# Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics

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## Abstract

**:**

## 1. Introduction

## 2. A General Framework for Lie Group Statistical Mechanics and Symmetries

#### 2.1. A Class of Generalized Gibbs Probability Densities, Its Associated Entropy and Fisher Metric

**Lemma**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4**

**Proof.**

**Koszul-Vinberg characteristic function.**We now quickly describe a particular case of the above setting, which is related to Hessian geometry and in which the characteristic function Equation (2) recovers the Koszul-Vinberg characteirstic function, see [17,18,19,20,21] and the references in [3]. In this case, the Fisher information metric of information geometry coincides with the canonical Koszul Hessian metric given by Koszul forms. Analogies between Koszul-Vinberg model and Souriau symplectic model of statistical mechanics were enlightened in [3]. Here we will show how these two models precisely arise as special cases of the general setting presented in Section 2.1.

#### 2.2. Equivariance with Respect to Lie Group Actions

**Proposition**

**5.**

**Proof.**

**Equivariance in the Koszul model.**For the Koszul model recalled above, see [3] and references therein, $G=Aut\left(\mathsf{\Omega}\right)$ is the group of linear isomorphism that preserves $\mathsf{\Omega}\subset E$. Given $g\in Aut\left(\mathsf{\Omega}\right)$, we have ${\rho}_{g}:\mathsf{\Omega}\to \mathsf{\Omega}$ and it is clear that the dual action ${\rho}_{g}^{\ast}$ preserves the dual cone ${\mathsf{\Omega}}^{\ast}$. In this very special case, $M={\mathsf{\Omega}}^{\ast}$ and the G action on M is chosen as ${\varphi}_{g}:={\rho}_{{g}^{-1}}^{\ast}$. Since $U:{\mathsf{\Omega}}^{\ast}\to {E}^{\ast}$ is the identity, there is no cocycle. However, we have $c\left(g\right)=J{\varphi}_{g}$ which is not equal to one in general and, for instance, the transformation Equation (17) of the Massieu potential reads

#### 2.3. Souriau Symplectic Model of Statistical Mechanics

**Remark**

**6**

#### 2.3.1. Souriau Symplectic Model of Satistical Mechanics

#### 2.3.2. Lie-Poisson Equations with Cocycle and Property of the Entropy in Souriau’s Model

**Remark**

**7**

**Corollary**

**8.**

**Proof.**

**Elementary examples.**A particularly simple case of Souriau symplectic model is when the symplectic manifold is a cotangent bundle $M={T}^{\ast}Q$ endowed with the canonical symplectic form. Let G be a Lie group acting on the left on Q. Then its cotangent lifted action on ${T}^{\ast}Q$ is symplectic and admits the momentum map $\mathbb{J}:{T}^{\ast}Q\to {\mathfrak{g}}^{\ast}$ given by

#### 2.3.3. Dynamics with Casimir Dissipation/Production

- (i)
- Energy conservation: taking $f=h$ in Equation (40), we obtain$$\frac{d}{dt}h={\{h,h\}}_{\mathsf{\Theta}}-\mathsf{\Lambda}\phantom{\rule{0.166667em}{0ex}}\gamma \left(\right)open="("\; close=")">\left(\right)open="["\; close="]">\frac{\delta h}{\delta \mu},\frac{\delta k}{\delta \mu}$$
- (ii)
- Casimir dissipation ($\mathsf{\Lambda}>0$) or production ($\mathsf{\Lambda}<0$): taking $f=s$ in Equation (40), and using ${\{s,f\}}_{\mathsf{\Theta}}=0$, we obtain$$\frac{d}{dt}s={\{s,h\}}_{\mathsf{\Theta}}-\mathsf{\Lambda}\phantom{\rule{0.166667em}{0ex}}\gamma \left(\right)open="("\; close=")">\left(\right)open="["\; close="]">\frac{\delta s}{\delta \mu},\frac{\delta h}{\delta \mu}$$

#### 2.3.4. Stochastic Hamiltonian Dynamics

#### 2.4. Polysymplectic Model of Statistical Mechanics

**Polysymplectic manifolds.**We only need a restricted amount of notions from polysymplectic geometry which are straighforward extensions of those recalled above in the symplectic context. We refer to [47] for more information. A polysymplectic manifold $(M,\omega )$ is a manifold M endowed with a closed nondegenerate ${\mathbb{R}}^{n}$-valued 2-form. We can identify $\omega $ with a collection $({\omega}^{1},\dots ,{\omega}^{n})$, of closed 2-forms with ${\bigcap}_{i=1}^{n}ker{\omega}^{i}=\left\{0\right\}$.

**Polysymplectic model.**The polysymplectic model of statistical mechanics is obtained by considering the following specific situation in the equivariant setting described in Section 2.2:

**Particular cases.**A particularly simple case of polysymplectic Souriau model is given by the manifold $M={T}^{\ast}Q\oplus \dots \oplus {T}^{\ast}Q$ (Whitney sum with n factors) endowed with the polysymplectic form $\mathsf{\Omega}=({\mathsf{\Omega}}^{1},\dots ,{\mathsf{\Omega}}^{n})$, with ${\mathsf{\Omega}}^{k}={\left({\pi}^{k}\right)}^{\ast}{\mathsf{\Omega}}_{\mathrm{can}}$. Here ${\pi}^{k}:{T}^{\ast}Q\oplus \dots \oplus {T}^{\ast}Q\to {T}^{\ast}Q$ is the projection onto the ${k}^{th}$ factor of the sum and ${\mathsf{\Omega}}_{\mathrm{can}}$ is the canonical symplectic form on ${T}^{\ast}Q$. Let G be a Lie group acting on the left on Q. Then its naturally induced action on ${T}^{\ast}Q\oplus \dots \oplus {T}^{\ast}Q$ is polysymplectic and admits the polysymplectic momentum map $\mathfrak{J}:{T}^{\ast}Q\oplus \dots \oplus {T}^{\ast}Q\to L(\mathfrak{g},{\mathbb{R}}^{n})$ given by

**Property of the entropy and polysymplectic Lie-Poisson equations with cocycle.**In the context of the polysymplectic model, a natural generalisation of the Lie-Poisson equations with cocycle Equation (36) are

#### 2.5. The Fisher Metric on Orbits and Equivariance

**Proposition**

**9.**

**Proof.**

**Souriau Lie group statistical model.**In this case, M is endowed with a symplectic structure $\omega $, we take $E=\mathfrak{g}$ and $U=\mathbb{J}:M\to {\mathfrak{g}}^{\ast}$ with nonequivariance cocycle $\theta \in {C}^{\infty}(G,{\mathfrak{g}}^{\ast})$, i.e., $\theta \left(g\right)=\mathbb{J}\left({\varphi}_{g}\left(m\right)\right)-{Ad}_{{g}^{-1}}^{\ast}\mathbb{J}\left(m\right)\in {\mathfrak{g}}^{\ast}$. The map $\mathsf{\Theta}$ defined in Equation (55) becomes here a two cocycle $\mathsf{\Theta}:\mathfrak{g}\times \mathfrak{g}\to \mathbb{R}$, see Equations (25)–(27), via the relation $\left(\right)open="\langle "\; close="\rangle ">\mathsf{\Theta}\left(\xi \right),\eta $. Proposition 9 immediately yields the following result as a corollary, which is obtained by noting that ${\xi}_{E}\left(\beta \right)={ad}_{\xi}\beta $ and ${\xi}_{{E}^{\ast}}\left(\nu \right)={ad}_{\xi}^{\ast}\nu $ and is a consequence of Equation (29).

**Corollary**

**10.**

**Polysymplectic Lie group statistical model.**In this case, M is endowed with a polysymplectic structure $\omega =({\omega}^{1},\dots ,{\omega}^{n})$, we take $E=L({\mathbb{R}}^{n},\mathfrak{g})$ and $U=\mathfrak{J}:M\to L(\mathfrak{g},{\mathbb{R}}^{n})$ with nonequivariance cocycle $\theta \in {C}^{\infty}(G,L(\mathfrak{g},{\mathbb{R}}^{n}))$:

**Corollary**

**11.**

**Koszul model.**For the Koszul model with $\mathsf{\Omega}=sym{\left(n\right)}^{+}$ the cone of positive definite matrices and the Lie group $G=GL\left(n\right)$, the actions Equations (21) and (22) have the associated infinitesimal generators

## 3. Applications

#### 3.1. Multivariate Gaussian Probability Densities

**Gaussian probability densities in generalized Gibbs form.**Consider a multivariate Gaussian density with symmetric and positive definite covariance matrix $R\in {sym}^{+}\left(n\right)$ and mean $m\in {\mathbb{R}}^{n}$. The Gaussian probability density is written in the generalized Gibbs form ${p}_{\beta}$ discussed above in Section 2.1 as follows:

**Characteristic function, thermodynamic heat, and entropy.**The Massieu potential is computed in terms of $\beta \in \mathsf{\Omega}$ as

**Fisher information metric.**We compute the generalized heat capacity $K\left(\beta \right):=-{\mathbf{D}}^{2}\mathsf{\Phi}\left(\beta \right)$ as follows, see Section 2.1:

**Equivariance with respect to the general affine group.**We consider the general affine group

**Lemma**

**12.**

**Proof.**

**Lemma**

**13.**

**Proof.**

**Geodesics on multivariate Gaussian densities and Noether theorem.**Let us consider the Lagrangian $L:T\mathsf{\Omega}=\mathsf{\Omega}\times E\to \mathbb{R}$ given by the kinetic energy of the Fisher metric

#### 3.2. Unitary Representations and Quantum Fisher Metric

**Casimir dissipation/production.**The general equations for Casimir dissipation/production Equation (40) applied here with $\mathfrak{g}\subset \mathfrak{u}\left(\mathcal{H}\right)$, ${\mathfrak{g}}^{\ast}=\mathfrak{g}$, and $\gamma (\nu ,\beta )=\left(\right)open="\langle "\; close="\rangle ">\nu ,\beta $, become

#### 3.3. Souriau Symplectic Model for $SE\left(2\right)$, Lie-Poisson Equations with Cocycle, and Casimir Dissipation

**Momentum map and cocycle.**Consider the special Euclidean group of the plane $SE\left(2\right)=SO\left(2\right)\phantom{\rule{0.166667em}{0ex}}\u24c8\phantom{\rule{0.166667em}{0ex}}{\mathbb{R}}^{2}$ with semidirect product group multiplication

**Gibbs densities, entropy, and Fisher metric.**The generalized Gibbs probability densities are here given on $M={\mathbb{R}}^{2}$ by

**Affine Lie-Poisson equations and Casimir dissipation.**The affine coadjoint action associated with Equation (70) is found as

## 4. Variational Principles and (Multi)Symplectic Integrators

#### 4.1. Preliminaries on Variational Lie Group Integrators

**Euler-Poincaré and Lie-Poisson equations.**We will be especially interested in the case where the configuration manifold is a Lie group, $Q=G$, and the Lagrangian $L:TG\to \mathbb{R}$ is right G-invariant. In this case, L induces a reduced Lagrangian ℓ on the quotient space $\left(TG\right)/G$ identified with the Lie algebra $\mathfrak{g}$, i.e., we get $\ell :\mathfrak{g}\to \mathbb{R}$ defined by the relation $L(g,\dot{g})=\ell \left(\dot{g}{g}^{-1}\right)$. The Euler-Lagrange equations for L are equivalent to equations on $\mathfrak{g}$ written in terms of the reduced Lagrangian $\ell :\mathfrak{g}\to \mathbb{R}$, called the Euler-Poincaré equations. They are obtained by computing the variational principle for ℓ induced by the Hamilton principle Equation (74). It is given by

**Variational integrators.**Let Q be a configuration manifold and let $L:TQ\to \mathbb{R}$ be a Lagrangian. Suppose that a time step $\mathsf{\Delta}t$ was fixed, denote by $\{{t}_{k}=k\mathsf{\Delta}t\mid k=0,\dots ,N\}$ the sequence of time, and by ${q}_{d}:{\left\{{t}_{k}\right\}}_{k=0}^{N}\to Q$, ${q}_{d}\left({t}_{k}\right)={q}_{k}$ a discrete curve. A discrete Lagrangian is a map ${L}_{d}:Q\times Q\to \mathbb{R}$, ${L}_{d}={L}_{d}({q}_{k},{q}_{k+1})$ that approximates the action integral of L along the curve segment between ${q}_{k}$ and ${q}_{k+1}$, that is, we have

**Discrete Euler-Poincaré equations.**For Lie groups, variational discretization and the associated discrete Lagrangian reductions, was started in [60,67], and referred to as Lie group variational integrators. The essential idea behind such integrators is to discretize Hamilton’s principle and to update group elements using group operations. For the case of invariant systems on Lie group, one chooses a discrete Lagrangian that inherits the invariance of the continuous Lagrangian, i.e., ${L}_{d}:G\times G\to \mathbb{R}$ satisfies ${L}_{d}({g}_{k}h,{g}_{k+1}h)={L}_{d}({g}_{k},{g}_{k+1})$, for all $h\in G$.

#### 4.2. Central Extensions and Variational Principle for the Lie-Poisson Equations with Cocycle

**Lie group operations on central extensions.**We shall focus on topologically trivial central extensions of finite dimensional Lie groups by $\mathbb{R}$. The central extended group is thus of the form $\widehat{G}=G\times \mathbb{R}$ with group multiplication

**Euler-Poincaré and Lie-Poisson equations on central extensions.**From Equation (95), the Euler-Poincaré equations for a reduced Lagrangian $\hat{\ell}:\widehat{\mathfrak{g}}=\mathfrak{g}\times \mathbb{R}\to \mathbb{R}$ take the form

#### 4.3. Variational Symplectic Integrators for the Lie-Poisson Equations with Cocycle

**Some useful identities.**Given a central extension $\widehat{G}=G\times \mathbb{R}$, we shall consider the retraction map $\tau :\widehat{\mathfrak{g}}\to \widehat{G}$ defined by

**Lemma**

**14.**

- (a)
- ${d}^{L}\tau (\xi ,u)\xb7(\eta ,v)=\left(\right)open="("\; close=")">{d}^{L}\overline{\tau}\left(\xi \right)\xb7\eta ,v-{D}_{2}B(\overline{\tau}\left(\xi \right),e)\xb7({d}^{L}\overline{\tau}\left(\xi \right)\xb7\eta )$
- (b)
- ${d}^{L}\tau {(\xi ,u)}^{\ast}\xb7(\mu ,a)=\left(\right)open="("\; close=")">{d}^{L}\overline{\tau}{\left(\xi \right)}^{\ast}(\mu -a{D}_{2}B(\overline{\tau}\left(\xi \right),e)),a$
- (c)
- ${d}^{L}{\tau}^{-1}(\xi ,u)\xb7(\zeta ,w)=\left(\right)open="("\; close=")">{d}^{L}{\overline{\tau}}^{-1}\left(\xi \right)\xb7\zeta ,w+{D}_{2}B(\overline{\tau}\left(\xi \right),e)\xb7\zeta $
- (d)
- ${d}^{L}{\tau}^{-1}{(\xi ,u)}^{\ast}\xb7(\mu ,a)=\left(\right)open="("\; close=")">{d}^{L}{\overline{\tau}}^{-1}{\left(\xi \right)}^{\ast}\xb7\mu +a{D}_{2}B(\overline{\tau}\left(\xi \right),e),a$,

**Proof.**

- (a)
- Using the definition of ${d}^{L}\tau $, we compute$$\begin{array}{cc}\hfill {d}^{L}\tau (\xi ,u)\xb7(\eta ,v)& =\tau {(\xi ,u)}^{-1}(\mathbf{D}\tau (\xi ,u)\xb7(\eta ,v))\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={(\overline{\tau}\left(\xi \right),u)}^{-1}(\overline{\tau}\left(\xi \right),\mathbf{D}\overline{\tau}\left(\xi \right)\xb7\eta ,u,v)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =(\overline{\tau}{\left(\xi \right)}^{-1}\mathbf{D}\overline{\tau}\left(\xi \right)\xb7\eta ,v+{D}_{2}B(\overline{\tau}{\left(\xi \right)}^{-1},\overline{\tau}\left(\xi \right))\xb7(\mathbf{D}\overline{\tau}\left(\xi \right)\xb7\eta ))\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =({d}^{L}\overline{\tau}\left(\xi \right)\xb7\eta ,v+{D}_{2}B(\overline{\tau}{\left(\xi \right)}^{-1},\overline{\tau}\left(\xi \right))\xb7(\overline{\tau}\left(\xi \right){d}^{L}\overline{\tau}\left(\xi \right)\xb7\eta ))\hfill \end{array}$$$${D}_{2}B({g}^{-1},g)\xb7\left(g\eta \right)=-{D}_{2}B(g,e)\xb7\eta ,$$
- (b)
- Taking the dual map and using (a), we get$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left(\right)open="\langle "\; close="\rangle ">{d}^{L}\tau {(\xi ,u)}^{\ast}\xb7(\mu ,a),(\eta ,v)\hfill \end{array}\hfill \phantom{\rule{1.em}{0ex}}& =\left(\right)open="\langle "\; close="\rangle ">\mu ,{d}^{L}\overline{\tau}\left(\xi \right)\xb7\eta +a\left(\right)open="("\; close=")">v-{D}_{2}B(\overline{\tau}\left(\xi \right),e)\xb7({d}^{L}\overline{\tau}\left(\xi \right)\xb7\eta )\hfill $$
- (c)
- It follows by (a) and by inverting the relation $(\zeta ,w)={d}^{L}\tau (\xi ,u)\xb7(\eta ,v)$$$(\zeta ,w)={d}^{L}\tau (\xi ,u)\xb7(\eta ,v)=\left(\right)open="("\; close=")">{d}^{L}\overline{\tau}\left(\xi \right)\xb7\eta ,v+{D}_{2}B(\overline{\tau}{\left(\xi \right)}^{-1},\overline{\tau}\left(\xi \right))\xb7(\overline{\tau}\left(\xi \right){d}^{L}\overline{\tau}\left(\xi \right)\xb7\eta )$$$$(\eta ,v)=\left(\right)open="("\; close=")">{d}^{L}{\overline{\tau}}^{-1}\left(\xi \right)\xb7\zeta ,w-{D}_{2}B(\overline{\tau}{\left(\xi \right)}^{-1},\overline{\tau}\left(\xi \right))\xb7\left(\overline{\tau}\left(\xi \right)\zeta \right)$$
- (d)
- This follows by taking the dual map and using (c) as earlier. $\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\square $

**Variational discretization of the Lie-Poisson equations with cocycle.**With the previous result, we first give below a symplectic integrator for the Euler-Poincaré equations on central extensions. Then we will show how this provides a symplectic integrator for the Lie-Poisson equations with cocycle.

**Proposition**

**15**

- (a)
- The discrete curve$({\xi}_{k},{u}_{k})$is critical for the discrete Euler-Poincaré variational principle$$\delta \sum _{k}\widehat{\ell}({\xi}_{k},{u}_{k})=0,$$$$\left(\right)$$
- (b)
- The discrete curve$({\xi}_{k},{u}_{k})$is a solution of the discrete Euler-Poincaré equations$$\left(\right)$$$$\left(\right)$$

**Proof.**

**Proposition 16.**

**Proof.**

**Remark**

**17**

**Example**

**17.**

#### 4.4. Multisymplectic Lie Group Variational Integrators

**Variational principle for the Lie-Poisson field equations with cocycle.**The goal of this paragraph is to obtain a variational principle for the Lie-Poisson field equations with cocycle Equation (54) associated with Souriau’s polysymplectic model. By considering the Euler-Poincaré field equations Equation (109) on a central extension, we get the system

**Remark**

**18**

**Multisymplectic Lie group integrators.**To present multisymplectic integrators, we shall focus on the two dimensional case and assume that the fields are defined on a rectangle $U=[0,A]\times [0,B]\subset {\mathbb{R}}^{2}$. We shall write $({x}_{1},{x}_{2})=(x,y)$. Let Q be a configuration manifold and let $L:TQ\oplus TQ\to \mathbb{R}$ be a Lagrangian. We shall consider the very special case of a discrete grid determined by $\{({x}_{k},{y}_{a})=(k\mathsf{\Delta}x,a\mathsf{\Delta}y)\mid k=0,\dots ,{N}_{1},\phantom{\rule{0.277778em}{0ex}}a=1,\dots ,{N}_{2}\}$ with given $\mathsf{\Delta}x$ and $\mathsf{\Delta}y$. We shall denote by ${q}_{d}:{\left\{({x}_{k},{y}_{a})\right\}}_{k=0}^{N}\to Q$, ${q}_{d}({x}_{k},{y}_{a})={q}_{k}^{a}$ a discrete field. A discrete Lagrangian is a map ${L}_{d}:Q\times Q\times Q\to \mathbb{R}$, ${L}_{d}={L}_{d}({q}_{k}^{a},{q}_{k+1}^{a},{q}_{a+1}^{k})$ that approximates the action integral of L on the rectangle $[{x}_{k},{x}_{k+1}]\times [{y}_{a},{y}_{a+1}]$ for a field interpolating the values ${q}_{k}^{a},{q}_{k+1}^{a},{q}_{a+1}^{k}$. The discrete Hamilton principle reads

**Multisymplectic variational discretization for Lie-Poisson field equations with cocycle.**Based on the previous result, we first give below a multisymplectic integrator for the Euler-Poincaré field equations on central extensions. Then we deduce a multisymplectic integrator for the Lie-Poisson field equations with cocycle appearing in the polysymplectic Souriau model. The next proposition is the multisymplectic version of Proposition 15.

**Proposition 19.**

- (a)
- The discrete curve$({\xi}_{k},{u}_{k})$is critical for the discrete Euler-Poincaré field variational principle$$\delta \sum _{a,k}\widehat{\ell}(({\xi}_{k}^{a},{u}_{k}^{a}),({\zeta}_{k}^{a},{w}_{k}^{a}))=0,$$$$\left(\right){\displaystyle \delta {w}_{k}^{a}=\frac{1}{\mathsf{\Delta}y}\left(\right)open="("\; close=")">{v}_{k}^{a+1}-{v}_{k}^{a}-{D}_{2}B(\overline{\tau}\left(\mathsf{\Delta}y{\zeta}_{k}^{a}\right),e)\xb7{\eta}_{k}^{a}+{D}_{1}B(e,\overline{\tau}\left(\mathsf{\Delta}y{\zeta}_{k}^{a}\right))\xb7{\eta}_{k}^{a+1}}\hfill $$
- (b)
- The discrete curve$({\xi}_{k}^{a},{u}_{k}^{a},{\zeta}_{k}^{a},{w}_{k}^{a})$is a solution of the discrete Euler-Poincaré field equations$$\left(\right){\displaystyle \frac{1}{\mathsf{\Delta}x}({a}_{k-1}^{a}-{a}_{k}^{a})+\frac{1}{\mathsf{\Delta}y}({b}_{k}^{a-1}-{b}_{k}^{a})=0}\hfill $$$$\left(\right)$$

**Proof.**

**Proposition 20.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Barbaresco, F.; Gay-Balmaz, F.
Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics. *Entropy* **2020**, *22*, 498.
https://doi.org/10.3390/e22050498

**AMA Style**

Barbaresco F, Gay-Balmaz F.
Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics. *Entropy*. 2020; 22(5):498.
https://doi.org/10.3390/e22050498

**Chicago/Turabian Style**

Barbaresco, Frédéric, and François Gay-Balmaz.
2020. "Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics" *Entropy* 22, no. 5: 498.
https://doi.org/10.3390/e22050498