Wee Ser Kah Jun Hong

Zhiping Lin
email: ezplin@ntu.edu.sg

David Chee

Guan Foo

Acoustic Detection of Excessive Lung Water using Sub-band Features

Keywords: Lung sounds, edema, sub-band feature extration, SVM, PCA, SVMRFE, truncation I

This paper proposes a fast and accurate feature extraction and classification algorithm to detect excess water in lungs using lung sounds. The proposed design uses a three-part Segmented Sub-band Feature Extractor to extract features. The first part extracts features by segmenting the frequencies found in these sounds into bins through sub-bands. The second part uses Principle Component Analysis and Support Vector Machine Recursive Feature Elimination for feature selection. In the third part, Support Machine Vector and k-Nearest Neighbor classification methods are used as classifiers and the accuracies are compared. Preliminary results obtained from the data collected

show that the proposed method can achieve up to 99% accuracy. The proposed method was tested with real patient samples. Truncation of data up to ten bits was also tested with up to 95% accuracy. The algorithm was implemented on a Digital Signal Processor. The proposed method may be further developed to detect excessive water in lungs readily out of hospitals as a preliminary diagnostic device.

INTRODUCTION

Research has been done on the detection of certain symptoms found in lung sounds such as fine and coarse crackles (FC and CC), rhonchus (RC), wheezes and squawks (SQ). For example, [START_REF] Taplidou | Analysis of Wheezes Using Wavelet Higher Order Spectral Features[END_REF] used a combination of continuous wavelet transform and third-order spectra to analyze lung sounds with monophonic (MW) and polyphonic wheezes (PW). Although there were significant statistical differences between these features extracted from MW and PW, no classifier was used. The data were drawn from an online database. [START_REF] Hadjileontiadis | A Texture-Based Classification of Crackles and Squawks Using Lacunarity[END_REF] used lacunarity to differentiate between lung sounds with FC, CC and SQ. It obtained accuracies of 100%. The data were obtained from databases usually used for education. [START_REF] İçer | Classification and analysis of non-stationary characteristics of crackle and rhonchus lung adventitious sounds[END_REF] proposed various feature extraction procedures to differentiate between crackles, RC, and normal lung sounds. The features used were frequency ratio from power spectral density (PSD), average instantaneous frequency (IF) and exchange time with Hilbert Huang Transform (HHT), and singular spectrum analysis (SSA). The classification was done using Support Machine Vector (SVM). The PSD frequency ratio and SSA demonstrated higher classification accuracy (between 90% and 100%) than the calculations of average IF and exchange time of HHT. The lung sounds were recorded from a medical faculty. Results from these papers were good but they classified or analyzed lung sounds that contained certain acoustic symptoms of disease (such as FC, CC, RC SQ etc.) but not classification of lung sounds of specific diseases.

In the case of [START_REF] Martinez-Hernandez | Computerized Classification of Normal and Abnormal Lung Sounds by Multivariate Linear Autoregressive Model[END_REF], the authors presented a classification algorithm for recordings of lung sounds with diffuse interstitial pneumonia against healthy subjects. The authors used multivariate auto-regressive model with supervised neural network as a classifier with 82.5% validated accuracy.

Pulmonary edema happens when water accumulates in the lungs. This water collects in air sacs in the lungs, making it difficult to breathe [START_REF] Kumar | Kumar & Clark's clinical medicine[END_REF] and can lead to diminished gas exchange. If left unchecked, it may lead to respiratory failure. There have been many systems that detect water in lungs such as dilution techniques, and the more commonly used X-ray machines and CT-scans [START_REF] Elings | Apparatus for Measuring Extravascular Lung Water[END_REF] [START_REF] Iskander | Apparatus and Method for Measuring Lung Water Content[END_REF]. However, X-ray machines and CT machines are bulky, and can only be used by trained professionals. The equipment is usually expensive and the results take some time to develop. Thus, these methods of diagnosis are restricted to be used in hospitals. As a result, physicians still rely on the use of a stethoscope for preliminary diagnosis.

Only one other paper was found to propose a method to detect excessive lung water as a result of pulmonary edema. Recently, Yang and Wee proposed a signal processing approach for the detection of water in lungs using sound based sensing [START_REF] Yang | Lung water detection using acoustic techniques[END_REF]. Several types of features, including the Mel-Frequency Cepstral Coefficients (MFCC), Perceptual Linear Predictive Coefficients, Linear Prediction Coefficients and Wavelet Transform-Based Features were studied as possible features and SVM and k-Nearest-Neighbor (k-NN) were studied as the possible classifier. This paper proposes a fast and accurate feature extraction and classification algorithm to detect excess water in lungs using lung sounds. It was implemented on the TMDX5505eZDSP digital signal processor (DSP). The proposed design uses a three-part Segmented Sub-band Feature Extractor (SSFE). The first part extracts features by segmenting the frequencies found in the sounds into bins through subbands. The second part uses Principle Component Analysis (PCA) and Support Vector Machine Recursive Feature Elimination (SVMRFE) for feature selection and compares their effectiveness. In the third part, SVM and k-NN classification methods are used as classifiers and the accuracies are compared. The intent of the work is to assist the monitoring of the presence of excessive lung water outside the hospital setting and in clinics without X-ray and other bulky and expensive machines.

II. DATA AND METHODS

A. Data Collection

The data were collected from 25 volunteers. 13 had excessive lung water due to pulmonary edema while 12 were healthy. Respiratory sounds were recorded from the back of the test subjects using an electronic stethoscope connected to a laptop computer. The sampling rate was 8000Hz. The audio files were first normalized such that the root-mean-squared value of their energy in all the audio files were equal. The majority of the recordings ranged from 40 seconds to a minute. The recordings were validated by specialists.

The feature extraction step of the algorithm was run on the DSP board, TMDX5505eZDSP (Texas Instruments). It has a dedicated 16-bit ADC, hardware enabled Fast Fourier Transform (FFT), is USB powered, and is small [START_REF]Texas Instruments[END_REF]. The algorithm took an average of 30 seconds on the DSP. SSFE was thus shown to be able to run on portable hardware. Fig. 1 shows how the DSP board interfaces with the computer. The recorded lung sounds were played from the computer and fed into the DSP board through a line-in cable.

B. Sub-band Based Feature Extractor

In this paper, a segmented sub-band feature extraction algorithm is proposed. K rectangular bins are uniformly spaced out across the frequency spectrum, segmenting frequencies of the sounds into bins though sub-bands. The SSFE consists of three steps: the feature extraction step, the feature selection step, and classification step.

1) Feature Extraction Step:

The Feature Extraction Step of the algorithm proposed in this paper extracts features from the frequency spectrum. The space of frequencies investigated is within 0-4000Hz. Lung sounds are known to be in the range of 0-2500Hz [START_REF] Moussavi | Fundamentals of Respiratory Sounds and Analysis[END_REF], and the feature extraction step extends up to 4000Hz in order to increase the feature space.

K number of rectangular bins are chosen so that energy in the frequency spectrum are segmented into uniform bins through sub-bands. This is contrary to using individual frequency components of the sounds as features. If the individual frequency components are used, the classification step will be highly complicated due to large number of features. However, if the sounds are segmented into bins, the classification would be much simpler with fewer features.

The Segmented Frequency Features are extracted from the sound signals as follows: a) FFT: Short-term Fourier Transforms are done on an Mpoint, 50% overlapped window filtered by a Hamming Window.

The frequency spectrum of each frame is calculated using FFT. The magnitude of the frequency spectrum can be represented in the vector form: Relevant and useful features are selected in the feature selection step. Two kinds of feature selection methods are used to compare accuracy. The first is PCA. PCA uses eigenvalue ranking of the correlation matrix to account for features with the highest variance [START_REF] Duda | Pattern Classification[END_REF]. In this case, the eigenvectors chosen are those with corresponding eigenvalues making 95% of the total eigenvalue weight. These eigenvalues are then transformed back into the correlation space and are used for further classification. This reduces the number of features with lower variance.

The second feature selection method used is the Support Vector Machine Recursive Feature Elimination (SVMRFE) method [START_REF] Guyon | Gene Selection for Cancer Classification using Support Vector Machines[END_REF]. This method uses the SVM weights as ranking criterion. This ranks the features according to their contribution to the final classification.

c = v T L
Where L is a K-by-K matrix with Lij = vi for i = j and Lij = 0 for i ≠ j. c) Remove feature with smallest c-value: The feature with the smallest value is removed and the new weight vector is computed. The removed feature is labelled as rK.

d) Repeat:

The process is repeated until all the original K features are ranked, with the first removed feature as the lowest ranking feature. The ranking can be presented as:

r = [r1, r2 … ri … rK] T
Where ri is the i th most important feature.

PCA projects the K features onto another feature space and distorts these data. The features selected using PCA are not the original features gotten from the feature extraction step, thus it is harder to find out which frequency bins were selected using PCA.

SVMRFE does not projects the features unto a new space.

Using SVMRFE will allow identification of sub-bands that contribute to high accuracies found out in the classification step. These bins reveal the frequencies that contain useful information. With these sub-bands, particular hardware filters can be designed to reduce algorithm complexity.

3) Classification Step:

The feature classification step is the final step in which classification accuracies are calculated and the different feature selection methods are compared. SVM and k-NN methods of classification are used. The sensitivity, specificity, and Fmeasure are used to evaluate the accuracy of the algorithm. SVM and k-NN methods are tested using 0.632 bootstrapping instead of cross-validation to reduce variability due to the small number of samples [START_REF] Braga-Neto | Is cross-validation valid for smallsample microarray classification?[END_REF].

The formula used for sensitivity (SN) was SN = TP/(TP+FN)

The formula used for specificity (SP) was SP = TN/(TN+FP)

The formula used for the F-measure (F1) was

F1 = 2TP/(2TP + FP + FN)
Where TP is the number of true positives, TN is the number of true negatives, FP is the number of false positives and FN is the number of false negatives.

III. EXPERIMENTAL RESULTS

K was chosen to be 39 in this experiment, such that the length of the bins were 200Hz each. 512 points of data were used for the FFT. Zero padding was done to 1024 points on the FFT, allowing finer frequencies to be covered.

As stated, the DSP used for testing the algorithm has a 16bit ADC. By reducing the resolution of the ADC, computational cost can be reduced. However by doing so, the accuracy may be comprised. Thus, this paper also presents the investigation of reduced resolution by truncating the 16-bit signal data in the DSP while using the same novel algorithm to classify the data. Results of the truncation to 12-bit and 10-bit resolutions are shown in Table I and Table II.

The PCA feature selection produced results of F-measure up to 92%. Even though the truncations increased, the accuracy did not decrease when k-NN was used. PCA combined with k-NN was able to preserve accuracy when with truncation of up to ten bits with the given data. The results can be seen in Table I.

Referring to the results of SVMRFE in Table II, the SVMRFE feature reduction with SVM classification produced results of up to 98%, and 99% when used with k-NN. It fared better than MFCC [START_REF] Yang | Lung water detection using acoustic techniques[END_REF]. This may be due to the fact that MFCC uses a finite impulse filter (FIR) filter and Discrete Cosine     Transforms (DCT) to extract its features. The FIR filter decreases the magnitudes of lower frequencies of the data. This may reduce the amount of useful information. Also, the DCT attempts to identify harmonics in the frequency spectrum as its final features. Sounds from lungs that have excessive lung water have no known harmonics. Thus the DCT may also have reduced the amount of useful information in its features. The SSFE does not use DCT and an FIR filter. This may be one of the reasons that it is more accurate than MFCC within its given dataset.

Even under 10-bit truncation, the accuracy of the SSFE was 95%. This shows that within the given dataset, the SSFE is able to tolerate truncation or corruption of up to six bits. Feature selection using SVMRFE consistently fared better than PCA, and classification using k-NN consistently fared better than SVM. Truncation of up to ten bits gave an accuracy of at least 95%. The algorithm shows promise with its preliminary results within the given data. Once further developed, the device aid doctors and other healthcare professionals in the diagnosis of pulmonary edema in a real-time setting. Home-care professionals can bring the device with them to home visits, saving time for doctor consultation and freeing them to attend to patients who need more urgent care.

f

  = [f0, f1, f3 … fn … fN-1, fN] T Where fn represents the magnitude in the n th discrete frequency and N represents the number of points in the FFT. b) Bin matrix representation: The K number of bins used are uniformly spaced out across the frequency spectrum. The bins can be represented in the matrix: Where wj represents the magnitude at j th discrete frequency and I represents the number of discrete frequencies within each bin. Since the bins are rectangular, w values are 1. c) Obtaining features: These bins are used to obtain K features for classification. This can be done with the following equation: b = Wf Where b is the vector for the K features, b = [b1, b2 … bK] T 2) Feature Selection Step:



Fig. 1 .

 1 Fig. 1. Configuration of Hardware with Computer

  SSFE also has less computational complexity than MFCC. If the data length is much longer than the sample size, then MFCC has (1+2O+2FO/N)  L more multiplications than SSFE, where O is the filter bank size in MFCC, N is the window length, and F is the feature size. More studies are worth being conducted for the verification of such results among a greater sample size. IV. CONCLUSION This paper proposes a fast and accurate feature extraction and classification implemented on the TMDX5505eZDSP for the detection of excessive lung water level. It uses a three-part algorithm called the Segmented Sub-band Frequency Extractor (SSFE). The first part is the feature extraction step. It segments the frequencies found in the sounds into bins through sub-bands. The second part is the feature selection step, where Principal Component Analysis (PCA) and Support Vector Machine Recursive Feature Elimination (SVMRFE) feature selections methods are used. The third part is the classification step, where k-Nearest Neighbor (k-NN) and Support Vector Machine (SVM) classification methods are used. Recorded data from hospitals were used to test the performance of the proposed design. Truncation of the data was investigated using SVM and k-NN classifiers. Preliminary results for SVMRFE feature selection combined with k-NN gave an accuracy of 99%.

TABLE I .

 I

				EXPERIMENT RESULTS FOR PCA
				SVM			k-NN
	Algorithm	Resolution
	Used	of ADC
			Sensitivity	Specificity	F-Measure	Sensitivity Specificity F-Measure
		16-bit	74%	96%	87%	84%	98%	92%
	SSFE	12-bit	83%	88%	86%	94%	89%	92%
		10-bit	80%	90%	86%	91%	92%	92%
	MFCC	24-bit	86%	96%	91%	96%	96%	96%

TABLE II .

 II EXPERIMENT RESULTS FOR SVMRFE

				SVM			k-NN
	Algorithm	Resolution
	Used	of ADC
			Sensitivity Specificity	F-Measure	Sensitivity	Specificity	F-Measure
	Segmented	16-bit	97%	99%	98%	100%	98%	99%
	Sub-band Feature	12-bit	97%	95%	96%	100%	97%	98%
	Extractor	10-bit	87%	93%	91%	95%	95%	95%
	MFCC	24-bit	86%	96%	91%	96%	96%	96%

ACKNOWLEDGMENT

The authors would like to express their thanks to Mr. Yu JuFeng for his effort in designing the data collection software and conducting the data collection. They would also like to thank the staff at Tan Tock Seng hospital for allowing and assisting in collection of data. Lastly, the authors would like to express their thanks to Mr. Joshua Tan for his help in writing the bootstrapping code.


Excessive lung water occurs when too much water accumulates in the lung, causing breathing difficulty. Current diagnosis methods include X-rays and CT-scans. However, because of their bulk and the need for trained professionals to operate, physicians rely on auscultation for preliminary diagnosis. Recent attempts have been made to automate the auscultation process and some degree of success has been reported. Thus, it would be useful to provide more analysis of such lung sounds. This paper attempts to study the characteristics of breathing sounds from lungs with excessive water and compare them with breathing sounds from healthy lungs. Using a modified empirical mode decomposition to split the signals, the root-mean-squared energy (RMS) and kurtosis were used as characteristics. These characteristics were extracted from the intrinsic mode functions (IMF) and were analyzed. Results showed that certain IMF were effective in characterizing both kinds of sounds due to their small spread in RMS or kurtosis. Results also (using p-values from statistical tests) showed that for certain intrinsic mode functions, lung sounds with excessive lung water exhibit different medians from sounds of healthy lungs. There was strong linear independence between each IMF of the two sounds. Empirical mode decomposition was shown to be able to extract useful information for analyses.

I. INTRODUCTION

There has been research on the acoustic classification of adventitious lung sounds. Many techniques have been proposed to classify both discontinuous and continuous adventitious lung sounds (DAS and CAS). Techniques such as lacunarity [START_REF] Hadjileontiadis | A Texture-Based Classification of Crackles and Squawks Using Lacunarity[END_REF], frequency ratio from power spectral density [START_REF] İçer | Classification and analysis of non-stationary characteristics of crackle and rhonchus lung adventitious sounds[END_REF], and wavelet transform [START_REF] Taplidou | Analysis of Wheezes Using Wavelet Higher Order Spectral Features[END_REF] have been used to more than 90% accuracy with different classifiers such as k-Nearest Neighbor (k-NN) and Support Vector Machine (SVM). [START_REF] Charleston-Villalobos | Crackle Sounds Analysis By EprclMode Decomposition[END_REF] used Empirical Mode Decomposition (EMD) to analyze fine and coarse crackles. Using EMD, there was better discrimination for both fine and soft crackles. Hilbert Huang Transform (HHT), an application of EMD for instantaneous frequency (IF), was used to analyze fine and course crackles in [START_REF] Reyes | Analysis of discontinuous adventitious lung sounds by Hilbert-Huang spectrum[END_REF]. [START_REF] Du | HHT based lung sound crackle detection and classification[END_REF] used average IF and thresholding to classify fine and coarse crackles, with "almost 100% correct detection and total accurate classification." However, most literature tend to focus on analysis of DAS and CAS, and not specific diseases. This paper attempts to analyze sounds of lungs with excessive lung water. Excessive lung water happens when too much water accumulates in the lung and causes breathing difficulty [START_REF] Kumar | [END_REF].

*Research supported by School of EEE, National Technological University, Singapore.

Hong Kah Jun (corresponding) and Wee Ser are with Nanyang Technological University, 50 Nanyang Ave, 639798, Singapore (phone: 65-9221-5676; e-mail: khong008@e.ntu.edu.sg; ewser@ntu.edu.sg) Dr David Chee-Guan Foo is with the department of Cardiology, Tan Tock Seng Hospital, Singapore 308433, email: (david_foo@ttsh.com.sg).

To detect excessive lung water, some techniques have been developed, such as dilution techniques, X-ray machines and CT-scans. Due to their bulk and requirement for trained personnel to operate, these machines can only be found in hospitals. As a result, physicians rely on the use of auscultation for preliminary diagnosis.

Yang and Wee first proposed a signal processing approach for detecting excessive lung water using sound based sensing [START_REF] Yang | Lung water detection using acoustic techniques[END_REF]. Different features and classifiers were explored, and Mel-Frequency Cepstral Coefficients (MFCC) together with k-NN produced the best result, with a reported 95.7% accuracy. [START_REF] Jun | Acoustic detection of excessive lung water using sub-band features[END_REF] proposed a sub-band feature extractor that segmented the frequencies of lung sound recordings into sub-bands. These sub-bands were used as features that were ranked using Principle Component Analysis (PCA) and Support Vector Machine Recursive Feature Elimination (SVMRFE). Classification was done using k-NN and SVM. Accuracy was reported to have an F-measure of up to 99%. Since these recent attempts to automate the process of auscultation for excessive lung water have been successful, it would then be useful to provide more detailed analysis of lung sounds containing excessive water.

This paper attempts to study the characteristics of lung sounds with excessive water and lung sounds with healthy lungs. The study utilizes empirical mode decomposition (EMD), which allows for dealing with nonlinear and nonstationary signals. Proposed by Huang et al. [START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis[END_REF], EMD was claimed to be an adaptive and automatic time-varying filter.

Using a modified EMD, the proposed method splits the signal into "intrinsic mode functions (IMF)"-like signals, and analyzes the root-mean-squared (RMS) energy and kurtosis as features of these IMF-like signals. The analysis includes a study of the similarities and characteristics of these IMFs. Differences are also investigated for breathe sounds from lungs with excessive water and healthy lungs using statistical testing. The Mann-Whitney U Test is used for statistical significance. The null hypothesis is tested at p < 0.05.

II. DATA AND METHODS

A. Data Collection

The data was collected from 17 volunteers. 7 volunteers had excessive lung water due to pulmonary edema while 10 volunteers had healthy lungs. Respiratory sounds were recorded from the back of the test subjects using a stethoscope sensor (Addest Technovation) connected to a laptop. The ground truth was determined by an expert physician. The sampling rate was 8000Hz. The recordings were normalized to have the same RMS energy. The recordings were validated by specialists. The majority of the recordings ranged from 30 seconds to a minute and were divided into 181 four-second long samples. They were then

An Analysis of Sounds for Lungs with Excessive Water

Kah Jun Hong, Member IEEE, Wee Ser, Senior Member, IEEE, David Chee-Guan Foo categorized into either lung sounds with excessive lung water (ELW) or healthy lungs (HL).

B. Analysis Method 1) Empirical Mode Decomposition

The original EMD empirically identifies the intrinsic oscillatory modes of the signal by particular time scales [START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis[END_REF]. These modes are called IMF, and they characterize the data through oscillating waves with zero local mean. Because of the sifting process, EMD is a decomposition that is complete, orthogonal, local and adaptive [START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis[END_REF].

Each IMF satisfies two conditions. The first condition states that the number of extrema and the number of zero crossings can only differ by at most one. The second condition states that the local mean of the IMF must be zero. The locality of the mean can be defined by the problem. The signal can be represented by the sum of its IMF by

 

Where N is the total number of IMF, and rN(t) is the residue signal that is constant or monotonic.

The sifting process is as follows:

1. Generate an upper envelope of the given signal s(t) by a cubic spline using the local maximums.

2. Generate a lower envelope of the given signal s(t) by a cubic spline using the local minimums.

3. Find the average of the envelopes to attain the mean envelope, m(t).

Compute the difference h(t) = s(t) -m(t).

This is regarded as one sift. Once the first IMF is generated, the second IMF can be generated by using the sifting procedure on the first residue r1(ts(t)IMF1(t). IMF stop generating when the n th residue rN(t) is either monotonic or constant.

It is known that for real signals, it is sometimes difficult for the sifting process to meet the criteria for IMF. Thus for the proposed algorithm, two modifications are made.

Instead of checking if h(t) is an IMF after every sift,

an IMF was generated after M sifts. This was decided after analyzing each sifting. Two analyses were made.

For all the samples, the difference between extrema and zero crossing dropped exponentially and stabilized after less than 100 siftings. Fig. 1 shows the difference between extrema and zero crossing for each IMF up to the 14 th IMF for one sample.

The standard deviation between each sifting was also determined. The standard deviation was calculated with the formula:

  
Where hi(t) is the signal one sifting after hi-1(t).

The standard deviation also dropped exponentially and stabilized after 200 siftings. Thus there were insignificant change between each sifting after 200 sifts. M was chosen in this paper to be 250.

2. The number of IMFs are fixed to be N unless before N IMFs are generated, the latest IMF is monotonic or constant, which then N' number of IMFs are generated, with the residue rN being the monotonic or constant IMF.

2) Statistical Analysis

The root-mean-squared (RMS) energy and kurtosis of the IMFs were studied as proposed features. The RMS energy was calculated with the formula and kurtosis was calculated using where N is the length of signal.

The medians and spread of features values were analyzed to determine the characteristics and similarities for sounds of lungs with ELW and HL.

Statistical analysis was also used to study the differences between lung sounds with excessive lung water and healthy lungs. First, the Lilliefors test was used to determine if the distribution of the proposed features of each IMF was from the normal family. The Lilliefors test uses the null hypothesis that the data comes from a distribution in the normal family, against the alternative that it does not come from such a distribution. It was tested at the 5% significance level. The results of the test showed that, for the data collected, both the RMS energy and kurtosis of each IMF were non-normal.

To analyze the differences between ELW and HL using IMF, the Mann-Whitney U test was used. The Mann-Whitney U test was chosen because of the non-normal distribution of the IMFs. It uses the null hypothesis that two sets of independent data have equal medians, against the alternative that they are not equal. In this case, it was used to test if the median of the proposed features within each IMF between ELW and HL were the same. The null hypothesis was tested at the 5% significant level. Thus if p < 0.05, the null hypothesis would be rejected, and one could conclude that the median of the tested feature value was equal between the two types of lung sounds. For the purpose of being succinct in Section III, a conclusion that feature values are equal would refer to their medians being equal.

Since sounds for both ELW and HL are breathe sounds, they will have independent as well as dependent characteristics. Thus, correlation between IMF of ELW and HL were also done to investigate linear independence.

III. ANALYSIS RESULTS

The number of siftings to obtain each IMF was chosen to be M=250. The number of IMFs obtained was chosen to be N=14. Thus 15 separate signals were attained from each sample. EMD and feature extraction were applied on the whole four-second sample, and no windowing was implemented.

The proposed features of each IMF was calculated. For each IMF (and rN), two datasets for each feature was obtained. One dataset belonged to ELW sounds and the other dataset belonged to HL sounds. Each IMF was investigated with these two datasets. The Mann-Whitney U test was used on datasets for each IMF to test if the value of the proposed features between the two categories were different. If p < 0.05, the null hypothesis (described in Section B.2) would be rejected and one could conclude that the values for the proposed features extracted from the IMF between the two categories were different.

First, the RMS energy of each IMF was calculated. IMF11 and IMF12 have the smallest spread of RMS energy for both lung sounds. Thus, among all the IMF, IMF11 and IMF12 were the most effective in characterizing ELW or HL lungs sounds using RMS. IMF1-IMF3 had the largest spreads thus were most ineffective in characterizing ELW or HL lung sounds using RMS. Comparisons between ELW and HL were also made. The p-values for each IMF showed that the RMS energy for IMF1, IMF2, IMF7, to IMF14 and rN between samples with ELW and samples with HL were different. Tests on IMF3 to IMF6 failed to reject the null hypothesis and thus the RMS energy of IMF3 to IMF6 were shown to be equal for the two types of lung sounds. Correlation coefficients for each IMF between the two datasets were generally close to zero, ranging from -0.111 to 0.242. Thus RMS of these datasets have strong linear independence. Fig. 2 shows the box plots for RMS energy. Table 1 shows the pvalues for the RMS energy of each IMF. Table 2 shows the correlation coefficients for RMS energy.

Second, the kurtosis of each IMF was calculated. IMF13 and IMF14 had the smallest spread of kurtosis for both lung sounds. Thus these IMFs were the most effective in characterizing ELW or HL lung sounds. IMF1-IMF5 had the largest spreads thus were most ineffective in characterizing ELW or HL lung sounds. The p-values for each IMF showed that kurtosis for IMF1, and IMF6 to IMF14 between samples for ELS and samples for HL were different. Tests for IMF2 to IMF6 and rN failed to reject the null hypothesis and thus kurtosis of these IMFs (and rN) were shown to be equal for the two types of lung sounds. Correlation coefficients for ach IMF between the two datasets were generally close to zero, ranging from -0.222 to 0.371. Thus kurtosis of these datasets have strong linear independence too. Fig. 3 shows the box plot for kurtosis. Table 3 shows the p-values for kurtosis. Table 4 shows the correlation coefficients for kurtosis.

Using the modified EMD to analyze lung sounds, the results show that some IMFs could be used to characterize lung sounds if RMS energy and kurtosis were used. The median of RMS energies and kurtosis for certain IMFs were revealed to be different and some were same between sounds from lungs with excessive water and healthy lungs. Using RMS energy and kurtosis, sounds of ELW and HL showed strong linear independence. EMD was shown to be an effective way for analysis. Further work include using these features in a classifier and evaluating its accuracy. More investigations can be done with other features using EMD. Also, more data can be collected to increase the inter-patient variability.

Figure 1 .

 1 Figure 1. The difference between extrema and zero crossing decreases exponentially, then remains constant.

Figure 2 .

 2 Figure 2. Box plots of RMS energy for IMF showing median, and 25th and 75th percentile. Outliers have been removed for presentation. Red: Plots for ELW. Blue: Plots for HL.

TABLE III .

 III

		P-VALUES OF KURTOSIS FOR EACH IMF
	IMF1	IMF2	IMF3	IMF4	IMF5
	2.06E-6*	0.507	0.836	0.192	0.629
	IMF6	IMF7	IMF8	IMF9	IMF10
	1.74E-2*	3.80E-3*	1.31E-5*	6.42E-5*	2.50E-3*
	IMF11	IMF12	IMF13	IMF14	rN
	2.80E-3*	4.90E-2*	9.10E-3*	8.50E-3*	5.570E-2
			Asterisks denotes statistical significance at 5% level

TABLE II

 II

	.	CORRELATION COEFFICIENTS OF RMS FOR EACH IMF
	IMF1	IMF2	IMF3	IMF4	IMF5
	-5.03E-2	8.40E-2	7.18E-2	-5.37E-2	7.33E-2
	IMF6	IMF7	IMF8	IMF9	IMF10
	-9.25E-2	-0.133	-0.111	4.60E-3	0.157
	IMF11	IMF12	IMF13	IMF14	rN
	0.160	0.242	9.02E-2	0.129	1.97E-2

TABLE I .

 I P-VALUES OF RMS FOR EACH IMF

	IMF1	IMF2	IMF3	IMF4	IMF5
	3.65E-7*	1.22E-5*	0.153	0.300	0.714
	IMF6	IMF7	IMF8	IMF9	IMF10
	0.102	0.0437*	5.84E-5*	1.46E-4*	9.65E-4*
	IMF11	IMF12	IMF13	IMF14	rN
	1.20E-3*	1.90E-3*	1.64E-2*	7.40E-3*	1.10E-3*
			Asterisks denotes statistical significance at 5% level

TABLE IV

 IV

	.	CORRELATION COEFFICIENTS OF KURTOSIS FOR
			EACH IMF
	IMF1	IMF2	IMF3	IMF4	IMF5
	8.76E-2	-0.222	0.245	0.371	0.174
	IMF6	IMF7	IMF8	IMF9	IMF10
	0.161	-6.90E-2	-2.82E-2	-0.147	-0.140
	IMF11	IMF12	IMF13	IMF14	rN
	-6.01E-2	0.211	0.155	0.109	0.129

ACKNOWLEDGMENT

The authors would like to express their thanks to Mr. Yu JuFeng for his effort in designing the data collection software and collecting data. They would also like to thank the staff at Tan Tock Seng hospital for allowing and assisting in collection of data.


Paola Boito
email: paola.boito@unilim.fr

Yuli Eidelman

Luca Gemignani
email: l.gemignani@di.unipi.it

Efficient Solution of Shifted Quasiseparable Systems and Applications

Keywords: quasiseparable matrices, shifted linear system, QR factorization, matrix function, matrix equation. MSC 2000: 65F05

We propose an efficient algorithm for the solution of shifted quasiseparable systems, which exploits the invariance of the quasiseparable structure under diagonal shifting and inversion. This algorithm is applied to compute various functions of matrices, and to solve a class of linear matrix equations. Numerical experiments show the effectiveness of our approach.

Introduction

In this work we propose a novel method for computing the solution of shifted quasiseparable systems of the form (A + σ i I N )x i = y, i = 1, . . . , ,

and of more general parameter dependent linear matrix equations with quasiseparable representations. Our approach also has a noticeable potential for effectively solving some large-scale algebraic problems that reduce to evaluating the action of a quasiseparable matrix function to a vector.

Shifted Quasiseparable Systems

Quasiseparable matrices find their application in several branches of applied mathematics and engineering. For instance, quasiseparable structure often arises in the discretization of continuous operators, due to the local properties of the discretization schemes and/or to the decay properties of the operator or of its finite approximations. As a consequence, in the last decade there has been considerable interest in the development of fast algorithms for working with quasiseparable matrices [START_REF] Eidelman | Separable type representations of matrices and fast algorithms[END_REF][START_REF] Eidelman | Separable type representations of matrices and fast algorithms[END_REF][START_REF] Vandebril | Matrix computations and semiseparable matrices[END_REF][START_REF] Vandebril | Matrix computations and semiseparable matrices[END_REF].

Main Algorithm

It is well known that several operations with quasiseparable matrices can be performed in linear time with respect to matrix size. In particular, the QR factorization algorithm presented in [START_REF] Eidelman | A modification of the Dewilde-van der Veen method for inversion of finite structured matrices[END_REF] computes in linear time a QR decomposition of a quasiseparable matrix

A ∈ C N ×N of the form A = V • U • R,
where R is upper triangular, whereas U and V are banded unitary matrices and -this is the crucial point -V only depends on the generators of the strictly lower triangular part of A. This implies that any shifted matrix A + σI N , σ ∈ C, can also be factored as

A + σI N = V • U σ • R σ for suitable U σ and R σ .
Relying upon this fact, we design and implement an efficient algorithm [START_REF] Boito | Efficient solution of parameter dependent quasiseparable systems and computation of meromorphic matrix functions[END_REF] for solving a sequence of shifted quasiseparable linear systems. Its arithmetic complexity is linear with respect to N and to the number of shifts, and it is halved with respect to straightforward application of structured QR factorization to each shifted linear system. The Matlab code is available at http://www.unilim.fr/pages perso/paola.boito/software.html.

Some applications are presented in the following sections; they include the computation of f (A)v via series expansion or contour integrals, and the solution of linear matrix equations.

A Model Problem for Boundary Value ODEs

Consider the non-local boundary value problem:

dv dt = Av, 0 < t < τ, 1 τ τ 0 v(t) dt = g,
where A is a linear operator in R N and g ∈ R N is a given vector [START_REF] Tikhonov | On the solvability of a problem with a nonlocal integral condition for a differential equation in a Banach space[END_REF][START_REF] Tikhonov | Uniqueness theorems in linear nonlocal problems for abstract differential equations[END_REF][START_REF] Eidelman | The resolving formulas in the model nonlocal problem for the evolution equation[END_REF]. If all the numbers µ k = 2πik/τ, k = ±1, ±2, ±3, . . . are regular points of the operator A, the problem has a unique solution, given by v(t) = q t (A)g, q t (z) = τ ze zt e zτ -1 .

P. Boito, Y. Eidelman, L. Gemignani

Without loss of generality one can assume τ = 2π. We show that, for a suitable truncation index , v(t) can be approximated by the finite sum

v (t) = 3 j=0 V j (t)A j g -2 k=1 1 k 2 (A cos kt + 1 k A 2 sin kt)(A 2 + k 2 I N ) -1 A 3 g, with 0 ≤ t ≤ 2π and V 0 (t) = 1, V 1 (t) = t -π, V 2 (t) = π 2 3 -πt + t 2 2 , V 3 (t) = π 2 3 t - π 2 t 2 + t 3 6 .
The computation of v(t i ), 0 ≤ i ≤ M + 1, requires the solution of a possibly large set of shifted systems and our algorithm proves to be effective for this task if A is quasiseparable.

More generally, a similar approach can be applied to the computation of a function of a quasiseparable matrix, multiplied by a vector, whenever the function can be represented as a series of partial fractions. The classes of meromorphic functions admitting such a representation were investigated for instance in [START_REF] Sherstyukov | Expansion of the reciprocal of an entire function with zeros in a strip in the Kreȋn series[END_REF]. Other partial fraction approximations of certain analytic functions can be found in [START_REF] Hale | Computing A α , log(A), and related matrix functions by contour integrals[END_REF].

Sylvester-type Matrix Equations

As a natural extension of the problem (1), the right-hand side y could also depend on the parameter, so that we have a different right-hand side for each linear system. This situation is common in many applications, such as control theory, structural dynamics and timedependent PDEs [START_REF] Gu | Numerical solution of parameter-dependent linear systems[END_REF]. The systems to be solved take the form of a linear matrix equation:

AX + XD = Y, A ∈ R N ×N , D = diag [σ 1 , . . . , σ ] , Y = [y 1 , . . . , y ] .
The extension to the case where D is lower triangular can be carried out via backsubstitution, and a further generalization to the case where D is a general matrix is possible using the classical Bartels-Stewart approach based on Schur decomposition. This approach is especially interesting when is significantly smaller than N . If A is quasiseparable we can apply our structured approach as outlined above, with computational advantages with respect to the widespread method that relies on Kronecker products.

c CMMSE ISBN: 978-84-617-8694-7


Marie-Céline Isaïa

The Bishop

D'après une communication

Among the works of Hincmar of Rheims, the Vita sancti Remigii 1 continues to be regarded as a minor text, a little folkloric recreation that the archbishop permitted himself in old age. 2 Written after his most serious theological and political treatises, the Life of Saint Remigius has been seen as a motley compilation, accumulating traditions, or inventing them, about a relatively obscure Merovingian bishop. Certainly Saint Remigius, who died around 533/535, did not enjoy a very widespread cult when Hincmar began his work. An extremely short, anonymous and archaic Vita Remedii credited the saint with the merit of having resurrected a dead man -but it is a work of slight importance, compared to the Vitae of other glories of the Gallic sanctoral, such as Germanus of Auxerre. 3 Hincmar toiled to learn more, but in vain, as he recalls in his preface addressed to Saint-Remi's monks : the older ones remembered that their predecessors "had seen a very great book, written in an antique script, that recounted Remigius's birth, life, virtues and death… but it had been destroyed… You rightfully wished that I should gather these scattered elements in a single oeuvre... I would have achieved this a long time ago, if I had not been delayed by a vain hope: I heard, from here and there, that I might succeed in retrieving the great book about the virtues and life of our lord and patron saint;… all was but lies". 4 So Hincmar filled the gaps as he could.

It would however be unfair to see nothing in this hagiographical text but a work of the imagination, isolated from the whole oeuvre of Hincmar. The Vita is clearly entirely coherent with his political thought -a coherence that for Bruno Krusch proved it had been written ad hoc, that is to say that it tells nothing but a (forged) truth that served Hincmar. 5 De facto, the Vita attests to and 1 Hincmarus Remensis,

spreads the legend of the Holy Ampulla, in the same years as this object became useful for the coronation of Charles the Bald as king of Lotharingia (869). 6 Historians recognized a long time ago that the Vita exposes, with its own words, Hincmar's theory about contractual monarchy and warns Carolingian kings against the temptation to hijack the Church's property. 7 Moreover, the more one explores its composition, the more one realises that Hincmar conceived his Vita as a kind of library capable of holding the quintessentials of his theological and pastoral beliefs: intermingled with saint Remigius's biographical outline, more theoretical developments on predestination or the Trinity directly reflect Hincmar's teachings on these subjects. 8 Furthermore, Hincmar evidently began to write the Vita before 852, even if only his death in 882 prevented him from supplementing it again and again: one must conclude that the Vita Remigii is not (only) the testament of Hincmar, but (also) the founding text of his episcopacy 9 . 10 Hincmar wanted it to be a constantly enhanced text of selfjustification that could warrant all his deeds, from his promotion at Rheims against Ebbo's rights (845) to his actions as kings' counsellor: its main meaning is that the authority of the archbishop of Rheims should not be challenged, since it is based on the precedent of Remigius of Rheims, the greatest apostle that Francia ever had.

The normative effects of a hagiographical text

Hincmar's project followed a long tradition that led him to believe that his Vita would contribute to the religious progress of its readers. So he constructed his Vita as a dialogue between past and present, between the 6 th century of Remigius and his 9 th century, for a moral purpose. After describing some deed of saint Remigius, Hincmar draws a lesson addressed to all his contemporaries, especially Carolingian bishops, monks and kings. This is why of all his own works he cited in the Vita, it is his moral treaty, "The sins one should avoid and the virtues one should cultivate", De cavendis vitiis et virtutibus exercendis, on which he drew most often [START_REF]De cavendis vitiis et virtutibus exercendis[END_REF] : Hincmar quotes this work on at least eighteen occasions, because the Vita is the means Hincmar created to exhort his audience to moral and religous improvement. [START_REF] Krusch | not note the presence of the De cavendis inside the Vita Remigii, so there may be more quotations than the 18 important ones I have noticed. Even so, I believe that the De cavendis[END_REF] So the Vita tends towards being a speculum, exactly as the De cavendis is a moral guide for King Charles the Bald, and Hincmar reuses in both useful patristic formulas he found. When for example Remigius had dinner at home, as Hincmar tells us following the Vita Remedii, he distributed his leftovers to birds, who ate out of his hand. "We can receive an excellent moral lesson considering this action of saint Remigius" 13 continues Hincmar: we can rejuvenate by prayer, by studying, by regretting our sins, even if our flesh grows older. 14 He does not admit that this is a teaching of Pope Gregory the Great [START_REF] Magnus | [END_REF] , that he had already used in the De cavendis. [START_REF][END_REF] It was not unusual to assign a moral aim to a hagiographical text, which, as many Merovingian Vitae repeat, should contribute to the edification of each of the fideles who hears it read. This is the normal parenetic or hortatory function of hagiography. The only remarkable thing here is the systematic and explicit moral lesson: Hincmar explains in what sense we must interpret the story he tells, he does not let the reader draw his own conclusions. For instance, in chapter 17, Remigius saves a certain Eulogius, who was accused of treason: thanks to the intercession of the bishop, Eulogius was not sentenced to death by King Clovis, and even his villa of Epernay was not confiscated. Eulogius wanted to thank the bishop by giving him the villa, but Remigius refused the gift and bought it instead. Hincmar comments: This is a good example that Remigius left all bishops, both his contemporaries in the flesh and those who will come after him in the priesthood: when they help the poor, widows, orphans and all those who seek the refuge of the Church and of their mercy, those who have been wronged, or are condemned to exile or banishment because of their sins, or who are within the scope of any judgment -when they bring them any kind of relief, they should not receive temporal reward in exchange. But, according to the word of the Lord 'Freely you received, freely give' [Matthew 10, 8] they must give generously what they have received by the grace of God to those around them, that is what they were given freely -freely indeed meaning without awaiting temporal counterpart.17 " From a human experience that provides an exemplum, and from a saying from the Scriptures, Hincmar generalizes: this is the most common homiletical method. The originality of the Vita Remigii lies elsewhere, in the definition of the episcopal office it assumes. When adopting a homiletic tone, like many other Vitae, it immediately justifies this method by portraying the bishop of Rheims as a master whose examples and teachings must be followed step by step. Hincmar actually blurs the difference between the actions of Remigius on one hand and his own commentaries on the other, assuming that there is a perfect, mystical continuity between them. The Vita thereby offers a complete portrait of what Hincmar thinks that a Carolingian bishop must be, a man whose responsibility is to tell others what they have to do. Now this does not only imply a moral standpoint: Hincmar extends the clerical responsibility of teaching to a much broader normative authority. As the example of Eulogius shows, he does not merely tell good from evil, but also defines what is fair, legitimate and permitted. Hence, the Vita Remigii must be seen as the very core of Hincmar's oeuvre: the man who spends his life telling others what they should do or say or believe, chose hagiography to teach dogmatic and theological truths, to impose rules and to reinforce his normative power.

Of course, the Life of St. Remigius cannot be reduced simply to a series of legal articles: Hincmar also composed his Vita to give the episcopal Church of Rheims a history, to justify the monastic community of Saint-Remi's heavy dependence on the successor of Remigius as bishop, to exalt the memory of the saint who baptized Clovis and who introduced the Franks into the history of salvation. It seems nevertheless possible to say that Hincmar conceived his Vita both as a historical and a normative text; or as a normative text because it is primarily about the past, a christianized and heavily rewritten past. Nowadays, what we call rules are straightforward, unambiguous texts that set norms, requirements, laws of general scope: these texts are objective ones, or supposed to be, written in the present. Hagiographical texts are exactly the opposite: stories, series of anecdotes, narratives of events, with witnesses, main characters and secondary ones, a biographical logic, sometimes with plots or twists, all necessarily presented in the past and, by definition, not of general scope since it is always the history of an individual, a particular case. Meetings between the two genres, however, may have been more numerous during the early Middle Ages: their first common point is that both the rules and the Vitae are supported by a community and a common acknowledgment of a shared past. This is the community that provides the inspiration for the customs as for the Vita, by a collective effort of recollection of its past, both of past uses and past events -and it is a community that emerges in the sharing of rules or in the commemoration of its peculiar patron saint. Therefore, Hincmar's choice of hagiography is not totally unexpected. He makes a visible effort to underline that he wrote nothing about Remigius but the story he received from his predecessors. This is not simply the commonplace tune of hagiographical prefaces: the unanimity of the clergy, their participation in collecting and believing the events Hincmar tells about Remigius, is the very foundation on which the archbishop will base his authority. It justifies the entirely unusual importance of a normative discourse within the Vita.

Homiletic moments set apart, the normative strength of the Vita sometimes comes from its capacity to include evidence seized elsewhere and embedded in the text, as in the case of a letter of Pope Hormisdas, an authentic letter (adapted by Hincmar): the archbishop inserted this document as a conspicuous proof that could not be debated. But most of this evidence is slid inside the Vita, sewn into it as secretly as possible. Hincmar 'honestly' announced part of this technique in the preface: he would compose his Vita from different and heterogeneous origins… but he almost never indicates which piece is borrowed, or from whom. The heterogeneity he acknowledges is due, if we believe him, to his desire not to modify the different texts he collected about Remigius. 18 There is however a more striking heterogeneity, due to the (sometimes awkward) way in which Hincmar put into the Vita words he took from his own previous works. We cannot blame a computer for this frequent copy/paste, so one must understand that Hincmar collected in his treatises the theological convictions he thought most precious, to disseminate them again to a larger audience. In the table of contents Hincmar included in the begining of the Vita, he explicitly states that episodes from the life of Remigius will be accompanied by lessons. For example, chapter 8 announces a long speech about predestination, and the heresy of those who misunderstand it:

Chapter 8 : How he [saint Remigius] rescued the city of Rheims from the fire, this city that the devil had set alight, and how we shall also be rescued by the means of his merits and prayers, if we ask without weakening, by releasing us from the vices and flames of eternal fire. Hence, by comparison, the brief summary of the error of the heretics, who claim that God has predestined some men to eternal fire, and, conversely, the demonstration of the truth of the Catholic faith". 19 We must conclude that there are three steps in the Vita: the narrative of the facts (Remigius stopped the burning of the city of Rheims), their spiritual understanding (Remigius can fight against our vices), but also their dogmatic interpretation (what is to be believed for one's salvation). Hincmar improves the technique of the sermon, constructing his text on the various senses of Scriptures, literal, moral and anagogical, and playing on similarities, sumpta similitudine, here on the word "fire", that can be real fire, spiritual or eschatological. This is noteworthy in a hagiographical text: the shift from the historical narrative to the admonition is common, but not to this degree, and the third meaning is never or rarely developed so insistently. In chapter 8, there are 30 lines on the "real" fire, and 215 lines of theological teaching. Why is Hincmar so talkative on the subject? Because his teaching on heresy and its refutation is to be found, in a different order, in his treaty on the same topic, De praedestinatione. 20 In a way, Hincmar summarised his treatise for popularisation through the Vita Remigii.

The Vita admonishes kings as a Speculum does 21 ; the Vita publicises truths a Christian must understand; the Vita gives Rheims the past it must have to legitimate its domination of political intrigues. All these achievements are possible because, above all, the Vita states that a bishop is a law-giver, and the very source of every catholic -that is universal -norm. Anyone who thinks of Hincmar as conservative will be surprised to discover him claiming, sotto voce, complete freedom for bishops in creating the norms the Vita promotes; even if he presents the council as an opportune moment for this creation, he does not assert that the definition of norms is a collective action. Indeed, Hincmar's Vita appears to represent the moment when Latin hagiography attempts to become a medium for norms rather than an edifying discourse; though perhaps a brief apogée, rather than a successful and durable begining.

The bishop as a law-giver

One of the major innovations of Hincmar's Saint Remigius is his alleged involvement in councils. A brief and positivist look at the saint's episcopacy must conclude that there is no sign that Remigius attended any known council in Gaul from 458 to 535, and especially not the exceptional council of Orléans which Clovis organized in 511. 22 He did not call for provincial meetings either, not even the simplest diocesan synod, during more than fifty years. This lacuna did not fit new Carolingian norms, a problem which Hincmar solved in a very subtle way. In the Vita's chapter 20, he inserted a letter written by Pope Hormisdas. 23 It is well-known since Krusch's work, followed by Devisse, that the archbishop based his own claims to be a kind of papal vicar for the kingdom on this (forged) letter, as he asserts for himself the privileges Hormisdas is said to have accorded to Remigius. 24 This letter sums up in a theoretical way what the bishop's mission must be: it is not christianization, nor the spiritual life, the correct celebration of the sacraments or the defense of the faith against heresy, it is the observance of rules. More precisely, preaching the faith and constructing the Church may be summarised in a single expression, that is keeping the tradition expressed in canonical law and confirmed by papal authority. In his letter, Pope Hormisdas begins by rejoicing in the election of Remigius as a bishop: "So we received by anticipation an omen that our choice [of Remigius as the papal vicar] is judicious, when we learned that you have accomplished what we order in all humility to all others to do, that is to say that you try to maintain the observance of the rule of the fathers and the authority of the apostolic See in regions whose unity is threatened by such a removal 25 ." Then the Pope makes Remigius his vicar, congratulates him on the baptism of the Franks, and continues with exhortations on the mission of a bishop, that is mainly to maintain the canons: Thus we give the order that the rules and prescriptions of the Fathers that the most holy councils have defined may be observed by all. Your vigilance on these matters, your concern about these cases, we underline them with our fraternal exhortations. How well it is that you are so respectful a keeper of these laws that no single opportunity of fault remains, nor obstacle to the holy observance! This is where we find specified what is lawful and what is unlawful, written what is forbidden and after what no one should have the audacity to aspire, there what is allowed, and that a mind should seek to please God26 .

The bishop is thus defined by the authentic letter of Hormisdas as a custodian of the canonical law, as it summarised the Roman Church tradition: no wonder that the pope insists on this point, because he writes to a bishop of Spain under Visigothic domination27 . But Hincmar selected this letter, and no other, to forge a brief correspondance between Rheims and Rome: and it is not only a question of contemporaneity, even if Hincmar relies on the fact that Hormisdas and Remigius could have exchanged letters. The letter fits both Hincmar's desire and practice to reinforce the normative power of the archbishops, as sustained by papal authority. The letter goes further : it underlines "Remigius"'s responsibility in maintaining Church tradition, but also his ability to exert normative power over the clergy of his diocese, by the mean of the council: "Whenever the defense of religion demands a general council, all your brothers should gather on your summons, and if a particular topic is the occasion of trouble between some of them, then restrain disputes born between them, and find solutions to conflicts by examining the holy Law [the Scriptures, lex sacra] 28 ." With such a definition of episcopal ministry, one should not be surprised if, in the Vita, Remigius's voice and that of Hincmar combine to formulate real rules and not only encouragement or exhortation. The bishop is a legislator, thanks to his knowledge of Scriptures, patristic tradition and canonical law. 29 What is the more important: to maintain the observance of canon law or to be able to interpret it? Hormisdas openly chose observance, but Hincmar insists on the archbishop's autonomy and ability to create new rules from old ones. Hence, in my opinion, his relatively restrained use of the figure of Moses in the Vita Remigii. In the Vita, Remigius is compared to many great male figures of Old and New Testament, including archangels, kings, prophets and apostles. 30 Moses appears in the list, but not at its top, and is mentioned for a fact that probably is not the major one of his life: Moses is described not as the one who received the Law, nor the lawgiver, but as the man whose face was radiant:

We read this fact about Moses: his face became radiant when the Lord looked upon him. We hear the same about Remigius, who was glorified by a radiant light. That means that Moses was ordained by God to be a legislator for the people of the Old Testament exactly as the blessed Remigius emerged as the man chosen, by the gift of Christ, to be the steward of the grace of the gospel to the people that had to be renewed in the baptismal font 31 .

Here Hincmar prefers to oppose the age of Law and the age of Grace rather than insist on the continuity between the two legislators -perhaps because he thinks that a Carolingian bishop does not have to deliver a Law he did not write, but to write laws that had not been transmitted to him.

The council therefore is the place where a bishop is able to reinforce his effective authority over other clergy and his intellectual authority by the commemoration of canonical law32 . So it is certainly no coincidence that in chapter 21, Hincmar imagined a council chaired by Remigius, a chapter he completely invented, without any previous attestation. 33 This is a council without place or date, because this is not a historical council, but a staging of the normative authority of the bishop. In the episode's conclusion, Hincmar joined a meditation on the law, from the De cavendis and the Moralia, with a quite spectacular miracle. The situation is very plain: Remigius chairs a council which also involved an Arian heretic. The heretic refuses to get up when Remigius arrives, so he is reduced to silence. Actually mute, he recovers only to confess the Catholic faith on the order of Remigius:

The heretic, once proud, now a humble Catholic, confessed the Catholic faith in the Catholic holy and inseparable Trinity and in the Incarnation of Christ and promised to persevere in this faith he had confess. In this way, Saint Remigius, with the power of God, restored the health of soul and body to the man who had lost his soul by his infidelity and was condemned to the loss of his physical voice by his pride, to show unambiguously to all the other priests who were present, as to those who would read this passage or hear it, what to do about those who sin by their perverse opinions about Christ, who by His humanity deigned to become our neighbour and our brother, how to act towards those who sin against Him or against the Church and who disobey, as against those doing penance after confessing their sins 34 .

The whole commentary is important, because it is not a mere question of spiritual guidance, a problem of edification of any Christian. From the example of Remigius, Hincmar asks all the priests to behave in a certain way, or rather imposes a norm, since the duty to forgive sins is addressed to a social or professional category, here the clergy. The incentive to forgive does not depend on the personal behaviour of one priest or another: it is the Archbishop of Rheims -that means Remigius as much as Hincmar -who speaks, as he would in a provincial council, and gives a rule to all the clergy of his diocese. Indeed, this discourse of Hincmar is staged on the occasion of a theoretical or ideal type council, in which the roles of Hincmar and Remigius are inseparable: there is the example of Remigius on one side, who is able to convince those who attended the scene, then the extension of the work of Remigius by Hincmar, though other means of communication. This is a first response, very clear, to the question of the transmission of norms: as he explained himself, Hincmar hoped that the hagiographical text will be very widespread, read in public, pondered upon in private; a Life of Saint Remigius is awaited and will circulate throughout the province, so it is a very important medium for the dissemination of rules. It is because of the Vita's anticipated wide dissemination that Hincmar chose to mingle more theoretical lessons with the events of the life of Remigius, as he explains in his preface: I shall take care to add passages from the words of the Fathers to the admonition of those who read or listen, to passages that describe the miracles that the Lord made by our patron saint ; and to the extent of my understanding, I shall try to walk in the footsteps of the blessed Gregory, although I can not equal him, who, while he described the actions of the saints and the fall of the wicked, drew out an admonition, according to the wisdom that God had communicated to him, and who inserted in his narrative many necessary and useful considerations to those who read them or listen to them 35 . Hincmar warns us that he will pass from the action of Remigius to exhortations "for those who read or listen". In chapter 21, about the speechless heretic, it is Remigius himself who teaches "to all the other priests who were present, as to those who would read this passage or hear it" ; we understand that Hincmar and Remigius are pretty much interchangeable.

The rule that is expressed here -every penitent must be reconciled-is a recurrent matter for Hincmar. What is worth noticing is that he does not justify it by canons, but by the Law, contained in the Scriptures on the one hand, and by the history of Remigius that embodies justice on the other. Hincmar succeeds in converting a personal belief into an objective norm by the peculiar staging the Vita authorizes, through an alliance of lived example and imperative discourse. Yet that does not totally explain why Hincmar does not quote any canon on this matter. It is only in his table of contents, on chapter 16, that he alludes to canonical law and promises he will devote time to a canonical explanation about the bishop of Laon, Genebaud: "How one should understand the canonical statement that specifies with extreme insistence that a man subject to penance because of a public sin must not receive ecclesiastical orders or be kept in orders or recover his previous situation 36 ." What is chapter 16 about ? A bishop of Laon, Genebaud, had two children with his wife, though they should have separated after his episcopal ordination. Remigius, to whom Genebaud confessed his fault, did not degrade the bishop, but submitted him to a strict penance for seven years before reconciling him. 37 Hincmar tells the story, then raises the question: did Remigius act in accordance with canonical rules? His answer is a typically Hincmarian one, loudly proclaiming that nothing is more essential that the observance of Church's traditions… before concluding that Remigius was well-inspired not to follow its sacred rules:

No one should be bold enough to enunciate novelties in contradiction with the Apostle, revelling in his own words, by claiming that the sacred rules have been established against this example of God's mercy, or that the blessed Remigius acted in this case with respect to Genebaud against the sacred rules after receiving an order from God, or that it is against the regular canons that Genebaud was kept in his rank after his sin, since the Catholic Fathers proclaimed by mutual agreement this statement which should be sufficient: "When it was established in the Church that no one can receive orders after having done penance for a crime, or re-enter the clergy or remain in the orders, 35 VR preface, 254, 3-9: "In his autem, quae de scriptis virtutum miraculis, a Domino per beatissimum patronum nostrum operatis, ad exortationem legentium sive audientium pro modulo intellectus mei de catholicorum dictis subiungere studebo, vestigia beati Gregorii, licet non valeam, prosequi moliar, qui describens sanctorum actus pravorumque casus, exortatione inde assumpta, secundum sapientiam sibi a Deo datam multa necessaria et utilia legentibus ac audientibus interposuit". 36 VR preface, 256, 21-23: "et qualiter tenenda sit sententia canonum, quae post lapsum publice paenitentem ad gradum ecclesiasticum non accedere vel in gradu manere aut ad gradum redire". 37 VR c. 16, 300, 33-304, 5. On Genebaud, the most important commentary is now in R. Stone, "Gender and Hierarchy : archbishop Hincmar of Rheims (845-882) as a religious man" this has not been done due to a lack of confidence in forgiveness, but as a result of rigorous discipline. Otherwise it would challenge the power of the keys given to the Church by these words, 'Whatever you loose will be loosed [Matthew 18;18]'." Nothing is excluded or left out by that 'whatever', meaning 'everything'38 .

Hincmar concludes this passage with a long though implicit quotation from Augustine. 39 Augustine, and Hincmar after him, finds in the Scriptures the two cases of King David and Peter the apostle, to prove that by true humility, they confessed their sins and obtained not only their pardon, but also maintenance in their functions: David remained king as Peter remained apostle. Moreover, the Vita directly refers to a canonical text that Hincmar wrote himself and which could be found in his Capitula Episcoporum. 40 This capitulum about the penance of clerics may seem slightly stricter than the Vita, insisting on the necessity of a long penance that teaches humility when the Vita underlines the perfection of the mercy granted -yet there is no deep contradiction between them, only a difference in the form. The gloss Hincmar added in the Vita about the potestas ligandi of the Church aims to guarantee her full freedom of decision. The capitulum had to be drafted by an accumulation of patristic, explicit and traceable evidences or citations, while patristic quotations are invisible and far fewer in the Vita. The important point here is that the hagiographical text comment on the canons: the canons are by definition quite synthesised texts, decisions that do not leave much room for diversity of cases. In the Vita, Hincmar restores the legal interpretation that should accompany the canons. A law should not be used blindly but interpreted, adapted to circumstances. The facts are right against the principles, especially when the historical example is that of a saint. The hagiographic narrative thus serves as an argument of authority to demonstrate that there may be a difference between laws and judgments.

Finally, we obtain an unexpected answer to the question: Why did Hincmar choose to deliver his most intimate convictions and his most crucial commands via a hagiographic text? Speaking about a mere enlargment from the moral improvement assigned to every hagiographical text to a more normative and collective progress is not sufficient. Through hagiography, Hincmar creates two levels of language: on a first level, he composes what appears to be a reminder about canonical rules -on penance, on death, on the organization of a synod -and about dogmatic truths, a reminder which confirms the indisputable authority of the bishop on all these subjects. At the same time, but on another level, Saint Remigius's life demonstrates the total independance of the same bishop vis-à-vis these previous rules. Hincmar illustrates, supports and embodies in concrete behavior the idea he does not want to put explicitly: a bishop of Rheims is the source of law and is little constrained by the rules that constrain others.

Back to a stricter definition of hagiography as history

Hincmar counted on the diffusion of the Vita, expecting that it would be read in private and in public, broadcast by liturgy and memorized, more effectively than any synodal acts. In this particular regard, Hincmar's expectations failed: as a brief conclusion, a short study of some specific manuscripts of the Vita Remigii proves that this text was not copied in the way the archbishop hoped it would be.

At first sight, Hincmar's hopes would seem to have been fulfilled. After the death of Hincmar in 882, the Vita Remigii was often copied in Francia occidentalis, and, no surprise, especially in the province of Rheims. In fact it had an extraordinary dissemination, with manuscripts that are extremely numerous for a work of the ninth century: Krusch read 31 of them, but there are at least 85. However, from these 31, Krusch had to devise six categories to produce a coherent stemma, because the transmission of the text is highly volatile: there are only three almost full manuscripts and very few manuscripts are alike. The Vita Remigii is indeed a monument, a Vita so long that it was almost never thoroughly copied after the 11 th century. It is understandable that all its annexes -the prologue, table of contents, preface or reading guide, and documents such as the testament of Remigius-were omitted by scribes. But that is also true of the more biographical chapters, which are almost never copied verbatim. All copyists seem to have been puzzled by the heterogeneity of the work. They first removed the appendices. But they also cut pieces in narrative chapters, to rid the Vita of all speculative passages. Each manuscript of the Life of Saint Remigius therefore provides the chance to examine the shift between the enunciation of rules and their effectiveness. This is an ongoing problem in the study of medieval norms: we know what the rules say, but not if they were implemented. Starting from the consideration of a small part of the dissemination of Vita Remigii's manuscripts, we can actually assess whether the most normative passages of this text have been preserved, or not. Carmen 2 -

The text has been copied with infinite respect by several copyists, without missing a single line.

There are even the signs in the margins that Hincmar provided to distinguish the more difficult passages that have to be pondered on, and the ones that everybody can listen to. [START_REF] Vr Altera Praefatio | …quantum inde, populo audiente, legantur, et que instructioribus et studiosioribus, quando sibi licuerit vel libuerit, legenda serventur, designare curavi[END_REF] However, did these indications, that make it so easy to identify the speculative moments of the Vita, help other scribes to write an expurgated text, simply showing them what they had to suppress? A careful reading of some abbreviated manuscripts actually shows that, though the scribes indeed did remove the more theoretical passages, they chose them themselves. For instance, in a manuscript now in Rouen (Rouen, BM 1381 (U67)), let us read the chapter 6 of the Vita. In this chapter, Hincmar tells how Remigius healed a man who was poor, blind and possessed; he concludes that Remigius, who can give a threefold remedy, manifests the Trinity. Then he underlines the necessity for a Christian to pray and beg constantly, to accord what he believes and what he practises, to obtain by prayer what is really good for him. In the edition of the text, two moments are clearly distinguished: first, the story of the healing 42 , second, Hincmar's meditation on the three evils that afflicted the man, reported in the margin by the Greek letter gamma. 43 But the Rouen manuscript scribe failed to identify this organization. He kept the story and its conclusion about the Trinity, but only a part of the meditation that follows. 44 We might suppose that the scribe simply interrupted his work when he thought that the anecdote of the miraculous healing was completed, rather than selecting the sentences according to a specific purpose.

Another manuscript, again now in Arras, presents a more interesting case, even if it is difficult to draw solid conclusions from a manuscript that has been damaged (Arras BM, ms 31 (0823), fol. 16-32). The Vita Remigii copied here is an abbreviated version of the Hincmarian text. A brief look at chapter 16 about Genebaud gives a good idea of the proportions of this abbreviation: the scribe considered useless more than half of the text [START_REF][END_REF] , at first sight the most speculative passages, the dogmatic and exegetical teachings. However, not all the theoretical passages are omitted, so the selection does not refer to a more popular or a more liturgical use of the Vita, when shortened. The scribe omitted in fact all passages written in the first person plural 46 . Hincmar often uses this rhetorical formula, as if talking directly to the monks of Saint-Remi as their abbot, or to the clergy as their archbishop: "We read in the sacred history", "We have with Bishop Genebaud…", "We heard… ", "So, dearest brothers, we have… confidence in the mercy of our Creator", "Think about what we do, think again about what we did… ", etc. 47 All these exhortations, which are brief and easy to understand, have been removed : so the abbreviation does not match, or not only, the desire to make the text more accessible for the simple. This abbreviation, stifling the voice of Hincmar and the historical context of the first enunciation of the Vita, is the price that was paid for the text to be more than a Rheims history for Rheims readers. As the manuscripts are shortened, the legend of Remigius becomes part of a national history that largely exceeded Hincmar's purpose, in which the memory of the baptism of Clovis becomes the essential moment of Remigius's career.

The hagiography is still considered as a story that can teach something, but it is no longer a question of behaviour or of faith: out of Hincmar's hands, through the copy of scribes, hagiography now teaches the meaning of political history of the Frankish realm.

  20 Hincmarus Remensis, De Praedestinatione Dei et liberio arbitrio dissertatio posterior, Patrologia Latina 125, c. 65-474. One can notice at least these repetitions, VR 281, 14-15 = De Praed., col. 421 ; VR 281, 33-35 = De Praed., col. 190 = Grég., Hom. Ez. I, 9, 42 ; VR 283, 22-26 and 28-30 = De Praed., col. 444 ; VR 283, 37-39 = De Praed., col. 82 ; VR 283, 42-284, 1-12 = Pope Leo the Great, Tractatus septem et nonaginta, ed. A. Chavasse, CCSL 138, Turnhout, 1973, n° LXII, c. 351-352 with VR 283, 42-45 = De Praed., col. 345 ; VR 284, 22-27 = De Praed., col. 368 ; VR 284, 27-29 = De Praed., col. 288 ; VR 284, 29-30 looks like De Praed., col. 283 ; VR 284, 30-32 = De Praed., multi loci = Pope Leo the Great, Epistola CLXV, c. 1161 ; VR 284, 34-37 = De Praed., col. 307. 21 I suspect that a first version of the Vita was addressed to Louis the German, as a sort of monition: see Remi de Reims, op. cit. p. 523. 22 We still need a fresh look on Orléans' council of 511, that cannot be reduced to the beginning of a secular and French alliance between the throne and the altar. Gregory Halfond began to answer this need, first in his Archaeology of Frankish Church Councils, AD 511-768, Leiden/Boston, Brill, 2010, then in his paper "Vouillé, Orléans (511) and the Origins of the Frankish Conciliar Tradition", The Battle of Vouillé, 507 CE. Where France Began, ed. R. W. Mathisen and D. Shanzer, Boston/Belrin, Wlater de Gruyter, 2012, pp. 151-166. 23 VR 311, 31-313, 20, especially 3111, 34-312, 30. 24 Br. KRUSCH, Vita Remigii, pp. 241-242 and J. DEVISSE, Hincmar, op. cit., p. 652. 25 VR c. 20, 312, 8-11: "Praerogativam igitur de nostri sumpsimus electione iudicii, quando id operatum te esse didicimus, quod ceteris agendum obnixius imperamus, ut in provinciis tanta longinquitate disiunctis et apostolicae sedis vigorem et patrum regulis studeas adhibere custodiam"

  Let us first recall the general structure of the Vita. Some faithful scribes kept all the prose and narrative chapters, as in the oldest manuscript, from Saint-Vaast and still in Arras today (Arras, BM 199 (189), from the very end of the 10 th century). It has been damaged, but once contained most of the Vita:

	BHL 7152	Prologue	Hincmar recounts the circumstances under which
			he wrote the Vita and by what methods.
	BHL 7153	Table of contents	A very detailed table of chapters (summaries rather
			than titles)
	BHL 7154	Altera praefatio	A guide for the reading of the text
	BHL 7155	Cap. 1-23 (Vita)
	BHL 7156	Cap. 24-28 (Miracula
		post mortem)
	BHL 7157	Cap. 29 (translatio 852)
	BHL 7158	Cap. 30	Eulogy of Remigius: he equals all the other saints
	BHL 7159	Cap. 31	Another eulogy of Remigius
	BHL 7160	Cap. 32	Remigius' Testament
	BHL 7161	Carmen 1	Verses written on the saint's shrine by Hincmar
	BHL 7162	Carmen 2	Other verses written on the saint's shrine
	BHL 7152	Prologue	Now missing
	BHL 7153	Table	Fol. 1-2v
	BHL 7154	Altera praefatio	Fol. 3
	BHL 7155	Cap. 1-23 (Vita)	Fol. 3v-28.
	BHL 7156	Cap. 24-28 (Miracula)	Fol. 28-31v, with an accidental omission
	BHL 7157	Cap. 29 (translatio 852) Fol. 31v
	BHL 7158	Cap. 30	Fol. 32
	BHL 7159	Cap. 31	Fol. 33
	BHL 7160	Cap. 32 : testament	-
	BHL 7161	Carmen 1	-
	BHL 7162

VR c. 15, 296, 31-297, 4.

VR c. 14, 296, 1-15.

See below. Some important teachings can be read in VR c. 8, 281-284 on predestination and in VR c.7, 275 about the right understanding of the dogma of the Trinity.

VR c. 17, 309, 2-10: "bonum exemplum omnibus episcopis, tam suo tempore in carne viventibus, quam post eum in ordine succedentibus, derelinquens, ut pauperibus aut viduis vel pupillis vel pro his qui ad misericordiam aecclesiae confugiunt, qui iniuriam patiuntur, aut qui peccantes in exilio vel in insulis damnantur, aut certe quamcumque sententiam suscipiunt, iuxta sacros canones subvenientes, vel quaecumque agentes bona pro temporali retributione non faciant; sed iuxta vocem dominicam: Gratis accepistis, gratis date, quod Dei gratia, id est gratis data, acceperunt, gratis etiam, id est sine retributione temporali, proximis largiantur."

VR preface, 253, 26-28: "In the following pages, let the reader not be disturbed by the diversity of style : I put, exactly as I found them, the facts I received from the histories of our ancestors, and those I noticed in old parchments."

VR table of contents, 255, 7-12: "Qualiter civitatem Remorum conflagratam per demonem ab incendio liberavit, et nos, si fideliter petierimus, eius meritis et orationibus a viciorum flammis et ab igne perpetuo liberari valebimus. Unde, sumpta similitudine, commemorata est breviter falsitas predestinatianorum hereticorum, qui dicunt, Deum quosdam ad ignem perpetuum predestinasse; et hinc catholicae fidei veritas demonstratur."

VR c. 20, 312, 19-24: "Paternas igitur regulas et decreta sanctissimis diffinita conciliis ab omnibus servanda mandamus. In his vigilantiam tuam, in his curam, fraternae monita exortationis ostendimus. His ea quanta dignum est reverentia custoditis, nullum relinquit culpae locum sanctae observationis obstaculum . Ibi fas nefasque prescriptum est, ibi prohibitum, ad quod nullus audeat adspirare, ibi concessum, quid debeat mens Deo placitura presumere.

For the context of the correspondance between Hormisdas and Salluste, bishop of Seville, see Th. Deswarte, Une chrétienté romaine sans pape. L'Espagne et Rome (586-1085), Paris, 2010. [Page numbers?]

VR c. 20, 312, 24-26: "Quotiens universale poscit religionis causa concilium, te cuncti fratres evocante conveniant; et si quos eorum spetialis negocii pulsat intentio, iurgia inter eos oborta compesce discussa sacra lege determinando certamina."

On the growing importance of the knowledge of law in the episcopal ministry during Hincmar's career, see L. Jégou, L'évêque, juge de paix (VIII e -XI e siècle), Turnhout, 2011 and, more precisely about Hincmar, my « Être historien au IX e siècle», Rerum gestarum scritor. Mélanges Sot, ed. M. Coumert, K. Krönert, M.-C. Isaïa and S. Shimahara, Paris, 2012, p. 67-76.

See Vita chapter 30 for a good summary of the figures Hincmar used : VR 326, 24-328, 5. Archangels and other types of angels are to be found in the beginning of the very rhetorical chapter

31, VR 328,[START_REF]De cavendis vitiis et virtutibus exercendis[END_REF], that heavily depends on Gregory the Great's Homelium in Evangelia 34.

VR c. 14, 296, 13-20: "Quod de Moyse scriptum legimus, quia splendida facta est facies eius, dum respiceret in eum Dominus, hoc et in beatum Remigium luce splendida illustratum factum fuisse audimus; quoniam, sicut Moyses legislator populo veteri erat a Domino constitutus, ita et beatus Remigius euvangelicae gratie lator populo in proximo per fontem baptismatis innovando extitit munere Christi electus."

About the importance of councils as social experiences and occasions for the creation of a self-conscious representation of the Carolingian episcopate, see especially St. Patzold, Episcopus. Wissen über Bischöfe im Frankenreich des späten 8. bis frühen 10. Jahrhunderts, Ostfildern, 2008.

VR 313, 10.

VR c. 21, 314, 7-16: "ante superbus hereticus, humilis iam et catholicus catholicam fidem de sancta et inseparabili Trinitate et de Christi incarnatione catholice confessus est et in eadem confessionis sue fide se permansurum professus est. Sicque anima per infidelitatem perdito et corporali voce propter superbiam condempnato virtute divina sanctus Remigius et animae et corporis reddidit sanitatem, cunctis qui aderant vel lecturi seu haec audituri erant sacerdotibus patenter ostendens de male sentiendo peccante in Christum , qui per humanitatem proximus et frater nobis fieri dignatus est, quomodo erga peccantes in se vel in aecclesiam atque rebelies et erga post recognoscentes et paenitentes debeant agere."

VR , c. 16, 306, 5-15: "Et ne quis etiam contra apostolum delectans vocum novitates dicere presumat, aut sacras regulas contra hoc misericordiae Domini exemplum constituisse, aut beatum Remigium divina preceptione in hoc opere contra sacras regulas de Genebaudo egisse, aut ipsum Genebaudum contra regularem constitutionem post lapsum in gradu mansisse, satisfaciat illi consona catholicorum patrum sententia: Ut, inquiens, constitueretur in aecclesia, ne quisquam post alicuius criminis penitentiam clericatum accipiat, vel ad clericatum redeat, vel in clericatu maneat, non desperatione indulgentiae, sed rigore factum est disciplinae. Alioquin contra claves datas aecclesiae disputabitur , de quibus dictum est: 'Quaecumque solveritis super terram, erunt soluta et in caelo'. In eo enim, quod dicitur: 'Quaecumque solveritis, erunt soluta,' nihil excipitur, nihil non comprehensum relinquitur.

VR c. 16, 306, 21-30 = Aug., Ep. n°185, § 10, in Augustin, Epistolae, ed. A.Goldbacher, CSEL 57, repr. New York, 1961, p. 39.

Hincmarus Remensis, Capitula 1-5, ed. R. Pokorny, M. Stratmann, MGH, Capit. episc. 2, 1995, p. 34-89, on pp. 62-66, 4.