolution (0.5 m/s). This fact affects both the optimal speed profile algorithm and the low-level control of the Cybus. Figure 2 shows the reference (red line) and measured speed (blue line), and the instant power consumed (green line), illustrating this problem. It shows how the reference speed is not being correctly followed by the lowlevel controller. This behavior is directly translated on an undesired energy consumption because of the compensation-that is, torque applied backwards to slow down the vehicle-carried out by the controller to follow the reference.

B. Proposed Algorithm

This paper considers the scenario of a Cybercar that works as a last-mile transportation service, covering a specific route between two points all day long. This work aims to solve the autonomy problem and extend the battery life when the vehicle is driving, maintaining the pre-defined route and taking the same time to cover it.

The proposed solution is an improved algorithm based on a previous RITS work [START_REF] Flores | Optimal energy consumption algorithm based on speed reference generation for urban electric vehicles[END_REF]. It depends on both the route and vehicle characteristics. It is also proposed a highlevel fuzzy system to deal with the constraints previously described; mainly, the low-level control that causes undesired power peaks which are translated on energy losses, reducing vehicle's autonomy.

III. SPEED PROFILE GENERATION

This section describes the optimal speed profile algorithm to increase vehicle's autonomy. Its goal is to send the optimal reference speed to the low-level control in function of the road slope, the vehicle properties and the fixed time to cover the route.

Figure 3 illustrates how the optimal speed profile algorithm is coupled to the control architecture, including the two main blocks: Partial and Fully Profile Generation. The former is in charge of splitting the whole route into segments according to their slope in order to estimate the energy that the vehicle will expend while traveling along the route. This calculation is done before the vehicle departure. Since the required instant power is calculated for each segment, its optimal speed can be obtained through the energetic model.

Once the cybercar starts its route, the algorithm constantly requires the current position in order to set the appropriate reference speed according to the road layout (speed profile obtained in the Partial Profile Generation). This block is also in charge of executing the speed transitions from one segment to the next one through the generation of speed ramps. Finally, the reference speed is sent to the low-level controller to move the motor driver according to the reference.

A. High-level fuzzy system

This high-level controller is implemented to modify the output of the Partial Profile Generation output in order to achieve a more efficient driving. The motivation of selecting a fuzzy controller for this work is mainly based on two reasons: 1) its ability of applying human reasoning to its tuning [START_REF] Rojas | Fuzzy logic techniques for cybercars: a control and decision approach[END_REF]; and 2) the fact that a fuzzy controller does not require a mathematical model, which is appropriate in this case due to the difficulty of modeling the power-speed behavior of the vehicle [START_REF] Perez | Cascade architecture for lateral control in autonomous vehicles[END_REF].

Figure 4 shows how the low-level speed controller behaves. The failures on its response are highlighted on yellow circles. The real measured speed oscillates around a reference given, which is translated to more energy consumption when trying to correct the speed error. The fact that the low level speed control was factory made, brings as a consequence that the controller tuning values are unreachable and their modifications are out of the bounds of this work.

The reason for introducing a high-level fuzzy controller is for dealing with the unappropriated behavior of the lowlevel controller. The proposed fuzzy controller takes into consideration two inputs: Speed error (in meters per second), defined as measured speed minus reference one, and Power error (in KWatts) defined as measured instant power minus current optimal power. The former avoids the excessive braking maneuver when the error between the actual and desired speed is low. The latter is in charge of trying to follow the maximum optimal energy consumption.

RITS team at INRIA has developed a fuzzy software called FEMOT (Fuzzy Embedded Motor) which is an inference motor with a natural language-based input language [START_REF] Rojas | Fuzzy logic techniques for cybercars: a control and decision approach[END_REF]. Specifically for this application, FEMOT operates using Mamdani's inference method, with singleton-type membership functions to codify the output variables, taking control actions with high accuracy and very short delays which are key factors for real-time control systems.

The Fuzzy logic controller is defined through the rule base presented on table I. A heuristic method was followed to define the controller according to human reasoning and driver's needs, taking advantage of fuzzy logic control.

Several tests showed that the situations in which the power error is negative (downgrade segments) are the ones that cause the low-level control failures. For this reason, the highlevel controller has been designed to modify the reference speed, following the measured one. This action avoids that the low level could appreciate a great speed error and, as a consequence, the execution of a correction procedure as well.

On the other hand, the positive speed error case requires a common controller behavior, such as increasing the reference speed when the error is getting negative as example.

Finally, the high level fuzzy controller describes a control surface as shown on figure 5. It shows the smoothness through the whole surface, validating the suitability of the fuzzy rules selected.

Figure 6 shows how the high-level fuzzy controller is coupled to the control architecture. The reference speed is generated by adding the algorithm and the controller outputs. In this way, the low level is able to receive the control commands from the high-level, acting in a more efficient way.

IV. EXPERIMENTAL RESULTS

The proposed algorithm was validated using an experimental platform at Inria's facilities. The selected scenario includes upgrade, flat and downgrade segments to evaluate the performance of the system in different layouts. The upper part of Figure 7 shows an aerial view with the selected route remarked on yellow whereas the bottom part depicts the elevation profile of the entire course. The reference speed, the real speed and the instant power consumed was measured in real time on each travel.

A. Comparative analysis

Using the trajectory showed on Fig. 7, four driving behavior were defined to compare the performance of the Fig. 6. Fully Speed Generation Architecture Fig. 7. Test route at Inria's facilities developed system with current speed control algorithms. The first test is the one explained on section II-B, where a commuter drives manually a vehicle trying to do it efficiently from a starting point to a destination. The second test shows the performance of a vehicle equipped with a cruise control system. The third case is when the intelligent speed profile algorithm is applied over the longitudinal control, to check the energy consumption. Finally, the fourth scenario applies the same algorithm, but adding the high-level fuzzy controller to improve the reference speed generation. It is important to remark that the conditions for every scenario are the same, either the trajectory or the time established to cover the route. On each of these results, the reference speed (red line), the measured speed (blue line) and the instant power consumed (green line) are plotted.

1) Manual Driving: Figure 8 illustrates the reference speed generated by the driver to the low level control, the measured speed and the instant power consumed by the vehicle. The power consumed presents multiple peaks, increasing the energy consumption. This is because the continuous action over the acceleration/braking interfaces of the vehicle when trying to follow the reference speed.

2) Cruise Control: As second case, a cruise control system was tested. It maintains the reference speed constant during the whole route. Figure 9 shows the result of this test. As it can be appreciated, the power consumed turns to be more stable although there are still undesirable power peaks and some stretches where the energy consumption is higher than it could be.

3) Speed Profile Algorithm: The route described on figure 7 was divided in different segments in order to apply the Intelligent Speed Profile Algorithm (ISPA). It permits to determine the optimal speed for each of the different stretches that the vehicle is covering through the route. Figure 10 shows how the reference speed changes along the trajectory, trying to pursuit the best energy consumption over every road layout. There can also be seen that the power peaks caused by the low level control have been significantly eliminated but there are still some of them present on the downgrade stretch.

4) Speed Profile Algorithm + High Level Fuzzy Controller: Figure 11 plots the result of adding the high level fuzzy controller to the ISPA. One can see how the new reference speed-formed by adding the algorithm output plus the fuzzy controller output-is modified according to the segment slope, fluctuating around the reference generated by the algorithm for avoiding peaks in downgrades. This modification causes by the fuzzy controller avoids the correction executed by the low-level, leading to a great energy saving.

B. Results Analysis

Table II shows a comparison between the energy saving of each of the proposed algorithm. It is interesting to notice that Cruise Control systems significantly improved energy consumption with respect to a human driver. However, no specific behavior for energy saving is carried out since the reference is set to the same value either during upgrades or downgrades. The proposed ISPA system brings this consideration into the equation, generating a specific reference speed for each of the stretch in the route. It causes an improvement in the energy consumption on the part of the Cybus. Finally, the high-level controller specifically designed to deal with the low-level problems improves all the previous algorithms.

The power peaks that were appreciated on the first, second and third case were totally removed by adding the high level fuzzy controller, which acts on downgrades reducing the speed error, to prevent a braking action from the low level to the motor. Otherwise, when the vehicle is going through an upgrade segment, the fuzzy controller corrects normally the error.

The comparison from Table II shows a saving percentage of 22.68% comparing the manually driven tests with the proposed algorithm. The amount of energy consumed was determined by integrating the instant power consumed while

C. Validation

Another route was selected in order to test the performance of the module developed using a different layout that the one used for adjusting the fuzzy controller. The new trajectorysee Figure 12-is formed by several upgrade, flat and downgrade segments. It has a total length of 766 meters, a top slope of 9.3% and a minimum slope of -7.4%, being different enough from the first scenario.

1) Result Analysis: Using the route presented on Fig. 12, two algorithms were set to compare the performance of the developed work. The first shows a car equipped with a cruise control system would cover the route (see Fig. 13 ), keeping the speed constant. The second case is where the proposed architecture is applied: the intelligent speed profile algorithm with the high level fuzzy controller generating the reference speed given to the low level (see Fig. 14).

Figure 14 exhibits some power peaks appear when the Cybus goes downgrade. This is because the length of the segment. The vehicle reaches the maximum allowed speed, causing a braking action from the low-level controller. Table III presents the energy consumption on each case, showing the big saving using the proposed algorithm compared with the classical cruise control systems. V. CONCLUSIONS This paper presents an intelligent control algorithm for reference speed generation, considering energy consumption. An energetic model able to generate the proper speed for each segment on the route was described. Additionally, a fuzzy controller is proposed to improve low-level limitations on the experimental platform.

The proposed method was compared with three other algorithms, showing its benefits in terms of energy savings. All algorithms was compared using the same road layout and the same time to cover the route. Results demonstrated that the proposed system significantly extends the battery life of CTS, showing promising results for future implementations on fleets.

As future works, real time traffic information will be considered in order to recalculate the reference speed on each segment on real time.

Fig. 1 .

 1 Fig. 1. Hall effect sensor, acquisition stage and the Analog Input to USB card required; and their place over the Cybus

Fig. 2 .

 2 Fig. 2. Example of speed low level control problems.

Fig. 3 .

 3 Fig. 3. Algorithm block diagram

Fig. 8 .

 8 Fig. 8. Manual driver test

Fig. 12 .

 12 Fig. 12. Validation route

TABLE I CONTROLLER

 I RULE BASE

	PowerError	SpeedError Positive Null Negative
	Positive	-0.2	0	0.2
	Null	-0.1	0	0.1
	Negative	0.7	0	-0.6
	Fig. 5. Fuzzy control surface

TABLE II ENERGY

 II CONSUMPTION PER ALGORITHM

	Evaluation	Consumed Energy	Savings
	Scenario	(KJoules)	Percentage (%)
	Manual Driving	243.23	-
	Cruise Control	213.74	12.12
	ISP Algorithm	201.68	17.08
	ISPA + HL Fuzzy Controller	188.07	22.68

TABLE III ENERGY

 III CONSUMPTION ON THE VALIDATION ROUTE

	Evaluation	Consumed Energy	Savings
	Scenario	(KJoules)	Percentage (%)
	Cruise Control	355.03	-
	ISPA + HL Fuzzy Controller	297.12	16.31

ACKNOWLEDGMENTS

Authors express their gratitude to the CoCoVeA (Coopération Conducteur-Vehicule Automatisé) ANR project for its support in the development of this work.


Jamel Ghanouchi
email: jamel.ghanouchi@live.com

A proof of the Riemann hypothesis

In this paper, we present the Riemann problem and define the real primes. It allows to generalize the Riemann hypothesis to the reals. A calculus of integral solves the problem. We generalize the proof to the integers.

The Riemann hypothesis

The Riemann conjecture is a conjecture which has been formulated in 1859 by Bernard Riemann in the subject of the Riemann function zeta or  . It is called the zeta Riemann function.

This function is defined as follows 1

1 1 1 ( ) ( ) 1 ... 23 n s s s n s n         
The first result is the divergence of the harmonic serie In the XVIII century, Leonard Euler has discovered the main proprieties of the  function.

In the 1730's he conjectured after numerical calculus the following equality, which is often called the Basel problem. for the positive evennumbers.

2 2 2 2 2 1 | | ( 2 ) 1 1 1 ( 2 ) ( ) 1 ... 2 3 2 ( 2 ) ! k n k k k k n B k nk            Where 2 k
B are the Bernoulli numbers.

Thereafter, he proved in 1744 the Euler idendity where prime numbers are related to the  function.

            
Consequently he deduced the divergence of the serie of the inverse of primes.

With Bernard Riemann, s can be complex number. Riemann proved the following formula

(1 ) 22 1 ( ) ( ) ( ) (1 ) 22 ss ss ss            
Where .

This conjecture is called the Riemann hypothesis.

They calculated numerically one billion zeros of the Riemann they are all located in the critical line.

Resolution of the Riemann hypothesis for the reals

Definition

A real number is compound if it can be written as

. Of course, it is a convention, because, we can consider [START_REF] Gourdon | The 10 13 first zeros of the Riemann zeta function, and zeros computation at very large height[END_REF]  as prime and  will be no more prime. It is equivalent in what will follow. Thus

1 q q pp  is compound. Also 1 11 q q pp   
is prime when p is prime and we have

1 1 1 1 2 2 2 1 ( 1) ( 1) ( 1) ...( 1
) i i i p p p p p          
compound for p prime, for example.

The approach of the Riemann hypothesis

The Rieman hypothesis states that the non trivial zeros of the Riemann zeta function For t integer, Euler has proved that

1 1 1 1 ( ) ( ) 1 z z z tN p r im e s p r im e s z t p p            
, it is the Euler identity. For t real, it is still true and it becomes

1 1 1 11 () 1 z zz Q p r im e s dt t p t z              But let 1 () z 
the Riemann function for the reals and

() z 
the Riemann function for the integers, we have

1 2 1 11 () 1 22 1 1 1 2 2 1 1 2 1 1 1 1 ( ) ( ) 1
                                     Let 1 1 1 2 2 2
                                           
                                                      2 2 1 2 2 1 1 2 21 2 1 2 1 21 2 ) 1 lim ( (1 ) ) lim ( ) 1 lim ( ) lim ( ) lim ( ) 1 1 ( ) (1 ) lim ( ) lim ( (1 )
                                                  1 2 1 1 2 1 1 2 2 ) ) lim ( ) lim ( ) lim ( ) lim (
                     
It is true for every m, particularly in the infinity, thus

                           
And it is impossible as it means that the Riemann function is equal to 0 in all the segment between 0 and 1. Thus m=2 defines the zeros of the Riemann function !

We have proved that the non trivial zeros of the Riemann function for the reals lie in the critical line ! So the hypothesis is proved for the real numbers. The Riemann hypothesis is important because it gives information about the zeros of the Riemann function and the distribution of those zeros are related to real primes !

The generalization to the integers

We have

1 11 ( ) ( ) ( ) ( ) ( ) ' ' ( ) z z A B z z z A B z            
The interpretationiswhatfollows

1 1 1 \ 1 11 1 1 1 1 1 ' ' 1 1 1 . ( ). z z z z z t Q Q N z z z t N t p r im e s dt t t z t t t B B A t p t                                     But 1 () ( ' 1) ( ' 1) '( 1) ( 1) (
) ' '( 1) '( 1) ( ' 1) ' 1 z A B B B B A B B z A B B B B A B B                 Thus 1 1 ( ' 1) ( ) ( ' 1) ( 1) ( ) ' ( 1)
( 1) ( )

'( ' 1) ( ) '( ' 1)

B z A B B z A B B B z A B B B z A B                    1 1 1 ( ) 0 ( ) 0 ' 2 ( ) ' ( ) 2 ( ) 0 ( ) 0 z B z B B z B z z z                  
We have the equivalence

Conclusion

We have generalized the concept of prime to the reals. It allowed to prove the conjecture for the reals and for the integers. Thus, we have proved the Riemann hypothesis.

The Bibliography



  It has been proved in the middle age by Nicole Oresme.



  Euler proved it in 1748 and introduced the  function. He calculatedits value

n

  are rationals. This decomposition in prime factors is unique. A prime real number or R-prime can be written only as p=p.1 . Thus we define other real prime numbers like  , e, ln

  Thus the Riemann hypothesis is proved ! Thus the non trivial zeros of the Riemann function zeta lie in the critical line like for the reals ! It is the proof of the Riemann hypothesis !

  

											1
	1 2 tt 0 lim ( ) lim (     t   2 1 ) 0 1 t 
												2
	Wededucethat
	lim (	t		1 2 		iy	) lim ( 	t	1 2		iy		1	) 0 ; 		y	| lim (	t		iy	) 0 
	00 1 t t     Hence t   iy 
												2
	11 22 (1 tt lim ( )) lim (     iy iy t     And for every m	 tt 1 	2 iy iy	)
																			2
																					1	2	m
																					iy	2	iy
																1					1 mm 2	m	1
																		iy			iy	2	iy	2	m
					1																t	(	t	(	iy	) 1)	t
																m					m	m	0
	lim ( And		m		iy	) lim (		t t			) lim (	t t	t	) lim (	m	)
	t									t		m			1				t	m	1	t	m	1
																		iy			iy	iy
													m						m	m
	1 2 iy iy  ) lim ( ) 1 t t  2 1 2 ) lim ( ) (1 ) iy t iy t t iy t 2 m 2 1 iy m m ) 1) iy t  ) lim ( 1 ( t 1 1 iy iy m m ( ( t      m    11 t tt 1 2 lim ( iy t    t   lim ( mm 2 2 iy	iy	1 2	iy
																mm
	Or
																1  1 iy iy  	1		iy	1		iy
	lim (	t t 	22 2 2 ) lim ((1 )( )) lim ( iy t t t t t     	iy		t	2		t	2	)
	t	 								t	 										t	 
	Consequently
				11 iy 	iy
	lim (	t	22 ) lim ( t  ) lim ( ) iy t
	t	 							t	 						t	 
	Or
				11
	lim (	22 ) lim ( ) 1 tt 
	tt  	 
	Thus

  (

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  )

					mm
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'( ') () 1 '( ' 1)

And if


Jamel Ghanouchi
email: jamel.ghanouchi@live.com

The Goldbach conjectures

We deal with two problems known by the name of Goldbach conjectures, weak and strong versions.

The Goldbach conjectures

They are two of the oldest and best-known problems of number theory and all mathematics. The first, known as the weak one, states that an odd number greater than 25 is always the sum of three prime numbers, when the strong Goldbach conjecture states that an even number greater than 22 is always the sum of two prime numbers. This last one has been shown to hold up through 28 4 .1 0 but remains unproven despite considerable effort. We present here a very elementary approach of the problems and prove them by the same way. For the strong version, we present two approaches !

The weak Goldbach conjecture

First of all, an odd number can be different of the sum of three powers of odds.

We have 2k=4p+2m+1+2k'-1 that describes all even numbers from 4p+2k' until infinity when m describes all the integers. Thus 2k+1=4p+2m+1+2k' describes all odd numbers from 4p+2k'+1 until infinity for the same m.

But 3p+2m describes all the odd numbers from 3p to infinity when m describes all the integers.

Hence, there always exists m' for which 3p+2m'=q is a prime number.

We deduce that 2k+1=p+3p+2m'+2(m-m')+2k'+1=p+q+2(m-m')+2k'+1 Describes all the odd numbers from 4p+2k'+1 until infinity and this for every k'.

But 2(m-m')+2k'+1 describes all the odd numbers from 2(m-m')+1 to infinty when k' describes all the integers.

There always exists k'=k'' for which 2(m-m')+2k''+1=r a prime number.

In conclusion :

2k+1=p+q+r which describes the odds from 4p+2k'+1 to infinty is always the sum of three prime numbers.

We want to calculate the first value of 2k+1 : Let p=5 then 5+17+3=2k+1 means k=12=2p+k'=10+k' and we know now that the conjecture is true for all the odds from 25 to infinity !

The strong Goldbach conjecture

First approach

First of all, we must notice that an even number is not always the sum of two powers of odd numbers. A counter example will show it : n>1, m=1 (or m=2 or 3) and the exponent n=2 n=2

          
There is an even number (6) which is not equal to the sum of the powers of two odd numbers ! It means that, in what follows, n=1.

For every even number 2m greater or equal to 12, there exists an integer I, a prime p and an integer k for which :

2 4 2 m ip k 
We want to suppose there exists an even which is not equal to the sum of two primes. We must have

2 4 q r k N p  
With q and r are primes. Thus it is rational :

2 4 q r k a pb   WithGCD(a,b)=1 But 4 ( 2 ) a p b q r k   
b does not divise a, and as p is prime, b does not divise p, hence it divises 4, consequently b=4 or b=2.

If b=4

2 a p q r k    It means aeven : impossible because GCD(a,b)=1 Thus b=2 q r k a p  
As i can take every value : let i=a

  2 4 2 4 2 2 2 m ip k a p k q r k        Or 2 2 m ip k a p k q r k        And 2 ( 2 ' 1) 2 ', q r k a i k w p k w N       As     2 4 2 2 '' 2 '' m ip k q r k k w ip k k          It means   2 22 q r n wi p p p    
Where w isinteger or

2 2 q r k w p p p    Let 2 ' 1 i w k   And 2 2 ( 2 ) q r n wi p p p     2 ( 4 ' 1) 2 ( 4 ' 1) ( 4 ' 1) 2 ( 4 ' 1) 2 ' 1 ( 2 ' 1) ( 2 ' 1) k i m k m k k m m k k k p k p p k p                 Thus           4 ' 1 2 ' 1 4 ' 1 4 ' m k k k q r k k k q r k k qr              Hence donc 2 k q r  Or 2 2 ( ) 2 m q r k q r     
But we have supposed the contrary : it means that it is impossible to suppose that there exists an even is different of the sum of two primes ! Remark that q and r can be only . We conclude that an even is always the sum of two primes !

Second approach

Also 2k=4p+2m+2k' describes all the even numbers from 4p+2k' to infinity when m describes all the integers and, we saw it, there always exists m' and q a prime number for which 3p+2m'=q.

Thus 2k=p+q+2(m-m')+2k' describes all the evens from 4p+2k' to infinity and this for every k'.

Particularly, there always exists k'=k'' for which : 2k'+2(m-m')=0 and 2k=p+qdescribes all the evens from 4p+2k' to infinty and is the sum of two primes.

Practically :

For p=5 and 5+17=2k or k=11=2p+k'=10+k'.

It means that the strong conjecture is true for all the evens from 22 to infinity !

Conclusion

Our approach was sufficient to demonstrate the Goldbach conjectures.

Bibliography

[1] Fliegel, Henry F.; Robertson, Douglas S. (1989). "Goldbach's Comet: the numbers related to Goldbach's Conjecture". Journal of Recreational Mathematics 21 (1): 1-7.

[2] Chen, J. R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes". Sci. Sinica 16: 157-176.