Ashesh Shah
email: ashesh.shah@fhnw.ch

Fabiola Alonso
email: fabiola.alonso@liu.se

Jean-Jacques Lemaire
email: jjlemaire@chu-clermontferrand.fr

Karin K Wårdell
email: karin.wardell@liu.se

Daniela Pison
email: daniela.pison@fhnw.ch

Jerome Coste
email: jcoste@chu-clermontferrand.fr

Erik Schkommodau
email: erik.schkommodau@fhnw.ch

Simone Hemm-Ode
email: simone.hemm@fhnw.ch

Wårdell K Coste

Hemm-Ode S Ashesh Shah

Jérôme Coste

3D Visualization of intraoperative stimulation test results for better target selection in DBS surgery

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

Despite an increasing use of deep brain stimulation (DBS), the fundamental mechanisms underlying therapeutic and adverse effects as well as the optimal stimulation site remain largely unknown. The aim of the present study was to develop a method to correlate electric field simulations for intraoperative stimulation tests with quantitatively evaluated symptom improvement and patient specific anatomy to estimate the optimal stimulation site. One essential tremor patient, bilaterally implanted in the ventro-intermediate nucleus (Vim) has been included. Preoperatively Vim and its anatomic neighbors were manually outlined according to spontaneous MRI contrasts using the commercial planning software from where the structures could be exported via a specifically designed interface. During the intervention, intraoperative stimulation tests were performed on two trajectories per hemisphere (8 positions per trajectory). The change in tremor at each stimulation position compared to baseline was evaluated using a 3-axis accelerometer. Based on accelerometer data, two stimulation amplitudes (low and high improvement) were identified per position and corresponding electricfield isosurfaces (0.2V/mm) were simulated. As each voxel in the region of interest may be part of several isosurfaces -each surface depicting one amplitude responsible for one improvement in tremor-the voxel was assigned to the isosurface representing the minimum improvement using Matlab. Data were imported into Paraview (VTK based 3D visualization software). Colorcoded minimum 3D-improvement maps were visualized on the patient's MR images together with the manually outlined anatomical structures. The resulting visualization was evaluated by clinicians. The software allowed 3D visualization as well as orthographic slices parallel to the trajectory. Clinicians confirmed that it enables the identification of the most effective stimulation areas with respect to the anatomy. This new concept based on quantitative symptom evaluation, electric field simulations, and patient specific anatomical data will allow the analysis of a high amount of intraoperative data which might help to elucidate the mechanism of action of DBS.


Ruiqing Yin

Hervé Bredin

Claude Barras Limsi

Neural speech turn segmentation and affinity propagation for speaker diarization

Keywords: speaker diarization, re-segmentation, LSTM, affinity propagation

Speaker diarization is the task of determining "who speaks when" in an audio stream. Most diarization systems rely on statistical models to address four sub-tasks: speech activity detection (SAD), speaker change detection (SCD), speech turn clustering, and re-segmentation. First, following the recent success of recurrent neural networks (RNN) for SAD and SCD, we propose to address re-segmentation with Long-Short Term Memory (LSTM) networks. Then, we propose to use affinity propagation on top of neural speaker embeddings for speech turn clustering, outperforming regular Hierarchical Agglomerative Clustering (HAC). Finally, all these modules are combined and jointly optimized to form a speaker diarization pipeline in which all but the clustering step are based on RNNs. We provide experimental results on the French Broadcast dataset ETAPE where we reach state-of-the-art performance.

Introduction

Speaker diarization is the task of determining "who speaks when" in an audio stream that usually contains an unknown amount of speech from an unknown number of speakers [START_REF] Tranter | An Overview of Automatic Speaker Diarization Systems[END_REF][START_REF] Anguera | Speaker diarization: A Review of Recent Research[END_REF]. While speech and speaker recognition systems have improved enormously thanks to deep learning approaches, speaker diarization systems have yet to fully take advantages of these new techniques. This may be explained by the fact that speaker diarization is an unsupervised classification task difficult to address with (mostly supervised) deep learning approaches.

Speaker diarization systems are usually built as the combination of four main stages. First, non-speech regions such as silence, music and noise are removed by speech activity detection (SAD). Next, speech regions are split into speaker-homogeneous segments by speaker change detection (SCD), later grouped according to the identity of the speaker thanks to unsupervised clustering approaches. Finally, speech turn boundaries and labels are (optionally) refined with a resegmentation stage.

In conventional speaker diarization systems [START_REF] Barras | Multi-Stage Speaker Diarization of Broadcast New[END_REF][START_REF] Rouvier | An Open-source State-of-the-art Toolbox for Broadcast News Diarization[END_REF], GMMbased SAD and sliding windows-based SCD are widely used.

In GMM-based SAD, GMMs for speech class and non-speech class are used to remove non-speech regions with Viterbi decoding. In sliding window-based SCD, one will use two adjacent sliding windows on the audio data and compute a distance between them, then decide (usually by thresholding the distance) whether the two windows originate from the same speaker. Gaussian divergence [START_REF] Siegler | Automatic Segmentation, Classification and Clustering of Broadcast News Audio[END_REF] and Bayesian Information Criterion (BIC) [START_REF] Chen | Speaker, Environment and Channel Change Detection and Clustering via the Bayesian Information Criterion[END_REF] have been used extensively in the literature to compute such a distance.

Since SAD and SCD can be modeled as supervised binary classification task (speech vs. non-speech for SAD, change vs. non-change for SCD), we did manage to improve those two stages using deep learning approaches. Both were addressed as frame-wise sequence labeling tasks using bi-directional LSTM on top of MFCC features [START_REF] Yin | Speaker Change Detection in Broadcast TV using Bidirectional Long Short-Term Memory Networks[END_REF][START_REF] Gelly | Minimum Word Error Training of RNN-based Voice Activity Detection[END_REF], leading to much better performance than traditional methods. The first contribution of this paper (introduced in Section 2) is to adapt this LSTM-based sequence labeling framework to the case of unsupervised resegmentation stage. Traditionally, a GMM is trained for each cluster. The audio is then re-segmented through a Viterbi decoding [START_REF] Barras | Improving Speaker Diarization[END_REF]. Several alternation of clustering and re-segmentation steps are usually performed. Post-processing segment boundaries using the output of a word or phone recogniser also improves the overall diarization output [START_REF] Barras | Improving Speaker Diarization[END_REF].

Speech activity detection

Input audio

Speaker change detection

Speech turn embedding

Affinity propagation

Re-segmentation output diarization result

Initial segmentation

Our second contribution concerns the clustering step and is described in Section 3. Most work rely on variations of Hierarchical Agglomerative Clustering (HAC) approaches [START_REF] Meignier | LIUM SpkDiarization: An Open Source Toolkit for Diarization[END_REF] and use BIC, CLR or i-vector to compute similarity between clusters. In this paper, we propose to use affinity propagation clustering on top of neural speaker embeddings introduced in [START_REF] Bredin | TristouNet: Triplet Loss for Speaker Turn Embedding[END_REF][START_REF] Wisniewski | Combining Speaker Turn Embedding and Incremental Structure Prediction for Low-Latency Speaker Diarization[END_REF]. This is similar to [START_REF] Wang | Speaker Diarization with LSTM[END_REF] in which the authors use spectral clustering instead. An affinity propagation variant has been introduced in [START_REF] Zhang | A Novel Speaker Clustering Algorithm via Supervised Affinity Propagation[END_REF] for speaker diarization but it is supervised by the number of speakers and relies on standard statistical models to compute speaker similarities.

Our third and final contribution is the combination and joint optimization of all these modules into a speaker diarization system where all but the clustering stage are based on RNNs: LSTM-based SAD, LSTM-based SCD, LSTM-based speaker embedding, and LSTM-based re-segmentation. The full pipeline is depicted in Figure 1. Experiments on the ETAPE dataset are summarized in Section 4 and state-of-the-art results are discussed in Section 5.

Sequence labeling based on LSTM

In this section, we generalize the approach used for speech activity detection in [START_REF] Gelly | Spoken Language Identification using LSTM-based Angular Proximity[END_REF] and speaker change detection in [START_REF] Yin | Speaker Change Detection in Broadcast TV using Bidirectional Long Short-Term Memory Networks[END_REF] and show how it can also be applied to re-segmentation.

Principle

Let x ∈ X be a sequence of feature vectors extracted from an audio recording (e.g. MFCC features): x = (x1, . . . , xT ) where T is the length of the sequence. Let y ∈ Y be the corresponding sequence of labels: y = (y1, . . . , yT ) and yi ∈ {0, . . . , K -1} where K is the number of classes.

The objective is to find a function g : X → Y that matches a feature sequence x to the corresponding label sequence y. We propose to model this function g using a stacked LSTM neural architecture (further described in Section 4.2) trained with cross-entropy loss. Short fixed-length sub-sequences (a few seconds, typically) of (otherwise longer and with variable length) audio files are fed into the model. This allows to increase the number of training samples and augment their variability.

At test time, audio files are processed using overlapping sliding windows of the same length as above. For each time step i, this results in several overlapping sequences of Kdimensional (softmax-ed) scores, which are averaged to obtain the final score of each class.

Initial segmentation

The initial segmentation step aims at removing non-speech regions and splitting it into speaker-homogeneous segments. It is composed of two stages: speech activity detection (SAD) and speaker change detection (SCD).

SAD is the direct application of the above principle with K = 2 classes: yi = 1 for speech, yi = 0 for non-speech. At test time, the sequence of speech scores is post-processed using two (θonset and θoffset) thresholds for the detection of the beginning and end of speech regions [START_REF] Gelly | Minimum Word Error Training of RNN-based Voice Activity Detection[END_REF].

SCD can also be addressed using the same principle with K = 2 classes: yi = 1 if there is a speaker change during the ith frame, yi = 0 otherwise. To balance the number of positive and negative samples during training, the positive class is increased artificially by labeling as positive every frame in the direct neighborhood (e.g. less than 50 milliseconds apart) of the actual change point. At test time, the sequence of speaker change scores is post-processed so that all local maxima on a sliding window of duration δpeak exceeding a threshold θpeak are marked as speaker change points [START_REF] Yin | Speaker Change Detection in Broadcast TV using Bidirectional Long Short-Term Memory Networks[END_REF].

Re-segmentation

Given the output of the clustering step (later discussed in Section 3), re-segmentation aims at refining speech segments boundaries and labels and is usually solved with a combination of GMMs cluster modeling and Viterbi decoding. Assuming the output of the clustering step predicts k different speakers, we propose (and this is our first contribution) to use the same principle as above with K = k + 1 classes: yi = 0 for nonspeech and yi = k for speaker k.

At test time, using the (unsupervised) output of the clustering step as its unique training file, the neural network is trained for a (tunable) number of epochs E and applied on the very same test file it has been trained on. The resulting sequence of K-dimensional scores is post-processed by choosing the class with maximum score for each frame. To stabilize the choice of the hyper-parameter E and make the prediction scores

(A) seg1 seg2 seg3 seg4 (B) (C) (D)
... smoother, scores from the m = 3 previous epochs are averaged when doing predictions at epoch E.

f f f f
While this re-segmentation step does improve the labeling of speech regions, it also has the side effect of increasing false alarms (i.e. non-speech regions classified as speech). Therefore, its output is further post-processed to revert speech/non-speech regions back to the original SAD output.

Clustering

In this section, we introduce our second contribution: the combination of neural embeddings and affinity propagation for speech turn clustering.

Neural embedding

First, a neural embedding network f : X → R D is trained using the triplet loss paradigm to embed speech sequences x into a D-dimensional space. The network architecture used is the one introduced in [START_REF] Bredin | TristouNet: Triplet Loss for Speaker Turn Embedding[END_REF] and further improved in [START_REF] Gelly | Spoken Language Identification using LSTM-based Angular Proximity[END_REF]. In the embedding space, two sequences xi and xj of the same speaker (resp. two different speakers) are expected to be close to (resp. far from) each other according to their angular distance:

∠(xi, xj) = arccos f (xi) • f (xj) f (xi) • f (xj ) (1)
Figure 2 depicts how this network f -initially meant to process fixed-length (a few seconds, typically) speech sequencescan be used to embed variable-length speech segments coming from the initial segmentation step (A). The idea is to slide a fixed-length window (B) over the duration of the file, embed each of these subsequences (C), and then sum the embedding of all overlapping subsequences to obtain one embedding per initial segment (D). The embedding of segment i is denoted as ωi in the next paragraph.

Clustering by Affinity propagation (AP)

The goal of SAD and SCD is to produce pure speaker segments containing a single speaker. The clustering stage is then responsible for grouping these segments based on speaker identities. We chose the affinity propagation (AP) algorithm [START_REF] Frey | Clustering by Passing Messages Between Data Points[END_REF] for clustering. AP does not require a prior choice of the number of clusters contrary to other popular clustering methods. All speech segments are potential cluster centers (exemplars). Taking as input the pair-wise similarities between all pairs of speech segments, AP will select the exemplars and associate all other speech segments to an exemplar. In our case, the similarity between i th and j th speech segments is the negative angular distance between their embeddings: s(i, j) = -∠(ωi, ωj)

On the diagonal of the similarity matrix, s(k, k) is set to the preference value θAP, a hyper parameter which influences the choice of ω k as exemplar and thus the final number of clusters. AP clustering can be viewed as a "message passing" process between speech segments with two kinds of message: responsibility and availability. Responsibility r(i, k) is a message sent from segment i to k that quantifies how well-suited x k is to serve as the exemplar for xi. Availability a(i, k) is a message sent from segment k to i that represents how appropriate it would be for segment i to pick segment k as its exemplar. Responsibilities and availabilities are first initialized to 0 and then updated iteratively with a damping factor λAP which is introduced to avoid numerical oscillations. At each iteration, AP combines the responsibilities and availabilities to control the selection of exemplars. For segment i, the segment k which maximizes r(i, k) + a(i, k) is the corresponding exemplar. The whole AP procedure terminates after a fixed number of iterations or after the exemplar stay unchanged for a chosen number of iterations. More details about AP can be found in [START_REF] Frey | Clustering by Passing Messages Between Data Points[END_REF]

Experiments

This section describes the experimental framework used in the rest of the paper. Open-source implementations of both the evaluation protocol and the proposed pipeline are available here: github.com/yinruiqing/diarization_ with_neural_approach 1 summarizes the data used for running the experiments. Both REPERE [START_REF] Giraudel | The REPERE Corpus: A Multimodal Corpus for Person Recognition[END_REF] and ETAPE TV [START_REF] Gravier | The ETAPE Corpus for the Evaluation of Speech-based TV Content Processing in the French Language[END_REF] datasets contain recording of French TV broadcast with news, debates, and entertainment. The REPERE corpus was used for training the neural networks used in SAD, SCD, and embeddings. The ETAPE TV development subset was used for hyper-parameter tuning, while the ETAPE TV test subset was used for evaluation.

Speaker diarization systems are usually compared using diarization error rate (DER). In order to account for manual annotation imprecision, it is common practice to not evaluate short collars centered on each speech turn boundary (usually 250ms on both sides) and speech regions with more than one simultaneous speaker. In the case of the ETAPE TV dataset, these skipped regions (which are likely to be the most difficult to correctly classify) represent more than 20% of the total speech duration.

Yet, precise "who speaks when" annotations are distributed alongside the ETAPE TV dataset. They were obtained using the following two-steps process: automatic forced alignment of the manual speech transcription followed by manual boundaries

Implementation details

Feature extraction. Each part of the diarization pipeline shares the same set of input features extracted every 10ms on a 25ms window using Yaafe toolkit [START_REF] Mathieu | YAAFE, an Easy to Use and Efficient Audio Feature Extraction Software[END_REF]: 19 mel-frequency cepstral coefficients (MFCC), their first and second derivatives, and the first and second derivatives of the energy (amounting to a total of 59 dimensions).

Sequence labeling. SAD, SCD and re-segmentation modules share a similar network architecture which stacks Bi-LSTM and a multi-layer perceptron. The architecture details are shown in Table 2. For example, the model for SCD is composed by two Bi-LSTM layers and 2 fully connected layers. Bi-LSTM1 has 64 outputs (32 forward and 32 backward). Bi-LSTM2 has 40 (20 each). The fully connected layers are 40-and 10-dimensional respectively. As described in Section 2, the final output layer dimension depends on the task.

Sequence embedding. Implementation details are identical to the ones used in [START_REF] Wisniewski | Combining Speaker Turn Embedding and Incremental Structure Prediction for Low-Latency Speaker Diarization[END_REF]. Trained on REPERE dataset, 192-dimensional embeddings are extracted every 0.8s on subsequences of duration 3.2s.

Contrastive approaches

The proposed speaker diarization pipeline is compared to two alternative approaches. The first one is a variant of the proposed approach where the affinity propagation module is replaced by standard hierarchical agglomerative clustering. The second one, dubbed "S4D" in the rest of the paper, is a state-of-the-art system developed at LIUM using sidekit [START_REF] Larcher | An Extensible Speaker Identification Sidekit in Python[END_REF] and S4D [START_REF] Broux | S4D: Speaker Diarization Toolkit in Python[END_REF] toolkits.

Hierarchical agglomerative clustering. This variant of the proposed pipeline relies on complete-link clustering [START_REF] Duda | Pattern Classification[END_REF] on top of the affinity matrix S based on angular distance defined in Section 3 and a distance threshold θHAC to stop merging clusters. Other linkages were tested (average, single) but found to lead to worse performance.

S4D system outputs were provided to us by LIUM and use the following pipeline. Segmentation based on Gaussian divergence first generates (short) pure segments. Adjacent segments from the same speaker are then fused based on the Bayesian Information Criterion (BIC), leading to (longer) speech turns. Hierachical clustering based on Cross-Likelihood Ratio then groups them into pure clusters, further grouped into larger clusters using another i-vector-based clustering. Three clustering approaches were compared for this last step: regular HAC, graph-based clustering, and integer linear programing. Graph- 3: Effect of re-segmentation on proposed pipeline based clustering was selected as the one leading to the best performance on the ETAPE TV development set. Finally, Viterbi decoding adjusts segment boundaries. Note that the trainable parts of this system were trained on a much larger dataset (ES-TER [START_REF] Galliano | The ESTER Phase II Evaluation Campaign for the Rich Transcription of French Broadcast News[END_REF], ETAPE [START_REF] Gravier | The ETAPE Corpus for the Evaluation of Speech-based TV Content Processing in the French Language[END_REF], REPERE [START_REF] Giraudel | The REPERE Corpus: A Multimodal Corpus for Person Recognition[END_REF], and NIST RT03) than the one we used (REPERE only).

Joint optimization

While speaker diarization modules are usually tuned empirically and independently from each other, our third contribution consists in the joint global optimization of the whole diarization pipeline. More precisely, we use the Tree-structured Parzen Estimator hyper-parameter optimization approach [START_REF] Bergstra | Making A Science of Model Search: Hyperparameter Optimization in Hundreds of Dimensions for Vision Architectures[END_REF] available in hyperopt toolkit [START_REF] Bergstra | Hyperopt: A Python Library for Optimizing the Hyperparameters of Machine Learning Algorithms[END_REF] to automatically select the set of hyperparameters that minimizes diarization error rate on ETAPE TV development set: θonset and θoffset for SAD, θpeak and δpeak for SCD, θAP and λAP for affinity propagation (or θHAC for hierarchical agglomerative clustering). Note that re-segmentation hyper-parameter E was tuned separately as we wanted to carefully analyze its behavior, but it should ideally be optimized with the rest of the pipeline.

Results and discussions

Figure 3 summarizes the main experimental results. The proposed pipeline reaches state-of-the-art performance on the ETAPE TV dataset, though the difference with LIUM's S4D is not statistically significant (24.2% vs. 24.5%). However, switching from affinity propagation to hierarchical agglomerative clustering does degrade significantly the performance (24.2% vs. 28.5%). Comparison to results obtained when tuned directly on test set shows that there is room for improvement regarding the joint optimization of the proposed pipeline: LIUM's S4D tends to generalize better (∆DER = 1.6% vs. 0.6%). Nevertheless, affinity propagation beats hierarchical agglomerative clustering under this criterion (∆DER = 1.6% vs. 2.8%).

Table 3 shows the effect of the proposed re-segmentation step on the output of affinity propagation clustering: it improves both cluster purity and coverage, leading to an absolute decrease of 1.4% in diarization error rate. Interestingly, the same conclusion holds when it is applied to the HAC-based pipeline or the LIUM S4D, even though the latter uses its own re-segmentation step in Figure 3.

Figure 4 is meant to analyze the behavior of the approach and to evaluate the robustness of its unique hyper-parameter E. The horizontal dashed line is the DER of the system before re-segmentation (i.e. the output of the clustering step). DER quickly decreases during the first few epochs, reaches an improved minimum value, then starts to over-fit to the original input and asymptotically converges to the original performance. This observation, combined with the fact that the optimal number of epochs on the test set is close to the one selected on the development set, leads us to the conclusion that the proposed LSTM-based re-segmentation is stable and very unlikely to degrade performance.

Conclusion

The proposed pipeline is a step toward an integrated end-to-end neural approach to speaker diarization. We show that both the initial segmentation and the final re-segmentation can be formulated as a set of sequence labeling problems, addressed using recurrent neural networks. However, in re-segmentation step, finding the best epoch E relies on a development set. We plan to investigate a way to automatically select the best epoch for each file. In addition, though neural networks can be used to embed and compare pairs of speech segments, it remains unclear how to do also cluster them in a differentiable manner. Our experiments also show that affinity propagation outperforms the standard agglomerative clustering with complete-link, when comparing speaker embeddings.

Figure 1 :

 1 Figure 1: Proposed diarization pipeline and hyper-parameters. Modules with gray background are based on neural approaches.

Figure 2 :

 2 Figure 2: Aggregation of fixed-length subsequence embeddings.

Figure 4 :

 4 Figure 4: Re-segmentation on development (top) and test sets (bottom). The best epoch on the development set is marked with an orange dot.

Table 1 :

 1 Datasets statistics with mean and standard deviation of speaker counts per file.

	4.1. Corpora and evaluation metric
	Use	Dataset	Hours (speech)	nb. of speakers Total Per file
	train	REPERE	59 (96%) 1758 9.6±6.1
	dev.	ETAPE TV (dev.)	4 (93%)	93	8.0±4.4
	test	ETAPE TV (test.)	4 (92%)	92	9.2±5.6
	Table

Table 2 :

 2 Network architectures for (re-)segmentation models. Note: K depends on each file.adjustment by trained phoneticians. Therefore, and unless otherwise stated, results reported in this paper do not use collars, nor do they skip overlapping regions. Practically, we use the open-source implementation of diarization error rate available in pyannote.metrics[START_REF] Bredin | pyannote.metrics: A Toolkit for Reproducible Evaluation, Diagnostic, and Error Analysis of Speaker Diarization Systems[END_REF].

		Bi-LSTM	MLP	Output
	SAD	16 × 2	16	2
	SCD	32 × 2, 20 × 2 40, 10	2
	Re-segm. 16 × 2, 16 × 2	16	K

Acknowledgements

This work was partly supported by ANR through the ODESSA (ANR-15-CE39-0010) and PLUMCOT (ANR-16-CE92-0025) projects. We would like to thank Sylvain Meignier for providing us with the output of the LIUM's S4D system. Following discussions with the authors of [13], we updated the paper after publication in the official proceedings and removed some incorrect statements about their approach.


Daniela Pison

Fabiola Alonso

Karin K Wårdell

Ashesh Shah

Jerome Coste

Jean-Jacques Lemaire

Erik Schkommodau

Simone Hemm-Ode

A method for side effect analysis based on electric field simulations for intraoperative test stimulation in deep brain stimulation surgery

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

Despite an increasing use of deep brain stimulation (DBS) the fundamental mechanisms underlying therapeutic and adverse effects remain largely unknown. The simulations of electric entities are increasingly used to evaluate stimulation effects. To our knowledge, so far no group has considered such simulations combined with a side effect analysis of data obtained during intraoperative test stimulations. The aim of the present paper is to introduce a method allowing patient-specific electric field simulations for stimulation amplitudes inducing side effects during deep brain stimulation surgery.

Two female patients presenting essential tremor, both bilaterally implanted in the ventral intermediate nucleus (VIM) region in the Clermont-Ferrand University Hospital were included in the study. Intraoperative test stimulations were performed on central and posterior trajectories in each hemisphere at 8-9 positions per trajectory. At each position in addition to the evaluation of the therapeutic effects, side effects such as pyramidal symptoms and paresthesia without localization indicator or paresthesia with localization indicated by the patient (in the hand or in the fingers) were noted. The anatomical structures such as VIM and its neighbors were preoperatively manually outlined using the iPlan software (Brainlab, Feldkirchen, Germany) according to spontaneous MRI contrasts [START_REF] Lemaire | MRI anatomical mapping and direct stereotactic targeting in the subthalamic region: functional and anatomical correspondence in Parkinson's disease[END_REF]. The so identified structures were exported via a specifically designed interface (VVLink, Brainlab, Feldkirchen, Germany). Whenever side effects occurred the inducing stimulation amplitude was chosen for electric field simulations. A finite element method [START_REF] Wårdell | Deep Brain Stimulation of the Pallidum Internum for Gilles de la Tourette Syndrome: A Patient-Specific Model-Based Simulation Study of the Electric Field[END_REF] was applied to calculate the electric field distribution. Conductivity values were deduced from the patient's T1 weighted MRI. An isofieldlevel of 0.2V/mm was chosen and the points of the isosurface were exported. They were visualized together with the extracted anatomical structures and the trajectories. The different structures present inside the volume defined by the isofield level and their appearance were determined. Combinations of structures always appearing together for a specific side effect were identified.

For both patients, 8 electric field simulations were successfully performed. A first analysis showed that pyramidal effects appear when parts of the ventrooral nucleus (VO) and the VIM were present inside the isosurface. The ventrocaudal lateral nucleus (VCL), the ventrocaudal medial nucleus (VCM) and the VIM were among the identified structures in hand paresthesia (VCL, VIM), finger paresthesia (VCL, VCM, VIM) and paresthesia with location not formally identified by the patient (VCM, VIM).

The application of our method to two patients has shown its feasibility. Our results are consistent with anatomical knowledge that stimulation of VCL and VCM induce paraesthesia in the body and the face respectively. Nevertheless, more patient data have to be analysed to draw any conclusions on the regions responsible for inducing the side effects. The present method will allow an optimised data exploration than by taking into account only the anatomical position of the center of the measurement electrode.

1 

 1 A method for side effect analysis based on electric field simulations for intraoperative test stimulation in deep brain stimulation surgery Pison D. 1 , Alonso F. 2 , Wårdell K. 2 , Shah A. 1 , Coste J. 3 , Lemaire J.-J. 3 , Schkommodau E. 1 ,

	Hemm-Ode S. 1
	1. Institute for Medical and Analytical Technologies, School of Life Sciences, Muttenz, SWITZERLAND;
	2. Department of Biomedical Engineering, Linköping University, Linkôping, SWEDEN;
	3. Image Guided Clinical Neuroscience and Connectomics (EA7282, UdA), Clermont-Ferrand, FRANCE.


Elisabeth Piotelat
email: elisabeth.piotelat@limsi.fr

Keywords: internationalization, SETI, web technologies Acronyms/Abbreviations Communication with Extraterrestrial Intelligence (CETI) Internet Protocol (IP) Nançay Radio Telescope

The Search for Extraterrestrial Intelligence (SETI) has been popularized in France by Jean Heidman and François Biraud since 1980. To increase public interest in SETI, a French version of the SETI League website has been created in 1995. In 1999 the SETI@Home website was the second one in French. Amateurs need help in their native language to build antennas, use software or access open data. Web2.0 technologies and social networking have changed the way people find information about SETI. Ten years of statistics of a blog in French about SETI, Space and education have been analysed. Even if translation tools are widely used, some journalists or students won't read a scientific paper, a web page or even a tweet in English. Statistics show the need for translation and adaptation with the appropriate vocabulary. Outreach public events, popular books or scientific workshops about the search for biological or technological signatures have been organized in France during last decade. None of them use the SETI acronym even if the main question remains "Are we alone?". The French vocabulary used in the titles of those events shows that translating SETI into French presents a social aspect.

Introduction

The recent discoveries of exoplanets have increased the public interest about life in the universe. A lot of academics find it difficult to keep a scientific approach when answering questions such as "are we alone?" [START_REF] Foucher | A Statistical Approach to Illustrate the Challenge of Astrobiology for Public Outreach[END_REF]. According to Tarter [START_REF] Tarter | The 21th century : The century of biology on earth an beyond[END_REF], the search for extraterrestrial intelligence (SETI) is in fact a search for technology, as a proxy for intelligence. The words chosen are important when translating information in other languages or speaking about life in the universe to general public from an other cultural environment. A great effort of internationalization has been done by the IAA. The multilingual space dictionary [START_REF]English, Chinese, French, German, Japanese and Russian Space Lexicon[END_REF] has entries such as "CETI" (p104), "radio astronomy" (p380) or "SETI" (p412).

Such a dictionary is useful to translate or adapt technical and scientific information in diverse languages being as accurate as possible. With 77 millions native speakers [START_REF]Ethnologue, language of the world[END_REF], French language is spoken by 2.8% of internet users [START_REF]Internet world users by language[END_REF]. They are located in Europe (52% are in France), Africa (Cote d'Ivoire 5,8%, Cameron 5,7%, Congo 4,7%, etc...) or America (Canada 4,7%) as shown in Fig 1. This paper focuses on two websites dealing with SETI in French. The first one [6] is only a translation of some of the pages of the SETI League website [7]. It has been created in 1995 and uses web1.0 technology. The second one [START_REF]La vie, l'univers et le reste[END_REF] is a blog created in 2007 entitled "Life, the universe and everything" after Douglas Adams science-fiction novel [START_REF] Adams | Life, the universe and everything[END_REF]. At 2018/09/01, 987 posts are published, 153 are labelled SETI (see Table 2). The other available labels are:

-education: posts regarding museum visits or debates about French educative politics.

-space: videos about rocket launches or pictures of ISS passing over my home.

-science-fiction: book or film reviews.

-science: astronomy, physics, biology, etc... -everything: pictures about travels or local events, social movements, personal thoughts, etc.. computers with IP address located in French speaking countries (France, Canada, Belgium, Switzerland) but also USA, Germany, Ukraine, etc... (see Table 7)

SETI public research in France 1970-1996

The cultural, historical and sociological context matters. The Nançay Radio Telescope (NRT) will be the host of a SETI survey at the end of 2018. The announcement [START_REF] Hellbourg | A SETI pilot survey with the Nançay Radio Telescope[END_REF] of the pilot program entitled "A Search for Technosignatures from Nearby Stars and Galaxies" by the Breakthrough Listen Team has not yet been shared by Paris Observatory on official websites or social networks.

The NRT was inaugurated the 17 th of May 1965 by the French president Charles De Gaulle. The country was rather proud to have one of the best instrument in the world. During the cold war, new maths had been taught to pupils in order to increase the number of researchers and engineers. Every children born in the 60's has watched the Kraus antenna on one of the three official TV channel.

The first popular book dealing with SETI in French was written by two astronomers, François Biraud and Jean-Claude Ribes in 1970 [START_REF] Biraud | Le dossier des civilisations extraterrestres[END_REF]. From 1970 to 1972, the first search for extraterrestrial intelligence at NRT targeted 5 OH Masers sources. [START_REF] Paschenko | A Search for OH Emission from Infrared Objects at 1612 1WE[END_REF][START_REF]SETI 2020[END_REF].

The second popular book in French about SETI was published in 1979 [START_REF] Ducrocq | La Chaîne bleue[END_REF]. The author, Albert Ducrocq, was a pioneer in cybernetics. He created a society called "SETI-France", changing the acronym's definition for "Société d'Etude pour une Télédétection de l'Intelligence" (see appendix A for translation). A first meeting was organised during the 33st Paris Air Show in Le Bourget the 11 th of Juny 1979 and a conference took place at UNESCO in November 1979. [START_REF] De La Cortardière | Communiquer avec d'autres intelligences?[END_REF] The next SETI observations in Nançay began in 1981 [START_REF] Biraud | SETI at the Nançay radiotelescope[END_REF]. François Biraud and Jill Tarter targeted 344 stars of type F,G,K. The runs were in June 1981, August 1984, April 1987, October 1987, January and April 1988. [START_REF] Biraud | Conclusion of the selected target search at the Nançay Observatory[END_REF].

Some technical improvements of the NRT were required in order to use the NASA megachannel spectrum analyser [START_REF] Gulkis | Technical considerations on using the large Nancay radio telescope for SETI[END_REF].

François Biraud and Sabine Airieau conducted a survey of 4 stars with giant planets in 1996 at NRT, using the same protocol than the 1981-1988 observations [START_REF] Biraud | A search for artificial signals from the newly discovered planetary systems[END_REF][START_REF]SETI 2020[END_REF].

The new focal and receiver system project for the NRT ended in 2000. [START_REF] Granet | A new dual-reflector feed system for the Nancay radio telescope[END_REF] Even if Nancay is mentioned in the list of places for the HRMS project, no target search took place. [START_REF] Tarter | HRMS: Where We've Been, and Where We're Going[END_REF] 1.2 Amateurs 1994-now Tim Berners Lee invented the World Wide Web in 1989 [START_REF] Berners-Lee | Information Management: A Proposal[END_REF]. The first website appeared in 1993 [START_REF]World Wide Web[END_REF]. The total number of websites now is around 1900 millions, but in 1994 there was only 2,738 online websites. [START_REF]Internet live stats[END_REF]. The idea of using web technologies to share data with citizen and ask them to be active in scientific projects came very early.

SETI League

The SETI League [7] was founded in 1994 in response to Congress terminating all NASA SETI funding in October of 1993. The setileague.org domain has been registered this same year. For the first time, people all around the world could be member of a SETI organization, and later participate in the Argus project.

The French version of the SETI League website was created in 1995 [6]. At a time where Google didn't exist, requests about SETI usually came by email or written letters. Inviting people or journalists to look at a web page written in English didn't work. Translating the technical manual or press releases was the only way to make it accessible to non English readers. For several years it has been the only website in French dealing with SETI.

SETI@HOME

The second website about SETI in French is also a translation written by an amateur wishing to promote the SETI@HOME software developed by the Berkeley University [START_REF] Sullivan | A new major SETI project based on Project Serendip data and 100,000 personal computers[END_REF].

The software did get a lot of media interest in France when it was released in 1999. A strong community of French speaking users shared there results using mailing lists. The BOINC software enable the creation of teams. One of them is called "Alliance Francophone" and has a very active forum.

Material and methods

The data can be downloaded for further studies [26]. The files will be updated after the IAC2018 conference. The following figures are those till end of august 2018.

Classification.

If all the posts of the Setileague website deal with SETI, it's not the case for the blog [START_REF]La vie, l'univers et le reste[END_REF]. Each post has been manually checked to fit into 1 to 3 labels described in introduction.

It was difficult to use algorithms that look for the word "SETI" in order to put the corresponding label. For instance only 20 titles among 153 contain the 4 letters of SETI. Some texts labelled SETI don't use the acronym. That's the case for the "Challenge Yourself" blog post relating one of the presentations of the workshop : "What's life? An extraterrestrial perspective" organized by METI international in March 2017 in Paris. The same text in an other context could have the label "science".

The classification of all texts can be found for each year in the statisticblog.ods file [26].

IAC-18-F1.2.3

Statistics 2.2.1 Country

The Webalizer software gives 12 months of statistics for the French version of the Setileague website [6]. The number of access by country has been reported in the setileague.ods file. Some data about language and internet access from [START_REF]English, Chinese, French, German, Japanese and Russian Space Lexicon[END_REF][START_REF]Internet world users by language[END_REF] have been added in the file [26], as well as country participation for the SETI WOW event organized each year in Germany [START_REF]Country statistics for the WOW[END_REF].

The blogger software gives only the 10 top countries by day, week, month and year. It has not been taken into account. [START_REF] Croll | Stats Displays Are Significant, In Their Own Context[END_REF]

Tools for statistics

According to Webalizer man page [START_REF]Webalizer man[END_REF], the definition of a "hit" is the following : "This number represents the total number of requests that were made to the server during the specified report period.". It represents the number of requests done by the user, no matter if it's for a picture, a style sheet or a file that doesn't exist on the server.

A bot, a spider or a spammer will generate a great number of hits. This could explain the high number of hits from Russia for instance. Webalizer can answer questions such has "from which country do visitor come" but not "how many real humans visited this website".

As the blogger tool is not a third party visitors meter depending on browser configuration, it's not filtered by readers. [START_REF] Croll | What Does Stats Provide, That Third Party Visitor Activity Meters Cannot Provide?[END_REF] It gives the number of access to each page. This will be use only for comparison between SETI labelled pages and the others since an automatic tool such as spider or bot will visit all the pages.

A python script is used to compute statistics.

Hypertext

The French version of the setileague website [6] is a bridge to the official website [7]. Each translated page has a link to the original version.

The blog [START_REF]La vie, l'univers et le reste[END_REF] is a kind of gate to some 1119 different other websites. More than 10% of the SETI related texts have links to scientific publications. There's also several posts to report or announce workshops or conferences.

There's also some links to major organizations contributing to the search for technosignatures.

Vocabulary

"SETI" is something not really clear for most people. It's not only a screen saver, it's not just an institute in the USA or a NASA programme ended in 1993. In France, journalists or translators often put an article before the acronym "Le" SETI. This expression "The SETI" shows that most people misunderstand the search for extraterrestrial life.

Things could change with a survey conducted in France.

As noted by J.M. Contant. the work of the IAA and the International Institute of Space Law about SETI has been formally and definitely recognized by the United Nations on December 8, 2000 and the the personal contribution of Jean Heidmann was essential [START_REF] Contant | IAA: 35 years of search for extraterrestrial intelligence, tribute to Jean Heidmann[END_REF] In popular books written before the use of the term astrobiology, J. Heidmann uses SETI to speak about all the surveys from Ozma to HRMS [START_REF] Heidman | La vie dans l'univers[END_REF], but introduced bioastronomy defined by IAU few years before [START_REF] Marx | Bioastronomy -search for extraterrestrial life[END_REF].

A popular book about Fermi paradox has been recently published by CNRS editions. Chapter 3 has for title "Short story of the radio survey of the galaxy". If we except the references to SETI Institute (3), SETI League (3), SETI@HOME and SETI Hacker, the "SETI" acronym appears only 13 times within the 200 pages. [START_REF] Agelou | Où sont-ils ? Les extraterrestres et le paradoxe de Fermi[END_REF] The June 2017 discussion about WOW signal is mentioned but there is not a single word about Breakthrough Listen.

To include interdisciplinary contributions, the IAA workshops organized since 2008 had for title "Search for life signatures" (see table 1). There have been a lot of email debate about the title. The SETI acronym was removed in order to increase the participation of biologists and astrophysicists. An analysis of the words used in IAA workshops titles according to first speaker nationality is done. The links are gathered in a links.txt file by using the following bash command :

grep http 20* | cut -d'/' -f3 | sort > links.txt

Word used by professional to speak about SETI

All the presentations of the six search for life symposiums have been gathered in a bibtex file. [36] . The php bibtexbrowser tool is used to look for particular words or authors. [37] The bibtexreader.py program looks for words used in the title of the presentations of the IAA search for life signature and counts the number of occurrences of each term. It has also been used with the list of paper of the first symposium published in Acta Astronautica [38]

Results

Analysis of blog visits

Table 3 represents the number of posts, the mean and the median values of the number of visits per topic.

The median and means value are similar for texts labelled SETI or not.

There's not enough texts with two labels (for instance SETI+ EDU) to conclude that it's better to mix topics and deals with SETI and education than just for writing about SETI. The median value is more significant than the mean value. The most successful post is about the Lego version of Margaret Hamilton and has more than 15000 visits. The distribution of posts doesn't show differences between texts about SETI and not. A quarter of posts of both categories has between 50 and 100 visits.

Links in the blog

The blog has 24 SETI related posts with references to scientific publications and 31 about conferences starting with the 2008 UNESCO search for life signature symposium and ending with Breakthrough Discuss 2018. The most scholar websites linked are represented in table 5. Table 6 shows the number of links to main SETI and research organizations or institutes.

French speaking amateur activity

The French BOINC users are gathered in a team called "Alliance Francophone". There's 21729 users listed in this team. 9893 of them participate in the SETI@HOME project and 529 are active [START_REF]SETIBzH statistics[END_REF].

The French version of the SETI League website is accessed by main French speaking countries (see fig 4). Visitors mainly come from France, Belgium, Canada and Switzerland.

There's also very active SETI amateurs in theses countries according to the SETI League roster [START_REF]La vie, l'univers et le reste[END_REF] or to the participation of the 2018 latest WOW event [START_REF]Country statistics for the WOW[END_REF].

Words used by French scientists

Figure 5 shows the words used in the title of the six Search for Life Signatures symposiums (see Table 1). The 180 titles represent 1533 words. The titles with first author from France represent 190 words (12,39%). The SETI acronym appear 65 times. If we only took titles from France we count 14 times "SETI" (21,54%).

Very common words such as "life" (20 times), "extraterrestrial" [START_REF] Paschenko | A Search for OH Emission from Infrared Objects at 1612 1WE[END_REF], "interstellar" [START_REF] Biraud | Le dossier des civilisations extraterrestres[END_REF], "radio" (9), "signal" (7) are used once or not by authors from Fance.

A selection of papers from the first symposium at UNESCO in 2008 have been published in Acta Astronautica [38]. Among the 18 published papers only 2 (11%) are from France [START_REF] Rospars | Terrestrial biological evolution and its implication for SETI[END_REF][START_REF] Raulin-Cerceau | The pioneers of interplanetary communication: From Gauss to Tesla[END_REF], whereas 25% of the Unesco talks have been presented by local speakers.

The bibtexreader.py programm has been used to analysed the 3452 words present in the titles, keywords and abstracts of the acta2010.bib file. [35] Among these, the citation from [START_REF] Rospars | Terrestrial biological evolution and its implication for SETI[END_REF][START_REF] Raulin-Cerceau | The pioneers of interplanetary communication: From Gauss to Tesla[END_REF] represent 533 words (15,44%). SETI appears 33 times but only 3 times in French papers.

Discussion

Classification.

The search for extraterrestrial intelligence is an interdisciplinary research. The classification of SETI related papers or blog posts would be different when made by people of different background. A further study could consist in presenting texts to citizens or scientists and asking if this has to do with search for technological signatures, biological signature, extraterrestrial life, science, space, science-fiction or something else.

Culture or vocabulary translation.

The word "alien" has been popularized in France by the film directed by Riddley Scott in 1979. For most people in France, it's a synonym of "extraterrestrial". It doesn't represent something else, the foreigner.

Steven Spielberg film "E.T. the extraterrestrial" is also very famous. When we ask to French people "If I say extraterrestrial, what picture do you see ?", the answer is not the same for several groups. On the "Alliance Francophone" BOINC forum users, 26% see a bacteria and 20% a little grey or green entity. On a Facebook group for gifted people, 79% choose "a planet, a star, everything beyond the Earth". This kind of survey should be done on a wider scale.

When asking young children "can you give me the name of a cartoon with an extraterrestrial ?" they hesitate. Most of them never saw "E.T." or the 1977 Steven Spielberg film "Close Encounters of the Third Kind".

François Biraud, Albert Ducrocq and Jean Heidmann have popularized SETI in France between 1970 and 1996. Even the Nançay radiotelescope is the place for regular SETI surveys [START_REF] Hellbourg | A SETI pilot survey with the Nançay Radio Telescope[END_REF] it may be harder to speak about a candidate signal to scientists, journalists or citizen without knowing what they imagine when reading word the extraterrestrial.

What is a country?

The webalizer software use the IP address to locate the visitors. Technologies such as proxy and virtual private network (VPN) could hide the real position of the user.

The SETI League roster uses postal address to find the country. The BOINC users have to choose a country . They can also prefer to stay "international user". The country could be different of the nationality and the links between countries and language spoken should be more accurate if done by sociologists or linguists.

Conclusion

Translating SETI into French means choosing the vocabulary and taking into account societal and cultural aspects.

IAC-18-F1.2.3

Page 6 of 8

3 Page 1 of 8 Fig 1 :

 381 Fig 1 : map of French speaking countries

Figure 2 :

 2 Figure 2 : number of posts regarding SETI and not according to the number of visits (0-5, 5-10, etc...)

Fig 3 :

 3 Fig 3: distribution of visits

Fig 5 :

 5 Fig 5: words in symposium titles. The yellow disks represent words used by French authors with a rate up to 12,39%

Table 1 :

 1 Search for Life Signature workshops

	Year	City, country		Talks	Local
						speakers
	2008 Paris, FR		59	25%
	2010 London, UK		32	28%
	2011 St Petersburg, RU	21	71%
	2012 San Marino, IT	37	29%
	2014 Paris, FR		16	12%
	2015 Paris, FR		16	13%
	3. Calculation
	3.1 Analysis of blog visits
	A python program called lecturecsv.py reads the
	staticsblog.ods file and computes the number of blog
	posts for each combination of items. [35]
	Table 2. Number of posts for each label
		SETI Space Science	Education	SF
	SETI	127
	Space		2	27
	Science		-	1	91
	Education	6	3	10	129
	SF		13	1	8	4	50

Table 3 .

 3 Statistical values for the blog[START_REF]La vie, l'univers et le reste[END_REF]

	label	posts	means	medians
	SETI	154	162	136
	Non SETI	826	214	124
	SETI + Edu	6	358	305
	SETI + Space	2	149	212
	SETI + SF	13	162	148
	Only SETI	132	152	81

Table 4 .

 4 Statistical values for the blog[START_REF]La vie, l'univers et le reste[END_REF]

	label	posts	means	medians
	SETI	154	162	136
	Scholar	24	168	194
	Conf	31	120	248

Table 5

 5

	. scholar websites
	domain	links
	arxiv.org	23
	Doi.org	7
	Nature.com	5
	Table 6. SETI and research domains
	domain	links
	Setileague.free.fr	21
	Setileague.org	10
	Seti.org	16
	Berkeley.edu	15
	Nasa.gov	9
	Iaaweb.org	8
	Esa.int	8
	nrao.edu	6
	Iaaseti.org	5

Table 7 .

 7 SETI activity of amateurs.

	Country	WOW	SETI	Visits
		2018	League	during last
			members	12 months
	France	30	7	10055
	Belgium	1	4	509
	Canada	14	43	467
	Switzerland	3	5	122
	It's impossible to know if visitors of Belgium,
	Canada or Switzerland speak French, Flemish, English,
	Italian or German.
	Few visits come from countries with very low
	internet penetration [5]. There's no SETI League
	members or WOW 2018 participation.
	Table 8. Countries with L1 French speakers and low
	internet penetration visiting [6].
	Country	L1 speakers	Internet access
	Madagascar	120000	5%
	Mali	15000	12%
	Benin	16700	12%
	Haiti		600	12%
	Burkina-Faso	170000	14%
	Senegal	47000	25%
	Côte d'Ivoire	45000	26%
	Guadeloupe	7300	46%

th International Astronautical Congress (IAC), Bremen, Germany, 1-5 October 201_. Copyright ©2018 by the International Astronautical Federation (IAF). All rights reserved.

Copyright ©2018 by the International Astronautical Federation (IAF). All rights reserved.

The statistics show a need for pages in French and of bridges between scholars and citizens. Popularizing the words such as "bioastronomy" or "technosignature" is important to clarify what SETI people really do. It could also help scientists looking for biological signatures to explain their research goal in outreach events or to talk in SETI workshops.

Amateur activities worldwide have to be promoted. The communities such as "Alliance Francophone" play an important role to maintain interest about BOINC and SETI@HOME. More than 20 years after the creation of the French version of the Setileague website, educative documents such as the technical manual is still downloaded.

Websites without advertisement, blogs and open forums enable the share of knowledge as dreamed by Tim Berners-Lee. The professional and amateurs working for SETI have to enhance internationalization.

Appendix A (French words used in the paper)

La vie, l'univers et le reste : Life, universe and everything. [START_REF]La vie, l'univers et le reste[END_REF] Le dossier des civilisations extraterrestres : The files of extraterrestrial civilisations. [START_REF] Biraud | Le dossier des civilisations extraterrestres[END_REF] La chaîne bleue : The bleue channel [START_REF] Ducrocq | La Chaîne bleue[END_REF] Société d'Etude pour une Télédétection de l'Intelligence: Society for the studies for a remote sensing of intelligence.


Christophe Chesneau
email: christophe.chesneau@unicaen.fr

Some tight polynomial-exponential lower bounds for an exponential function

Keywords: Algebraic bounds, exponential function. 2000 MSC: 33B10, 26D07

This note is devoted to new sharp lower bounds for exp(x 2 ). We first introduce and study a new lower bound defined with polynomial of degree 2 and exponential (or hyperbolic) functions. Then we propose two improvements of this lower bound by using two different approaches; the first approach consists in adding well-chosen polynomial term to it, whereas the second approach aims to transform it for large values of |x|. We show that they are better to well-known lower bounds. The analytic results are supported by some numerical studies and graphics. A part of the study is devoted to some integral methods having the ability to generate new lower bounds for exp(x 2 ).

Introduction

Inequalities involving exponential functions are useful in all the areas of mathematics. The most famous of them can be found in [START_REF] Mitrinović | Elementary Inequalities[END_REF], [START_REF] Bullen | A Dictionary of Inequalities[END_REF] and [START_REF] Kuang | Applied Inequalities, 2nd edition[END_REF]. See also [START_REF] Qi | A method of constructing inequalities about e x[END_REF], [START_REF] Bae | Optimal polynomial lower bounds for the exponential function[END_REF], [START_REF] Kim | Densely algebraic bounds for the exponential function[END_REF] and [START_REF] Bae | On a generalization of an upper bound for the exponential function[END_REF] for current developments on lower and upper bound for exp(x). The purpose of this note is to provide simple and tight lower bounds for exp(x 2 ). Such bounds are important tools to evaluate lower or upper bounds of mathematical terms involving exp(x 2 ). Basic examples include the functions cosh(x 2 ), sinh(x 2 ) and tanh(x 2 ), the integral x 0 exp(y 2 )dy, the sum ∞ k=0 exp(-k 2 ), the cumulative distribution function of the Gaussian or Kolmogorov distributions. Well-known lower bounds for exp(x 2 ) are cosh( √ 2x), sinh √ 6x /( √ 6x), exp(x) -x, 1 + x 2 + x 4 /2 and 1 + x 2 /a a with a > 0. The last one is sharp for |x| large only for large values of a, corresponding to a polynomial with a high degree and large coefficients when a is an integer. Recent sharp lower bounds can be found in [START_REF] Bagul | Inequalities involving circular, hyperbolic and exponential functions[END_REF] for x is a an interval of the form [0, b] with a precise value for b.

The motivation of this paper is to introduce new sharp lower bounds for exp(x 2 ) defined with simple functions, at least uniformly better to the two benchmarks: cosh( √ 2x) and 1 + x 2 + x 4 /2, for all x ∈ R. In a first part, a first lower bound is introduced. It is defined as an even function on R with simple polynomial of degree 2 and exponential functions (without power of x 2 ). Comparison to cosh( √ 2x), exp(x) -x and 1 + x 2 + x 4 /2 are made analytically and with the used of graphics. Then we propose two significant improvements of this lower bound via two different approaches. The first approach aims to add well chosen polynomial terms to the former lower bound. The second approach adopts the transformation suggested in [START_REF] Chesneau | On two simple and sharp lower bounds for exp(x 2 )[END_REF]. It consists in weighting and translating the former lower bound when |x| is large enough. Only polynomial of degree 2 and exponential functions are used. In each case, the theoretical results are supported by a short numerical study and some graphics, illustrating the tightness of the new lower bounds.

1 Finally, we present new integration approaches to generate lower bounds for exp(x 2 ) using existing lower bounds. The link existing with this approach and the main lower bound of the study is discussed.

The rest of this note is structured as follows. Section 2 presents our main lower bound. Section 3 is devoted to an improvement of this lower bound. Another improvement is developed in Section 4. Section 5 presents some integral approaches to determine lower bounds for exp(x 2 ). All the proofs of our results are postponed in Section 6.

Lower bound

The main lower bound for exp(x 2 ) is presented in the proposition below.

Proposition 1. Let us define the function f (x) by

f (x) = 1 2 exp √ 2x (7 -4 √ 2x + 2x 2 ) + exp - √ 2x (7 + 4 √ 2x + 2x 2 ) -12 .
(1)

Then, for all x ∈ R, exp(x 2 ) ≥ f (x) ≥ 1.
Let us observe that f (x) is a continuous even function on R using simple polynomial-exponential functions. It is of the form f

(x) = (1/2)(G(x) + G(-x) -12), where G(x) = exp √ 2x (7 -4 √ 2x + 2x 2 ).
It can be also express in terms of hyperbolic cosh and sinh functions as:

f (x) = (7 + 2x 2 ) cosh √ 2x -4 √ 2x sinh √ 2x -6. (2)
A visual comparison between exp(x 2 ) and f (x) is performed in Figure 1, for x ∈ [-1, 1] for the first graphic and x ∈ [-2.5, 2.5] for the second graphic. We can see that the two curves are relatively close, specially for small value for |x|. This comment is also supported by Table 1 which Table 1 The tightness of f (x) is highlighted in Proposition 2 below; we proves that f (x) is uniformly greater to most well-known simple lower bounds for exp(x 2 ): cosh

: Numerical evaluations of R(b) for b ∈ {0.5, 1, 1.5, 2, 2.5, 3}. b = 0.5 b = 1 b = 1.5 b = 2 b = 2.5 b = 3 R(b) 1.
√ 2x , exp(x) -x, 1 + x 2 + x 4 /2.
Proposition 2. Let f (x) be the function given by (1). We have, for all x ∈ R,

f (x) ≥ max cosh √ 2x , exp(x) -x, 1 + x 2 + x 4 2 .
Hence f (x) can be viewed as the best lower bounds among cosh √ 2x , exp(x) -x and 1 + x 2 + x 4 /2. Figure 2 illustrates this result by considering two intervals of values for x : [-1, 1] and [-3, 3], one for each graphics. It shows that f (x) is closer to exp(x 2 ) is comparison to cosh √ 2x , exp(x) -x and 1 + x 2 + x 4 /2.

-1.0 -0.5 0.5 1.0 1.5 2.0 2.5 f(x) 1] in the first graphic, and for

1 + x 2 + x 4 2 cosh 2 x exp(x) -x -3 -2 -1 1 2 3 10 20 30 40 50 f(x) 1 + x 2 + x 4 2 cosh 2 x exp(x) -x Figure 2: Superimposed curves of f (x), 1 + x 2 + x 4 /2, cosh √ 2x and exp(x) -x for x ∈ [-1,
x ∈ [-3, 3] in the second graphic.

Since cosh

√ 2x ≥ 1 or exp(x) -x ≥ 1 or 1 + x 2 + x 4 /2 ≥ 1, Proposition 2 implies that f (x) ≥ 1, which is the second inequality in Proposition 1.
If we consider the polynomial of degree 6: 1 + x 2 + x 4 /2 + x 6 /6, which is also a loser bound for exp(x 2 ), we have f (x) ≥ 1 + x 2 + x 4 /2 + x 6 /6 for some x, but their exists x such that the reverse holds. This motivates the study of some improvements of f (x), which concerns the rest of the study. Two modifications are proposed: adding a well-chosen polynomial term to f (x) or transforming f (x) for |x| large.

Remark 1. Let us mention that some continuous even upper bounds for exp(x 2 ) using hyperbolic cosh and sinh functions can be proved. An example with |x| bounded is the following: For any a > 0 and |x| ≤ a, we have exp(x 2 ) ≤ cosh (ax) + (x/a) sinh (ax) (≤ exp(a|x|)).

First improvement of the lower bound

First of all, let us consider an intermediary result which can be viewed as an improvement of the wellknown inequality: for all x ∈ R, exp(x 2 ) ≥ cosh √ 2x .

Lemma 1. For all x ∈ R and n ∈ N, we define the polynomial P n (x) by

P n (x) = 2x 4 n k=0 x 2k k!(2k + 3)(k + 2)
Then, for all x ∈ R and n ∈ N, we have exp(x 2 ) ≥ cosh √ 2x + P n (x).

Since P n (x) ≥ 0 for all x ∈ R and n ∈ N, it is clear that exp(x 2 ) ≥ cosh √ 2x + P n (x) ≥ cosh √ 2x . Let us observe that P n (x) satisfies the recurrence relation:

P n (x) = P n-1 (x) + 2x 2(n+2) /(n!(2n + 3)(n + 2)), with P 0 (x) = x 4 /3. Expressions of P n (x) for several values of n are given in Table 2.
Lemma 1 is a key tool to the proof of the following proposition in which we determine a tight lower bound of exp(x 2 ) uniformly greater to f (x).

n = 0 n = 1 n = 2 n = 3 P n (x) 1 3 x 4 1 3 x 4 + 2 15 x 6 1 3 x 4 + 2 15 x 6 + 1 28 x 8 1 3 x 4 + 2 15 x 6 + 1 28 x 8 + 1 135 x 10 n = 4 n = 5 1 3 x 4 + 2 15 x 6 + 1 28 x 8 + 1 135 x 10 + 1 792 x 12 1 3 x 4 + 2 15 x 6 + 1 28 x 8 + 1 135 x 10 + 1 792 x 12 + 1 5460 x 14
Proposition 3. For all x ∈ R and n ∈ N, we define the polynomial Q n (x) by

Q n (x) = 4x 6 n k=0 x 2k k!(2k + 3)(k + 2) 1 2(2k + 5)(k + 3) + x 2 (k + 4)(2k + 7)
.

Let f (x) be the function given by (1). We define the function f * (x; n) by

f * (x; n) = f (x) + Q n (x).
Then, for all x ∈ R and n ∈ N, we have

exp(x 2 ) ≥ f * (x; n) ≥ f (x). Thus f * (x; n) is a better lower to f (x) for exp(x 2 ), for all x ∈ R. Remark that Q n (x) satisfies the recurrence relation: Q n (x) = Q n-1 (x)+4x 2(n+3) /(n!(2n+3)(n+2))[1/(2(2n+5)(n+3))+x 2 /((n+4)(2n+7))], with Q 0 (x) = x 6 /45 + x 8 /42. Expressions of Q n (x)
for several values of n are given in Table 3. The result of Proposition 3 is illustrated in Figure 3. The two graphics consider the two intervals for x: [-1, 1] and [-2.5, 2.5].

n = 0 n = 1 n = 2 Q n (x)
-1.0 -0.5 0.5 1.0 Proposition 4. Let f (x) be the function given by (1). For any a ≥ 0, we define the function f (x; a) by

1.5 2.0 2.5 exp x 2  f (x, 0) f(x) -2 -1 1 2 20 40 60 80 100 exp x 2  f (x, 0) f(x)
f • (x; a) = f (x)1 {|x|<a/2} (x) + f (|x| -a) exp(2a|x| -a 2 )1 {|x|≥a/2} (x),
where 1 A (x) denotes the indicator function over A, i.e. 1 A (x) = 1 if x ∈ A and 0 elsewhere. Then, for all a ≥ 0 and x ∈ R, we have

exp(x 2 ) ≥ f • (x; a) ≥ f (x).
Another look of the function f

• (x; a) is given by f • (x; a) = max f (x), f (|x| -a) exp(2a|x| -a 2 ) . Note that f • (x; a) is an even continuous function on R. It follows from Proposition 4 that f • (x; a) is a better lower to f (x) for exp(x 2
), for all a ≥ 0 and x ∈ R. Figure 4 proposes a graphical illustration of Proposition 4. The two graphics consider the two intervals respectively: [-1, 1] and [-3, 3]. We see that exp(x 2 ) and f • (x, a) with a = 1 are near confounded for the considered values for x, showing the sharpness of the lower bound.

Owing to Proposition 2, we have

f • (x; a) ≥ max cosh √ 2x , 1 + x 2 + x 4 2 1 {|x|<a/2} (x) + max cosh √ 2(|x| -a) , 1 + (|x| -a) 2 + (|x| -a) 4 2 exp(2a|x| -a 2 )1 {|x|≥a/2} (x).
Therefore, defining with the same a, this lower bound is sharper to the lower bounds exhibited in Chesneau (2017).

-1.0 -0.5 0.5 1.0 Table 5 shows the numerical values of the error measure: R

1.5 2.0 2.5 exp x 2  fo x, 1) f x) -3 -2 -1 1 2 3 100 200 300 400 exp x 2  fo x, 1) f x)
• (b) = b -b f • (x; a) -exp(x 2 ) 2 dx, for a = 1
and several values for b.

The value a = 1 in the numerical study is arbitrary chosen; one can find a more optimal value for the problem for a given criteria of optimization. A comparison of the two lower bounds f * (x; n) and f • (x; a) is now discussed. When |x| is small, say |x| < a/2 for a fixed a, we have 4, and Tables 4 and5, we claim that f • (x; n) is better for some a and n.

f * (x; n) ≥ f (x) = f • (x; a), so f * (x; n) is better. When |x| is large, in view of Figure
As a direct applications of our results, let us mention that sharp polynomial-exponential lower bounds for exp(x 2 ) give sharp polynomial-exponential lower bounds for cosh(x 2 ) or sinh(x 2 ); using the inequality exp(y) ≥ 1 + y, for all y ∈ R, for any ψ

(x) ∈ {f (x), f * (x; n), f • (x; a)}, for all x ∈ R, we have cosh(x 2 ) = exp(x 2 ) + exp(-x ) 2 ≥ 1 2 ψ(x) + max(1 -x 2 , 0) .
On the other hand, using the inequality: exp(-y) ≤ 1 -y + y 2 /2, for all y ≥ 0, for any x ∈ R, we have

sinh(x 2 ) = exp(x 2 ) -exp(-x 2 ) 2 ≥ 1 2 ψ(x) -max 1 -x 2 + x 4 2 , 1 .
These inequalities can be usefull in various mathematical settings.

On some generators of lower bounds

We now present and discuss some general approaches based on integration to generate new lower bounds for exp(x 2 ) from existing lower bounds. When it is possible, conditions are mentioned to improved the tightness of the former lower bounds.

First integral approach

The main result is described in the proposition below.

Proposition 5. Let θ(x) be a positive function on R and ω(x) be the function defined by

ω(x) = |x| 0 y 0 2(1 + 2t 2 )θ(t)dt dy + 1. • Suppose that exp(x 2 ) ≥ θ(x) for all x ∈ R. Then exp(x 2 ) ≥ ω(x) for all x ∈ R.
• Suppose that θ(x) is even, two times differentiable with θ(0) = 1, θ (0) = 0 and 2(1+2x 2 )θ(x)-θ (x) ≥ 0 for all x ≥ 0. Then, for all x ∈ R, we have

ω(x) ≥ θ(x).
Under the assumptions of the two points above, we have

exp(x 2 ) ≥ ω(x) ≥ θ(x).
So ω(x) is a better lower bound to θ(x).

Connections between Proposition 5 and 1 exist. Indeed, let us consider the well-known lower bound for exp(x 2 ): θ(x) = cosh √ 2x . We have θ (x) = √ 2 sinh √ 2x and θ (x) = 2 cosh √ 2x . Therefore θ(0) = 1, θ (0) = 0 and 2(1 + 2x 2 )θ(x) -θ (x) = 4x 2 cosh √ 2x ≥ 0. It follows from Proposition 5 that a better lower bound of θ(x) = cosh √ 2x is given by

ω(x) = |x| 0 y 0 2(1 + 2t 2 )θ(t)dt dy + 1 = |x| 0 -4y cosh √ 2y + 3 √ 2 sinh √ 2y + 2 √ 2y 2 sinh √ 2y dy + 1 = (7 + 2x 2 ) cosh √ 2x -4 √ 2x sinh √ 2x -6.
We thus obtain the hyperbolic expression of the lower bound f (x) given by ( 2). Naturally, the first point of Proposition 5 can be used to generate new lower bounds for exp(x 2 ). For instance, using the inequality exp(y) ≥ 1 + y for all y ∈ R, we have exp(

x 2 ) = exp(x 2 -|x|) exp(|x|) ≥ (1 + x 2 -|x|) exp(|x|). Let us set θ(x) = (1 + x 2 -|x|) exp(|x|).
Hence a new lower bound for exp(x 2 ) is given by

ω(x) = |x| 0 y 0 2(1 + 2t 2 )θ(t)dt dy + 1 = |x| 0 exp(y) 136 -134y + 66y 2 -20y 3 + 4y 4 -136 dy + 1 = exp(|x|) 618 -482|x| + 174x 2 -36|x| 3 + 4x 4 -617 -136|x|.
However, note that this lower bound is not better to θ(x). In particular, the assumption 2(1 + 2x 2 )θ(x)θ (x) ≥ 0 for all x ∈ R is not satisfied. This bound is tight but not shaper than f (x) for all x ∈ R. Moreover, from a mathematical point of view, it is more difficult to manipulate to f (x).

Let us mention that the well-known lower bounds: 1 + x 2 + x 4 /2 and exp(|x|) -|x|, also satisfy θ(0) = 1, θ (0) = 0 and 2(1 + 2x 2 )θ(x) -θ (x) ≥ 0 for all x ≥ 0, yielding more sharp lower bounds ω(x) for exp(x 2 ). However, one can show that they are not better to f (x) for all x ∈ R (and the presented improvements).

Generalization

Proposition 6 below presents a generalization of Proposition 5. From two lower bounds θ 1 (x) and θ 2 (x) of exp(x 2 ), one can construct a lower bound better to θ 1 (x) or θ 2 (x), under some assumptions. Proposition 6. Let θ 1 (x) and θ 2 (x) be two positive functions on R and κ(x) be the function defined by

κ(x) = |x| 0 y 0 2(θ 1 (t) + 2t 2 θ 2 (t))dt dy + 1. • Suppose that exp(x 2 ) ≥ max [θ 1 (x), θ 2 (x)] for all x ∈ R. Then exp(x 2 ) ≥ κ(x) for all x ∈ R.
• Suppose that θ 1 (x) and θ 2 (x) are even, two times differentiable with

-θ 1 (0) = 1, θ 1 (0) = 0, 2(θ 1 (x) + 2x 2 θ 2 (x)) -θ 1 (x) ≥ 0 for all x ≥ 0. Then, for all x ∈ R, we have κ(x) ≥ θ 1 (x).
This inequality holds with θ 2 (x) by inverting the role of θ 1 (x) and θ 2 (x) in the definition of κ(x) and the conditions.

-θ 1 (0) = 1, θ 2 (0) = 1, θ 1 (0) = 0, θ 2 (0) = 0, 2(θ 1 (x) + 2x 2 θ 2 (x)) -θ 1 (x) ≥ 0 and 2(θ 1 (x) + 2x 2 θ 2 (x)) -θ 2 (x) ≥ 0 for all x ≥ 0.
Then, for all x ∈ R, we have

κ(x) ≥ max [θ 1 (x), θ 2 (x)] .
Under the assumptions of the first point and the second item of the second point, we have

exp(x 2 ) ≥ κ(x) ≥ max [θ 1 (x), θ 2 (x)] .
So κ(x) is a better lower bound to θ 1 (x) for exp(x 2 ), or both of them.

Taking θ 1 (x) = θ 2 (x) in Proposition 6, we obtain Proposition 5 with θ(x) = θ 1 (x).

Note that, taking θ 1 (x) = cosh √ 2x and θ 2 (x) = n k=0

x 2k /(k!), the first point and the first item of the second point of Proposition 6 are satisfied; we thus obtain Lemma 1.

Another example is given by chosing θ 1 (x) = f (x) and θ 2 (x) = cosh √ 2x . After some calculus, we have

κ(x) = 27 cosh √ 2x -2x 2 3 -2 cosh √ 2x -12 √ 2x sinh √ 2x -26.
Also, one can show that the first point and the second item of the second point of Proposition 6 are satisfied; κ(x) is a better lower bound to f (x). However, the comparison with the proposed improvements for f (x) need further investigations.

Another integral approach

We conclude this section by presenting another generator of lower bounds for exp(x 2 ).

Proposition 7. Let θ(x) be a positive function on R and γ(x) be the function defined by

γ(x) = exp(-|x|) |x| 0 exp(t) [(1 + 2t)θ(t) -1] dt + 1. If exp(x 2 ) ≥ θ(x) for all x ∈ R. Then exp(x 2 ) ≥ γ(x) for all x ∈ R.
For instance, if θ(x) = cosh √ 2x , we have

γ(x) = 8 exp(-|x|) -(7 + 2|x|) cosh √ 2x + √ 2(5 + 2|x|) sinh √ 2|x| .
However, one can show that γ(x) is not better to f (x) for all x ∈ R.

Proofs

Proof of Proposition 1. Let us set g(x) = exp(x 2 ) -f (x). We aim to study this function. It follows from several algebraic manipulations that

g (x) = 2x exp(x 2 ) - 1 2 exp √ 2x (3 √ 2 -4x + 2 √ 2x 2 ) -exp - √ 2x (3 √ 2 + 4x + 2 √ 2x 2 ) . (3)
In order to study the sign of g (x), let us investigate g (x). Algebraic manipulations and simplifications give

g (x) = 2(1 + 2x 2 ) exp(x 2 ) -cosh √ 2x . (4)
Owing to the elementary inequality: exp(x 2 ) ≥ cosh √ 2x , we have g (x) ≥ 0. Thus g (x) is increasing. Since g (0) = 0, the minimum of g(x) is attained in x = 0: for all x ∈ R, we have g(x) ≥ g(0) = 0, implying that exp(x 2 ) ≥ f (x).

The second inequality can be prove in a similar manner. Let us set h(x) = f (x) -1. It follows from several algebraic manipulations that

h (x) = 1 2 exp √ 2x (3 √ 2 -4x + 2 √ 2x 2 ) -exp - √ 2x (3 √ 2 + 4x + 2 √ 2x 2 ) and h (x) = 2(1 + 2x 2 ) cosh √ 2x .
So h (x) ≥ 0 and h (x) is increasing. Since h (0) = 0, for all x ∈ R, we have h(x) ≥ h(0) = 0. Hence f (x) ≥ 1. This ends the proof of Proposition 1.

Proof of Proposition 2. Let us prove that, for all

x ∈ R, f (x) ≥ cosh √ 2x , f (x) ≥ exp(x) -x and f (x) ≥ 1 + x 2 + x 4 /2, in turn. • Proof for f (x) ≥ cosh √ 2x . Let us set k(x) = f (x) -cosh √ 2x .
After calculus and simplifications, we obtain

k (x) = ( √ 2 -2x + √ 2x 2 ) exp √ 2x -( √ 2 + 2x + √ 2x 2 ) exp - √ 2x and k (x) = 4 cosh √ 2x x 2 .
Since k (x) ≥ 0, k (x) is increasing. Since k (0) = 0, for all x ∈ R, we have k(x) ≥ k(0) = 0, implying the desired inequality.

• Proof for f (x) ≥ exp(x) -x. Let us set (x) = f (x) -exp(x) + x. After calculus and simplifications, we obtain

(x) = 1 2 exp √ 2x (3 √ 2 -4x + 2 √ 2x 2 ) -exp - √ 2x (3 √ 2 + 4x + 2 √ 2x 2 ) -2 exp(x) + 2 and (x) = 2(1 + 2x 2 ) cosh √ 2x -exp(x).
We have (x) ≥ 2 cosh √ 2x -exp(x) ≥ 0, so (x) is increasing. Since (0) = 0, for all x ∈ R, we have (x) ≥ (0) = 0, implying the desired inequality.

• Proof for f (x) ≥ 1 + x 2 + x 4 /2. Let us set m(x) = f (x) -(1 + x 2 + x 4 /2). After calculus and simplifications, we obtain

m (x) = 1 2 exp √ 2x (3 √ 2 -4x + 2 √ 2x 2 ) -exp - √ 2x (3 √ 2 + 4x + 2 √ 2x 2 ) -4x(1 + x 2 ) and m (x) = 2(1 + 2x 2 ) cosh √ 2x -2 -6x 2 . Since cosh √ 2x = +∞ k=0 √ 2x 2k /(2k)! > 1+x 2 , we have m (x) ≥ 2(1+2x 2 )(1+x 2 )-2-6x 2 = 4x 4 ≥ 0.
Thus m (x) is increasing. Since m (0) = 0, for all x ∈ R, we have m(x) ≥ m(0) = 0, implying the desired inequality.

The proof of Proposition 2 is complete.

On Remark 1. Let us set ρ(x) = (1/2)[1 + (x/a)]. Since |x| ≤ a, observe that ρ(x) ∈ [0, 1]. Also, we can write x 2 = ρ(x)(ax) + (1 -ρ(x))(-ax).
Owing to the convexity of the function exp(y), we have

exp(x 2 ) = exp [ρ(x)(ax) + (1 -ρ(x))(-ax)] ≤ ρ(x) exp(ax) + (1 -ρ(x)) exp(-ax) = cosh (ax) + x a sinh (ax) .
This completes Remark 1.

Proof of Lemma 1. Let us set p(x) = exp(x 2 ) -cosh √ 2x -P n (x). Then we have

p (x) = 2x exp(x 2 ) - √ 2 sinh √ 2x -4 n k=0 x 2k+3 k!(2k + 3) and p (x) = 2 exp(x 2 ) + 4x 2 exp(x 2 ) -2 cosh √ 2x -4x 2 n k=0 x 2k k! = 2 exp(x 2 ) -cosh √ 2x + 4x 2 exp(x 2 ) - n k=0
x 2k k! .

It follows from the well-know inequalities: exp(x 2 ) ≥ cosh √ 2x and exp(x 2 ) = +∞ k=0

x 2k /k! > n k=0

x 2k /k! that p (x) ≥ 0. So p (x) is increasing and since p (0) = 0, for all x ∈ R, we have p(x) ≥ p(0) = 0. This concludes the proof of Lemma 1.

Proof of Proposition 3. Set g(x) = exp(x 2 ) -f (x) and q(x) = exp(x 2 ) -f * (x; n) = g(x) -Q n (x). Let us recall that g (x) and g (x) have been determined in ( 3) and ( 4) respectively. We have

q (x) = g (x) -4x 5 n k=0 x 2k k!(2k + 3)(k + 2) 1 2k + 5 + 2x 2 2k + 7
and, by (4),

q (x) = g (x) -4(1 + 2x 2 )x 4 n k=0 x 2k k!(2k + 3)(k + 2) = 2(1 + 2x 2 ) exp(x 2 ) -cosh √ 2x -2x 4 n k=0
x 2k k!(2k + 3)(k + 2) .

It follows from Lemma 1 that q (x) ≥ 0, implying that q (x) is increasing and since q (0) = 0, for all x ∈ R, we have q(x) ≥ q(0) = 0. So exp(x 2 ) ≥ f * (x; n). Since Q n (x) ≥ 0 for all x ∈ R and n ∈ N, it is clear that f * (x; n) ≥ f (x). The proof of Proposition 3 is complete.

Proof of Proposition 4. Using x 2 = (|x| -a) 2 + 2a|x| -a 2 and Proposition 1, we have

exp(x 2 ) = exp((|x| -a) 2 ) exp(2a|x| -a 2 ) ≥ f (|x| -a) exp(2a|x| -a 2 ).
Let us set v(x; a) = f (|x| -a) exp(2a|x| -a 2 ). Hence we have exp(x 2 ) ≥ max [f (x), v(x; a)]. We will now show that v(x; a) can be a better lower bound to f (x) for some x. Let us study it according to the variable a. After several algebraic calculus and simplifications, we obtain

∂ ∂a v(x; a) = exp 2a|x| -a 2 w(|x| -a),
where w(y) is the function defined by

w(y) = 1 2 exp - √ 2y (3 √ 2 + 18y + 10 √ 2y 2 + 4y 3 ) + exp √ 2y (-3 √ 2 + 18y -10 √ 2y 2 + 4y 3 ) -12y,
Since exp 2a|x| -a 2 > 0, we need to determine the sign of w(|x| -a). Let us show that w(y) is increasing on R. After several algebraic calculus and simplifications, we have

w (y) = exp - √ 2y (6 + √ 2y -4y 2 -2 √ 2y 3 ) + exp √ 2y (6 - √ 2y -4y 2 + 2 √ 2y 3 ) -12, w (y) = exp - √ 2y (-5 √ 2 -10y -2 √ 2y 2 + 4y 3 ) + exp √ 2y (5 √ 2 -10y + 2 √ 2y 2 + 4y 3 ) and w (y) = 2y exp - √ 2y (3 √ 2 + 8y -2 √ 2y 2 ) + exp √ 2y (-3 √ 2 + 8y + 2 √ 2y 2 ) .
Let us observe that this last function is even and of the form w (y) = 2y (R(y) -R(-y)), with R(y) = exp √ 2y (-3 √ 2 + 8y + 2 √ 2y 2 ). So it is enough to study its sign on [0, ∞), which corresponds to the sign of R(y) -R(-y) on [0, ∞). We have, for all y ≥ 0,

(R(y) -R(-y)) = 2 exp - √ 2y (1 -6 √ 2y + 2y 2 ) + exp √ 2y (1 + 6 √ 2y + 2y 2 ) = 4(1 + 2y 2 ) cosh √ 2y + 24 √ 2y sinh √ 2y ≥ 0. So R(y) -R(-y) is increasing on [0, ∞), we have R(y) -R(-y) ≥ R(0) -R(-0)
= 0 for all y ≥ 0, and, a fortiori, w (y) = 2y (R(y) -R(-y)) ≥ 0 for all y ≥ 0. Since w (y) is even, we have w (y) ≥ 0 for all y ∈ R. This implies that w (y) is increasing on R. Since w (0) = 0, w (y) is decreasing (-∞, 0] and increasing on (0, ∞). So w (y) ≥ w (0) = 0 implying that w(y) is increasing on R.

It follows from this result that, if a < |x|, we have w(|x| -a) > w(0) = 0, implying that ∂ ∂a v(x; a) > 0, so v(x; a) is strictly increasing according to a. If a > |x|, we have w(|x| -a) < w(0) = 0, implying that ∂ ∂a v(x; a) < 0, so v(x; a) is strictly decreasing according to a. Clearly, we have ∂ ∂a v(x; a) = 0 if and only if a = |x|. Since f (x) is an even function and v( Hence, for all x ∈ R, we have

x; 0) = v(x; 2|x|) = f (x), for all |x| ≥ a/2, we have exp(x 2 ) = v(x; |x|) = sup a∈[0,2|x|] v(x; a) ≥ v(x; a) ≥ inf a∈[0,2|x|] v(x; a) = inf [v(x; 0), v(x; 2|x|)] = f (x).
exp(x 2 ) ≥ max [f (x), v(x; a)] = f (x)1 {|x|<a/2} (x) + v(x; a)1 {|x|≥a/2} (x) = f • (x; a) ≥ f (x).
Proposition 4 is proved.

Proof of Proposition 5. Let us prove the two points in turn.

• Let us remark that, by two successive integrations, we have Thanks to the assumption 2(1 + 2x 2 )θ(x) -θ (x) ≥ 0, we have β (x) ≥ 0, implying that β (x) is increasing. Since θ (0) = 0, we have β (0) = 0 and the minimum of β(x) is attained in x = 0. Since θ(0) = 1, for all x ≥ 0, we have β(x) ≥ β(0) = 0, implying that β(|x|) ≥ 0 for all x ∈ R. Since θ(x) is even, we have The proof of Proposition 5 is complete.

Proof of Proposition 6. We proceed as the proof of Proposition 5.

• If follows from the equality (5), exp(x 2 ) ≥ θ 1 (x) and exp(x 2 ) ≥ θ 2 (x) that exp(x 2 ) -κ(x) = |x| 0 y 0 2 (exp(t 2 ) -θ 1 (t)) + 2t 2 (exp(t 2 ) -θ 2 (t)) dt dy ≥ 0.

Hence exp(x 2 ) ≥ κ(x).

• Let us define the function φ(x) by φ(x) =

x 0 y 0 2(θ 1 (t) + 2t 2 θ 2 (t))dt dy + 1 -θ 1 (x).

Two differentiations give φ (x) =

x 0 2(θ 1 (t) + 2t 2 θ 2 (t))dt -θ 1 (x), φ (x) = 2(θ 1 (x) + 2x 2 θ 2 (x)) -θ 1 (x).

Since 2(θ 1 (x) + 2x 2 θ 2 (x)) -θ 1 (x) ≥ 0, we have φ (x) ≥ 0, implying that φ (x) is increasing. Since θ 1 (0) = 0, we have φ (0) = 0 and the minimum of φ(x) is attained in x = 0. Since θ(0) = 1, for all x ≥ 0, we have φ(x) ≥ φ(0) = 0, implying that φ(|x|) ≥ 0 for all x ∈ R. Since θ(x) is even, we have Proposition 7 is proved.

Figure 1 :

 1 Figure 1: Superimposed curves of exp(x 2 ) and f (x) for x ∈ [-1, 1] in the first graphic, and for x ∈ [-2.5, 2.5] in the second graphic.

Figure 3 :

 3 Figure 3: Superimposed curves of exp(x 2 ), f * (x; n) for n = 0 and f (x) for x ∈ [-1, 1] in the first graphic, and for x ∈ [-2.5, 2.5] in the second graphic.

Table 4 :R

 4 Numerical evaluations of R * (b) for b ∈ {0.5, 1, 1.5, 2, 2.5, 3}. * (b) 4.89107 × 10 -11 0.0000226587 0.1001 73.0819 26273.5 9.5921 × 10 6 4. Second improvement of the lower bound We now investigate a transformation of f (x) for large |x|, based on a multiplicative exponential weight and translation. It is an adaptation of the method developed by Chesneau (2017) to f (x).

Figure 4 :

 4 Figure 4: Superimposed curves of exp(x 2 ), f•(x; a) for a = 1 and f (x) for x ∈ [-1, 1] in the first graphic, and for x ∈ [-2.5, 2.5] in the second graphic.

Table 5 :R

 5 Numerical evaluations of R•(b) for b ∈ {0.5, 1, 1.5, 2, 2.5, 3}. • (b) 1.57828 × 10 -8 3.39908 × 10 -8 7.93014 × 10 -7 0.151264 1081.86 2.42278 × 10 6

For

  |x| < a/2, we have f (x) = inf a∈[0,2|x|] v(x; a) ≥ sup a>2|x| v(x; a) ≥ v(x; a).

  2t 2 ) exp(t 2 ) -θ(t) dt dy ≥ 0.Hence exp(x 2 ) ≥ ω(x).• Let us define the function β(x) byβ2t 2 )θ(t)dt dy + 1 -θ(x). 2t 2 )θ(t)dt -θ (x), β (x) = 2(1 + 2x 2 )θ(x) -θ (x).

  2t 2 )θ(t)dt dy + 1 ≥ θ(x).

  1 (t) + 2t 2 θ 2 (t))dt dy + 1 ≥ θ 1 (x).By exchanging the role of θ 1 (x) and θ 2 (x), we obtain1 (t) + 2t 2 θ 2 (t))dt dy + 1 ≥ θ 2 (x).This ends the proof of Proposition 6.Proof of Proposition 7. We have|x| 0 exp(t) (1 + 2t) exp(t 2 ) -1 dt = exp(t 2 + t) -exp(t) |x| 0 = exp(|x|) exp(x 2 ) -1 , so exp(x 2 ) = 1 + exp(-|x|) |x| 0 exp(t) (1 + 2t) exp(t 2 ) -1 dt. If exp(x 2 ) ≥ θ(x) for all x ∈ R, we have exp(x 2 ) ≥ 1 + exp(-|x|) |x| 0 exp(t) [(1 + 2t)θ(t) -1] dt = γ(x).

Table 2 :

 2 Analytic expressions for Pn(x) with n ∈ {0, . . . , 5}.

Table 3 :

 3 Analytic expressions for Qn(x) with n ∈ {0, . . . , 5}.