F G Schmitt

P Chainais
email: pchainai@isima.fr

On causal stochastic equations for log-stable multiplicative cascades

Keywords: 02, 50, Ey Stochastic processes -47, 27, E-Turbulence simulation and modelling -5, 45, Df Fractals

We reformulate various versions of infinitely divisible cascades proposed in the literature using stochastic equations. This approach sheds a new light on the differences and common points of several formulations that have been recently provided by several teams. In particular, we focus on the simplification occurring when the infinitely divisible noise at the heart of such model is stable: an independently scattered random measure becomes a stable stochastic integral. In the last section we discuss the D-dimensional generalization.

Introduction

Multifractals have been introduced more than two decades ago in the fields of turbulence, geophysics and chaos theory. Since then, the multifractal framework has been used widely in many fields including turbulence [START_REF] Frisch | Turbulence; the legacy of A. N. Kolmogorov[END_REF][START_REF] Schertzer | [END_REF], precipitations [3,4], oceanography [5,6], biology [7][8][9], chemistry [10] astrophysics [11], finance [12][13][14], etc.

The main properties of multifractal models or data is to possess high variability on a wide range of spatial or temporal scales, associated to scaling intermittent fluctuations and long-range power-law correlations. Scale invariance is usually associated to a power-law spectrum (e.g. the k -5/3 Kolmogorov law in fully developed turbulence) which is the signature of the absence of a characteristic scale. At large scales, scale invariance may be connected to long range dependence due to power law correlations. At small scales, scale invariance is rather interpreted as a singular behaviour (e.g., non differentiable) of the form |f (x)f (x o )| ∼ |xx o | h connected to a high variability on small scales. It appears that these intepretations of the scale invariance property can be linked to the statistics of the multiscale fluctuations of some physical data (e.g., the velocity increments or the dissipation field in a turbulent flow). Indeed, there exists a parallel between the description of the scale invariant behaviour of a field by a set of multiscaling exponents and the description of the statistics of its fluctuations at various scales [15][16][START_REF] Chainais | IEEE Trans. on Pattern Analysis and Machine intelligence[END_REF].

For a long time, research in this framework was mainly devoted to data analysis and to the estimation of some parameters, but no continuous model was proposed to re-produce the captured properties in the general case, and there were only discrete scale constructions like Mandelbrot binomial cascades [START_REF] Mandelbrot | [END_REF] or cascades on wavelet trees [19]. Only recently, research has developed on the definitions and properties of versatile multifractal stochastic processes. In this framework, various recent works [20][21][22][23][24]16] have presented similar objects in different ways, involving multifractal stochastic processes depending on a continuous scale parameter. These objects belong to the family of infinitely divisible cascades, a large class of multifractal scalar fields. We propose to review these presentations of infinitely divisible cascades which rely on the use of rather abstract mathematical objects such as additive stochastic measures. Then we show how a specific family of stochastic processes, namely stable processes (including Gaussian processes), connects to a related family of processes that is much more easier to describe. In this approach, we will use descriptions based on the framework of multiplicative cascades as well as the framework of stochastic integrals as proposed for the Gaussian case in 1 dimension in [22].

First, we consider cascades in 1 dimension only. We recall the main definitions and properties of infinitely divisible (ID) cascades. We also recall the definition using a stochastic integral proposed in [22] in the special Normal case. We extend the result of [22] describing explicitly how the ID framework is related to a simpler framework for Lévy stable random variables in 1 dimension. Then we show that most of our arguments are easier to understand in 1 dimension but they generalize rather naturally to D ≥ 2 dimensions. Moreover, a nice formulation based on the use of a stochastic integral with respect to some αstable noise is given. Finally, we propose to consider separately D spatial dimensions and 1 time dimension within a unique D +1 dimensional cascade so that causal animated scalar fields can be considered. Again a formulation using a stochastic integral with respect to some α-stable noise is proposed that makes explicit the main properties of the resulting field: multifractality both in time and space, homogeneity, stationarity and causality.

From cascades to stochastic integrals: the 1D causal case

Examples of realizations of 1D, 2D and 3D infinitely divisible cascades are shown on figure 1. Let us first recall the definition of a scale invariant infinitely divisible cascade.

Definitions

Let G(X) be an infinitely divisible distribution with moment generating function G(q) that can be written in the following form G(q) = IE(e qX ) = e -ρ(q) . In full generality, let dm(t, r) = g(r)dtdr a positive measure on the time-scale half-plane P + := R × R + . We will see below that scale invariance imposes the special choice dm(t, r) = 1 r 2 dtdr. Let M denote an infinitely divisible (ID), additive independently scattered random measure (called an "ID random measure" in the following) distributed by G, and supported on the time-scale halfplane P + and associated to its so-called control measure dm(t, r). For any subset E of P + , the random measure M is such that

IE[exp [qM (E)]] = exp [-ρ(q)m(E)] . (1)
For all disjoints subsets E 1 and E 2 , M (E 1 ) and M (E 2 ) are independent random variables and

M (E 1 ∪ E 2 ) = M (E 1 ) + M (E 2 )
. For more information on random measures, see [25][START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes[END_REF][START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF], and for a first use for cascades see [20].

Definition 1

For a given resolution 0 < ℓ ≤ 1, let C ℓ (t) the cone of influence defined below for every t ∈ R. An Infinitely Divisible Cascading measure density (IDC measure density) is a family of processes Q ℓ (t) parametrized by ℓ of the form

Q ℓ (t) = exp [M (C ℓ (t))] IE[exp M (C ℓ (t))] . (2)
Possible choices for distribution G are the Normal distribution, Poisson distribution, compound Poisson distributions, Gamma laws, Lévy-stable laws, and infinitely many others, so that a large variety of choices is available for modelling and applications (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]).

Different definitions of the cone C ℓ (t) have been recently proposed in the literature. To get causal definitions, we adapt the expressions given in [21,23,24], and consider only integration on positions (t ′ , r ′ ) for which t ′ ≤ t. The corresponding choices are 1 : [21,23], see fig. 2(a); [24], see fig. 2(b).

           C BM ℓ (t) = {(t ′ , r ′ ) : ℓ ≤ r ′ ≤ 1, t -r ′ ≤ t ′ ≤ t} ∪{(t ′ , r ′ ) : t -1 ≤ t ′ ≤ t, r ′ ≥ 1} in
C CRA ℓ (t) = {(t ′ , r ′ ) : ℓ ≤ r ′ ≤ 1, t -r ′ ≤ t ′ ≤ t} in
(

) 3
In this contribution, we will focus on stable laws. Indeed, it appears that for stable cascades the ID stochastic measure M may be simplified, and expressed using a stochastic integral versus a stable noise. Such a formulation was proposed by Schmitt and Marsan [20] and its properties in the Gaussian case studied in Schmitt [22]:

Definition 2 ǫ λ (t) = λ -σ 2 /2 exp σ t t+1-λ (t + 1 -u) -1/2 dB(u) (4)
where λ plays the role of the inverse of a resolution as 1/ℓ in definition 1, dB(u) is a Gaussian noise and σ 2 is a variance parameter. We will use the notation Qℓ (t) = ǫ λ (t) in the sequel.

Our first aim is to gather the various definitions above within consistent notations and to precisely formulate the link between approaches using respectively "multiplicative cascades" like (2) and "stochastic integrals" like (4). To bring all approaches within the same context, we will mainly use causal definitions.

Scaling properties

The scaling properties of IDCs in 1 dimension have been studied in Refs. [20][21][22][23][24][START_REF] Barral | [END_REF]30]. We recall here some basic results. The measure M , the distribution G, the control measure m and the geometry of the cone of influence C ℓ (x) control the scaling structure as well as marginal distributions of the cascade. One major property of IDCs is the following [20,21,24]:

IE[Q q ℓ ] = exp [-ϕ(q) m(C ℓ )] (5)
where

ϕ(q) = ρ(q) -qρ(1), ϕ(1) = 0, (6)
for all q for which ρ(q) =log G(q) is defined. We may also remark that, turning to local averages κ r over a box of size r ≥ ℓ

κ r (t) = 1 r |t ′ -t|<r/2 Q ℓ (t ′ ) dt ′ , (7)
one gets:

IEκ r (t) q ∝ exp [τ (q) m(C r (t))] (8)
1

Note that the large scale in the definition of C ℓ (t) has been arbitrarily set to 1 without loss of generality. Choosing a different large scale L would simply reduce to a change of units where in general τ (q) = ϕ(q) at least within some limited range of values of q [START_REF] Barral | [END_REF]31]. This corresponds to observables and is therefore denoted "dressed cascades" as opposed to "bare cascades" represented by Q ℓ [START_REF] Mandelbrot | [END_REF]3]. We do not consider this question further here, and mainly focus on the formalism describing the construction of bare cascades.

t → t • L, ℓ → r • L.
C BM ℓ (t) C CRA ℓ (t) C SM ℓ (t) ℓ ℓ ℓ t -1 t -ℓ t t -1 t -ℓ t t -1 t -ℓ t (a) (b) (c)

The Normal case

In this section, the distribution G of definition 1 is the Normal law N (µ, σ 2 ) with average value µ and variance σ 2 . It was shown in [21,23] that an exact power law scaling is observed if (see figure 2(a)):

         dm(t, r) = dt dr r 2 , C ℓ (t) = C BM ℓ (t) = {(t ′ , r ′ ) : ℓ ≤ r ′ ≤ 1, t -r ′ ≤ t ′ ≤ t} ∪{(t ′ , r ′ ) : t -1 ≤ t ′ ≤ t, r ′ ≥ 1}. (9)
Then, the key quantity is the control measure of the cone of influence C ℓ :

m(C BM ℓ (t)) = 1 ℓ t t-r dt ′ =r dr ′ r 2 + ∞ 1 t t-1 dt ′ =1 dr ′ r 2 = log(1/ℓ) + 1. (10)
The essential term that ensures a multifractal behaviour is the log(1/ℓ) term ; the last"+1" term only ensures exact power law scaling behaviours for scales ranging from ℓ to 1. In [24], only the 1st term was used. Indeed, as ℓ → 0, the differences due to the choice of the cone of figure 2(a) or (b) asymptotically disappear, except around t ≃ 1. The contribution of the upper part

{(t ′ , r ′ ) : t -1 ≤ t ′ ≤ t, r ′ ≥ 1} is minor indeed.
In [21,23] as in [24,16], the presentations of infinitely divisible cascades are based on multiplicative cascades and their generalizations, as in definition 1. In [22], another viewpoint is proposed that evokes random walks through the use of a stochastic integral as in definition 2. To clarify the link between these apparently different approaches, let us note that the integrals in (10) were written integrating first w.r. to t and then w.r. to r. Inverting this order and

0 1 t r t-1 0 1 t r t-1 (a) (b)
Fig. 3. (a) Integrating first w.r. to t, and then w.r. to r: the process is described as a multiplicative cascade; (b) Integrating first w.r. to r, then w.r. to t: the process is described as the exponential of a stochastic integral.

computing the same integrals, we obtain (see figure 3):

m(C ℓ (t)) = t-ℓ t-1 ∞ t-t ′ dr r 2 (t-t ′ ) -1 dt ′ + t t-ℓ ∞ ℓ dr r 2 1/ℓ dt ′ = t-ℓ t-1 (t -t ′ ) -1 dt ′ + 1 (11)
This writing can be used to suggest a rewriting of eq. ( 2) using stochastic integrals. Eq. ( 11) implicitly gives a decomposition of the stochastic measure M (C ℓ (t)) as a sum of successive elementary terms in time (see figure 3) which corresponds to a sum of independent identically distributed normal variables associated to the vertical slices on figure 3(b). This is due to the fact that M is an independent and additive random measure. This way, the use of a 2D random measure is replaced by a 1D stochastic integration. In the Normal case, one has ρ(q) = -µqσ 2 q 2 2 in (1). Thus, from (11), the random variable associated to an elementary slice E of width dt ′ around t ′ has mean

∝ µm(E) and variance ∝ σ 2 m(E) where m(E) = dt ′ /(t-t ′ ) if ℓ ≤ t -t ′ ≤ 1 and m(E) = 1/ℓ if t -t ′ ≤ ℓ.
As a consequence, for some given instant t we get the following new expression for M (C ℓ (t)), written as a stochastic integral (see also Appendix A):

M (C ℓ (t)) µ • m(C ℓ (t)) + σ t-ℓ t-1 (t -t ′ ) -1/2 dB(t ′ ) + σ t t-ℓ ℓ -1/2 dB(t ′ ). ( 12
)
where means "equality in distributions" and m(C ℓ ) is given by (10) when C ℓ = C BM ℓ . Equivalently, this can be written in the form (13) with

M (C ℓ (t)) µ • m(C ℓ (t)) + σ t t-1 K ℓ (t -t ′ ) dB(t ′ ),
K ℓ (τ ) = τ -1/2 , ℓ ≤ τ ≤ 1, ℓ -1/2 , 0 ≤ τ ≤ ℓ, (14)
when C ℓ = C BM ℓ as in (9). Thus, for any instant t, the random variable Q ℓ (t) associated to an infinitely divisible cascade as defined by (2) can be described thanks to some Brownian motion dB(t ′ ) by

Q ℓ (t) e µm(C ℓ ) e (µ+σ 2 /2)m(C ℓ ) exp σ t t-1 K ℓ (t -t ′ ) dB(t ′ ) e -σ 2 2 m(C ℓ ) exp σ t t-1 K ℓ (t -t ′ ) dB(t ′ ) (15)
where e -σ 2 2 m(C ℓ ) is a normalization factor. On the way back to the formulation (4) proposed by Schmitt and Marsan in [20], let ℓ = 1/λ. The change of variable t ′ = t -1 t+1-u in (4) yields:

Qℓ (t) = ǫ λ (t) = ℓ σ 2 /2 exp σ t-ℓ t-1 (t -t ′ ) -1/2 dB(t ′ ) , (16)
where one recognizes exactly the exponential first terms of (12) with

C SM ℓ (t) = {(t ′ , r ′ ) : t -1 ≤ t ′ ≤ t -ℓ, r ′ ≥ t -t ′ }. (17)
Then, m(C SM ℓ ) = log(1/ℓ) and the normalization factor e -σ 2 m(C ℓ )/2 in (15) becomes ℓ σ 2 /2 in (16). The cone C SM ℓ is illustrated on fig. 2(c). The difference between this cone and the cone of figure 2(a) is a thin strip{(t ′ , r ′ ) : tℓ ≤ t ′ ≤ t, r ′ ≥ ℓ} that becomes infinitely thin and negligible as ℓ → 0. We have indeed shown that this formulation is closely related to [START_REF] Schertzer | [END_REF].

Last, note that choosing the cone proposed by Chainais et al. in [24] -see figure 2(b)-yields

M (C CRA ℓ (t)) µ•log(1/ℓ)+σ t-ℓ t-1 1 t -t ′ -1 1/2 dB(t ′ ) + σ t t-ℓ 1 ℓ -1 1/2 dB(t ′ ). ( 18
)
Again, the difference between ( 12) and ( 18) is sensitive for ℓ ≃ 1 only, and asymptotically disappears as ℓ → 0.

It is important to note that the equalities in distribution in equations ( 12) to ( 15) are valid for one given instant t only. Indeed, (15) could not be written simultaneously with the same Brownian motion for two distinct instants t 1 = t 2 . Therefore, Qℓ (t) = ǫ λ (t) in ( 16) and Q ℓ (t) in (15) are not identical processes. One can simply say that, for a given instant t, the random variables Q ℓ (t) and Qℓ (t) are equal in distribution. Something has been "lost" from Q ℓ (t) to Qℓ (t): the independence of the (time, scale) stochastic measure M in scale is no more present in the definition of ǫ λ where only the integration over time remains. Given a Brownian motion dB(t ′ ), the correlations of the process Qℓ (t) are controlled by the kernel K ℓ (τ ). However, it was shown in [22] that the essential multifractal properties have been preserved. In particular, one recovers the characteristic property of multiplicative cascades cov(log Qℓ (0), log Qℓ (τ )) ∼ σ 2 log(1/τ ). This important remark remains valid for all the generalizations considered below.

The stable non-Gaussian case

Previous results generalize easily to the stable case, i.e., for the choice G = S(α, σ, µ, β) [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes[END_REF], provided the law G is asymmetrical with β = -1, this condition being imposed by the fact that we need some moments of the exponential of the stable stochastic process to exist (see Schertzer and Lovejoy [3,[START_REF] Schertzer | [END_REF] and Kida [32]). Then, the function in (1) takes the form ρ(q) = -µq-σ α q α with 0 < α < 2 (α = 1).

The same framework as before gives eq. ( 13) with a Lévy stable measure dL α with parameter 0 ≤ α ≤ 2. From (11), the random variable associated to an elementary slice E of width dt ′ around t ′ has now mean ∝ µm(E) and scale parameter ∝ σ α m(E) where again

m(E) = dt ′ /(t- t ′ ) if ℓ ≤ t -t ′ ≤ 1 and m(E) = 1/ℓ if t -t ′ ≤ ℓ. Therefore,
provides the following expression:

M (C ℓ (t)) µ • m(C ℓ (t)) + σ t t-1 K ℓ (t -t ′ ) dL α (t ′ ), ( 19
)
where the kernel is here:

K ℓ (τ ) = τ -1/α , ℓ ≤ τ ≤ 1, ℓ -1/α , 0 ≤ τ ≤ ℓ. (20)
The choice α = 2 gives again the Gaussian case.

The 2-dimensional case

A generalization of infinitely divisible cascades to D dimensions was proposed in [16]. Examples of realizations in 2D and 3D generated using the algorithms proposed in this reference are shown on figure 1. For pedagogical reasons, we first consider the 2D case in this section before going to D-dimensional generalization in the next section. For D = 2 dimensions we have for

X ∈ R 2 dm(X, r) = C(2) dr r 3 dX (21)
where C( 2) is a constant that will be estimated below, and X = (x, y) is a point in the 2D (x, y) space. The first subsection below deals with the 2D spatial case and the second subsection deals with the causal case where the y dimension is replaced by time t ≡ y.

The two-dimensional spatial case

In the 2D spatial case, the quantity M (C ℓ ) results from an integral over a conical volume defined by:

C ℓ (X) = {(X ′ , r ′ ) : ℓ ≤ X ′ -X ≤ 1, r ′ ≥ X ′ -X } ∪{(X ′ , r ′ ) : X ′ -X ≤ ℓ, r ′ ≥ ℓ}. ( 22
)
as described on figure 4. Here the cone belongs to a 3 dimensional space: 2 dimensions for the vector X = (x, y) and 1 dimension for the scale r. We must also underline that the norm . in the above equation concerns the 2dimensional projection of a point having a 3D position (X, r).

The key quantity to compute is m(C ℓ (X)).

m(C ℓ (X)) = ℓ≤ X ′ -X ≤1 ∞ X ′ -X C(2)dr r 3 dX ′ + X ′ -X ≤ℓ ∞ ℓ C(2)dr r 3 dX ′ (23)
where we have again integrated w.r. to r first as in (11). We then note that the integration over the conic volume can be decomposed into a sum of thin cylindrical tubes. We will therefore use below cylindrical coordinates with ρ = X ′ -X . Let S D the surface of the D dimensional unitary sphere (e.g. 2π in 2 dimensions, 4π in 3 dimensions...). This yields in 2 dimensions:

m(C ℓ (X)) = 1 ℓ ∞ ρ C(2)dr r 3 S 2 ρdρ (24) + ℓ 0 ∞ ℓ C(2)dr r 3 S 2 ρdρ = C(2)S 2 2 log 1 ℓ + 1 2 = log 1 ℓ + 1 2 if C(2)S 2 2 = 1. ( 25
)
which does not depend on X. This last condition prescribes the choice C(2) = 1/π. Then, self-similarity is ensured in the same conditions as in dimension due to the choice of the control measure in (21). By computing the integrals w.r. to r (23), we get from arguments similar to those used in 1 dimension (see also Appendix 2) the following expression for any given position X: [START_REF] Samorodnitsky | Stable Non-Gaussian Random Processes[END_REF] where the new kernel is:

M (C ℓ (X)) µ • m(C ℓ ) + σ X ′ -X ≤1 K (2) ℓ ( X ′ -X ) dL α (X ′ ),
K (2) ℓ (ρ) = 2 -1/α ρ -2/α , for ℓ ≤ ρ ≤ 1, 2 -1/α ℓ -2/α , for ≤ ρ ≤ ℓ, (27)
for 0 < α ≤ 2. For α < 2, dL α is, as before, an α-stable random measure with asymmetric parameter β = -1. Section 4 presents the general version of this result in D dimensions.

3.2

The 2D space-time causal case: X = (x, t)

It was also proposed in [16] to use multi-dimensional infinitely divisible cascades to generate time evolving processes. Here we consider the 2D space-time causal case. The y dimension is now considered as a time variable and denoted t ≡ y so that X = (x, t). Our purpose is to show that the previous construction can be used to build some time evolving scalar field. The resulting field can be seen as a 1D spatial multifractal field obeying some multifractal time evolution. Causality is guaranteed by performing a stochastic integration over a cylinder analogous to the one in previous section but with the further condition t ′ ≤ t (or y ′ ≤ y), see fig. 5. The previous approach applies with only some changes due to the introduction of causality. For instance, the normalizing constant C(2) becomes 2/π in place of 1/π. The previous modifications of the stochastic Fig. 5. Spatial domain of integration corresponding to a stable random measure for a 2D stable process on a space (x, t).

integration for stable random measures still apply, and we obtain here the following expression for any given position X:

M (C ℓ (X)) µ • m(C ℓ ) + σ F K (2) ℓ ( X ′ -X ) dL α (X ′ ), ( 28
) where X = (x, t) and F = {X ′ = (x ′ , t ′ ) : X ′ -X ≤ 1; t ′ ≤ t}: see Fig 5.
The 2D causal kernel is still given by equation [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF]. Section 4 presents the general version of this result in D dimensions.

Then, the origin of temporal, respectively spatial, correlations in the resulting process Qℓ (x, t) can simply be explained geometrically. The correlations are governed by the measure of the intersection of half-hyperspheres in the (x, t) domain. Figure 6(a) illustrates the origin of shared information between successive values of the process at some given position, Qℓ (x, t 1 ) and Qℓ (x, t 2 ). Figure 6(b) shows the origin of spatial correlations between Qℓ (x 1 , t) and Qℓ (x 2 , t) at some given instant t. Figure 6(c) shows the most general situation for different time instants t 1 = t 2 and space positions

x 1 = x 2 .
Such a time dependent positive valued process Qℓ (x, t) exhibits a multifractal behaviour both in time and space. Moreover, it can be considered as more realistic than the approach proposed in [33][34][35]. Indeed, in these works, temporal correlations at large time intervals are controlled by the same quantities that control spatial correlations over small distances. This is inherent to their construction. We emphasize that there is not such a paradox in the present definition: temporal correlations over large time intervals are not dominated by spatial correlations over small distances.

4 The D-dimensional case with D ≥ 2

The spatial case

As shown in [16], the definitions naturally extends from 1 to D > 1 dimensions by using where C(D) is a constant linked to the surface of a hypersphere of dimension D, and chosen to ensure the adequate law. The definition of C ℓ (X), the conical volume in D-dimension is still given by equation (22), where the cone belongs to a dimension D + 1: D dimensions for the vector X and 1 dimension for the scale r. As before, the norm . concerns the D-dimensional projection of a point of position (X, r).

dm(X, r) = C(D) dr r D+1 dX (29)
Computations similar to those of ( 23) & ( 25) performed in D dimensions yield:

m(C ℓ (x)) = 1 ℓ ∞ ρ C(D)dr r D+1 S D ρ D-1 dρ + ℓ 0 ∞ ℓ C(D)dr r D+1 S D ρ D-1 dρ (30) so that C(D) = D/S D = DΓ (D/2)/(2π D/2 ) since S D = 2π D/2 /Γ (D/2)
where Γ is the gamma function. This choice finally yields

ℓ (X)) = log 1 ℓ + 1 D . (31)
which does not depend on X. Therefore, the previous approach for D = 2 for an α-stable stochastic measure generalizes to D dimensions providing an expression with a polar isotropic kernel:

M (C ℓ (X)) µ • m(C ℓ ) + σ X ′ -X ≤1 K (D) ℓ ( X ′ -X ) dL α (X ′ ), (32)
where the new kernel is for 0 < α ≤ 2:

K (D) ℓ (a) = D -1/α ρ -D/α , for ℓ ≤ ρ ≤ 1, D -1/α ℓ -D/α , for 0 ≤ ρ ≤ ℓ. (33)
For D = 1 and D = 2, we recover the kernel described respectively by (20) and [START_REF] Applebaum | Lévy Processes and Stochastic Calculus[END_REF]. As a result, we get a new description of a D-dimensional process Qℓ that is very similar to Q ℓ (X) (see discussion in section 2.3) as defined by ( 2) by using a stochastic integral over the unitary disc {X ′ : X ′ -X ≤ 1} (in D dimension). Interestingly, this can be compared to the heuristic derivation of the Ddimensional stable case presented in [3,[START_REF] Schertzer | [END_REF]. These authors also introduced a power-law integration kernel over some volume, but the exponent was not the same: K(ρ) ≈ ρ -γ with γ = 1 -D + D/α. An important remark concerns the simulation of logstable multiplicative cascades. They are usually rather painful to simulate because they do not belong to the family of compound Poisson cascades, see [24]. The results presented here and eq. ( 32) in particular give us a much more simple way to simulate not exactly identical but very similar processes. One must first simulate the independent random measure dL α (X), which remains to simulate a set of i.i.d. stable variables over some regular discrete sampling of the X space. Then, one simply uses eq. ( 32) to get M (C ℓ (X)). Taking the exponential yields Qℓ (X).

Once again, it is important to note that equation (32) holds for one given position X only to which dL α (X ′ ) is indeed associated. Therefore, we emphasize that the process Qℓ (X) obtained by taking the exponential of ( 32) is not identical to Q ℓ (X) as defined by [START_REF] Schertzer | [END_REF]. However, as already mentioned in section 2.3, the essential multifractal properties are preserved.

The causal spatio-temporal case

We come back to the use of D-dimensional infinitely divisible cascades to build time evolving processes as already evoked in section 3.2 and introduced in [16]. Some "multifractal films" can be synthesized this way 2 . In this spirit, the same approach as above for 2-dimensional causal cascades can be easily generalized to D dimensions by using X = (x, t). Again, we will use half of a cone C ℓ as defined above: the half for which t ′ ≤ t. The resulting field can be seen as a D -1-dimensional multifractal field, with a time evolution. The previous simplifications of the random integration for stable random measures still apply, and we obtain the following final expression: (34) where X = (x, t) and F = {X ′ = (x ′ , t ′ ) : (x ′ , t ′ ) -(x, t) ≤ 1; t ′ ≤ t}. The D-dim causal kernel is still given by equation (33).

M (C ℓ (X)) µ • m(C ℓ ) + σ F K (D) ℓ ( X ′ -X ) dL α (X ′ ),
The comments on figure 6 are still relevant in this general context. As a consequence, this construction inherits some nice geometrical understanding. Such a description makes explicit:

the stationarity of the process at fixed position x o ; -a causal time evolution ; -a description in terms of a stochastic integral with respect to a stable noise dL α ; -a correspondence between this description where time and space variables are explicitely distinguished and the global description exp M (C ℓ (x, t)) where x ∈ R d and t ∈ R are indeed merged.

Such a construction can be compared to the fractionally integrated fields described in [3] and later extended to causal (D+1) space-time multifractal processes in [36]. In the latter, a Fourier approach was proposed to impose the fundamental property cov(log ǫ λ (0), log ǫ λ (x)) ∼ σ 2 log(1/ x ) thanks to k -1 filtering in the Fourier domain. The resulting processes are causal, but, due to the use of a Fourier transform, they could not be synthesized in a causal manner. Moreover, such a Fourier approach is memory demanding. The definitions in terms of stochastic integrals proposed here overcome such drawbacks and provide an explicit description directly in the space-time domain.

Moreover, note that definition (34) could be formulated in the framework of Generalized Scale Invariance [37,[START_REF] Pecknold | Cellular Automata: Prospects in astrophysical applications[END_REF] as well. This framework permits to introduce some controled anisotropy by replacing the classical self-similarity property in D + 1 dimensions by a matrix operator selfsimilarity. For instance, this can be achieved by using (see equation ( 44) in [36])

(x, t) = (|x| d el -D/α + |t| d el -D/α 1-H ) 1 d el -D/α
where H characterizes the anisotropy (H = 0 in the isotropic case) and d el = D + 1 -H. This could be helpful to understand better how large-scale anisotropy in turbulent flows have influence at small scales. Indeed, in Kolmogorov's homogeneous turbulence framework, it is often assumed that the cascade processes "washes" the details of the large-scale flow, so that after few cascade steps, small scales become locally isotropic. In fact experimental results indicate that this may not be true, and the present approach may provide a theoretical framework to develop continuous anisotropic and scaling models.

Conclusion and discussion

In this paper, we have first given a consistent presentation of various definitions and presentations of stable cascades, a special family of infinitely divisible cascades, in 1 dimension. To this aim, we have used both the multiplicative cascade viewpoint as in [20,21,23,24,16] and the stochastic integral viewpoint as in [22]. We have made explicit the differences between definitions mainly in terms of the chosen cone C ℓ when using the multiplicative cascade approach or in terms of the chosen integration domain when using the stochastic integral approach. In the latter, the integration kernel has been identified. We have also clarified the similarities and differences between the definitions based respectively on multiplicative cascades and on stochastic integrals.

We have also considered a D dimensional generalization, showing that the simplification arising in the 1D case for stable random measures can also be introduced in D-dimension, using cylindrical coordinates. We have provided explicit expressions, using an integration over a D-dimensional space, with a kernel whose general expression was provided. We have first given the 2D case before generalizing straightforwardly to D ≥ 2. In each case we have also considered the spatio-temporal situation where X = (x, t), which is built using the same kernel as for the purely spatial situation; only the integration domain is different, to ensure causality.

A general formulation in terms of a stochastic integral with respect to a Lévy noise allowed us to make clear the fundamental properties of such fields: multifractality both in time and space, homogeneity, stationarity and causality. The extension to the framework of generalized scale invariance as in [36] is considered. Such a time dependent positive valued process Qℓ (x, t) exhibits a multifractal behaviour both in time and space. Moreover, it can be considered as more realistic than the approach proposed in [33][34][35], since in these works, the same quantities control both spatial correlations over small distances and temporal correlations over large time intervals. We emphasize that this is not the case in our definition and that the correlation function receives a quite intuitive geometrical interpretation (see fig. 6).

Finally, an important remark concerns the simulation of log-stable multiplicative cascades. They are usually rather painful to simulate because they do not belong to the family of compound Poisson cascades and cannot be obtained from a simple marked point Poisson process. The present results provide a much more simple way to the simulation of very similar α-stable processes. The framework presented here corresponds to D-dimensional scalar multifractal processes. This is yet useful for practical applications. However this cannot be directly used for the simulation of real flows, since turbulence is fully tensorial. For real flow situations, one would need to generalize this continuous scalar framework to continuous multifractal tensorial processes. The corresponding theoretical framework is still to be developed. If turbulent cascade models could be adapted to vectorial or tensorial frameworks, it could lead to much more reliable predictive models for in- dustrial flows, than the eddy-viscosity models, which are usually only able to correctly predict moments of order 1 and 2.

Fig. 1 .

 1 Fig. 1. Examples of realizations of infinitely divisible cascades in 1, 2 and 3 dimensions respectively.

Fig. 2 .

 2 Fig. 2. Cone C ℓ (t) chosen by (a) Bacry & Muzy [21], (b) Chainais, Riedi & Abry [24]; (c) Schmitt and Marsan in [20] once the correspondence is established (see text).

1 AFig. 4 .

 14 Fig. 4. Cone C ℓ (x = (x, y)) for D = 2 decomposed in two parts A ℓ (x) and B ℓ (x).

Fig. 6 .

 6 Fig.6. Illustration of the origin of (a) temporal, respectively (b) spatial, correlations in the resulting process Qℓ (x, t). The correlations can be described in terms of the measure of the intersection of half-hyperspheres in the (x, t) domain.

Fig. 7 .

 7 Fig. 7. Vertical plate denoted T (r0, a, b) in the text, in the (t, r) space.

See http://www.isima.fr/∼chainais/PUB/software.html; an example of a multifractal film, obtained by simulating a 2+1 dimensional field and considering 2 spatial dimensions and 1 time dimension.

A Stability property for random measures on thin strips: the 1D case

This appendix is aimed at giving a hint to the computations used to link the stable stochastic measure of some set to some sotchastic integral with respect to some stable noise in dimension 1.

We consider here a simple case, illustrated by Figure 7. We denote this set for b > a, T (r 0 , a, b) = {(t ′ , r ′ ) : a ≤ t ′ ≤ b; r 0 ≤ r ′ }. We have m(T (r 0 , a, b)) = (ba)/r 0 . And we consider here a Gaussian process with ρ(q) = -1 2 q 2 , so that:

Let us note here ω a Gaussian random measure of variance σ and integrated on an interval (a, b):

We then have:

(37) Comparing equations (35) and (37), we see that the stable random variable M (T (r 0 , a, b)), which is a random measure on a 2D space, is then equal in distribution to a 1D process, following the stability property. This gives the following relation for a Gaussian random measure:

where means "equality in distribution". Applying this relation to the thin plate T (r 0 , t, t+dt) gives the following:

For the stable case of basic index 0 ≤ α ≤ 2, we get:

B Stability property for random measures on a cylindrical corona: the 2D case

We consider here the case of a void cylinder, illustrated by Figure 8. We denote this set for b > a, V C(r

On one hand, the measure of this set is given by:

where C(2) is the normalizing constant in 2 dimensions. We consider here a Gaussian process M (V C(r 0 , X 0 , a, b)) depending on X 0 and such that

(42) On the other hand, let us note here ω a Gaussian random measure of variance σ and integrated over a cylindrical corona of inner radius a and outer radius b, centered at point

Comparing equations ( 42) and (43), we see that the random variable M (V C(r 0 , X 0 , a, b)), which is a random measure on a 3D volume, is then equal in distribution to a random measure on a 2D surface, with variance σ = ( √ 2r 0 ) -1 . This gives the following relation:

For the stable case of basic index 0 ≤ α ≤ 2, we have:

C Stability property for random measures on hypercylindrical coronas: the D-dimensional case

The previous approach for dimension 2 can be generalized for D dimension. In this case, the only difference is the introduction of the term r -D-1 instead of r -3 in the integral of eq. ( 41) We still consider a void cylinder V C(r 0 , X 0 , a, b) = {(X ′ , r ′ ) : a ≤ X ′ -X 0 ≤ b; r 0 ≤ r ′ } and provide here directly the stable case of index α:

M (V C(r 0 , X 0 , a, b)) Dr D 0 1/α M (A(X 0 , a, b)) (46)

The author gratefully acknowledge Nicolas Perpete for stimulating discussions and comments.


Fatma Fezai

Childérick Séverac

Pierre Gros

Martine Meireles

David Evrard

A new sensor with increased lifetime based on a mixed diazonium thick film/gold nanoparticles interface for Hg(II) trace detection

A new sensor with increased lifetime based on a mixed diazonium thick film/gold nanoparticles interface for Hg(II) trace detection.


Brian M De Silva

Kathleen Champion

Markus Quade

Jean-Christophe Loiseau

J Nathan Kutz

Steven L Brunton

PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data

. Maximizing the impact of these model discovery methods requires tools to make them widely accessible to scientists across domains and at various levels of mathematical expertise.

PySINDy is a Python package for the discovery of governing dynamical systems models from data. In particular, PySINDy provides tools for applying the SINDy approach to model discovery [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF]. Given data in the form of state measurements x(t) ∈ R n , the SINDy method seeks a function f such that

d dt x(t) = f (x(t)).
SINDy poses this model discovery as a sparse regression problem, wherein relevant terms in f are selected from a library of candidate functions. Thus, SINDy models balance accuracy and efficiency, resulting in parsimonious models that avoid overfitting while remaining interpretable and generalizable. This approach is straightforward to understand and can be readily customized using different sparse regression algorithms or library functions.

The PySINDy package is aimed at researchers and practitioners alike, enabling anyone with access to measurement data to engage in scientific model discovery. The package is designed to be accessible to inexperienced practitioners, while also including options that allow more advanced users to customize it to their needs. A number of popular SINDy variants are implemented, but PySINDy is also designed to enable further extensions for research and experimentation. The package follows object-oriented design and is scikit-learn compatible.

The SINDy method has been widely applied for model identification in applications such as chemical reaction dynamics [START_REF] Hoffmann | Reactive SINDy: Discovering governing reactions from concentration data[END_REF], nonlinear optics [START_REF] Sorokina | Sparse identification for nonlinear optical communication systems: SINO method[END_REF], thermal fluids [START_REF] Loiseau | Data-driven modeling of the chaotic thermal convection in an annular thermosyphon[END_REF], plasma convection [START_REF] Dam | Sparse identification of a predator-prey system from simulation data of a convection model[END_REF], numerical algorithms [START_REF] Thaler | Sparse identification of truncation errors[END_REF], and structural modeling [START_REF] Lai | Sparse structural system identification method for nonlinear dynamic systems with hysteresis/inelastic behavior[END_REF]. It has also been extended to handle more complex modeling scenarios such as partial differential equations [START_REF] Rudy | Data-driven discovery of partial differential equations[END_REF][START_REF] Schaeffer | Learning partial differential equations via data discovery and sparse optimization[END_REF], systems with inputs or control [START_REF] Kaiser | Sparse identification of nonlinear dynamics for model predictive control in the low-data limit[END_REF], corrupt or limited data [START_REF] Schaeffer | Extracting sparse high-dimensional dynamics from limited data[END_REF][START_REF] Tran | Exact recovery of chaotic systems from highly corrupted data[END_REF], integral formulations [START_REF] Reinbold | Using noisy or incomplete data to discover models of spatiotemporal dynamics[END_REF][START_REF] Schaeffer | Sparse model selection via integral terms[END_REF], physical constraints [START_REF] Loiseau | Constrained sparse Galerkin regression[END_REF], tensor representations [START_REF] Gelß | Multidimensional approximation of nonlinear dynamical systems[END_REF], and stochastic systems [START_REF] Boninsegna | Sparse learning of stochastic dynamical equations[END_REF]. However, there is not a definitive standard implementation or package for applying SINDy. Versions of SINDy have been implemented within larger projects such as sparsereg [START_REF] Quade | Sparsereg -collection of modern sparse regression algorithms[END_REF], but no specific implementation has emerged as the most widely adopted and most versions implement only a limited set of features. Researchers have thus typically written their own implementations, resulting in duplicated effort and a lack of standardization. This not only makes it more difficult to apply SINDy to scientific data sets, but also makes it more challenging to benchmark extensions to the method against the original and makes such extensions less accessible to end users. The PySINDy package provides a dedicated central codebase where many of the basic SINDy features are implemented, allowing for easy use and standardization. This also makes it straightforward for users to extend the package in a way such that new developments are available to a wider user base.

Features

The core object in the PySINDy package is the SINDy model class, which is implemented as a scikit-learn estimator. This design was chosen to make the package simple to use for a wide user base, as many potential users will be familiar with scikit-learn. It also expresses the SINDy model object at the appropriate level of abstraction so that users can embed it into more complicated pipelines in scikit-learn, such as tools for parameter tuning and model selection.

Applying SINDy involves making several modeling decisions, namely: which numerical differentiation method is used, which functions make up the feature library, and which sparse regression algorithm is applied to learn the model. The core SINDy object uses a set of default options but can be easily customized using a number of common approaches implemented in PySINDy. The package provides a few standard options for numerical differentiation (finite difference and smoothed finite difference), feature libraries (polynomial and Fourier libraries, as well as a class for creating custom libraries), and sparse regression techniques (sequentially thresholded least squares [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF], LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], and sparse relaxed regularized regression [START_REF] Zheng | A unified framework for sparse relaxed regularized regression: SR3[END_REF]). Users can also create their own differentiation, sparse regression, or feature library objects for further customization.

The software package includes tutorials in the form of Jupyter notebooks. These tutorials demonstrate the usage of various features in the package and reproduce the examples from the original SINDy paper [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF].

Acknowledgments

This project is a fork of sparsereg (Quade, 2018). SLB acknowledges funding support from the Air Force Office of Scientific Research (AFOSR FA9550-18-1-0200) and the Army Research Office (ARO W911NF-19-1-0045). JNK acknowledges support from the Air Force Office of Scientific Research (AFOSR FA9550-17-1-0329). This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant Number DGE-1256082.


Peter Adeniyi Alaba

Ching Shya Lee

Faisal Abnisa

Mohamed Kheireddine Aroua

Yousif Abdalla Abakr

Mustapha Danladi

Patrick Cognet

Yolande Peres-Lucchese

Wan Mohd

Wan Daud

Adeniyi

Faisal Abnisa Shya

Mohamed Kheireddine

Ibrahim Abdalla

Yousif Abdalla Abakr8

Yolande Pérès

Investigating the electrocatalytic oxidation of glycerol on simultaneous nitrogen-and fluorine-doped on activated carbon black composite

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

ABSTR AC T

To develop non-metallic electrocatalyst for glycerol electrooxidation, simultaneous co-doping of nitr og en and fluorine into activated carbon black (ACB) composite was explored to investigate the physical and electro chemical characteristics. The ACB was prepared by mixing activated carbon and carbon black. The N and F were incorporated using aniline and polytetrafluoroethylene as the precursors. The morphol og ies of the prepared samples were analyzed and the electrochemical behavior, as well as the electrocatalytic performance, was in vestigated in acid and alkaline environrnent. Porosity analysis sho11vs that 20% N and F co-doped ACB (ACB N 2 F 2 ) reduced the surface area (491.64 m 2 g-1 ) and increased the electroactive surface area, which could contribute to faster mass transport and electron transfer process to enhance the catalytic activity the electrode.

The doping defect also reduced the charge transfer resistance, 11vhich could increase the spin densities and maximize charge re-distribution to generate more electroactive surface. The electrodes N-doped ACB (ACB-Ni) and ACB-N2 F 2 exhibited a remarlcable electrocatalytic activity in alkaline medium, 11vhile only 30% N and F co doped ACB electrode are suitable in acidic medium. Moreover, the catalyst displayed remarkable long-term stability/durability in alkaline environment.

With the progress in advanced materials technolog y , development of electrocatalysts for alcohol electrooxidation is one of the big issues towards electrochernical energy conversion, value-added chemical production, and storage devices, like metal-air batteries, anion ex change membrane fuel cell (AEMFC) and polymer electrolyte fuel cells (PEFCs) [START_REF] Cheng | Metal-air batteries: from oxygen reduction electrochemistry to cathode catalysts[END_REF][START_REF] Liang | Strongly coupled inorganic/nanocarbon hybrid materials for advanced electrocatalysis[END_REF][START_REF] Liu | Anion exchange membranes with well-developed conductive channels: effect of the functional groups[END_REF]. Thus far, Pt-based materials are the most popularly utilized electrocatalyst for alcohol oxidation with high exchange cur rent density [START_REF] Gasteiger | Activity benchmarks and requirements for Pt, Pt-alloy, and non-Pt oxygen reduction catalysts for PEMFCs[END_REF]. Nevertheless, large scale application of Pt-based cat alysts for production of value-added chemicals, AEMFCs and PEFCs is prevented by scarcity, high cost, poisoning by carbonaceous reaction intermediate species and CO contaminants of the H2 fuel are the major limitations of Pt catalysts [START_REF] Wu | Co/Co3O4/C-N, a novel nanostructure and excellent catalytic system for the oxygen reduction reaction[END_REF]. Consequently, rational design of cost effective Pt-free catalysts with high alcohol electrooxidation activity constitutes a great challenge. Therefore, several efforts have been channeled towards replacing Pt-based electrocatalysts in alcohol elec trooxidations and fuel cell sciences [START_REF] Liang | Oxygen reduction electrocatalyst based on strongly coupled cobalt oxide nanocrystals and carbon nanotubes[END_REF][START_REF] Zhang | CeO2 promoted Au/C catalyst for Scheme 1. Reaction mechanism proposed for the electrooxidation of glycerol on ACBN 2 F 2 . glycerol electro-oxidation in alkaline medium[END_REF]. Cheap earth-abundant metals like transition metal oxides, perovskites, and spinel based materials have been fabricated with the aim of achieving extremely active and durable non-precious metal as alternative electrocatalysts [8-10]. For instance, several authors have studied the transition metal oxides

Table 1

The textural parameters of ACB and heteroatom doped ACB.

Samples

SBET (m 2 g -1 )

Ex-SBET (m 2 g -1 )

PSD (nm) V micro (cm 3 g -1 )

V total (cm 3 g - supported by carbon materials like graphene, Activated carbon and Multiwalled carbon nanotubes (MWCNTs) [START_REF] Yang | Boron-doped carbon nanotubes as metal-free electrocatalysts for the oxygen reduction reaction[END_REF][START_REF] Gupta | Highly purified CNTs: an exceedingly efficient catalyst support for PEM fuel cell[END_REF][START_REF] Kruusenberg | Cobalt-nitrogen Co-doped carbon nanotube cathode catalyst for alkaline membrane fuel cells[END_REF][START_REF] Amyab | Platinum nanoparticles with superaciddoped polyvinylpyrrolidone coated carbon nanotubes: electrocatalyst for oxygen reduction reaction in high-temperature proton exchange membrane fuel cell[END_REF][START_REF] Beltrán-Gastélum | Evaluation of PtAu/MWCNT (multiwalled carbon nanotubes) electrocatalyst performance as cathode of a proton exchange membrane fuel cell[END_REF]. Their reports show that carbon materials are viable support for non-precious metal catalysts since they possess high chemical and mechanical stability, inherent elasticity, corrosion resistance, surface area and outstanding conductivity. Although Hummers' methods or/and chemical vapor deposition (CVD) are generally implemented for the synthesis of graphene oxide/graphene sheets. The feasibility of graphene-based materials for various industrial applications is limited because of high cost and environmental concerns since hazardous reagents are used. On the other hand, MWCNT is easy to prepare, and their geometrical and morphological features are tunable via well-established and adaptable preparation procedures. Moreover, Weber et al. [START_REF] Weber | Enhanced catalytic glycerol oxidation activity enabled by activated-carbon-supported palladium catalysts prepared through atomic layer deposition[END_REF] utilized activated carbon as a support for Pd using Atomic Layer Deposition and reported a remarkable performance towards glycerol electrooxidation.

Lately, carbon matrix functionalized with heteroatoms like fluorine (F) phosphorous (P), sulfur (S), boron (B) and nitrogen (N) individually or in combination has been examined as a promising electrocatalyst capable of replacing the Pt-based [START_REF] Weber | Enhanced electrocatalytic performance triggered by atomically bridged boron nitride between palladium nanoparticles and carbon fibers in gas-diffusion electrodes[END_REF]. Prominent of all the heteroatom doped carbon electrocatalysts is N doped carbon, which displays better activity. For instance, several authors have reported that the catalytic activity of N doped carbon during ORR is on par with conventional Pt/C in alkaline medium [START_REF] Li | An oxygen reduction electrocatalyst based on carbon nanotube-graphene complexes[END_REF][START_REF] Chen | A nitrogen-doped graphene/carbon nanotube nanocomposite with synergistically enhanced electrochemical activity[END_REF][START_REF] Chen | Nitrogendoped carbon nanocages as efficient metal-free electrocatalysts for oxygen reduction reaction[END_REF]. However, N doped carbon exhibits much lower catalytic performance in comparison with P t/C in acidic solutions. Furthermore, simultaneous co-doped N and F into carbon matrix are proposed to induce maximum charge delocalization on the adjacent carbon atoms. This significantly improves adsorption of oxygen and its subsequent reduction than the single N doped carbon [START_REF] Sun | Fluorine-doped carbon blacks: highly efficient metal-free electrocatalysts for oxygen reduction reaction[END_REF][START_REF] Sun | Fluorine-doped BP 2000: highly efficient metal-free electrocatalysts for acidic oxygen reduction reaction with superlow H2O2 yield[END_REF]. The simultaneous doping of N and F greatly increased the bond distance of O 2 due to its superior electron density and more polarization in the CeN, CeF, and CeC bonds helped to increase the kinetics of ORR. Moreover, Wood et al. [START_REF] Wood | Effect of halide-modified model carbon supports on catalyst stability[END_REF] and Peera et al. [START_REF] Peera | Nitrogen and fluorine co-doped graphite nanofibers as high durable oxygen reduction catalyst in acidic media for polymer electrolyte fuel cells[END_REF] reported that F doping also enhanced the long-time stability of N-doped carbons in acidic solutions. Recently, P eera et al. [START_REF] Peera | Nitrogen and fluorine co-doped graphite nanofibers as high durable oxygen reduction catalyst in acidic media for polymer electrolyte fuel cells[END_REF] reported the synthesis graphite nanofibers (N-F/GNFs) simultaneously co-doped N and F. The synergistic effect of N and F on the morphology of N-F/GNFs and catalytic activity in acidic media during ORR were studied. They observed that the interlayer spacing between the GNF layers increases with increasing electronegativity of F when combined with N. Moreover, structural transformation occurred to the crumpled graphene sheets with several openedge active sites, which facilitates a remarkable catalytic activity during ORR. This level of catalytic activity is not achievable with single doping on GNF matrix using N [START_REF] Peera | Nitrogen and fluorine co-doped graphite nanofibers as high durable oxygen reduction catalyst in acidic media for polymer electrolyte fuel cells[END_REF].

With this impetus, this study explored, simultaneous co-doping of N and F into activated carbon composite by a simple blending, and heat treatment method and investigated the morphological evolution. The electrocatalytic activity during glycerol oxidation was studied. The effect of N and F co-doping on the maximum charge delocalization on the carbon matrix due to electronegativity difference between N and F and the interlayer spacing between the sample layers were studied.

Materials and methods

Materials 2.4. Physicochemical characterizations

X-ray diffractometer (Seifert X-ray Diffractometer JSO 2002) was used to record the XRD pattern for all the prepared heteroatom doped carbon composite using nickel filtered Cu Kα radiation (λ = 1.544 A) ranging from 5.018 to 69.966°(2θ) and 0.026°as the step size. The Nitrogen adsorption/desorption analysis was performed with the aid of surface area and porosity analyzer (Micrometrics ASAP 2020) under -----------------, --ACB ---------------- analysis bath Temperature of 77.350 K as described by [START_REF] Alaba | Synthesis and characterization of hierarchical nanoporous HY zeolites from acid-activated kaolin[END_REF][START_REF] Alaba | Synthesis and application of hierarchical mesoporous HZSM-5 for biodiesel production from shea butter[END_REF]. Perkin Elmer Spectrum RX Fr-IR was employed for the infrared spectroscopy (IR) to check the existing functional groups in the prepared heteroatom doped carbon composite of 77.350 K described by [START_REF] Alaba | Synthesis of hierarchical nanoporous HY zeolites from activated kaolin, a central composite design optimization study[END_REF]. Field Emission Scanning electron microscope/Energy Dispersive X-ray (FESEM/EDX), FEI Quanta 400 FE-SEM as the accelerating voltage to check the mor phology of the prepared heteroatom doped carbon composite is 20 kV. Prior to the scanning, the samples were coated with gold to enhance the electrical conductivity as described by Alaba et al. [START_REF] Alaba | Synthesis and characterization of sulfated hierarchical nanoporous faujasite zeolite for efficient transesterification of shea butter[END_REF].

Electrochemical measiuements

To evaluate the catalytic activities of these electrocatalysts, cyclic voltammograms analysis was conducted in an electrolyte comprising 1.0 M KOH and 1.0 M glycerol at a potential range of -0.5 to 1 V (vs SHE) using 50 m Vs -l as the sweeping rate. Prior to the electrochemical measurements, the catalyst pastes were prepared by blending the composite samples with para.ffin oil (70:30, w%) in a mortar and pestle for 30 min. About 25 mg of the paste was compactly filled into a PTFE Teflon rod ( 4.5 mm internai diameter with 2 mm depth). A 3 mm stainless rod was connected to the paste through the other end of the rod. Ali the electrochemical tests were performed using BioLogic Science instrument with EC-Lab Software. The electroactive surface area (ECSA) of anodes was determined using the Randles-Sevcik for mula [START_REF] Alaba | Synthesis and characterization of sulfated hierarchical nanoporous faujasite zeolite for efficient transesterification of shea butter[END_REF] l p = 2.69 X 10 5 X AD 1 1 2 n 31 2y 1l2C where n is the number of the electrons taking part in the oxidation reaction, A is the electrode surface area of the ( cm 2 ), y is the scan rate of the potential perturbation (Vs-1 ), Di s the diffusion coefficient of the molecules in the electrolyte ( cm 2 •sl ), and C is the concentration of the probe molecule in the bulk solution (mol cm-3 ). -----------

Results and discussion

Characterization of NF-ACC nanocomposites

Electrochemical characterization

The value ECSA was determined using the Randles-Sevcik formula and 12.42, 122.53, 13.72, 146.64 , and 31.95 cm 2 g-1 were obtained for ACB, ACB-N2, ACB-N1F1, ACB-N2F2, and ACB-N3F3. These results reveal that rational heteroatom doping can significantly boost the ECSA value of ACB composite electrocatalyst. The best ECSA is exhibited by ACB N2F2, which is over 10 times higher than that of ACB.

The nove! electrodes were characterized using consecutive CV be tween -0. 5 and 1.0Vat 50mvs-1 for l0cycles was performed in 1 M KOH solution (Fig. 4). Fig. 4

Electrooxidation of glycerol

Electrooxidation of glycerol was studied in a CV experiment be tween -0. 5 and 1.0V at 50mvs-1 using 1 M KOH as the supporting electrolyte. Fig. 6 presents the CV results of the electrode performance with and without 1 M glycerol. The addition of 1 M glycerol to the al kaline solution significantly boosts the electrochemical response of ACB-N2 and ACB-N2F2 electrodes. lt is noteworthy that based on a specific activity (mNcm 2 ), ACB-N2 exhibits the highest glycerol oxi dation activity (Fig. 6b), while based on mass activity (mNmg), ACB-4.. .------------------, § C"' 0.03 � 1000 500 0 promote glycerol oxidation [START_REF] Liu | Influence of Iron source type on the electrocatalytic activity toward oxygen reduction reaction in Fe-N/C for Al-air batteries[END_REF][START_REF] Zhan | Non noble metal catalyst for oxygen reduction reaction and its characterization by simulated fuel cell test[END_REF][START_REF] Li | Electrocatalytic activity and design principles of heteroatom-doped graphene catalysts for oxygen-reduction reaction[END_REF].

3 (a ) / 1 • • • • • • • • ACB-N3F3 --ACB . /! è 2 •••••••• ACB-NIFI --ACB-N2F2 / j • � •••••••• ACB
••••••••• ACB-N3F3 --ACB g 0.02 •••••••••ACB-NIFI --ACB-N2F2 è :> -• -ACB-N2 •,ai
• ••• • • • • • eACB • • • • • • • • 0 0 0 • • • • � . • 5000 10000 15000 20000 Z'/ ohm eACBN IFI • •• • • • • • • l. • 20000 40000 Z'/ohm • • • 60000 80000 eACBN3F3 • • • • • • •• • • • •
The linear sweep voltammetry results also confinn the observation regarding the specific activity and mass activity of ACB-N 2 and ACB N 2 F 2 electrodes (Fig. S3). The glycerol oxidation activity of the elec trodes was also tested in the presence of 1 M H 2 S0 4 . The performance of the electrodes in 1 M H 2 S0 4 and 1 M glycerol + 1 M H 2 S0 4 were pre sented in Fig. 6 (c) and(d). Of ail the electrodes, only ACB-N:if 3 exhibits a little significant electrocatalytic activity towards glycerol oxidation. Furthermore, ACBN 2 and ACBN 2 F 2 showed an immense performance improvement towards glycerol electrooxidation when compared with ACB. The improved activity could be ascribed to the defect created on the carbon structure by the optimal heteroatom doping, which improved the hydrophilicity of its surface with an increase in the hydrophilic functional groups, thereby promoting the mass transfer between the electrolyte and the heteroatom doped ACB composites [START_REF] Le | Nitrogen-doped graphitized carbon electrodes for biorefractory pollutant removal[END_REF]. Furthermore, heteroatom doping supplies additional electron to the ACB substrates readily, meaning that the resistance to the flow of electron is minimized [START_REF] Zheng | High lithium anodic performance of highly nitrogendoped porous carbon prepared from a metal-organic framework[END_REF]. The expected products of glycerol electrooxidation using ACBN 2 F 2 include valuable compounds like glycerate, dihydroxyacetone, tartronate, mesoxalate, etc. (Scheme 1), which could effectively minimize the costs of production of H 2 [START_REF] Weber | Enhanced catalytic glycerol oxidation activity enabled by activated-carbon-supported palladium catalysts prepared through atomic layer deposition[END_REF].

Electrocatalytic performance using chronoamperomet,y

Electrochemical impedance spectroscopy (EIS)

Conclusion

Nitrogen-and fluorine-doped ACB composite were successfully synthesized using aniline and PTFE as precursors in the presence of 1,3 propanediol and distill water. Previously, heteroatom doped carbon materials have been successfully used for oxygen reduction reaction. We have revealed that rationally designed nitrogen and fluorine simultaneously doped ACB composite is suitable for glycerol oxidation. The results of the XRD, FTIR BET, and FESEM of the doped samples show that the doping was successful, and the doping effect reflects in the electrochemical behavior. The sample ACB-N 2 F 2 has the lowest SBET (491 m 2 g -1 ) and the highest electroactive surface area (146.64 cm 2 g -1 ), which could contribute to faster mass transport and electron transfer process to enhance the catalytic activity. The doping defect also reduced the charge transfer resistance, which could increase the spin densities and maximize charge re-distribution to produce the highest number electroactive sites.

ACB-N 2 and ACB-N 2 F 2 electrodes exhibit a remarkable activity in alkaline medium, while only the ACB-N 3 F 3 electrode is suitable in acidic medium. The superior mass activity of ACB-N 2 F 2 is due to high electronegativity difference of nitrogen and fluorine, which can synergistically tune the electronic structure to favor glycerol oxidation reaction. The activity order is ACB < ACB-N 1 F 1 < ACB-N 3 F 3 < ACB-N 2 < ACB-N 2 F 2 in the alkaline environment, while the activity order is ACB-N 3 F 3 > ACB-N 2 F 2 in acidic environment. These results show that ACB-N 2 F 2 is the best electrocatalyst and alkaline media is the most suitable for glycerol oxidation, using heteroatom-doped carbon composite.

Fig. 1 .

 1 Fig. 1. N 2 adsorption/desorption isotherms (a) and the corresponding pore size distributions (b) of ACB and heteroatom doped ACB.

Fig. 2 .

 2 Fig. 2. FESEM images of (a) ACB; (b) ACB-N 2 ; (c) ACB-N 1 F 1 ; (d) ACB-N 2 F 2 ; (e) ACB-N 3 F 3 .

  -N2 68 +----�----�----

Fig. 3 .

 3 Fig. 3. Ff-m spectra of ACB and heteroatom doped ACB.

Fig. 1

 1 Fig. 1 presents the N2 adsorption/desorption isotherms and the corresponding pore size distributions of ACB and heteroatom doped ACB. Ali the samples exhibit a typical type IV isotherm with an obvious

1 E 1 - 2 Fig. 4 .Potential

 1124 Fig. 4. A se ries of CVs during 10 consecutive potential cycles of (a) ACB; (b) ACB-N2; (c) ACB-N1F1; (d) ACB-N2F2; (e) ACB-N3F3 electrode in a 1.0M KOH solution between -0.5 and 1.0V at o = somvs -1 .

  reveals the peak current density of the forward scan increases with an increasing number of cycles, while that of the backward scan shifts towards the negative current. Doping of the heteroatoms improves the oxygen evolution potential of the activated carbon composite material. Of ail the doped carbon materials, ACB-N2 shows the highest activity followed by ACB-N2F2 and ACB-N3F3, while ACB-N1F1 has the lowest activity in a solution containing 1 M of KOH. A different set of electrodes was used in a solution containing 1 M of H 2 SO 4 • The pH of the electrolyte solution l arg ely influences the redox behavior of the film [34,35]. Fig. 5 reveals a comparative illustration of the redox activity in both acidic and alkaline supporting electrolytes. The alkaline environment is a more promising redox electrode. This is contrary to other reports of carbon paste electrodes [35]. While the electrodes prove to be bifunctional in alkaline solution, it is only pro mising for oxygen reduction reaction (ORR) in acidic medium, except for ACB-N 3 F 3 , which shows a well-defined oxidation peak.

u 2 - 1 -Fig. 6 .

 216 Fig. 6. Electrochemical response of the electrodes in (a) 1 M KOH; (b) 1 M glycerol + 1 M KOH; (c) 1 M H2SO.; (d) 1 M glycerol + 1 M H 2 SO •. 0.03 (a) 0.025 e 0.02

Fig. 7 .Fig. 8 .

 78 Fig. 7. Mass activity of the electrodes in (a) 1 M KOH; (b) 1 M glycerol + 1 M KOH.

Fig. 9 .

 9 Fig. 9. Nyquist plots of the carbon composite electrocatalysts operated in 1 M glycerol + 1 M KOH solution at the potential of -0.1 V.

Fig. 8

 8 Fig. 8 presents the chronoamperometric measurements of the elec trodes in 1 M glycerol + 1 M KOH and 1 M glycerol + 1 M H2S04. This measurement reveals the electrochemical activity and stability of the electrodes. Initially, the conversion of glycerol occurred rapidly in both acidic and alkaline environment and then reach a constant state for ail the electrodes. A higher current density at the end of the experiment indicates better catalytic performance. The activity order is ACB < ACB-N1F1 < ACB-N 3 F 3 < ACB-N 2 < ACB-N 2 F 2 in the alkaline en vironment, while the activity order is ACB-N:if 3 > ACB-N 2 F 2 in acidic environment. These results show that ACB-N2F2 is the best electro catalyst and alkaline media is the most suitable for glycerol oxidation, using heteroatom-doped activated carbon black composite. lt is also

Fig. 9

 9 Fig. 9 presents the electrochemical impedance spectroscopy (EIS) at the potential of -0.1 V, which unravels the resistance to the electro chemical reaction processes of the electrocatalysts. The results show that doping with heteroatom helps to decrease the x-intercept of ACB com posite, thereby minimizi the bulk resistance (Rs), which is the resistance of electrolyte solution, inllerent resistance of the active swface of the electrode, and the contact resistance at the interface between the current collector electrocatalyst [39). The double-layer capacitance (Ccu) in par allel with the charge transfer resistance CRcJ at the interface of the elec trode/electrolyte is reflected by the small semicircle (quasi-semicircle or Diameter impedance arc (DIA)) in the plots [40,41). The quasi-semicircle is an important parameter for determining the electrical conductivity of the fabricated activated carbon black composite electrocatalysts. The order of diameter of the quasi-semicircle of these composite electrocatalysts is as follows: ACBN1F1 > ACB > ACBN 3 F 3 > ACBN 2 F 2 > ACBN 2 . This re sult shows that ACBN 2 followed by ACBN 2 F 2 exhibits the smallest electron transfer resistance and the best electrical conductivity. However, ACBN2F2 exhibits a better electrocatalytic performance than ACBN 2, showing that nitrogen doping is responsible for the declined electron transfer resistance,

Acknowledgments

The authors acknowledge the Fundamental Research Grant Scheme (FRGS) from the Ministry of Education, Malaysia (Department of Higher Education), for funding this work through Project No. "FP046-2017A".

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.diamond.2019.107626.