Mohammad Abu-Zaineh
email: mohammad.abuzaineh@univ-amu.fr

BREAD AND SOCIAL JUSTICE: MEASUREMENT OF SOCIAL WELFARE AND INEQUALITY USING ANTHROPOMETRICS

Keywords: I14, I15, O5 anthropometrics, health achievement and inequality, survival thresholds, Arab countries

We address the question of the measurement of health achievement and inequality in the context of variables exhibiting an inverted-U relation with health and well-being. The chosen approach is to measure separately achievement and inequality in the health increasing range of the variable, from a lower survival bound a to an optimum value m, and in the health decreasing range from m to an upper survival bound b. Because in the health decreasing range, the equally distributed equivalent value associated with a distribution is decreasing in progressive transfers, the paper introduces appropriate relative and absolute achievement and inequality indices to be used for variables exhibiting a negative association with well-being. We then discuss questions pertaining to consistent measurement across health attainments and shortfalls, as well as the ordering of distributions exhibiting an inverted-U relation with well-being. An illustration of the methodology is provided using a group of five Arab countries.

intRoduction

The improvement of key health indicators has been a major concern of the development debate for many decades, and remains so today, as formulated for instance in the Millennium Development Goals (2000Goals ( -2015) ) and Sustainable Development Goals (2015)(2016)(2017)(2018)(2019)(2020)(2021)(2022)(2023)(2024)(2025)(2026)(2027)(2028)(2029)(2030). Beyond improving the average value of key indicators, it has increasingly been recognized that the shape of the distribution is also in need of attention. There are several reasons for turning our attention to 2 inequality in the distribution of a health variable. In the case of calorie intake for instance, low levels of nutrition are associated with stunting in infants and certain severe deficiencies for adults. High levels of energy intake are also problematic, as they increase the risk of cardiovascular disease and type II diabetes (WHO, 2011). Additionally, there are the usual normative concerns for preferring less to more inequality in health, in relation to two distributions with the same mean value. In this context, [START_REF] Wagstaff | Inequality Aversion, Health Inequalities and Health Achievement[END_REF] forcefully argues that average levels of attainment should not be the prime focus of policy, and instead introduces health achievement indices, the analogues of equally distributed equivalent incomes of the Atkinson-Kolm-Sen (AKS) approach, as the appropriate metrics of health policy. Thus, greater emphasis and interest by researchers in the last 20 years has placed the measurement of achievement and inequality in health at the center of the development debate [START_REF] Wagstaff | Inequality Aversion, Health Inequalities and Health Achievement[END_REF]Erreygers, 2013).

In this paper we are interested in the context of anthropometric health indicators such as body mass, which exhibit an inverted U relation with health status.

Other leading examples of anthropometrics include hip, and mid-upper arm, circumferences. One point we emphasize in this paper, following [START_REF] Aristondo | Measuring the Inequality of Bounded Distributions: A Joint Analysis of Attainments and Shortfalls[END_REF] and [START_REF] Kjellsson | On Correcting the Concentration Index for Binary Variables[END_REF] is that survival considerations place lower and upper bounds on the range of values an anthropometric indicator takes. Following [START_REF] Apablaza | When More Does Not Necessarily Mean Better: Healthrelated Illfare Comparisons with Nonmonotone Wellbeing Relationship[END_REF], we accommodate the non-monotonic relation between a health indicator y and well-being by measuring social welfare when the health variable is reported on the union of two intervals (a, m], and (m, b), where a and b are respectively lower and higher critical bounds beyond which survival is no longer likely, while m is the optimum level of the health indicator. In the context of anthropometrics such as body mass, the lower bound is generally taken to be a = 10 kilograms per square meter, b is approximately equal to 60, while m can generally be any value chosen in the interval of 18.5 to 24.9 (WHO, 2017).

The main emphasis of this paper is on the upper tail of the distribution of the health indicator, the interval from m to b, as the measurement of inequality and achievement is generally well understood in the context of variables that exhibit a positive association with well-being. With this observation in mind, we state the four objectives of this paper. Firstly, we enrich the [START_REF] Wagstaff | Inequality Aversion, Health Inequalities and Health Achievement[END_REF] framework in deriving health achievement indices in the context of variables that are negatively associated with welfare, from both the perspectives of the distributions of health attainments and shortfalls. Because, in the present context, welfare is decreasing in the health indicator, we show in Proposition 1 that the equally distributed equivalent (EDE) value is a Schur convex function; that is, the EDE value is decreasing in Pigou-Dalton transfers (when the EDE is typically increasing in progressive transfers in the context of variables positively associated with health and well-being). We then address the question of the robustness of the achievement index to changes in the survival thresholds, and we are led to derive a new family of achievement indices that are translation invariant. [START_REF] Lambert | On the Consistent Measurement of Attainment and Shortfall Inequality[END_REF] provide a family of theorems clarifying the relation between the inequality ordering of a pair of distributions in terms of health attainments and shortfalls. In the formulation of Lambert and Zheng, y denotes health attainment, and m -y denotes health shortfall. Accordingly, the second objective of this paper is to extend Theorems 1 and 2 of these authors, in the context of our study of the upper tail of the distribution of an anthropometric variable, where health attainment is b -y and health shortfall is y -m. A related literature has examined various forms of the consistency property in relation to inequality indices of health attainments and shortfalls [START_REF] Erreygers | Can a Single Indicator Measure Both Attainment and Shortfall Inequality?[END_REF][START_REF] Lambert | On the Consistent Measurement of Attainment and Shortfall Inequality[END_REF][START_REF] De La Vega | Proposing Indicators to Measure Achievement and Shortfall Inequality Consistently[END_REF][START_REF] Aristondo | Measuring the Inequality of Bounded Distributions: A Joint Analysis of Attainments and Shortfalls[END_REF][START_REF] Bosmans | Consistent Comparisons of Attainment and Shortfall Inequality: A Critical Examination[END_REF][START_REF] Chakravarty | On a Family of Achievement and Shortfall Inequality Indices[END_REF][START_REF] Yalonetzky | Inequality of Ratios[END_REF]. Hence, we furthermore explore the consistency property in relation to the health achievement indices introduced in the paper.

The third purpose of this paper is to reflect on how to compare a pair of distributions in terms of social welfare and inequality, over the entire domain of a health indicator y that exhibits a non-monotonic relation with well-being. One approach developed in [START_REF] Apablaza | When More Does Not Necessarily Mean Better: Healthrelated Illfare Comparisons with Nonmonotone Wellbeing Relationship[END_REF] is to aggregate shortfalls and excess values from the optimum. The approach we take in this paper however is different, in that we do not view the contribution to social welfare of an undernourished person i and an overweight person j as being comparable. Instead, the approach we pursue in this paper is to construct an order relation on the entire domain of the health indicator y, using as inputs two separate order relations-one of which is defined on the lower tail of the distribution, and the other on the upper tail. We suggest two such construction methods, that we call a product order and a lexicographic order, and explain the implicit value judgments needed to justify the adoption of each of these order relations.

In a recent study, Alvaredo et al. (2019, p. 686), conclude that in terms of income concentration, the Middle East and North Africa (MENA) "appears to be the most unequal region in the world." The final purpose of this paper is to illustrate the methodology we develop, in the context of a group of five countries from the MENA region (Comoros, Egypt, Jordan, Morocco and Yemen) where we study variations in social welfare and inequality in relation to an anthropometric variable, namely body mass.

The structure of the paper is as follows. In Section 2 we examine issues arising from the measurement of health achievement and inequality in the context of variables exhibiting a negative association with well-being. In Section 3 we focus our discussion on the derivation of absolute achievement and inequality indices, for such variables. In Section 4 we extend the Lambert and Zheng (2011) theorems, in the context of variables that exhibit a negative association with well-being. Section 5 discusses the ordering of distributions exhibiting an inverted-U relation with well-being. Section 6 illustrates the methodology of the paper using a group of five Arab countries, and Section 7 concludes the paper. Further results and proofs are relegated to two appendices.

Relative achieveMent and inequality MeasuReMent with an indicatoR negatively associated with health

We consider anthropometric measures of health such as the body mass index (BMI), which have two defining properties: (i) survival places lower and upper bounds on their domain of variation, and (ii) they exhibit a non-monotonic, inverted U, relation with health. 1 Let Ω = 1 , …, n be a vector of anthropometric observations where i is defined on one of two intervals: either i ∈ (a, m] or i ∈ (m, b ). The distribution Ω = 1 , …, n can, thus, be partitioned into two separate vectors defined, respectively, over the intervals (m, b) and (a, m].

Let n 1 and n 2 = nn 1 denote respectively, the number of observations that belong to the two intervals (m, b) and (a, m]. We may now define two vectors Y : = y 1 , …, y n 1 and X : = x 1 , …, x n 2 such that Ω = [XY]. Throughout, we work with vectors X and Y that exhibit some variation. Accordingly, we define the following sets where Y ↑ = y (1) , …, y (n) and X ↑ are the increasing rearrangements of the vector Y and X; and we restrict our attentions to vectors X and Y defined respectively on D am and D mb .

In the Atkinson-Kolm-Sen (AKS) approach, the derivation of inequality indices is approached in relation to a social welfare function taken to capture a preference for higher health, and less inequality. The inequality index is derived via a comparison of the mean of a variable with the equally distributed equivalent (EDE) value of the distribution. Health achievement indices [START_REF] Wagstaff | Inequality Aversion, Health Inequalities and Health Achievement[END_REF] are also derived as equally distributed values. One purpose of this section is to show that for an anthropometric indicator exhibiting a negative association with health, the equally distributed equivalent value is decreasing in progressive transfers (when we would expect the opposite from such a summary statistic). We then dwell further on the implications of this finding for alternative specifications of achievement and inequality indices.

Fundamental Axioms

We provide an axiomatic formulation of the measurement of social welfare in relation to health attainment, and we discuss more briefly the form of the underlying social welfare function in relation to health shortfall. With respect to health attainment, we measure welfare with reference to individual i's position from the upper survival threshold b using a social valuation function, more simply a utility function, i (by) for i = 1, …, n 1 , and we let W Y : D mb → ℝ denote a social welfare function in relation to an anthropometric variable y negatively related with health. We let n 1 denote an n 1 -dimensional vector of ones, n 1 = (1, …,1), and we consider several axioms commonly used for social welfare functions. In what follows therefore b n 1 -Y is a compact notation for the vector by 1 , …, by n 1 . We begin by stating some elementary transformations of the data.

Definition 1 Let Y A = y A 1 , …, y A n 1 and Y B = y B 1 , …, y B n 1 denote two distribu- tions in D mb .
(i) We say that Y B is obtained from Y A using a single increment on D mb if for some person i and ∈ 0, by A i , y B i = y A i + and y B j = y A j for all j ≠ i. (ii) We say that Y B is obtained from Y A using a single decrement on D mb if for some person i and ∈ 0, y A im , y B i = y A iand y B j = y A j for all j ≠ i. (iii) We say that Y B is obtained from Y A using a single Pigou-Dalton transfer on D mb if there are individuals i and j with y A i < y A j , and 𝛿 < y A jy A i ∕2, y B i = y A i + , y B j = y A jand y B l = y A l for all l ≠ i, j. We next consider the set of social welfare functions on D mb that are anonymous, and defined as the average of welfare levels experienced by individuals:

• ADD (Additivity): The social welfare function is the average of the utility levels of the n 1 individuals:

where the function : (m, b) → ℝ is the anonymous utility function.

The monotonicity axiom MON requires that decrements of Definition 1.ii increase social welfare; that is social welfare increases when individual endowments y i are reduced. Preference for greater equality is introduced via the axiom EQUAL, requiring that social welfare strictly increases with Pigou-Dalton transfers (see Definition 1.iii).

• MON (Monotonicity): The social welfare function W Y is strictly decreasing in y 1 , …, y n 1 .

• EQUAL (Social aversion to inequality): Let Y * = y * 1 , ⋯, y * n 1 be obtained from Y = y 1 , ⋯, y n 1 via one or several Pigou-Dalton transfers, then

W Y b -y * 1 , ⋯, b -y * n 1 > W Y b -y 1 , ⋯, b -y n 1 . That is, W Y is strictly increasing in Pigou-Dalton transfers.
The monotonicity axiom MON restricts the derivative of the function to have a negative sign on the interval (m, b). On the other hand, the social welfare function satisfies the social aversion to inequality axiom, EQUAL if is concave on (m, b). Finally, we discuss two invariance axioms capturing certain transformations of the data that leave the ordering of distributions by the social welfare function unchanged. The first of these, SCALINV, guarantees that the ranking of a pair of distributions does not change when units of measurement are modified in a particular manner. The second, TRANSINV, is similarly used to capture the notion that the ranking of distributions is invariant to translational shifts in the data.

• SCALINV (Scale invariance): For any scalar λ > 0, and

(1)

W Y b -y 1 , …, b -y n 1 : = 1 n 1 n 1 ∑ i=1 b -y i for any pair of distributions Y A , Y B ∈ D mb , there holds W Y b n 1 -Y A ≥ W Y b n 1 -Y B ⇔ W Y b n 1 -Y A ≥ W Y b n 1 -Y A .
• TRANSINV (Translation invariance): For any admissible value of λ and for any pair of distributions

Y A , Y B ∈ D mb , W Y b n 1 -Y A ≥ W Y b n 1 -Y B ⇔ W Y b n 1 -Y A + n 1 ≥ W Y b n 1 -Y B + n 1 . 2,3
Together the axioms ADD, MON, EQUAL and SCALINV restrict the choice of (by) to the family u (by) of power functions (see for instance [START_REF] Kolm | Unequal Inequalities I[END_REF][START_REF] Skiadas | Scale-invariant Uncertainty-averse Preferences and Source-dependent Constant Relative Risk Aversion[END_REF]:

Accordingly, the family of social welfare functions that satisfies the above five axioms is of the form:

We shall return to the axiom TRANSINV in the next section of the paper.

Anthropometric Relative Achievement Index

Let ŷ ∈ (m, b) denote the equally distributed health level in the distribution Y: [START_REF] Wagstaff | Inequality Aversion, Health Inequalities and Health Achievement[END_REF], this equally distributed equivalent value is known as the achievement index associated with the distribution of health attainments. In the income inequality literature, the equally distributed equivalent income is increasing in Pigou-Dalton progressive transfers. The context of health indicators exhibiting a negative association with welfare produces a marked difference:

it then follows that u b -ŷ = W Y b -y 1 , …, b -y n 1 . Following
Proposition 1 Let u() denote any real-valued function that is strictly decreasing, and concave on some closed interval m 0 , b 0 ⊆ (m, b). Then, for any distribution Y ∈ D mb , with mean y, the equally distributed equivalent value

2 In this context, a value of λ is admissible if Y + n 1 is an element of D mb , whenever Y is an element of D mb .
3 Consider the following formulation of the translation invariance axiom: for any admissible value of λ and for any pair of distributions

Y A , Y B ∈ D mb , W Y b n 1 -Y A ≥ W Y b n 1 -Y B ⇔ W Y (b + ) n 1 -Y A + n 1 ≥ W Y (b + ) n 1 -Y B + n 1 .
Upon simplifying the right-hand side of the equivalence, we would have that

W Y b n 1 -Y A ≥ W Y b n 1 -Y B ⇔ W Y b n 1 -Y A ≥ W Y b n 1 -Y B .
Therefore, this formulation of the axiom would not entail any particular restrictions on the form of the social welfare function.

(2)

u 𝛽 (b -y) = ⎧ ⎪ ⎨ ⎪ ⎩ (b -y) 1-𝛽 1 -𝛽 , 𝛽 > 0, 𝛽 ≠ 1 ln (b -y) , 𝛽 = 1 (3) W Y b -y 1 , …, b -y n 1 = 1 n 1 n 1 ∑ i=1 u b -y i is a Schur-convex function; that is, ŷ is decreasing in Pigou-Dalton transfers.
Furthermore, ŷ satisfies the inequality m ≤ y ≤ ŷ.

We shall make repeated use of the above result (see Corollaries 2 and 3 below).

Because, on the one hand, the equally distributed equivalent value is a Schurconvex function, and on the other hand, the inequality between the mean income and the equally distributed equivalent is reversed, we shall have to write AKS inequality indices based on ŷ in a different form in the context of distributions defined on D mb .

Returning to the specific context of equations ( 2) and ( 3), the equally distributed equivalent value is of the form:

We refer to ŷa R (Y ; ) as the anthropometric relative achievement index of the dis- tribution of health attainments. The superscript a in ŷa R is introduced to denote that the equally distributed equivalent value is associated with a concept of health attainment. The subscript R in ŷa R is introduced to denote relative achievement indices; i.e. indices that satisfy the scale invariance axiom SCALINV. Note that because an increase in y is associated with a decrease in health and well-being, larger values of ŷa R are associated with lower levels of social welfare. As an alternative to measuring social welfare in relation to health attainment, we briefly discuss the context of shortfall y -m. We associate a social welfare function W Y y 1m, …, y n 1m with the vector of shortfalls Ym n 1 . The axioms of additivity, monotonicity, preference for equality and scale invariance are easily defined in this context. They imply that the social welfare function is the average of utilities, where the individual utility function is of the form:

The function u () is decreasing in y and concave. The associated equally distributed equivalent value in terms of health shortfalls is of the present form

(4) ŷ = u -1 1 n 1 n 1 ∑ i=1 u y i (5) � y a R (Y ;𝛽) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ b - � 1 n 1 n 1 � i=1 � b -y i � 1-𝛽 � 1 1-𝛽 , 𝛽 > 0, 𝛽 ≠ 1 b -exp � 1 n 1 n 1 � i=1 ln � b -y i � � , 𝛽 = 1 (6) u 𝛾 (y -m) = - (y -m) 1-𝛾 1 -𝛾 ; 𝛾 < 0 (7) � y s R (Y ;𝛾) = m + n 1 ∑ i=1 1 n 1 ∑ i y i -m 1-𝛾 1 1-𝛾 ; 𝛾 < 0.
As an application of Proposition 1 we state the following result:

Corollary 2 For Y ∈ D mb , each of the equally distributed equivalent values ŷa R and ŷs R associated respectively with the distribution of health attainments and shortfalls is a Schur-convex function. Furthermore, the following inequalities are satisfied:

m ≤ y ≤ ŷa R and m ≤ y ≤ ŷs R .

The AKS Family of Relative Inequality Indices

It follows from Proposition 1 and Corollary 2 that application of the standard AKS inequality index introduced by [START_REF] Atkinson | On the Measurement of Inequality[END_REF], namely the function 1 -ŷa R ∕y in the context of attainment, and 1 -ŷs R ∕y in the context of short- falls, will provide the data user with two challenges. Firstly, in the light of the inequalities stated in Proposition 1 each of these two inequality indices will take on negative values. More importantly, Pigou-Dalton transfers will increase the value taken by these indices. It is thus important to adapt the relative AKS index in the context of anthropometric indicators exhibiting a negative association with health, so as to achieve these desired properties (positive-valued function, and decreasing in Pigou-Dalton transfers). Consider in particular the following forms:

Because of the inequalities m ≤ y ≤ ŷa R and m ≤ y ≤ ŷs R , each of the indices (8a) and (8b) takes on positive values. Furthermore, because both equally distributed equivalent values ŷa R and ŷs R are decreasing in Pigou-Dalton transfers, the inequality indices (8a) and (8b) are now increasing functions of ŷa R and ŷs R . Finally, it is clear that the inequality indices above are invariant to rescaling b, m and the distribution Y by the same constant λ > 0.

attainMents, shoRtfalls and the absolute anthRopoMetRic achieveMent index

The scale invariance axiom guarantees that changing the units of measurement of y, and the two thresholds m and b does not result in any change in social welfare and inequality (equations 5 and 8). A separate concern however may have to do with disagreement about the level of the thresholds m and b. Medical research (8a)

I a R (Y ; ) : = 1 - b - ŷa R b -y (8b) I s R (Y ; ) : = 1 - y -m ŷs R -m
can inform about the values of m and b. But in most cases, such information comes in the form of a range of values for the parameters.

Recall that the threshold m enters the calculation of health shortfalls, while the upper survival threshold plays a similar role in the calculation of health attainments. Consider therefore, the effect of changing the value assigned to the upper threshold b,

from b o to b 1 = b o -. Then, observing that b 1 n 1 -Y = b o -n 1 -Y = b o n 1 -Y + n 1
, changing the value assigned to the upper threshold is equivalent to obtaining a new distribution Y + n 1 , while maintaining the upper threshold at the initial value b o . It is clear that such translational shifts in the distribution of resources will result in a shift in the relative Lorenz curve. Following [START_REF] Kolm | Unequal Inequalities I[END_REF] and [START_REF] Moyes | A New Concept of Lorenz Domination[END_REF], it is however possible to work with inequality indices that are invariant to changes in the thresholds m and b.

The key to deriving indices that are robust to changes in the upper threshold b is to replace the scale invariance axioms by translation invariance axioms. Consider first the measurement of social welfare in relation to health attainment. Following [START_REF] Kolm | Unequal Inequalities I[END_REF], or [START_REF] Skiadas | Scale-invariant Uncertainty-averse Preferences and Source-dependent Constant Relative Risk Aversion[END_REF], together the axioms ADD, MON, EQUAL and TRANSINV restrict the choice of (by) to the family u (by) of exponential functions:

Accordingly, the family of social welfare functions defined on D mb that satisfies the above four axioms is of the form The equally distributed equivalent value ŷa A pertaining to the above family of social welfare functions satisfies the identity u � b -ŷa

A � = 1 n 1 ∑ n 1 i=1 u � b -y i � .
Specifically,

We refer to ŷa A as the anthropometric absolute achievement index of the distribution of health attainments. As such, the index (11) complements the work of [START_REF] Wagstaff | Inequality Aversion, Health Inequalities and Health Achievement[END_REF] that pertained to the derivation of the relative achievement indices in relation to health attainment.

Similarly, switching to shortfalls, ym, the utility function u : D mb → ℝ that satisfies the axioms discussed above is of the form

(9) u 𝜅 (b -y) = 1 -exp ( -𝜅 (b -y)) , 𝜅 > 0 (10) W Y b -y 1 , …, b -y n 1 ; m = 1 n 1 n 1 ∑ i=1 u b -y i (11) ŷa A (Y ; ) = b + 1 ln 1 n 1 n 1 ∑ i=1 exp -b -y i (12) u 𝛿 (y -m) = 1 -exp ( -𝛿 (y -m)) , 𝛿 < 0
The resulting anthropometric absolute achievement index of the distribution of health shortfalls is given by the expression

As a result of imposing the underlying translation invariance axiom, it is to be noted that the absolute achievement index for health attainment ŷa A is invariant to changes in the value assigned to the upper survival threshold b, and likewise, ŷs

A is invariant to changes in the value taken by the parameter m. Furthermore, the following result is a direct application of Proposition 1. 4. fRoM health attainMent to shoRtfalls: the laMbeRt and Zheng consistency theoReMs

One important question that [START_REF] Lambert | On the Consistent Measurement of Attainment and Shortfall Inequality[END_REF] address is to what extent the inequality ordering of two health distributions is robust to the specifications of outcomes in terms of attainments and shortfalls.

The inequality indices (8 and 14) associated with the social welfare functions underlying the derivation of the relative and absolute forms of the anthropometric achievement index do not satisfy the consistency property across the distribution of attainments and shortfalls. That is, as demonstrated by [START_REF] Lambert | On the Consistent Measurement of Attainment and Shortfall Inequality[END_REF], in the class of rank independent inequality indices that are additively decomposable, only the variance satisfies the consistency property.

As forcefully demonstrated by de la Vega and Aristondo (2012), Bosmans (2016), [START_REF] Chakravarty | On a Family of Achievement and Shortfall Inequality Indices[END_REF] and [START_REF] Yalonetzky | Inequality of Ratios[END_REF], it is certainly possible to dispense with some of the axioms of this paper in order to obtain indices of ( 13)

ŷs A (Y ; ) = m - 1 ln 1 n 1 n 1 ∑ i=1 exp -y i -m . (14a) I a A (Y ; ) = ŷa A -y (14b) I s A (Y ; ) = ŷs A -y
inequality that are consistent across health attainments and shortfalls. 4 As the emphasis in this paper is on social welfare and achievement in relation to anthropometric variables, we first explore how the Lambert and Zheng (2011) theorems, as originally formulated in the context of variables y in positive association with health, need to be extended in the context of variables defined on D mb .

For this purpose, it is convenient to define the [START_REF] Zoli | A Generalized Version of the Inequality Equivalence Criterion: A Surplus Sharing Characterization, Complete and Partial Orderings[END_REF] two-parameter family of Lorenz orderings ≺ 𝜏,𝜃 on D mb , with 0 ≤ , ≤ 1. For a given distribution Y : = y 1 , …, y n 1 ∈ D mb , define: and where, for ease of exposition, from here on Y will denote the mean of the vector Y. For given parameter values τ and θ, and for two distributions, Y A and Y B in D mb , we will say that Y A Zoli-dominates Y B , written Y A ≻ 𝜏,𝜃 Y B if the following n 1 inequalities are satisfied:

Of particular relevance in this paper are the case = = 0, which produces the ordering by absolute Lorenz curves [START_REF] Moyes | A New Concept of Lorenz Domination[END_REF], and the case = = 1, which produces the relative Lorenz ordering. 5 Let Z and S respectively denote the mean health attainment and shortfall associated with a vector Y ∈ D mb . In the context of health attainments and shortfalls, we also define the following functions of the vectors of shortfalls and attainments

The following results provide an adaptation of Lambert and Zheng's (2011) Theorems 1 and 2 in the context of this paper (see Appendix B for proofs). 4 Typically, the definition of consistency one adopts can be relaxed (cf. Propositions 5 and 6). It is also possible to address the issue of consistency in relation to relative shortfalls and attainments. See [START_REF] Bosmans | Consistent Comparisons of Attainment and Shortfall Inequality: A Critical Examination[END_REF] and [START_REF] Yalonetzky | Inequality of Ratios[END_REF]. ( 15)

d (Y ; , ) = Y + (1 -) (16) y i ;Y , , = 1 d (Y ; , ) y i -Y (17) 1 n 1 j ∑ i=1 l y A (i) ;Y A ; , ≥ 1 n 1 j ∑ i=1 l y B (i) ;Y B ; , ∀j = 1, …, n 1
5 For the case where 0 < 𝜏 < 1 and = 1, we obtain in equation ( 17) intermediate inequality orderings (see [START_REF] Lambert | On the Consistent Measurement of Attainment and Shortfall Inequality[END_REF] for further discussion). ( 18)

Ẑ (Y ; , ) = 1 d (Z; , ) z 1 -Z , …, z n 1 -Z (19) Ŝ (Y ; , ) = 1 d (S; , ) s 1 -S , …, s n 1 -S Proposition 4
In the context of anthropometric variables y i that exhibit a negative as- sociation with health, and are subject to survival bounds, define attainment as z i = by i and shortfall as s i = y im.

= 0), Ŝ Y A ; , ∼ , Ŝ Y B ; , . b If ≠ 0, then � S Y A ;𝜃, 𝜏 ≻ 𝜏,𝜃 � S Y B ;𝜃, 𝜏 if 𝜇 Z A < 𝜇 Z B and � S Y A ;𝜃, 𝜏 ≺ 𝜏,𝜃 � S Y B ;𝜃, 𝜏 if 𝜇 Z A > 𝜇 Z B.
As discussed in the literature, ranking distributions in terms of health shortfalls versus attainments does not in general produce consistent comparisons within the Zoli parametric family of orderings. Only in the case of the absolute Lorenz order, statement (a) above reveals, that as in [START_REF] Lambert | On the Consistent Measurement of Attainment and Shortfall Inequality[END_REF], in the present formulation, two distributions Y A and Y B are ranked equivalent in terms of health attainment if and only if they are equivalent in terms of health shortfalls. On the other hand, as point (b) reveals, outside of the context of the absolute Lorenz order, if the two distributions are equivalent in terms of attainments, the relative magnitude of the average attainments in the two distributions determines the ranking of the distributions of shortfalls. Proposition 5 has the following implication for the empirical section of the paper: if say the distribution of health attainments in Morocco is more egalitarian (in the sense of the absolute Lorenz order) than the distribution of, say Jordan, then it is also the case that in the context of health shortfalls, the Morocco distribution is more egalitarian than the Jordan distribution.

Because we shall make extensive use of the generalized Lorenz curve in the empirical section of the paper, we have found it useful to state an appropriate consistency result across health attainment and shortfalls in the context of this order relation. For i = A, B define the attainment and shortfall vectors [START_REF] Shorrocks | Ranking income distributions[END_REF], Z B is generalized Lorenz dominated by Z A , written as Z B ≺ w Z A if the following n 1 inequalities are satisfied:

Z i = b n 1 -Y i and S i = Y i -m n 1 . For the increasing rearrangement Z ↑ = z (1) , …, z (n1) , following
(20) j ∑ i=1 z B (i) ≤ j ∑ i=1 z A (i) ; j = 1, 2, …, n 1
Then, the distribution of attainments Z A generalize-Lorenz dominates Z B if Z A is obtained from Z B via a sequence of increments6 and Pigou-Dalton transfers. Accordingly, we further introduce a relation ≺ SM on the distribution of shortfalls.

For Y A , Y B ∈ D mb we shall write S A ≺ SM S B , if S B is obtainable from S A using a sequence of increments and regressive transfers. We view the relation ≺ SM , defined on shortfall distributions, as the dual to the relation ≺ w defined on attainments dis- tributions. We refer to ≺ SM as shortfall majorization.

7 Define Y ↓ = y [1] , y [2] , …y [n1]
as the decreasing arrangement of the vector Y . Following Marshall et al. (2011, chapter 1), S A ≺ SM S B if the following n 1 inequalities are satisfied:

We are now ready to state the following result:

Proposition 6 Let Y A , Y B ∈ D mb .
Then, there holds S A ≺ SM S B if and only if Z B ≺ w Z A . Equivalently, the distribution of health shortfalls S A is shortfall-majorized by the distribution of health shortfalls S B , if and only if the distribution of health attainments Z A generalize-Lorenz dominates the distribution of health attainments Z B .

Observe that unlike the Lambert and Zheng ( 2011) consistency theorems that are formulated in relation to a fixed ordering across shortfalls and attainments, we have used two separate orderings, namely, ≺ SM in relation to shortfalls, and ≺ w in relation to health attainments, in order to state the above consistency result. As such, we have followed a more flexible approach as advocated by [START_REF] Bosmans | Consistent Comparisons of Attainment and Shortfall Inequality: A Critical Examination[END_REF] in deriving a practically relevant consistency result for the distributional analysis of health attainment and shortfalls in relation to social welfare. 8We next state a further consistency result, in relation to the anthropometric absolute achievement index. For ease of exposition, instead of Y A , Y B , we now write Y 1 , Y 2 for the pair of distributions to be compared.

(Y 1 ; ) ≥ ŷa A (Y 2 ; ) if and only if ŷs A (Y 1 ; -) ≥ ŷs A (Y 2 ; -).
Observe that unlike Theorem 4 of [START_REF] Lambert | On the Consistent Measurement of Attainment and Shortfall Inequality[END_REF], we obtain the above consistency result in relation to a pair of distributions and a pair of absolute anthropometric indices. Note furthermore, that while we have stated Proposition 7 in ( 21)

j ∑ i=1 (y A [i] -m) ≤ j ∑ i=1 (y B [i] -m); j = 1, 2, …, n 1 A A
terms of achievement indices, from equations ( 14a) and (14b) it is possible to obtain a comparable result in association with the absolute inequality indices for health attainments and shortfalls: that is, for any pair of distributions Y 1 , Y 2 ∈ D mb and any value 𝜅>0 we have that I a (Y 1 ; ) ≥ I a (Y 2 ; ) if and only if I s A (Y 1 ; -) ≥ I s A (Y 2 ; -). To conclude this section, we summarize the implications of the above consistency results for empirical work. Firstly, in terms of inequality orderings, to achieve consistency in distributional comparisons across attainments and shortfalls, we must work with the absolute Lorenz ordering (Proposition 5). To achieve consistency in terms of welfare orderings, the generalized Lorenz ordering in association with health attainments, is consistent with the ordering of distributions by the relation ≺ SM in association with health shortfalls. Finally, every absolute achievement index ŷa A (Y ; ) defined on attainments is paired with an achievement index ŷs A (Y ; -) defined on shortfalls, rendering the ordering of distributions of health attainments and shortfalls consistent, in the sense of Proposition 7.

fRoM oRdeRs on the loweR and uppeR tails to full distRibutional coMpaRisons

We have thus far discussed the question of distributional comparisons in D mb . In Appendix A, we briefly discuss the derivation of relative and absolute achievement indices pertaining to the lower tail of the distribution, D am . As argued earlier, we do not assume that well-being is meaningfully comparable below and above the optimum value, m, in the context of anthropometrics. Consider then two individuals i and j with x i < m and y j > m such that x ia = by j . In contrast with [START_REF] Apablaza | When More Does Not Necessarily Mean Better: Healthrelated Illfare Comparisons with Nonmonotone Wellbeing Relationship[END_REF], we take the view that despite the fact that x ia = by j , the welfare levels of two individuals i and j, one being undernourished, and the other overweight are not meaningfully comparable. Therefore, we suggest to construct the social welfare and inequality relations on the entire domain, from two separate order relations defined on respectively the lower and upper tails (a, m] and (m, b) of the distribution of the anthropometric variable. We detail two such procedures. We first begin with the following definitions (see for instance [START_REF] Davey | Introduction to Lattices and Order[END_REF].

Definition 2 Let P, ≼ P and Q, ≼ Q denote two pre-ordered sets. For all

(X A , Y A ), X B , Y B ∈ P × Q, (i) we define the lexicographic order relation P × Q, ≼ LEX by (X A , Y A ) ≼ LEX X B , Y B ifX A ≺ P X B , or ifX A ∼ P X B and Y A ≼ Q Y B ,
(ii) we define the product order relation

P × Q, ≼ 𝜋 by (X A , Y A ) ≼ 𝜋 X B , Y B if bothX A ≼ P X B and Y A ≼ Q Y B .
Under (i), the lexicographic order, there are two ways to achieve an ordering of two distributions. Firstly, distribution (X A , Y A ) has lower social welfare than (X B , Y B ) if it is the case that in the relation ≺ P ,X A is dominated by X B , (regard- less of how Y A and Y B relate in the relation ≺ Q ). Secondly, the two distributions may be lexicographically ordered if it is the case that in the ≺ P relation, the social planner is indifferent between X A and X B , and that in the relation ≺ Q , Y B is socially preferred over Y A . Under (ii), the product order, distribution X B , Y B has higher social welfare than (X A , Y A ) if it is the case that both X B is preferred to X A accord- ing to the relation ≺ P and similarly that

Y A ≺ Q Y B .
The above lexicographic order assumes implicitly that one domain of the distribution has priority over the other. If reducing undernutrition is a more important social objective than tackling overweight, the set P in part (i) of the definition would denote the lower tail of the distribution, while Q would denote the upper tail of the distribution. If tackling overweight is a social priority over undernutrition, this would amount to swapping around the definitions of the two sets P and Q. We consider both perspectives in our empirical illustrations below. It is finally possible that society is indecisive about the order of priority of tackling inequality in undernutrition and overweight. In this case the product order is an appropriate aggregation procedure for the separate orders, in the sense that it does not consider one tail of the distribution to be more important than the other in terms of society's preferences.

an eMpiRical illustRation

The purpose of this section is to assess health achievement and inequality using anthropometric data on body mass, pertaining to five Arab countries. In Section 6.1 we compare the five countries in terms of the anthropometric achievement index for health attainments and shortfalls, in its relative and absolute variants, as developed in Sections 2 and 3. Next in Section 6.2, we explore social welfare and inequality orderings pertaining to the upper tail of the distribution of body mass. In Section 6.3 we illustrate the two orderings that we discussed in Section 5, namely the product and lexicographic orders. Our application makes use of anthropometric data on adult (non-pregnant) women of reproductive age (15 to 49). The analysis is performed using data from Demographic and Health Surveys (DHS) conducted in five Arab countries: Egypt (2015), Yemen (2013), Jordan (2012), Comoros (2012) and Morocco (2004). The anthropometric indicator of interest here is taken as the body mass index (BMI), calculated by the authors as weight in kilograms divided by squared height (2017), for the purpose of the present analysis, we set the value of a to be equal to 10 and b to be equal to 60, while the cut-off value m is fixed at 24.90. 9 After cleaning the data for missing and miscoded values on the variable of interest, the respective sample sizes are as follows: Egypt (n 1 = 5226, n 2 = 1962 ), Jordan (n 1 = 6336, n 2 = 4740 ), Morocco (n 1 = 6238, n 2 = 10677), Yemen (n 1 = 5666, n 2 = 17276) and Comoros (n 1 = 1926, n 2 = 3156).

Anthropometric Achievement and Inequality Indices: The Upper Tail of the Distribution

Consider first the measurement of achievement and inequality in relation to health attainments and shortfalls in the upper tail of the distribution. We begin by examining the relative achievement indices ŷa R and the related inequality indices I a R in the five countries. We report in Table 1 calculations pertaining to inequality and achievement indices in relation to three values for the inequality aversion parameter: = 0.5, 1 and 3. To begin with, it is worth noting that the mean of the distribution is highest in Egypt (32.4) and lowest in Morocco (28.8). Consider first the results 9 For most anthropometrics, m could be defined as a range of values. The WHO guidelines for nonpregnant women define an optimum interval with values of m ranging between 18.5 and 24.9. As this does not raise unresolved conceptual problems in the context of our paper, we shall simplify our exposition by assuming that m is a single point. ). Increasing the social aversion to inequality parameter ( = 3 ), results in lower health achievement (that is, higher values) in all countries. We note nonetheless that this does not change the ranking of the countries.

Turning now to inequality of attainments (equation 8a), we find that the relative inequality index, I a R , evaluated at = 1, takes the highest value in Egypt: 2.9 percent. In contrast, this figure is the lowest in Morocco (0.8 percent), while inequality is between these two values in the context of the other three countries. For = 3, inequality in Egypt remains highest, at 30.7 percent. This is followed by 8.9 percent in Jordan, 5.3 percent in the Comoros and 4.1 percent in Yemen, while it is the lowest in Morocco (2.9 percent).

Turning to the distribution of shortfalls, we report in Table 2 results pertaining to the achievement and inequality indices ŷs R (equation 7) and I s R (equation 8b). The results of Table 2 show that the achievement rankings of the countries in terms of the distribution of health shortfall are similar to the results of Table 1, pertaining to the distribution of health attainments. However, when we examine inequality indices, there are substantial variations in the rankings of countries when moving from the distribution of health attainments to the distribution of health shortfalls. For example, consider in Table 2 the results pertaining to = -1 . We find that Egypt exhibits the lowest level of inequality in the distribution of shortfalls. On the other hand, the results of Table 1 indicate that Egypt exhibits the highest level of inequality in the distribution of health attainments.

Recall that the relative indices of inequality emerged to be inconsistent in their rankings of countries in the distributions of health attainments and shortfalls. We next compare the group of five countries using the absolute achievement and inequality indices; namely ŷa A (equation 11) and I a A (equation 14.a). We report in Table 3 findings pertaining to the distribution of health attainments. We discuss briefly the findings related to the inequality aversion parameter = 1. Health achievement remains the lowest in Egypt (ŷ a A = 51.7) and the highest in Morocco (ŷ a A = 42.9). Similarly, inequality remains highest in Egypt (I a A = 19.3) and lowest in Morocco (I a A = 14.0 ). Recall that absolute indices of achievement and inequality are consistent-in the sense of Proposition 7-in their orderings of distributions of attainments and shortfalls. The results pertaining to the absolute achievement We next report the ordering of countries according to the absolute inequality criteria. In accordance with the Lambert and Zheng (2011) theorems as well as Propositions 4 and 5, we limit ourselves to examining the distribution of health attainments. We additionally recall that the absolute Lorenz order is robust to the choice of the upper survival threshold b. With these observations in mind, we find that the five countries exhibit no intersection between curves, with Morocco exhibiting the most egalitarian distribution, and Egypt the least egalitarian distribution. Between Morocco and Egypt, we find that Yemen comes second followed by Comoros, followed by Jordan (Figure 1).

Social Welfare Orderings

Recall that the generalized Lorenz curve is consistent across rankings of distributions of health attainments and shortfalls in the sense of Proposition 6. In what follows therefore we focus entirely on the distribution of health attainments. In the upper tail of the distribution, the generalized Lorenz curve of a hypothetical optimum health distribution Y * = (m, …, m), would take the form of a straight line starting at zero with a slope equal to bm. We observe on the basis of Figure 2 that the attainment distributions pertaining to the five countries are ordered, in that we do not observe crossing curves. In terms of social welfare, Morocco ranks best, followed by Yemen, then by Comoros. The Jordan distribution is second from the bottom while the Egypt distribution exhibits the least level of social welfare.

We next turn to an examination of the lower tail of the distribution, where health and well-being are positively associated. Figure 3 reveals that Egypt ranks highest in terms of social welfare and Yemen exhibits the least level of social welfare. While the generalized Lorenz curves pertaining to Morocco, Jordan and Comoros appear to overlap, the Morocco distribution generalize-Lorenz dominates the other two distributions. For Jordan and Comoros, we find however that the two curves intersect once, with the Jordan curve lying above the Comoros curve in the first four deciles of the distribution.

Health Achievement and Social Welfare: Orderings on the Entire Domain of the Distribution

We focus here our discussion on social welfare, where we first consider the lexicographic order. Firstly, if we are of the view that tackling undernutrition should have precedence over tackling overweight, then the ordering of the five countries is as depicted in the Hasse diagram on the left-hand side of Figure 4. Conversely, if we take the view that tackling overweight should have precedence over tackling undernutrition, the ordering of the distributions is as depicted in the Hasse diagram on the right-hand side of Figure 4. This figure reveals that the approach one takes in the lexicographic order matters for the ranking of countries: Egypt ranks highest and Yemen ranks lowest when the lower tail of the distribution has precedence over the upper tail. Conversely, Morocco ranks best and Egypt ranks lowest when the upper tail has precedence over the lower tail of the distribution.

In Figure 5, we report the Hasse diagram of the five countries from the perspective of the product order. As the product order treats the two domains of the distribution symmetrically, we expect, that overall, it enables us to compare fewer pairs of distributions than either of the lexicographic orders. Indeed, we find that Egypt is not comparable to the other four countries. Furthermore, Yemen, Jordan and Comoros are pairwise incomparable. Nonetheless, these three countries are dominated by the Morocco distribution, from the perspective of the product order.

conclusion

The purpose of the paper was to address the question of the measurement of health achievement and inequality in the context of variables exhibiting an inverted-U relation with well-being. We adopted a general framework whereby we measure separately achievement and inequality in the health-increasing range of the variable, from a lower survival bound a to an optimum value m, and in the health decreasing range from m to an upper survival bound b.

We have shown in Proposition 1 that for variables exhibiting a negative association with well-being, the equally distributed equivalent value is a Schur-convex function: that is, a function that is decreasing in progressive transfers. This has meant that the [START_REF] Wagstaff | Inequality Aversion, Health Inequalities and Health Achievement[END_REF] health achievement index, and relative Atkinson-Kolm-Sen inequality indices available from the income inequality literature, as well as the Kolm type absolute indices required some adaptation in the context of variables exhibiting a decreasing relation with well-being.

In the empirical illustration, it was found that in the lower tail of the distribution health achievement is highest in Egypt and lowest in Yemen; while in the upper tail of the distribution achievement is highest in Morocco, but lowest in Egypt. That is, while Egypt was found to achieve the highest level of social welfare in relation to the phenomenon of undernutrition, it ranked worst in terms of the distribution of excess body mass. This result points to the need of disaggregating the distributional analysis of an anthropometric health indicator that exhibits a non-monotonic relation with well-being, as we have suggested in this paper.

Corollary 3

 3 For a distribution Y ∈ D mb , each of the equally distributed equivalent values ŷa A and ŷs A associated respectively with the distributions of health attainments and shortfalls is a Schur-convex function. Furthermore, the following inequalities are satisfied: m ≤ y ≤ ŷa A and m ≤ y ≤ ŷs A . Because the equally distributed equivalent values ŷa A and ŷs A are decreasing in Pigou-Dalton transfers, we shall presently write the Kolm absolute inequality indices for health attainments and shortfalls in the following forms: Written in the form (14a and 14b), the above inequality indices take on positive values, and are decreasing in Pigou-Dalton transfers.

  shortfalls. That is, for the absolute inequality ordering ( = 0 and 0 ≤ ≤ 1), and for any Y A and Y B in D mb we have � S Y A ;0, 𝜏 ≻ 𝜏,𝜃 � S Y B ;0, 𝜏 ⇔ � Z Y A ;0, 𝜏 ≻ 𝜏,𝜃 � Z Y B ;0, 𝜏 .

Figure 1 .

 1 Figure 1. Absolute Lorenz Curves: The Upper Tail of the BMI Distribution in Five Arab Countries (b = 60 and m = 24.9)

Figure 2 .Figure 3 .

 23 Figure 2. Generalized Lorenz Curves: The Upper Tail of the BMI Distribution in Five Arab Countries (b = 60 and m = 24.9)

Figure 4 . 1 .

 41 Figure 4. Hasse Diagram of the Lexicographic Order Note: An arrow pointed from country A to country B indicates that social welfare is higher in country B according to the lexicographic order.

Figure 5 .

 5 Figure 5. Hasse Diagram of the Product Order Note: An arrow pointed from country A to country B indicates that social welfare is higher in country B according to the product order.

  Consider two distributions, Y A and Y B in D mb with respec- tive means Y A and Y B and with associated functions Ẑ Y A ; , and Ẑ Y B ; , of the vectors of attainments, and associated functions Ŝ Y A ; , and Ŝ Y B ; , of the vectors of shortfalls. Fori = A, B let Z i = b -Y i.Then, for 0 ≤ ≤ 1 and 0 ≤ ≤ 1, and Ẑ Y A ; , , Ẑ Y B ; , ,

a in the context of the absolute Lorenz order (that is where

Proposition 7

  Let Y 1 , Y 2 ∈ D mb denote two distributions. Associate with Y j , an an-; for the distribution of health shortfalls . Then, for any value 𝜅 > 0 we have that ŷa A

	thropometric absolute achievement index ments and an index ŷs A Y j	ŷa A Y j ; for the distribution of health attain-

Note that this methodology could be adapted in other settings. Consider the context of routinely used biomarkers of health status, such as sugar level. Then, the lower bound for survival is a = 40 milligrams of glucose per decilitre of blood, and the corresponding critical upper bound is b = 450 milligrams per deciliter.D mb = Y ∈ (m, b) n 1 : y (1) ≤ y (2) ≤ ⋯ ≤ y (n 1) and y (1) < y (n 1 ) D am = X ∈ (a, m

) n 2 : x (1) ≤ x (2) ≤ ⋯ ≤ x (n 2 ) and x (1) < x (n 2 )

Note that in D mb , attainment increases when y decreases.

We note nonetheless that in other disciplines ≺ SM is known under the name of weak submajorization (seeMarshall et al., 2011, chapter 1).

Note furthermore that because the n 1 inequalities defining the generalized Lorenz ordering are linear in the distributions Y A and Y B , when working with health attainments, the ordering of distributions is robust to changes in the values taken by the upper bound b. For the same reason, when working with shortfalls, the generalized Lorenz ordering of a pair of distributions is likewise robust to changes in the values taken by the optimum health threshold m.

This work was supported by the French National Research Agency Grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University-A*MIDEX. The authors would like to thank two anonymous reviewers, Sameera Awawda and Gaston Yalonetzky for constructive comments and suggestions. -Marseille Université.

Supporting information

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Gao Feng

Amine Bermak

Sarah Benchabane-Gaiffe

Abdelkrim Laurent Robert

Acoustic-radiation free phononic biosensor for in-liquid low-noise gravimetric detection


Nikolay Bessonov

Anne Beuter

Sergei Trofimchuk

Vitaly Volpert

Estimate of the travelling wave speed for an integro-differential equation

Keywords: nonlocal reaction-diffusion equation, wave speed, minimax representation, estimates

Introduction

We consider the nonlocal reaction-diffusion equation

∂u ∂t = D ∂ 2 u ∂x 2 + ∫ ∞ -∞ ϕ(x -y)S (u (y + a, t -τ )) dy -ku, (1.1)
where ϕ(x) is a non-negative integrable function with a bounded support, S(u) is a smooth non-negative and non-decreasing function such that the equation

S(u) = ku (1.2)
has three solutions, u + < u 0 < u -, and S ′ (u ± ) < k. Some additional conditions and examples will be considered below. The diffusion coefficient D can be positive or zero, τ ≥ 0, a is a real number. This equation describes various applications including propagation of neural waves [START_REF] Ermentrout | Existence and uniqueness of traveling waves for a neural network[END_REF][START_REF] Modolo | Using a virtual cortical module implementing a neural field model to modulate brain rhythms in Parkinsons disease[END_REF]. In this case S(u) is neuron response function, ϕ(x) the connectivity function showing how the connection between neurons depends on the distance between them, time delay τ is an average time of signal transmission between connected neurons. A positive constant k characterizes the rate of signal decay, and a real number a is introduced for convenience, it will be used in the analysis of the problem. Equation (1.1) is considered on the whole axis. We will be interested in traveling waves solutions of this equations introduced in Section 2 below. One of the important their characteristics is the speed of wave propagation. Though, in general, it cannot be found explicitly, various various minimax and variational representations of the wave speed can be used to get some estimates and to study its asymptotics. The first mini-max representation of the wave speed was suggested in [START_REF] Hadeler | Travelling fronts in nonlinear diffusion equations[END_REF] for the scalar equation on the basis of the phase plane analysis. A similar approach was used in [START_REF] Vit | Propagation velocity estimation for condensed phase combustion[END_REF] for a system of two equations in combustion theory. Another mini-max representation, based on the global stability of travelling waves, was developed for monotone reaction-diffusion systems (see [START_REF] Volpert | Elliptic partial differential equations[END_REF] and the references therein). It was used to estimate the speed of blood coagulation waves in [START_REF] Galochkina | Reaction-diffusion waves of blood coagulation[END_REF]. This approach was generalized for some integro-differential equations [START_REF] Ducrot | Spectrum of some integro-differential operators and stability of travelling waves[END_REF][START_REF] Coville | Propagation speed of travelling fronts in non local reactiondiffusion equations[END_REF]. Let us also mention variational representations of the wave speed obtained in [START_REF] Benguria | Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation[END_REF][START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF]. In this work, we will use the method based on the global stability argument in order to obtain a minimax representation of the wave speed for another integro-differential equation. We will use it to get analytical estimates and asymptotic values of the speed.

Minimax representation of the wave speed in the monotone case

Profile w(x) of a travelling wave solution u(x, t) = w(x -ct) of equation (1.1) is a solution of the problem

Dw ′′ + cw ′ + ∫ ∞ -∞ ϕ(x -y)S(w(y + a + cτ ))dy -kw = 0, (2.1)
w(±∞) = u ± . (2.2)
Existence and global stability of travelling wave solutions of integro-differential equations are studied under various conditions on the functions ϕ(x), S(w), and constants D > 0 or D = 0, τ > 0 or τ = 0 [START_REF] Ermentrout | Existence and uniqueness of traveling waves for a neural network[END_REF][START_REF] Ducrot | Spectrum of some integro-differential operators and stability of travelling waves[END_REF][START_REF] Chen | Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations[END_REF][START_REF] Ma | Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation[END_REF]. Moreover, these waves are monotonically decreasing functions of x.

The wave is globally asymptotically stable if the following convergence

sup x∈R |u(x, t) -w(x -ct -h)| → 0 , t → ∞ (2.3)
occurs for some constant h and for the solution u(x, t) of equation (1.1) considered on the whole axis with the initial condition u(x, t) = u 0 (x), -τ ≤ t ≤ 0. Here u 0 (x) is a continuous decreasing function with the limits u 0 (±∞) = u ± . Note that more general stability results are available. This assumption is sufficient for what follows. If the function S(u) is increasing and ϕ(x) is non-negative, then the wave stability follows from the results of the work [START_REF] Ma | Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation[END_REF] (see also [START_REF] Chen | Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations[END_REF] for the nonlocal equation without delay). We suppose these conditions to be satisfied.

Proposition 2.1. Suppose that τ = 0, S(u) is increasing, and there exists a monotonically decreasing solution w(x) of problem (2.1), (2.2). Then for any function ρ(x) continuous together with its second derivative, ρ ′ (x) < 0 for all x ∈ R, ρ(±∞) = u ± , the following estimate holds:

inf x∈R F (ρ) ≤ c ≤ sup x∈R F (ρ) , (2.4)
where

F (ρ) = ( Dρ ′′ + ∫ ∞ -∞ ϕ(x -y)S(ρ(y + a))dy -kρ ) /(-ρ ′ ) .
Proof. Let us verify the right-hand side inequality in (2.4). Suppose that it does not hold. Then for some function ρ(x), we have c > sup x∈R F (ρ). Then there exists such value c 0 that

c > c 0 > sup x∈R F (ρ),
and, consequently, the inequality

Dρ ′′ + c 0 ρ ′ + ∫ ∞ -∞ ϕ(x -y)S(ρ(y + a))dy -kρ < 0 (2.5)
holds for all x ∈ R. Consider the Cauchy problem

∂v ∂t = D ∂ 2 v ∂x 2 + c 0 ∂v ∂x + ∫ ∞ -∞ ϕ(x -y)S (v (y + a, t)) dy -kv (2.6)
with the initial condition v(x, 0) = ρ(x). Then by virtue of the comparison theorems [START_REF] Ducrot | Spectrum of some integro-differential operators and stability of travelling waves[END_REF] and (2.5), the solution v(x, t) decreases as a function of t for every x. Hence, we get the estimate

v(x, t) ≤ ρ(x) , x ∈ R, t ≥ 0. (2.7)
Let u(x, t) be solution of equation (1.1) with the same initial condition. Then v(x, t) = u(x + c 0 t, t). Convergence (2.3) implies:

sup x∈R |v(x, t)-w(x-(c-c 0 )t-h)| = sup x∈R |u(x+c 0 t, t)-w(x-(c-c 0 )t-h)| → 0 , t → ∞. (2.8)
Since c > c 0 and w(x) is a decreasing function, then v(x, t) → u -for each x fixed, as t → ∞. This convergence contradicts (2.7). This contradiction proves the right inequality in (2.4).

The left inequality can be proved similarly.

Theorem 2.2. Suppose that τ = 0, S(u) is increasing, and there exists a monotonically decreasing solution w(x) of problem (2.1), (2.2). Then

c = inf ρ sup x F (ρ) = sup ρ inf x F (ρ), (2.9)
where infimum in the first equality and supremum in the second equality are taken with respect to all functions ρ(x) continuous together with their second derivatives, and such that ρ ′ (x) < 0 for all x ∈ R, ρ(±∞) = u ± .

Proof. The inequalities

sup ρ inf x F (ρ) ≤ c ≤ inf ρ sup x F (ρ)
follow from (2.4). It remains to note that if we take the wave solution as a test function, ρ(x) ≡ w(x), then we obtain the equalities.

Theorem 2.3. Suppose that τ > 0, a = 0, S(u) is increasing, and there exists a monotonically decreasing solution w(x) of problem (2.1), (2.2). Then

c = inf ρ sup x Φ(ρ, c) = sup ρ inf x Φ(ρ, c), (2.10)
where

Φ(ρ, c) = ( Dρ ′′ + ∫ ∞ -∞ ϕ(x -y)S(ρ(y + cτ ))dy -kρ ) /(-ρ ′ ) .
This equation has a unique solution.

Proof. Set a = στ in (2.9), and denote by c(σ) the left-hand side of this equality in order to show its dependence on σ. Then 3 Non-monotone case

c(σ) = inf ρ sup x Φ(ρ, σ) = sup ρ inf x Φ(ρ, σ) (2.11) since Φ(ρ, σ) = F (ρ) (for a = στ ). Since ρ(x)
The results presented above essentially use the wave monotonicity and global stability. They are obtained in the case where the function S(u) is monotonically increasing and the function ϕ(x) is non-negative. These assumptions provide the applicability of the maximum principle and comparison theorems for equation (1.1). We will now obtain some estimates of the wave speed for a non-monotone function S(u). The case of ϕ(x) with a variable sign can be studied similarly.

Lemma 3.1. Let S(u) ≤ S + (u) for all u ≥ 0 and some smooth monotonically increasing function S + (u). Assume, moreover, that ϕ(x) ≥ 0. Then the solution v(x, t) of the equation

∂v ∂t = D ∂ 2 v ∂x 2 + ∫ ∞ -∞ ϕ(x -y)S + (v (y + a, t -τ )) dy -kv (3.1)
with an initial condition v(x, t) = v 0 (x, t), -τ ≤ t ≤ 0 satisfies the inequality

v(x, t) ≥ u(x, t), (3.2)
where u(x, t) is the solution of equation (1.1) with the same initial condition.

Proof. By virtue of the conditions on the functions S + and S, we have

S + (v) -S(u) = (S + (v) -S + (u)) + (S + (u) -S(u)) = b(x, t)(v -u) + g(x, t), where b(x, t) = 0 if v(x, t) = u(x, t) and b(x, t) = S + (v) -S + (u) v -u ≥ 0
otherwise, g(x, t) = S + (u) -S(u) ≥ 0. Set z = v -u and subtract equation (1.1) from equation (3.1). We get

∂z ∂t = D ∂ 2 z ∂x 2 + ∫ ∞ -∞ ϕ(x -y) (b(y, t)z(y + a, t -τ ) + g(y, t)) dy -kz. (3.3)
Since z(x, t) = 0 for -τ ≤ t ≤ 0, then z(x, t) ≥ 0 for all t ≥ 0. The lemma is proved.

Suppose, next, that S(u) ≥ S -(u) for u ≥ u, where S -(u) is a monotonically increasing function. Then a similar assertion holds with inequality (3.2) replaced by an opposite inequality.

Theorem 3.2. Suppose that S -(u) ≤ S(u) ≤ S + (u) for u ≥ 0, where S -(u) and S + (u) are smooth monotonically increasing functions. Suppose, next, that each of the equations S(u) = ku, S -(0) = ku, S + (u) = ku has three solutions: u ± , u 0 ; u

(+) ± , u (+) 0 ; u (-) ± , u (-) 0 , respectively, such that u + < u 0 < u -, u (+) + < u (+) 0 < u (+) -, u (-) + < u (-) 0 < u (-) -, S ′ (u ± ) < k, S ′ ± (u (±) ± ) < k, and u + < u (±) 0 < u -.
Assume that problem (2.1), (2.2) has a travelling wave solution with a speed c. Then the following estimate holds:

c -≤ c ≤ c + , (3.4)
where c ± are the wave speed in the problems

Dw ′′ + cw ′ + ∫ ∞ -∞ ϕ(x -y)S ± (w(y + a + cτ ))dy -kw = 0, (3.5)
w(±∞) = u (±) ± . (3.6)
Proof. Consider equation (3.1) with the initial condition v(x, t) = w(x), -τ ≤ t ≤ 0, where w(x) is a solution of problem (2.1), (2.2). The solution of this problem converges to the travelling wave solution, that is, solution of problem (3.5), (3.6) with c = c + . From Lemma 3.1 it follows that c ≤ c + . The left-hand side inequality in (3.4) can be proved similarly.

4 Estimates of the wave speed

Analytical solution

We will now apply the minimax representation in order to obtain analytical estimates of the wave speed. Accuracy of the estimates depends on the choice of test functions ρ(x). If the estimates from above and from below are close to each other, then they are also close to the exact value of the speed. We will construct test functions as exact solutions of some approximate models. Let

S 0 (u) = { 0 , u < u 0 b , u ≥ u 0 ,
and ϕ(x) = ϕ 0 exp(-r|x|) for some positive constants u 0 , b, ϕ 0 , and r, ku 0 < b. In this case, equation (2.1) (a = τ = 0) can be solved explicitly. In order to simplify the formulas below, we set ϕ 0 = r. If w(x) is a decreasing function and w(0) = u 0 , then

J(w) ≡ ∫ ∞ -∞
ϕ(x -y)S 0 (w(y))dy = b (2 -e rx ) , x < 0, J(w) = be -rx , x ≥ 0.

Equation (2.1) is reduced to the problem

Dw ′′ + cw ′ -kw + b (2 -e rx ) = 0 , x < 0, Dw ′′ + cw ′ -kw + be -rx = 0 , x ≥ 0, (4.1) w(-0) = w(+0), w ′ (-0) = w ′ (+0). (4.2)
If Dr 2 ± cr ̸ = k, its solution writes:

w(x) = c 1 e µ 1 x + c 2 e rx + 2b k , x ≤ 0, w(x) = c 3 e µ 2 x + c 4 e -rx , x ≥ 0, (4.3)
where

µ 1 = - c 2D + √ c 2 4D 2 + k D , µ 2 = - c 2D - √ c 2 4D 2 + k D , c 1 = αµ 2 -β µ 2 -µ 1 , c 3 = αµ 1 -β µ 2 -µ 1 , c 2 = b (Dr 2 + cr -k) , c 4 = - b (Dr 2 -cr -k) , α = c 4 -c 2 - 2b k , β = -(c 2 + c 4 )r .
Condition w(0) = u 0 provides the equation for the wave speed c = c 0 :

αµ 1 -β µ 2 -µ 1 - b (Dr 2 -cr -k) = u 0 . (4.4)
In the limiting case D = 0, after solving (4.4), we obtain the expression for the wave speed:

c 0 = 1 r ( bϕ 0 ru 0 -k ) . (4.5)
In order to compare with numerical simulations, this formula is given for the case where ϕ 0 can be different from r.

In the limiting case of large r (ϕ 0 = r), which corresponds to the local equation, after solving (4.4), we obtain:

c 0 = 2σ √ kD √ 1 -σ 2 , σ = 1 - ku 0 b . (4.6)
Numerical simulations of equation (1.1) with the step-wise constant function S 0 (u) and an exponential function ϕ(x) are shown in Figure 1. In the case of small diffusion coefficients (curves 1 and 2), the wave speed practically coincides with the analytical formula (4.5) obtained for D = 0. The influence of the diffusion coefficient becomes more essential for D = 10 -3 and D = 10 -2 . It is interesting to note that the dependence of the wave speed on diffusion coefficient for the integro-differential equation is quite different in comparison with the reaction-diffusion equation where the wave speed grows proportionally to √ D (see also Figure 2). Numerical simulations were carried out with an implicit finite difference method based on the Tomas algorithm for the inversion of the tridiagonal matrix. Numerical accuracy was verified by the convergence of the results under the decrease of time and space steps and by the comparison with the analytical results.

Estimates and asymptotics

Consider now a smooth increasing function S(u) such that S(u) ≡ S 0 (u) for |u -u 0 | ≥ ϵ. We will use the estimate of the wave speed (Proposition 2.2) with the test function w(x) given by (4.3). We have

F (w) = c 0 + H(x),
where The right-hand side of the last inequality converges to 0 as ϵ → 0 if the function ϕ(x) has, for example, a bounded support or decays at infinity faster than w ′ (x). Thus, inequalities (4.7) provide not only the estimates of the wave speed but also its convergence to c 0 as ϵ → 0.

H(x) = 1 w ′ (x) ∫ ∞ -∞ ϕ(x -y)(S 0 (w(y)) -S(w(y)))dy. Hence c 0 + inf x H(x) ≤ c ≤ c 0 + sup x H(x).
We finally consider one more numerical example with the function

S(u) = 2b π arctan((u -u 0 )/ϵ) + b
in order to illustrate asymptotics of the wave speed. Figure 2 (left) shows the dependence of the wave speed on ϵ. The limiting value for ϵ → 0 coincides with the value found for S 0 (u).

Conclusions

Integro-differential equations describe various processes in biological and biomedical applications. In many cases, these equations have travelling wave solutions. The speed of wave propagation characterizes these processes. In particular, the speed of cortex waves should belong to a certain range providing normal brain functioning. It depends on the response and connectivity functions of populations of neurons. In the damaged cortical tissue, these parameters can change following a stroke for example, resulting in a decrease of the cortical wave speed. External stimulation with the characteristics calculated from the properties of healthy and partially damaged cortical tissue can be introduced in order to restore the wave speed.

Though several methods are developed to determine the speed of wave propagation for reaction-diffusion equations, we are not aware whether other representations are applicable for the problem under consideration. The estimates obtained in this work are applicable under some restrictive conditions on the function S(u). Further studies are necessary to generalize them for a broader class of functions.

  is a decreasing function, then for any ρ fixed and for any x ∈ R, Φ(ρ, σ)(x) is a non-increasing function of σ. Hence c(σ) is a nonincreasing continuous function of σ. Therefore, equation c(σ) = σ has a unique solution. It provides equality (2.10) with c = σ.

Figure 1 :

 1 Figure 1: Numerical simulation of wave propagation. A snapshot of solution (left) and the wave speed dependence on the parameter u 0 (right) found numerically and by the analytical formula (4.5). The curves are obtained for different values of the diffusion coefficient: 1. D = 10 -6 , 2. D = 10 -4 , 3. D = 10 -3 , 4. D = 10 -2 . The values of other parameters are as follows: k = 0.1, b = 0.1, r = 20, ϕ 0 = 10. The analytical curve coincides with curve 1.

(4. 7 )

 7 Since w(0) = u 0 and |w(y) -u 0 | ≥ ϵ for |y| ≥ ϵ/β, where β ∼ |w ′ (0)|, then |H(x)| ≤ b |w ′ (x)| ∫ ϵ/β -ϵ/β ϕ(x -y)dy.

Figure 2 :

 2 Figure 2: Dependence of the wave speed on ϵ (left) and on D (right). The values of parameters are as follows: k = 0.1, r = 20, ϕ 0 = 10, b = 0.05, u 0 = 0.1, 1. D = 10 -6 , 2. D = 10 -4 , 3. D = 10 -3 (left); k = 0.1, b = 0.05, u 0 = 0.1, ϵ = 10 -5 , r = 2ϕ 0 ; 1. analytical formula (4.6), 2. ϕ 0 = 30, 3. ϕ 0 = 20, 4. ϕ 0 = 10, 5. ϕ 0 = 5 (right).

Acknowledgments N. Bessonov was supported by Russian Foundation of Basic Research grant 16-01-00068, 2016-2018. S. Trofimchuk was supported by FONDECYT (Chile) project 1150480, and V. Volpert by the "RUDN University Program 5-100".


Feng Gao
email: fgao@hbku.edu.qa

Amine Bermak

Sarah Benchabane

Marina Raschetti

Abdelkrim Khelif

Nonlinear effects in locally-resonant nanostrip phononic metasurface at GHz frequencies

In this paper, we report on the observation of nonlinear effects in a nanostrip phononic metasurface (NPM) that enable the tuning of resonance frequencies at 1.42 GHz. The NPM resonator made of periodic nanostrip array is fabricated on a lithium niobate substrate. Each of the nanostrip is 250-nm wide and is made of 680-nm-thick SiO2 layer stacking on 50-nm Al metal electrodes. Finite element analysis reveals that the device operates in a vertically polarized (compression) mode with substantial acoustic energy confined in the nanostrips, leading to a local resonance at low acoustic velocity. Due to the nonlinearity, the resonance frequency of the device decreases with the increase of stimulation power. The underlying mechanism of the nonlinearity is found to be the power-dependent coupling of the adjacent nanostrips. This coupling induces softening of the substrate surface region, which reduces the acoustic velocity and hence the bulk radiation. As a result, the quality factor of the NPM resonator is found to improve with the increase of stimulation power. The power-dependent coupling of nanostrips in the NPM resonator demonstrates a reliable method for the realization of nonlinearity in phononic metasurfaces, which would significantly enrich the mechanisms for the manipulation of surface acoustic waves at high frequencies.

Nonlinearity in phononic crystals, acoustic/elastic metamaterials and phononic metasurfaces is a desired effect for the implementation of many useful acoustic phenomena including breathers 1 , bifurcation 2 , tunability [START_REF] Cha | Electrical tuning of elastic wave propagation in nanomechanical lattices at MHz frequencies[END_REF] , localization [START_REF] Campbell | Localizing energy through nonlinearity and discreteness[END_REF] , and chaos [START_REF] Strogatz | Nonlinear dynamics and chaos[END_REF] . Phononic nonlinearities were previously exploited in nanomechanical waveguides [START_REF] Cha | Electrical tuning of elastic wave propagation in nanomechanical lattices at MHz frequencies[END_REF][START_REF] Kurosu | Mechanical Kerr Nonlinearity of Wave Propagation in an On-Chip Nanoelectromechanical Waveguide[END_REF] , magnetic lattices [START_REF] Molerón | Nonlinear excitations in magnetic lattices with long-range interactions[END_REF][START_REF] Palermo | Tuning of surface-acoustic-wave dispersion via magnetically modulated contact resonances[END_REF] , surface acoustic wave (SAW) gratings [START_REF] Lu | Nonlinear Surface Acoustic Wave Grating for Parametric Amplification[END_REF] and lamb wave devices [START_REF] Schneider | Frequency conversion through nonlinear mixing in acoustic waves[END_REF] . Nevertheless, most of the demonstrations were in the kHz or MHz frequency ranges. Nonlinearities of phononic metasurfaces operating in GHz frequencies are still to be explored.

A good option for achieving nonlinearity in phononic crystal is to build a phononic pillar array whose composing elements are coupled nanomechanical resonators that exhibit nonlinearity. As an example, Midtvedt et al. exploited this idea by introducing coupling through an atomically thin graphene membrane transferred to the top of a phononic crystal [START_REF] Midtvedt | Nonlinear phononics using atomically thin membranes[END_REF] . But coupling nano or micro-resonators can also be achieved through the substrate. Recently, it was reported that cylindrical pillars very close to each other exhibit substrate-induced coupling that enables phase modulation of the pillars [START_REF] Raguin | Dipole states and coherent interaction in surface-acoustic-wave coupled phononic resonators[END_REF] . Utilizing this phenomenon, we built a nanostrip phononic metasurface (NPM) resonator with high-aspect-ratio nanostrips that exhibit substrate coupling induced nonlinearity. The coupling can be controlled by the stimulation power, which results in a power-dependent frequency shift of the NPM resonators. The majority of the stack consists of a 680-nm-thick silicon dioxide (SiO2) layer, serving as the structural material. A metal electrode lies beneath the SiO2 layer for electrical conduction, which is made of 20-nm titanium (Ti), 50-nm aluminum (Al) and 5-nm Ti from bottom to top. The width of the nanostrip is 250 nm, leading to an aspect ratio of 2.96. Due to the high aspect ratio, a substantial amount of the acoustic energy is stored in the nanostrips, which forms a locally confined resonance. The interdigitated transducers (IDTs), consisting of 30 pairs of nanostrips, are located in the center of the NPM resonator for stimulating the phononic resonance. On the two sides of the IDTs, acoustic reflectors formed by 20 nanostrips are used to enhance the resonance. The pitch of the nanostrip array is 500 nm, which leads to an acoustic wavelength (𝜆) of 1 𝜇𝑚.

The device was fabricated by firstly depositing Ti, Al and SiO2 on the LiNbO3 substrate. A complementary resist pattern of the device was then produced by e-beam lithography. After that, chromium etching mask was formed by e-beam evaporation and lift-off process. Reactive ion etching (RIE) was then used to remove the stacked materials between the high-aspect-ratio nanostrips. At last, conventional photolithography and another RIE were used to remove the silicon dioxide above the bus bars of the IDTs to expose the metal. Extra aluminum lines were deposited and patterned to extend the connection from the bus bars for external electrical connection. A scanning electron microscope (SEM) image of the fabricated device is shown in Fig. 2a.

Fig. 2b shows a cross-section of the high-aspect-ratio nanostrips obtained by focused ion beam (FIB). False color overlays are used to identify the different materials. The green overlay shows the platinum deposited during the FIB cross-sectioning for protecting the top surface of the device. The metal electrodes (Ti and Al), SiO2 and LiNbO3 are marked in yellow, blue and red, respectively.

To obtain the resonance behavior of the NPM resonator, we measured its impedance (Z) using a network analyzer setup shown in Fig. S1 of the Supplementary Material. The impedance curves under 0, 10, 15 and 19.5 dBm stimulation power are shown in Fig. 3a. The minimum and maximum of impedance amplitude correspond to the resonance (𝑓 ! ) and anti-resonance frequencies (𝑓 " ) of the resonator. At 0-dBm stimulation power, 𝑓 ! and 𝑓 " are 1.422 GHz and 1.453 GHz, respectively. The corresponding phase velocity (𝑣 = 𝑓 ! 𝜆) of the NPM wave is thus 1422 m/s, which is much lower than the velocity of the Rayleigh wave (𝑣 ! = 3988 𝑚/𝑠). This reduced velocity is because of the slowing-down effect induced by the high-aspect-ratio nanostrips [START_REF] Laude | Surface acoustic wave trapping in a periodic array of mechanical resonators[END_REF][START_REF] Gao | Towards Acoustic Radiation Free Lamb Wave Resonators for Highresolution Gravimetric Biosensing[END_REF] . Based on the actual geometry and materials of the NPM resonator, we simulated its mode shape and slowingdown effect in a simplified one-wavelength unit cell using finite element method (FEM). The details of the simulation model and the relevant material parameters are given in Fig. S2 and Table. S1, respectively. Fig. 3b shows the simulated mode shape of the NPM resonator. In contrast to conventional SAW resonators, in which the particle displacement is the largest on the substrate surface, the NPM resonator exhibits the largest particle displacement within the high-aspect-ratio nanostrips. The resonance frequency of the device is thus mostly determined by the geometrical dimensions and by the material composition of the nanostrips. It should be noted that the simulated resonance frequency (1.53 GHz) is slightly higher than the measured one due to the discrepancies between the simulation model and the actual device. The slowing-down curve obtained by FEM is shown by the blue-dashed curve in Fig. 3c. The acoustic velocity of the NPM resonator decreases with the increase of SiO2 thickness due to the classical mass loading effect and the hybridization between the surface acoustic wave and the elastic resonance of the highaspect-ratio nanostrips [START_REF] Gao | Acoustic radiation-free surface phononic crystal resonator for in-liquid low-noise gravimetric detection[END_REF] . Besides the slowing-down effect, the nanostrips also result in confinement of elastic energy to the surface region of the substrate. The orange-dashed curve in Fig. 3c shows the ratio between the simulated elastic energy stored in the first 0.2𝜆-depth substrate (𝐸 #.%& ) and the first 1𝜆-depth substrate (𝐸 '& ). The locations of 𝐸 #.%& and 𝐸 '& are marked by green-dashed and red-dashed boxes in Fig. 3b. The increase of 𝐸 #.%& /𝐸 '& means that more energy is concentrated to the surface region when the SiO2 thickness increases.

To investigate the effect of stimulation power on the coupling induced nonlinearity, we measured the impedances of the NPM resonators under various stimulation power. Fig. 3a shows that the impedance curves shift to the lower frequencies with the increase of stimulation power. The measured frequency shift (∆𝑓 = 𝑓 -𝑓 #()* ) of the NPM resonators in the power range of 0 to 19.5 dBm is shown in Fig. 4a. The data points marked by different colors are from four devices fabricated in the same batch, which shows good repeatability. The dashed green line is the mean frequency shift of the four devices. As the frequency shift is negative, |∆𝑓| is shown in the figure for clarity. The impedance at higher stimulation power is not measured as it exceeds the 1 dB compression point (P1dB) of the amplifier in the test setup. Fig. 4a shows that the frequency shift increases with the increase of stimulation power.

FIG. 4. Power-dependent frequency shifts of (a) NPM resonators and (b). 10-𝜇𝑚 VP-PM resonators

To investigate if the nonlinearity also exists in larger scale vertical polarized phononic metasurface (VP-PM) [START_REF] Gao | Acoustic radiation-free surface phononic crystal resonator for in-liquid low-noise gravimetric detection[END_REF] and the conventional Rayleigh SAW [START_REF] Gao | Dual transduction surface acoustic wave gas sensor for VOC discrimination[END_REF] resonators, we measured their resonance frequency under different stimulation power. The wavelengths of the VP-PM and the Rayleigh SAW resonator are both 10 𝜇𝑚. The VP-PM is an enlarged equivalent of the NPM but uses 7.8-𝜇𝑚-thick nickel to form the high aspect ratio electrode array. It operates at 130 MHz and was previously reported in [15]. The single port Rayleigh SAW resonator operating at 384 MHz is made of 40 pairs of IDTs and 100 reflector strips on 128° Y-cut LiNbO3 substrate. Its electrodes are made of 200-nm Al. When the stimulation power increases, the resonance frequency of the Rayleigh SAW remains constant as shown in Fig. S3. In contrast, the resonance frequency of the VP-PM was found to shift with the stimulation power (Fig. 4b). We used a scanning laser heterodyne interferometer set-up [START_REF] Kokkonen | Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations[END_REF] to obtain the resonance frequency of VP-PM by tracking its maximum electrode displacement point. The set-up supports a maximum stimulation power of 30 dBm. However, due to the poor impedance match at high frequencies, this setup cannot track the resonance frequency of the NPMs with good resolution. We thus have to obtain the resonance frequency shift of the NPM using the network analyzer set-up, which has a smaller power range (Fig. 4a). Two VP-PMs were measured as shown in Fig. 4b. Their resonance frequency shifts increase with the stimulation power but saturate when the stimulation power exceeds 24 dBm.

Various mechanisms could lead to the nonlinearity of individual nanomechanical resonators. Typical ones are geometrical nonlinearity, material nonlinearity, actuation nonlinearity, etc [START_REF] Schmid | Fundamentals of nanomechanical resonators[END_REF] . The well-known geometrical nonlinearity could be a potential cause [START_REF] Jia | Resonance frequency response of geometrically nonlinear micro-switches under electrical actuation[END_REF] . However, a detailed analysis rules out this possibility. Geometrical nonlinearities of mechanical resonators appear when the displacement amplitude is sufficiently large compared to the device dimensions [START_REF] Schmid | Fundamentals of nanomechanical resonators[END_REF][START_REF] Catalini | Soft-clamped phononic dimers for mechanical sensing and transduction[END_REF] . In order to evaluate this effect in our devices, we measured the out-of-plane displacements of the nanostrips at two different locations (Point A and B as marked in Fig. 2a) using the scanning laser heterodyne interferometer. The results in Fig. 5a show that the displacement amplitude increases linearly with the square root of the power until 400 mW (26 dBm). This is an expected result as the square of the displacement corresponds to the mechanical energy in the system, which is proportional to the electrical power of the stimulation. However, the displacement amplitude saturates, then decreases slightly after the stimulation power exceeds 400 mW. The VP-PM resonators exhibit similar behavior. with the frequency saturation point in Fig. 4b, which implies the saturation in vibration amplitude leads to the saturation in frequency shift. The differences in frequency shift and electrode displacement of the two VP-PM resonators are due to the process variations. The saturation of frequency shift is not observed for the NPM resonator in Fig. 4a because the stimulation power of the network analyzer setup did not reach the saturation point (26 dBm) shown in Fig. 5a. The maximum displacements of the NPM and VP-PM are both too small compared to their device dimensions to trigger geometrical nonlinearity [START_REF] Schmid | Fundamentals of nanomechanical resonators[END_REF] .

The NPM can also be considered as an array of singlyclamped beams. The geometrical nonlinearity of singlyclamped beam generally occurs in flexural modes when the displacement is comparable to the beam thickness [START_REF] Lifshitz | Nonlinear dynamics of nanomechanical and micromechanical resonators[END_REF] . However, the nanostrips constituting the NPM operate in vertical polarized compression mode and have a displacement amplitude much smaller than their thickness. Moreover, geometrical nonlinearities usually result in a hardening effect of the fundamental mode that increases the mechanical resonator's resonance frequency [START_REF] Schmid | Fundamentals of nanomechanical resonators[END_REF][START_REF] Jia | Resonance frequency response of geometrically nonlinear micro-switches under electrical actuation[END_REF] . In contrast, the NPM exhibits a resonance frequency decrease with the increase of stimulation power that corresponds to a softening effect.

It is previously reported that coupling induced by a covering graphene membrane on a phononic crystal can result in nonlinearity [START_REF] Midtvedt | Nonlinear phononics using atomically thin membranes[END_REF] . The NPM resonator actually shares a similar mechanism in which the coupling between adjacent nanostrips is through the substrate instead of an external membrane. The resonance frequency decrease can be explained by the softening effect in the surface region of the substrate. Fig. 3c shows that the high aspect ratio nanostrips help the concentration of elastic energy to the surface region of the substrate. The increased coupling between adjacent nanostrips at high stimulation power further improves this energy confinement effect, leading to the softening of the surface region and the reduction of acoustic velocity (resonance frequency). In fact, this energy confinement effect is supported by both the quality factor (Q) and the nanostrip vibration amplitude measurement results. As the confinement of elastic energy to the surface region will suppress the energy leakage into the bulk substrate, the Q factor of the NPM resonator should improve with the increase of stimulation power, which is indeed observed by the Q factor measurement shown in Fig. 6. Besides, the energy confinement along the substrate surface could also reduce the energy allocated to the nanostrips, which explains the saturation of nanostrip vibration amplitude at high stimulation power in Fig. 5a. This amplitude saturation can in turn leads to the saturation of frequency shift as observed in the 10-𝜇𝑚 VP-PM (Fig. 4b). It should be noted that the difference of the Q factors between the four devices in Fig. 6 is due to the variation of the fabrication process. The low-Q devices have more process defects. However, the process defects do not have a significant impact on the operating frequency and the nonlinearity (Fig. 4a) as they are determined by the device geometries.

In summary, nonlinear effects causing a resonance frequency downshift in an NPM operating at 1.42 GHz were observed. Similar behavior can also be observed in VP-PM resonators with 10-𝜇𝑚 wavelength operating at 130 MHz. The analysis shows that the nonlinearity is most likely originated from the coupling between adjacent nanostrips rather than by geometrical nonlinearity. By increasing the stimulation power, the coupling between the nanostrips is enhanced, which results in a softening effect of the substrate surface. The softening effect not only reduces the resonance frequency but also enhances the energy confinement along the substrate surface and thus improves the quality factor. The power-dependent coupling in the NPM demonstrated a reliable method for the implementation of controllable nonlinearity in GHz phononic devices, which can enrich the mechanisms for the manipulation of surface acoustic waves at high frequencies.

Supplementary material

See the supplementary material for more details of the FEM simulation model and relevant materials properties.

Acknowledgement

This work is funded by NPRP grant No. NPRP10-0201-170315 from the Qatar National Research Fund (a member of Qatar Foundation). This work is also supported by the EIPHI Graduate School (contract "ANR-17-EURE-0002") and the French RENATECH network with its FEMTO-ST technological facility. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 865724). The findings herein reflect this work and are solely the responsibility of the authors.

Fig. 1

 1 Fig. 1 shows a diagram of the NPM made of periodic high-aspect-ratio nanostrips on a 128° Y-cut lithium niobate (LiNbO3) substrate. Each of the nanostrip is made of a stack of materials as shown in the insert of Fig. 1.The majority of the stack consists of a 680-nm-thick silicon dioxide (SiO2) layer, serving as the structural material. A metal electrode lies beneath the SiO2 layer for electrical conduction, which is made of 20-nm titanium (Ti), 50-nm aluminum (Al) and 5-nm Ti from bottom to top. The width of the nanostrip is 250 nm, leading to an aspect ratio of 2.96. Due to the high aspect ratio, a substantial amount of the acoustic energy is stored in the nanostrips, which forms a locally confined resonance. The interdigitated transducers (IDTs), consisting of 30 pairs of nanostrips, are located in the

FIG. 1 .

 1 FIG.1. NPM resonator with periodic high-aspect-ratio nanostrips covering the IDTs.

FIG. 2 .

 2 FIG. 2. (a) Top view SEM image of the NPM resonator (b) Cross-sectional view of the high-aspect-ratio electrodes.

  Fig.5breports the electrode displacements of the two VP-PM resonators measured at their IDT centers. The turning point in Fig.5b agrees

FIG. 5 .

 5 FIG. 5. Out-of-plane electrode displacements of the (a). NPM and (b). VP-PM resonators.

Data Availability

The data that supports the findings of this study are available within the article.