dx\right),$ $\displaystyle=\oint_{C}\frac{1}{\rho}\left(\rho
d\left(\frac{\delta\ell}{\delta\rho}\right)\right),$

Then invoking Stoke’s theorem, the last integral is equal to the integral of
the exterior derivative of the potential $\nabla\frac{\delta\ell}{\delta\rho}$
and it amounts to zero, i.e.

$\displaystyle\frac{d}{dt}I=\oint_{C}d\left(\frac{\delta\ell}{\delta\rho}\right)=\int
d^{2}\left(\frac{\delta\ell}{\delta\rho}\right)=0.$ (6.13)

∎

###### Remark 6.2.

An alternative derivation of conservation laws is by using the approach of
[19], where it uses the multisymplectic conservation law

$\displaystyle\partial_{t}\omega+\partial_{x_{i}}\vartheta_{i}=0,$ (6.14)

with the pre-symplectic forms

$\displaystyle\omega(W,V)=\left\langle
MW,V\right\rangle,\quad\text{and}\quad\vartheta_{i}(W,V)=\left\langle
K_{i}W,V\right\rangle,$

where vector fields $W,V$ are arbitrary vector fields on the phase space and
$M$ and $K_{i}$ are degenerate skew-symmetric matrices defined in (3.18). The
formula for conservation symplecticity (6.14), in this form can be derived
from the multisymplectic form formula (2.12) as shown in [86]._The local
conservation of energy_ is defined by

$\displaystyle\partial_{t}E(z)+\partial_{x_{i}}F_{i}(z)=0,\quad\text{where}\quad
E(z)=H(z)-\frac{1}{2}\vartheta_{i}(z_{x_{i}},z),\quad
F(z)=\frac{1}{2}\vartheta_{i}(z_{t},z).$

The corresponding global energy conservation law is obtained by first taking
the integral of $E(z)$ and then computing the total derivative of the
integral, i.e.

$\displaystyle\frac{d}{dt}\int E(z)\,dx=0.$ (6.15)

Likewise, _the local conservation of momentum_ is when
$E(z)=\frac{1}{2}\omega(z_{x_{i}},z)$ and
$F(z)=H(z)-\frac{1}{2}\omega(z_{t},z)$.

###  6.1 Hamiltonian formulation of deep learning

In the section 3, we have introduced the Euler–Poincaré equation (3.23) for
deep learning. The equation uses the Lagrangian description of dynamics,
however, for studying the nonlinear stability of deep learning, it is required
to use the Hamiltonian formulation. The main difference is that in the
Lagrangian picture the dynamics are defined on the Lie algebra, while the
Hamiltonian dynamics belong to the dual of a Lie algebra, which is a Poisson
manifold [85, 123]. For this particular deep learning problem, the Hamiltonian
point of view was used in [40]. The manifold is the space of densities on
manifold $Q$, thus the technical details are left out of this note. Let $G$ be
a Lie group with $\mathfrak{g}$ as the associated Lie algebra and
$\mathfrak{g}^{\ast}$ is the dual of the Lie algebra. On $\mathfrak{g}^{\ast}$
we have the following bracket

$\displaystyle\left\\{F,G\right\\}_{\pm}(\mu)=\pm\left\langle\mu,\left[\frac{\delta
F}{\delta\mu},\frac{\delta G}{\delta\mu}\right]\right\rangle,$ (6.16)

where $F,G:\mathfrak{g}^{\ast}\to\mathbb{R}$ are functions on
$\mathfrak{g}^{\ast}$, and it the bracket $\left\\{F,G\right\\}_{\pm}(\mu)$ is
known as _Lie–Poisson bracket_. The deep learning Hamiltonian density
$\mathcal{H}:J^{1}(E)^{\ast}\to\Lambda^{n+1}(Q)$, first mentioned in (3.6) and
it is given by

$\displaystyle\mathcal{H}=\left(\frac{1}{2}\rho\|w\|^{2}+\frac{\nu^{4}}{8}\rho\left\|\nabla\mathrm{log}(\rho)\right\|^{2}\right)d^{n}x\,dt.$
(6.17)

The main assumption here is that $\nabla\cdot w=0$. Here, we consider the
Hamiltonian with the variables the momentum, $\mu=\rho
w+\frac{\nu^{2}}{2}\nabla\rho$, and the probability density function, $\rho$.
The Lie–Poisson bracket of deep learning

$\displaystyle\\{F,G\\}(\mu,\rho)$
$\displaystyle=\int_{Q}\mu\cdot\left[\left(\frac{\delta
G}{\delta\mu}\cdot\nabla\right)\frac{\delta F}{\delta\mu}-\left(\frac{\delta
F}{\delta\mu}\cdot\nabla\right)\frac{\delta G}{\delta\mu}\right]d^{n}x$ (6.18)
$\displaystyle+\int_{Q}\rho\left[\left(\frac{\delta
G}{\delta\mu}\cdot\nabla\right)\frac{\delta F}{\delta\rho}-\left(\frac{\delta
F}{\delta\mu}\cdot\nabla\right)\frac{\delta G}{\delta\rho}\right]d^{n}x$
$\displaystyle=\left\langle(\mu,\rho),\left[\frac{\delta
F}{\delta(\mu,\rho)},\frac{\delta G}{\delta(\mu,\rho)}\right]\right\rangle.$

This bracket is in fact a Poisson bracket on the dual of a semidirect product
Lie algebra $\mathfrak{s}=\mathfrak{X}(Q)\times\mathcal{F}(Q)$. The bracket
(6.18) does not only appear in deep learning, but originally it appeared in
hydrodynamics [15, 94, 53, 87] and in semidirect reduction [88]. With that the
Euler–Poincaré equation (3.23) is written as the Lie–Poisson on semidirect
product space
$\mathfrak{s}^{\ast}=\mathfrak{X}^{\ast}(Q)\times\mathrm{Dens}(Q)$

$\displaystyle\frac{d}{dt}\mu$
$\displaystyle=\pm\mathrm{ad}^{\ast}_{\frac{\delta
H}{\delta\mu}}\mu\mp\frac{\delta H}{\delta\rho}\diamond\rho,$ (6.19)
$\displaystyle\frac{d}{dt}\rho$ $\displaystyle=\mp\frac{\delta
H}{\delta\mu}\rho,$

and it can be written in terms of Lie–Poisson bracket (6.18) as

$\displaystyle\partial_{t}(\mu,\rho)$ $\displaystyle=\pm\\{(\mu,\rho),H\\}.$
(6.20)

###  6.2 Review of the energy–Casimir method

Up to this point, terms like linearised and nonlinear stability were used
indistinguishably, and in fact, stability can refer to various ideas. There
are different concepts of stability such as: spectral stability, linear
stability, formal stability and nonlinear stability, and the difference
between each of them is detailed in [55] and references therein. They all
describe how the system responds to perturbations around its equilibrium
solution. In general, stability requires a measure of distance, $\mathrm{d}$,
and a trajectory $\mu_{e}$ is said to be stable if a trajectory that starts
close to $\mu_{e}$, it remains close for all $t>0$. Each of these meanings
require different stability criteria, and in some cases one implies the other.
However, the system at hand here is nonlinear and infinite-dimensional, and,
for example, formal stability does not imply nonlinear stability. In this
article, the concept of stability that is of concern is nonlinear stability
obtained by the energy–Casimir method [55].

###### Definition 6.3 (Nonlinear Stability [55]).

For the equilibrium trajectory $\mu_{e}$, we define a neighbourhood of
$\mu_{e}$, $U_{1}$, by $\mathrm{d}(\mu(t),\mu_{e})<\delta$, where
$\delta\in\mathbb{R}_{>0}$. We also define another neighbourhood of $\mu_{e}$,
$U_{2}$, by $\mathrm{d}(\mu(t),\mu_{e})<\epsilon$, where
$\epsilon\in\mathbb{R}_{>0}$. If a trajectory $\mu(t)$, whose initial value
$\mu(0)\in U_{2}$, for every $\epsilon$ used to define $U_{2}$, there is
$\delta$, such that $\mu(t)\in U_{1}$ for all $t>0$, then $\mu(t)$ is said to
be nonlinearly stable.

What is different from linearised stability is that here, the trajectory
$\mu(t)$ satisfies a nonlinear differential equation. The goal here is to find
the norm used in defining the distance $\mathrm{d}$. Energy–Casimir method
provides a way of doing that. Traditionally, stability conditions are based on
a linearised dynamics of the system being examined, and the definiteness of
the Lyapunov function. One popular choice for the Lyapunov function is the
energy function. For nonlinear stability, the energy function being the
Hamiltonian is augmented with conserved quantities arising from symmetries.
From there a norm is constructed using $H$ and $C$ in which the perturbations
to the equilibrium are bounded. Here, the conserved quantity $C$ is chosen to
be the _Casimir function_ , which satisfies $\\{C,F\\}=0$ for any function $F$
that maps from the phase-space to $\mathbb{R}$. Using a function of single
variable $\varphi$, we can construct a family of Casimir functions
$C_{\varphi}=\varphi\circ C$. Proceeding forward, given an fixed-point
$\mu_{e}$ solution of (6.19), the energy–Casimir method requires the following
steps to determine the stability of $\mu_{s}$:

  * •

Find $\varphi$, such that when evaluated at $\mu_{s}$, the first variation of
$H+C$ vanishes.

  * •

Check the convexity of $H+C$ by first finding the quadratic forms $Q_{1}$, and
$Q_{2}$ such that they satisfy the following inequalities

$\displaystyle Q_{1}(\Delta\mu)$ $\displaystyle\leq
H(\mu_{s}+\Delta\mu)-H(\mu_{s})-\left\langle\delta
H(\mu_{s}),\Delta\mu\right\rangle,$ $\displaystyle Q_{2}(\Delta\mu)$
$\displaystyle\leq
C_{\varphi}(\mu_{s}+\Delta\mu)-C_{\varphi}(\mu_{s})-\left\langle\delta
C_{\varphi}(\mu_{s}),\Delta\mu\right\rangle,$

where $\Delta\mu=\mu-\mu_{s}$. Further, an additional requirement is that

$\displaystyle Q_{1}(\Delta\mu)+Q_{2}(\Delta\mu)>0,$

for all $\Delta u\neq 0$.

  * •

Ensure the continuity of $H_{C}$ by asserting that

$\displaystyle\left|H(\mu_{s}+\Delta\mu)-H(\mu_{s})-\left\langle\delta
H(\mu_{s}),\Delta\mu\right\rangle\right|$ $\displaystyle\leq
C_{1}\|\Delta\mu\|^{\alpha},$
$\displaystyle\left|C_{\varphi}(\mu_{s}+\Delta\mu)-C_{\varphi}(\mu_{s})-\left\langle\delta
C_{\varphi}(\mu_{s}),\Delta\mu\right\rangle\right|$ $\displaystyle\leq
C_{2}\|\Delta\mu\|^{\alpha},$

where $C_{1},C_{2},\alpha>0$.

Earlier, a distance $\mathrm{d}$ is used to define nonlinear stability, and
once the aforementioned steps are fulfilled, we set the the distance as

$\displaystyle\mathrm{d}(w_{1},w_{2})=\|w_{1}-w_{2}\|^{2}=Q_{1}(w_{1}-w_{2})+Q_{2}(w_{1}-w_{2})>0,$
(6.21)

for $w_{1}-w_{2}\neq 0$. Moreover, the existence of $C_{1},C_{2}$ imply that
the distance between the trajectory $\mu(t)$ and $\mu_{s}$ is bounded

$\displaystyle\mathrm{d}(\mu(t),\mu_{s})\leq(C_{1}+C_{2})\mathrm{d}(\mu(0),\mu).$

One consequence of directly examining the nonlinear stability of the system
(6.19) is that an equilibrium solution $(\mu_{s},\rho_{s})$ might not be
meaningful for deep learning. This necessitates a different approach. Here, we
aim to use analysis that is based on both forward and backward error analysis.
The latter is used to derive the modified system and use it to derive an
evolution equation for the error between the numerical and the exact solution.
This way, the error equation’s equilibrium solution is the trivial solution
and the aim of the analysis here becomes the nonlinear stability when the
error is close to zero.

###### Remark 6.4.

For equations in the Euler–Poincaré family, especially the dispersionless
Camassa-Holm equation [23], choosing the zero solution as the initial
condition and perturbing the equation results in the emergence of nonzero
solution.

Assume that there exists a solution (6.19), which solves the training problem
4.1. The numerical solution of obtained by the multisymplectic integrator is
Lagrangian and it can be formulated as a Hamiltonian system using Legendre
transform $\mathbb{F}\mathcal{L}_{m}:J^{3}(E)\to(J^{3}(E))^{\ast}$ defined by

$\displaystyle H_{m}$ $\displaystyle=\left(\frac{\partial
L}{\partial\phi_{\mu_{1}}^{A}}-D_{\mu_{2}}\left(\frac{\partial
L}{\partial\phi_{\mu_{1}\mu_{2}}^{A}}\right)+D_{\mu_{3}\mu_{2}}\left(\frac{\partial
L}{\partial\phi_{\mu_{1}\mu_{2}\mu_{3}}^{A}}\right)\right)\frac{\partial\phi^{A}}{\partial{\mu_{1}}}$
(6.22) $\displaystyle+\left(\frac{\partial
L}{\partial\phi_{\mu_{1}\mu_{2}}^{A}}-D_{\mu_{3}}\left(\frac{\partial
L}{\partial\phi_{\mu_{1}\mu_{2}\mu_{3}}^{A}}\right)\right)\frac{\partial\phi^{A}}{\partial{\mu_{1}\mu_{2}}}+\left(\frac{\partial
L}{\partial\phi_{\mu_{1}\mu_{2}\mu_{3}}^{A}}\right)\frac{\partial\phi^{A}}{\partial{\mu_{1}\mu_{2}\mu_{3}}}-L,$

and the Hamiltonian analogue of the modified Lagrangian (5.6) is

$\displaystyle H_{m}$
$\displaystyle=\frac{1}{2}\rho(w^{1})^{2}+\frac{1}{2}\rho(w^{2})^{2}+\frac{(\Delta
x^{1})^{2}}{4}\rho w^{1}\partial_{x^{1}x^{1}}w^{1}+\frac{(\Delta
x^{1})^{2}}{4}\rho w^{2}\partial_{x^{1}x^{1}}w^{2}+\frac{(\Delta
x^{1})^{2}}{8}\rho\left(\partial_{x^{1}}w^{1}\right)^{2}$ (6.23)
$\displaystyle+\frac{(\Delta
x^{1})^{2}}{8}\rho\left(\partial_{x^{1}}w^{2}\right)^{2}+\frac{(\Delta
x^{2})^{2}}{4}\rho w^{1}\partial_{x^{2}x^{2}}w^{1}+\frac{(\Delta
x^{2})^{2}}{4}\rho w^{2}\partial_{x^{2}x^{2}}w^{2}+\frac{(\Delta
x^{2})^{2}}{8}\rho\left(\partial_{x^{2}}w^{1}\right)^{2}$
$\displaystyle+\frac{(\Delta
x^{2})^{2}}{8}\rho\left(\partial_{x^{2}}w^{2}\right)^{2}+\frac{(\Delta
t)^{2}}{4}\rho w^{1}\partial_{tt}w^{1}+\frac{(\Delta t)^{2}}{4}\rho
w^{2}\partial_{tt}w^{2}+\frac{(\Delta
t)^{2}}{8}\rho\left(\partial_{t}w^{1}\right)^{2}$ $\displaystyle+\frac{(\Delta
t)^{2}}{8}\rho\left(\partial_{t}w^{2}\right)^{2}+\frac{\nu^{4}\left(\partial_{x^{1}}\rho\right)^{2}+\nu^{4}\left(\partial_{x^{2}}\rho\right)^{2}}{8\rho+(\Delta
x^{1})^{2}\partial_{x^{1}x^{1}}\rho+(\Delta
x^{2})^{2}\partial_{x^{2}x^{2}}\rho+(\Delta t)^{2}\partial_{tt}\rho}$

Unlike the case with linear dynamical systems where the error is taken as the
difference between the numerical and exact trajectories, here the error is
defined as a Lie group element. The reason for that is that we are comparing
two trajectories on the same manifold, but their dynamics live on different
spaces, which is an obstacle in deriving an evolution equation for the error.
Moreover, the error on Lie group allows the derivation of the Lie–Poisson
equation for the error in order to apply the energy–Casimir method. The first
step is to define the _configuration error function_ [21], denoted by
$\Psi:G\times G\to\mathbb{R}$. The configuration error function need to
satisfy a the following properties: it needs to be smooth, symmetric, i.e. for
$g_{0},g_{1},\in G$ we have $\Psi(g_{0},g_{1})=\Psi(g_{1},g_{0})$, positive
definite and $\Psi(g,g)=0$ for any $g\in G$. One candidate for configuration
error function is

$\displaystyle\Psi(g,h)=\mathrm{Tr}\left(\|\mathrm{Id}-g^{-1}h\|^{2}\right)=\mathrm{Tr}\left(\|\mathrm{Id}-h^{-1}g\|^{2}\right),$

where $g,h\in G$ and $\mathrm{Id}$ is the identity element of $G$. Here,
$g^{-1}h$ is the left action of $g^{-1}$ on $h$. We set
$g_{e}=g_{s}^{-1}g_{m}$ as _the group error_ , where $g_{s}\in G$ is the
analytic solution of deep learning training problem and $g_{m}$ is the
numerical solution obtained using the variational multisymplectic integrator,
whose modified Hamiltonian is given by (6.23). The idea is the closest $g_{m}$
to the analytic solution, the smaller is $\Psi(g_{r},g)$ value. Before
computing the evolution equation of the error on $TG$, we use the fact the
solution of the Lie–Poisson equation on $\mathfrak{g}^{\ast}$ can be used to
reconstruct the curves on $T^{\ast}G$. Part of this is the reconstruction
equation and, for the analytic solution $\mu_{s}$ of the training problem, we
assume that with the Hamiltonian, $H:\mathfrak{s}^{\ast}\to\mathbb{R}$, it
satisfies

$\displaystyle\partial_{t}\left(\mu_{s},\rho_{s}\right)=\left(\mathrm{ad}^{\ast}_{\frac{\delta
H_{s}}{\delta\mu_{s}}}\mu_{s}+\frac{\delta
H_{s}}{\delta\rho_{s}}\diamond\rho_{s},\frac{\delta
H_{s}}{\delta\mu_{s}}\rho_{s}\right),$

The reconstruction equation on $T^{\ast}S$, where $S$ is the group whose Lie
algebra is $\mathfrak{X}(Q)\times\mathcal{F}(Q)$, for the analytic solution is
then

$\displaystyle\frac{dg_{s}}{dt}$ $\displaystyle=L_{g_{s}}\frac{\delta
H_{s}}{\delta\mu_{s}},$ $\displaystyle\text{and}\quad\frac{dv_{s}}{dt}$
$\displaystyle=L_{g_{s}}\frac{\delta H_{s}}{\delta\rho_{s}}.$

Similarly, the numerical solution of the training problem satisfies the
modified equation with Hamiltonian $H_{m}:(J^{3}(E))^{\ast}\to\mathbb{R}$ and
its Lie–Poisson and corresponding reconstruction equation

$\displaystyle\partial_{t}\left(\mu_{m},\rho_{m}\right)$
$\displaystyle=\left(\mathrm{ad}^{\ast}_{\frac{\delta
H_{m}}{\delta\mu_{m}}}\mu+\frac{\delta
H_{m}}{\delta\rho_{m}}\diamond\rho_{m},\frac{\delta
H_{m}}{\delta\mu_{m}}\rho_{m}\right),$ $\displaystyle\quad\frac{dg_{m}}{dt}$
$\displaystyle=L_{g_{m}}\frac{\delta H_{m}}{\delta\mu_{m}},$
$\displaystyle\text{and}\quad\frac{dv_{m}}{dt}$
$\displaystyle=L_{g}\frac{\delta H_{m}}{\delta\rho_{m}}.$

Now, with $(g_{e},v_{e})=(g_{s}^{-1}g_{m},-g_{s}^{-1}v_{m}+g_{s}^{-1}v_{s})$,
its time evolution is

$\displaystyle\frac{d}{dt}(g_{e},v_{e})$
$\displaystyle=\left(-g_{s}^{-1}\frac{dg_{s}}{dt}g_{s}^{-1}g_{m}+g^{-1}_{s}\frac{dg_{m}}{dt},-g_{s}^{-1}\frac{dg_{s}}{dt}g_{s}^{-1}v_{m}+g_{s}^{-1}\frac{dv}{dt}+g_{s}^{-1}\frac{dg_{s}}{dt}g_{s}^{-1}v_{s}-g_{s}^{-1}\frac{dv_{s}}{dt}\right)$
$\displaystyle=\left(-\left[\frac{\delta
H_{s}}{\delta\mu}-\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right]g_{e},-\frac{\delta
H_{s}}{\delta\mu}v_{e}+g_{e}\frac{\delta H_{m}}{\delta\rho}-\frac{\delta
H_{s}}{\delta\rho}\right).$

Defining the coadjoint action of the error group action $(g_{e},v_{e})$ on
$(\mu_{e,0},\rho_{0})$, by
$(\mu_{e},\rho_{e})=\mathrm{Ad}^{\ast}_{(g_{e},v_{e})^{-1}}(\mu_{e,0},\rho_{0})\in\mathfrak{g}^{\ast}$,
its time evolution is governed by

$\displaystyle\partial_{t}\left(\mu_{e},\rho_{e}\right)$
$\displaystyle=\left(\mathrm{ad}^{\ast}_{\left(\frac{\delta
H_{s}}{\delta\mu}-\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right)}\mu_{e}+\left(-\left[\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right]v_{e}+g_{e}\frac{\delta
H_{m}}{\delta\rho}-\frac{\delta
H_{s}}{\delta\rho}\right)\diamond\rho_{e},\right.$ (6.24)
$\displaystyle\left.\left(\frac{\delta
H_{s}}{\delta\mu}-\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right)\rho_{e}\right),$

Further, let us define a Hamiltonian function $H_{\mathrm{err}}$, as

$\displaystyle H_{\mathrm{err}}(\mu_{e},\rho_{e},t)$
$\displaystyle=H_{s}(\mu_{e},\rho_{e})+H_{m}(\mathrm{Ad}^{\ast}_{g_{e}^{-1}}(\mu_{e},\rho_{e}),t).$
(6.25)

and the functional derivative are

$\displaystyle\frac{\delta H_{\mathrm{err}}}{\delta\mu_{e}}$
$\displaystyle=\frac{\delta H_{s}}{\delta\mu}-\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu},$ $\displaystyle\frac{\delta
H_{\mathrm{err}}}{\delta\rho_{e}}$
$\displaystyle=-\left[\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right]v_{e}+g_{e}\frac{\delta
H_{m}}{\delta\rho}-\frac{\delta H_{s}}{\delta\rho},$

and a candidate for that is
$H_{\mathrm{err}}=\frac{1}{2}\left\langle\mu_{e},\frac{\delta
H_{s}}{\delta\mu}-\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right\rangle$. Here, for the deep learning training, $H_{s}$
is (6.17) and $H_{m}$ is exactly (6.23). In order to carry out with stability
analysis, we need to write $H_{\mathrm{err}}$ in terms of $(\mu_{e},\rho_{e})$
and that is done by using
$(\mu_{e},\rho_{e})=\mathrm{Ad}^{\ast}_{(g_{e},v_{e})^{-1}}(\mu_{e,0},\rho_{e,0})$.
Let $(\mu_{e,0},\rho_{e,0})$ be the error between the exact solution and the
numerical one at time $t=0$ and it is set as
$(\mu_{e,0},\rho_{e,0})=(\mu_{s,0}-\mu_{m,0},\rho_{s,0}-\rho_{m,0})$. At the
same time, we have the following
$(\mu_{s},\rho_{s})=\mathrm{Ad}^{\ast}_{(g_{s},v_{s})^{-1}}(\mu_{s,0},\rho_{s,0})$
and
$(\mu_{m},\rho_{m})=\mathrm{Ad}^{\ast}_{(g,v)^{-1}}(\mu_{m,0},\rho_{m,0})$.
Then, we can write $(\mu_{e},\rho_{e})$ in terms of $(\mu_{s},\rho_{s})$ and
$(\mu_{m},\rho_{m})$ using

$\displaystyle\left(\mu_{e},\rho_{e}\right)$
$\displaystyle=\mathrm{Ad}^{\ast}_{(g_{e},v_{e})^{-1}}\left(\mathrm{Ad}^{\ast}_{(g_{s},v_{s})}(\mu,\rho)-\mathrm{Ad}^{\ast}_{(g_{m},v_{m})}(\mu,\rho)\right)$
(6.26)
$\displaystyle=\mathrm{Ad}^{\ast}_{(g_{e},v_{e})^{-1}}\left(\mathrm{Ad}^{\ast}_{g_{s}}\mu-\mathrm{Ad}^{\ast}_{g_{m}}\mu+v_{s}\diamond((g_{s}^{-1})^{\ast}\rho)\right.$
$\displaystyle+\left.v_{m}\diamond((g_{m}^{-1})^{\ast}\rho),((g_{s}^{-1})^{\ast}\rho-((g_{m}^{-1})^{\ast}\rho\right)$
$\displaystyle=\left(\mathrm{Ad}^{\ast}_{g_{e}^{-1}}\mathrm{Ad}^{\ast}_{g_{s}}\mu-\mathrm{Ad}^{\ast}_{g_{e}^{-1}}\mathrm{Ad}^{\ast}_{g_{m}}\mu+\mathrm{Ad}^{\ast}_{g_{e}^{-1}}\left(v_{s}\diamond((g_{s}^{-1})^{\ast}\rho)\right)\right.$
$\displaystyle\left.+\mathrm{Ad}^{\ast}_{g_{e}^{-1}}\left(v_{m}\diamond((g_{m}^{-1})^{\ast}\rho)\right)+v_{e}\diamond\left(((g_{m}^{-1})^{\ast}\rho-((g_{s}^{-1})^{\ast}\rho\right),\right.$
$\displaystyle\left((g_{m}^{-1})^{\ast}\rho-((g_{s}^{-1})^{\ast}\rho\right).$

###  6.3 Nonlinear stability of deep learning

From the Lie–Poisson structure of the deep learning problem defined on the
semidirect product group of diffeomorphisms and densities,
$\mathrm{Diff}(Q)\circledS\mathrm{Dens}(Q)$, the Casimir function should be
the same as for the two-dimensional barotropic flow [54] and it is given by

$\displaystyle
C_{\varphi}=\int_{Q}\rho\varphi\left(\frac{\omega}{\rho}\right)d^{n}x,$

where $\varphi:\mathbb{R}\to\mathbb{R}$ is an arbitrary function and

$\displaystyle\omega=d\left(\frac{1}{\rho}\frac{\delta\ell}{\delta w}\cdot
dx\right).$ (6.27)

and the quantity $\frac{\omega}{\rho}$ is conserved according to equation
(6.8). Another conserved quantity that, stated in lemma 6.1, is the Kelvin’s
circulation

$\displaystyle I_{i}(\mu,\rho)=\oint_{(\partial
Q)_{i}}\frac{1}{\rho}\frac{\delta\ell}{\delta w}\cdot dx,$

where $(\partial Q)_{i}$ is a component of the boundary of the manifold $Q$.
Now, we have the Casimir $C_{\varphi}$ and Kelvin’s circulations $I$, we
augment to the error Hamiltonian $H_{\mathrm{err}}$, to get the following new
Hamiltonian

$\displaystyle
H_{C}(\mu,\rho)=H_{\mathrm{err}}(\mu,\rho)+C_{\varphi}(\mu,\rho)+\sum\limits_{i}a_{i}I_{i}(\mu,\rho).$
(6.28)

We assume that the equilibrium solution of (6.24) is $(\mu_{e},\rho_{e})$.
Applying the the energy–Casimir method, where the calculation it closely
follows [54] and for the sake of self-containing of this article, it is
reproduced in Appendix A. For this particular application, the nonlinear
stability is measured with the metric $\mathrm{d}(\mu,\rho)$, the same as in
[54], defined by

$\displaystyle\mathrm{d}(\mu,\rho)=\left\|(\Delta\mu,\Delta\rho)\right\|^{2}=Q_{1}(\Delta\mu,\Delta\rho)+Q_{2}(\Delta\mu,\Delta\rho),$

where

$\displaystyle Q_{1}(\Delta\mu,\Delta\rho)$
$\displaystyle=\frac{1}{2}\int_{Q}\|\Delta(\mu\rho)\|^{2}+\left(\frac{c_{1}^{2}}{\rho_{\mathrm{max}}}-\frac{|\mu_{e}|^{2}}{\rho_{\mathrm{min}}}\right)\left\|\Delta\rho\right\|^{2}d^{n}x,$
$\displaystyle\text{and}\quad Q_{2}(\Delta\mu,\Delta\rho)$
$\displaystyle=\frac{c_{2}}{2}\rho_{\mathrm{min}}\int_{Q}\left(\Delta\left(\frac{\omega}{\rho}\right)\right)^{2}d^{n}x.$

and $c_{1}$ and $c_{2}$ must chosen so that

$\displaystyle c_{1}\leq H_{\mathrm{err}}^{\prime\prime},\quad
c_{2}\leq\frac{1}{z}K^{\prime}(z),$

where

$\displaystyle
K\left(\frac{\omega}{\rho}\right)=-\left[\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right]v_{e}+g_{e}\frac{\delta
H_{m}}{\delta\rho}-\frac{\delta H_{s}}{\delta\rho}.$ (6.29)

###### Remark 6.5.

In fluid dynamics, the function $K$ is the Bernoulli Law.

###### Theorem 6.6 (Nonlinear stability theorem for deep learning).

Let $\Sigma=(Q,f,\mathfrak{X}(Q))$ be a stochastic control system, which
describe the limit when the number of hidden layers of a residual neural
networks approaches infinity, and
$\Sigma_{\rho}=(\mathrm{Dens}(Q),f,\mathfrak{X}(Q))$ be the mean–field type
control equivalent. Given the training dataset
$\\{(x_{0}^{(i)},c^{(i)}\\}_{i=0}^{N}$, and the cost functional

$\displaystyle\mathcal{S}_{\Sigma_{\rho}}=\sum\limits_{i=0}^{N}\int_{Q}\Phi(\pi(\rho(x,T)),c^{(i)})dx+\int_{0}^{T}\int_{Q}\rho
L(w+\nabla\mathrm{log}(\rho))dxdt,$

where $L:\mathfrak{X}(Q)\to\mathbb{R}$ is the cost function. The network
parameters $\theta^{\ast}$ and trajectory $\rho^{\ast}$ that satisfy

$\displaystyle\mathcal{S}_{\Sigma_{\rho}}(\rho^{\ast},\theta^{\ast})<\mathcal{S}_{\Sigma_{\rho}}(\rho,\theta),\quad\forall(\rho,\theta)\in\mathrm{Dens}(Q)\times\mathfrak{X}(Q).$

Let $(\widetilde{\rho},\widetilde{\theta})$ be the numerical solution obtained
by the multisymplectic variational integrator whose Lagrangian is (4.1), and
it satisfies the modified Euler–Poincaré equation (3.21). Given the norm
defined by

$\displaystyle\|(\mu,\rho)\|^{2}=\frac{1}{2}\int_{Q}\|\Delta(\mu\rho)\|^{2}+\left(\frac{c_{1}^{2}}{\rho_{\mathrm{max}}}-\frac{|\mu_{e}|^{2}}{\rho_{\mathrm{min}}}\right)\left\|\Delta\rho\right\|^{2}d^{n}x+c_{2}\int_{Q}\left\|\Delta\left(\frac{\omega}{\rho}\right)\right\|^{2}d^{n}x.$
(6.30)

The error between the numerical error $(\widetilde{\rho},\widetilde{\theta})$
and the exact analytical solution $(\rho^{\ast},\theta^{\ast})$, the following
inequality holds

$\displaystyle\|(\widetilde{\mu}(t)-\mu^{\ast},\widetilde{\rho}(t)-\rho^{\ast})\|^{2}\leq(C_{1}+C_{2})\|(\widetilde{\mu}(0)-\mu^{\ast},\widetilde{\rho}(0)-\rho^{\ast})\|^{2},$
(6.31)

provided it satisfies

$\displaystyle 0$
$\displaystyle<c_{1}<H_{\mathrm{err}}^{\prime\prime}<\infty,$ (6.32)
$\displaystyle 0$ $\displaystyle<c_{2}<K^{\prime}<\infty,$

where $K$ is (6.29) and the Hamiltonian $H_{\mathrm{err}}$ defined by (6.25).

From the second line in (6.32), the bound depends on the derivative of the
function, $K$, and the its interest for deep learning is that parameters such
as the number of nodes per hidden layers, number of hidden layers and learning
rate appear directly. Thus, such parameters directly influence the nonlinear
stability of the training’s error. Consider the numerical solution of the deep
learning training problem $(\widetilde{\mu},\widetilde{\rho})$ and assume that
there exists an exact solution of the training problem
$(\mu^{\ast},\rho^{\ast})$. Here, we consider the case when
$(\widetilde{\mu},\widetilde{\rho})$ is close to $(\mu^{\ast},\rho^{\ast})$,
and making the approximation that $(g_{e},v_{e})\approx(\mathrm{Id},0)$. As
$K$ is in terms of $z$, we introduces the vector valued map
$\zeta:\mathbb{R}\to Q$. Thus, we can parametrise $w$ and density $\rho$ in
$K$ by $z$. The reduced function (6.29) is

$\displaystyle K\left(z\right)$ $\displaystyle\approx\frac{\delta
H_{m}}{\delta\rho}-\frac{\delta H_{s}}{\delta\rho}$
$\displaystyle\approx\frac{(\Delta
x^{j})^{2}}{4}\left(w^{j}\partial_{x^{j}x^{j}}w^{j}+w^{i}\partial_{x^{j}x^{j}}w^{j}+\frac{1}{2}\left(\partial_{x^{j}}w^{j}\right)^{2}+\frac{1}{2}\left(\partial_{x^{j}}w^{i}\right)^{2}\right)$
$\displaystyle+\frac{(\Delta
t)^{2}}{4}\left(w^{j}\partial_{tt}w^{j}+\frac{1}{2}\left(\frac{\partial}{\partial
t}w^{j}\right)^{2}+\frac{1}{2}\left(\frac{\partial}{\partial
t}w^{i}\right)^{2}\right)$ $\displaystyle+\frac{\nu^{4}(\Delta
x^{j})^{2}}{8}\frac{(\partial_{x^{j}}\rho)(\partial_{x^{j}x^{j}x^{j}}\rho)+(\partial_{x^{i}}\rho)(\partial_{x^{i}x^{j}x^{j}}\rho)}{\left(\rho+\frac{(\Delta
x^{j})^{2}}{8}\partial_{x^{j}x^{j}}\rho+\frac{(\Delta
t)^{2}}{8}\partial_{tt}\rho\right)^{2}}+\frac{\nu^{4}(\Delta
t)^{2}}{8}\frac{(\partial_{x^{j}}\rho)\partial_{x^{j}tt}\rho+(\partial_{x^{i}}\rho)\partial_{x^{i}tt}\rho}{\left(\rho+\frac{(\Delta
x^{j})^{2}}{8}\partial_{x^{j}x^{j}}\rho+\frac{(\Delta
t)^{2}}{8}\partial_{tt}\rho\right)^{2}}$ $\displaystyle-\frac{\nu^{4}(\Delta
x^{j})^{2}}{4}\frac{\rho^{2}(\partial_{x^{j}}\rho)(\partial_{x^{j}x^{j}x^{j}}\rho)+\rho^{2}(\partial_{x^{i}}\rho)(\partial_{x^{i}x^{j}x^{j}}\rho)+\rho\left(\partial_{x^{j}}\rho\right)^{2}(\partial_{x^{j}x^{j}}\rho)+\rho(\partial_{x^{j}x^{j}}\rho)\left(\partial_{x^{i}}\rho\right)^{2}}{\left(\rho+\frac{(\Delta
x^{j})^{2}}{8}\partial_{x^{j}x^{j}}\rho+\frac{(\Delta
t)^{2}}{8}\partial_{tt}\rho\right)^{4}}$ $\displaystyle-\frac{\nu^{4}(\Delta
t)^{2}}{4}\frac{\rho^{2}(\partial_{x^{j}}\rho)\partial_{x^{j}tt}\rho+\rho(\partial_{tt}\rho)\left(\partial_{x^{j}}\rho\right)^{2}+\rho(\partial_{tt}\rho)\left(\partial_{x^{i}}\rho\right)^{2}}{\left(\rho+\frac{(\Delta
x^{j})^{2}}{8}\partial_{x^{j}x^{j}}\rho+\frac{(\Delta
t)^{2}}{8}\partial_{tt}\rho\right)^{4}}$

Taking the derivative of $K$ with respect to $z$, requires the chain rule, and
for now we write it abstractly we get

$\displaystyle K^{\prime}(z)$
$\displaystyle=\frac{\partial}{\partial\zeta^{j}}\left(\frac{\delta
H_{m}}{\delta\rho}-\frac{\delta
H_{s}}{\delta\rho}\right)\left(\partial_{z}\zeta^{j}\right).$

The derivative $\partial_{z}\zeta^{j}$ is computed by taking the derivative of
$\zeta$ with respect to $x^{j}$ using the chain rule and then solve for
$\partial_{z}\zeta^{j}$ to obtain

$\displaystyle\partial_{z}\zeta^{j}=\frac{\frac{1}{\rho}\frac{\partial\omega}{\partial
x^{j}}-\frac{\omega}{\rho^{2}}\frac{\partial\rho}{\partial
x^{j}}}{\sum\limits_{i=1}^{n}(\frac{1}{\rho}\frac{\partial\omega}{\partial
x^{i}}-\frac{\omega}{\rho^{2}}\frac{\partial\rho}{\partial x^{i}})^{2}}.$
(6.33)

In order to ensure that the inequality is valid, we need
$\|\partial_{z}\zeta^{j}\|\neq 0$ and it is true only if

$\displaystyle\frac{1}{\rho}\frac{\partial\omega}{\partial
x^{j}}-\frac{\omega}{\rho^{2}}\frac{\partial\rho}{\partial x^{j}}\neq 0.$
(6.34)

The reason for that is because $0<c_{2}<K^{\prime}$ ceases to be valid when
$\partial_{z}\zeta^{j}=0$ and this is easy to see when writing it in terms of
chain rule and Hölder inequality

$\displaystyle
c_{2}<\|\partial_{\zeta^{j}}K\|^{2}\|\partial_{z}\zeta^{j}\|^{2}.$

The exterior derivative of the one for
$\omega=d\left(\frac{1}{\rho}\frac{\delta\ell}{\delta w}\cdot dx\right)$ when
expanded, we obtain

$\displaystyle\omega=d\left(w\cdot dx+\nabla\mathrm{log}{\rho}\cdot
dx+\frac{(\Delta x^{j})^{2}}{8}\partial_{x^{j}x^{j}}w\cdot dx+\frac{(\Delta
t)^{2}}{8}\partial_{tt}w\cdot dx\right)$

We substitute it into (6.34) and it becomes

$\displaystyle\frac{1}{\rho}\frac{\partial\omega}{\partial x^{j}}$
$\displaystyle+\frac{(\Delta
x^{j})^{2}}{8}\frac{1}{\rho}\frac{\partial\omega_{1}}{\partial
x^{j}}+\frac{(\Delta
t)^{2}}{8}\frac{1}{\rho}\frac{\partial\omega_{2}}{\partial x^{j}}$ (6.35)
$\displaystyle-\frac{\omega}{\rho^{2}}\frac{\partial\rho}{\partial
x^{j}}-\frac{(\Delta
x^{j})^{2}}{8}\frac{\omega_{1}}{\rho^{2}}\frac{\partial\rho}{\partial
x^{j}}-\frac{(\Delta
t)^{2}}{8}\frac{\omega_{2}}{\rho^{2}}\frac{\partial\rho}{\partial x^{j}}\neq
0.$

and here we impose that this relation always positive and we write it as the
following inequality

$\displaystyle\partial_{x^{j}}\left(\frac{\omega}{\rho}\right)+\frac{(\Delta
x^{j})^{2}}{8}\partial_{x^{j}}\left(\frac{\omega_{1}}{\rho}\right)+\frac{(\Delta
t)^{2}}{8}\partial_{x^{j}}\left(\frac{\omega_{2}}{\rho}\right)>0.$

Multiplying both sides by a function $\chi$, such that
$\partial_{x^{j}}\chi\neq 0$ and $\|-\partial_{x^{j}}\chi\|_{\infty}>0$, we
get the following inequality

$\displaystyle 0$
$\displaystyle<\int_{Q}\left(\partial_{x^{j}}\left(\frac{\omega}{\rho}\right)\right)\chi+\frac{(\Delta
x^{j})^{2}}{8}\left(\partial_{x^{j}}\left(\frac{\omega_{1}}{\rho}\right)\right)\chi+\frac{(\Delta
t)^{2}}{8}\left(\partial_{x^{j}}\left(\frac{\omega_{2}}{\rho}\right)\right)\chi$
$\displaystyle<\int_{Q}\frac{\omega}{\rho}\left(-\partial_{x^{j}}\chi\right)+\frac{(\Delta
x^{j})^{2}}{8}\frac{\omega_{1}}{\rho}\left(-\partial_{x^{j}}\chi\right)+\frac{(\Delta
t)^{2}}{8}\frac{\omega_{2}}{\rho}\left(-\partial_{x^{j}}\chi\right)$
$\displaystyle<\|\frac{\omega}{\rho}+\frac{(\Delta
x^{j})^{2}}{8}\frac{\omega_{1}}{\rho}+\frac{(\Delta
t)^{2}}{8}\frac{\omega_{2}}{\rho}\|_{1}\|-\partial_{x^{j}}\chi\|_{\infty}.$

Utilising the Hölder inequality again, the above inequality can turns into

$\displaystyle 0$ $\displaystyle<\|\frac{\omega}{\rho}+\frac{(\Delta
x^{j})^{2}}{8}\frac{\omega_{1}}{\rho}+\frac{(\Delta
t)^{2}}{8}\frac{\omega_{2}}{\rho}\|_{1}$ $\displaystyle<\|\omega+\frac{(\Delta
x^{j})^{2}}{8}\omega_{1}+\frac{(\Delta
t)^{2}}{8}\omega_{2}\|_{\infty}\|\frac{1}{\rho}\|_{1}$

and with from the Triangle inequality, we have the following relationship

$\displaystyle 0<\|\omega\|_{\infty}+\frac{(\Delta
x^{j})^{2}}{8}\|\omega_{1}\|_{\infty}+\frac{(\Delta
t)^{2}}{8}\|\omega_{2}\|_{\infty},$

which results in

$\displaystyle\frac{\frac{(\Delta
x^{j})^{2}}{8}\|\omega_{1}\|_{\infty}+\frac{(\Delta
t)^{2}}{8}\|\omega_{2}\|_{\infty}}{\|\omega\|_{\infty}}<1.$

###### Corollary 6.7.

Problem (4.1) for training of stochastic neural network is nonlinearly stable,
if the number of of hidden layers $N_{t}$ and the number of nodes per layer
$\prod\limits_{k=0}^{d}N_{x^{k}}$ satisfy

$\displaystyle\frac{\|\omega_{1}\|_{\infty}}{8(N_{x^{j}})^{2}\|\omega\|_{\infty}}+\frac{\|\omega_{2}\|_{\infty}}{8(N_{t})^{2}\|\omega\|_{\infty}}<1.$
(6.36)

##  7 Conclusion

In this paper, we have formulated deep learning training as a multisymplectic
system that appears in hydrodynamics. Using mean–field type control to solve
the training of a stochastic deep network, the resultant necessary conditions
for optimality yield a system of equations which we have shown that it can be
reduced to the Euler-Poincaré equation. The family of equations are the link
between deep learning and fluids and they have rich geometric properties. The
underlying geometric structure of the reduced equations have allowed us to
derived a structure–preserving multisymplectic integrator to solve the
training problem. We have derived the modified equation for the integrator and
we have applied energy–Casimir method and we have shown that the error of the
training procedure is nonlinearly stable. Using other related variational
principles and different cost Lagrangian that yield interesting singular
solutions is the focus of future work.

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##  Appendix A Energy–Casimir method for deep learning: calculation

Here, we carry out the calculation of the energy–Casimir method and it follows
many of the steps in the energy–Casimir calculation for the two-dimensional
barotropic flow [54]. Starting with the error Hamiltonian with integrals of
motions

$\displaystyle
H_{C}(\mu,\rho)=H_{\mathrm{err}}(\mu,\rho)+C_{\varphi}(\mu,\rho)+\sum\limits_{i}a_{i}I_{i}(\mu,\rho).$
(A.1)

The equilibrium solution makes the first variation of the Hamiltonian $H_{C}$
vanish when the following condition for all $\delta\mu,\delta\rho$ holds:

$\displaystyle 0$ $\displaystyle=\left\langle\frac{\delta
H_{C}}{(\delta\mu,\delta\rho)},(\delta\mu,\delta\rho)\right\rangle$ (A.2)
$\displaystyle=\int_{\mathrm{Q}}\left\langle\rho_{\mathrm{err}}\left[\frac{\delta
H_{s}}{\delta\mu}-\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}-\sum\limits_{i_{1},i_{2},i_{3}=1}^{d+1}\epsilon_{i_{1},i_{2},i_{3}}\mathbf{e}_{i_{1}}\delta_{i_{2},d+1}\partial_{x^{i_{3}}}\varphi^{\prime}\left(\frac{\omega}{\rho}\right)\right],\delta\mu\right\rangle$
$\displaystyle+\int_{\mathrm{Q}}\left[-\left(\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right)v_{e}+g_{e}\frac{\delta
H_{m}}{\delta\rho}-\frac{\delta
H_{s}}{\delta\rho}+\varphi\left(\frac{\omega}{\rho}\right)-\varphi^{\prime}\left(\frac{\omega}{\rho}\right)\frac{\omega}{\rho}\right]\delta\rho$
$\displaystyle+\sum\limits_{i}\oint_{\partial
Q_{i}}\varphi^{\prime}\left(\frac{\omega}{\rho}\right)\delta\mu\cdot
dx+a_{i}\delta\mu\cdot dx.$

From the first variation of $H_{C}$, computed in (A.2), holds provided the
following are satisfied

$\displaystyle
a_{i}=-\varphi^{\prime}\left(\left.\frac{\omega_{e}}{\rho_{e}}\right|_{(\partial
Q)_{i}}\right),$ (A.3)

and

$\displaystyle K(z)+\varphi(z)-z\varphi^{\prime}(z)=0,$ (A.4)

where $K$ is the variational derivative of $H_{\mathrm{err}}$ with respect to
$\rho_{\mathrm{err}}$ and it is

$\displaystyle
K\left(\frac{\omega}{\rho}\right)=-\left[\mathrm{Ad}_{g_{e}}\frac{\delta
H_{m}}{\delta\mu}\right]v_{e}+g_{e}\frac{\delta
H_{m}}{\delta\rho}-\frac{\delta H_{s}}{\delta\rho}.$ (A.5)

Solving the differential equation (A.4), we obtain

$\displaystyle\varphi(z)=z\left(\int^{z}\frac{K(t)}{t^{2}}\,dt+C\right),$

where $C\in\mathbb{R}$ is a constant. For the convexity estimates, we only
perform it for $Q_{2}$, as $Q_{1}$ is set

$\displaystyle Q_{1}(\Delta\mu,\Delta\rho)$
$\displaystyle=\frac{1}{2}\int_{Q}\|\Delta(\mu\rho)\|^{2}+\left(\frac{c_{1}^{2}}{\rho_{\mathrm{max}}}-\frac{|\mu_{e}|^{2}}{\rho_{\mathrm{min}}}\right)\left\|\Delta\rho\right\|^{2}d^{n}x,$

where $c_{1}\in\mathbb{R}$ and it should satisfy $c_{1}<\delta^{2}H$. The
$L^{2}$ norm for $\Delta\rho$ is suitable as the terms in $H$ are derivatives
of $\Delta\rho$ and according to Poincaré inequality, then $L^{2}$ is bounded
by Sobolev norms. On the other hand, $Q_{2}$ is required to satisfy

$\displaystyle
Q_{2}(\Delta\mu,\Delta\rho)\leq\int_{Q}\varphi\left(\frac{\omega}{\rho}+\Delta\left(\frac{\omega}{\rho}\right)\right)-\varphi\left(\frac{\omega}{\rho}\right)-\varphi^{\prime}\left(\frac{\omega}{\rho}\right)\cdot\left(\Delta\left(\frac{\omega}{\rho}\right)\right)d^{n}x.$

One candidate is the $L^{2}$ norm

$\displaystyle
Q_{2}\left(\Delta\left(\frac{\omega}{\rho}\right)\right)=\frac{c_{2}}{2}\rho_{\mathrm{min}}\int_{Q}\left(\Delta\left(\frac{\omega}{\rho}\right)\right)^{2}d^{n}x,$
(A.6)

where $c_{2}\in\mathbb{R}$ is a constant and it is required to satisfy
$c_{2}\leq\varphi^{\prime\prime}(z)$ and $c_{2}>0$ for all $z$. Thus, the
convexity holds when $c_{1},c_{2}>0$, $c_{1}\leq
H_{\mathrm{err}}^{\prime\prime}$ and $c_{2}\leq\varphi^{\prime\prime}$. Next,
we determine a priori estimates of $Q_{1}$ and $Q_{2}$ and using the
inequalities for them we arrive at

$\displaystyle
Q_{1}\left(\Delta\mu(t),\Delta\rho(t)\right)+Q_{2}\left(\Delta\mu(t),\Delta\rho(t)\right)$
$\displaystyle\leq H_{C}(\mu+\Delta\mu(t),\rho+\Delta\rho(t))$ $\displaystyle-
H_{C}(\mu,\rho)-\left\langle\delta H_{C},(\delta\mu,\delta\rho)\right\rangle$
$\displaystyle\leq
H_{C}(\mu+\Delta\mu(t),\rho+\Delta\rho(t))-H_{C}(\mu,\rho).$

The term $\left\langle\delta H_{C},(\delta\mu,\delta\rho)\right\rangle$
vanishes as a result of setting the first variation of $H_{C}$ to be zero at
the equilibrium. Since $H_{C}$ is conserved by the Lie–Poisson flow, it is
conserved, hence

$\displaystyle
Q_{1}\left(\Delta\mu(t),\Delta\rho(t)\right)+Q_{2}\left(\Delta\mu(t),\Delta\rho(t)\right)$
$\displaystyle\leq
H_{C}(\mu+\Delta\mu(0),\rho+\Delta\rho(0))-H_{C}(\mu,\rho).$

This means that $Q_{1}+Q_{2}$ is bounded by the initial values
$(\Delta\mu(0),\Delta\rho(0))$. After establishing the convexity and a priori
estimate for $Q_{1}$ and $Q_{2}$, we set the norm

$\displaystyle\left\|(\Delta\mu,\Delta\rho)\right\|^{2}=Q_{1}(\Delta\mu,\Delta\rho)+Q_{2}(\Delta\mu,\Delta\rho),$
(A.7)

and with

$\displaystyle c_{1}\leq H_{\mathrm{err}}^{\prime\prime},\quad
c_{2}\leq\frac{1}{z}K^{\prime}(z),$

we have Lyapunov stability and the error is bounded.


230 | B1 IV | 25 | 4.59 | 4.0 | $-8.7$ | 0.05 | $-4.6$ | 10.5 | 2400 | 180 | 170 | 41 | 46.7 | $-0.5$
231 | O3 V((f*))z | 46 | 5.61 | 4.1 |  $-7.5$∗ | 0.08 | $-5.1$ | 10.1 | 2900∗ | 100 | 170 | 47 | 49.4 |  $1.3$∗
232 | B0.5 V | 27 | 4.03 | 4.4 | $-9.3$ | 0.05 | $-2.5$ | 4.7 | 2600 | 110 | 170 | 21 | 46.3 | $-1.5$
233 | B1.5 (IV)e | 23 | 4.37 | 3.8 | $-8.9$ | 0.15 | $-3.8$ | 9.7 | 1800 | 120 | 170 | 22 | 46.2 | $-0.9$
234 | B3 IV | 18 | 3.62 | 3.8 | $-9.7$ | 0.05 | $-2.6$ | 6.7 | 800 | 100 | 170 | 10 | 44.6 | $-2.3$
235 | B2 (IV)e | 21 | 4.34 | 3.6 | $-9.0$ | 0.15 | $-3.7$ | 11.2 | 1500 | 120 | 170 | 18 | 45.8 | $-1.0$
236 | B2.5 V | 19 | 3.72 | 4.2 | $-9.6$ | 0.05 | $-2.4$ | 6.7 | 1200 | 200 | 170 | 26 | 44.6 | $-2.1$
237 | B0.7 V | 27 | 4.26 | 4.2 | $-9.1$ | 0.05 | $-3.3$ | 6.2 | 2300 | 200 | 170 | 22 | 46.5 | $-1.1$
238 | O8.5 V | 34 | 4.88 | 4.2 | $-8.4$ | 0.05 | $-4.6$ | 8.0 | 2300 | 140 | 170 | 37 | 48.2 | $0.0$
239 | O9 V | 34 | 4.73 | 4.2 | $-8.6$ | 0.05 | $-3.4$ | 6.7 | 2100 | 75 | 170 | 26 | 48.0 | $-0.3$
240 | B2 V | 22 | 4.04 | 4.2 | $-9.3$ | 0.05 | $-3.4$ | 7.2 | 2500 | 90 | 170 | 30 | 45.5 | $-1.5$
241 | O7 V | 37 | 5.1 | 4.2 | $-8.2$ | 0.09 | $-4.8$ | 8.7 | 2300 | 150 | 170 | 43 | 48.6 | $0.4$
242 | B1.5 V | 23 | 3.69 | 4.4 | $-9.6$ | 0.05 | $-1.8$ | 4.4 | 2500 | 85 | 170 | 18 | 45.2 | $-2.1$
243 | B2.5 V | 19 | 3.82 | 4.0 | $-9.5$ | 0.05 | $-3.4$ | 7.5 | 1000 | 110 | 160 | 21 | 44.8 | $-1.9$
244 | B2 V | 21 | 3.86 | 4.0 | $-9.4$ | 0.05 | $-2.4$ | 6.4 | 1900 | 100 | 170 | 15 | 45.1 | $-1.8$
245 | B1.5 (IV)e | 22 | 3.84 | 3.8 | $-9.5$ | 0.10 | $-2.3$ | 5.7 | 1400 | 300 | 160 | 8 | 45.5 | $-1.9$
246 | B1 V | 25 | 4.25 | 4.2 | $-9.1$ | 0.03 | $-3.4$ | 7.1 | 2500 | 180 | 180 | 29 | 46.2 | $-1.1$
247 | B5 V | 16 | 3.42 | 3.8 | $-9.9$ | 0.05 | $-2.2$ | 6.7 | 800 | 90 | 160 | 10 | 44.3 | $-2.6$
248 | B0.2 V | 30 | 4.2 | 4.4 | $-9.1$ | 0.05 | $-2.5$ | 4.7 | 2500 | 120 | 170 | 20 | 46.8 | $-1.2$
249 | B2 (III)e | 20 | 4.61 | 3.2 | $-8.7$ | 0.10 | $-4.4$ | 16.9 | 600 | 300 | 170 | 16 | 45.9 | $-0.5$
250 | B2 V | 22 | 3.81 | 3.8 | $-9.5$ | 0.05 | $-2.2$ | 5.5 | 1400 | 120 | 170 | 7 | 45.4 | $-1.9$
251 | B1.5 V | 23 | 3.76 | 4.4 | $-9.5$ | 0.05 | $-1.7$ | 4.8 | 2600 | 80 | 170 | 21 | 45.2 | $-2.0$
252 | B9 Ia | 10 | 3.92 | 2.6 | $-9.4$ | 0.05 | $-4.4$ | 30.5 | 400 | 100 | 170 | 13 | 43.4 | $-1.7$
253 | B1.5 V | 23 | 3.92 | 4.4 | $-9.4$ | 0.05 | $-1.9$ | 5.8 | 2900 | 130 | 170 | 30 | 45.4 | $-1.7$
254 | B1.5 V | 21 | 4.06 | 4.0 | $-9.3$ | 0.05 | $-3.4$ | 8.1 | 2100 | 120 | 170 | 24 | 45.3 | $-1.5$
255 | B2 V | 20 | 3.56 | 4.2 | $-9.7$ | 0.05 | $-2.2$ | 5.0 | 1100 | 80 | 140 | 15 | 44.6 | $-2.4$
256 | B1.5 V | 23 | 3.86 | 4.4 | $-9.4$ | 0.05 | $-2.4$ | 5.4 | 2800 | 80 | 170 | 27 | 45.3 | $-1.8$
257 | B0.5 V | 27 | 4.46 | 4.2 | $-8.9$ | 0.05 | $-4.0$ | 7.8 | 2600 | 100 | 170 | 35 | 46.8 | $-0.8$
258 | B0.5 (III)e | 25 | 4.69 | 3.4 | $-8.6$ | 0.12 | $-4.8$ | 11.8 | 1100 | 300 | 150 | 13 | 47.0 | $-0.4$
259 | B0.7 V | 27 | 4.03 | 4.4 | $-9.3$ | 0.05 | $-2.5$ | 4.7 | 2600 | 110 | 175 | 21 | 46.3 | $-1.5$
260 | B0.5 V | 27 | 4.26 | 4.2 | $-9.1$ | 0.05 | $-2.9$ | 6.2 | 2300 | 160 | 160 | 22 | 46.5 | $-1.1$
261 | B0.7 V | 26 | 3.82 | 4.4 | $-9.5$ | 0.05 | $-2.3$ | 4.0 | 2400 | 80 | 170 | 15 | 46.0 | $-1.9$
262 | B2 (III)e | 20 | 4.28 | 3.2 | $-9.0$ | 0.10 | $-3.6$ | 11.5 | 500 | 180 | 140 | 8 | 45.6 | $-1.1$
263 | B0.2 V | 29 | 4.44 | 4.2 | $-8.9$ | 0.05 | $-3.5$ | 6.6 | 2300 | 80 | 170 | 25 | 47.0 | $-0.8$
264 | B2.5 III | 17 | 3.95 | 3.2 | $-9.4$ | 0.05 | $-3.3$ | 10.9 | 500 | 80 | 170 | 7 | 45.0 | $-1.7$
265 | B0.5 IV | 27 | 4.75 | 3.8 | $-8.6$ | 0.08 | $-4.4$ | 10.9 | 1800 | 120 | 180 | 27 | 47.2 | $-0.3$
266 | B1 V | 25 | 4.35 | 4.0 | $-9.0$ | 0.02 | $-3.6$ | 8.0 | 2100 | 150 | 160 | 23 | 46.4 | $-1.0$
267 | B2 V | 20 | 3.64 | 4.2 | $-9.7$ | 0.05 | $-2.2$ | 5.5 | 1100 | 80 | 170 | 18 | 44.7 | $-2.2$
268 | B2 V | 20 | 3.64 | 4.2 | $-9.7$ | 0.05 | $-2.1$ | 5.5 | 1100 | 80 | 170 | 18 | 44.7 | $-2.2$
269 | O8 Vz | 35 | 4.64 | 4.2 | $-8.7$ | 0.05 | $-3.5$ | 5.7 | 2000 | 160 | 120 | 19 | 48.0 | $-0.4$
270 | B0.7 V | 26 | 3.76 | 4.2 | $-9.5$ | 0.05 | $-2.1$ | 3.7 | 1800 | 70 | 170 | 8 | 45.9 | $-2.0$
271 | B3 (IV)e | 18 | 3.62 | 3.8 | $-9.7$ | 0.07 | $-2.5$ | 6.7 | 800 | 120 | 170 | 10 | 44.6 | $-2.3$
272 | B2.5 IV | 19 | 3.67 | 3.8 | $-9.6$ | 0.05 | $-2.4$ | 6.3 | 700 | 90 | 170 | 9 | 44.7 | $-2.2$
273 | B1.5 V | 23 | 3.66 | 4.4 | $-9.6$ | 0.05 | $-1.9$ | 4.3 | 2500 | 200 | 170 | 17 | 45.1 | $-2.2$
274 | B0.5 V | 28 | 4.03 | 4.4 | $-9.3$ | 0.05 | $-2.6$ | 4.4 | 2500 | 120 | 160 | 18 | 46.4 | $-1.5$
275 | B0.7 V | 26 | 4.06 | 4.2 | $-9.3$ | 0.05 | $-2.8$ | 5.3 | 2100 | 210 | 170 | 16 | 46.2 | $-1.5$
276 | B0.5 V | 26 | 4.14 | 4.2 | $-9.2$ | 0.05 | $-2.8$ | 5.8 | 2200 | 100 | 170 | 20 | 46.3 | $-1.3$
277 | B1 IV | 25 | 4.4 | 3.6 | $-8.9$ | 0.05 | $-3.2$ | 8.5 | 1200 | 200 | 170 | 10 | 46.6 | $-0.9$
278 | B1 IV | 26 | 4.73 | 3.6 | $-8.6$ | 0.05 | $-4.2$ | 11.5 | 1400 | 300 | 170 | 19 | 47.1 | $-0.3$
279 | B2 V | 22 | 3.96 | 4.2 | $-9.3$ | 0.05 | $-2.6$ | 6.6 | 2400 | 220 | 180 | 25 | 45.4 | $-1.7$
280 | B1 V | 26 | 3.94 | 4.2 | $-9.4$ | 0.05 | $-2.1$ | 4.6 | 2000 | 50 | 170 | 12 | 46.1 | $-1.7$
281 | B0.2 V | 29 | 4.11 | 4.4 | $-9.2$ | 0.05 | $-2.8$ | 4.5 | 2500 | 70 | 170 | 19 | 46.6 | $-1.4$
282 | B2.5 (IV)e | 19 | 3.87 | 3.8 | $-9.4$ | 0.06 | $-2.7$ | 8.0 | 800 | 150 | 170 | 15 | 44.9 | $-1.8$
283 | B0.5 (III)e | 27 | 5.01 | 3.4 | $-8.3$ | 0.38 | $-4.6$ | 14.7 | 1100 | 550 | 140 | 20 | 47.7 | $0.2$
284 | B1.5 IV | 20 | 4.33 | 3.8 |  $-9.1$∗ | 0.02 | $-3.8$ | 12.2 | 752∗ | 140 | 170 | 34 | 45.9 |  $-1.0$∗
285 | B1.5 V | 23 | 3.76 | 4.4 | $-9.5$ | 0.05 | $-2.1$ | 4.8 | 2600 | 100 | 170 | 21 | 45.2 | $-2.0$
286 | B7 IV | 16 | 3.72 | 3.8 | $-9.6$ | 0.08 | $-3.1$ | 9.5 | 900 | 120 | 170 | 21 | 44.6 | $-2.1$
287 | O6 V((f))z | 37 | 4.75 | 4.2 |  $-8.6$∗ | 0.05 | $-3.6$ | 5.8 | 2500∗ | 40 | 160 | 19 | 48.3 |  $-0.3$∗
288 | O8.5 III | 32 | 5.4 | 3.6 |  $-7.9$∗ | 0.06 | $-5.7$ | 16.4 | 1900∗ | 100 | 170 | 39 | 48.7 |  $0.9$∗
289 | B1.5 (IV)e | 23 | 3.96 | 3.8 | $-9.3$ | 0.20 | $-2.6$ | 6.0 | 1400 | 300 | 160 | 8 | 45.8 | $-1.7$
290 | B1 V | 24 | 3.54 | 4.2 | $-9.8$ | 0.05 | $-2.0$ | 3.4 | 1700 | 150 | 170 | 7 | 45.4 | $-2.4$
291 | B1 V | 26 | 3.84 | 4.2 | $-9.5$ | 0.02 | $-2.6$ | 4.1 | 1900 | 300 | 170 | 10 | 46.0 | $-1.9$
292 | O7 V((f))z | 37 | 5.15 | 4.2 |  $-8.3$∗ | 0.05 | $-4.8$ | 9.2 | 2500∗ | 55 | 170 | 49 | 48.7 |  $0.5$∗
293 | B2 V | 19 | 3.52 | 4.2 | $-9.8$ | 0.05 | $-2.1$ | 5.3 | 1100 | 150 | 170 | 16 | 44.4 | $-2.4$
294 | B1 V | 24 | 3.94 | 4.2 | $-9.4$ | 0.05 | $-2.5$ | 5.4 | 2200 | 180 | 170 | 17 | 45.8 | $-1.7$
295 | B2 V | 21 | 3.86 | 4.0 | $-9.4$ | 0.05 | $-2.7$ | 6.4 | 1900 | 100 | 170 | 15 | 45.1 | $-1.8$
296 | B5 V | 19 | 3.42 | 4.2 | $-9.9$ | 0.05 | $-2.1$ | 4.7 | 1000 | 100 | 170 | 13 | 44.3 | $-2.6$
297 | B1.5 V | 23 | 3.76 | 4.4 | $-9.5$ | 0.05 | $-2.4$ | 4.8 | 2600 | 170 | 170 | 21 | 45.2 | $-2.0$
298 | O9.5 V | 32 | 4.77 | 4.2 |  $-8.7$∗ | 0.08 | $-3.8$ | 7.9 | 1700∗ | 180 | 170 | 36 | 47.8 |  $-0.2$∗
299 | B1.5 V | 23 | 3.76 | 4.4 | $-9.5$ | 0.09 | $-2.5$ | 4.8 | 2600 | 200 | 170 | 21 | 45.2 | $-2.0$
300 | B1.5 V | 22 | 3.86 | 4.2 | $-9.4$ | 0.05 | $-2.8$ | 5.9 | 2300 | 80 | 160 | 20 | 45.3 | $-1.8$
301 | B1.5 V | 23 | 3.99 | 4.4 | $-9.3$ | 0.01 | $-2.6$ | 6.2 | 3000 | 230 | 170 | 36 | 45.5 | $-1.6$
302 | B0 V | 31 | 4.57 | 4.2 | $-8.7$ | 0.02 | $-3.7$ | 6.7 | 2200 | 80 | 170 | 26 | 47.5 | $-0.6$
303 | B1.5 V | 23 | 3.86 | 4.4 | $-9.4$ | 0.05 | $-2.6$ | 5.4 | 2800 | 230 | 180 | 27 | 45.3 | $-1.8$
304 | B0.5 V | 27 | 4.29 | 4.2 | $-9.0$ | 0.05 | $-3.4$ | 6.4 | 2300 | 120 | 150 | 24 | 46.6 | $-1.1$
305 | B1.5 V | 23 | 3.66 | 4.4 | $-9.6$ | 0.05 | $-2.1$ | 4.3 | 2500 | 150 | 170 | 17 | 45.1 | $-2.2$
306 | B0 V | 31 | 4.49 | 4.4 | $-8.8$ | 0.05 | $-3.3$ | 6.1 | 2800 | 150 | 170 | 34 | 47.3 | $-0.7$
307 | B0.5 V | 28 | 3.81 | 4.4 | $-9.5$ | 0.05 | $-2.1$ | 3.4 | 2200 | 150 | 170 | 11 | 46.2 | $-1.9$
308 | B5 V | 17 | 3.15 | 3.8 | $-10.1$ | 0.06 | $-1.9$ | 4.3 | 600 | 100 | 160 | 4 | 44.1 | $-3.1$
309 | B2.5 V | 20 | 3.56 | 4.2 | $-9.7$ | 0.05 | $-2.1$ | 5.0 | 1100 | 180 | 170 | 15 | 44.6 | $-2.4$
310 | O7.5 In(f)p | 30 | 5.27 | 3.2 |  $-6.6$∗ | 0.06 | $-5.3$ | 16.0 | 600∗ | 300 | 170 | 15 | 48.8 |  $0.7$∗
311 | B1 V | 26 | 3.8 | 4.4 | $-9.5$ | 0.05 | $-2.2$ | 3.9 | 2300 | 80 | 170 | 14 | 46.0 | $-1.9$
312 | O9 IV | 33 | 4.72 | 4.0 | $-8.6$ | 0.05 | $-3.6$ | 7.0 | 1700 | 40 | 170 | 18 | 48.0 | $-0.3$
313 | B2.5 V | 20 | 3.76 | 4.2 | $-9.5$ | 0.05 | $-2.1$ | 6.3 | 1200 | 160 | 170 | 23 | 44.8 | $-2.0$
314 | B2 V | 19 | 3.52 | 4.0 | $-9.8$ | 0.05 | $-2.4$ | 5.3 | 900 | 180 | 170 | 10 | 44.5 | $-2.4$
315 | O9.7 (III)e | 31 | 5.16 | 3.6 | $-8.2$ | 0.22 | $-4.8$ | 13.2 | 1300 | 440 | 170 | 25 | 48.4 | $0.5$
316 | B2 (V)e | 19 | 3.62 | 4.0 | $-9.7$ | 0.09 | $-2.4$ | 6.0 | 900 | 100 | 170 | 13 | 44.6 | $-2.3$
317 | B0 V | 31 | 4.25 | 4.2 | $-9.1$ | 0.05 | $-2.5$ | 4.6 | 1900 | 75 | 170 | 12 | 47.2 | $-1.1$
318 | B2 V | 19 | 3.42 | 4.2 | $-9.9$ | 0.05 | $-1.8$ | 4.7 | 1000 | 150 | 170 | 13 | 44.3 | $-2.6$
319 | B1.5 V | 22 | 3.94 | 4.2 | $-9.4$ | 0.15 | $-2.2$ | 6.4 | 2400 | 40 | 170 | 24 | 45.4 | $-1.7$
320 | B2 V | 20 | 3.66 | 4.2 | $-9.6$ | 0.05 | $-2.0$ | 5.6 | 1100 | 100 | 220 | 18 | 44.7 | $-2.2$
Table 2: continued. 555 $1$$1$footnotetext: $\log\,M_{\odot}$ values with ∗ are determined from UV P-Cygni profiles. We derived a relation for these measured $\log\,M_{\odot}$ and $\log\,L/L_{\odot}$ and applied for rest of the stars. $2$$2$footnotetext: $\varv_{\infty}$ values with ∗ are determined from UV P-Cygni profiles. Other values are theoretically calculated from $\varv_{\mathrm{esc}}$. $3$$3$footnotetext: $\log\,L_{\mathrm{mec}}$ values with ∗ are calculated for nine stars with UV spectra. For the rest of the stars we adopted values from the derived relation of these nine stars. Table 3: Candidate runaway OB stars in the SMC supergiant shell. SMCSGS-FS | $\varv_{\rm rad}$ | $\varv_{\rm rad}-\bar{\varv}_{\rm rad}$ | spectral type
---|---|---|---
# | (km s-1) | (km s-1) |
203 | 130 | $-39$ | B0.5 (III)e
209 | 150 | $-19$ | B1.5 V
210 | 145 | $-24$ | B0.5 III
212 | 150 | $-19$ | B2.5 V
255 | 140 | $-29$ | B2 V
258 | 150 | $-19$ | B0.5 (III)e
262 | 140 | $-29$ | B2 (III)e
269 | 120 | $-49$ | O8 Vz
283 | 140 | $-29$ | B0.5 (III)e
304 | 150 | $-19$ | B0.5 V
320 | 220 | $51$ | B2 V
8 | 120 | $-43$ | B2 V
16 | 200 | $37$ | B1.5 V
19 | 120 | $-43$ | B5 V
50 | 200 | $37$ | B9 Ib
55 | 100 | $-63$ | B1.5 IV
64 | 120 | $-43$ | O9 V
90 | 200 | $37$ | O9.7 V
93 | 130 | $-33$ | B2.5 V
109 | 110 | $-53$ | B2.5 V
111 | 240 | $77$ | O8.5 V
113 | 110 | $-53$ | B7 V
161 | 130 | $-33$ | B5 IV
165 | 130 | $-33$ | B5 V
190 | 200 | $37$ | B0.7 IV
Table 4: Ages and evolutionary masses of the OB stars determined from stellar evolutionary tracks and isochrones.  SMCSGS-FS | Age | $M_{\rm ev}$
---|---|---
# | [Myr] | [$M_{\odot}$]
1 | 6.0 | 9.3
2 | 8.6 | 12.0
3 | 38.5 | 6.4
4 | 22.0 | 10.1
5 | 36.9 | 6.6
6 | 21.4 | 8.3
7 | 21.0 | 7.9
8 | 32.4 | 7.0
9 | 22.4 | 8.9
10 | 101.0 | 5.3
11 | 32.4 | 7.0
12 | 100.0 | 5.1
13 | 17.6 | 9.4
14 | 38.5 | 6.4
15 | 32.9 | 6.8
16 | 18.9 | 9.5
17 | 63.3 | 5.9
18 | 13.4 | 12.1
19 | 72.9 | 5.1
20 | 80.0 | 5.2
21 | 72.9 | 5.1
22 | 10.6 | 13.1
23 | 135.0 | 5.4
24 | 80.0 | 5.4
25 | 56.6 | 5.4
26 | 55.2 | 5.8
27 | 38.5 | 6.4
28 | 43.4 | 6.5
29 | 13.1 | 10.3
30 | 5.1 | 8.9
31 | 5.6 | 20.3
32 | 10.7 | 14.9
33 | 43.4 | 6.5
34 | 42.4 | 6.3
35 | 21.0 | 7.9
36 | 6.1 | 23.0
37 | 13.0 | 10.6
38 | 10.1 | 12.8
39 | 12.0 | 11.7
40 | 14.0 | 11.1
41 | 37.2 | 5.6
42 | 24.7 | 8.7
43 | 23.2 | 8.6
44 | 7.9 | 14.3
45 | 16.7 | 11.3
46 | 6.3 | 13.2
47 | 24.6 | 8.0
48 | 22.6 | 8.6
49 | 39.8 | 6.9
50 | 47.2 | 6.4
51 | 19.2 | 9.7
52 | 7.6 | 11.1
53 | 38.5 | 6.4
54 | 10.4 | 7.9
55 | 18.0 | 9.2
56 | 0.9 | 9.4
57 | 75.8 | 5.1
58 | 46.3 | 5.7
59 | 8.3 | 13.6
60 | 5.2 | 10.2
61 | 17.5 | 9.7
62 | 8.3 | 13.9
63 | 10.0 | 10.8
64 | 5.0 | 22.2
65 | 5.9 | 11.7
66 | 27.6 | 7.9
67 | 6.3 | 15.7
68 | 36.9 | 6.6
69 | 7.3 | 19.7
70 | 8.0 | 15.3
71 | 21.0 | 7.4
72 | 21.8 | 9.4
73 | 6.4 | 20.4
74 | 38.5 | 6.4
75 | 24.6 | 8.0
76 | 12.2 | 9.9
77 | 15.7 | 10.4
78 | 30.9 | 6.6
79 | 38.5 | 6.4
80 | 38.5 | 6.4
81 | 20.0 | 8.9
82 | 28.9 | 7.2
83 | 17.5 | 8.2
84 | 17.5 | 8.2
85 | 13.1 | 10.6
86 | 82.0 | 5.0
87 | 15.3 | 7.6
88 | 6.0 | 8.9
89 | 6.3 | 17.7
90 | 6.2 | 18.6
91 | 13.5 | 11.6
92 | 1.0 | 9.5
93 | 41.9 | 6.1
94 | 14.8 | 9.6
95 | 17.5 | 8.2
96 | 16.9 | 10.6
97 | 18.5 | 7.4
98 | 13.0 | 10.7
99 | 11.3 | 14.3
100 | 33.9 | 6.1
101 | 36.9 | 6.6
102 | 63.0 | 5.3
103 | 41.9 | 6.1
104 | 83.0 | 5.3
105 | 29.2 | 7.4
106 | 21.5 | 8.3
107 | 7.2 | 16.0
108 | 32.7 | 6.4
109 | 35.5 | 6.4
110 | 30.2 | 7.8
111 | 4.0 | 19.1
112 | 34.5 | 6.4
113 | 150.0 | 5.3
114 | 38.5 | 6.4
115 | 20.8 | 6.9
116 | 35.5 | 6.4
117 | 38.4 | 7.0
118 | 37.2 | 6.8
119 | 6.1 | 16.7
120 | 8.5 | 13.7
121 | 6.1 | 17.6
122 | 32.7 | 6.4
123 | 26.1 | 8.1
124 | 25.3 | 8.7
125 | 11.1 | 11.4
126 | 65.7 | 5.6
127 | 27.0 | 7.2
128 | 4.3 | 18.6
129 | 46.0 | 6.3
130 | 38.5 | 6.4
131 | 22.8 | 6.9
132 | 24.6 | 8.0
133 | 24.6 | 8.0
134 | 36.9 | 6.6
135 | 26.1 | 8.1
136 | 14.1 | 10.9
137 | 17.7 | 10.2
138 | 25.0 | 7.9
139 | 26.0 | 7.4
140 | 29.3 | 7.1
141 | 22.6 | 8.6
142 | 27.7 | 7.7
143 | 4.3 | 25.5
144 | 36.5 | 6.1
145 | 24.6 | 8.0
146 | 32.4 | 7.0
147 | 11.1 | 9.4
148 | 27.7 | 7.7
149 | 8.3 | 10.8
150 | 48.0 | 6.2
151 | 21.0 | 7.9
152 | 38.5 | 6.4
153 | 18.6 | 9.9
154 | 22.6 | 8.6
155 | 11.7 | 10.9
156 | 8.2 | 12.0
157 | 25.9 | 8.3
158 | 9.2 | 15.2
159 | 7.9 | 9.0
160 | 14.0 | 11.1
161 | 35.9 | 7.2
162 | 6.7 | 8.4
163 | 36.0 | 5.4
164 | 10.1 | 10.4
165 | 72.9 | 5.1
166 | 4.6 | 26.0
167 | 8.3 | 18.4
168 | 21.0 | 8.9
169 | 14.7 | 11.3
170 | 10.8 | 11.0
171 | 25.7 | 10.2
172 | 6.0 | 12.5
173 | 42.5 | 6.4
174 | 32.5 | 6.4
175 | 90.0 | 5.1
176 | 102.9 | 5.3
177 | 62.2 | 5.8
178 | 23.0 | 7.1
179 | 25.3 | 8.7
180 | 16.0 | 10.5
181 | 15.3 | 7.6
182 | 8.2 | 8.9
183 | 72.9 | 5.1
184 | 26.0 | 7.4
185 | 10.5 | 10.9
186 | 11.1 | 9.9
187 | 9.6 | 13.6
188 | 61.8 | 6.0
189 | 16.8 | 11.1
190 | 9.2 | 16.2
191 | 9.1 | 11.3
192 | 17.5 | 9.7
193 | 6.2 | 9.9
194 | 74.0 | 5.2
195 | 6.1 | 18.3
196 | 46.0 | 6.3
197 | 22.3 | 7.3
198 | 13.9 | 11.7
199 | 14.4 | 10.0
200 | 26.0 | 7.9
201 | 10.1 | 10.8
202 | 12.9 | 11.0
203 | 6.9 | 21.6
204 | 12.4 | 9.8
205 | 32.5 | 6.4
206 | 38.0 | 6.9
207 | 26.6 | 8.2
208 | 32.9 | 7.3
209 | 17.5 | 8.2
210 | 9.8 | 13.5
211 | 33.8 | 7.3
212 | 32.4 | 7.0
213 | 17.5 | 8.4
214 | 18.5 | 9.1
215 | 5.9 | 18.4
216 | 11.3 | 12.5
217 | 13.1 | 12.1
218 | 57.8 | 6.0
219 | 31.3 | 7.9
220 | 18.2 | 9.0
221 | 35.6 | 6.3
222 | 8.7 | 11.1
223 | 11.1 | 12.9
224 | 12.6 | 13.6
225 | 22.6 | 8.6
226 | 13.9 | 11.7
227 | 25.5 | 8.3
228 | 28.5 | 6.4
229 | 18.3 | 8.1
230 | 12.6 | 13.1
231 | 1.4 | 46.7
232 | 10.2 | 10.8
233 | 15.3 | 11.3
234 | 46.0 | 6.3
235 | 16.7 | 10.8
236 | 32.4 | 7.1
237 | 11.9 | 11.5
238 | 4.8 | 20.1
239 | 4.2 | 18.7
240 | 20.5 | 9.1
241 | 3.8 | 25.0
242 | 20.0 | 8.0
243 | 34.7 | 7.2
244 | 27.7 | 7.7
245 | 28.0 | 7.8
246 | 14.3 | 10.8
247 | 65.7 | 5.6
248 | 5.8 | 12.3
249 | 14.9 | 12.9
250 | 25.2 | 7.8
251 | 17.5 | 8.2
252 | 39.8 | 6.8
253 | 21.0 | 8.6
254 | 22.7 | 9.0
255 | 36.9 | 6.6
256 | 21.5 | 8.3
257 | 11.3 | 13.2
258 | 10.8 | 14.5
259 | 7.2 | 10.8
260 | 8.9 | 11.5
261 | 11.4 | 9.1
262 | 17.8 | 10.3
263 | 8.5 | 13.7
264 | 29.4 | 7.9
265 | 9.0 | 16.0
266 | 13.1 | 11.6
267 | 29.3 | 7.1
268 | 29.3 | 7.1
269 | 3.1 | 18.6
270 | 8.1 | 9.0
271 | 46.0 | 6.3
272 | 33.2 | 6.8
273 | 16.4 | 7.9
274 | 6.0 | 11.1
275 | 13.0 | 10.6
276 | 13.0 | 10.6
277 | 13.6 | 12.0
278 | 9.5 | 15.4
279 | 22.6 | 8.6
280 | 11.8 | 9.7
281 | 6.0 | 11.7
282 | 30.1 | 7.6
283 | 7.3 | 19.2
284 | 17.2 | 10.6
285 | 17.5 | 8.2
286 | 46.2 | 6.7
287 | 2.0 | 20.1
288 | 4.6 | 29.3
289 | 20.1 | 8.9
290 | 10.1 | 7.5
291 | 10.0 | 9.1
292 | 3.9 | 25.8
293 | 58.5 | 6.4
294 | 14.0 | 9.0
295 | 27.7 | 7.7
296 | 41.9 | 6.1
297 | 17.5 | 8.2
298 | 6.2 | 17.9
299 | 17.5 | 8.2
300 | 24.6 | 8.0
301 | 20.4 | 9.0
302 | 6.6 | 15.3
303 | 21.5 | 8.3
304 | 11.3 | 11.7
305 | 10.4 | 7.9
306 | 6.2 | 14.7
307 | 2.0 | 9.4
308 | 64.2 | 5.0
309 | 36.9 | 6.6
310 | 5.5 | 25.4
311 | 9.6 | 9.1
312 | 3.9 | 18.1
313 | 29.2 | 7.4
314 | 58.5 | 6.4
315 | 5.6 | 22.5
316 | 42.5 | 6.4
317 | 4.0 | 13.2
318 | 41.9 | 6.1
319 | 22.7 | 8.5
320 | 32.4 | 7.0