text:
Jaguar Land Rover ramps-up visor production for NHS staff
The protective visors are being produced in Warwickshire
Enda Mullen
Jaguar Land Rover is scaling-up production of protective visors for NHS staff
Coventry car maker Jaguar Land Rover is massively scaling-up production of protective visors for NHS staff fighting coronavirus.
Thanks to a partnership with a Birmingham plastics firm the company now has the capacity to produce a further 14,000 visors each week for key workers working in the front line across the UK.
Earlier this month, Jaguar Land Rover announced the manufacture of 3D-printed face visors at its Advanced Product Creation Centre in Gaydon, Warwickshire.
Visors have been produced and shipped to NHS Trusts across the UK, including South Warwickshire NHS Foundation Trust and the Great Western Hospitals NHS Foundation Trust.
Jaguar Land Rover is able to scale-up production using injection mould tooling supplied by WHS Plastics of Sutton Coldfield.
The operation will produce in excess of 2,000 visors per day – with one polypropylene headband generated every 30 seconds.
Following the success of its 3D-printing efforts, Jaguar Land Rover is also releasing its NHS-approved CAD design in the hope that businesses with Additive Manufacturing facilities will join the global call for more PPE.
How can Jaguar Land Rover bounce back from the coronavirus crisis?
Jaguar Land Rover plans to 'gradually resume' production next month
Ben Wilson, Jaguar Land Rover additive manufacturing and prototype design manager, said: "We hope that by providing the files for download, many more companies will be inspired to use their 3D-printing facilities to print vital PPE equipment for our key workers.
"Every effort, however small it may seem, is a contribution to a collective effort that will ensure our emergency services have what they need to continue their vital work."
Each face visor has been designed to be reusable and can be easily dismantled and cleaned before being used again, thus helping NHS trusts with future shortages.
Paul Nicholson, group managing director for WHS Plastics, said: "Each and every day, our NHS heroes are risking their lives to ensure that those affected by coronavirus receive urgent medical attention.
"In manufacturing, we are uniquely placed to support our essential services, to help deliver critical equipment to those on the frontline here in the Midlands. It's a small effort, but by working with Jaguar Land Rover we can help turn the tide."
Jaguar Land Rover continues to work closely with the UK government and has offered its research and engineering expertise as part of the fight against coronavirus.
It is also supporting the fight in other ways, including supplying Jaguar and Land Rover vehicles to organisations such as the British Red Cross and the emergency services.
South Warwickshire


As they work together, they uncover another mystery behind a wall in Charlotte's home—a diary penned by a woman who disappeared from the small town decades ago. Within the diary's pages, Charlotte believes she has found answers to the lifelong questions about her own past. The only problem—someone wants those secrets to remain buried and will go to murderous lengths to keep it that way.
When Charlotte is faced with a reality much different than what she had envisioned, she must decide what the past means for her present. Finding Franklin explores the nature of family and demonstrates how God can transform our greatest tragedies into lessons of love and grace.
Click here to learn more about her work.
We'd love for you to consider Finding Franklin for your next book club read. To focus your meeting and faciliate discussion, we've provided a question guide. Happy reading!
To order a signed copy of Finding Franklin at the best price available online, click either the hardback or paperback "Add to Cart" button below. You can pay via PayPal or credit card.
If you want a personalized inscription, please send your request to [email protected] . We will be happy to accomodate.
Signed copies of Finding Franklin are also available at Landmark Booksellers, Bound Booksellers in West Haven, and Handy Hardware in downtown Franklin. You can purchase it online at Amazon , Barnes & Noble , BOOKS-A-MILLION , and other retailers. An eBook is offered at Amazon and Barnes & Noble for $4.99.
If you love Southern small towns or historic homes like those featured in Finding Franklin, then you'll definitely want to follow Katie on Facebook or Instagram. With weekly photos of gorgeous old homes and peeks at quaint downtown Franklin (dubbed "America's Favorite Mainstreet" ), her feed is packed with all sorts of Southern charm.


text:
Timeline of Samson
More Locomotives
Built in 1838 by renowned locomotive pioneer Timothy Hackworth. Samson was shipped to Nova Scotia in 1839 by the General Mining Association (GMA) in the company of two other Hackworth engines, Hercules and the John Buddle. Also on board were engineer George Davidson and Hackworth's shop foreman John Stubbs. Once landed, Stubbs prepared the locomotives for use and returned to England. Davidson spent the rest of his career running Samson on the 10 kilometer route from Albion Mines to the loading ground at the mouth of the East River where the coal was loaded onto ships for market.
Samson spent its working life very near where it is now exhibited. The GMA locomotive shed was on the site now occupied by the museum.
Samson ran coal to the loading ground from 1839 to 1867 and then was "semi-retired" as a shunter in the rail yard until 1883. By 1883 it was old enough to be a curiosity. It was taken to Chicago and exhibited at the National Exhibition of Railway Appliances and ten years later at the Chicago World's Fair. Until 1927 it was at the Baltimore and Ohio Railway Museum when it finally came home to Nova Scotia. It was on display at Halifax and New Glasgow and came to its present home at the Museum of Industry in 1990.
Samson Facts:
Vertical cylinder, standard gauge
Top speed: 8 mph
Boiler capacity: 540 gallons (2400 litres)
Weight: 18 tons
Timeline of Samson »
Learn about conserving Samson »

redpajama_set_name: RedPajamaCommonCrawl


text:
Mobutu Sese Seko (1930-1997), né Joseph-Désiré Mobutu, homme politique zaïrois, président de la République du Zaïre de 1965 à 1997
 Mobutu Nzanga (1970-), né François-Joseph Mobutu, fils du précédent, homme politique kino-congolais
 Commune de Mobutu, commune de Bandundu.

redpajama_set_name: RedPajamaWikipedia


text:
This is a list of minor warships of World War II. It contains minor combat vessels that are generally under 1,000 t standard displacement, and includes fast attack craft, submarine chasers, gunboats, missile boats, torpedo boats and patrol boats. It also contains similar vessels, over 1,000 t, such as patrol vessels or patrol ships.

The List of ships of the World War II contains military vessels of the war, arranged alphabetically and by type. The list includes armed vessels that served during the war and in the immediate aftermath, inclusive of localized ongoing combat operations, garrison surrenders, post-surrender occupation, colony re-occupation, troop  and prisoner repatriation, to the end of 1945. For smaller vessels, see also List of World War II ships of less than 1000 tons. Some uncompleted Axis ships are included, out of historic interest. Ships are designated to the country under which they operated for the longest period of the World War II, regardless of where they were built or previous service history.

See also
 List of major World War II warships built by minor powers

References

Bibliography

World War II naval ships
Minor warships


text:
\section{Introduction}\label{sec:intro}

Affine Toda field theories \cite{OlPerMikh,BowCor2,BowCor3,Olive,Olove2} received considerable attention of the Mathematical physics community in the past three decades and
are one of the best understood integrable massive field theories at classical and quantum levels in $1+1$ dimensions \cite{BBT}. The interest in Toda field theories was inspired by  the work of A. Zamolodchikov \cite{Zamol} on deformation of conformal field theories  preserving
integrabilty. It was shown in \cite{Zamol} that the resultant theory is characterised by eight masses
related to the Cartan matrix of $E_8$, and by integrals of motion with spins given by the
exponents of $E_8$ modulo its Coxeter number.

Affine Toda field theories (ATFT) are integrable models of real scalar field ${\bf q}=(q_1, \dots q_n)$, in one space
dimension  with exponential interactions. The Lagrangian of the theory is given by
\begin{equation}\label{eq: ATFT-Lagr}
{\cal L}[{\bf q}] = {1\over 2}(\partial_\mu {\bf q} \cdot \partial^\mu {\bf q})- {m^2\over \beta^2}\sum_{k=0}^{n} n_k \left({\rm e}^{\beta (\alpha_k \cdot{\bf q})} -1\right),
\end{equation}
where the field ${\bf q}(x,t)$ is an $n$-dimensional vector. This corresponds to equations of motion of the form
\begin{equation}\label{eq:ATFT}
{\partial ^2 {\bf q}  \over \partial x \partial t } =
\sum_{j=0}^{r} n_j \alpha_j e^{-(\alpha_j ,{\bf q}(x,t))},
\end{equation}
Each ATFT is associated to a (finite) simple Lie algebra ${\frak g}$ \cite{AMV,BBT}. Here $n$ is the rank of ${\frak g}$, $\alpha_k$s ($k = 1,\dots,n$) are the simple roots of ${\frak g}$ and $\alpha_0$ is the minimal root.
Thus,  \eqref{eq: ATFT-Lagr} are described by the extended Dynkin diagrams of the associated affine
algebra $\hat{\frak g}$. The fields $q_k$ can be rescaled so that $\beta$ only appears in ${\cal L}$ through a common factor $1/\beta^2$ (expanding in powers of $\beta^2$ is equivalent in  Quantum theory to expanding in $\hbar$) \cite{BowCor2,BowCor3}.

Non-simply laced theories are obtained from the affine Toda theories based on simply
laced algebras by folding the corresponding Dynkin diagrams. The same process, called
classical ``reduction'', provides solutions of a non-simply laced theory from the classical
solutions with special symmetries of the parent simply laced theory.

One of the important features of affine Toda field theories is their integrability by the inverse scattering method (ISM) \cite{FaTa,ZMNP}. The starting point is the existence of the so-called Lax operator (see \eqref{eq:2.1} below). The interpretation of the ISM as a generalized Fourier transforms \cite{AKNS,IP2,GeYa} allows one to study all the
fundamental properties of the corresponding  nonlinear evolutionary equations (NLEE's): i)~the description of the class of NLEE related to a~given Lax operator
$L(\lambda)$ and solvable by the ISM; ii)~derivation of the infinite family of integrals of motion;  iii)~their hierarchy
of Hamiltonian structures \cite{GVYa}; and iv) description  of the gauge equivalent systems \cite{GGMV,Grah,Grah2}.

Real Hamiltonian forms (RHF) are another type of ``reductions'' of Hamiltonian systems.
The extraction of RHFs is similar to the obtaining a real forms of
a semi-simple Lie algebra. The Killing form for the later is indefinite in general (it is negatively-
definite for the compact real forms). So one should not be surprised of getting RHF's with
indefinite kinetic energy quadratic form. Of course this is an obstacle for their quantization.

The purpose of this paper is to outline the spectral theory of the Lax operators of real Hamiltonian forms of affine Toda field theories related to complex untwisted exceptional affine Lie algebras.

The structure of the paper is as follows: In Section 2 we provide a brief summary of the Lax representation of ATFTs and all necessary Lie algebraic background knowledge. As to the root systems of the exceptional algebras we are following the conventions in \cite{Bourb1}. In Section 3 first we describe the general method for constructing RHFs. Then we briefly describe the sets of admissible roots of the exceptional Lie algebras, and the explicit constructions of their RHFs.
Section 4 is devoted to the spectral theory of the Lax operators of ATFTs with ${\Bbb Z}_h$-reductions, where $h$ is the Coxeter number of ${\frak g}$. We first provide the general construction of the fundamental analytic solutions $\xi_\nu (x,t,\lambda)$ (FAS) of the Lax operator and show their relevance to the Riemann-Hilbert problem (RHP) on a set of $2h_g$ rays $l_\nu$ closing angles $\pi /h_g$. Then we introduce the asymptotics of the FAS  $\xi_\nu (x,t,\lambda)$ for $x\to \pm \infty$ and $\lambda \in l_\nu$ which determine the scattering data of $L$. This Section ends with a theorem specifying the minimal sets of scattering data for generic choice of the simple Lie algebra $\mathfrak{g}$. Section 5 contains the specific data, concerning each of the exceptional Lie algebras that allows one to determine the minimal set os scattering data for each case. The paper ends up with Conclusions.

\section{Affine Toda field theories: preliminaries}\label{sec:prelim}

To each simple Lie algebra $\mathfrak{g} $ one can relate a Toda field
theory  in $1+1 $ dimensions. It allows a Lax representation in a zero-curvature form:
\begin{equation}\label{eq:0.1}
[L,M]=0,
\end{equation}
where $L $ and $M $ are first order ordinary differential (Lax)
operators:
\begin{eqnarray}\label{eq:2.1}
L\psi \equiv \left(  i{d  \over dx } - iq_x(x,t) - \lambda
J_0\right)
\psi (x,t,\lambda )=0, \\
M\psi \equiv \left(  i{d  \over dt } -  {1\over \lambda}
I(x,t)\right) \psi (x,t,\lambda )=0.
\end{eqnarray}
whose potentials take values in $\mathfrak{g} $.
Here also $q(x,t) \in \mathfrak{h}$ - the Cartan subalgebra of
$\mathfrak{g}$, $\q(x,t)=(q_1,\dots , q_r) $ is its dual $r
$-component vector  ($r=\mbox{rank}\,\mathfrak{ g}$). The potentials of the Lax operators are chosen as follows
\begin{equation}\label{eq:2.2}
J_0 = \sum_{\alpha \in \pi}^{} E_{\alpha },\qquad I(x,t) =
\sum_{\alpha \in \pi}^{} e^{-(\alpha ,\q(x,t))} E_{-\alpha }.
\end{equation}
Here $\pi_{\mathfrak{g}} $ stands for the set of admissible roots of
$\mathfrak{g} $, i.e.  $\pi_{\mathfrak{g}} = \{\alpha _0, \alpha
_1,\dots, \alpha _r\} $, with $\alpha _1,\dots, \alpha _r $ being
the simple roots of $\mathfrak{ g}$ and $\alpha _0 $ being the
minimal root of $\mathfrak{ g} $.  The corresponding Toda field theory is known
as  affine Toda field theory (ATFT). The Dynkin graph corresponding to the set of
admissible roots $\pi_{\mathfrak{g}}=\{\alpha _0, \alpha _1,\dots,\alpha _r\} $ of ${\frak g}$ is called extended Dynkin diagram (EDD). The equations of motion are of the form:
\begin{equation}\label{eq:2.3}
{\partial ^2 \q  \over \partial x \partial t } =
\sum_{j=0}^{r} n_j \alpha_j e^{-(\alpha_j ,\q(x,t))},
\end{equation}
where $n_j $ are the minimal positive  integer coefficients $n_k $ that
provide the decomposition of the $\alpha _0 $ over the simple roots of
$\mathfrak{g} $:
\begin{equation}\label{eq:n_k}
-\alpha _0 = \sum_{k=1}^{r} n_k\alpha _k.
\end{equation}
It is  well known that ATFT models are an infinite-dimensional Hamiltonian system.  The (canonical)
Hamiltonian structure is given by:
\begin{eqnarray}\label{eq:H-g}
H_{\mathfrak{g}} &=& \int_{-\infty }^{\infty }dx\, \mathcal{H}_{\mathfrak{g}}(x,t), \qquad
\mathcal{H}_{\mathfrak{g}}(x,t)= { 1\over 2} ({\bf p}(x,t),
{\bf p}(x,t))+\sum_{k=0}^{r} n_k(e^{-({\bf q}(x,t),\alpha _k)}-1)  ,\\
\label{eq:ome-g}
\Omega _{\mathfrak{g}} &=& \int_{-\infty }^{\infty } dx\,
\omega_{\mathfrak{g}} (x,t), \qquad
\omega_{\mathfrak{g}} (x,t)=( \delta {\bf p}(x,t)\wedge \delta {\bf q}(x,t)),
\end{eqnarray}
where $H_{\mathfrak{g}} $ is the canonical Hamilton function and $\Omega _{\mathfrak{g}}$ is the canonical symplectic structure.
Here also ${\bf p} = d{\bf q}/dt $ are the canonical momenta
and coordinates satisfying canonical Poisson brackets:
\begin{equation}\label{eq:ATFT-PB}
\{ q_k(x,t) , p_j(y,t)\} = \delta_{jk} \delta (x-y).
\end{equation}
The infinite-dimensional phase space $\mathcal{M}=\{{\bf q}(x,t),{\bf p}(x,t)\}$ is spanned by the canonical coordinates and momenta.

\section{Real Hamiltonian forms and affine Toda field theories}\label{sec:RHF}

The Lax representations of the ATFT models widely discussed in the
literature (see e.g. \cite{Mikh,OlPerMikh,Olive,SasKha} and the
references therein) are related mostly to the normal real form of the Lie
algebra $\mathfrak{ g} $, see \cite{Helg}. Here we will study real Hamiltonian forms of ATFT models. The notion of real Hamiltonian forms was introduced in \cite{2} and used to study reductions of ATFTs in \cite{GG}. After a brief outline of the basic theory (following \cite{2}), in this Section we will describe the real Hamiltonian forms for ATFT models related to exceptional untwisted complex Kac-Moody algebras \cite{Kac,VinOni,Xu}.

\subsection{Real Hamiltonian forms}\label{ssec:rHF}

The starting point in the construction of real Hamiltonian forms (RHF) is the complexification of the field involved in the Hamiltonian of the model. First for dynamical variables we consider complex-valued fields
\[
{\bf q}^\bbbc = {\bf q}^0 + i{\bf q}^1, \qquad {\bf p}^\bbbc =
{\bf p}^0 + i{\bf p}^1.
\]
Next we introduce an involution $\mathcal{ C} $ acting on the
phase space $\mathcal{ M} \equiv \{q_k(x), p_k(x)\}_{k=1}^{n} $ as
follows:
\begin{eqnarray}\label{eq:Cc}
&& \mbox{1)} \qquad \mathcal{ C}(F(p_k,q_k)) = F(\mathcal{ C}(p_k),
\mathcal{ C}(q_k)),  \nonumber\\
&& \mbox{2)} \qquad \mathcal{ C}\left( \{ F(p_k,q_k), G(p_k,q_k)\}\right) =
\left\{ \mathcal{ C}(F), \mathcal{ C}(G) \right\} , \\
&& \mbox{3)} \qquad \mathcal{ C}(H( p_k,q_k)) = H(p_k,q_k) . \nonumber
\end{eqnarray}
Here $F(p_k,q_k) $, $G(p_k,q_k) $ and the Hamiltonian $H(p_k,q_k) $ are
functionals on $\mathcal{M} $ depending analytically on the fields
$q_k(x,t) $ and $p_k(x,t) $.

The complexification of the ATFT is rather straightforward. The resulting
complex ATFT (CATFT) can be written down as standard Hamiltonian system with
twice as many fields ${\bf q}^a(x,t) $, ${\bf p}^a(x,t)  $, $a=0,1 $:
\begin{equation}\label{eq:qp-c}
{\bf p}^\bbbc (x,t) = {\bf p}^0(x,t)+i {\bf p}^1(x,t), \qquad
{\bf q}^\bbbc (x,t)= {\bf q}^0(x,t)+i {\bf q}^1(x,t),
\end{equation}
\begin{equation}\label{eq:qp-pb}
\{{q}_{k}^0(x,t), {p}_{j}^0(y,t) \}= - \{{q}_{k}^1(x,t),
{p}_{j}^1(y,t) \} = \delta _{kj} \delta (x-y).
\end{equation}
The densities of the corresponding Hamiltonian and symplectic form
equal
\begin{eqnarray}\label{eq:H_0}
\mathcal{H}_{\rm ATFT}^\bbbc &\equiv & \re \mathcal{H}_{\rm ATFT}
({\bf p}^0+i {\bf p}^1, {\bf q}^0+i {\bf q}^1) \nonumber\\
&=&  {1\over 2 } ({\bf p}^0,{\bf p}^0) -{1\over 2 }
({\bf p}^1,{\bf p}^1) + \sum_{k=0}^{r} e^{-({\bf q}^0,\alpha _k)}
\cos (({\bf q}^1,\alpha _k)) ,  \\
\label{eq:ome_0}
\omega^\bbbc &=& (d{\bf p}^0\wedge  i d{\bf q}^0) - (d{\bf p}^1\wedge
d {\bf q}^1).
\end{eqnarray}
The family of RHF then are obtained from the CATFT by imposing an
invariance condition with respect to the involution
$\tilde{\mathcal{ C}} \equiv \mathcal{ C}\circ \ast $ where by
$\ast $ we denote the complex conjugation. The involution
$\tilde{\mathcal{ C}} $ splits the phase space $\mathcal{ M}^\bbbc
$ into a direct sum $\mathcal{ M}^\bbbc \equiv {\cal M}_+^\bbbc
\oplus \mathcal{M}_-^\bbbc$ where
\begin{equation}\label{eq:M-c}\begin{aligned}
\mathcal{M}_+^\bbbc &= \mathcal{ M}_0 \oplus i \mathcal{ M}_1, &\quad \mathcal{M}_-^\bbbc &= \mathcal{ M}_0 \oplus -i \mathcal{ M}_1, \\
 \mathcal{C}(\q^+ + i \q^-) &= (\q^+ - i \q^-), &\quad  \mathcal{C}(\p^+ + i \p^-) &= (\p^+ - i \p^-).
\end{aligned}\end{equation}
Each involution $\mathcal{C} $ induces an involution
(involutive automorphism) $\mathcal{C}^\# $ also of $\mathfrak{g} $.
Below we will choose $\mathcal{C}^\# $ in a special way, which is related to the Coxeter
automorphism defined below in its dihedral form (\ref{eq:DCox-E6}), see \cite{Bourb1, Helg, Cart}.
Indeed, we will define $\mathcal{C}^\# $ using the reflection $S_1$ with respect to the set of
black roots of the Dinkin dyagram (see Figure \ref{fig:Dynkin-E6}). More precisely:
\begin{equation}\label{eq:alj}\begin{aligned}
\mathcal{C}^\# \alpha_j &=  \alpha_j \quad \mbox{for}\quad  \alpha_j \in W_g, &\qquad
\mathcal{C}^\# \beta_j &=  -\beta_j \quad \mbox{for} \quad \beta_j \in B_g , \\
{\bf p}^0(x,t) &= \sum_{\alpha_j\in W_g} {\bf p}_j(x,t) \alpha_j,    &\qquad   {\bf p}^1(x,t) &= \sum_{\beta_j \in B_g} {\bf p}_j(x,t) \beta_j, \\
{\bf q}^0(x,t) &= \sum_{\alpha_j\in W_g} {\bf q}_j(x,t) \alpha_j,    &\qquad   {\bf q}^1(x,t) &= \sum_{\beta_j \in B_g} {\bf q}_j(x,t) \beta_j,
\end{aligned}\end{equation}
where $W_g$ (resp. $B_g$) is the set of white roots (resp. black roots) of the Dinkin diagram.

Thus to each involution $\mathcal{ C} $ one can relate a RHF of the ATFT.  Note that $\mathcal{C}^\# $
preserves the system of admissible roots of $\mathfrak{g} $ and the
extended Dynkin diagrams of $\mathfrak{ g} $ studied in \cite{SasKha}.
 For a sake of brevity below we will skip the dependence of ${\bf p} $ and ${\bf q} $ on
$x $ and $t $. Indeed, the condition 3) in \eqref{eq:Cc} requires that:
\begin{equation}\label{eq:*1}
(\mathcal{C}(\q ),\alpha ) = ({\bf q} , \mathcal{C}^\# (\alpha )), \qquad \alpha \in \pi_{\mathfrak{g}},
\end{equation}
and therefore we must have $\mathcal{C}(\pi_{\mathfrak{g}}) = \pi_{\mathfrak{g}} $.
As a result:
\begin{equation}\label{eq:palj}\begin{aligned}
\mathcal{C}( {\bf p}^0(x,t), \alpha_j) &= ({\bf p}^0(x,t),\alpha_j), &\qquad \mathcal{C}( {\bf p}^1(x,t), \beta_j) &= -({\bf p}^1(x,t),\beta_j), \\
( {\bf p}^0(x,t), \beta_j) &= 0, &\qquad ( {\bf p}^1(x,t), \alpha_j) &= 0, \\
\mathcal{C}( {\bf q}^0(x,t), \alpha_j) &= ({\bf q}^0(x,t),\alpha_j), &\qquad \mathcal{C}( {\bf q}^1(x,t), \beta_j) &= -({\bf q}^1(x,t),\beta_j), \\
( {\bf q}^0(x,t), \beta_j) &= 0, &\qquad ( {\bf q}^1(x,t), \alpha_j) &= 0,
\end{aligned}\end{equation}
where $\alpha_j \in W_g$, and $\beta_j \in B_g$.
Then applying the ideas of \cite{Mikh} we obtain the following result
for the RHF of the ATFT related to $\mathfrak{g} $:
\begin{equation}\label{eq:*H-g}\begin{aligned}
\mathcal{H}^{\bbbr}_{\mathfrak{g}} &= {1 \over 2} ({\bf p}^+ , {\bf p}^+ ) -{1 \over 2} ({\bf p}^- , {\bf p}^- ) +
\sum_{\alpha_k\in W_g} n_k' (e^{-({\bf q}^+ ,\alpha _k)} -1) + \sum_{\beta_k\in B_g} n_k'' ( \cos ({\bf q}^- ,\beta_k)-1),
\end{aligned}\end{equation}
\begin{equation}\label{eq:*ome-g}\begin{split}
\omega ^{\bbbr}_{\mathfrak{g}} = (\delta {\bf p}^+  \wedge \delta {\bf q}^+) -(\delta {\bf p}^-  \wedge \delta {\bf q}^-) ,
\end{split}\end{equation}
The Hamiltonian along with the terms related to the simple roots, contains also the minimal root $-\alpha_0$.
It is well known that the maximal root $\alpha_0$ can be expanded as sum of the simple roots with integer nonnegative coefficients:
\begin{equation}\label{eq:a0}\begin{split}
 \alpha_0 = \sum_{j=1}^{r} n_j \alpha_j = \sum_{\alpha_k\in W_g} n_k' \alpha_k + \sum_{\beta_j\in B_g} n_k'' \beta_j .
\end{split}\end{equation}
In the second expression above we have separated the terms with the white and black roots.

The RHF of ATFT are more general integrable systems than the models
described in \cite{Evans, Evans2, Evans3, SasKha} which involve only the fields ${\bf q}^+ $, ${\bf p}^+
$ invariant with respect to $\mathcal{C} $.

\subsection{Affine Toda field theories related to $E_6^{(1)}$}\label{ssec:E6-1}

The set of admissible roots for this algebra is
\begin{equation}\label{eq:e6roots}\begin{aligned}
\alpha_1&={1\over 2}(e_1-e_2-e_3-e_4-e_5-e_6-e_7+e_8), &\quad \alpha_2 &=e_1+e_2, \\
\alpha_3 &=e_2-e_1,\qquad \alpha_4=e_3-e_2, \qquad \alpha_5=e_4-e_3, &\quad \alpha_6 &=e_5-e_4,\\
\alpha_0 &=-{1\over 2}(e_1+e_2+e_3+e_4+e_5-e_6-e_7+e_8).
\end{aligned}\end{equation}
Here $\alpha_1,\dots , \alpha_6$ form the set of simple roots of
$ E_6$ and $\alpha_0$ is the minimal root of the algebra. The extended Dynkin diagram of $E_6^{(1)}$ is shown on Figure \ref{fig:Dynkin-E6}.
This is the standard definition of the root system of $E_6
$ embedded into the 8-dimensional Euclidean space $\bbbe^8 $. The
root space $\bbbe_6 $ of the algebra $E_6$ is the
6-dimensional subspace of $\bbbe^8 $ orthogonal to the vectors
$e_7+e_8 $ and $e_6+e_7+2e_8 $. Thus any vector ${\bf q} $
belonging to $\bbbe_6 $ has only 6 independent coordinates and can
be written as:
\begin{gather}\label{eq:vec-q}
{\bf q} = \sum_{k=1}^{5} q_k e_k + q_6 e'_6, \qquad e'_6 = {1\over
\sqrt{3} } (e_6 + e_7 - e_8).
\end{gather}
The fundamental weights of $E_6$ are
\begin{eqnarray*}
\omega_1&=&{2\over 3} (e_8-e_7-e_6), \qquad \omega_2= {1\over 2}(e_1+e_2+e_3+e_4+e_5-e_6-e_7+e_8);\\
\omega_3 &=& {1\over 2}(-e_1+e_2+e_3+e_4+e_5)+{5\over 6}(e_8-e_7-e_6), \qquad
\omega_4 = e_3+e_4+e_5-e_6-e_7+e_8,\\
\omega_5 &=& e_4+e_5 + {2\over 3} (e_8-e_7-e_6), \qquad \omega_6=e_5+ {1\over 3}(e_8-e_7-e_6).
\end{eqnarray*}

\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[thick] (0,0) --(6,0);
\draw[thick] (3,0) --(3,3);
\draw[thick,fill=white] (0,0) circle (1.5 mm);
\draw[thick,fill=black] (1.5,0) circle (1.5 mm);
\draw[thick,fill=white] (3,0) circle (1.5 mm);
\draw[thick,fill=black] (4.5,0) circle (1.5 mm);
\draw[thick,fill=white] (6,0) circle (1.5 mm);
\draw[thick,fill=black] (3,1.5) circle (1.5 mm);
\draw[thick,fill=white] (3,3) circle (1.5 mm);
\draw (0,-0.5) node {\small $\alpha_1$};
\draw (1.5,-0.5) node {\small $\alpha_3$};
\draw (3,-0.5) node {\small $\alpha_4$};
\draw (4.5,-0.5) node {\small $\alpha_5$};
\draw (6,-0.5) node {\small $\alpha_6$};
\draw (3.5,1.5) node {\small $\alpha_2$};
\draw (3.5,3) node {\small $\alpha_0$};
\draw (0.0,3) node {$E_6^{(1)}$};

\end{tikzpicture}
\end{center}
  \caption{\small The extended Dynkin diagram of the  complex untwisted affine Kac-Moody algebra $E_6^{(1)}$. The white roots are invariant with respect to the automorphism $S_2$ in \eqref{eq:DCox-E6}.}\label{fig:Dynkin-E6}
\end{figure}

\noindent
We will use the dihedral realization of the Coxeter automorphism for $E_6^{(1)}$:
\begin{equation}\label{eq:DCox-E6}
{\rm Cox}\, (E_6^{(1)})=S_1\circ S_2, \qquad S_1=  S_{\alpha_2}\circ S_{\alpha_3}\circ S_{\alpha_5}, \qquad S_2= S_{\alpha_1}\circ S_{\alpha_4}\circ S_{\alpha_6}.
\end{equation}
If we require an invariance of the Hamiltonian and the symplectic form \eqref{eq:*H-g} with respect to $S_1$,
and restrict on the set of admissible roots $\beta_k$, then we get a real Hamiltonian form  of the $E_6^{(1)}$ ATFT, described by:
\begin{equation}\label{eq:E6-rhf}\begin{split}
\mathcal{H}^{\bbbr}_{E_6} &= {1 \over 2} ({\bf p}^+ , {\bf p}^+ ) -{1 \over 2} ({\bf p}^- , {\bf p}^- ) +
\sum_{k =0,1,4,6} n_k' (e^{-({\bf q}^+ ,\alpha _k)} -1) + \sum_{k= 2,3,5} n_k'' ( \cos ({\bf q}^- ,\beta_k)-1),
\end{split}\end{equation}
where $n_1'=n_6'=1$, $n_4'=3$ and $n_2'' =n_3'' =n_5'' =2$ \cite{Bourb1}, and
\begin{equation}\label{eq:*ome-g6}\begin{split}
\omega ^{\bbbr}_{E_6^{(1)}} =
(\delta {\bf p}^+  \wedge \delta {\bf q}^+) -(\delta {\bf p}^-  \wedge \delta {\bf q}^-) .
\end{split}\end{equation}

\subsection{Affine Toda field theories related to $E_7^{(1)}$}\label{ssec:E7-1}

The extended root system of  this algebra is given by
\begin{equation}\label{eq:e7roots}\begin{aligned}
\alpha_1 &={1\over 2}(e_1-e_2-e_3-e_4-e_5-e_6-e_7+e_8), &\quad \alpha_2 &=e_1+e_2, \\
\alpha_3 &=e_2-e_1,\qquad \alpha_4=e_3-e_2, &\quad \alpha_5 &=e_4-e_3, \\
\alpha_6 &=e_5-e_4,\qquad \alpha_7=e_6-e_5, &\quad \alpha_0 &=e_7-e_8,
\end{aligned}\end{equation}
where $\alpha_1,\dots , \alpha_7$ form the set of simple roots of
$E_7$ and $\alpha_0$ is the minimal root of the algebra. The extended Dynkin diagram for $E_7^{(1)}$ is shown on Figure \ref{fig:Dynkin-E7}.
This is the standard def\/inition of the root system of $E_7
$ embedded into the 8-dimensional Euclidean space $\bbbe^8 $. The
root space $\bbbe_7 $ of the algebra $E_7$ is the
7-dimensional subspace of $\bbbe^8 $ orthogonal to the vector
$e_7+e_8 $. Thus any vector $\vec{q} $ belonging to $\bbbe_7 $ has
7 independent coordinates and can be written as:
\begin{gather}\label{eq:vec-q'}
\vec{q} = \sum_{k=1}^{6} q_k e_k + q_7 e'_7, \qquad e'_7 = {1\over
\sqrt{2} } (e_7 - e_8).
\end{gather}

\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[thick] (-1.5,0) --(7.5,0);
\draw[thick] (3,0) --(3,1.5);
\draw[thick,fill=white] (-1.5,0) circle (1.5 mm);
\draw[thick,fill=black] (0,0) circle (1.5 mm);
\draw[thick,fill=white] (1.5,0) circle (1.5 mm);
\draw[thick,fill=black] (3,0) circle (1.5 mm);
\draw[thick,fill=white] (4.5,0) circle (1.5 mm);
\draw[thick,fill=black] (6,0) circle (1.5 mm);
\draw[thick,fill=white] (7.5,0) circle (1.5 mm);
\draw[thick,fill=white] (3,1.5) circle (1.5 mm);

\draw (-1.5,-0.5) node {\small $\alpha_0$};
\draw (0,-0.5) node {\small $\alpha_1$};
\draw (1.5,-0.5) node {\small $\alpha_3$};
\draw (3,-0.5) node {\small $\alpha_4$};
\draw (4.5,-0.5) node {\small $\alpha_5$};
\draw (6,-0.5) node {\small $\alpha_6$};
\draw (7.5,-0.5) node {\small $\alpha_7$};
\draw (3.5,1.5) node {\small $\alpha_2$};

\draw (-1.5,1.5) node {$E_7^{(1)}$};

\end{tikzpicture}
\end{center}
  \caption{\small The extended Dynkin diagram of the  complex untwisted affine Kac-Moody algebra $E_7^{(1)}$. The white roots are invariant with respect to the automorphism $S_1$ in \eqref{eq:DCox-E7}.}\label{fig:Dynkin-E7}
\end{figure}

\noindent
Using the dihedral realization of the Coxeter automorphism for $E_7^{(1)}$:
\begin{equation}\label{eq:DCox-E7}
{\rm Cox}\, (E_7^{(1)})=S_1\circ S_2, \qquad S_1= S_{\alpha_1}\circ S_{\alpha_4}\circ S_{\alpha_6}, \qquad S_2=  S_{\alpha_2}\circ S_{\alpha_3}\circ S_{\alpha_5}\circ S_{\alpha_7},
\end{equation}
we  impose an invariance of the Hamiltonian \eqref{eq:*H-g} with respect to the automorphism $S_1$ in \eqref{eq:DCox-E7}.
Then the root space ${\Bbb E}_7$ will split into direct sums: ${\Bbb E}_7={\Bbb E}_{7,+}\oplus {\Bbb E}_{7,-}$, with
\[
\alpha_2, \alpha_3, \alpha_5, \alpha_7 \in {\Bbb E}^{7,+}, \qquad \alpha_1, \alpha_4, \alpha_6 \in {\Bbb E}^{7,-}.
\]
If we again require an invariance of the Hamiltonian and the symplectic form \eqref{eq:*H-g} with respect to $S_1$ from \eqref{eq:DCox-E7}, and restrict on the set of admissible roots $\beta_k$, then we get a real Hamiltonian form  of the $E_7^{(1)}$ ATFT, described by:
\begin{equation}\label{eq:eq:E7-rhf}\begin{split}
\mathcal{H}^{\bbbr}_{E_7} &= {1 \over 2} ({\bf p}^+ , {\bf p}^+ ) -{1 \over 2} ({\bf p}^- , {\bf p}^- ) +
\sum_{k =0,2,3,5,7} n_k' (e^{-({\bf q}^+ ,\alpha _k)} -1) + \sum_{k= 1,4,6} n_k'' ( \cos ({\bf q}^- ,\beta_k)-1),
\end{split}\end{equation}
where $n_1'=n_6'=2$, $n_4'=4$  and $n_2'' =2$, $n_3''=n_5''=3$, $n_7'' =1$  \cite{Bourb1}, and
\begin{equation}\label{eq:*ome-g7}\begin{split}
\omega ^{\bbbr}_{E_7^{(1)}} =
(\delta {\bf p}^+  \wedge \delta {\bf q}^+) -(\delta {\bf p}^-  \wedge \delta {\bf q}^-) .
\end{split}\end{equation}

\subsection{Affine Toda field theories related to $E_8^{(1)}$}\label{ssec:E8-1}

The set of admissible roots for this algebra is
\begin{equation}\label{eq:e8roots}\begin{aligned}
\alpha_0 &=\frac{1}{2}(e_1 + e_2 + e_3 + e_4 + e_5 - e_6 - e_7 + e_8) \quad
\alpha_1 =\frac{1}{2}(e_1 - e_2 - e_3 - e_4 - e_5 - e_6 - e_7 + e_8) , \\
\alpha_2 &= e_1 + e_2, \quad
\alpha_3 = e_2 - e_1, \quad
\alpha_4 = e_3 - e_2, \\
\alpha_5 &= e_4 - e_3, \quad
\alpha_6 = e_5 - e_4, \quad
\alpha_7 = e_6 - e_5 \quad
\alpha_8 = e_7 - e_6.
\end{aligned}\end{equation}
where $\alpha_1,\dots , \alpha_8$ form the set of simple roots of
$E_8$ and $\alpha_0$ is the minimal root of the algebra.
If we exclude  $\alpha_8$  from \eqref{eq:e8roots}, we will get the set of positive roots \eqref{eq:e7roots} form $E_7$. The extended Dynkin diagram for $E_8^{(1)}$ is shown on Figure \ref{fig:Dynkin-E8}.

The fundamental weights of $E_8$ are
\begin{eqnarray*}
\omega_1&=& 2e_8, \qquad \omega_2 = {1\over 2}(e_1+e_2+e_3+e_4+e_5+e_6+e_7+5e_8),\\
\omega_3 &=& {1\over 2}(-e_1+e_2+e_3+e_4+e_5+e_6+e_7+7e_8), \qquad \omega_4= e_3+e_4+e_5+e_6+e_7+5e_8,\\
\omega_5 &=& e_4+e_5+e_6+e_7+4e_8, \qquad \omega_6 = e_5+e_6+e_7+3e_8,\\
\omega_7 &=& e_6+e_7+2e_8, \qquad \omega_8=e_7+e_8=-\alpha_0.
\end{eqnarray*}

\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[thick] (0,0) --(10.5,0);
\draw[thick] (3,0) --(3,1.5);

\draw[thick,fill=black] (0,0) circle (1.5 mm);
\draw[thick,fill=white] (1.5,0) circle (1.5 mm);
\draw[thick,fill=black] (3,0) circle (1.5 mm);
\draw[thick,fill=white] (4.5,0) circle (1.5 mm);
\draw[thick,fill=black] (6,0) circle (1.5 mm);
\draw[thick,fill=white] (7.5,0) circle (1.5 mm);
\draw[thick,fill=black] (9,0) circle (1.5 mm);
\draw[thick,fill=white] (10.5,0) circle (1.5 mm);
\draw[thick,fill=white] (3,1.5) circle (1.5 mm);

\draw (0,-0.5) node {\small $\alpha_1$};
\draw (1.5,-0.5) node {\small $\alpha_3$};
\draw (3,-0.5) node {\small $\alpha_4$};
\draw (4.5,-0.5) node {\small $\alpha_5$};
\draw (6,-0.5) node {\small $\alpha_6$};
\draw (7.5,-0.5) node {\small $\alpha_7$};
\draw (9,-0.5) node {\small $\alpha_8$};
\draw (10.5,-0.5) node {\small $\alpha_0$};
\draw (3.5,1.5) node {\small $\alpha_2$};

\draw (0,1.5) node {$E_8^{(1)}$};

\end{tikzpicture}
\end{center}
  \caption{\small The extended Dynkin diagram of the  complex untwisted affine Kac-Moody algebra $E_8^{(1)}$. The white roots are invariant with respect to the automorphism $S_2$ in \eqref{eq:DCox-E8}.}\label{fig:Dynkin-E8}
\end{figure}

\noindent
We will use the dihedral realization of the Coxeter automorphism for $E_8^{(1)}$:
\begin{equation}\label{eq:DCox-E8}
{\rm Cox}\, (E_8^{(1)})=S_1\circ S_2, \qquad S_1= S_{\alpha_1}\circ S_{\alpha_4}\circ S_{\alpha_6}\circ S_{\alpha_8}, \qquad S_2= S_{\alpha_2}\circ S_{\alpha_3}\circ S_{\alpha_5}\circ S_{\alpha_7}.
\end{equation}
We  impose an invariance of the Hamiltonian \eqref{eq:*H-g} with respect to the automorphism $S_1$ in \eqref{eq:DCox-E7}.
Then the root space ${\Bbb E}_8$ will split into direct sums: ${\Bbb E}_8={\Bbb E}_{8,+}\oplus {\Bbb E}_{8,-}$, with
\[
\alpha_2, \alpha_3, \alpha_5, \alpha_7 \in {\Bbb E}_{8,+}, \qquad \alpha_1, \alpha_4, \alpha_6 ,\alpha_8\in {\Bbb E}_{8,-}.
\]
The invariance of the Hamiltonian and the symplectic form \eqref{eq:*H-g} with respect to $S_1$ from \eqref{eq:DCox-E8}, and restricting on the set of admissible roots $\beta_k$, will result in a real Hamiltonian form  of the $E_8^{(1)}$ ATFT, described by:
\begin{equation}\label{eq:eq:E8-rhf}\begin{split}
\mathcal{H}^{\bbbr}_{E_8} &= {1 \over 2} ({\bf p}^+ , {\bf p}^+ ) -{1 \over 2} ({\bf p}^- , {\bf p}^- ) +
\sum_{k =0,2,3,5,7} n_k' (e^{-({\bf q}^+ ,\alpha _k)} -1) + \sum_{k= 1,4,6,8} n_k'' ( \cos ({\bf q}^- ,\beta_k)-1),
\end{split}\end{equation}
where $n_1''=n_8''=2$, $n_4''=6, n_6''=4$ and $n_2'=n_7'=3$, $n_3' =4$, $n_5' =5$ \cite{Bourb1}, and
\begin{equation}\label{eq:*ome-g8}\begin{split}
\omega ^{\bbbr}_{E_8^{(1)}} =
(\delta {\bf p}^+  \wedge \delta {\bf q}^+) -(\delta {\bf p}^-  \wedge \delta {\bf q}^-) .
\end{split}\end{equation}

\subsection{Affine Toda field theories related to $F_4^{(1)}$}\label{ssec:F4-1}

The extended roots of this algebra are
\begin{equation}\label{eq:f4roots}\begin{aligned}
\alpha_1&= e_2-e_3, \qquad \alpha_2 = e_3-e_4,  &\quad \alpha_3&= e_4, \\
 \alpha_4&= {1\over 2}(e_1-e_2-e_3-e_4), &\quad \alpha_0&=-e_1-e_2,
\end{aligned}\end{equation}
with $\alpha_0$ being the minimal root. The extended Dynkin diagram of $F_4^{(1)}$ is shown on Figure \ref{fig:Dynkin-F4}.
The fundamental weights of $F_4$ are
\begin{equation}\label{eq:weif4}\begin{aligned}
  \omega_1 &= e_1+e_2, &\quad \omega_2 &= 2e_1+e_2+e_3, \\
  \omega_3 &= {1\over 2}(3e_1+e_2+e_3+e_4), &\quad \omega_4 &=e_1.
\end{aligned}\end{equation}

\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[thick] (0,0) --(3,0);
\draw[thick] (4.5,0) --(6,0);
\draw[thick] (3,0.15) --(4.5,0.15);
\draw[thick] (3,-0.15) --(4.5,-0.15);

\draw[thick] (3.5,-0.4) --(3.75,0);
\draw[thick] (3.5,0.4) --(3.75,0);

\draw[thick,fill=white] (0,0) circle (1.5 mm);
\draw[thick,fill=black] (1.5,0) circle (1.5 mm);
\draw[thick,fill=white] (3,0) circle (1.5 mm);
\draw[thick,fill=black] (4.5,0) circle (1.5 mm);
\draw[thick,fill=white] (6,0) circle (1.5 mm);

\draw (0,-0.5) node {\small $\alpha_0$};
\draw (1.5,-0.5) node {\small $\alpha_1$};
\draw (3,-0.5) node {\small $\alpha_2$};
\draw (4.5,-0.5) node {\small $\alpha_3$};
\draw (6,-0.5) node {\small $\alpha_4$};

\draw (-1.4,0) node {$F_4^{(1)}$};

\end{tikzpicture}
\end{center}
  \caption{\small The extended Dynkin diagram of the  complex untwisted affine Kac-Moody algebra $F_4^{(1)}$. The white roots are invariant with respect to the automorphism $S_2$ in \eqref{eq:DCox-F4}.}\label{fig:Dynkin-F4}
\end{figure}

\noindent
 Using the dihedral realisation of the Coxeter automorphism for $F_4^{(1)}$:
\begin{equation}\label{eq:DCox-F4}
{\rm Cox}\, (F_4^{(1)})=S_1\circ S_2, \qquad S_1=  S_{\alpha_1}\circ S_{\alpha_3}, \qquad S_2= S_{\alpha_2}\circ S_{\alpha_4},
\end{equation}
and imposing  an invariance of the Hamiltonian \eqref{eq:*H-g} with respect to the automorphism $S_1$ in \eqref{eq:DCox-F4}, will extract a real Hamiltonian form of $F_4^{(1)}$ ATFT of the form
\begin{equation}\label{eq:F4-RHF}\begin{split}
\mathcal{H}^{\bbbr}_{F_4^{(1)}} =
{1 \over 2} \sum_{k=1}^{4} \left((p_k^+)^2- (p_k^-)^2\right) +
\sum_{k =0,2,4} n_k' (e^{-({\bf q}^+ ,\alpha _k)} -1) + \sum_{k= 1,3} n_k'' ( \cos ({\bf q}^- ,\beta_k)-1),
\end{split}\end{equation}
where $n_2'=3$,  $n_4'=2$ and  $n_1''=2$,  $n_3''=4$. In addition
\begin{equation}\label{eq:*ome-g4}\begin{split}
\omega ^{\bbbr}_{F_4^{(1)}} =
(\delta {\bf p}^+  \wedge \delta {\bf q}^+) -(\delta {\bf p}^-  \wedge \delta {\bf q}^-) .
\end{split}\end{equation}
Then the root space ${\Bbb E}_4$ will split into direct sums: ${\Bbb E}_4={\Bbb E}_{4,+}\oplus {\Bbb E}_{4,-}$, with
$\alpha_2, \alpha_4  \in {\Bbb E}_{4,+}$ and $\alpha_1, \alpha_3 \in {\Bbb E}_{4,-}$.

\subsection{Affine Toda field theories related to $G_2^{(1)}$}\label{ssec:G2-1}

The set of admissible roots for this algebra is given by
\begin{eqnarray}\label{eq:g2roots}
\alpha_1&=& e_1-e_2, \qquad \alpha_2= -e_1+e_2+e_3, \qquad \alpha_0=-e_1-e_2+2e_3.
\end{eqnarray}
The fundamental weights of $G_2$ are
\begin{eqnarray*}
  \omega_1 &=& e_1-e_2+2e_3, \qquad \omega_2= -e_1-e_2+2e_3.
\end{eqnarray*}
The extended Dynkin diagram of $G_2^{(1)}$ is shown on Figure \ref{fig:Dynkin-G2}.

Let us take $C_1=S_{\alpha_2}$.
The invariance of the Hamiltonian \eqref{eq:*H-g} with respect to the automorphism $C_1$ in will give a real Hamiltonian form of $G_2^{(1)}$ ATFT described by
\begin{eqnarray}\label{eq:G2-RHF}
\mathcal{H}^{\bbbr}_{G_2^{(1)}} &=&
{1 \over 2} \sum_{k=1}^{r} \left((p_k^+)^2- (p_k^-)^2\right) + (e^{-({\bf q}^+ ,\alpha_0)}-1) +
 3(e^{-({\bf q}^+ ,\alpha_1)}-1)  + 2 (\cos ({\bf q}^-,\alpha_2)-1),\\
\label{eq:*ome-g2}
\omega ^{\bbbr}_{G_2^{(1)}} &=&
(\delta {\bf p}^+  \wedge \delta {\bf q}^+) -(\delta {\bf p}^-  \wedge \delta {\bf q}^-) ,
\end{eqnarray}
The root space ${\Bbb E}_2$ will split into direct sums: ${\Bbb E}_2={\Bbb E}_{2,+}\oplus {\Bbb E}_{2,-}$, with
$\alpha_0, \alpha_1  \in {\Bbb E}_{2,+}$ and $\alpha_2 \in {\Bbb E}_{2,-}$.

\begin{figure}
\begin{center}
\begin{tikzpicture}

\draw[thick] (0,0.15) --(1.5,0.15);
\draw[thick] (0,0) --(1.5,0);
\draw[thick] (0,-0.15) --(1.5,-0.15);
\draw[thick] (1.5,0) --(3,0);

\draw[thick] (0.85,0.4) --(0.6,0);
\draw[thick] (0.85,-0.4) --(0.6,0);

\draw[thick,fill=white] (0,0) circle (1.5 mm);
\draw[thick,fill=black] (1.5,0) circle (1.5 mm);
\draw[thick,fill=white] (3,0) circle (1.5 mm);

\draw (0,-0.5) node {\small $\alpha_1$};
\draw (1.5,-0.5) node {\small $\alpha_2$};
\draw (3,-0.5) node {\small $\alpha_0$};

\draw (-1.4,0) node {$G_2^{(1)}$};

\end{tikzpicture}
\end{center}
  \caption{\small The extended Dynkin diagram of the  complex untwisted affine Kac-Moody algebra $G_2^{(1)}$.

  }\label{fig:Dynkin-G2}
\end{figure}

\section{On the spectral properties of the Lax operators with $\mathbb{Z}_h $-reduction}

\subsection{General theory}

The  operator $L$ (\ref{eq:2.1}) is very convenient for deriving the
2-dimensional TFT. However this formulation is not convenient  for
constructing the fundamental analytic solutions (FAS) of $L$.  The first
step we will do will be to apply on $L$ a similarity transformation
\begin{equation}\label{eq:Ltil}\begin{split}
 \tilde{L}\tilde{\psi} = g_0^{-1} L g_0 \tilde{\psi} =  i \frac{\partial \tilde{\psi}}{ \partial x } -
\left( i \sum_{k=0}^{6} q_k \mathcal{E}_k + \lambda \tilde{J}_0 \right) \tilde{\psi} =0
\end{split}\end{equation}
where $\tilde{J}_0 \in \mathfrak{h}$; i.e.
\[ \tilde{J}_0 = \sum_{ k=1}^{6}y_j H_{\alpha_j} =\sum_{ k=1}^{6}z_j H_{\omega_j}  \]
where $y_j$ and $z_j$ must be expressed in terms of $\omega = \exp( 2\pi i/h_g)$.

Let us somewhat simplify the notations in the two Lax operators

\begin{equation}\label{eq:LtL}\begin{split}
L\psi &\equiv \frac{\partial \psi}{ \partial x } + (\vec{q}_x - \lambda J_0 )\psi(x,t,\lambda) =0, \\
\tilde{L}\tilde{\psi}  &\equiv \frac{\partial \tilde{\psi} }{ \partial x } + (Q(x,t) - \lambda \tilde{J} )\tilde{\psi} (x,t,\lambda) =0,
\end{split}\end{equation}
where $J_0 = \sum_{\alpha \in \delta_0}^{} E_\alpha$, $\delta_0$ is the set of simple roots $\alpha_j$ and the minimal root
$\alpha_0$, and $ \tilde{J} \in \mathfrak{h}$ and $\omega = \exp(2\pi i/h)$.

The spectral theory of the Lax operators with $\mathbb{Z}_h$ Mikhailov reduction groups \cite{Mikh}
for the classical series of simple Lie algebras are by now well developed, see \cite{GeYa,SIAM*14,VG-Ya-13}
Here we will formulate them taking into account when necessary the peculiarities of the exceptional algebras.

In \cite{TJ22} the problem for the interrelation between $L$ and $\tilde{L}$ was solved by using the Chevallie
basis for $A_5^{(1)}$ \cite{DriSok}. Indeed, for the classical series Chevallie basis for the typical representations are well
known so it is not difficult to diagonalize $J_0$ and then evaluate $\alpha_j(\tilde{J}_0)$. The same procedure for the exceptional algebras
(except $G_2$) is rather more complicated. Indeed, the bases for the typical representations of $F_4$, $E_6$ and $E_7$ are well
known, see \cite{Howlett} where one can find all root-vectors of $F_4$ as $26\times 26$ matrices. For $E_6$
and $E_7$ the simple roots have been given in \cite{Howlett}; it is possible that all root vectors for all exceptional algebras
have been evaluated in the framework of MAGMA project. So in general one can find $J_0$ for the typical representations
of all exceptional algebras. Then the construction of $\tilde{J}_0$ becomes the next task. The eigenvalues of $J_0$
and the eigenvectors may be calculated, though the latter would be rather involved expressions.
The next challenge would be to constructing out of the properly normalized eigenvectors $g_0$ and to ensure that it
is an element of the corresponding exceptional group.

The easiest way to get a realization of the Coxeter automorphism $C_g$ is to represent it as an element
of the Cartan subgroup:
\begin{equation}\label{eq:Cox0}\begin{split}
C_g = \exp \left( \frac{2\pi i}{h_g} \sum_{j=1}^{r} H_{\omega_j} \right),
\end{split}\end{equation}
where $h_g$ is the Coxeter number and $\omega_j$ are the fundamental weights of the algebra $\mathfrak{g}$. Indeed,
using the Cartan-Weyl commutation relations \cite{Helg} one can easily check that
\begin{equation}\label{eq:Cox3}\begin{split}
 C_g J_0 C_g^{-1} = \omega J_0, \qquad \omega = \exp \left( \frac{2\pi i}{h_g} \right).
\end{split}\end{equation}
This construction is compatible with the Lax operator $L$, since $C_g $ commutes with $\vec{q}_x$ and therefore
\begin{equation}\label{eq:CoxL}\begin{split}
 C_g L (\lambda) C_g^{-1} =  L (\omega\lambda).
\end{split}\end{equation}
Treating the operator $\tilde{L}$ requires the use of the Coxeter automorphism $\tilde{C}_g$ realized as
element of the Weyl group  $\mathcal{W}_g$. We already mentioned the difficulties involved with the standard approach based on the
use of the Chevallie basis. Therefore, treating the exceptional algebras  we will use a different approach.
Our idea is to work only in the Euclidean space $\mathbb{E}_r$ in which the root systems are embedded.
Indeed, there we can construct $\tilde{C}_g$ using its dihedral form \cite{Helg, Bourb1, Cart}:
\begin{equation}\label{eq:Cox1}\begin{split}
\tilde{C}_g = S_w \circ S_b, \qquad S_w = \prod_{\alpha \in W_g}^{} S_\alpha, \qquad S_b = \prod_{\beta \in B_g}^{} S_\beta,
\end{split}\end{equation}
where $W_g$ (respectively $B_g$) are the sets of `white` roots (resp. `black` roots) of the Dinkin diagram of the
algebra $\mathfrak{g}$. Note that all elements of $W_g$ (resp. $B_g$) are all orthogonal, so $S_w$ and $S_b$ are
Weyl reflections, i.e.  $S_w^2 =\openone$ and $S_b^2 =\openone$. Then $C_g^{h_g} =\openone$, where $h_g$ is the Coxeter
number of the algebra $\mathfrak{g}$. Thus we will find the realization of the Coxeter automorphism as element of the
Weyl group $\mathcal{W}_g$ of the algebra.

The next step should be to construct the Cartan subalgebra element $\tilde{J}_0$. In fact we will replace
this step, and instead of constructing $\tilde{J}_0$ we will construct the vector $\vec{v}_g \in \mathbb{E}_r$ which is
dual to $\tilde{J}_0$. Then we will us the fact that $\alpha_j(\tilde{J}_0) = (\alpha_j, \vec{v}_g)$. This is all we need
in order to construct the continuous spectrum and the spectral data for $\tilde{L}$. On the other hand, since  $\tilde{L}$
is related to $L$ by the similarity transformation  $\tilde{L} =g_0 L g_0^{-1}$ then both operators will have the
same spectral data.

The construction of the vector $\vec{v}_g$ is based on its basic property of invariance under the $\mathbb{Z}_h$
Mikhailov reduction group; like in (\ref{eq:CoxL}) we must have:
\begin{equation}\label{eq:CoxtL}\begin{split}
\tilde{C}_g \tilde{L} (\lambda) \tilde{C}_g^{-1} =  \tilde{L} (\omega\lambda), \qquad \mbox{i.e.} \qquad
  \tilde{C}_g(\vec{v}_g) = \omega \vec{v}_g.
\end{split}\end{equation}
Such invariant vectors can be obtained by taking weighted averaged action by $\tilde{C}_g$:
\begin{equation}\label{eq:vg}\begin{split}
 \vec{v}_{0} = \sum_{s=0}^{h_g-1} \omega^{-s} \tilde{C}_g^s (\vec{v}_g).
\end{split}\end{equation}
Obviously there is an arbitrariness in the choice of $\vec{v}_g$: it can be a root, or fundamental weight  of the algebra
$\mathfrak{g}$. At the same time we can use the fact that the spectral parameter $\lambda$ can be redefined, so
this arbitrariness can be taken care of.

\subsection{The FAS of the Lax operators $L$}\label{sec:2.3}

Here we just outline the procedure of constructing the fundamental analytic solutions
(FAS) of $L$ \cite{BeCo1, BeCo2,  SIAM*14, GeYa}. First we have to determine the regions of analyticity. For
smooth potentials $Q(x) $ that fall off fast enough for $x\to \pm \infty$ these regions
are the $2h $ sectors $ \Omega _\nu  $ separated by the rays
$l_\nu  $ on which $\re \lambda (\alpha, \vec{v}_g) =0 $, where by $\alpha$ is a root of $\mathfrak{g}$.  The rays $l_\nu $ and the sectors $\Omega_\nu$ are given by:
\begin{eqnarray}\label{eq:l-nu}
l_\nu \colon \arg(\lambda) ={\pi (\nu -1) \over h_g } , \qquad \Omega_\nu \colon \frac{\pi (\nu -1)}{h_g} \leq \arg(\lambda) \leq \frac{\nu}{h_g},
\qquad \nu =1,\dots , 2h,
\end{eqnarray}
and close angles equal to $\pi/h $.

The main result is that in each of the sectors $\Omega_\nu$ we can construct fundamental analytic solutions
(FAS) $\xi_\nu (x,t,\lambda)$  of the Lax operator $\mathcal{L}$:
\begin{equation}\label{eq:calL}\begin{split}
 \mathcal{L}\xi_\nu (x,t,\lambda) \equiv i \frac{\partial \xi_\nu }{ \partial x } +Q_x \xi_\nu (x,\lambda) - \lambda [\tilde{J},\xi_\nu]=0, \\
 Q_x = \sum_{j=1}^{r} q_{j,x} \mathcal{O}_j, \qquad \mathcal{O}_j = \sum_{s=0}^{h_g-1} C_g^s(E_{\alpha_j}).
\end{split}\end{equation}

 The asymptotics of $\xi^\nu (x,\lambda ) $ and $\xi^{\nu -1} (x,\lambda )  $ along the
ray $l_\nu  $ can be written in the form \cite{67,GeYa}:
\begin{equation}\label{eq:xi-as}\begin{aligned}
\lim_{x\to -\infty } e^{-\lambda \tilde{J} x} \xi^\nu (x,\lambda e^{i0} )
e^{\lambda \tilde{J} x} &= S_\nu ^+ (\lambda ), &\quad \lambda \in l_\nu ,\\
\lim_{x\to -\infty } e^{-\lambda \tilde{J} x} \xi^{\nu -1} (x,\lambda e^{-i0} )
e^{\lambda \tilde{J} x} &= S_{\nu} ^- (\lambda ), &\quad \lambda \in l_{\nu },\\
\lim_{x\to \infty } e^{-\lambda \tilde{J} x} \xi^\nu (x,\lambda e^{i0} )
e^{\lambda \tilde{J} x} &= T_\nu ^-D_\nu ^+ (\lambda ), &\quad \lambda \in l_\nu ,\\
\lim_{x\to \infty } e^{-\lambda \tilde{J} x} \xi^{\nu -1} (x,\lambda e^{-i0} )
e^{\lambda \tilde{J} x} &= T_{\nu} ^+D_{\nu } ^-(\lambda ), &\quad \lambda \in l_{\nu } ,
\end{aligned}\end{equation}
where the matrices $S_\nu ^+ $, $T_\nu ^+ $ (resp. $S_\nu ^- $, $T_\nu ^-
$) are upper-triangular (resp. lower-triangular) with respect to the $\nu
$-ordering. They provide the Gauss decomposition
of the scattering matrix with respect to the $\nu  $-ordering, i.e.:
\begin{equation}\begin{split}\label{eq:nu-gauss}
T_\nu (\lambda ) &= T_\nu ^-(\lambda ) D_\nu ^+(\lambda ) \hat{S}_\nu ^+(\lambda ), \qquad \lambda \in l_\nu e^{i0}, \\
&=  T_\nu ^+(\lambda ) D_\nu ^-(\lambda ) \hat{S}_\nu ^- (\lambda ) ,  \qquad \lambda \in l_\nu e^{-i0}.
\end{split}\end{equation}
More careful analysis shows \cite{GeYa} that in fact $T_\nu
(\lambda ) $ belongs to a subgroup ${\cal G}_\nu$ of $SL(N,\mathbb{C})$.
In particular we can choose $\lambda$ in such a way that the simple roots of $\mathfrak{g}$
will be related to the rays $l_0$ and $l_1$; more precisely the `black` roots will be related to
$l_0$ while the `white` roots will be related to $l_1$. In particular this means that with
$l_0$ and $l_1$ we can relate subalgebras of $\mathfrak{g}$ which will be direct sums of several
copies of $sl(2)$. In addition we can introduce the analogs of the reflection coefficients $\rho^\pm_{\nu,j}$
and $\tau^\pm_{\nu,j}$ with $\nu=0,1$ as follows:
\begin{equation}\label{eq:58}\begin{aligned}
 S_0^\pm(\lambda) &=\exp \left( \sum_{\alpha \in B_g}^{}\tau_{0,\alpha}^\pm (\lambda) E_{\pm \alpha}\right), &\quad
 T_0^\mp(\lambda) &=\exp \left( \sum_{\alpha \in B_g}^{}\rho_{0,\alpha}^\pm (\lambda) E_{\mp \alpha}\right),  \\
 S_1^\pm (\lambda) &=\exp \left( \sum_{\alpha \in W_g}^{}\tau_{0,\alpha}^\pm (\lambda) E_{\pm \alpha}\right), &\quad
 T_1^\mp (\lambda)&=\exp \left( \sum_{\alpha \in W_g}^{}\rho_{0,\alpha}^\pm (\lambda) E_{\mp \alpha}\right),&\quad \\
 D_0^\pm(\lambda) &= \exp \left( \sum_{j=1}^{r} \frac{2d_{0,j}^\pm (\lambda) }{(\alpha_j,\alpha_j)}H_{\alpha_j}\right), &\quad
 D_1^\pm(\lambda) &= \exp \left( \sum_{j=1}^{r} \frac{2d_{0,j}^\pm (\lambda)}{(\alpha_j,\alpha_j)}H_{\alpha_j}\right).
\end{aligned}\end{equation}
Next the $\mathbb{Z}_h $-symmetry allows us to extend these scattering data to the other rays as well. Indeed applying the Coxeter
automorphism we get first the relations between FAS:
\begin{equation}\label{eq:Z_n-cons}\begin{aligned}
 C_g(\xi^\nu (x,\lambda)) &= \xi^{\nu +2} (x,\lambda\omega ) ,
&\quad C_g( T_\nu^\pm (\lambda  ))  &= T_{\nu +2}(\lambda\omega ), \\
C_g (S^\pm_\nu (\lambda  ))  &= S^\pm_{\nu +2}(\lambda \omega),
&\quad C_g( D^\pm_\nu (\lambda)) &= D^\pm_{\nu +2}(\lambda \omega) ,
\end{aligned}\end{equation}
where the index $\nu +2 $ should be taken modulo $2h $.
Consequently we can view as independent only the data on two of the
rays, e.g. on $l_1 $ and $ l_0 $; all the rest will be
recovered using (\ref{eq:Z_n-cons}).

\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[->] (-5,0) -- (5,0);
\draw[->] (0,-5) -- (0,5);

\draw (4,4) circle (3mm);
\draw (4,4) node {\small $\lambda$};

\draw [blue,thick] (-4,0) -- (4,0);
\draw [blue,thick,latex-latex] (-2.25,0) -- (2.25,0);

\draw [blue,thick,rotate=15] (-4,0) -- (4,0);
\draw [blue,thick,-latex,rotate=15] (-4,0) -- (-2,0);
\draw [blue,thick,latex-,rotate=15] (2,0) -- (4,0);

\draw [blue,thick,rotate=30] (-4,0) -- (4,0);
\draw [blue,thick,latex-latex,rotate=30] (-2.25,0) -- (2.25,0);

\draw [blue,thick,rotate=45] (-4,0) -- (4,0);
\draw [blue,thick,-latex,rotate=45] (-4,0) -- (-2,0);
\draw [blue,thick,latex-,rotate=45] (2,0) -- (4,0);

\draw [blue,thick,rotate=60] (-4,0) -- (4,0);
\draw [blue,thick,latex-latex,rotate=60] (-2.25,0) -- (2.25,0);

\draw [blue,thick,rotate=75] (-4,0) -- (4,0);
\draw [blue,thick,-latex,rotate=75] (-4,0) -- (-2,0);
\draw [blue,thick,latex-,rotate=75] (2,0) -- (4,0);

\draw [blue,thick,rotate=90] (-4,0) -- (4,0);
\draw [blue,thick,latex-latex,rotate=90] (-2.25,0) -- (2.25,0);

\draw [blue,thick,rotate=105] (-4,0) -- (4,0);
\draw [blue,thick,-latex,rotate=105] (-4,0) -- (-2,0);
\draw [blue,thick,latex-,rotate=105] (2,0) -- (4,0);

\draw [blue,thick,rotate=120] (-4,0) -- (4,0);
\draw [blue,thick,latex-latex,rotate=120] (-2.25,0) -- (2.25,0);

\draw [blue,thick,rotate=135] (-4,0) -- (4,0);
\draw [blue,thick,-latex,rotate=135] (-4,0) -- (-2,0);
\draw [blue,thick,latex-,rotate=135] (2,0) -- (4,0);

\draw [blue,thick,rotate=150] (-4,0) -- (4,0);
\draw [blue,thick,latex-latex,rotate=150] (-2.25,0) -- (2.25,0);

\draw [blue,thick,rotate=165] (-4,0) -- (4,0);
\draw [blue,thick,-latex,rotate=165] (-4,0) -- (-2,0);
\draw [blue,thick,latex-,rotate=165] (2,0) -- (4,0);

\draw[blue] (4.25,0.25) node {\small $l_0$};
\draw[blue] (-0.25,4.25) node {\small $l_6$};
\draw[blue] (-4.25,-0.25) node {\small $l_{12}$};
\draw[blue] (0.25,-4.25) node {\small $l_{18}$};

\draw[blue] (15:4.25) node {\small $l_{1}$};
\draw[blue] (30:4.25) node {\small $l_{2}$};
\draw[blue] (45:4.25) node {\small $l_{3}$};
\draw[blue] (60:4.25) node {\small $l_{4}$};
\draw[blue] (75:4.25) node {\small $l_{5}$};

\draw[blue] (105:4.25) node {\small $l_{7}$};
\draw[blue] (120:4.25) node {\small $l_{8}$};
\draw[blue] (135:4.25) node {\small $l_{9}$};
\draw[blue] (150:4.25) node {\small $l_{10}$};
\draw[blue] (165:4.25) node {\small $l_{11}$};

\draw[blue] (195:4.25) node {\small $l_{13}$};
\draw[blue] (210:4.25) node {\small $l_{14}$};
\draw[blue] (225:4.25) node {\small $l_{15}$};
\draw[blue] (240:4.25) node {\small $l_{16}$};
\draw[blue] (255:4.25) node {\small $l_{17}$};

\draw[blue] (285:4.25) node {\small $l_{19}$};
\draw[blue] (300:4.25) node {\small $l_{20}$};
\draw[blue] (315:4.25) node {\small $l_{21}$};
\draw[blue] (330:4.25) node {\small $l_{22}$};
\draw[blue] (345:4.25) node {\small $l_{23}$};

\draw[magenta] (7.5:3.25) node {\small $\Omega_{0}$};
\draw[magenta] (22.5:3.25) node {\small $\Omega_{1}$};
\draw[magenta] (37.5:3.25) node {\small $\Omega_{2}$};
\draw[magenta] (52.5:3.25) node {\small $\Omega_{3}$};
\draw[magenta] (67.5:3.25) node {\small $\Omega_{4}$};
\draw[magenta] (82.5:3.25) node {\small $\Omega_{5}$};

\draw[magenta] (97.5:3.25) node {\small $\Omega_{6}$};
\draw[magenta] (112.5:3.25) node {\small $\Omega_{7}$};
\draw[magenta] (127.5:3.25) node {\small $\Omega_{8}$};
\draw[magenta] (142.5:3.25) node {\small $\Omega_{9}$};
\draw[magenta] (157.5:3.25) node {\small $\Omega_{10}$};
\draw[magenta] (172.5:3.25) node {\small $\Omega_{11}$};

\draw[magenta] (187.5:3.25) node {\small $\Omega_{12}$};
\draw[magenta] (202.5:3.25) node {\small $\Omega_{13}$};
\draw[magenta] (217.5:3.25) node {\small $\Omega_{14}$};
\draw[magenta] (232.5:3.25) node {\small $\Omega_{15}$};
\draw[magenta] (247.5:3.25) node {\small $\Omega_{16}$};
\draw[magenta] (262.5:3.25) node {\small $\Omega_{17}$};

\draw[magenta] (277.5:3.25) node {\small $\Omega_{18}$};
\draw[magenta] (292.5:3.25) node {\small $\Omega_{19}$};
\draw[magenta] (307.5:3.25) node {\small $\Omega_{20}$};
\draw[magenta] (322.5:3.25) node {\small $\Omega_{21}$};
\draw[magenta] (337.5:3.25) node {\small $\Omega_{22}$};
\draw[magenta] (352.5:3.25) node {\small $\Omega_{23}$};

\end{tikzpicture}

\end{center}
  \caption{\small The continuous spectrum of the Lax operators related to the algebras $E_6$ and $F_4$
  whose Coxeter nubers equal 12. Obviously the continuous spectra of for $E_8$, $E_7$ and $G_2$
  will have different number of rays equal to 60, 36 and 12 respectively.}\label{fig:Spectrum}
\end{figure}

\subsection{The inverse scattering problem and the Riemann-Hilbert problem}\label{sec:ism-rhp}

The next important step is the possibility to reduce the solution of the
inverse scattering problem (ISP) for the generalized Zakharov-Shabat system  to a (local) RHP. More precisely, we have:
\begin{equation}\label{eq:*-nu}\begin{aligned}
\xi^\nu (x,t,\lambda ) &= \xi^{\nu -1}(x,t,\lambda ) G_\nu (x,t,\lambda ), &\; \lambda &\in l_\nu ,\\
G_\nu (x,t,\lambda ) &= e^{\lambda \tilde{J} x -\lambda^{-1} \tilde{I} t} G_{0,\nu }(\lambda ) e^{-\lambda \tilde{J} x +\lambda^{-1} \tilde{I} t},  &\; G_{0,\nu
}(\lambda ) &= \left. \hat{S}_\nu ^- S_\nu ^+(\lambda )\right|_{t=0} .
\end{aligned}\end{equation}
where $ \tilde{I} = g_0^{-1} I(x,t) g_0$
The collection of all relations (\ref{eq:*-nu}) for $\nu =1,2,\dots,2h $
together with
\begin{equation}\label{eq:*-nu-norm}
\lim_{\lambda \to\infty } \xi^\nu (x,t,\lambda ) = \openone ,
\end{equation}
can be viewed as a local RHP posed on the collection of rays
$\Sigma \equiv \{l_\nu \}_{\nu =1}^{2h} $ with canonical normalization.
Rather straightforwardly we can  prove that if $\xi^\nu (x,\lambda ) $ is a solution of the
RHP (\ref{eq:*-nu}), (\ref{eq:*-nu-norm}) then $\chi ^\nu (x,\lambda
)=\xi^\nu (x,\lambda ) e^{-\lambda U_1 x} $ is a FAS of $L$ with potential
\begin{equation}\label{eq:q-CBC}
Q_x(x,t) = \lim_{\lambda \to\infty } \lambda \left( \tilde{J} - \xi^\nu (x,t,\lambda ) \tilde{J} \hat{\xi}^\nu (x,t,\lambda ) \right).
\end{equation}
For the classical Lie algebras the analyticity properties of  $d_j^\pm(\lambda ) $ allow one to
reconstruct  them from the sewing function $G(\lambda ) $ (\ref{eq:*-nu}) and from the and from the locations of their  zeroes and poles
\cite{GeYa, GVYa}. The idea was to consider the relations (\ref{eq:nu-gauss}) in the $j$-th fundamental representation of $\mathfrak{g}$
and take the matrix elements between the highest $|\omega_j^+ \rangle$ (resp. lowest  $|\omega_j^- \rangle$) weight vectors. This can be
done also for exceptional groups with the results
\begin{equation}\label{eq:dpm}\begin{aligned}
 d_{\nu,j}^+(\lambda) -  d_{\nu,j}^-(\lambda) & = -\ln \langle \omega_j^+ | \hat{T}_\nu^+ T_\nu^-(\lambda) | \omega_j^+\rangle \\
 &= -\ln \langle \omega_j^- | \hat{S}_\nu^- S_\nu^+(\lambda) | \omega_j^+\rangle .
\end{aligned}\end{equation}
If $\mathfrak{g}$ is one of the classical Lie algebras, then the right hand sides of (\ref{eq:dpm}) can be related to the
principal upper and lower minors of the scattering matric $T(\lambda)$ in the
typical representation.  For the exceptional algebras the fundamental representations can not so easy be related to the typical
(or lowest dimensional) one. So recovering the generating functionals of integrals of motion from their analyticity properties is an
open problem.

\section{The minimal sets of scattering data}

\subsection{The minimal sets of scattering data from the RHP}

As a consequence of the results in Section \ref{sec:ism-rhp}  we conclude that the following theorem holds true:
\begin{theorem}\label{thm:1}
Let us assume that the  Lax operator $\mathcal{L}$ (\ref{eq:calL}) is constructed along the
ideas in Section 4 above, i.e.  it is related to a simple Lie algebra $\mathfrak{g}$ and possesses
$\mathbb{Z}_h$ symmetry where $h$ is the Coxeter number of $\mathfrak{g}$.
Let us also assume that its potential $Q_x$ is such that $\mathcal{L}$ has
no discrete eigenvalues. Then its FAS $\xi^\nu (x,t,\lambda)$ in the sector $\Omega_\nu$
will provide a regular solution of the Riemann-Hilbert Problem  (\ref{eq:*-nu}).
Let us denote by $\delta_0$ and  $\delta_1$ the subsets of roots of the algebra $\mathfrak{g}$ which satisfy the equations:
\begin{equation}\label{eq:del10}\begin{aligned}
\alpha \in \delta_0 \qquad \mbox{if} \qquad \im \;\lambda (\alpha, \vec{v}_g) =0  \qquad \mbox{for} \qquad \lambda \in l_0, \\
\alpha \in \delta_1 \qquad \mbox{if} \qquad \im \; \lambda (\alpha, \vec{v}_g) =0  \qquad \mbox{for} \qquad \lambda \in l_1.
\end{aligned}\end{equation}
Let also
\begin{equation}\label{eq:}\begin{split}
\mathcal{T}_1 &= \{ \tau^\pm_{0,\alpha}(\lambda), \quad \alpha\in \delta_0, \; \lambda \in l_0\} \cup
\{ \tau^\pm_{1,\alpha}(\lambda), \quad \alpha\in \delta_1, \; \lambda \in l_1\} ; \\
\mathcal{T}_2 &= \{ \rho^\pm_{0,\alpha}(\lambda), \quad \alpha\in \delta_0, \; \lambda \in l_0\} \cup
\{ \rho^\pm_{1,\alpha}(\lambda), \quad \alpha\in \delta_1, \; \lambda \in l_1\} .
\end{split}\end{equation}
Then: A) $\mathcal{T}_1 $ (resp. $\mathcal{T}_2$) provide a minimal set of scattering data which allow one to recover:
\begin{enumerate}
  \item the Gauss factors $S_\nu^\pm (\lambda)$ for $\nu =0, 1$ (resp. $T_\nu^\pm (\lambda)$ for $\nu =0, 1$;

  \item the Gauss factors $D_\nu^\pm (\lambda)$ for $\nu =0, 1$ (resp. $T_\nu^\pm (\lambda)$ for $\nu =0, 1$;

  \item the sewing functions $G_\nu(x,t,\lambda)$ and the scattering matrices $T_\nu (\lambda)$ for each ray $l_\nu$, $\nu =0, \dots, 2h_g -1$;

\end{enumerate}

B)  $\mathcal{T}_1 $ (resp. $\mathcal{T}_2$) allows one to construct the regular solution $\xi^\nu (x,t,\lambda)$ of the RHP;

C)  $\mathcal{T}_1 $ (resp. $\mathcal{T}_2$) allows one to recover the potential $Q_x$ of the Lax operator.

\end{theorem}

\begin{proof}
A) Given $\mathcal{T}_1$ (respectively $\mathcal{T}_2$) and using (\ref{eq:Z_n-cons}) we recover the Gauss factors, i.e.,
the group-valued functions $S_\nu^\pm(\lambda,t)$ (respectively $T_\nu^\pm(\lambda,t)$)  for $\nu=0,1$.
Next, using the $\mathbb{Z}_h$-symmetry we construct $S_\nu^\pm(\lambda,t)$ (respectively $T_\nu^\pm(\lambda,t)$)  for $\nu=2,
\dots , 2h_g-1$, see eq.   (\ref{eq:Z_n-cons}).

The next step consists in recovering the functionals of the integrals of motion $d_{\nu, j}^\pm (\lambda)$. The relations
(\ref{eq:dpm}) provide in principle this possibility. Indeed, the sets $\mathcal{T}_1$ and $\mathcal{T}_1$ then
we know the Gauss factors $S_\nu^\pm(\lambda,t)$ and  $T_\nu^\pm(\lambda,t)$ not only in the typical representations,
but also in all fundamental representations. Of course there are serious difficulties in evaluating the right hand sides
of (\ref{eq:dpm}). However there is no doubt that  they are determined uniquely by any of the sets $\mathcal{T}_1$
and $\mathcal{T}_1$. As a result we have determined all Gauss factors in eq. (\ref{eq:58}), i.e. we obtained all sewing
functions $G_\nu(x,t,\lambda)$ and the scattering matrices $T_\nu (t,\lambda)$.

B) Since the operator $\mathcal{L}$ has no discrete eigenvalues, then the corresponding FAS satisfy a regular RHP.
Therefore the sewing functions $G_\nu(x,t,\lambda)$ determine uniquely the solution $\xi_\nu^\pm(\lambda,t)$ of the
RHP.

C) Since the solution $\xi_\nu^\pm(\lambda,t)$ of the RHP is uniquely determined then the corresponding potential of
$\mathcal{L}$ is determined from eq. (\ref{eq:q-CBC}).

\end{proof}

\subsection{Minimal sets of scattering data for the exceptional algebras}

Here we will formulate the results about the spectral data for the Lax operators related to the
exceptional algebras. The calculations were performed by Maple. The allowed us to calculate the
corresponding vectors $\vec{v}_g$ as well as the scalar products $(\vec{v}_g, \alpha_j)$ and
to establish for each of the exceptional algebras the results in Subsection 4.2.
In particular we found that the subsets of roots $\delta_0$ and $\delta_1$ are
given by the sets of `black` and `white` roots, i.e:
\begin{equation}\label{eq:del01}\begin{split}
\delta_0 \equiv B_g, \qquad \delta_1 \equiv W_g .
\end{split}\end{equation}
Below we list, besides the sets of roots $\delta_0$ and $\delta_1$, also Coxeter automorphism as a
linear operator (matrix) acting on the Euclidean space ${\Bbb E}^r$.

\subsubsection{$\mathfrak{g}\simeq E_6^{(1)}$}
Let $\varepsilon_6 = \frac{1}{\sqrt{3}}(- e_6 - e_7 + e_8)$. Then the set of admissible roots \eqref{eq:e6roots} ca be rewritten as:
\begin{equation}
\label{eq:del0E6}
\begin{aligned}
\alpha_0 &= \frac{1}{2}(e_1 + e_2+ e_3 + e_4 + e_5 + \sqrt{3}\varepsilon_6), \quad
\alpha_1 = \frac{1}{2}(e_1 - e_2 - e_3 - e_4 - e_5 + \sqrt{3}\varepsilon_6), \\
\alpha_2 &= e_1 + e_2, \quad
\alpha_3 = e_2 - e_1, \quad
\alpha_4 = e_3 - e_2, \quad
\alpha_5 = e_4 - e_3, \quad
\alpha_6 = e_5 - e_4.
\end{aligned}
\end{equation}
There are two rays \eqref{eq:l-nu}, corresponding tho the Coxeter automorphism of $E_6^{(1)}$. Roots related to the rays $l_0$ and $l_1$:
\begin{equation}
\label{eq:laE6}
\begin{aligned}
 l_0 &= \arg \lambda =0, &\qquad B_g &\equiv \{ \alpha_1, \alpha_4, \alpha_6 \}; \\
 l_1 &= \arg \lambda =\frac{\pi}{12}, &\qquad W_g &\equiv \{ -\alpha_2, -\alpha_3, -\alpha_5 \};
 \end{aligned}
\end{equation}
The Coxeter number of $E_6^{(1)}$ is $h=12$ and the exponents are $1,4, 5, 7, 8, 11$.

The Coxeter automorphism as a linear operator on the dual Euclidean space $\mathbb{E}_6$:
\begin{equation}
\label{eq:Ce6}
\begin{split}
C_{E_6} = \frac{1}{4}
\left(\begin{array}{ccccccc}
 -3 & -1 & -1 & -1 & -1 & \sqrt{3} \\
 -1 & 1 & -3 & 1 & 1 & -\sqrt{3} \\
 1 & -1 & -1 & -1 & 3 &  \sqrt{3} \\
1 & 3 & -1 & -1 & -1 &  \sqrt{3} \\
  1 & -1 & -1 & 3 & -1 & \sqrt{3} \\
 -\sqrt{3} & \sqrt{3} & \sqrt{3} & \sqrt{3} & \sqrt{3} & 1
 \end{array}\right),
\quad C_{E_6}^{12} = \openone_6, \quad \det C_{E_6} = 1.
\end{split}
\end{equation}
Here  $e_j$, $j=1, \dots, 5$ and $\varepsilon_6$ is the basis of $\mathbb{E}_6$.
The characteristic polynomial of $C_{E_6}$ reads:
\begin{equation}
\label{eq:Pe6}
\begin{split}
P_{E_6}= z^6+ z^5  - z^3 +z +1.
\end{split}
\end{equation}

\subsubsection{$\mathfrak{g}\simeq E_7^{(1)}$}

Introducing the notation
\[
\varepsilon_7=-{e_7-e_8\over \sqrt{2}},
\]
the set of admissible roots \eqref{eq:e7roots} can be rewritten as:
\begin{equation}\label{eq:del0}\begin{aligned}
\alpha_0 &= -\sqrt{2}\varepsilon_7 = e_7-e_8, &\quad \alpha_1 &= \frac{1}{2}(e_1 - e_2 -e_3 -e_4 - e_5 -e_6 + \sqrt{2}\varepsilon_7), \\
\alpha_2 &= e_1 + e_2, \qquad \alpha_2 = e_1 + e_2, &\quad \alpha_3 &= e_2 - e_1, \quad \alpha_4 = e_3 - e_2, \\
\alpha_5 &= e_4 - e_3, \qquad \alpha_6 = e_5 - e_4, &\quad \alpha_3 &= e_6 - e_5.
\end{aligned}\end{equation}
Roots related to the rays $l_0$ and $l_1$ are respectively:
\begin{equation}\label{eq:laE7}\begin{aligned}
 l_0 &= \arg \lambda =0, &\qquad B_g &\equiv \{ \alpha_1, \alpha_4, \alpha_6 \}; \\
 l_1 &= \arg \lambda =\frac{\pi}{18}, &\qquad W_g &\equiv \{ -\alpha_2, -\alpha_3, -\alpha_5, -\alpha_7 \};
 \end{aligned}\end{equation}
The Coxeter number for $E_7^{(1)}$ is $h=18$ and the exponents are $1, 5, 7, 9, 11, 13, 17$.
The Coxeter automorphism acts as a linear operator on the space $\mathbb{E}_7$ as:
\begin{equation}\label{eq:Ce7}\begin{split}
C_{E_7} = \frac{1}{4} \left(\begin{array}{ccccccc} -3 & -1 & 1 & 1 & 1 & 1 & \sqrt{2} \\ -1 & 1 & -1 & 3 & -1 & -1 & -\sqrt{2} \\
 -1 & -1 & -3 & -1 & -1 & -1 & -\sqrt{2} \\  -1 & 1 & -1 & -1 & -1 & 3 & -\sqrt{2} \\  -1 & 1 & -1 & -1 & 3 & -1 & -\sqrt{2} \\
-\sqrt{2} & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & -\sqrt{2} & 2
 \end{array}\right), \quad C_{E_7}^9 = -\openone_7, \quad \det C_{E_7} = -1.
\end{split}\end{equation}
Here   $e_j$, $j=1, \dots, 6$ and $\varepsilon_7$ forms the basis in the dual Euclidean space ${\Bbb E}_7$.
The characteristic polynomial of $C_{E_7}$ is:
\begin{equation}\label{eq:Pe7}\begin{split}
P_{E_7}= z^7 + z^8 - z^4 - z^3 +z +1
\end{split}\end{equation}

\subsubsection{$\mathfrak{g}\simeq E_8^{(1)}$}
The set of admissible roots is given by \eqref{eq:e8roots}.
Roots related to the rays $l_0$ and $l_1$ are:
\begin{equation}
\label{eq:laE8}
\begin{aligned}
 l_0 &= \arg \lambda =0, &\qquad B_g &\equiv \{ \alpha_1, \alpha_4, \alpha_6, \alpha_8 \}; \\
 l_1 &= \arg \lambda =\frac{\pi}{30}, &\qquad W_g &\equiv \{ -\alpha_2, -\alpha_3, -\alpha_5, -\alpha_7 \};
 \end{aligned}
\end{equation}
The Coxeter number  for $E_8^{(1)}$ is $h=30$ and the exponents are $1, 7, 11, 13, 17, 19, 23, 29$.
The action of the  Coxeter automorphism as a linear operator on the dual Euclidean space  $\mathbb{E}_7$ is given by
\begin{equation}
\label{eq:CE8}
\begin{split}
C_{E_8} = \frac{1}{4}
\left(\begin{array}{cccccccc}
-3 & -1 & 1 & 1 & 1 & 1 & 1 & -1 \\
-1 & 1 & -1 & 3 & -1 & -1 & -1 & 1 \\
-1 & -3 & -1 & -1 & -1 & -1 & -1 & 1 \\
-1 & 1 & -1 & -1 & -1 & 3 & -1 & 1 \\
-1 & 1 & 3 & -1 & -1 & -1 & -1 & 1 \\
-1 & 1 & -1 & -1 & -1 & -1 & 3 & 1 \\
-1 & 1 & -1 & -1 & 3 & -1 & -1 & 1 \\
1 & -1 & 1 & 1 & 1 & 1 & 1 & 3
 \end{array}\right),
\quad C_{E_8}^{15} = -\openone_8, \quad \det C_{E_8} = -1.
\end{split}
\end{equation}
Here  $e_j$, $j=1, \dots, 6$ and $\varepsilon_7$ form up the basis of ${\Bbb E}_7$.
The characteristic polynomial of $C_{E_8}$:
\begin{equation}
\label{eq:PE8}
\begin{split}
P_{E_8}= z^8 + z^7 - z^5 - z^4 - z^3 +z + 1.
\end{split}
\end{equation}

\subsubsection{$\mathfrak{g}\simeq F_4^{(1)}$}

The set of admissible roots is given by \eqref{eq:f4roots}.
Roots related to the rays $l_0$ and $l_1$ are as follows:
\begin{equation}
\label{eq:laF4}
\begin{aligned}
 l_0 &= \arg \lambda =0, &\qquad B_g &\equiv \{ \alpha_1, \alpha_3 \}; \\
 l_1 &= \arg \lambda =\frac{\pi}{12}, &\qquad W_g &\equiv \{ -\alpha_2, -\alpha_4 \};
 \end{aligned}
\end{equation}
The Coxeter number of $F_4^{(1)}$ is $h=12$ and the exponents are $1, 5, 7, 11$.
The Coxeter automorphism acts as a linear operator on the dual space $\mathbb{E}_4$ via:
\begin{equation}
\label{eq:CF4}
\begin{split}
C_{F_4} = \frac{1}{2}
\left(\begin{array}{cccccccc}
1 & -1 & 1 & -1  \\
-1 & -1 & 1 & 1  \\
1 & -1 & -1 & 1 \\
1 & 1 & 1 & 1
 \end{array}\right),
\quad C_{F_4}^6 = -\openone_4, \quad \det C_{F_4} = 1.
\end{split}
\end{equation}
Here  $e_j$, $j=1, \dots, 4$ is the basis of ${\Bbb E}_4$.
The characteristic polynomial of $C_{F_4}$:
\begin{equation}
\label{eq:PF4}
\begin{split}
P_{F_4}= z^4  - z^2 + 1.
\end{split}
\end{equation}

\subsubsection{$\mathfrak{g}\simeq G_2^{(1)}$}

This comes to be  both the simplest and the most peculiar case. The peculiarity is in the fact, that
though the rank of $G_2$ equals 2, its root system is traditionally written as a set of vectors in the
3-dimensional space $\mathbb{E}_3$ which are all orthogonal to the vector $e_1+e_2+e_3$. The Coxeter
automorphism here is the composition of the Weyl reflection $C_{G_2} = S_{\alpha_1} S_{\alpha_2}$.
One can check that $C_{G_2}$ is expressed as the following linear operator in $\mathbb{E}_3$:
\begin{equation}\label{eq:0G2}\begin{split}
C_{G_2} = \frac{1}{3} \left(\begin{array}{ccc} 2 & -1 & 2 \\  2 & 2 & -1 \\  -1 & 2 & 2  \end{array}\right),
\end{split}\end{equation}
which has the following properties:
\begin{equation}\label{eq:1G2}\begin{split}
\qquad C_{G_2}^2 = \left(\begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right) , \qquad
C_{G_2}^4 = \left(\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right) , \qquad
C_{G_2}^6  = \openone_3.
\end{split}\end{equation}
The characteristic polynomial is
\begin{equation}\label{eq:2G2}\begin{split}
P_{G_2} = z^3 -2z^2 + 2z -1 = (z-1) (z^2 -z +1).
\end{split}\end{equation}
Another way to approach the problem is to project the root system of $G_2$ onto the 2-dimensional plane
orthogonal to the vector $e_1 + e_2 + e_3$. Indeed, let
\[
\varepsilon_1 = \frac{1}{\sqrt{2}}(e_2 - e_1),\qquad \varepsilon_2 = \frac{1}{\sqrt{2}}(e_3 - e_2).
\]
Then the set of admissible roots \eqref{eq:g2roots} can be rewritten as:
\begin{equation}
\label{eq:del0G2}
\begin{aligned}
\alpha_0 &= \sqrt{2}(\varepsilon_1 + 2\varepsilon_2) = -e_1 -e_2 + 2e_3, \quad
\alpha_1 = \sqrt{2}\varepsilon_1 = e_2 - e_1, \\
\alpha_2 &= \sqrt{2}(\varepsilon_2 - \varepsilon_1) = e_1 - 2e_2 + e_3,
\end{aligned}
\end{equation}
Roots related to the rays $l_0$ and $l_1$ are:
\begin{equation}
\label{eq:laG2}
\begin{aligned}
 l_0 &= \arg \lambda =0, &\qquad B_g &\equiv \{ \alpha_1\}; \\
 l_1 &= \arg \lambda =\frac{\pi}{12}, &\qquad W_g &\equiv \{ -\alpha_2 \};
 \end{aligned}
\end{equation}
The Coxeter number for $G_2^{(1)}$ is $h=6$ and the exponents are $1, 5$.
This Coxeter automorphism induces an action on the dual space $\mathbb{E}_2$ with  a matrix:
\begin{equation}
\label{eq:CG2}
\begin{split}
C'_{G_2} =
\left(\begin{array}{cccccccc}
0 & 1  \\
-1 & 1
 \end{array}\right),
\quad C'_{G_2}{}^3 = -\openone_2, \quad \det C'_{G_2} = 1.
\end{split}
\end{equation}
Here  $\{\varepsilon_1, \varepsilon_2\}$ form up a basis in ${\Bbb E}_2$. Finally, the characteristic polynomial of $C_{G_2}$ reads:
\begin{equation}
\label{eq:PG2}
\begin{split}
P_{G_2}= z^2  - z + 1.
\end{split}
\end{equation}

\section{Conclusions}

We presented here real Hamiltonian forms of affine Toda field theories related  exceptional untwisted complex Kac-Moody algebras. We
established that the special properties of these models allow us to relate
the construction of the RHF to the study of ${\Bbb Z}_h $ symmetries of the
extended Dynkin diagrams (with $h$ being the Coxeter number of ${\frak g}$). The general construction is illustrated by
several examples of such models related to the exceptional Kac-Moody algebras $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$,$D_4^{(1)}$ and $G_2^{(1)}$. We used the  ``dihedral realisation'' of the Coxeter automorphism ${\rm Cox}\, ({\frak g})=S_1\circ S_2$, where $S_1, S_2$ are ${\Bbb Z}_2$-automorphisms of ${\frak g}$. One of these automorphisms extracts the RHF and the other one acts as additional ${\Bbb Z_2}$-reduction on it.

Each of the real Hamiltonian forms obtained above has its dual one.
Indeed, we could define the involution $\mathcal{C}$ using the Weyl reflection $S_2$. The consideration is quite analogous:
simply we interchange the black roots with the white ones. The reformulation of all the above results is rather
obvious. Note that the dual real Hamiltonian forms of 2DTFT {\it are not} equivalent to the ones obtained above.
Indeed, it is easy to check that in all cases treated above the number of the white roots is different from
the number of black roots.

Some additional problems are natural extensions to the results presented
here:

\begin{itemize}
  \item The complete classification of all nonequivalent RHF of ATFT.

\item The description of the hierarchy of Hamiltonian structures of RHF of ATFT (for a review of the infinite-dimensional cases see e.g.
\cite{DriSok,67} and the references therein) It is also an
open problem to construct the RHF for ATFT using some of its
higher Hamiltonian structures.

\item The extension of the dressing Zakharov-Shabat method \cite{ZaSha2}  to the above classes of Lax operators is also an open problem. One of the difficulties is due to the fact that the $\mathbb{Z}_h$ reductions requires dressing factors with $2h$ pole singularities \cite{VSG-88}. This  makes the relevant linear algebraic equations rather involved \cite{Zhu}.

\item Another open problem is to study types of boundary conditions and boundary effects of ATFT's and their RHF \cite{Caudrelier,Doikou1}. This includes types of boundary defects \cite{Doikou2}.

\item The last and technically more involved
problem is to solve the inverse scattering problem for the Lax operator
and thus prove the complete integrability of all these models. The ideas of \cite{AKNS, GVYa} about the interpretation of the inverse scattering method as a generalized Fourier
transform holds true also for the $\mathbb{Z}_h$ reduces Lax operators \cite{SIAM*14, GeYa, Yan2013a, VG-Ya-13,YanVi*2012b}. This may allow one to derive
the action-angle variables for these classes of NLEE.

\end{itemize}

\section*{Acknowledgements}

This work is  supported by the Bulgarian National Science Fund, grant KP-06N42-2.

redpajama_set_name: RedPajamaArXiv


text:
Stăuceni este satul de reședință al comunei cu același nume din sectorul Rîșcani, municipiul Chișinău, Republica Moldova. Aici activează Colegiul Național de Viticultură și Vinificație din Chișinău.

Demografie

Structura etnică
Structura etnică a satului Stăuceni conform recensământului populației din 2004:

Referințe

Legături externe

 Google Maps
 Stauceni vazut pe harta OSM

Localități din municipiul Chișinău


A Patent Trial and Appeal Board decision holding that assignor estoppel may not bar an assignor or his or her privies from petitioning for inter partes review was not judicially reviewable, the U.S. Court of Appeals for the Federal Circuit has decided. The court lacked jurisdiction under 35 U.S.C. §314 to review the Board's decision to institute inter partes review of a molding machine patent owned by Husky Injection Molding Systems, despite Husky's claim that assignor estoppel barred the Board from considering Athena Automation Ltd.'s petition. On a cross appeal filed by Athena, the Federal Circuit vacated the Board's decision that four challenged claims were not anticipated (Husky Injection Molding Systems Ltd. v. Athena Automation Ltd., September 23, 2016, Lourie, .).
U.S. Patent 7,670,536 (the '536 patent) discloses a molding machine having a clamp assembly comprising a stationary platen, a movable platen, tie bars, tie bar locks that couple the tie bars to the movable platen, and clamp actuators that supply a clamping force to the tie bars.
A co-inventor of the '536 patent and owner and president of Husky assigned the patent to Husky before selling Husky to a private equity group, and leaving to form Athena. Athena filed a petition for inter partes review of the '536 patent.
The Patent Trial and Appeal Board found that several claims (1, 4–16, 18, and 20–22) of the '536 patent were anticipated, while claims 2, 3, 17, and 19 were not anticipated. Both parties appealed.
Husky's appeal. The court dismissed Husky's appeal for lack of jurisdiction based on a lack of authority to review the Board's decision on assignor estoppel. Husky challenged the Board's determination during the institution phase that assignor estoppel cannot bar an assignor or his or her privies from petitioning forinter partes review. Assignor estoppel prevents an assignee from later arguing that what was assigned lacked value.
Section 314(d) of the America Invents Act provides that decision on instituting inter partes review are nonappealable. The Supreme Court left open the possibility that § 314(d) allows for review of appeals that (1) implicate constitutional questions, (2) depend on other less closely related statutes, or (3) that present other questions of interpretation that reach well beyond Section 314. Husky's argument that the third exception applied in this case was rejected. Before instituting review, the Board considers whether there is a likelihood of success on the patentability grounds. Because assignor estoppel bars an assignor from challenging the patentability of the patent earlier assigned, assignor estoppel cannot be divorced from the scope of Section 314 and is, thus, beyond review.
The court also found that the question did not relate to the Board's ultimate invalidation authority. The Board's invalidation authority derives from the patent challenged and the type of the review requested; it is not a question of who petitions for review. A bar preventing particular petitioners from challenging a patent does not impact the Board's invalidation authority. Ultimately, any question concerning assignor estoppel implicates who may petition for review, and that issue falls outside of the narrow exceptions to the otherwise broad ban on our review of the decision whether to institute.
Athena appeal. Athena asserted that various claims of the '536 patent were anticipated either by U.S. Patent Application 2004/0208950 (Glaesener) in combination with its incorporation by reference of U.S. Patent 5,753,153 (Choi). The Board found that Athena failed to explain what Glaesener incorporates from Choi, much less how Choi's clamping device would secure the components of Glaesener's molding machine.
The court vacated the Board's finding of no anticipation with respect to claims 2, 3, 17, and 19, and remanded the case for further consideration. Athena argued on appeal that the Board's determination that the prior art Glaesener did not incorporate the prior art Choi by reference, and thereby the Board's finding that claims 2, 3, 17, and 19 are not unpatentable as anticipated. The court agreed with Athena that a skilled artisan would understand with sufficient particularity what the "pineapple and toothed-ring mechanism" language referred to in Choi. Also, the court agreed that the second statement effects a broad incorporation that cannot be negated by the use of a more precise incorporation statement elsewhere.
Dissenting opinion. CircuitJudge Plagerdissented from the majority opinion regarding its lack of jurisdiction to review the determination by the Board regarding whether assignor estoppel precluded it from instituting inter partes review. Judge Plager suggested that the majority's view may contribute to the confusion regarding which matters the Federal Circuit can review on appeal from a final decision by the Board. Given the long tradition of judicial review of administrative decision-making, it is anomalous to say that the application of a rule like the assignor estoppel doctrine to proceedings before the Board, a unit of an administrative agency, is exclusively for the agency to decide. Rather, the doctrine is in the nature of a common law doctrine, and not itself based on statute. Because the majority refused to undertake the proper review of the Board's decision, Judge Plager respectfully dissented.
The cases are Nos. 2015-1726 and 2015-1727.
Attorneys: Matthew L. Cutler (Harness, Dickey & Pierce, PLC) for Husky Injection Molding Systems Ltd. Marshall John Schmitt (Michael Best & Friedrich, LLP) for Athena Automation Ltd. Joseph Matal, Office of the Solicitor, United States Patent and Trademark Office, Alexandria, VA, for Michelle K. Lee.
Companies: Husky Injection Molding Systems Ltd.; Athena Automation Ltd.


text:
Rintis is een bestuurslaag in het regentschap Pekanbaru van de provincie Riau, Indonesië. Rintis telt 6300 inwoners (volkstelling 2010).

Plaats in Riau


text:
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The ability of their mind is outstanding and smoking weed could have significant long term changes to your mind. I believe that it really is like anything, even if you are using it too much it's got the potential to damage your body, in the event that you can learn to restrain yourself then you shouldn't have a problem.
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Remember if you desire to cannabis free, then you will be using affirmations! Stop your chronic marijuana habit today and try it out. Repeat the procedure above every morning and night for a minimum of 30 days. I guarantee you will not desire bud.


text:
<?php
namespace f2r\Fydrate\Adapter\Pgsql;

use f2r\Fydrate\Adapter\ConnectionInterface;
use f2r\Fydrate\Adapter\Exception;

class Connection implements ConnectionInterface
{
    protected $connection;

    public function __construct($host, $user, $password = null, $database = null, $port = null)
    {
        $dsn = 'host=' . $host . ' user=\'' . addslashes($user) . '\'';
        if (isset($password)) {
            $dsn .= ' password=\'' . addslashes($password) . '\'';
        }
        if (isset($database)) {
            $dsn .= ' dbname=\'' . addslashes($database) . '\'';
        }
        if (isset($port)) {
            $dsn .= ' port=' . $port;
        }

        $this->connection = pg_connect($dsn);
        pg_set_error_verbosity($this->connection, PGSQL_ERRORS_TERSE);
    }

    public function escapeString($string)
    {
        return pg_escape_string($this->connection, $string);
    }

    public function beginTransaction()
    {
        return @pg_query($this->connection, 'BEGIN');
    }

    public function commit()
    {
        return @pg_query($this->connection, 'COMMIT');
    }

    public function rollback()
    {
        return @pg_query($this->connection, 'ROLLBACK');
    }

    public function query($sql)
    {
        return @pg_query($this->connection, $sql);
    }

    public function exec($sql)
    {
        return @pg_query($this->connection, $sql);
    }

    public function getError()
    {
        return pg_last_error($this->connection);
    }

    public function setOption($name, $value)
    {
        throw new Exception('Could not set options for postgres adapter');
    }

    public function setCharset($encoding)
    {
        pg_set_client_encoding($this->connection, $encoding);
        return $this;
    }

    public function getAffectedRows()
    {
        return pg_affected_rows($this->connection);
    }

    public function getLastInsertId()
    {
        return pg_last_oid($this->connection);
    }
}

redpajama_set_name: RedPajamaGithub