Christophe Courbage
email: christophe.courbage@hesge.ch

Guillem Montoliu-Montes

Béatrice Rey

How vulnerable is risk

Keywords: risk aversion, (cross-) DARA, (cross-) risk vulnerability, background risk, health risk JEL codes: D01, D81, I12 risk aversion, (cross-) DARA, (cross-) risk vulnerability, background risk, health risk. JEL classification: D01, D81, I12

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Introduction

Risk aversion behaviours drive many economic decisions, whether this is related to insurance, investment, portfolio allocation, health care or housing. Better understanding the determinants of risk aversion behaviours is therefore key to many areas in economics.

Risk aversion has been shown to be influenced by various factors amongst which individual's wealth, individual's health and the presence of other risks faced by individuals. [START_REF] Arrow | Aspects of the Theory of Risk Bearing[END_REF] was the first to make the assumption of risk aversion decreasing in wealth, also known as decreasing absolute risk aversion (DARA). Other works, e.g. [START_REF] Malevergne | On cross-risk vulnerability[END_REF], made reference to the concept of cross-DARA, i.e. risk aversion to wealth decreasing with individual's health. While DARA and cross-DARA deal with how vulnerable risk aversion is to a loss of wealth or health, another strand of literature investigated how risk aversion towards financial risk is vulnerable to the presence of another risk, i.e. a background risk, being either financial or non-financial. This has given rise to the concept of risk vulnerability [START_REF] Gollier | Risk vulnerability and the tempering effect of background risk[END_REF] and cross-risk vulnerability [START_REF] Malevergne | On cross-risk vulnerability[END_REF], respectively. Both strands of literature are theoretically linked to each other. For instance, under the expected utility framework, DARA along with the convexity of absolute risk aversion is a sufficient condition for risk vulnerability [START_REF] Gollier | Risk vulnerability and the tempering effect of background risk[END_REF]. In addition, (cross-) DARA is a necessary condition for (cross-) risk vulnerability [START_REF] Gollier | Risk vulnerability and the tempering effect of background risk[END_REF][START_REF] Malevergne | On cross-risk vulnerability[END_REF].

The aim of this paper is to empirically assess how financial risk aversion reacts to a change in individuals' wealth and health and to the presence of both financial and health risk using the Survey of Health, Ageing, and Retirement in Europe (SHARE) database. This allows to test whether individuals' preferences exhibit DARA, cross-DARA, risk vulnerability and cross-risk vulnerability. In addition, this paper also tests the hypotheses of convexity of financial risk aversion in wealth and in health. While the DARA and cross-DARA hypotheses have been empirically studied rather extensively, few papers have investigated the other hypotheses directly. When they did, they often indirectly tested risk aversion through the composition of a financial portfolio in terms of risky assets, they did consider specific background risks or they considered these hypothesis separately. This paper takes advantage of a specific question in SHARE that makes it possible to measure directly risk aversion in terms of individual preferences in the spirit of the definition by [START_REF] Arrow | Aspects of the Theory of Risk Bearing[END_REF] and [START_REF] Pratt | Risk aversion in the small and the large[END_REF]. By using the SHARE database, we can also study these hypothesis altogether and then investigate the links that exist amongst themselves.

This paper uses various variables to define the level of wealth, the level of health as well as financial and health risks so as to take into account potential endogeneity concerns regarding risk aversion. In particular, we once again take advantage of the SHARE database that makes it possible to define precise measures of wealth and health status as well as of financial and health risks. Moreover, the dataset we use allows us to address different econometric issues potentially biasing our analysis and results. In particular, we address the issue of endogeneity in the variables defining wealth and financial background risk and the problem of measurement error in the variable quantifying individuals' health.

Our results show that, indeed, individuals exhibit DARA and cross-DARA, i.e. financial risk aversion decreases in wealth and health. They also show that individuals exhibit risk vulnerability and cross-risk vulnerability, i.e. financial risk aversion increases with the presence of a background financial and health risk. Furthermore, we also find risk aversion to be convex with respect to wealth while linear with respect to health. These findings complement the literature on risk aversion behaviours. They can shed light on how various economic decisions under risk should be impacted by changes in the external environment of individuals regarding their wealth and health.

The article is organised as follows. In section 2, we briefly summarise the theoretical and empirical literature related to the hypothesis of risk aversion. Section 3 is devoted to the presentation of the database and the variables used. Section 4 presents and discuss the empirical strategy and the results of various econometric specifications. Finally, the last section is devoted to some concluding remarks.

Literature review

The literature of risk aversion stems from the seminal works of [START_REF] Arrow | Aspects of the Theory of Risk Bearing[END_REF] and [START_REF] Pratt | Risk aversion in the small and the large[END_REF] who defined an individual as risk-averse if he always prefers the expected outcome of a lottery with certainty rather than the lottery itself. [START_REF] Arrow | Aspects of the Theory of Risk Bearing[END_REF] was the first to hypothesise that risk aversion is likely to be decreasing with individual wealth, a hypothesis know as DARA. DARA has been extensively empirically studied. Most empirical works confirm this hypothesis, for instance whether this is for a sample of rice growers on small farms in Nepal (Hamal and Anderson, 1987), for Italy [START_REF] Guiso | Risk aversion, wealth and background risk[END_REF], or using experiments [START_REF] Levy | Absolute and relative risk aversion: An experimental study[END_REF]. DARA explains why wealthier people should buy less insurance and invest more in risky assets than people with less wealth. Yet, a few studies have found increasing absolute risk aversion based on life insurance decisions (see e.g. [START_REF] Eisenhauer | Risk aversion, wealth and the DARA hypothesis: A new test[END_REF].

Cross-DARA is a much more recent hypothesis and stems from the literature on multivariate risk aversion where preferences depend both on wealth and health. The term cross-DARA was first coined in [START_REF] Malevergne | On cross-risk vulnerability[END_REF] as risk aversion to wealth being decreasing in health even if this hypothesis was studied earlier on. Empirically, there is currently a debate on the effects of a deterioration in health on risk aversion. Cross-DARA has been studied mostly in the financial literature in the context of the determinants of individuals' asset allocation without making explicit reference to the term cross-DARA. [START_REF] Rosen | Portfolio choice and health status[END_REF] using American data from the 1992-1998 waves of the Health and Retirement Study (HRS) show that poor health is associated with a smaller share of financial wealth held in risky assets and a larger share in safe assets, therefore supporting the cross-DARA hypothesis. This is confirmed by [START_REF] Cardak | The determinants of household risky asset holdings, Australian evidence on background risk and other factors[END_REF] who find that risky asset holding is discouraged by poor health using Australian data. Yet, [START_REF] Love | Does health affect portfolio choices?[END_REF], using the 1992-2006 waves of the HRS and controlling for unobserved household heterogeneity, find that health does not appear to significantly affect portfolio choice among single households. [START_REF] Fratantoni | Home ownership and investment in risky assets[END_REF], using the U.S. 1989 Survey of Consumer Finances, finds that self-perceived health is positively correlated with the share in risky assets. [START_REF] Atella | Household portfolio choices, health status and health care systems: A cross-country analysis based on SHARE[END_REF], using the SHARE database, find mixed evidence of cross-DARA as they show that worse current perceived health status force households to reduce the exposure to financial risk in their portfolio allocation only in countries without a protective full-coverage national health service. A few other papers directly test cross-DARA (still without making reference to this term) by looking at the effect of health shocks on various measures of risk aversion. [START_REF] Hammit | Effects of health and longevity on financial risk tolerance[END_REF], using data from a stated-preference survey, find that financial risk tolerance, measured in terms of choices over risky gambles, is positively associated with health, supporting the cross-DARA hypothesis. [START_REF] Schurer | Lifecycle patterns in the socioeconomic gradient of risk preferences[END_REF] finds that health shocks significantly increase individual risk aversion using data from waves 2006-2012 of the German Socio-Economic Panel and where risk aversion is measured by self-assessed risk willingness. [START_REF] Decker | Health shocks and risk aversion[END_REF] find that individuals, after a shock in their health measured as a sharp reduction in their grip strength, declare themselves to be less willing to take risks in general.

Risk vulnerability stems from [START_REF] Gollier | Risk vulnerability and the tempering effect of background risk[END_REF] and means that risk aversion increases with the presence of an independent unfair background risk. Empirical studies on risk vulnerability are rather scarce. Once again, most papers use portfolio asset allocation decisions as a measure of risk aversion. [START_REF] Guiso | Income risk, borrowing constraints and portfolio choice[END_REF] show that investment in risky financial assets responds negatively to income risk using the 1989 Bank of Italy Survey of Household Income and Wealth and where income risk is defined by the subjective variance of the expected future real income, supporting the risk vulnerability hypothesis. Similar results are found for U.S. data [START_REF] Haliassos | Why do so few hold stocks?[END_REF]Frantatoni, 1998), German data [START_REF] Hochguertel | Precautionary motives and portfolio decisions[END_REF] and Australian data (Cardak and Wilkins, 2008). Very few papers define risk aversion in terms of willingness to take risks and not in terms of portfolio allocation decision. The only one we are aware of is [START_REF] Guiso | Risk aversion, wealth and background risk[END_REF]. Using a macroeconomic indicator of background risk and defining risk aversion by the willingness to take risks, they find that risk aversion is positively affected by background risk for Italian households, supporting also the risk vulnerability hypothesis. Their results also lead support to convexity of risk aversion in wealth while not directly testing such assumption.

Finally, regarding cross-risk vulnerability, this concept was theoretically introduced by [START_REF] Malevergne | On cross-risk vulnerability[END_REF]. Very few papers have tested this hypothesis and they strongly rely on both the definition of health risks and of risk aversion. [START_REF] Edwards | Health risk and portfolio choice[END_REF] shows that individuals with higher probabilities of medical expenses lower their risky portfolio shares. This is confirmed by [START_REF] Goldman | Medical expenditure risk and household portfolio choice[END_REF] showing that Medicare beneficiaries who face less medical expenditure risk are more likely to hold risky financial assets. [START_REF] Atella | Household portfolio choices, health status and health care systems: A cross-country analysis based on SHARE[END_REF] using SHARE data show that health risks, as measured by an index taking into account risky behaviour and asymptomatic disease, affect portfolio choices only in countries without a protective full-coverage national health service.

Data and variables

We use the Survey of Health Ageing and Retirement in Europe (SHARE) database to empirically assess how financial risk aversion react to various exogenous shocks on individual's wealth and health. SHARE is a multidisciplinary, longitudinal and cross-national micro-database containing information on health-related variables, labour market variables, economic variables and other variables (including education, housing, social support and risk attitudes) of a representative sample of European individuals aged 50 or older and their spouses. A first wave of SHARE was released in 2004. Since then, five other waves of the survey were released. SHARE follows the design of the U.S. Health and Retirement study and the English Longitudinal Study of Ageing. In this study, we use the second, fourth and fifth waves of the survey which were conducted in 2006, 2011 and 2013 respectively 1 . For more details on SHARE, readers should refer to Börsch-Supan and Jürges (2005).

Financial risk aversion

In SHARE, individuals evaluate their financial risk aversion by answering the following question:

When people invest their savings they can choose between assets that give low return with little risk to lose money, for instance a bank account or a safe bond, or assets with a high return but also a higher risk of losing, for instance stocks and shares. Which of the statements on the card comes closest to the amount of financial risk that you are willing to take when you save or make investments? 1.

Take substantial financial risks expecting to earn substantial returns 2.

Take above average financial risks expecting to earn above average returns 3.

Take average financial risks expecting to earn average returns 4.

Not willing to take any financial risks

Given the categorical nature of this variable, we recoded it as a binary variable, with 0 corresponding to individuals reporting to be willing to take financial risks (i.e., individuals answering to be willing to take substantial, above average or average financial risks) and 1 corresponding to individuals not willing to take any financial risks.

As it often happens with self-assessed variables, one could question the reliability of selfreported risk aversion in predicting future real-life behaviour. For instance, are incentives correctly aligned in order to ensure that interviewed people are going to reveal their true preferences? [START_REF] Dohmen | Individual risk attitudes: Measurement, determinants and behavioural consequences[END_REF] address this particular issue for risk aversion and conclude that the answers to that kind of question reflect in a reliable way the behaviour of people in a laboratory experiment.

Financial wealth

We consider financial wealth, i.e. the amount the respondent has in bank accounts, bonds, stocks, mutual funds and retirement accounts, measured in euros, as the variable to test the DARA hypothesis. We decided to focus on wealth and not on income as we expect, for our sample of relatively aged individuals, changes in wealth to be more relevant in determining 1 Waves 1 and 3 are not used because some of the variables of interest were missing. Wave 6 is not used due to the very low response rate for the variable defining financial risk aversion. risk aversion than changes in income. Similarly, we decided to focus only on financial wealth, excluding real assets, as we expect liquid wealth to have a higher impact on risk aversion than less liquid forms of wealth such as the value of owned property.

Potential endogeneity in the marginal effect of financial wealth on risk aversion, coming from simultaneity bias, is likely to be present in our study. Indeed, an increase in financial assets might cause a decrease in risk aversion (DARA property), but the relationship can go the other way round as well if a decrease in risk aversion increases financial wealth. This happens, for example, via stock returns if they are larger than for safer investments' returns. In this case, a negative estimated marginal effect of wealth on risk aversion might not reflect DARA behaviour in our sample, but rather a negative correlation between risk aversion and financial wealth coming from conservative investment.

Health

In SHARE, two categories of questions can be used to measure the respondents' health status. The first category relates to the standard self-reported measure of health for which individuals are asked how they would define their health status between excellent, very good, good fair and poor. However, drawbacks of self-reported measures of health have largely been documented in the literature. In particular, [START_REF] Bound | The dynamic effects of health on the labour force transitions of older workers[END_REF] acknowledge measurement bias, i.e. given that respondents are asked for subjective judgments, those judgments might not be entirely comparable across respondents, leading to the so-called error of measurement bias which tends to underestimate the impact of health on the variable of interest.

The second category of questions relates to objective and detailed measures of health for which individuals are asked whether they have ever been diagnosed to have had a series of diseases 2 . The inclusion as an explanatory variable of each detailed measure of health defined through diagnosis of a specific disease can be problematic because of the difficulties of interpreting the marginal effect of a change in a given health condition on the outcome of interest and because of multicollinearity [START_REF] Bound | The dynamic effects of health on the labour force transitions of older workers[END_REF]. By aggregating the individual health conditions in an index, for example by using the weighted number of chronic diseases, measurement error can also be present if this aggregate objective measure does not reflect correctly the severity of the respondents' diseases or the worsening in their quality of life. For instance, measurement error bias might be the reason why Attella et al. (2012) find that selfreported health affects stockholding contrary to an objective measure of health status. Other studies in health economics such as [START_REF] Van Houtven | Family structure and long-term care insurance purchase[END_REF] come to the same conclusion, finding significant effects of self-reported health but not of objective measures of health on their variable of interest.

In order to circumvent the problems of measurement error and multicollinearity, we create a health index such as the one suggested by [START_REF] Bound | The dynamic effects of health on the labour force transitions of older workers[END_REF]. This index is constructed by using an ordered probit model with the categorical self-reported measure of health as the dependent variable and a set of significant dummy variables indicating the diagnosis of a chronic disease as independent variables. As a way to investigate the convexity of risk aversion with respect to health, we additionally estimate with the same methodology a second health index using the square of the categorical self-reported measure of health as the dependent variable.

Finally, in order to facilitate the interpretation of the results, we re-scaled the original selfreported health question in SHARE so as to have zero as the modal answer and we inversed 2 Individuals are asked if they have ever had a heart attack, high blood pressure or hypertension, high blood cholesterol, a stroke, diabetes or high blood sugar, chronic lung disease, cancer, stomach or duodenal ulcer, Parkinson, cataracts, hip or femoral fracture, other fractures, arthritis, Alzheimer or any other non-mentioned condition. the order of the answers i.e., answer 2 corresponds to an excellent level of health and answer -2 to a poor level of health. The estimates of the two ordered probit models are displayed in appendix A.1. From the results of the two ordered probits, we compute the predicted expected latent variable and use its value as our measure of health.

Financial background risk

Different measures of financial background risk have been used in the literature, with labour income uncertainty being the dominant one. For instance, [START_REF] Guiso | An empirical analysis of earnings and employment risk[END_REF] and [START_REF] Hochguertel | Precautionary motives and portfolio decisions[END_REF] model financial background risk by using self-perceived measures of income uncertainty. [START_REF] Cardak | The determinants of household risky asset holdings, Australian evidence on background risk and other factors[END_REF] measure financial background risk as the realized variation of the household past labour income controlled with other variables such as the fact of being multiple earners in the household, the occupation and the mortgage over income ratio. [START_REF] Haliassos | Why do so few hold stocks?[END_REF] consider observed individual characteristics such as the fact of working in a profession with high risk of unemployment. Finally, [START_REF] Guiso | Risk aversion, wealth and background risk[END_REF] use a macroeconomic indicator of financial background risk, i.e. the variance of the residuals obtained by regressing the GDP in the province of the surveyed individual on a time trend.

We follow this literature and define financial background risk in the form of income risk through a binary variable equal to 1 if the individual feels his/her job security is poor and equal to zero if he/she does not have this feeling. As for financial wealth, we suspect this definition of background risk to be endogenous and to suffer from simultaneity bias. Indeed, risk aversion is likely to determine the individuals' exposition to financial background risk, if more risk averse individuals are less prone to choose risky professions in terms of job security than less risk averse individuals.

Health background risk

Defining a health risk is not so obvious. Attella et al. (2012), studying the effect of future health risk on the probability of stock holding, define health risk through an index including an average number of risky behaviours (smoking, drinking and sedentary lifestyle), an average number of asymptomatic diseases (high blood pressure, high blood cholesterol and osteoporosis) and grip strength. However, such a measure could be potentially biased. Indeed, risky behaviours might be endogenous as financial risk aversion could be correlated with unobserved determinants of risky healthy behaviour. Endogeneity could also affect the number of asymptomatic diseases as this variable is dependent on risky behaviours according to different WHO reports (WHO;2009, 2013) studying the risk factors leading to different diseases such as hypertension or diabetes.

In order to avoid these potential endogeneity concerns, we define health risk through a binary variable equal to 1 if the interviewed qualifies his/her current job as physically demanding and zero if he/she does not. Indeed, amongst individuals having the same current health status, the fact of working in a physically demanding job can be seen as a health risk factor because of the increased and continuous exposition to different physical hazards, physical load or fatigue. Empirical evidence corroborates this hypothesis. For instance, [START_REF] Leino Arjas | Leisure time, physical activity and strenuousness of work as predictors of physical functioning: a 28 year follow up of a cohort of industrial employees[END_REF] find high physical work strenuousness to predict poor physical functioning 28 years later in a cohort of metal industry employees. In the same line, Russo et al. (2006) find that individuals having had a history of manual work are more likely to have worse physical function than those with a history of non-manual work.

Other variables

In the econometric analysis, we control the effect of wealth, health and background risks on financial risk aversion with a series of additional variables. We first include four personal characteristics studied by [START_REF] Dohmen | Individual risk attitudes: Measurement, determinants and behavioural consequences[END_REF] as plausibly exogenous determinants of risk aversion which are gender (equal to 1 if the respondent is a woman, 0 otherwise), age, height and education in years. Additionally, we include a measure of cognitive ability created from three different tests, i.e. numeracy questions, verbal fluidity and memory following [START_REF] Bonsang | Risk attitude and cognitive aging[END_REF] who show that this variable is an important determinant of risk attitudes for old people. We include a dummy variable equal to 1 if the individual is self-employed as this might be simultaneously correlated with risk aversion, wealth and background financial risk. We also include two measures of household composition, a dummy variable which equals one if the surveyed is married or has a registered partnership and another variable specifying the number of people living in the same household as the interviewed individual, including co-residential children and parents but excluding lodgers (i.e. persons subletting the respondent's apartment). Indeed, being married or having a large family might act as a hedge for background financial and health risks if home production and market production are substitutes [START_REF] Edwards | Health risk and portfolio choice[END_REF]. Besides, we include in the econometric specification a series of behaviours potentially induced by risk aversion and likely to simultaneously influence the effects of the variables of interest. Those behaviours are smoking, drinking, defined as having drunk six or more drinks in one occasion at least once a month in the last three months, and physical activity defined as performing a vigorous physical activity at least once a month. We also control for the fact of being overweight, defined as having a BMI greater than 30.

Empirical analysis and results

Sample selection criteria and descriptive statistics

Given that our definitions of financial background risk and health background risk are closely related to labour market participation, we selected for our analysis only those respondents declaring themselves to be employed or self-employed, excluding those declaring to be housemakers, retired, unemployed or permanently sick or disabled. We additionally discarded some observations with unreliable values for the variable quantifying financial wealth. These restrictions together with missing values for some variables left us with an unbalanced panel containing 18'030 observations corresponding to 14'141 different individuals. The following table reports some descriptive statistics for the variables used in the empirical analysis. The majority of our sample (61.3% of the respondents) reports to be unwilling to take any financial risk, i.e. are financial risk averse. The sample average financial wealth is 64'146 €.

Looking at the health index, its sample mean is equal to 0.356. Regarding the exposure to financial and health background risks, 21.5% of individuals in our sample declare to be in a job with poor security and 42.7% consider their current profession to be a physical job. We additionally computed the explanatory variables means conditioned on risk aversion by taking advantage of using a dichotomous risk aversion measure. The more risk averse group, i.e. declaring to be unwilling to take any financial risk has, on average, 2.7 times less financial wealth than the less risk averse group, i.e. reporting to be willing to take financial risks. Moreover, individuals in the first group have, as well, a lower level of health on average and have higher probabilities of working in a precarious or physical job.

Econometric specifications

In order to test, in our sample, the hypotheses of DARA, cross-DARA, risk vulnerability and cross risk vulnerability, we assume our binary measure of financial risk aversion (𝑅𝐴𝑖𝑡) to be a function of the natural logarithm of financial wealth (𝑤𝑖𝑡), health (ℎ𝑖𝑡) and health squared, financial background risk (𝑓𝑟𝑖𝑡), health background risk (ℎ𝑟𝑖𝑡), a vector of controls (𝑋𝑖𝑡), a vector of country dummies (𝐶𝑖) and a vector of wave dummies (𝑊𝑖𝑡). These assumptions correspond to the following model:

𝑃𝑅𝐴𝑖𝑡=1𝛺= 𝐹𝛼+ 𝛽1𝑙𝑛𝑤𝑖𝑡+𝛽2ℎ𝑖𝑡+𝛽3ℎ𝑖𝑡2+𝛽4𝑓𝑟𝑖𝑡+ 𝛽5ℎ𝑟𝑖𝑡+𝑋𝑖𝑡𝛽𝑥+𝐶𝑖𝛽𝑐+ 𝑊𝑖𝑡𝛽𝑊
Assuming 𝐹. to be the cumulative distribution function of a standard normal distribution, the model described by Eq. ( 1) can be estimated using a probit model. However, in the framework of a probit model, it is very difficult to study the convexity of financial risk aversion with respect to wealth and to health. Moreover, given that financial wealth (𝑤𝑖𝑡) and financial background risk (𝑓𝑟𝑖𝑡) are potentially endogenous and the fact that the instrumental variables approach applied to non-linear models leads to inconsistent estimates [START_REF] Terza | Two-stage residual inclusion estimation: addressing endogeneity in health econometric modelling[END_REF], we additionally study the case where 𝐹(.) is a linear function. This corresponds to a linear probability model which can be estimated with the ordinary least squares (OLS) and two steps least squares (2SLS) methods.

Probit and OLS results

The following table presents the results of different estimations assuming exogeneity of financial wealth and financial background risk. In the first two columns, we present the results of two probit models. The specification of the first column does not include the health squared index and the second model includes this variable. In the third and fourth columns, we present the results of two linear probability specifications estimated by OLS. As in the previous two columns, the third column does not include the health squared index while the fourth column includes it. The effect of the logarithm of financial wealth on risk aversion is negative and highly significant, being by far the coefficient with the largest t-statistic in all regressions. Moreover, as wealth enters in the model as the natural logarithm of financial wealth, a convex instead of a linear relationship between wealth and financial risk aversion is found in the OLS models of columns 3 and 4. Like DARA, convexity of risk aversion is a common assumption in the theoretical literature (e.g. [START_REF] Gollier | Risk vulnerability and the tempering effect of background risk[END_REF].

Looking at the relationship between health and risk aversion, the health index coefficient is negative and statistically significant at the 1% level in the first three cases and significant at the 5% level in the last case. The coefficient corresponding to health squared is negative but not significant in any case and no relevant changes in the rest of the coefficients are observed in the models including the quadratic term. Therefore, even if a slight reduction in the coefficient corresponding to health is observed in these models, the models not including the health squared index are preferred. The non-significance of the reduction in the coefficients corresponding to the health index is confirmed by a one parameter Hausman test 3 on these estimates in the probit and OLS specifications. Indeed, the results of these tests equal 0.446 and 0.367 (p-values equal to 50.41% and 54.48%) for the probit and the OLS models respectively. This shows that the inclusion of the quadratic term does not significantly alter the effect of the health index on risk aversion.

The fact of feeling to be in a job with poor security, i.e. facing a background financial risk, is positively and significantly related to more risk aversion. Working in a physical job, which is a proxy of facing a health background risk, is also positively and significantly linked to a higher level of risk aversion.

Thus, our first empirical results support the DARA, cross-DARA, risk vulnerability and cross-risk vulnerability hypotheses. The effect of the health background risk is found to be the one having the largest magnitude in the three models, as this is, in absolute value, the largest among the four coefficients of interest in the three specifications. Our results also support the hypothesis of convexity of risk aversion with respect to wealth but not with respect to health as the health squared index coefficient in the model of column 4 is not statistically significant at the usual levels.

Concerning the set of controls, the exogenous determinants of risk aversion suggested by [START_REF] Dohmen | Individual risk attitudes: Measurement, determinants and behavioural consequences[END_REF] have the expected signs and are also highly significant. Women and older people are expected to be more risk-averse while tall individuals, highly educated individuals and people with good results in the test of cognitive ability are less risk-averse on average. Looking at the effects of risky behaviours used by [START_REF] Atella | Household portfolio choices, health status and health care systems: A cross-country analysis based on SHARE[END_REF] to define health risks, results are non-significant, as neither drinking nor smoking are significantly associated with risk aversion. Finally, physical activity, which can be seen as an activity reducing health risk, is related to lower risk aversion. However, causal interpretations of the control variables, especially those defining a behaviour, should be done with caution as their OLS estimates, without appropriate robustness checks, might not capture causality anyhow.

Instrumental variables approach and results

In this section, we follow an instrumental variables strategy and estimate the OLS model of the column 3 of section 4.3 by the two Steps Least Squares (2SLS) method 4 . This is done as we suspect the variables quantifying wealth and financial background risk to be endogenous. Concretely, we estimate two different models: the first one (column 1) assuming wealth is the endogenous variable and the second one (column 2) assuming financial background risk to be the inconsistent variable. In both models we discard the inclusion of the health squared index, as it was found to be non-significant in the previous section.

For the design of the instrumental variables (IV) strategy, we follow Mullahy (1999) by assuming there are some control variables highly correlated with the endogenous variables, included in our definition of 𝑋, that might be properly excluded from 𝑋 but included in 𝑍, the set of instrumental variables. Concretely, given that the variables smoking, drinking and overweight as well as some country dummies and the time dummies are not significant in any of the models of section 4.3, they can potentially be excluded from these specifications.

In the first model, we use two country dummies, Switzerland and Estonia, equal to one if the interviewed lives in Switzerland and Estonia respectively and zero otherwise, as instruments for the logarithm of financial wealth. Indeed, given the international dimension of our sample, the assumption of relevance for these two instruments is a plausible one, as in 2015, the Swiss GDP per capita was 66% higher than the EU-28 average, while the Estonian was 25% lower (Eurostat, 2017). The weak instruments F-statistic (see Table 3) and R 2 of the first-stage regression (see appendix A.2.1) show the high relevance of these instruments.

In the second model, we instrument the variable financial background risk with the three binary control variables smoking, drinking and overweight, with four country dummies Switzerland, Estonia, Portugal and Belgium and with the time dummy wave 4, equal to one if the individual was interviewed in the wave 4 of the survey and zero otherwise. Given that the available instruments for financial background risk are much less powerful than the instruments for financial wealth, in order to maximize the predictive power of the first stage regression, we have to include in 𝑍 all the available non-significant control variables in the OLS model significantly associated with financial background risk. Intuitively, the three variables smoking, drinking and overweight are likely to be correlated with poor job security as cigarette, alcohol use and high biometric personal values are found to be positively associated with presenteeism (i.e. working while sick) and absenteeism (i.e. absence from work) [START_REF] Goetzel | The relationship between modifiable health risk factors and medical expenditures, absenteeism, short-term disability, and presenteeism among employees at Novartis[END_REF]. The four country dummies might be exogenous determinants of poor job security because these countries have very different job market institutions. Lastly, wave 4 might be a good predictor for job insecurity as this wave field work was done in 2011, a period of relative low economic growth in Europe (Eurostat, 2017). The weak instruments statistic p-value (see Table 3), the F-statistic and the R 2 of the first-stage regression (see appendix A.2.2) show that the set of instruments used in the second 2SLS model is still relevant but to a much lower extent than the instruments for financial wealth. 4 The following models are estimated with the ivreg function included the library AER of the R statistical software.

The following table presents the results of the IV regressions: When financial wealth is instrumented by Switzerland and Estonia (column 1), the marginal effect of wealth on financial risk aversion increases in absolute value, from -0.047 (see column 3 of Table 2) to -0.051, and still being significant at the 1% level. However, the Wu-Hausman test does not reject the consistency of the OLS coefficient and therefore, we prefer the OLS estimate as it is more efficient. Excluding the two country dummies and using them as instruments has no effect on the remaining coefficients, as no additional significant changes are observed between the results of the OLS model of the third column of section 4.3 and the results of the first IV model of this section. Consequently, the Sargan test does not reject the null of validity of the overidentifying restrictions.

In the second model of Table 3, the estimated marginal effect of a financial background risk on risk aversion still has a positive sign and increases sharply, from 0.024 (see column 3 of Table 2) to 0.157. However, the coefficient's precision is dramatically reduced as the estimate's standard error increases much more, from 0.008 to 0.102. As a consequence, even if the IV coefficient is around 6.5 times larger than the OLS coefficient, it becomes nonsignificant at the usual confidence levels. Concretely, the two-tailed hypothesis test with the null hypothesis 𝐻0 : 𝛽3=0 displays a t-statistic equal to 1.541, corresponding to a p-value of 12.34%. However, the Wu-Hausman test does not reject the validity of the OLS estimate for financial background risk. Finally, the remaining coefficients remain stable and the only additional remarkable difference observed with respect to the OLS model of the previous section is a moderate reduction of the health index coefficient, which is still anyway significant at the 5% level. As a consequence, the Sargan test does not reject the validity of this model overidentifying restrictions.

Hence, the IV approach validates the DARA hypothesis and the hypothesis of convexity of risk aversion with respect to wealth. The coefficient corresponding to financial wealth instrumented by Switzerland and Estonia (column 1) still has the expected negative sign and is highly significant. Moreover, the Wu-Hausmann test does not reject the consistency of the OLS coefficient. As for the risk vulnerability hypothesis, the IV results are less conclusive as the unbiased estimate is close, but not significant at the usual levels (p-value of 12.34%). Nevertheless, the IV parameter of column 2 still has the expected positive sign, its magnitude is greater than the OLS parameter and the Wu-Hausman test does not reject the validity of the OLS estimate.

Conclusion

This paper empirically assesses how financial risk aversion reacts to a change in individuals' wealth and health and to the presence of both financial and health risks using the SHARE database. We take advantage of a specific question in SHARE that makes it possible to measure directly risk aversion in terms of individual preferences in the spirit of the definition by [START_REF] Arrow | Aspects of the Theory of Risk Bearing[END_REF] and [START_REF] Pratt | Risk aversion in the small and the large[END_REF]. The measure of financial risk aversion is based on a question on the unwillingness by individuals to take any financial risk when hypothetically saving or making an investment. By using the SHARE database, we can also study many hypothesis about risk aversion altogether and then investigate the links that exist amongst themselves.

Our findings support the hypothesis of DARA, cross-DARA, risk vulnerability and crossrisk vulnerability. Moreover, financial risk aversion is found to be convex with respect to wealth while linear with respect to health.

We also consider the potential endogeneity of wealth and financial background and develop an instrumental variable strategy. In that latter case, the DARA hypothesis still holds while the results for the risk vulnerability hypothesis are only partially validated.

These results can offer some valuable recommendations regarding various economic decisions in the face of shocks on health and wealth as they are often driven by risk aversion behaviours with respect to wealth. We provide a few illustrations. For instance, people facing risky wealth or risky health should increase their purchase of insurance. Those facing health risks should invest less in risky financial assets than those not facing health risks. Marriage and childhood have also been shown to be influenced by risk aversion [START_REF] Schmidt | Risk preferences and the timing of marriage uncertainty[END_REF]; more risk-averse women being less likely to delay marriage and more likely to delay birth or give birth at later ages. So women facing financial risks or health risks should marry young and/or give birth at later ages than women facing no financial or health risk.

Finally, our results are also in accordance with various links that have been theoretically shown in relation to risk vulnerability and cross-risk vulnerability. (Cross-) DARA is known to be a necessary condition for (cross-) risk vulnerability [START_REF] Gollier | Risk vulnerability and the tempering effect of background risk[END_REF][START_REF] Malevergne | On cross-risk vulnerability[END_REF]. Our results do not contradict these relationships as the hypothesis of DARA, cross-DARA, risk vulnerability and cross-risk vulnerability are found within our sample. In addition, convexity of risk aversion along with DARA are sufficient conditions for risk vulnerability [START_REF] Gollier | Risk vulnerability and the tempering effect of background risk[END_REF]. Hence, as convexity of risk aversion with respect to wealth is also found in our sample, our results empirically validate these theoretical links. Lastly, we find that risk aversion is linear in health which is not incompatible with cross-risk vulnerability. Indeed, Malvergne and Rey (2009) show that cross-DARA together with the convexity or linearity of risk aversion in health and a negative cross derivative of the utility function are sufficient conditions for cross-risk vulnerability. Unfortunately, our analysis could not identify empirically the sign of the cross derivative of the utility function5 as well as whether the condition to obtain (cross-) risk vulnerability are sufficient or necessary conditions. Such analysis which requires additional data beyond the ones we use in this paper are left to further empirical research.

A.2. First stage regressions of the IV models

Table 2

 2 Results of the probit and the OLS models Robust standard errors are recorded in parentheses. The significance levels of the two-tailed hypothesis test are coded as follows: * significance at 10% level, ** significance at 5% level, *** significance at 1% level. † Cox and Snell pseudo-R 2

	Dependent variable:	Probit	Probit	OLS	OLS
	Financial Risk Aversion	(i)			(ii)		(iii)	(iv)
	Intercept	2.668	***	2.717	***	1.386	***	1.399	***
		(0.321)	(0.339)	(0.106)	(0.112)
	Financial wealth (log)	0.148	***	0.148	***	0.047	***	0.047	***
		(0.007)	(0.007)	(0.002)	(0.002)
	Health index	0.094	***	0.088	***	0.027	***	0.025	**
		(0.027)	(0.030)	(0.009)	(0.010)
	Health squared index	--			0.043 (0.096)	--		0.012 (0.032)
	Financial Background Risk	0.078	***	0.078	***	0.024	***	0.024	***
		(0.026)	(0.026)	(0.008)	(0.007)
	Health Background Risk	0.197	***	0.197	***	0.067	***	0.067	***
		(0.022)	(0.022)	(0.007)	(0.007)
	Female	0.283	***	0.283	***	0.097	***	0.097	***
		(0.028)	(0.028)	(0.009)	(0.009)
	Age	0.015	***	0.015	***	0.005	***	0.004	***
		(0.002)	(0.002)	(0.001)	(0.001)
	Height (in cm)	0.006	***	0.006	***	0.002	***	0.002	***
		(0.002)	(0.002)	(0.001)	(0.001)
	Education (in years)	0.024	***	0.024	***	0.008	***	0.008	***
		(0.003)	(0.003)	(0.001)	(0.001)
	Cognitive Ability	0.041	***	0.041	***	0.013	***	0.013	***
		(0.007)	(0.007)	(0.002)	(0.002)
	Self Employed	0.278	***	0.278	***	0.096	***	0.096	***
		(0.029)	(0.029)	(0.010)	(0.010)
	Married	0.041	0.041	0.015	*	0.015	*
		(0.026)	(0.026)	(0.009)	0.009
	Household Members	0.011	0.011	0.004	0.004
		(0.011)	(0.011)	(0.004)	(0.004)
	Smoking	0.024	0.024	0.011	0.011
		(0.024)	(0.024)	(0.008)	(0.008)

Table 3

 3 Results of the instrumental variables modelsRobust standard errors are recorded in parentheses. The significance levels of the two-tailed hypothesis test are coded as follows: * significance at 10% level, ** significance at 5% level, *** significance at 1% level.

	Country
	dummies
	(Italy as
	referenc
	e
	country)
	and
	wave
	dummies
	(wave 5
	as
	referenc
	e wave)
	are
	included
	but not
	reported
	.

  Robust standard errors are recorded in parentheses. The significance levels of the two-tailed hypothesis test are coded as follows: * significance at 10% level, ** significance at 5% level, *** significance at 1% level. † Cox and Snell pseudo-R 2

	Dependent variable:	Model:				Dependent variable:	Model:
	Self-Reported Health	Ordered probit	Self-Reported Health squared	Ordered probit
	Heart Disease		0.717	***	Heart Disease	0.086	**
	Hypertension		(0.039) 0.486 ***	Hypertension	(0.040) 0.197 ***
	Cholesterol		(0.019) 0.154 ***	Cholesterol	(0.020) 0.148 ***
	Stroke		(0.022) 0.425 ***	Stroke	(0.023) 0.284 ***
	Diabetes		(0.079) 0.556 ***	Arthritis	(0.081) 0.005 **
	Lung Disease		(0.035) 0.650 ***	0 | 1	(0.028) 0.375 ***
	Arthritis		(0.048) 0.694 ***	1 | 4	(0.011) 0.900 ***
			(0.027)	(0.013)
	Cancer		0.614	***	Observations	18'030
	Ulcer		(0.050) 0.577 ***	Pseudo R 2 (%)	†	0.96
			(0.044)
	Parkinson		1.293	***
			(0.265)
	Cataracts		0.133	**
			(0.057)
	Hip Fracture		0.247	**
			(0.102)
	Other Fracture		0.170	***
			(0.035)
	Alzheimer		0.668	**
			(0.265)
	2 | 1		2.616	***
			(0.026)
	1 | 0		1.321	***
			(0.014)
	0 | 1		0.137	***
			(0.011)
	1 | 2		0.817	***
			(0.013)
	Observations		18'030
	Pseudo R 2 (%)	†	17.33

Table A .

 A Robust standard errors are recorded in parentheses. The significance levels of the two-tailed hypothesis test are coded as follows: * significance at 10% level, ** significance at 5% level, *** significance at 1% level.

	2.1				Table A.2.2
	First stage IV model (Wealth)	First stage IV model (Financial background risk)
	Dependent variable:	IV model	Dependent variable:	IV model
	Wealth (log)	(First stage)	Financial background risk	(First stage)
	Intercept	9.309	***	Intercept	0.181	***
		(0.017)		(0.005)
	Switzerland	1.647	***	Smoker	0.027	***
		(0.046)		(0.007)
	Estonia	1.701	***	Drinking	0.019	**
		(0.043)		(0.009)
	Observations	18'030	Overweight	0.024	***
	Adjusted R 2 (%) F-statistic	15.5 1'653.625	Switzerland	(0.008) ** 0.021 (0.010)
					Estonia	0.083	***
						(0.009)
					Portugal	0.070	***
						(0.025)
					Belgium	0.084	***
						(0.010)
					Wave 4	0.046	***
						(0.006)
					Observations	18'030
					Adjusted R 2 (%)	1.68
					F-statistic	39.499

[START_REF] Evans | Estimation of state dependent utility function using survey data[END_REF] and[START_REF] Finkelstein | Approaches to estimating the health state dependence of the utility function[END_REF] present empirical evidence suggesting that the marginal utility of wealth is higher when healthy than when sick, supporting the negative sign of the cross derivative of the utility function.

In multinomial probit models, the t-statistics' critical values are not the usual ones.

Appendix

A.1. Multinomial probit model


Nyoman Juniarta
email: nyoman.juniarta@loria.fr

Miguel Couceiro

Amedeo Napoli

Chedy Raïssi

Application of Biclustering to the Discovery of Constant and Gradual Patterns

Introduction

Biclustering is a simultaneous grouping of rows and columns of a matrix. Here we are focusing on constant-column and coherent-sign-changes biclustering. We show that those types of biclustering can be used in the problem of gradual pattern mining.

Clustering 2. Biclustering

Gradual Patterns Biclustering

A B C D E 1 2 3 4 5 A B C D E 1 2 3 4 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Constant-value bicluster

Constant-column (CC) bicluster 2 5 1 6 2 5 1 6 2 5 1 6 2 5 1 6

+ + -+ + + -+ --+ - + + -+
Coherent-sign-changes (CSC) bicluster

Difference between clustering and biclustering

Some types of bicluster

All rows jointly are in a given "agreement" with the columns m 1 m 2 m 3 m 4 m 5 g 1 1 5 3 2 7 g 2 1 1 4 2 7 g 3 2 5 4 5 3 g 4 2 5 4 5 7 A partition 𝑑 = 𝑝 𝑖 of a set of objects 𝐺 is a collection of 𝑝 𝑖 ⊆ 𝐺 such that there is no overlap among 𝑝 𝑖 . Given 𝐺 as the set of objects and 𝑀 as the set of attributes, then a partition function 𝛿: 𝑀 → 𝐷 maps an attribute 𝑚 to a partition 𝑑. Example:

Relation of CC and CSC biclusters

𝛿 𝑚 1 = 𝑔 1 , 𝑔 2 , 𝑔 3 , 𝑔 4
Similarity and order between two partitions Partition (of object)

𝑑 1 = 𝑝 𝑖 and 𝑑 2 = 𝑝 𝑗 𝑑 1 ⊓ 𝑑 2 = ራ 𝑖,𝑗 𝑝 𝑖 ∩ 𝑝 𝑗 𝑑 1 ⊑ 𝑑 2 ⟺ 𝑑 1 ⊓ 𝑑 2 = 𝑑 1 Example: 𝛿 𝑚 1 = 𝑔 1 , 𝑔 2 , 𝑔 3 , 𝑔 4 𝛿 𝑚 2 = 𝑔 1 , 𝑔 3 , 𝑔 4 , 𝑔 2 𝛿 𝑚 1 ⊓ 𝛿 𝑚 2 = 𝑔 1 , 𝑔 2 , 𝑔 3 , 𝑔 4

Partition pattern concept and CC bicluster

Given a set of attribute 𝑀 and partition space 𝐷, a pair 𝐴, 𝑑 is a partition pattern concept iff 𝐴 ′ = 𝑑 and 𝑑 ′ = 𝐴, where: Four pairs of hotels (out of six) show the relation among city population, hotel's distance from city center, and hotel's price.

𝐴′ =⊓ 𝑚∈𝐴 𝛿 𝑚 𝐴 ⊆ 𝑀 𝑑 ′ = 𝑚 ∈ 𝑀|𝑑 ⊑ 𝛿 𝑚 𝑑 ∈ 𝐷 Example: A concept 𝑚 1 , 𝑚 4 ,
Attribute h 1 h 2 h 1 h 3 h 1 h 4 h 2 h 3 h 2 h 4 h 3 h 4 Pop. > > < < < < Dist. < > > > > < Price > < < < < < 5.

Grouping of rows Grouping of rows and columns

The hotels in a big cities are more expensive than those in a smaller cities.

The hotels near city center are more expensive than those far from center.

CSC bicluster

  Pattern Structures for CC Biclustering

  𝑔 1 , 𝑔 2 , 𝑔 3 , 𝑔 4Two CC biclusters: 𝑔 1 , 𝑔 2 , 𝑚 1 , 𝑚 4 and 𝑔 3 , 𝑔 4 , 𝑚 1 , 𝑚 4

	Hotel	City population	Distance from center	Price
	h 1	1000	5	45
	h 2	500	10	25
	h 3	600	1	50
	h 4	40000	2	100


Ludivine Garcia

Federico Cisnetti

Natacha Gillet

Reǵis Guillot

Magali Aumont-Nicaise

Jean-Philip Piquemal

Michel Desmadril

‡,#,∇ Francois Lambert

Clotilde Policar

Entasis through Hook-and-Loop Fastening in a Glycoligand with Cumulative Weak Forces Stabilizing Cu I

The idea of a possible control of metal ion properties by constraining the coordination sphere geometry was introduced by Vallee and Williams with the concept of entasis, which is frequently postulated to be at stake in metallobiomolecules. However, the interactions controlling the geometry at metal centers remain often elusive. In this study, the coordination properties toward copper ionsCu II or Cu I of a geometrically constrained glycoligand centered on a sugar scaffold were compared with those of an analogous ligand built on an unconstrained alkyl chain. The sugar-centered ligand was shown to be more preorganized for Cu II coordination than its open-chain analogue, with an unusual additional stabilization of the Cu I redox state. This preference for Cu I was suggested to arise from geometric constraints favoring an optimized folding of the glycoligand minimizing steric repulsions. In other words, the Cu I d 10 species is stabilized by valence shell electron pair repulsion (VSEPR). This idea was rationalized by a theoretical noncovalent interactions (NCI) analysis. The cumulative effects of weak forces were shown to create an efficient buckle as in a hook-and-loop fastener, and fine structural features within the glycoligand reduce repulsive interactions for the Cu I state. This study emphasizes that monosaccharide platforms are appropriate ligand backbones for a delicate geometric control at the metal center, with a network of weak interactions within the ligand. This structuration availing in glycoligands makes them attractive for metallic entasis.

The possible control of metal properties by the coordination geometry was an idea introduced by Vallee and Williams 1 with the concept of "entatic state" 2,3 and refers to proteins in which metal cations are in an unusual energized state. 4 Entatic states can be achieved using structures enabling the coordination but with some constraints leading to a geometry that does not match the geometrical intrinsic requirements of the metal ion. 2,4 In the case of copper(II/I) redox couple, as Cu II and Cu I cations display different geometrical requirements, the geometry can be used to control the redox potential, 5 as shown in a few examples in the literature. [6][7][8] Indeed, the control of the redox state for the Cu II/I couple can be achieved by a structural constraint favoring Cu II over Cu I , as explained in the following. 5 Cu I is a d 10 ion, and therefore, it has no structural preference based on its electronic structure. Hence, Cu I complexes often adopt 4-coordinate tetrahedral geometries, organizing Lewis bases at the farthest possible positions, as expected from the VSEPR (valence shell electron pair repulsion) theory. Cu II d 9 ion undergoes Jahn-Teller distortion 9 with a strong preference for axially distorted over perfectly octahedral structures. Then, a ligand forcing an axial distortion would favor Cu II over Cu I . Since the pioneering works in the 1980s, 10,11 geometrically constrained ligands have been described in the literature and used to control the properties of various metal ions, such as ligands bearing bulky substituents, 12 in particular to control the geometry of the coordination sphere and redox potential, 6 ligands with rigid spacers between Lewis bases, 11 substituted macrocycles, 13 capped calixarenes, 14 and nonmacrocyclic ligands. 15,16 An original and still scarcely used strategy is to take advantage of ligands built on a sugar central platform that acts as a constrained central platform to control the physicochemical properties of the derived complexes. [17][18][19][20][21] We developed glycoligands as a family of monosaccharides with appended chelating units, which could embed different ions with control of the chelation site [22][23][24][25][26][27][28][29][30][31][32] and of physicochemical properties such as magnetic properties, 23 stereochemistry, 24,31 and binding constant. 32 In the present article, the coordination properties of a ligand centered on a ribo scaffold (L) are compared to those of an analogous ligand built on an open unconstrained alkyl chain (L′). Interestingly, a preorganization of the ribo-centered ligand (L) for Cu II coordination with a higher stabilization of the Cu I redox state in the case of L was evidenced experimentally. This has been rationalized by a theoretical noncovalent interactions (NCI) analysis. 33,34 This approach, focusing at the weak VSEPR interactions in an extended network involving the ligand at the immediate metal neighborhood, showed a clear loss in repulsive interactions in the Cu I state both in the glycocomplex and in its analogue, with some significant additional stabilization in the case of the ribobased system. Possible fine tuning of redox potential by noncovalent interactions in Cu-proteins has arisen some interest lately, 35,36 and the study reported here shows that this can be achieved in small complexes. Interestingly, it was shown that the cumulative effects of several weak interactions, identified by the NCI method, can be a source of efficient stabilization for Cu, as in a hook-and-loop fastener.

■ RESULTS AND DISCUSSION

To study the influence of the constraints of the sugar platform on the properties of Cu II complexes, we synthesized a glycoligand (L) based on a ribo-furano scaffold with an isopropylidene ketal that acts as a conformational lock and its unconstrained analogue (L′) based on an open alkyl chain (see Figure 1, Material and Methods, and Supporting Information for synthesis).

The structure of the Cu II complexes was solved by X-ray crystallography in the solid state. As shown in Figure 1 (see also Supporting Information), the structure of [Cu(L)] 2+ is closer to the ideal trigonal bipyramid (TBP) than that of [Cu(L′)] 2+ , which is distorted toward the square pyramid (SP) (τ = 0.84 and 0.67, respectively, τ = 1 being for an ideal TBP and τ = 0 for an ideal SP). 37 Overall, the spectroscopic properties in solution, both in X-band EPR as well as UV-vis, are consistent with the X-ray structures (see Supporting Information, Figures S3 andS4). Interestingly, the redox potential of the glycocomplex couple was significantly higher than that for the acyclic analogue (E°g lyco = -39 mV/NHE, E°a cy = -121 mV/ NHE, ΔE glyco-acy = 82 mV) (see Supporting Information, Figure S5).

Complexation of Cu II by L and L′ was studied in EtOH by isothermal titration calorimetry (ITC) to determine the various thermodynamic parameters, including the association constant (K), the binding enthalpy (ΔH), and the entropy (ΔS) at 298 K. Figure S6, Supporting Information, depicts the series of exothermic heat profiles for the titration of L and L′ with Cu II ion, and the parameters from the simulation (see Material and Methods and Supporting Information) are reported in Table 1.

The binding enthalpy ΔH reflects classical steric terms such as van der Waalsas well as electrostatic interactions and hydrogen bonding but also angular and torsional strains and electronic contributions. The entropy ΔS is associated with the reorganization of the ligand upon complexation. [START_REF] Martell | Determination and Use of Stability Constants[END_REF][START_REF] Hancock | [END_REF][40] For the reaction [Cu II (OH 2 ) n ] 2+ + Lgd aq = [Cu II (Lgd)] 2+ + n′H 2 O (Ldg = ligand, n′ can be greater than n if the ligand was solvated), solvation effects, both at the metal ion and at the ligand level, are also important enthalpic and entropic factors. The release of water molecule(s) to solvent upon binding of a polydentate ligand induces an increase in disorder and hence a favorable entropic contribution. [START_REF] Bertini | Biological Inorganic Chemistry, Structure and reactivity[END_REF] In the present study, the two ligands offer a similar 3N im -2N amine coordination sphere. Despite the difference in steric constraints at the central scaffold and in the ligands folding, the Lewis bases are likely to be similarly exposed to the solvent, with similar solvation and similar changes in solvation upon binding the Cu II ion. Hence, by comparing the two ligands, alkyl (L′) and glyco centered (L), the solvation contribution compensate for each other, and they are therefore not considered in the following discussion.

The binding constant of Cu II to L was found to be 20 times higher than that for L′, which was paradoxically associated with a less negative binding enthalpy (Table 1). This suggests that the constraints from the ribo platform prevent optimization of the Cu-N distances. This is particularly true in the case of Cu-N alkyl (N3 and N5) included in the metallacycle (Cu-N3-C3-C4-C5-N5), which is fused with the furanoid cycle in the case of L (see Table S2, Supporting Information). In addition, the observation that |ΔH([Cu(L′)] 2+ )| > |ΔH([Cu(L)] 2+ )| indicates a higher covalency for the Cu-N bonds in [Cu(L′)] 2+ , which is also suggested by the comparison of the molar extinction coefficients in UV-vis spectra, the higher value for [Cu(L′) 2+ ] being indicative of a higher covalency (see Supporting Information). The higher stability of the complex [Cu(L)] 2+ is clearly of an entropic origin. Indeed, the entropy loss observed in the case of [Cu(L)] 2+ is so small that this contribution overcompensates the enthalpy and leads to an increased K value in comparison with [Cu(L′)] 2+ . This smaller entropy loss is the consequence of weaker reorganization of the ligand structure upon coordination 4 in the case of L than in the case of L′, suggesting a higher preorganization of the glycoligand L for the coordination of Cu II . Preorganization is a concept that usually avails for macrocyclic ligands and cryptates that refers to the entropic loss associated with the freezing of the ligand in a conformation close to that encountered in the complex itself. 15,[START_REF] Martell | [END_REF] Preorganization includes the overcoming of repulsive forces between polar groups coming toward each other in the complexation process: if they are already close in the ligand, this unfavorable interaction preexists in the ligand itself and is not working against complexation. [START_REF] Martell | [END_REF] The chelating part in both L and L′ is an open-chain ligand with constraints induced by the glyco scaffold in L. This study is fully consistent with the idea according to which the concept of preorganization [START_REF] Martell | [END_REF] can be an important factor governing the stabilities of the complexes derived from open-chain ligands.

To confirm this suggested preorganization of the glycoligand L for Cu II coordination, the structure of the sugar ring in the metal-free ligand in solution was compared with its structure in the coordinated ligand in the solid state. To do so, the coupling constants 3 J calcd , calculated using the Haasnoot-Altona equation 43 and H-C-C-H torsion angles in the sugar from the solid state structure of [Cu(L)](PF 6 ) 2 were compared with the experimental coupling constant 3 J exp obtained by 1 H NMR of L in solution (see details of the Haasnoot and Altona analysis in the Supporting Information). Interestingly, a good match was obtained indicating that the structure of ribo platform is unchanged upon complexation. This result clearly supports the preorganization of L for Cu II coordination.

Rorabacher and co-workers have shown in the case of a series of tripodal ligands containing N and S donor atoms that an increase in Cu II/I redox potentials was associated with the relative destabilization of the Cu II complexes rather than with stabilization of the Cu I species. 5 In the present case, [Cu II (L)] 2+ was shown to be 20 times more stable than [Cu II (L′)] 2+ (see above). The higher redox potential observed for the Cu II/I complexes derived from L can then only be explained by a much higher stability of the Cu I complex derived from L in comparison with L′. Indeed, the potential values of the Cu II/I redox couples are dependent on the stability constants of both Cu II and Cu I complexes according to

°= °- E E R Tn F K K (2.303 / ) log( / )
glyco or acy Cu(II)/Cu(I)

Cu(II)(Lgd) Cu(I)(Lgd) (1)
where E o Cu(II)/Cu(I) is the standard potential of the Cu II/I redox couple in ethanol (E°g lyco for Lgd = L and E°a cy for Lgd = L′) Generally speaking, the folding of a ligand to chelate a metal ion induces unfavorable steric interactions. 4,[START_REF] Hancock | [END_REF][START_REF] Martell | [END_REF] As Cu(I) d 10 has no structural preference, its complexes will adopt a geometry that minimizes these steric repulsions. Their geometry should therefore be dictated by the VSEPR theory: for a pentadentate ligand, the optimal spatial distribution of the five donor atoms should be TBP, which occupies the whole 3D space, and not SP, which occupies only half-space. The ligand L in [Cu(L)] 2+ was shown to offer a preorganized coordination sphere closer to TBP than to SP (see Figure 1, bottom). Hence, the higher stabilization of Cu I by L could be understood as a VSEPR-controlled stabilization.

To support this suggestion of a VSEPR-controlled stability for the Cu I complex with L we performed a gas-phase quantum chemical study at the B3LYP/6-31G++** level [44][45][46] of the noncovalent interactions (NCI) at work within the Cu II and Cu I complexes of both L and L′ ligands. To do so, we used the NCI plot program 34 recently introduced by one of us which allows one to perform a NCI analysis (noncovalent interaction). 33,34 Analysis of the electron density enables the visualization and quantification of local weak interactions. To the best of our knowledge, this method has still seldom been used for transition metal complexes and this is its first application to investigate the influence of the metal redox state. Details about the NCI method are provided in the Supporting Information (see also refs 33, 47, and 48). This analysis provides 3D representations of the interaction surfaces meant to locate the repulsive and attractive troughs within the atomic structure. In these representations, the nature of the interaction is color coded: red for repulsive, green for weakly attractive or repulsive, and blue for attractive interactions.

The NCI approach is a powerful tool to study weak interactions. 33,34 We compared the interaction network within [Cu(L)] 2+ , [Cu(L)] + , [Cu(L′)] 2+ , and [Cu(L′)] + structures. Two kinds of domains were considered to understand the stability of transition metal complexes: (a) the metal subvalence core corresponding to the partially covalent coordination bonding and (b) the ligand interaction network including destabilizing interactions due to steric packing. 47,48 For Cu II complexes, the NCI analyses were performed using the crystallographic structures. Overall, the optimized coordination sphere geometries are very close for L and L′ and to the experimental values (see Table S6, Supporting Information). Table 1 shows thermodynamic parameters computed using the Gaussian software on optimized structures based on crystallographic data at the same level of computation used for the NCI computation (see Supporting Information). A good qualitative agreement between experimental and computed values was obtained. The interaction network shown in the NCI-3D plot in Figure 2 involves stabilizing interactions (in blue) that clearly correspond to ligand-Cu bonds or subvalence core. Within the ligand interaction network of both Cu II complexes two features are noteworthy.

(a) The ligand interaction network leads to a weak overall stabilization because the involved interactions are dual, with an attractive domain (blue to green) that is compensated by a repulsive contribution (red) from steric effects (see Figure 2). This indicates overpacking for both [Cu(L)] 2+ and [Cu(L′)] 2+ . This is the case, for instance, for interaction between imidazole rings (see Figure 2 insert (b)). (b) Overall, there is a larger number of these interactions for the glycoligand L than for the acyclic ligand L′. For instance, an additional H bond is clearly observed in [Cu(L)] 2+ between O 2 and H N3 as depicted in Figure 2a, which appeared dual with both attractive and repulsive features. Because of this compensation, the H bond is only weakly attractive. Clearly, [Cu(L)] 2+ exhibits an extended network of weak interactions ranging from H-bond to van der Waals interactions that is much richer than in [Cu(L′)] 2+ . This is an important main difference. The present analysis shows that the overall stabilization provided by this network is weak due to the compensation of attractive interactions and repulsive interactions, with the repulsive part due to steric crowding.

For the analysis of the Cu I complexes, the structures complexes were optimized at the B3LYP/6-31G++** level. The Cu II → Cu I reduction globally relaxes the coordination spheres with an increase in N-Cu with both ligands (see Table S6, Supporting Information). The geometry is TBP with a strong axial distortion, leading to an environment close to trigonal, with a global loosening of the steric crowding and repulsions within the ligand interaction network (see Figure 3 and movies 3 ([Cu(L)] + ) and 4 ([Cu(L′)] + ), Supporting Information).

In other words, the reduction to Cu I induces a dilatation of the subvalence core, which was expected from the ionic radii of Cu I and Cu II . 49 This dilatation leads to a weakening of the repulsive steric contributions, and therefore, the attractive contributions of the ligand interaction network prevail in the Cu I structures. As this attractive network is richer in the case of the L complex, the stabilizing effect upon reduction is stronger.

Two specific examples can be given to illustrate this richer network leading to an enhanced stabilization of [Cu(L)] + .

(a) The steric repulsion between imidazole rings and generally in the Cu-ion surroundings are reduced in the Cu I complexes: as seen on the NCI-3D representations, the number of red/repulsive troughs between imidazole rings is smaller in the case of the Cu I with a higher number of green/attractive troughs than in the case of the Cu II compounds (compare Figure 3 with Figure 2, see also figures in the Supporting Information and movies). (b) The hydrogen bond between O 2 and H N3 strengthens with a weaker repulsive contribution than in the case of the Cu II structure and an additional attractive interaction with H Cl with a smaller O 2 H Cl distance in the Cu I complex (see Table S6, Supporting Information). The NCI analyses provide a fine understanding of the effect of the ribo platform. Most of the interactions are similar for the L and L′ complexes, but importantly, the ribo platform induces additional weak interactions. The relaxation of the coordination sphere upon copper reduction decreases the steric congestion for both L and L′. The NCI analysis reveals that, in the case of L, the overall interaction network, because it involves a higher number of weak interactions than in L′, becomes significantly more attractive in the Cu I complex: as in a hook-and-loop fastener, the combination of several weak attractive interactions leads to a significantly higher stability of [Cu(L)] + . The present NCI analysis focused at the VSPER interactions network can be directly linked to both experiment and computed thermodynamic values (see Table 1). The extended network identified for L induces a rigidity higher for L than for L′, which organizes L in a conformation suitable for coordination. For the ribo scaffold, a large part of the loss in entropy associated with the reduction of conformational space upon complexation preexists in uncomplexed L. The same trends are observed with both oxidation states. For Cu I , free energies are larger in values, giving also a preference for L (ΔΔG(Cu I vsCu II )= -3.1 kcal mol -1 for L, see Supporting Information). Differences between L and L′ complexes are mainly due to ΔS values, which are found to be also similar to Cu II with a value lower by 11.2 kcal mol -1 for L. Thus, there is an additional cohesion within L, which is similar in its Cu II and Cu I complexes, associated with multiple interactions that have been unraveled by NCI. Overall stabilities of the networks were verified by performing NCI on limited ab initio Born-Oppenheimer molecular dynamics trajectories (225 fs simulation at the same level). 50 These analyses were performed in the gas phase, but similar interaction patterns have been observed in the presence of a continuum solvation model (PCM, see Supporting Information). 44 These computational results from the NCI analysis highlight the role of the ribo platform in the extended network bridging the metal density to the ligand and provide an extended view of the VSEPR argument based on the experimental observations.

■ CONCLUSION

The coordination properties of a glycoligand based on a ribo platform (L) and its open-chain analogue have been investigated by several physicochemical techniques, including UV-vis and EPR spectroscopies, ITC, and cyclic voltammetry. The preorganization of the glycoligand L for coordination of Cu II has been evidenced by ITC and by comparison of structural data in the solid state (X-ray structure) and in solution (NMR). The comparison of the electrochemical properties of the two couples [Cu(L)] 2+ /[Cu(L)] + and [Cu(L′)] 2+ /[Cu(L′)] + has suggested that L has a preference for Cu I over Cu II compared with the acyclic ligand. This preference for Cu I can be rationalized by the geometric constraints imposed by the sugar, which favor an optimized arrangement of L folding in space, minimizing steric repulsions. In other words, the stability of the two Cu I complexes is controlled by VSEPR.

This proposition was questioned using the NCI method. This approach takes into account an extended VSEPR network, including both the close-by ligands and the metal domain. A network of weak interactions is at stake and must be considered for an adequate description of the metal complexes stabilization. The steric constraints imposing an extended coordination sphere in the glycoligand are clearly associated with the stabilization of the Cu I glycocomplex. The net cumulative stabilization effect is induced by the weakening of the repulsive interactions due to the dilatation of the coordination core. Interestingly, this weak interaction network is shown to be the source of an efficient stabilization, as in a hook-and-loop fastener where the cumulative effects of weak forces create an efficient buckle. This study is thus an original and unprecedented demonstration of a VSEPR stabilization of Cu I complexes through efficient relaxation of repulsive interactions. Clearly, fine structural features in the glycoligand contribute to the stabilization of the Cu II complex and also, to a larger extent, of the Cu I complex. This study thus emphasizes that monosaccharide platforms are appropriate ligand backbones for a delicate geometric control of the properties of the complexes with a network of weak interactions within the ligand mimicking in a close shell interaction at stake in biomolecules structuration. This structuration availing in glycoligand makes them attractive for metallic entasis.

■ MATERIAL AND METHODS

Ligands Synthesis. The ligands were synthesized by classical reductive amination. Treatment of 3,5-diaminodideoxy-1,2-O-isopropylidene-α-D-ribo-furanose, synthesized as previously described 51 with 3 equiv of 2-(N-methyl)-imidazolyl-carboxaldehyde using NaBH-(OAc) 3 as a reductive agent in 1,2-dichloroethane, gave selectively L in 41% yield. Selectivity for the 5N,5N,3N-trisubstitution can be rationalized by the steric hindrance from the cyclic isopropylidene acetal preventing disubstitution of N3 (see numbering in Figure 1). Similar synthesis from N,N′-bis((1-methylimidazol-2-yl)methyl)-1,3diaminopropane provided L′ (yield 57%). Details and full characterization are provided in the Supporting Information.

Complexes. The complexes were obtained by addition of an equimolar amount of Cu(NO 3 ) 2 •3H 2 O in EtOH. Addition of NH 4 PF 6 (2.5 equiv) resulted in an immediate precipitation. The precipitate was redissolved using acetone, and crystals suitable for X-ray diffraction were grown by slow evaporation. Details of the complexes characterization, crystal data, data collection, and refinement are given in the Supporting Information.

Computations were performed using the Gaussian package 44 The B3LYP functional 45,46 was used along with the 6-31++G** basis set 52 for all computations.

Experimental Details. Materials and methods including chemicals, synthesis of the ligands, crystal data of the complexes, physicochemical characterization of the complexes, and NCI analyses can be found in the Supporting Information.

■ ASSOCIATED CONTENT * S Supporting Information Synthesis of the ligands and preparation of the complexes, crystal data of the complexes, physico-chemical characterization of the complexes (UV-vis spectra, cyclovoltammograms, thermograms, Haasnoot and Altona's analyses, computations, and NCI analyses. This material is available free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATION

Figure 1 .

 1 Figure 1. Structures of L and L′. Representation of the [Cu(L)] 2+ unit in the solid state structure of [Cu(L)](PF 6 ) 2 and of the [Cu(L′)] 2+ unit in the solid state structure of [Cu(L′)] 2 (NO 3 )(PF 6 ) 3 •2CH 3 OH, showing anisotropic vibration ellipsoids (probability 50%). (Bottom) τ axis showing the distortion of the two structures with regard to TBP and SP.

Figure 2 .

 2 Figure 2. NCI-3D of [Cu(L)] 2+ (A) and [Cu(L′)] 2+ (B): (blue domains) Strong attractive noncovalent interactions, (green) weak interactions, and (red) repulsive interactions. O 2 -H N3 interaction is highlighted by the green circle and enlarged in a. (b) Dual interactions between imidazole rings. Note that the color scale depends on the value of the electronic density chosen for the representation of the interactions. It can be modulated to highlight the red/repulsive or blue/attractive character of usually green/weak interactions. In this figure, the electronic density scale is different for the global complexes and for each insert. The corresponding color scale is given next to each picture. Figures are available as movies in the Supporting Information, see movies 1 ([Cu(L)] 2+ ) and 2 ([Cu(L′)] 2+ ).

Figure 3 .

 3 Figure 3. NCI-3D of [Cu(L)] + (A) and [Cu(L′)] + (B): (blue domains) strong attractive noncovalent interactions, (green domains) weak interactions, and (red domains) repulsive interactions (see also movies 1 and 3, Supporting Information). The electronic density scale is different for the global complexes and for the inset of the O 2 -H N3 interaction. The corresponding color scale is given next to each picture.

Table 1 .

 1 Thermodynamic Parameters of the Association Constant and Computed Values of ΔG, ΔH, and ΔS a Thermodynamic parameters of the association constant [Cu II (OH 2 ) n ] 2+ + Lgd aq = [Cu II (Lgd)] 2+ + n′H 2 O (Lgd = L or L′) Determined by ITC at 298 K in EtOH (see Supporting Information). Calcd: computed values of ΔG, ΔH, and ΔS using Gaussian in the gas phase (Cu 2+ + Lgd = [Cu(Lgd)] 2+ (see Supporting Information)).

	L/L′		K Cu(II)Ligand	ΔG b	ΔH b	-TΔS b
	L	exp	(2.9 ± 0.05) × 10 7	-10.20	-12.78 ± 0.05	2.59
		calcd		-11.98	-13.69	1.71
	L′	exp	(1.5 ± 0.2) × 10 6	-8.47	-20.77 ± 0.16	12.30
		calcd		-9.02	-21.14	12.13

a Exp: b ΔG, ΔH, and TΔS in kcal mol

-1

  F.C.: Clermont Universite, UniversitéBlaise Pascal ICCF, UMR CNRS 6296 Aubiere, France.

	Corresponding Author
	clotilde.policar@ens.fr
	Present Address

⊥

Author Contributions

All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

L.G. and C.P. thank the Ecole Normale Supeŕieure de Cachan for a Ph.D. fellowship for L.G. The French ministry of research is gratefully acknowledged for financial support (ACI-JCJC4044).


David Demmer
email: david.demmer@cea.fr

Robin Gerzaguet
email: robin.gerzaguet@irisa.fr

Jean-Baptiste Doré
email: jean-baptiste.dore@cea.fr

Didier Le Ruyet
email: didier.leruyet@cnam.fr

Analytical study of 5G NR eMBB co-existence

Keywords: 5G NR, PHY layer, OFDM, inter-numerology interference

3GPP release 15 focusing on 5G general outline has been published in December 2017. The major difference with respect to currently deployed LTE is the support of various physical layer numerologies. Making the physical layer scalable allows to properly address new services such as low latency or millimeter communications. However it poses the problem of numerology coexistence. Indeed the orthogonality of the OFDM waveform is broken by the use of different subcarrier spacings and therefore multiplexed communications may interfere with each others. A first and simple solution to limit the distortion is to consider guard bands. In this paper, the authors develop analytical metrics to quantify the level of distortion induced by 5G multi-service multiplexing. Besides, general comments and guard band dimensioning are carried out. It is shown that high interference rejection needs to be associated with side lobe reduction techniques to favor an efficient bandwidth use.

I. INTRODUCTION

Cellular networks have evolved a lot since their creation. They were initially designed to support low-data rate Humanto-Human communications. Nowadays 4G LTE also provides broadband Internet access with a huge data consumption on the downlink. Tomorrow, the cellular networks are expected to support Machine-Type-Communications (MTC) as well. It brings a set of new typical applications to be supported by the network from sporadic communications with a long idlestate to critical communications with low latency constraints. The currently deployed 4G LTE shows intrinsic limits to support the heterogeneous services [START_REF]Next Generation Mobile Network Alliance[END_REF] [START_REF]View on 5G Archicture[END_REF]. It thus pushes the forthcoming wireless technologies to favor superior flexibility which paves the way to 5G New Radio (NR).

Release 15 of the 3GPP has been published in December 2017 and is mainly dedicated to 5G NR Enhanced Mobile Broadband (eMBB) and Fixed Wireless Access (FWA) [START_REF]3rd Generation Partnership Project (3GPP)[END_REF]. It defines the complete framework for the physical layer and exhibits structural differences with legacy LTE. First, it addresses higher frequencies with targeted carrier frequencies up to 52.6 GHz. Then, the typical bandwidth is increased from 20 MHz to 400 MHz. Last but not least, the PHY layer is designed to support various configurations. Multiplexing different numerologies allows the service to choose between a set of supported subcarrier spacing (SCS) and symbol durations that better fit its needs. However the problem of service coexistence arise. Indeed, multiplexing various SCS/symbol durations breaks the OFDM orthogonality and thus is source of interference. One approach to limit the interference is the use of large guard bands between distinct-numerology transmissions which however lowers the bandwidth use. This practice is not compatible with the initial vision of 5G which aims at maximizing the bandwidth use to properly support the huge expected traffic [START_REF]View on 5G Archicture[END_REF].

The main contribution of this paper is the examination of the 5G NR numerology coexistence issue. The level of induced distortion is analytically expressed which leads to an analysis and an evaluation of the guard band size required to satisfy given targets quality of signals.

The remainder of the paper is structured as follows. Section II is dedicated to the system model. The 5G NR numerologies are introduced and the OFDM transmissions techniques as well. Section III is dedicated to the derivation of the distortion level indicator. The Mean Square Error (MSE) has been chosen as indicator. Then, the analysis and guard band size evaluation is conducted in Section IV. Section V gives the conclusion and perspectives.

II. SYSTEM MODEL

A. 5G NR standard and numerologies

Regarding physical layer specifications, the major innovation of release 15 with respect to former standards is the support of various numerologies [START_REF]3rd Generation Partnership Project (3GPP)[END_REF] which allows scalable SCS and symbol duration. By doing so, challenges of 5G can be properly addressed. Indeed shortening symbols reduces the latency required for some Ultra Reliable and Low Latency Communications [START_REF] Luo | 5G Standard Development: Technology and Roadmap[END_REF]. Moreover, enlarging the SCS increases the robustness against the Doppler effect occurring in mobility scenarios and strong phase noise of millimeter wave frequencies [START_REF] Zaidi | Evaluation of Waveforms for Mobile Radio Communications Above 6 GHz[END_REF].

Therefore, five different numerologies will be supported as shown in Table I. They are identified with a numerology Symbol alignment in time can still be ensured as the symbol duration linearly decreases with µ as depicted in Figure 1. Resource allocation in time can therefore be performed according to a frame system as in LTE. Downlink and uplink communications are organized into frames of 10 ms [START_REF]3rd Generation Partnership Project (3GPP)[END_REF].

The total cell bandwidth is split into Bandwidth Parts (BWP). Bandwidth adaptation is performed by configuring the user equipment (UE) with the BWPs. By doing so, the cell bandwidth can be larger than the maximum UE achievable bandwidth [START_REF]3rd Generation Partnership Project (3GPP)[END_REF]. Besides, the different services can be supported in the same cell bandwidth. Indeed, even if each BWP supports an unique numerology 1 , adjacent BWPs can address distinct services.

B. Notations

From now and for the rest of the paper, we propose the following notations to respectively express the useful symbol N µ , the cyclic prefix (CP) N µ CP and entire symbol lengths N µ e as function of µ.

N µ = N 2 -µ = 4096 × 2 -µ (1) N µ CP = N CP 2 -µ = 288 × 2 -µ (2) N µ e = N µ + N µ CP (3)

C. Transmitter scheme

The conventional CP-OFDM transmitted signal can be expressed as in (4) where a f,n denote the transmitted constellation symbol over the f th subcarrier at time instant n, Ω the set of the active subcarriers, T = (N µ + N µ CP )T s the length of one complete symbol where T s denotes the sampling period,

T µ CP = N µ CP T s the CP length, ∆ µ f = (T s N µ ) -1 the subcarrier 1 multiple-numerology BWP is not addressed in current releases
s µ (t) = 1 √ N µ n f ∈Ω a µ f,n Π T (t -nT )e j2π(t-T µ CP )f ∆ µ f (4)

D. Receiver scheme

The incoming signal is first discretized with a sampling period T s . By setting s µ [l] = s µ (lT s ), the conventional CP-OFDM received symbols can be expressed as in [START_REF] Zaidi | Evaluation of Waveforms for Mobile Radio Communications Above 6 GHz[END_REF].

âµ f,n = 1 √ N µ N µ -1 k=0 s µ [nN µ e + N µ CP + k] e -j 2π N µ kf (5)
The expression ( 5) is given for distortion-free and noiseless propagation. Indeed, as the main purpose of this study is to analyze the interference induced by 5G NR numerologies, other sources of distortion such as multi-path channels are omitted. The study of a realistic 5G transmission and the comparison with the results derived in this paper is left free for future studies.

Besides, the paper only addresses CP-OFDM communications and therefore focuses on the downlink. Nonetheless, the authors strongly believe that the results can be easily extended and verified for DFT-s-OFDM and therefore the uplink as well.

Last but not least, power control is not considered neither in the proposed study. There are already many transmission parameters that are taken into consideration and therefore the authors have preferred limiting their number and omitting power control impact. The paper somehow considers the worst case where the User Equipment (UE) is at equal distance of Base Stations (BS) delivering different services. It can be either the same BS addressing two distinct services (i.e. numerologies) over adjacent BWPs or an UE at the edge of cells of same size providing distinct services.

III. DISTORTION ANALYSIS

In LTE, received signals remain orthogonal at the receiver sde from each other as long as they are synchronized within the CP duration and share the same numerology (subcarrier spacing, symbol length). According to 5G NR specifications, adjacent BWPs may address distinct services (different parameters µ). Hence, they may interfere with each others even if proper frequency allocation and perfect synchronization at the receiver side are ensured. Indeed, frequency tones may not properly fit in zeros of sine cardinal (sinc) response of adjacent tones as illustrated in Figure 2. Ensuring that frequency tones always corresponds to the zeros of the sinc response of all other tones would significantly compromise the bandwidth use efficiency and is therefore not acceptable.

In this paper, we propose to express and quantify the level of distortion induced by the multiplexing of different services. The MSE has been chosen as distortion indicator.

For the sake of the analysis, let us assume a single-tone (i.e. on subcarrier) interferer working at frequency f i and with numerology µ i and a single-tone user of interest working at frequency f u and with numerology µ u . Extension to multiple allocated subcarriers will be provided later on.

The resulting MSE is defined in [START_REF] Yin | OFDMA: A Broadband Wireless Access Technology[END_REF] and developed in (7).

MSE µi→µu [∆ gb ] = E n âµu fu,n -a µu fu,n 2 σ 2 a ( 6
)
Regarding the notations, ∆ gb = |f i 2 µif u 2 µu | is the guard band, Q = 2 µi-µu is the number of OFDM symbols with numerology µ i interfering with one symbol of numerology µ u , E n [.] is the expectation operator applied in time, . is the ceiling operator, L = N µi e -N µu CP and D L (f ) are the partial sum of the Dirichlet Kernel expressed in f defined by:

D L (f ) = L-1 l=0 e j 2π N lf (8)
The development of the expression (7) relies on the independence in time of the transmitted symbols and the separation of the Q interfering OFDM symbols.

The MSE expression (7) expresses the distortion level induced by a one-tone interferer over a unique-subcarrier band of interest with a guard band ∆ gb . One can observe than the distortion can be decomposed into Q terms corresponding to the Q interfering symbols overlapping with each symbol of interest.

It is now possible to determine the level of distortion induced by N int subcarriers over a sub-carrier spaced by a guard band ∆ gb , the contributions of each interfering subcarrier can be summed (assuming that constellation symbols are independent). The resulting expression is given in (9).

MSE µi→µu Nint [∆ gb ] = Nint-1 k=0 MSE µi→µu [∆ gb + kQ] (9)
It is also possible to determine the average level of distortion induced by N int subcarriers over a full RB spaced by a guard band ∆ gb as given in (10). Even if this indicator can be interesting at the system level, it won't be used for this study. Indeed information is lost by averaging the level of distortion over multiple subcarriers which compromises the analysis. Generally speaking, the closest subcarrier to the interferer RBs suffers from the biggest distortion. Therefore, to the authors' point of view, considering only one useful subcarrier is more relevant.

MSE µi→µu

A. Interference level

The MSE expression developed in (9) is first verified and compared to simulation results for a whole RB (N int = 12) as shown in Figure 3.

To begin with, one can observe that the analytical expression perfectly matches to the simulation results. Besides, when the interferer and the user of interest share the same numerology (whatever it is), it exists a guard band (and all its multiple) which guarantees the orthogonality between the two users. It is a well-known result used in LTE and OFDMA to multiplex several users [START_REF] Yin | OFDMA: A Broadband Wireless Access Technology[END_REF]. The minimum guard band that ensures the orthogonality is defined by the ratio between the sampling frequency and the total number of sub-carriers.

As expected, the result no longer holds when different numerologies are considered. They may even significantly interfere from each other even when large guard band spaces the two streams. It is worth pointing out that the distortion induced by lower numerologies (Q < 1) impacts less than higher numerologies. Moreover, for higher numerologies, the bigger Q is, the more distortion occurs. Indeed, Q corresponds to the number of interfering waveform symbols, therefore it makes sense that the higher Q is, the more distortion is induced.

It seems worth observing that it is possible to simplify the model by reducing the number of degrees of freedom by one. Indeed by looking at Figure 3, one can observe that the approximation (11) is valid for any α such as

(µ i + α) × (µ u + α) ∈ [0, 5] 2 .
It can be verified that the approximation still holds for N int = 12. It means that if the MSE is expressed for a normalized guard band (for instance the number of subcarrier instead of in frequential units), the MSE does not depend on the two numerology indexes µ i and µ u but only on their ratio Q. By considering this approximation, the MSE is function of N int the number of interferer subcarriers, Q the numerology index ratio and the guard band expressed in number of subcarriers.

MSE µi+α→µu+α 12 [2 α ∆ gb ] ≈ MSE µi→µu 12 [∆ gb ] (11)
The impact of N int on the MSE is now studied. The results are depicted in Figure 4 for five different interferer bands. The obtained results confirm that lower numerologies impact less than higher. It seems interesting to observe that for a MSE target, the guard band logarithmically increases with the interfering size band.

B. Guard band dimensioning

As an application of previous analytical results, it seems interesting to determine the guard band size in order to ensure a given MSE target. The idea is somehow to inverse the expression (9) in order to express the guard band as function of the interferer band given a MSE target and a value of Q. However, one can observe with Figure 4 , that the function (7) is not bijective. Therefore it can not be inversed. We thus propose to numerically determine the minimum guard band to ensure a given level of distortion for each configuration. The results are depicted in Figure 5 for Q = {2, 4}.

MSE µ i →µu [∆ gb ] = 1 N µu σ 2 a En   N µu -1 l=0 sµ i [l + N µu CP + nN µu e ]e -j 2π N µu lfu 2   = 1 N µu N µ i    L-1 l=0 e j 2π N l∆ gb 2 + Q -2 k=0 L+(k+1)N µ i e -1 l=L+kN µ i e e j 2π N l∆ gb 2    = 1 N µ i N µu |D L ∆ gb | 2 + ( Q -1)|D N µ i e ∆ gb | 2 (7
Those results confirm the logarithmic trend of the function with respect to the number of interfere subcarriers. As a reminder, the guard band (y axis) is expressed for the user of interest. Therefore the effective guard band [in kHz] is found by multiplying the value by 180 × 2 µu . Regarding the x axis, the effective interferer band [in kHz] can be found by multiplying the corresponding value with 15 × 2 µi . As expected, the required guard band for Q = 4 is larger than for Q = 2. One can observe that the required guard band length levels off quickly for low MSE targets (around N int = 50 for MSE target = 25 dB for both Q = 2 and Q = 4) but much later for higher MSE targets. It means that a 3-RB-long guard band is enough to ensure 25 dB MSE for any interferer band with Q = 2 . However, the required guard band to ensure higher quality becomes significant even for a few RBs of interferer. Indeed, a 9-RB -long guard band ensures a MSE of 35 dB for 2 RB of interferer with Q = 2 while only 30 dB MSE for Q = 4. It means that ensuring low level of interference (high MSE target) is only possible at a significant bandwidth band efficiency compromise even for Q = 2.

In this last paragraph, we evaluate the bandwidth use efficiency based on a simple scenario as depicted in Figure 6. Three services µ = {0, 1, 2} are considered and placed as shown such as the numerology ratio between adjacent services is Q = {1/2, 2}. All the services have the same number of allocated RBs. The total bandwidth defined as the sum of the three service bandwidths and the minimum guard band to satisfy the MSE target and the corresponding bandwidth use efficiency are then computed according to the previous results. The results are depicted in Table II band observed in Figure 5. Besides, the efficiency decreases with the MSE target. It leads to unacceptable bandwidth efficiencies for high quality of service that may be required for ultra-reliable communications. These results justifies the need of side lobe rejection techniques at the transmitter and the receiver side so as to ensure both high inter-numerology interference rejection and acceptable bandwidth use.

Q = 2 Q = 4

V. CONCLUSION AND PERSPECTIVES

The paper investigates the 5G NR eMBB numerology coexistence problem. Based on analytical indicator expressions, general comments and a guard band evaluation have been carried out. Generally speaking, it seems that the key of 5G NR coexistence is to avoid strong multiplexing of the proposed services and to concentrate each service in large dedicated BWPs. However, even by doing so, high inter-numerology interference rejection can only be ensured at the expense of a significant bandwidth use efficiency compromise. The proposed interference metric could be part of the future 5G radio resource management algorithm to optimize the user allocation and manage the quality of service given a level of signal to noise plus interference ratio.

Eventually, side lobe rejection techniques should therefore be considered in 5G physical layer in order to efficiently support the huge and diverse traffic. Recent literature has focused on filtering and/or windowing to deal with asynchronous communications. Those solutions may be applied and extended to the 5G NR numerology coexistence scenario.

Fig. 1 .

 1 Fig. 1. Time frequency illustration of the 5G NR frame structure with different numerologies

Fig. 2 .

 2 Fig. 2. Sine cardinal of two distinct numerologies interfering with each other (a.u. stands for arbitrary units)

Fig. 3 .

 3 Fig. 3. Level of distortion induced by 1-RB interferer over 1 sub-carrier user of interest according to guard band spacing the two users for µ = {0, 1, 2, 3, 4, 5}.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Guard band of function of N int

.

  One can observe that the bandwidth use efficiency increases with the number of allocated RBs. It is induced by the logarithm trend of the guard band as function of the interferer

										Q = 2														Q = 4
			-10				1RB		3RBs	5RBs	10RBs	20RBs		-10					1RB		3RBs	5RBs	10RBs	20RBs
		MSE [dB]	-30 -20													MSE [dB]	-30 -20
			-40	0	1	2	3	4	5	6	7	8	9	10	11	12	-40	0	1	2	3	4	5	6	7	8	9	10	11	12
									Guard Band [RBs]												Guard Band [RBs]
									Q = 1/2														Q = 1/4
			0				1RB		3RBs	5RBs	10RBs	20RBs		0					1RB		3RBs	5RBs	10RBs	20RBs
		MSE [dB]	-20													MSE [dB]	-20
			-40	0	1	2	3	4	5	6	7	8	9	10	11	12	-40	0	1	2	3	4	5	6	7	8	9	10	11	12
									Guard Band [RBs]												Guard Band [RBs]
							Fig. 4. Level of distortion obtained for different interfering bands respectively for Q = {4,2,1/2,1/4}.
					MSE Target 25 dB MSE Target 35 dB		MSE Target 30 dB MSE Target 40 dB
		25
	Guard Band [RBs]	5 10 15 20
		0	0		12	24	36		48	60		72	84		96
									N int