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home Economic Statistics Database Economic Statistics by Country Nigeria Nationality, Ethnic Groups, Languages and Religions for Year 2015
Nigeria Nationality, Ethnic Groups, Languages and Religions
Country or Region: Nigeria › Change country
Economic Indicator: Nationality, Ethnic Groups, Languages and Religions › See all Nigeria indicators, › See Nationality, Ethnic Groups, Languages and Religions data for all countries
National or Regional Currency: Naira
Currency Symbol: NGN
Nationality: noun: Nigerian(s) ; adjective: Nigerian
Ethnic Groups: Nigeria, Africa's most populous country, is composed of more than 250 ethnic groups; the following are the most populous and politically influential: Hausa and Fulani 29%, Yoruba 21%, Igbo (Ibo) 18%, Ijaw 10%, Kanuri 4%, Ibibio 3.5%, Tiv 2.5%
Religions: Muslim 50%, Christian 40%, indigenous beliefs 10%
Languages: English (official), Hausa, Yoruba, Igbo (Ibo), Fulani, over 500 additional indigenous languages
Other Aggregate Indicators for Nigeria
Geography (Location, geographic coordinates, map references, area, terrain, elevation extremes, borders, coastline and maritime claims) for Nigeria
Climate and Environment (climate, environmental issues, environmental agreements, natural hazards) for Nigeria
Natural Resources, Agricultural Products, Land Use and Water Use for Nigeria
Nationality, Ethnic Groups, Languages and Religions for Nigeria
Telephone Statistics: Number of Fixed Lines, Mobile Phone Penetration Rate for Nigeria
Age Structure, Median Age, Gender and Urbanization for Nigeria
Population Growth Rate, Birth Rate, Death Rate, Net Migration Rate for Nigeria
HIV, AIDS and Other Major Infectious Disease Rates for Nigeria
Literacy Rate, Educational Expenditure and School Life Expectancy for Nigeria
Industries and Industrial Growth Production Rate for Nigeria
Labor Force by Occupation for Nigeria
Poverty Line, Gini Index, Household Income and Consumption for Nigeria
National Budget and Public Debt for Nigeria
Bank Rates (Commercial Bank Prime Lending Rate, Central Bank Discount Rate) for Nigeria
Stock of Money, Stock of Quasi Money, Stock of Domestic Credit for Nigeria
Foreign Exchange Reserves and Gold Reserves for Nigeria
FDI Statistics (Foreign Direct Investment at Home/ Inbound FDI, Foreign Direct Investment Abroad/ Outbound FDI) for Nigeria
Exchange Rate Statistics for Nigeria
Electricity Production, Electricity Consumption, Electricity Exports and Electricity Imports for Nigeria
Oil Production, Proven Oil Reserves, Oil Consumption, Oil Exports and Oil Imports for Nigeria
Natural Gas Production, Proven Natural Gas Reserves, Natural Gas Consumption, Natural Gas Exports and Natural Gas Imports for Nigeria
Television and Radio Statistics (TV Stats, Radio Stats) for Nigeria
Internet Statistics (Internet Users, Internet Hosts, Country TLD) for Nigeria
Transport Statistics (Railways, Roadways, Waterways, Ports and Terminals, Airports, Merchant Marine) for Nigeria
Military Statistics (Military Expenditure, Military Branches, Service Age and Obligation, Manpower Available and Fit for Military Service) for Nigeria
Disputes, Refugees, Human Trafficking and Illicit Drug Production for Nigeria
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Kate Geraghty/Fairfax Media via Getty Images
Making Migration Work for Everyone
Jun 20, 2019 Md. Shahidul Haque
As different categories of migration have begun to overlap, governments have struggled to adapt, pointing to the need for a new framework to manage this key domain of the global economy. With the right norms and mechanisms in place, the far-reaching benefits of immigration can be realized, and the attendant social concerns alleviated.
GENEVA – In a globalized world, migration is a fact of life that should be governed accordingly. To that end, it is time to establish what I call "Migration Order 3.0," a new framework that would make migration work for everyone.
Until World War II, transnational human mobility was subject to Migration Order 1.0: immigration controls were established solely at the national level by governments. After the war, international institutions such as the United Nations High Commissioner for Refugees and the forerunner to the International Organization for Migration (IOM) were created to manage refugee and migrant flows, primarily from and within Europe. During this era of Migration Order 2.0, the movement of people across national borders was governed by mutually agreed norms, standards, and practices.
Large-scale cross-border movements of people, however, have exposed fault lines in that post-war framework. We now know that Migration Order 2.0 is inadequate to the task at hand. According to the UN Department of Economic and Social Affairs, most migrants tend to move within their own continents, and not necessarily toward the destinations that one would assume.
Md. Shahidul Haque
5 Commentaries
Md. Shahidul Haque, a former foreign secretary of Bangladesh, is a senior fellow at North South University in Dhaka.
Dr.Tajul Islam Aug 5, 2019
Well said sir.Speciallythe main four reason for migration U've sort out.But in my sense if we could scape the capping of a country or so much restriction,war & over nationalish & make the world free for every one then someday, obviouslyit will take same decades to get the desired result,but one day we may able to find a refugee free world.
Henk Crop Jun 21, 2019
"far reaching benefits of immigration can be realised" only if there is a vacancy in a host country and a qualified migrant to fill it.
In reality mass-immigration can often be very costly to a host country. The host country has to offer housing, social security, medicare, etc. to an immigrant who never paid for it.
It is an illusion to believe that migration can offer a solution to alleviate the exploding world population of 83 million per year, 96 % of which is in developing regions.
The only solution is that countries with a birthrate exceeding 2,1 child per woman reduce their population growth to that level.
How far the Elite is out of touch with reality was demonstrated by the GCM ( Marrakesh Pact).
It excessively pampered the migrant but it did not hold the home countries responsible to take their illegal immigrants back, which is a major problem. Also the host countries were recommended to curtail the freedom of their press, and abandon a fundamental right.
vivek iyer Jun 21, 2019
Indeed! The only outcome of the pact was that it brought down the Belgium Govt.! Since that is not particularly hard to do, one wonders why tax payer money was wasted on giving bureaucrats a nice holiday.
Because of bogus Asylum claims the undertaking to take back migrants- which could be enforced in any case by reciprocal action denying visas- is completely worthless.
It is the poorest and most desperate migrants who pay the price for the gesture politics of the bien pensant elite. They are told they will get everything if they go and pay through the nose to do so. But their horrendous suffering is instrumentalized by the elite to show that the ordinary person in the target country is 'deplorable' for not demanding the scrapping of border controls.
Even Hilary Clinton is telling Europe to get tough on migration. St. Angela, of course, can do what she likes now she need face no more election battles.
Our experience of the last two decades shows the bien pensant elite- especially Economists- were completely wrong in their fundamental assumptions re. the levels and effects of migration both 'regular' and 'irregular'. These same elites had no problem bailing themselves out while imposing austerity on the 'deplorables'. Sadly, it does not appear that elites have been replaced by pragmatic people. Rather the democratic deficit has been capitalised on by financially unscrupulous cliques.
This GCM is based on the mostly false assumption that migration is beneficial to all, migrant and host country. It neglects the difficulty of illegal migrants returning to their home countries. Either the migrant or the home country refuse to cooperate. Notorious are in this respect: Moroccco, Afghanistan, Iraq, Iran, Algeria, Ivory Coast and Ghana.
The UN and the EU have made regulations so complex that it is very difficult and costly to deport an illegal migrant and it also invites human trafficking. On average more than 80 % of the immigrants are illegal.
The GCM suggests that it can be a solution to the population explosion.
Africa has now 1250 million people and it is expected to double in the next 35 years. How can migration bring any significant solutions ? When you consider mrs. Merkel was already having trouble to integrate 1,2 million migrants.
With the greatest respect to the erudite author, this appears to be a wholly delusive project. It is time that the UN recognize that its own vision of itself is not sustainable- never mind its 'Sustainability Goals'.
P. Kamath Jun 20, 2019
Very sad but true. There are too many human beings in our planet and still multiplying. No where to go. Unless leaders of nations get together and plan to solve this problem, one can expect only conflicts and nothing else in the future.
Nobody can isolate themselves by building even mile high walls around nations.
https://prosyn.org/JDR8QMg

redpajama_set_name: RedPajamaCommonCrawl


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text:
\section{Introduction}

\medskip

The Lawson correspondence states \cite{Lawson} that for any minimal surface in ${\mathbb S}^3$ there exists an isometric constant mean curvature surface in ${\mathbb R}^3$. It is an important tool for the investigation and construction of CMC surfaces. In particular it was a crucial tool for the classification of trinoids in \cite{KGBKS} and for the numerical construction of examples of CMC surfaces with higher topology in \cite{KGBP}. For the last purpose it was important to integrate it once and to formulate it terms of the corresponding frames \cite{OP} (see also Theorem~\ref{thm:Lawson_smooth}).

Although discrete CMC surfaces in ${\mathbb R}^3$ have been known for a longtime already \cite{BP}, as well as discrete minimal surfaces in ${\mathbb S}^3$ \cite{BHL, BuHRS}, the discrete Lawson correspondence has remained a challenge. The main problem was to define a proper discrete analogue of isometry.

The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in ${\mathbb R}^3$ and discrete minimal surfaces in ${\mathbb S}^3$ formulated in Theorem~\ref{thm:discrete_Lawson}. This is a correspondence between two discrete isothermic surfaces. We show that it is an isometry in the following sense: it preserves the metric coefficients for isothermic nets introduced previously in \cite{BS}. Exactly as in the smooth case, this is a correspondence between nets with the same Lax matrices, and the immersion formulas also coincide with the smooth case. As necessary intermediary results, we show that commuting Lax pairs generate discrete CMC and minimal surfaces via an immersion formula as in the smooth case (Theorems \ref{thm:CMCR3} and \ref{thm:CMCS3}), and, conversely, that all discrete CMC and minimal surfaces are generated by Lax pairs (Theorem~\ref{thm:reconstruction}).

Another approach to discrete isothermic surfaces in spaceforms via conserved quantities
and discrete line bundles has been proposed in \cite{BuHRS,BuHR}, encompassing constant mean
curvature nets as a special case. A Calapso transformation is defined therein,
which generalizes the Lawson correspondence between the relevant spaceforms and preserves
geometric quantities. In contrast, our more pedestrian method focuses on the immersion formulas
which are proven to be identical as in the smooth case, and provide a more explicit
definition of the correspondence and its metric invariance. Still, both definitions agree,
as we prove in Remark~\ref{Calapso}.
\medskip

We shall consider in this paper only meshes with quadrilateral planar faces,
known as quad-nets or Q-nets for short (also called PQ-meshes), whose
theoretical properties mimic those of their smooth counterparts. In the
particular case of nets indexed by $\mathbb{Z}^2$, indices play the same role
as coordinates of an immersion, and specific choices of Q-nets correspond to
specific parametrizations of surfaces.

Throughout the text, we will use the shift notation to describe the local
geometry: When $F$ is a net, $F = F_0 = F (0, 0)$ will denote a base point,
while $F_1, F_2, F_{12}$ will stand for $F (1, 0), F (0, 1), F (1, 1)$, so
that indices $1, 2$ correspond to shift in the first and second variables
respectively. The same holds for any vertex-based function. The edges of the
face $(F, F_1, F_{12}, F_2)$ are labeled $(0, 1)$, $(1, 12)$, $(12, 2)$ and
$(2, 0)$ and the values of an edge-based function $u$ will be denoted by
$u_{01}$, $u_{1, 12}$, etc. If $(i, j)$ is a pair of indices corresponding to
an edge, $\mathop{}\!\mathrm{d} \varphi_{i j}$ is by definition $\varphi_j - \varphi_i$.

\medskip

\emph{Acknowledgments}: the Authors wish to thank Udo Hertrich-Jeromin and
Wayne Rossman for the fruitful discussions on this topic, and Tim Hoffmann for
his judicious remarks.

\section{The smooth theory}	\label{sec:smooth-theory}

\subsection{Constant mean curvature surfaces in $\mathbb{R}^3$}

We recall here a well known (see for example \cite{B, BP} for more details) description of CMC surfaces in $\mathbb{R}^3$ in terms of loop groups and quaternionic frames. The normalizations used in the present paper coincide with the normalizations in \cite{BP}.

In the sequel we identify the Euclidean three space $\mathbb{R}^3$ with imaginary quaternions ${\rm Im}\ {\mathbb{H}}$, and the standard imaginary quaternions with an orthonormal basis of $\mathbb{R}^3$,
and use the following matrix representation:
\begin{eqnarray}
\i=\left( \begin {array}{cc} 0&-i\\-i&0 \end{array}\right),
\j=\left( \begin {array}{cc} 0&{-1}\\operatorname{\mathbbl{1}}&0 \end{array}\right),
\k=\left( \begin {array}{cc} -i&0\\0&i \end{array}\right),
\operatorname{\mathbbl{1}} =\left( \begin {array}{cc} 1&0\\0&1 \end{array}\right).     \label{AC1.3}
 \end{eqnarray}
This results in the following matrix representation of vectors in $\mathbb{R}^3$:
\begin{eqnarray}
X = (X_1,X_2,X_3) \ \longleftrightarrow \ \left(
\begin{array}{cc}
-i X_3 & -i X_1 -X_2 \\
-i X_1 +X_2 & i X_3 \end{array} \right) \cdotp                        \label{AC1.5}
\end{eqnarray}

Umbilic free CMC-surfaces are isothermic (conformal curvature line parametrization). Let $(x,y)\mapsto F(x,y)$ be a CMC isothermically parametrized surface. Without loss of generality, one can normalize the mean curvature $H=1$ and the Hopf differential $Q=\langle F_{xx}-F_{yy}+2iF_{xy},N\rangle=\frac{1}{2}$. Let $e^u$ be the corresponding conformal metric: $\langle dF,dF \rangle = e^u (dx^2+dy^2)$. It satisfies the elliptic sinh-Gordon equation
\begin{equation} \label{eq:sinh-Gordon}
u_{xx}+u_{yy}+\sinh u=0.
\end{equation}
The quaternionic frame $\Phi$ is defined as a solution of the system
\begin{eqnarray}
&\Phi_x = U\Phi,\qquad
\Phi_y= V\Phi,              \label{H1.6}
\end{eqnarray}
where
\begin{eqnarray}
&U=\displaystyle {1\over 2}\left( \begin {array}{cc}
-\displaystyle{\frac {i}{2}}u_y & -\lambda e^{-u/2}-\displaystyle{1\over \lambda}e^{u/2}\\
\lambda e^{u/2}+\displaystyle{1\over \lambda}e^{-u/2}  &  \displaystyle{\frac {i}{2}}u_y
\end{array} \right),                \nonumber\\
&                                    \label{H1.7}\\
&V=\displaystyle {1\over 2}\left( \begin {array}{cc}
\displaystyle{\frac {i}{2}}u_x & -i\lambda e^{-u/2}+\displaystyle{i\over \lambda}e^{u/2}\\
i\lambda e^{u/2}-\displaystyle{i\over \lambda}e^{-u/2}  &  -\displaystyle{\frac {i}{2}}u_x
\end{array}\right).                 \nonumber
\end{eqnarray}
The matrices \eqref{H1.7} belong to the loop algebra
$$
g_H[\lambda ] =\{ \xi:S^1\to su(2):
\xi (-\lambda)=\sigma_3 \xi (\lambda)\sigma_3 \} \, ,
\text{ where }
\sigma_3 = i \k = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},
$$
and $\Phi$ in \eqref{H1.6} lies in the corresponding loop group
\begin{equation}
G_H[\lambda ] =\{ \phi:S^1\to SU(2):
\phi (-\lambda)=\sigma_3 \phi (\lambda)\sigma_3 \}.     \label{H1.8}
\end{equation}
Here $S^1$ is the set $| \lambda |=1$.

The system \eqref{H1.7} is the Lax representation for \eqref{eq:sinh-Gordon}, where the parameter $\lambda$ is called the spectral parameter.

\begin{theorem} \label{th:smooth_R^3}
The formulas
\begin{equation}   \label{eq:smooth_immersion_R3}
	\quad \hat{N} =-\check{N}= - \Phi^{- 1} \k \Phi, \quad
	\left\{ \begin{array}{ll}
	\hat{F} & = - \Phi^{- 1}  \frac{\partial \Phi}{\partial \gamma} - \frac{1}{2}  \hat{N}
   \\
   \check{F} & = - \Phi^{- 1} \frac{\partial \Phi}{\partial \gamma} +
   \frac{1}{2}  \hat{N} = \hat{F} + \hat{N}
   \end{array} \right.,
\end{equation}
where $\lambda=e^{i\gamma}$, describe two parallel surfaces $\hat{F}, \check{F}$ with constant
mean curvature $H=1$ and their Gauss maps $\hat{N}, \check{N}$.
Variation of $\gamma$ is an isometry, and the corresponding one parameter family of CMC surfaces is called the {\em associated family}. For $\gamma=0$, i.e. $\lambda=1$, the parametrizations of $\hat{F}$ and  $\check{F}$ are isothermic.
\end{theorem}

\subsection{Constant mean curvature and minimal surfaces in $\mathbb{S}^3$}

The same Lax pair yields a CMC net in $\mathbb{S}^3$ through the immersion formula obtained in \cite{B}.
We identify $S^3$ with unitary quaternions
\begin{eqnarray}
X = (X_1,X_2,X_3,X_4) \ \longleftrightarrow \ \left(
\begin{array}{cc}
-i X_3 +X_4 & -i X_1 -X_2 \\
-i X_1 +X_2 & i X_3+X_4 \end{array} \right) \cdotp                        \label{eq:quaternionic_S^3}
\end{eqnarray}
If we gauge the frame into
$\Psi = \begin{pmatrix}  e^{i \gamma / 2} & 0 \\ 0 & e^{- i \gamma / 2}
\end{pmatrix} \Phi = \exp \left( - \frac{\gamma}{2} \k \right) \Phi$,
then for any pair $\lambda_1 = e^{i \gamma_1}, \lambda_2 = e^{i
\gamma_2}$ in the unit circle,
\begin{equation*}
F = \Psi (\lambda_1)^{- 1} \Psi (\lambda_2), \quad
N = - \Psi (\lambda_1)^{- 1} \k \Psi (\lambda_2)
\end{equation*}
are an orthogonal pair of vectors in $\mathbb{S}^3$. They describe a surface $F$ with constant mean curvature $H=\cot(\gamma_1-\gamma_2)$ and its Gauss map $N$.
In terms of the original frame the formulas look as follows:
\begin{equation}   \label{eq:smooth_S^3}
	F = \Phi (\lambda_1)^{- 1} M \Phi (\lambda_2), \;
	N = - \Phi (\lambda_1)^{- 1} \k M \Phi (\lambda_2),
\end{equation}
where $M = \exp (\frac{\gamma_1 - \gamma_2}{2} \k)$.

\begin{theorem} \label{th:smooth_S^3}
Let $\Phi(\lambda)$ be a solution of \eqref{H1.6}. Formulas \eqref{eq:smooth_S^3} describe a surface $F$ with constant mean curvature $H=\cot(\gamma_1-\gamma_2)$ and its Gauss map $N$. The parametrization $F(x,y)$ is isothermic if and only if $\lambda_2 = \pm \lambda_1^{-1}$ ($\gamma_1 + \gamma_2 \equiv 0 \mod \pi$).
In particular, for $\gamma_1=-\gamma_2=\frac{\pi}{4}$ one obtains an isothermically parametrized minimal surface $F$ with the Gauss map $N$ and the conformal metric $e^{-u}$. Equivalently they can be treated as an isothermically parametrized minimal surface $N$ with the Gauss map $F$ and the conformal metric $e^{u}$.
\end{theorem}

\subsection{The Lawson correspondence}

As we have indicated already in Theorem~\ref{th:smooth_S^3}, CMC surfaces in $\mathbb{R}^3$ and minimal surfaces in $\mathbb{S}^3$ corresponding to the same Lax pair are isometric. This correspondence can be lifted to the frames without referring to the Lax representation. The corresponding formulas were obtained in \cite{OP}. We will derive them from the immersion formulas \eqref{th:smooth_R^3}, \eqref{eq:smooth_S^3}.

\begin{theorem}\label{thm:Lawson_smooth}
Let $F$ and $N$ be a pair of Christoffel dual ($F^*=N$) isothermic surfaces in $\mathbb{S}^3$. They can be treated equivalently as the minimal surface $F$ with the Gauss map $N$ or the minimal surface $N$ with the Gauss map $F$.
Then there exist surfaces $\hat{F}$ and $\check{F}$ in $\mathbb{R}^3$ with constant mean curvature $H=1$ isometric to $F$ and $N$ respectively. They and their Gauss maps $\hat{N}$ and $\check{N}$  are given by the following formulas:
\begin{eqnarray}
d\hat{F}= F^{-1}*dF, \quad \hat{N}=F^{-1}N=-NF^{-1} \nonumber\\
d\check{F}=N^{-1}*dN, \quad \check{N}=-F^{-1}N=NF^{-1} \label{eq:Lawson_smooth}.
\end{eqnarray}
Here $*$ is the Hodge star defined by ($*f_x=-f_y, *f_y=f_x$).
The parametrization of surfaces $\hat{F}, \check{F}$ given by \eqref{eq:Lawson_smooth} inherited from the isothermic parametrization of $F$ and $N$ is not isothermic. Formulas \eqref{eq:Lawson_smooth} give the surfaces from the associated family \eqref{eq:smooth_immersion_R3} corresponding to $\lambda=\lambda_2= e^{-i\frac{\pi}{4}}$.

Surfaces $F,N$ in $\mathbb{S}^3$ and surfaces $\hat{F},\check{F}$ in $\mathbb{R}^3$ with the Gauss maps $\hat{N},\check{N}$ are described by the formulas \eqref{eq:smooth_S^3} and \eqref{eq:smooth_immersion_R3} with the same Lax matrices \eqref{H1.7}.
\end{theorem}

\begin{proof}
Follows from direct computation. In the minimal surface case \eqref{eq:smooth_S^3} becomes
\begin{eqnarray}
F = \Phi (\lambda_1)^{- 1} M \Phi (\lambda_2), \;
	N =  \Phi (\lambda_1)^{- 1} M^{-1}\Phi (\lambda_2),\;
	M=\left( \begin {array}{cc}
e^{-i\frac{\pi}{4}} & 0\\
0  &  e^{i\frac{\pi}{4}}
\end{array} \right).
\end{eqnarray}

Formulas for the Gauss maps follow immediately:
$$
F^{-1}N=\Phi^{-1}(\lambda_2)M^{-2}\Phi(\lambda_2)=\hat{N}(\lambda_2).
$$
Computations for $d\hat{F}$ and $d\check{F}$ are slightly more involved,
$$
d\hat{F}=-d(\Phi)^{-1}\frac{\partial\Phi}{\partial\gamma}-\Phi^{-1}\frac{\partial d\Phi}{\partial\gamma}=
\Phi^{-1}\left(
-\frac{\partial}{\partial\gamma}(Udx+Vdy)+\frac{1}{2}[ \k,Udx+Vdy]
\right)\Phi.
$$
Calculating at $\lambda_2=e^{-i\frac{\pi}{4}}$ we get
\begin{equation} \label{eq:dF^}
d\hat{F}=e^{-u/2}\Phi^{-1}(\lambda_2)\left(
\left( \begin {array}{cc}
0 & e^{i\frac{\pi}{4}}\\
-e^{-i\frac{\pi}{4}}  & 0
\end{array} \right)dx+
\left( \begin {array}{cc}
0 & -e^{-i\frac{\pi}{4}}\\
e^{i\frac{\pi}{4}}  & 0
\end{array} \right)dy
\right)\Phi(\lambda_2).
\end{equation}
On the other hand from the formulas for surfaces in $\mathbb{S}^3$ we obtain
$$
dF=\Phi^{-1}(\lambda_1)\left(
(U(\lambda_1)dx-V(\lambda_1)dy)M+M(U(\lambda_2)dx+M(\lambda_2)dy)
\right)\Phi(\lambda_2),
$$
which implies
\begin{eqnarray*}
&*dF=\Phi^{-1}(\lambda_1)\left(
(V(\lambda_1)M-MV(\lambda_2))dx+(-U(\lambda_1)M+MU(\lambda_2))dy
\right)\Phi(\lambda_2), \\
&F^{-1}*F=\Phi^{-1}(\lambda_2)\left(
(M^{-1}V(\lambda_1)M-V(\lambda_2))dx+(-M^{-1}U(\lambda_1)M+U(\lambda_2))dy
\right)\Phi(\lambda_2)
\end{eqnarray*}
A direct computation shows that the last expression coincides with \eqref{eq:dF^}.
The identity for $d\check{F}$ follows in the same way.
\end{proof}

\section{Discrete CMC and minimal surfaces in $\mathbb{R}^3$ and~$\mathbb{S}^3$}
\label{sec:discrete_CMC_Q-nets}

We will now define the discrete analogs of constant mean curvature and minimal surfaces,
following the Steiner formula approach of \cite{BPW} and \cite{BHL}.

\medskip
Let $(F, N)$ be a pair of edge-parallel
maps from $\mathbb{Z}^2$ to $\mathbb{R}^4$, with planar faces, where either
\begin{itemize}
  \item  $F$ lies in $\mathbb{R}^3$ and $N$ takes values in $\mathbb{S}^2$, or,

  \item $F$ and $N$ lie in $\mathbb{S}^3$ and $F \bot N$ at each vertex.
\end{itemize}
The map $N$ is treated as the Gauss map of $F$. Since $N$ is planar and
constrained to a sphere, its faces are circular, and so are those of $F$, by
parallelism. Note that circular implies planar.

The area $\mathcal{A} (f)$ of a planar face $f$ being a quadratic form in its coordinates, we
define the mixed area $\mathcal{A} (f, f')$ of two edge-parallel faces to be the polar
form applied to $f, f'$:
\[ \mathcal{A} (f, f') = \frac{1}{4}  (\mathcal{A} (f + f') -\mathcal{A} (f -
   f')) \]
where $\mathcal{A} (f) = \mathcal{A} (f, f)$. This allows us to write a Steiner formula for the
area of the parallel face $f + \varepsilon f'$ as
\[
\mathcal{A} (f + \varepsilon f') =\mathcal{A} (f) + 2 \varepsilon
\mathcal{A} (f, f') + \varepsilon^2 \mathcal{A} (f')\, .
\]
Applying this formula to the mesh pair $(F, N)$ on the face $f$, we identify
the mean and Gaussian curvature by $\mathcal{A} (F (f) + \varepsilon N (f))
= (1 - 2 \varepsilon H_f + \varepsilon^2 K_f) \mathcal{A} (F (f))$,
so that
\[ H_f = - \frac{\mathcal{A} (F (f), N (f))}{\mathcal{A} (F (f))} \text{ and }
   K_f = \frac{\mathcal{A} (N (f))}{\mathcal{A} (F (f))} \;\cdot
\]
\begin{definition}
  A circular Q-net $(F, N)$, with $F,N$ as above, is of \emph{constant mean curvature}
  $H \neq 0$ (CMC) if $H_f = H$ on all faces $f$. It is \emph{minimal} if $H_f$ vanishes
  identically.
\end{definition}

Such a net is automatically {\emph{Koenigs}} (see \cite{BS}), i.e. it possesses a Christoffel
dual $F^{\ast}$ such that (i) $F^{\ast}$ is edge-parallel to $F$ and (ii) $A
(F, F^{\ast})$ vanishes identically. Indeed, \ if $(F, N)$ has constant mean
curvature $H$ (resp. is minimal), then $F^{\ast} = F + \frac{1}{H} N$ (resp.
$F^{\ast} = N$) is the dual.
Being Koenigs and circular is equivalent for $F$ to be {\emph{discrete
isothermic}}. Discrete isothermic nets were originally defined in \cite{BP_isothermic} as nets with factorisable cross ratios, i.e. the cross ratio $\text{cr} (F, F_1, F_{12}, F_2)$ is of the
form $A / B$, with $A$ depending
on the first coordinate and $B$ on
the second. Such functions $A, B$ are called {\emph{edge labelings}} and are uniquely defined up to a common factor.

In \cite{BS, BS-Koenigs} a discrete analogue of conformal metric was introduced for discrete isothermic surfaces.
It was shown that Koenigs nets possess a function $s:{\mathbb Z}^2\to{\mathbb R}^+$ defined at vertices,
called the {\emph{(discrete conformal) metric coefficient}}. Consider black and white sublattices of ${\mathbb Z}^2$ so that every elementary quad contains two vertices of each displaced diagonally.

The conformal factor $s$ is defined up to a so called black-white rescaling: $s\mapsto \lambda s$ at black points, and $s\mapsto \mu s$ at white points.
In particular $s$ relates the net to its Christoffel dual:
\begin{equation}\label{eq:dual_through_s}
 F^{\ast}_i - F^{\ast} = \frac{1}{s_i s} (F_i - F), \quad i=1,2.
\end{equation}
Moreover, for discrete isothermic nets the edge labeling\footnote{Edge labelings are unique up to global
multiplication of $A$ and $B$ by a constant. The choice mentioned here is canonical.}
is linked to the discrete conformal factor $s$ and the edge lengths as follows (see \cite{BS, BS-Koenigs}):
\begin{equation}\label{eq:AB_labeling}
A= \frac{\| F_1-F\|^2}{s s_1}, \ B= \frac{\| F_2-F\|^2}{s s_2}.
\end{equation}

One can approximate smooth isothermic surfaces by discrete isothermic surfaces \cite{BuM}. Probably this is also the case with minimal and CMC surfaces, although this is not yet proven.

\section{Loop group description}

Here following \cite{BP} we present the loop group description of discrete CMC surfaces in ${\mathbb R}^3$. We will show also that discrete CMC surfaces in ${\mathbb S}^3$ are described by the same discrete Lax representation and the immersion formula (\ref{eq:smooth_S^3}) of the smooth case.

\subsection{Discretization in the loop group}

As in the smooth case, we consider a frame $\Phi : \mathbb{Z}^2 \to G_H[\lambda ]$.
The discrete Lax pair $\mathcal{U} (\lambda) = \Phi_1(\lambda) \Phi (\lambda)^{- 1}$,
$\mathcal{V} (\lambda) = \Phi_2(\lambda) \Phi (\lambda)^{- 1}$
are maps from the edges into the loop group.
By analogy with the smooth immersions $\mathcal{U} (\lambda)$, $\mathcal{V} (\lambda)$ are defined of the following form:
on each edge,
\begin{equation}   \label{eq:Lax-pair}
\begin{split}
  \mathcal{U} (\lambda) & = \frac{1}{\alpha (\lambda)}  \begin{pmatrix}
    a & - \lambda u - \lambda^{- 1} u^{- 1}\\
    \lambda u^{- 1} + \lambda^{- 1} u & \bar{a}
  \end{pmatrix},
\\
	\mathcal{V} (\lambda) &= \frac{1}{\beta
  (\lambda)}  \begin{pmatrix}
    b & - i \lambda v + i \lambda^{- 1} v^{- 1}\\
    i \lambda v^{- 1} - i \lambda^{- 1} v & \bar{b}
  \end{pmatrix},  \end{split}
\end{equation}
where complex valued $a, b$ and real valued $u, v$ do not depend on $\lambda$, $u, v$ are positive and
$\alpha (\lambda)$ and $\beta (\lambda)$ are real such that the determinants are equal to $1$:
\begin{equation}\label{eq:alpha2beta2}
  \alpha (\lambda)^2 = | a |^2 + \lambda^2 + \lambda^{- 2} + u^2 + u^{- 2},
  \quad \beta (\lambda)^2 = | b |^2 - \lambda^2 - \lambda^{- 2} + v^2 +
  v^{- 2}.
\end{equation}
Furthermore, $\alpha$ and $\beta$ on the opposite edges coincide, i.e. they are edge labeling for the
first and second indices respectively.

\medskip

We will now focus on a single quad $(F, F_1, F_{12}, F_2)$, and let
$\mathcal{U}, \mathcal{V}$ be the Lax matrices associated to the edges $(F, F_1)$
and $(F, F_2)$ respectively; for the sake of simplicity, we will mark with a prime
the corresponding quantities on the opposite edges $(F_2, F_{12})$ and $(F_1,
F_{12})$: $\mathcal{U}'$, $\mathcal{V}'$, $a'$, $u'$, etc. In particular $\Phi_{12}=\mathcal{U}' \Phi_2$, $\Phi_{12}=\mathcal{V}' \Phi_1$. Note that $\alpha' = \alpha$ and $\beta' = \beta$.

The Lax pair satisfies
\begin{equation} \label{eq:discrete_Lax}
\mathcal{V}'(\lambda) \, \mathcal{U}(\lambda) = \mathcal{U}'(\lambda) \, \mathcal{V}(\lambda)
\end{equation}
on any quad and gives rise to a frame $\Phi (\lambda) : \mathbb{Z}^2
\rightarrow G_H[\lambda ]$.

This commutation property yields
\begin{equation}
  uu' = vv' \label{eq:uuprimevvprime}
\end{equation}
\begin{equation}
  b' a - ba' = i (u' v + uv' - u'^{- 1} v^{- 1} - u^{- 1} v'^{- 1})
\end{equation}
\begin{equation} \label{eq:commut1}
  \bar{b}u' - b' u = i (\bar{a}v' - a' v )
\end{equation}
\begin{equation} \label{eq:commut2}
  \bar{b}u'^{- 1} - b' u^{- 1} = i (a' v^{- 1} - \bar{a} v'^{- 1})
\end{equation}
As noticed in \cite[(4.23)]{BP}, equation \eqref{eq:uuprimevvprime} is equivalent
to the existence of a vertex function $w$ such that
\begin{equation}
  u = w w_1, \quad u' = w_2 w_{12}, \quad v = w w_2, \quad
  v' = w_1 w_{12} \label{eq:w}
\end{equation}

The function $w$ turns out to be essentially the discrete conformal metric $s$, as we will show in the next section.

\subsection{Discrete CMC nets in Euclidean three space}
\label{sec:discrete_CMC_R3}

Let $\Phi(\lambda)$ be a frame defined from commuting Lax pairs as above, and let
$\lambda = e^{i \gamma} \in \mathbb{S}^1$ be a spectral parameter.
We define two nets $\hat{F}, \check{F}$ and a unit Gauss map~$\hat{N}$ as follows:
\begin{equation}   \label{eq:discrete_immersion_R3}
	\quad \hat{N} = - \Phi^{- 1} \k \Phi, \quad
	\left\{ \begin{array}{ll}
	\hat{F} & = - \Phi^{- 1}  \frac{\partial \Phi}{\partial \gamma}_{| \gamma = 0
   } - \frac{1}{2}  \hat{N}
   \\
   \check{F} & = - \Phi^{- 1} \frac{\partial \Phi}{\partial \gamma}_{| \gamma = 0 } +
   \frac{1}{2}  \hat{N} = \hat{F} + \hat{N}
   \end{array} \right.
\end{equation}
where all the matrices are evaluated at $\gamma = 0$ (i.e. $\lambda = 1$).

\begin{theorem}[\cite{BP}] \label{thm:CMCR3} \ \\
  The pair $(\hat{F}, \hat{N})$ given by~\eqref{eq:discrete_immersion_R3}
  is a CMC net in $\mathbb{R}^3$ with $H=1$. Its Christoffel dual is
  $\check{F}$.

  On any quad the discrete conformal metric $s$ is given by
  \begin{equation}
    ss_1 = - u^2 \text{ and } ss_2 = v^2, \label{eq:vertexmetricR3}
  \end{equation}
  and the cross ratio  $\text{cr} (\hat{F}, \hat{F}_1, \hat{F}_{12}, \hat{F}_2)$ is equal to $- \beta(1)^2 / \alpha(1)^2$.
  The edge lengths are equal
  \begin{align*}
  \| \hat{F}_1 - \hat{F} \|^2 &= \frac{4 u^2}{\alpha(1)^2}
  = - \frac{4 s s_1 }{| a |^2 + 2 - ss_1 - s^{- 1} s_1^{- 1}} \, ,
  \\
  \| \hat{F}_2 - \hat{F} \|^2 &= \frac{4 v^2}{\beta(1)^2}
  = \frac{4 ss_2 }{| b |^2 - 2 + s s_2 + s^{- 1} s_2^{- 1}}
  \, \cdot
  \end{align*}
\end{theorem}

\begin{proof}
	Note that \cite{BP} use a slightly different, albeit equivalent notation.
	For the sake of completeness and compatibility with the spherical case,
	we shall sketch the proof here using our notations.
	Note: all $\lambda$-dependent quantities are evaluated at $\lambda = 1$,
	and we shall not write the variable $\lambda$ for greater legibility,
	so $\alpha$ stands for $\alpha(1)$, etc.

  Evaluating the edge vectors one obtains
  \begin{align}  \label{eq:def-U}

    \hat{F}_1 - \hat{F}
    & = - \Phi^{- 1} \mathcal{U}^{- 1}  \left( \dot{\mathcal{U}} -
    \frac{1}{2}  [\k, \mathcal{U}] \right) \Phi,
    &
    \hat{F}_2 - \hat{F} & = - \Phi^{- 1} \mathcal{V}^{- 1}  \left(
    \dot{\mathcal{V}} - \frac{1}{2}  [\k, \mathcal{V}] \right)
    \Phi,
    \\
    \hat{N}_1 - \hat{N} & =  - \Phi^{- 1} \mathcal{U}^{- 1}  [\k, \mathcal{U}] \Phi,
    &
    \hat{N}_2 - \hat{N} & = - \Phi^{- 1} \mathcal{V}^{- 1}  [\k, \mathcal{V}] \Phi,

  \end{align}
  where $\dot{\mathcal{U}}$ the derivative at $\gamma = 0$, and similar formulas for $\check{F}$.
Because $\alpha$ and $\beta$ have an extremum at $\lambda = 1$, $\dot{\mathcal{U}} = \frac{u - u^{- 1}}{\alpha} \i$ and $\dot{\mathcal{V}} = - \frac{v + v^{- 1}}{\beta} \j$. Since $[\k, \mathcal{U}] = - \frac{2 (u + u^{- 1})}{\alpha} \i$ and $[\k, \mathcal{V}] = \frac{2 (v - v^{- 1})}{\beta} \j$,
parallelism between the nets $\hat{F}$, $\hat{N}$ (and thus $\check{F}$) is clear,
as are the lengths of the edges; we also derive the
proportionality factors
  \[ \mathop{}\!\mathrm{d} \check{F}_{01} = - u^{- 2} \mathop{}\!\mathrm{d} \hat{F}_{01}, \quad
     \mathop{}\!\mathrm{d} \check{F}_{02} = v^{- 2} \mathop{}\!\mathrm{d} \hat{F}_{02}, \]
  and using \eqref{eq:w}
  \[ \mathop{}\!\mathrm{d} \check{F}_{01} = - \frac{\mathop{}\!\mathrm{d} \hat{F}_{01}}{w^2 w_1^2},
     \quad \mathop{}\!\mathrm{d} \check{F}_{02} = \frac{\mathop{}\!\mathrm{d} \hat{F}_{02}}{w^2
     w_2^2} \cdot \]
  That proves that $\hat{F}$ is Koenigs with Christoffel dual $\check{F}$ and
  a discrete conformal metric given by $s = \pm w^2$ (see \cite{BS} for details). The sign can be chosen constant in one direction and alternates in other direction.

The claim about the cross ratios is proven by direct computation using the quaternionic formulas for cross ratios (see \cite{BP}).
\end{proof}

\medskip

As noted in the proof, the choice of $\lambda=1$ for the spectral parameter
in~\eqref{eq:discrete_immersion_R3} is
crucial (though not unique, any extremum of $\lambda^2 + \lambda^{-2}$ will do, i.e.
$\lambda \in \{ \pm 1, \pm i \}$). Other values of the spectral parameter,
as in the smooth case (see Theorem~\ref{th:smooth_R^3}), will not satisfy the parallelism condition
between the edges, nor the planarity of the faces, so that these nets are
neither circular nor Koenigs.
Such nets
retain interesting properties, in particular they are
\emph{edge-constraint nets}, as defined in~\cite{HSFW}. Edge-constraint nets
are quadrilateral nets $F$ with non necessarily planar faces, and vertex normals $N$ such that, on any edge $e = (F F_i)$,
the average of the normals at the endpoints is orthogonal to the edge:
$F_i - F \perp N+N_i$.

Moreover, in this case the mean curvature can also be defined via a discrete version of Steiner's formula. For that purpose the area functional for non-planar quads is defined by projecting along the normal direction which is orthogonal to the diagonals of the quad. In was shown in \cite{HSFW} that all surfaces of the {\emph associated family} (i.e. those defined by (\ref{eq:discrete_immersion_R3}) with general unitary $\lambda$) possess constant mean curvature.

\subsection{Discrete CMC and minimal nets in the three sphere}

As in the case of smooth CMC surfaces, the same Lax pair leads yields a discrete CMC
surface in $\mathbb{S}^3$ through the immersion formula (\ref{eq:smooth_S^3}).
We gauge again the frame into
$\Psi = \begin{pmatrix}  e^{i \gamma / 2} & 0 \\ 0 & e^{- i \gamma / 2}
\end{pmatrix} \Phi = \exp \left( - \frac{\gamma}{2} \k \right) \Phi$,
and define, for any pair $\lambda_1 = e^{i \gamma_1}, \lambda_2 = e^{i
\gamma_2}$ in the unit circle,
\begin{equation*}
F = \Psi (\lambda_1)^{- 1} \Psi (\lambda_2), \quad
N = - \Psi (\lambda_1)^{- 1} \k \Psi (\lambda_2)
\end{equation*}
which are an orthogonal pair of vectors in $\mathbb{S}^3$.
Equivalently,
\begin{equation}  \label{eq:discrete_immersion_S3}
	F = \Phi (\lambda_1)^{- 1} M \Phi (\lambda_2), \;
	N = - \Phi (\lambda_1)^{- 1} \k M \Phi (\lambda_2),
\end{equation}
where $M = \begin{pmatrix} e^{i \frac{\gamma_2 - \gamma_1}{2}} & 0\\
0 & e^{- i \frac{\gamma_2 - \gamma_1}{2}} \end{pmatrix}
= \exp (\frac{\gamma_1 - \gamma_2}{2} \k)$.
\begin{theorem} \label{thm:CMCS3}
   The pair $(F, N)$ given by~\eqref{eq:discrete_immersion_S3}
  is a discrete isothermic CMC surface in $\mathbb{S}^3$ if, and only if, $\lambda_2 = \pm \lambda_1^{-1}$ ($\gamma_1 + \gamma_2 \equiv 0 \mod \pi$). Its mean curvature is equal to
  \[ H = \frac{\Re \lambda_1^2}{\Im \lambda_1^2 } = \cot (2
     \gamma_1) = \cot (\gamma_1 - \gamma_2) \, .
  \]
  Its Christoffel dual is $F^{\ast} = F + \frac{1}{H} N$ if $H\neq 0$, and
  $F^{\ast} = N$ if $H = 0$.

  On any quad the discrete conformal metric $s$ of $F$ is given by
  \begin{equation}
    \begin{array}{ll}
      ss_1 = - u^2  \sqrt{\frac{H^2}{1 + H^2}} \text{ and } ss_2 = v^2
      \sqrt{\frac{H^2}{1 + H^2}} & \text{ if } H \neq 0,\\
      ss_1 = - u^2 \text{ and } ss_2 = v^2 & \text{ if } H = 0,
    \end{array} \label{eq:vertex-metric-S3}
  \end{equation}
  and the cross ratio is equal to $- \beta^2 / \alpha^2$
  evaluated at $\lambda_1$.
  The edge lengths satisfy, when $H \neq 0$,

  \begin{eqnarray} \label{eq:edge_lengths_H}
    \| \mathop{}\!\mathrm{d} F_{01} \|^2  =\frac{4 u^2 \sin^2  (2 \gamma_1)}{\alpha^2} =
    \frac{4 | ss_1 |}{\alpha^2 H \sqrt{1 + H^2}}
    = \frac{4 | ss_1 |}{\left( |
    a |^2 + u^2 + u^{- 2} + \sqrt{\frac{H^2}{1 + H^2}} \right) H \sqrt{1 +
    H^2} },\nonumber\\
    \| \mathop{}\!\mathrm{d} F_{02} \|^2  = \frac{4 v^2 \sin^2  (2 \gamma_1)}{\beta^2} =
    \frac{4 | ss_2 |}{\beta^2 H \sqrt{1 + H^2}}
    = \frac{4 | ss_2 |}{\left( | b
    |^2 + v^2 + v^{- 2} - \sqrt{\frac{H^2}{1 + H^2}} \right) H \sqrt{1 + H^2}},
  \end{eqnarray}
  and for minimal nets,
  \begin{equation} \label{eq:edge_lengths_0}
  \| \mathop{}\!\mathrm{d} F_{01} \|^2 = \frac{4 u^2}{\alpha^2} = - \frac{4 ss_1}{| a
     |^2 - ss_1 - s^{- 1} s_1^{- 1}}, \; \| \mathop{}\!\mathrm{d} F_{02} \|^2 =
     \frac{4 v^2}{\beta^2} = \frac{4 s s_2}{| a |^2 + ss_2 + s^{- 1} s_2^{-
     1}} \cdot
    \end{equation}
\end{theorem}

\begin{proof} is given by direct computation.
  Let us check that the edges of $F$ and $N$ are parallel.
  \begin{align*}
    F_1 - F
    & = \Phi (\lambda_1)^{- 1} \mathcal{U} (\lambda_1)^{- 1}  (M\mathcal{U}
    (\lambda_2) -\mathcal{U} (\lambda_1) M) \Phi (\lambda_2),
    \\
    N_1 - N
    & = - \Phi (\lambda_1)^{- 1} \mathcal{U} (\lambda_1)^{- 1}
    (\k M\mathcal{U} (\lambda_2) -\mathcal{U} (\lambda_1) M\k) \Phi (\lambda_2).
  \end{align*}
  Note that, given two unitary matrices $U_1, U_2$ in $SU (2)$, $U_2 -
  U_1$ is a real multiple of $\k U_2 - U_1 \k$ iff their
  diagonals coincide. Applying it to $U_1 =\mathcal{U} (\lambda_1) M$, $U_2 =
  M\mathcal{U} (\lambda_2)$, we conclude in particular that $\alpha
  (\lambda_1) = \alpha (\lambda_2)$; so by \eqref{eq:alpha2beta2}, $\Re
  \lambda_2^2 = \Re \lambda_1^2$.
The case $\lambda_2 = \pm  \lambda_1$ leads to constant maps. The case $\lambda_2 = \pm \lambda_1^{- 1}$ leads to non-trivial discrete surfaces.
  Let us focus on $\lambda_2 = \lambda_1^{- 1}$, i.e. $\gamma_2 + \gamma_1 = 0$.
  We have then
  \[ M\mathcal{U} (\lambda_2) -\mathcal{U} (\lambda_1) M = - \frac{2 u \sin (2
     \gamma_1)}{\alpha} \i
     \text{ and }
      -\k M\mathcal{U} (\lambda_2) +\mathcal{U} (\lambda_1)
     M\k= \frac{2}{\alpha} (u^{- 1} + u \cos (2 \gamma_1))
     \i,
     \]
  and similarly
  \[ M\mathcal{V} (\lambda_2) -\mathcal{V} (\lambda_1) M = \frac{2 v}{\beta}
     \sin (2 \gamma_1) \j
  \text{ and }
  -\k M\mathcal{V} (\lambda_2) +\mathcal{V} (\lambda_1) M\k
  = \frac{2}{\beta}  (v^{- 1} - v \cos (2 \gamma_1)) \j
  \]
  which proves the parallelism of the corresponding edges. If we set $F^{\ast} = F +
  \frac{1}{H} N$, with $H = \cot (2 \gamma_1) \neq 0$, then
  \[
  \mathop{}\!\mathrm{d} F^{\ast}_{01} = \mathop{}\!\mathrm{d} F_{01} + \tan (2 \gamma_1) \mathop{}\!\mathrm{d} N_{01} =
     - \frac{u^{- 2}}{\cos (2 \gamma_1)} \mathop{}\!\mathrm{d} F_{01} = - u^{- 2}
     \sqrt{\frac{1 + H^2}{H^2}} \mathop{}\!\mathrm{d} F_{01}, \]
  \[ \mathop{}\!\mathrm{d} F^{\ast}_{02} = \mathop{}\!\mathrm{d} F_{02} + \tan (2 \gamma_1) \mathop{}\!\mathrm{d} N_{02} =
     \frac{v^{- 2}}{\cos (2 \gamma_1)} \mathop{}\!\mathrm{d} F_{02} = v^{- 2}  \sqrt{\frac{1
     + H^2}{H^2}} \mathop{}\!\mathrm{d} F_{02}. \]
  We infer the existence of a discrete conformal metric $s$, as in the proof of Theorem
  \ref{thm:CMCR3}. Note that
  \[
  s = \pm w^2  \sqrt[4]{\frac{H^2}{1 + H^2}} \, \cdot
  \]
  When $\gamma_1 = \pi / 4$, $H = 0$; we set $F^{\ast} = N$ and obtain
  \[ \mathop{}\!\mathrm{d} N_{01} = - u^{- 2} \mathop{}\!\mathrm{d} F_{01} \text{ and } \mathop{}\!\mathrm{d} N_{02} = v^{-2}
     \mathop{}\!\mathrm{d} F_{02} . \]
  For all values of $H$, this proves that $F^{\ast}$ is a Christoffel dual of $F$,
  and therefore $(F, N)$ has constant mean curvature $H$.

  To compute quaternionic cross ratio we assume that $\Phi(\lambda_1)=\operatorname{\mathbbl{1}}$ and use $\mathcal{U}' \mathcal{V}=\mathcal{V}' \mathcal{U} $:
  \begin{multline*}
  \text{cr} (F, F_1, F_{12}, F_2) = (F - F_1)  (F_1 - F_{12})^{- 1} (F_{12} - F_2)  (F_2 - F)^{- 1}
   \\
  \begin{split}
  	& = (\mathcal{U}^{} (\lambda_1)^{- 1} M\mathcal{U} (\lambda_2) - M) \mathcal{U} (\lambda_2)^{- 1}
    (\mathcal{V}'^{} (\lambda_1)^{- 1} M \mathcal{V}' (\lambda_2) - M)^{- 1}
    \\
    & = \left( - \frac{2 u \sin (2 \gamma_1)}{\alpha} \i \right)
    \left( \frac{2 v'}{\beta} \sin (2 \gamma_1) \j \right)^{- 1}
    \left( - \frac{2 u' \sin (2 \gamma_1)}{\alpha} \i \right)
    \left( \frac{2 v}{\beta} \sin (2 \gamma_1) \j \right)^{- 1}
    \\
    & =  - \frac{\beta^2}{\alpha^2}  \frac{uu'}{vv'}  \operatorname{\mathbbl{1}} = - \frac{\beta^2}{\alpha^2}  \operatorname{\mathbbl{1}}.
  \end{split}
  \end{multline*}

Formulas (\ref{eq:edge_lengths_H}, \ref{eq:edge_lengths_0}) follow directly from the quaternionic formulas for the corresponding edges derived above.
\end{proof}

\section{From  discrete minimal and CMC surfaces to the Lax pair}

We have seen that the (same) frame $\Phi$ integrating the Lax pair
in~\eqref{eq:Lax-pair}, gives rise to CMC or minimal quad-nets in $\mathbb{R}^3$
and $\mathbb{S}^3$. We shall now prove the converse.

\begin{theorem} \label{thm:reconstruction}
  For any Q-net of constant mean curvature in $\mathbb{R}^3$ or $\mathbb{S}^3$,
  or minimal in $\mathbb{S}^3$, there exists a Lax pair satisfying
  \eqref{eq:Lax-pair}, such that the immersion formula  \eqref{eq:discrete_immersion_R3}
  or \eqref{eq:discrete_immersion_S3} holds.
\end{theorem}

We will prove this theorem in several steps. First we identify geometric quantities $s, A, B$ and algebraic quantities $u, v, \alpha, \beta$ appearing in the corresponding descriptions (\ref{eq:dual_through_s}, \ref{eq:AB_labeling}) and (\ref{eq:Lax-pair}, \ref{eq:alpha2beta2}). We will further denote the edges by $\mathop{}\!\mathrm{d} \hat{F}_{0i}=F_i-F$.

Let us start with the case of discrete CMC surfaces $\hat{F}$ with $H=1$. Its Christoffel dual is given by $\check{F}=\hat{F}^*=\hat{F}+\hat{N}$. The corresponding edges are related by (\ref{eq:dual_through_s}), where $s$ is the conformal metric coefficient of $\hat{F}$. We assume that $s$ alternates its sign along the first direction and does not change the sign along the second direction, i.e. $s_1 s<0,\ s_2 s>0$. Geometrically this means that the quads $(\hat{F},\hat{F}_1,\check{F}_1,\check{F})$ are crossing trapezoids, and the quads $(\hat{F},\hat{F}_2,\check{F}_2,\check{F})$ are embedded trapezoids.

Comparing the analytic formulas of Theorem~\ref{thm:CMCR3} with the edge length formulas (\ref{eq:AB_labeling}) we obtain the following identification:
\begin{eqnarray} \label{eq:identification}
u^2=-s_1s,\quad v^2=s_2 s \nonumber \\
\alpha(1)=\frac{2}{\sqrt{-A}}, \quad \beta(1)=\frac{2}{\sqrt{B}}.
\end{eqnarray}

Let us compute the angle between an edge $\mathop{}\!\mathrm{d} \hat{F}_{0i}$ of a discrete CMC surface and its unit normal $\hat{N}$. Since $\mathop{}\!\mathrm{d} \hat{F}_{01}$ and $\mathop{}\!\mathrm{d}
  \hat{F}^{\ast}_{01}$ are parallel (though in opposite directions) and the other
  sides have length 1, one derives
\begin{equation} \label{eq:<dF,N>_1}
<\mathop{}\!\mathrm{d} \hat{F}_{01},\hat{N}> = \frac{1}{2}\left(\| \mathop{}\!\mathrm{d} \hat{F}_{01} \|+ \| \mathop{}\!\mathrm{d} \hat{F}^{\ast}_{01} \|\right)\| \mathop{}\!\mathrm{d} \hat{F}_{01} \|= \frac{1}{2}A(s_1s-1),
\end{equation}
and similarly,
\begin{equation} \label{eq:<dF,N>_2}
<\mathop{}\!\mathrm{d} \hat{F}_{02},\hat{N}> = \frac{1}{2}\left(\| \mathop{}\!\mathrm{d} \hat{F}_{02} \|- \| \mathop{}\!\mathrm{d} \hat{F}^{\ast}_{02} \|\right)\| \mathop{}\!\mathrm{d} \hat{F}_{02} \|= \frac{1}{2}B(s_2s-1).
\end{equation}
Here we have used (\ref{eq:dual_through_s},\ref{eq:AB_labeling}).

We will use the following technical lemma, which can be easily checked.
 \begin{lemma} \label{lemma:commutation-lambda}
    The commutation property (\ref{eq:discrete_Lax}) for all $\lambda$ is

    equivalent to
    the condition $u u' = v v'$ (equation \eqref{eq:uuprimevvprime}),
    together with the commutation property for matrices and its derivative
    evaluated \emph{only} at $\lambda = 1$:
    \[
    \mathcal{V}'(1) \mathcal{U}(1) = \mathcal{U}'(1) \mathcal{V}(1) \; \text{ and } \;
    \dot\mathcal{V}'(1) \mathcal{U}(1) + \mathcal{V}'(1) \dot\mathcal{U}(1) = \dot\mathcal{U}'(1) \mathcal{V}(1) + \mathcal{U}'(1) \dot\mathcal{V}(1)
    \]
    It is also equivalent to $u u' = v v'$ together with the commutation
    property evaluated at two values $\lambda,\lambda^{-1}$, provided
    $\lambda \notin {\pm 1, \pm i}$.
  \end{lemma}

The following lemmas prove the existence and uniqueness of the Lax pair for a given quad of a discrete CMC surface.

\begin{lemma}[$\mathbb{R}^3$ version]  \label{lemma:quad-reconstruction-R3}
  Let $Q = (\hat{F}, \hat{F}_1, \hat{F}_{12}, \hat{F}_2)$ be a CMC-1 quad in
  $\mathbb{R}^3$ with Gauss map $(\hat{N}, \hat{N}_1, \hat{N}_{12},
  \hat{N}_2)$. Let $\Phi$ be any frame for $\hat{N}$, namely $\hat{N} = -
  \Phi^{- 1} \k \Phi$ (such a frame at the point $\hat{F}$ is determined up to $U (1)$
  action).
  Then there exist $\mathcal{U}(\lambda), \mathcal{V}(\lambda), \mathcal{U}'(\lambda),
  \mathcal{V}'(\lambda)$ satisfying (\ref{eq:discrete_Lax}), and thus generating the quad, together with its Gauss map by formulas (\ref{eq:discrete_immersion_R3}).
\end{lemma}

\begin{proof}
  In the following, $\mathcal{U}$, $\alpha$, etc. will denote the usual quantities evaluated
  at $\lambda=1$, and we will write specifically $\mathcal{U}(\lambda)$ when considering the loop.

  As we have demonstrated above (geometric) CMC-1 quad $Q$ in $\mathbb{R}^3$, together with its Gauss map determines
  $\alpha, \beta, u, v, u', v'$ so only $a, b$ remain to be found\footnote{Although $| a |$ and $| b |$ are given by
  \eqref{eq:alpha2beta2}.}. The Lax matrix $\mathcal{U}$ can be determined uniquely
  from $\Phi$, through equation~\eqref{eq:def-U}:
  \[
  	  \Phi \mathop{}\!\mathrm{d} \hat{F}_{01} \Phi^{- 1} = - \frac{2 u}{\alpha} \i \mathcal{U}
	  = \frac{2i u}{\alpha^2} \begin{pmatrix}
       u + u^{- 1} &  \bar{a}\\
       a & - (u + u^{- 1})
     \end{pmatrix}.
  \]
  Indeed, the length of the edge is $| 2 u / \alpha |$ by (\ref{eq:identification}), so that $a$ is here
  only to fix the directions in the $(\i, \j)$ plane, provided the coordinate along
  the $(-\k)$ axis (i.e. the projection of $\mathop{}\!\mathrm{d} \hat{F}_{01}$ along $\hat{N}$) is
  \[
  \frac{2 u}{\alpha}  \frac{u + u^{- 1}}{\alpha}.
  \]
  This fact follows from  (\ref{eq:<dF,N>_1}).
  So that $a$ is now fixed, though it depends on the $\Phi$ gauge. Similarly,
  $\mathop{}\!\mathrm{d} \hat{F}_{02}$ determines $\mathcal{V}$  and $b$.

  Let us now check that $\Phi_1$ and $\Phi_2$ defined through $\mathcal{U}$ and $\mathcal{V}$
  are frames for $\hat{N}_1$ and $\hat{N_2}$ respectively. By property of the Koenigs net,
  $\hat{N}_1 - \hat{N} =  - (1+u^{-2}) \mathop{}\!\mathrm{d} \hat{F}_{01}$, and, conjugating by $\Phi$,
  we need to check that
  \[
  - \mathcal{U}^{-1} \k \mathcal{U} + \k = - \frac{2}{\alpha} \frac{u^2+1}{u} \i \mathcal{U}
  \]
  which holds due to the specific form of $\mathcal{U}$, and similarly for $\mathcal{V}$.
  From $\Phi_1$ and $\Phi_2$, we derive $\mathcal{U}',\mathcal{V}'$ as above, with their specific form.

  Having defined all their coefficients, the Lax matrices are fully determined,
  and there remains only to check the commutation property (\ref{eq:discrete_Lax}) for all $\lambda$.
  This can be done using Lemma~\ref{lemma:commutation-lambda}.

  %
  By (\ref{eq:identification}) $u u' = v v'$ holds. Comparing the cross ratio written in terms of
  $\mathcal{U}$, $\mathcal{V}$, $\mathcal{U}'$, $\mathcal{V}'$ and its known value $-\beta^2/\alpha^2$ proves
  the commutation for $\lambda=1$ (reverse the proof of Theorem~\ref{thm:CMCR3}).
  This shows also that $\hat{N}_{12} = - \Phi_{12}^{-1} \k \Phi_{12}$,
  where $\Phi_{12} = \mathcal{V}' \mathcal{U} \Phi = \mathcal{U}' \mathcal{V} \Phi$.
  The derived commutation property is a consequence of the additive commutation relation
  \[
  \mathop{}\!\mathrm{d} \hat{F}_{01} + \mathop{}\!\mathrm{d} \hat{F}_{1, 12} = \mathop{}\!\mathrm{d} \hat{F}_{02} + \mathop{}\!\mathrm{d} \hat{F}_{2, 12} \, .
  \]
  For simplicity, we apply it to $G = \hat{F} + \frac{1}{2}  \hat{N}
  = - \Phi^{- 1}  \frac{\partial \Phi}{\partial \gamma}_{| \gamma = 0 }$, which is a well-defined
  quad, and hence closes:
  \begin{eqnarray*}
  0 & = & \Phi (\mathop{}\!\mathrm{d} G_{01} + \mathop{}\!\mathrm{d} G_{1, 12} - \mathop{}\!\mathrm{d} G_{02} - \mathop{}\!\mathrm{d}
  G_{2, 12}) \Phi^{- 1}\\
  & = & \mathcal{V}^{- 1}  \dot{\mathcal{V}} +\mathcal{V}^{- 1}
  \mathcal{U}'^{- 1}  \dot{\mathcal{U}}' \mathcal{V}-\mathcal{U}^{- 1}
  \dot{\mathcal{U}} -\mathcal{U}^{- 1} \mathcal{V}'^{- 1}  \dot{\mathcal{V}}'
  \mathcal{U}\\
  & = & \mathcal{U}^{- 1} \mathcal{V}'^{- 1}  (\mathcal{U}'
  \dot{\mathcal{V}} + \dot{\mathcal{U}}' \mathcal{V}-\mathcal{V}'
  \dot{\mathcal{U}} - \dot{\mathcal{V}}' \mathcal{U})
  \end{eqnarray*}
  where we use twice the commutation at order zero.

  We have thus determined the Lax matrices (uniquely, once a frame $\Phi$ at $\hat{F}$ is set).
  The CMC-1 quad they generate is the one we started from.
  Note that, although we have started as usual with the lower left vertex, this
choice plays no role, and we might have fixed $\Phi_1$, $\Phi_2$ or $\Phi_{12}$
and recovered the rest similarly.
\end{proof}

\begin{lemma}[$\mathbb{S}^3$ version]
  \label{lemma:quad-reconstruction-S3}
  Let $Q = (F, F_1, F_{12}, F_2)$ be a CMC or minimal quad in $\mathbb{S}^3$
  with Gauss map $(N, N_1, N_{12}, N_2)$. Let $(\phi, \phi')$ be any frame
  at the vertex $F$, i.e. any couple in $SU (2)$ such that $F = \phi'^{- 1} M \phi$
  and $N = - \phi'^{- 1} \k M \phi$ (such a pair is determined up to $U(1)$
  action). Then there exist $\mathcal{U}(\lambda), \mathcal{V}(\lambda), \mathcal{U}'(\lambda),
  \mathcal{V}'(\lambda)$ satisfying (\ref{eq:discrete_Lax}), and thus generating the quad, together with its Gauss map by formulas (\ref{eq:discrete_immersion_S3}).
\end{lemma}

\begin{proof}
	The proof goes along the same lines as in Lemma \ref{lemma:quad-reconstruction-R3},
	with a few specificities; in particular, whereas in $\mathbb{R}^3$ scaling allows
	us to freely set the mean curvature to $1$, in $\mathbb{S}^3$, we use
	the mean curvature to determine $\lambda_1 = e^{i \gamma_1}, \lambda_2 = e^{i \gamma_2}$
	as in theorem \ref{thm:CMCS3}: $\gamma_1 = - \gamma_2 = \frac{1}{2} \mathrm{arccot} H$
  ($\gamma_1 = \pi / 4$ if $H = 0$), $\gamma_1$ being taken in $[0,\pi/2]$.
	A (geometric) CMC-$H$ quad $Q$ in $\mathbb{S}^3$,
  together with its Gauss map, comes with a discrete conformal metric $s$ and canonical
  edge labelings $A, B$, such that the edge lengths in both directions are
  $Ass_1$ and $Bss_2$ respectively. This allows us to determine $u, v$ by
  \begin{equation}  \label{eq:u_v_s}
   u^2 = - ss_1  \sqrt{\frac{1 + H^2}{H^2}} = - \frac{ss_1}{\cos (2
     \gamma_1)}, \quad v^2 = ss_2  \sqrt{\frac{1 + H^2}{H^2}} =
     \frac{ss_2}{\cos (2 \gamma_1)} \text{ if } H \neq 0,
 \end{equation}
  \[ u^2 = - ss_1, \quad v^2 = ss_2 \quad \text{ if } H = 0,
  \]
  and similarly $u',v'$. We set positive $\alpha, \beta$ such that
  \begin{equation}	\label{eq:alpha_A}
  \alpha^2 = - \frac{4 \sin^2
  (2 \gamma_1)}{A \cos (2 \gamma_1)},\quad \beta^2 = \frac{4 \sin^2 (2
  \gamma_1)}{B \cos (2 \gamma_1)}
  \text{ if } H \neq 0,
  \end{equation}
  $$
 \alpha^2 = - \frac{4}{A},\quad \beta^2
  = \frac{4}{B} \text{ if } H = 0.
  $$
  In the following, we will favor the
  notations in terms

  of $\gamma_1$:
  %
  \begin{align*}
  \phi' \mathop{}\!\mathrm{d} F_{01} \phi^{- 1}
  &= \mathcal{U} (\lambda_1)^{- 1} M \mathcal{U} (\lambda_2) - M
  = \mathcal{U} (\lambda_1)^{- 1}  \left( - \frac{2 u \sin (2 \gamma_1)}{\alpha} \i \right)
  \\
  & = \frac{2 u \sin (2 \gamma_1)}{\alpha}  \frac{1}{\alpha}  \begin{pmatrix}
       i (\lambda_1 u + \lambda_1^{- 1} u^{- 1}) & \bar{a}\\
       a & - i (\lambda_1 u^{- 1} + \lambda_1^{- 1} u)
     \end{pmatrix}
  \end{align*}
  %
  We recognize the length of the edge, which proves incidentally that the right
  hand side matrix is unimodular, and so is $\mathcal{U} (\lambda_1)$. As in
  $\mathbb{R}^3$, $a$ is given by the $(\i,
  \j)$ component, provided the $(\operatorname{\mathbbl{1}},
  \k)$ component is correct. The latter is equivalent to the following two conditions:
  \begin{enumerate}
    \item the angle $\theta$ between $\mathop{}\!\mathrm{d} F_{01}$ and $N$ satisfies
    \begin{align} \label{eq:cos_theta}
      \cos \theta
      & =  \frac{\langle \mathop{}\!\mathrm{d} F_{01}, N \rangle}{\| \mathop{}\!\mathrm{d} F_{01} \|}
      = \frac{1}{\alpha}  (u^{- 1} + \cos (2 \gamma_1) u) \, ,
    \end{align}
    \item the angle $\chi$ between $\mathop{}\!\mathrm{d} F_{01}$ and $F$ satisfies
    \begin{align} \label{eq:cos_xi}
      \cos \chi
      &= \frac{\langle \mathop{}\!\mathrm{d} F_{01}, F\rangle }{\| \mathop{}\!\mathrm{d} F_{01} \|}
      = - \frac{u \sin (2 \gamma_1)}{\alpha} \, \cdotp
    \end{align}
  \end{enumerate}

  Geometric derivation of $\cos\theta$ is analogous to (\ref{eq:<dF,N>_1}).
  In particular in the CMC case $H=\cot(2\gamma_1)\neq 0$ the dual isothermic surface
  of $F$ is $F+\frac{1}{H}N$, and (\ref{eq:<dF,N>_1}) is modified to
  $$
  \frac{2}{H}\cos \theta=\|\mathop{}\!\mathrm{d} F_{01}\|+\|\mathop{}\!\mathrm{d} F^*_{01}\|=\|\mathop{}\!\mathrm{d} F_{01}\|(1+\frac{1}{s_1s})=\sqrt{s_1sA}(1+\frac{1}{s_1s}).
  $$
 Substituting (\ref{eq:u_v_s}),(\ref{eq:alpha_A}) we arrive at (\ref{eq:cos_theta}).

  Identity (\ref{eq:cos_xi}) follows directly from $\| F \| = \| F_1 \| = 1$:
  \[ \cos \chi = \langle F, \mathop{}\!\mathrm{d} F_{01} \rangle = - \frac{1}{2}  \| \mathop{}\!\mathrm{d}
     F_{01} \|^2 = - \frac{u \sin (2 \gamma_1)}{\alpha}. \]
  Along the other coordinate, we have $v$ and non-crossing trapezoids, but the
  reasoning is analogous, and fixes $b$.

	Setting $\Phi(\lambda_1) = \phi'$ and $\Phi(\lambda_2) = \phi$,

	 we check that
	$N_1 = - \Phi_1(\lambda_1)^{- 1} \k M \Phi_1(\lambda_2)$, where
	$\Phi_1(\lambda_i) = \mathcal{U}(\lambda_i) \Phi(\lambda_i)$. Indeed, this amounts to reversing
	the proof in Theorem \ref{thm:CMCS3}. We have the analogous result for $N_2$,
	which allows us to compute $a',b'$ and therefore $\mathcal{U}'(\lambda)$ and $\mathcal{V}'(\lambda)$.

	To prove (\ref{eq:discrete_Lax}) we reverse the calculation in the proof of
	Theorem~\ref{thm:CMCS3}, which shows that $\mathcal{V}'(\lambda_1) \mathcal{U}(\lambda_1)
	= \mathcal{U}'(\lambda_1) \mathcal{V}(\lambda_1)$.

	To prove the analogous result at $\lambda_2$,
	we consider the symmetric (see (\ref{eq:quaternionic_S^3})) surface $F^{-1}$ with the normal $N^{-1}$. This exchanges $\lambda_1$ with $\lambda_2$ while
	preserving all the metric and affine properties. We conclude with
	Lemma~\ref{lemma:commutation-lambda}.

\end{proof}

\begin{proof}[Proof of the Theorem]~\\
  The same strategy works for the Euclidean and spherical nets, so we will
  describe it in the Euclidean case. We start by constructing the Lax pair
  on one quad $Q = (\hat{F}, \hat{F}_1, \hat{F}_{12}, \hat{F}_2)$.
  For any choice of compatible
  frame $\Phi$, i.e. $\hat{N} = - \Phi^{- 1} \k \Phi$ at the base
  vertex, we can find a (unique) Lax pair generating $Q$, according to the
  Lemmas \ref{lemma:quad-reconstruction-R3} or
  \ref{lemma:quad-reconstruction-S3}. This in turn determines a Lax pair on
  adjacent faces, which generates the corresponding quads, and therefore the
  whole Q-net.

  This reasoning holds provided that we do not obtain a contradiction when
  the Gauss map is given at more than one vertex. E.g. once $Q = (\hat{F},
  \hat{F}_1, \hat{F}_{12}, \hat{F}_2)$ has been constructed, we may construct
  the adjacent quad $Q_1 = (\hat{F}_1, \hat{F}_{11}, \hat{F}_{112},
  \hat{F}_{12})$ by starting with $\hat{N}_1$ or with $\hat{N}_{12}$. However
  the two quads constructed this way must agree, because $\hat{N}_1$ fully
  determines $\hat{N}_{12}$ (knowing the corresponding edge), see lemma
  \ref{lemma:quad-reconstruction-R3}.
\end{proof}

\medskip

\section{The discrete Lawson correspondence}

The results shown above allow us to define a discrete Lawson correspondence between
Q-nets in $\mathbb{S}^3$ and $\mathbb{R}^3$.
\begin{theorem}   \label{thm:discrete_Lawson}
Let $F$ be a minimal Q-net in $\mathbb{S}^3$ with discrete conformal metric $s$.
Then there exists a constant mean curvature Q-net $\hat{F}$ in $\mathbb{R}^3$
with constant mean curvature $H = 1$ and the same discrete conformal metric $s$.
More generally, this correspondence takes any Q-net of constant mean curvature $H$
lying in the sphere of curvature $\kappa$ to a Q-net of constant mean curvature $H'$
in the sphere of curvature $\kappa'$, provided $H^2 + \kappa = H'^2 + \kappa'$.
The Euclidean case corresponds to $\kappa'=0$. Additionally, the two Q-nets
$F$ and $\hat{F}$ are given by formulas (\ref{eq:discrete_immersion_S3}) and
(\ref{eq:discrete_immersion_R3}) with the same Lax matrices $\mathcal{U}(\lambda)$ and $\mathcal{V}(\lambda)$.
\end{theorem}
\begin{proof}
The construction is a direct consequence of the previous theorems,
but to see it clearly, we shall state a slightly modified version of
Theorems~\ref{thm:CMCS3} and \ref{thm:reconstruction}.

Let us start with the claim about Q-nets in spheres with different curvatures.
Any Lax pair as defined in (\ref{eq:Lax-pair}), and any choice
$\lambda_1 = e^{i \gamma_1}$ of the spectral parameter gives rises
to an constant mean curvature Q-net $(F^{\gamma_1},N)$ in the sphere
of curvature $\kappa = \sin^2(2 \gamma_1)$, obtained as a a scaled up version
of the spherical net $F$ defined in Theorem~\ref{thm:CMCS3}:
\[
F^{\gamma_1} = \frac{1}{\sin (2 \gamma_1)} F
= \frac{1}{\sin (2 \gamma_1)} \Phi (\lambda_1)^{- 1} M \Phi (\lambda_1^{- 1}),
\]
and $N$ remains the same. The mean curvature is $H = \cos (2 \gamma_1)$.
If $\gamma_1 = \pi/4$, this is the minimal Q-net in $\mathbb{S}^3$.
\\

By the same reasoning according to Theorems~\ref{thm:CMCS3} and \ref{thm:reconstruction} there exists infinitely many other
CMC Q-nets $(F^{\gamma'_1},N)$ of constant mean curvature
$H' = \cos (2 \gamma'_1)$ in the sphere of curvature $\kappa'=\sin^2(2 \gamma'_1)$ with the same coefficients $u,v$ in the Lax pair.

One checks easily that
\[
H'^2 + \kappa' = \cos^2 (2 \gamma'_1) + \sin^2(2 \gamma'_1) = 1
= H^2 + \kappa.
\]
The most general case, where $H^2+\kappa \neq 1$ is obtained from the latter by
direct scaling.
Finally, since the discrete conformal metric coefficient is defined geometrically up to general factor (see Sect.~\ref{sec:discrete_CMC_Q-nets}), the coincidence of the Lax matrix coefficients $u,v$ is equivalent to the preservation of the discrete conformal metric coefficient $s$.

The case of $\kappa'=0$ is dealt in exactly the same way, except that
the immersion formula in Theorem~\ref{thm:CMCR3} is used for $\mathbb{R}^3$.
We note now that the factorization formulas \eqref{eq:vertexmetricR3} and \eqref{eq:vertex-metric-S3} for minimal nets in $\mathbb{S}^3$ and CMC-$1$ nets in $\mathbb{R}^3$ coincide. The coincidence of the coefficients $u,v$ implies that the discrete conformal metric factors $s$ determined from these formulas are identical, in the same way as, in the smooth case, both surfaces are isometric.

\end{proof}
\begin{remark}
Furthermore, the Q-net in $\mathbb{R}^3$ is the limit of the spherical CMC
nets, when the radius of the sphere increases to infinity, keeping the point
$\operatorname{\mathbbl{1}}$ fixed. Indeed, let $\gamma_1$ tend to zero. Then $H = \cos (2 \gamma_1)$
goes to $1$, and
\begin{align*}
  F - \operatorname{\mathbbl{1}} & = \Phi (\lambda_1)^{- 1} M \Phi (\lambda_1^{- 1})
  = (\Phi + \gamma_1  \dot{\Phi} + o (\gamma_1))^{- 1}  (\operatorname{\mathbbl{1}} +
  \gamma_1  \k + o (\gamma_1))  (\Phi - \gamma_1  \dot{\Phi} + o
  (\gamma_1)) - \operatorname{\mathbbl{1}}
  \\
  & = \gamma_1  (- 2 \Phi^{- 1}  \dot{\Phi} + \Phi^{- 1}  \k \Phi) + o(\gamma_1)
\end{align*}
so
\[
\frac{1}{\sin (2 \gamma_1)}  (F - \operatorname{\mathbbl{1}}) \sim \frac{\gamma_1}{2 \gamma_1}
(- 2 \Phi^{- 1}  \dot{\Phi} + \Phi^{- 1}  \k \Phi)
= - \Phi^{- 1}  \dot{\Phi} + \frac{1}{2} \Phi^{- 1}  \k \Phi = \hat{F} \, .
\]
\end{remark}

\bigskip

\begin{remark} \label{Calapso}
\em
At last, let us remark that this definition of the Lawson correspondence
coincides with the seemingly very different one proposed in \cite{BuHRS,BuHR},
based on the Calapso transform and the conserved quantities formalism.
We will not go into the details of the latter, but we will show that
both definitions agree. Indeed, and despite their very different formulations,
it suffices to show that the mean curvature, the discrete conformal metric and the labelings
change in the same way.

\medskip

Let $H=\cos(2 \gamma_1)$ and $H' = \cos(2 \gamma'_1)$ be the two corresponding mean curvatures
as defined above. Since $u,v$ are common to both immersions, and $s s_1 = - u^2 H$
(resp. $s s_2 = v^2 H$), then $r = s H^{-1/2}$ takes the same values for both surfaces.
The Reader may check that this vertex function $r$ is the one used in \cite[\S 3 and \S 4.2]{BuHR},
which is invariant under the Calapso transform.
The edge labelings are $-\alpha(\gamma_1)^{-2},\beta(\gamma_1)^{-2}$ and
$-\alpha(\gamma_1)^{-2},\beta(\gamma_1)^{-2}$, up to a multiplicative constant~$c$.
From~\eqref{eq:alpha2beta2}, we see that
\[
\alpha(\gamma'_1)^2 - \alpha(\gamma_1)^2 = \cos(2 \gamma'_1) - \cos(2 \gamma_1) = H' - H,
\]
and similarly $\beta(\gamma'_1)^2 - \beta(\gamma_1)^2 = H - H'$. Choosing $c=-1$,
the edge labeling satisfy
\[
a_{01}' = \frac{1}{\alpha'^2} = \frac{1}{\alpha^2 + H' - H} =
\frac{1}{\frac{1}{a_{01}} + H' - H} = \frac{a_{01}}{1 + (H' - H) a_{01}},
\]
and similarly $a_{02}' = \frac{a_{02}}{1 - a_{02}  (H - H')}$, which is again the prescribed
behavior of the Calapso transform. Therefore both correspondences agree.
\end{remark}

\medskip

\paragraph{Conclusion and open questions}
~\\

In this paper we have established a discrete Lawson isometry between discrete isothermic minimal surfaces in ${\mathbb S}^3$ and discrete isothermic CMC surfaces in ${\mathbb R}^3$. The isometry is understood in the sense that both corresponding isothermic surfaces have the same discrete conformal metric coefficient. It is appealing to lift this correspondence to the level of frames as in the smooth case (Theorem~\ref{thm:Lawson_smooth}). However the isothermic parametrizations in \eqref{eq:Lawson_smooth} do not correspond: an isothermic surface in ${\mathbb S}^3$ corresponds to a CMC surface from the associated family.

One way to reach that, and an important achievement by itself, would be to introduce geometrically a discrete metric for the associated families in ${\mathbb R}^3$ and ${\mathbb S}^3$ that generalizes the conformal metric coefficient $s$ of isothermic surfaces. It should be the same for the whole associated family. On the level of the Lax representation it is the coefficient $w$ in this paper. The next step then would be to find a discrete analogue of \eqref{eq:Lawson_smooth}.

Some progress in this direction has been achieved in \cite{HSFW} where the associated family in ${\mathbb R}^3$ was described as edge-constraint net with non-planar faces (see Section~\ref{sec:discrete_CMC_R3}). The curvatures were defined there but not the conformal metric. Here we are dealing with discrete isothermic surfaces in non-isothermic parametrization. Let us mention that such triangulated surfaces were recently introduced in \cite{LP}.

redpajama_set_name: RedPajamaArXiv


Born in 1989, she works and lives in New York. She has gone to portray large scale paintings of black men which depict the surroundings that she has been part of. She is more interested in sharing the sensitive along with humanistic stories of her community. All her paintings are part of a group show in Manchester ,and most of them revolve around the plight of young men and boys.
She was born in Los Angles in the year 1984 and currently works in California. In recent times she has gone on to feature in two groups shows in California. In fact, she has garnered a lot of attention in the portray of human beings particularly being cress-crossed gates. All the pictures in her characters are a reflection of the modern times and you can say that it was invented in her wildest imaginations. In her paintings, she went on to relate an abstract subject to a ground idea.
Born in the year 1982, she currently works in New York. Personal experience and dreams inspire her work to a large extent. All her paintings depict an aura of color, class, and race. All the subject matter that is chosen by her is truly contemporary in nature. In her recent works, a lot of brutalities has been exposed by the police forces and you can compare it to the Black lives Matter movement.
She was born in the year 1980 in Botswana and currently resides in South Africa. In her work she often tries to respond to stories and the figure is one of the best mechanisms to achieve that drive. As she was trained as a dancer, she has gone on to model her body while challenging the wide-ranging perceptions that are associated with the human body in general.


text:
package org.apache.derby.impl.services.bytecode;

import org.apache.derby.iapi.services.cache.Cacheable;
import org.apache.derby.iapi.services.cache.CacheManager;

import org.apache.derby.shared.common.sanity.SanityManager;

import org.apache.derby.iapi.services.classfile.ClassHolder;

/**
 * This class implements a Cacheable for a Byte code generator cache of
 * VMTypeIds.  It maps a Java class or type name to a VM type ID.
 */
class VMTypeIdCacheable implements Cacheable {
	/* The VM name of the Java class name */
	// either a Type (java type) or a String (method descriptor)
	private Object descriptor;

	/* This is the identity */
	private Object key;

	/* Cacheable interface */

	/** @see Cacheable#clearIdentity */
	public void clearIdentity() {
	}

	/** @see Cacheable#getIdentity */
	public Object getIdentity() {
		return key;
	}

	/** @see Cacheable#createIdentity */
	public Cacheable createIdentity(Object key, Object createParameter) {
		if (SanityManager.DEBUG) {
			SanityManager.THROWASSERT("VMTypeIdCacheable.create() called!");
		}
		return this;
	}

	/** @see Cacheable#setIdentity */
	public Cacheable setIdentity(Object key) {

		this.key = key;
		if (key instanceof String) {
			/* The identity is the Java class name */
			String javaName = (String) key;

			/* Get the VM type name associated with the Java class name */
			String vmName = ClassHolder.convertToInternalDescriptor(javaName);
			descriptor = new Type(javaName, vmName);
		}
		else
		{
			descriptor = ((BCMethodDescriptor) key).buildMethodDescriptor();
		}

		return this;
	}

	/** @see Cacheable#clean */
	public void clean(boolean remove) {
		/* No such thing as a dirty cache entry */
		return;
	}

	/** @see Cacheable#isDirty */
	public boolean isDirty() {
		/* No such thing as a dirty cache entry */
		return false;
	}

	/*
	** Class specific methods.
	*/

	/**
	 * Get the VM Type name (java/lang/Object) that is associated with this Cacheable
	 */

	Object descriptor() {
		return descriptor;
	}
}

redpajama_set_name: RedPajamaGithub


text:
Off-Kilter Cube Houses Creatively Renovated Into Homes for Ex-Cons
Geoff Manaugh
Filed to:architecture
Parts of the famously off-kilter "cube houses" project constructed back in 1984 to a design by architect Piet Blom in Rotterdam, Holland, have been transformed into new homes for 21 former prison inmates by Personal Architecture. The old, oddly angled complex has been updated with skylights and internal light-wells to allow more natural illumination, offering stronger "visual connections" between floors.
Alas, internal safety netting on the newly opened vertical shafts lend the interior an ironically prison-like feel, and the radiating hexagons of the building's original floorplan—about which Personal Architecture could do very little, if anything, in their renovation of the structure—cruelly resembles the cell-like layout of a penitentiary. But the rehab nonetheless does a lot with Blom's flawed source material, and it will be interesting to watch over the next few years to see how the newly refit building performs. [Personal Architecture via designboom]
Lead image by Ossip van Duivenbode, used with permission; all other photographs by René de Wit, via Personal Architecture


Home / TOEFL Learning Videos / TOEFL Articles / Rowena scores 101/120 on the TOEFL iBT.
Early this year, Rowena feared she would not be able to continue her college education. Because of her bad English, Rowena struggled to further improve her grammar and reading proficiency. Rowena was required by her university to score at least 100/120 on the TOEFL IBT to attain enrollment. However, Rowena felt her grammar and reading abilities in English were not where they needed to be in order for this goal to be met. Rowena noted she read many books in basic English, although she could still not get a good score on her English exam. Rowena's most recent score was 92/120 with: R-21, L-25, S-24, and W-22. Only 8 marks away from meeting her target score, Rowena reluctantly joined Michael's Online TOEFL Course "The 7 Step System to Pass the TOEFL IBT".
By the help of Michael's vocabulary section, Rowena learned to be able to support her statements during the speaking and writing sections of the TOEFL IBT with detailed descriptions. How was this possible? With Michael's vocabulary section, Rowena built a strong vocabulary foundation by reviewing vocabulary exercises, lesson plans, and practice tests for making use of 1,700 vocabulary words. Under the circumstances, Rowena developed an excellent vocabulary foundation.
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By the help of Michael's reading section, Rowena engaged in critical thinking reading exercises. These reading exercises, prepared Rowena not only for the reading section, but also for the integrated sections of the TOEFL IBT. How was this possible? Rowena learned understandable reading skills. For example, Rowena learned beneficial reading habits, such as: understanding important ideas and details, comprehending terms from reading passages, and being able to reach a conclusion.
In the end as a result of these sections, Rowena credited Michael for his website to giving her the benefit of developing her performance in English. Reluctantly, Rowena scored 101/120 on her next TOEFL IBT exam with R-26, L-26, S-24, and W-25. All in all, like Rowena, considering subscribing to develop your English proficiency in order to attain your English goals.


text:
package org.apache.causeway.commons.functional;

import java.util.function.BiFunction;
import java.util.function.Function;

import org.apache.causeway.commons.internal.base._Refs;

import lombok.val;

/**
 * Similar to a {@link BiFunction}, except that the first argument is an {@code int}.
 *
 * @since 2.x [@index}
 */
@FunctionalInterface
public interface IndexedFunction<T, R> {

    // -- INTERFACE

    public R apply(int index, T t);

    // -- UTILITY

    /**
     * Converts an {@link IndexedFunction} into a {@link Function},
     * having its index start at {@code offset},
     * and incremented after each call.
     */
    static <T, R> Function<T, R> offset(final int offset, final IndexedFunction<T, R> indexed){
        val indexRef = _Refs.intRef(offset);
        return t->indexed.apply(indexRef.getAndInc(), t);
    }

    /**
     * Converts an {@link IndexedFunction} into a {@link Function},
     * having its index start at 0,
     * and incremented after each call.
     */
    static <T, R> Function<T, R> zeroBased(final IndexedFunction<T, R> indexed){
        return offset(0, indexed);
    }

}


text:
Heritage Life Story Funeral Homes®
Honoring Tradition.
Celebrating Life.
Planning & Costs
//cdn.lifestorynet.com/obituaries/01b/134352/134352-life-panel.jpg
David Zylstra
July 15, 1924 - December 30, 2020
Life Story / Obituary
Visitation / Service / Other
Contributions & Flowers
Life Story Video
Poetry & Eulogies
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David Byron Zylstra lived a life rich in faith and family. A man who centered his life in his unwavering faith, David embraced each day and person as a gift. Naturally called to a life of service, David devoted himself to his family, community, and country, never hesitating to give of his talents and resources. David embodied the principles of his faith; in doing so, he proved a powerful role model. Revered by many, David's legacy will continue to live in the hearts of those he loved.
The 1920s were an age of dramatic social and political change. Post-war peace, a booming economy, rising wages for most Americans, and falling consumer prices gave rise to a higher standard of living for most. The world quickly changed with the inventions of wash machines, vacuum cleaners, the band-aid, and the first automobile with a combustion engine. Tremendous confidence, prosperity, and previously unknown comforts marked this roaring decade when a person's success was largely determined by their ability to identify their desires and then work to make them a reality. In East Grand Rapids, Michigan, this vibrant decade grew even brighter for Albert and Luwenda (DenHartigh) Zylstra as they welcomed their son David into their family on July 15, 1924, at Blodgett Hospital.
The third eldest of the eight Zylstra children, David grew up in a loving and busy home. His father, who had immigrated from the Netherlands, owned a sheet metal shop, and his mother devoted herself to homemaking. David's siblings, Paul, Lucille, Thomas, Jonathan, James, Carol, and Mary Lou, shared close bonds and proved good company for one another as they grew up in the house their father built in 1917 on Muskegon St NW. This was especially true when four of the children had scarlet fever within a single year. Memorial Day festivities were especially memorable as they were spent with his father's family. The extended family, which included over 100 cousins, usually gathered at David's Uncle Gerrit's 40-acre farm on the corner of Clyde Park and 54th. The family also made great memories at the cottage they rented on Silver Lake in Rockford each summer.
Blessed with an abundance of musical talent, the family formed an orchestra; the Zylstra Family Orchestra performed more than 325 times, mostly at local churches, including their home church, Broadway Christian Reformed. From the age of 11, David played trombone while attending each school and also with a gospel ensemble. He also enjoyed fishing, playing softball for Crystal Flash, and swimming at Richmond Park City Pool. David attended West Side Christian School, Grand Rapids Christian High, Brooklyn College, BIARRITZ University in France, Calvin College and the University of Michigan, where he earned his Bachelor's Degree.
From 1943 through 1946, David served in the US Army, 75th Division. He proudly served in four battles, including the Battle of the Bulge. He suffered a back injury during his service when the Jeep he was riding in was hit with a shell hole, and he was thrown from the vehicle.
In 1941, David was blessed to be set up on a blind date with Carolyn "Cookie" Mulder, a band director and teacher at Hagars School of Music in Grand Rapids. The pair dated for five years, during which time they played at a Youth For Christ Hymn Sing on the Ramona Boat on Ramona Lake. On June 7, 1946, David and Carolyn married at Bethel Christian Reformed Church. After exchanging their vows and celebrating their nuptials with friends and family, they borrowed Cookie's father's new 1936 Chevy and took a trip around Lake Michigan.
After returning home from their honeymoon, the newlyweds settled in a home on Thomas SE and joined Highland Hills CRC, which later became Westend CRC. As their family grew to include six children, David and Cookie consciously built their home upon the firm foundation of their faith. Judith, Mark, Randall, Scott, Timothy, and Jeffrey were the centers of their lives, and they worked together to create a loving and supportive home. David worked as an insurance adjuster for eight years before establishing his own business, the Dave Zylstra Insurance Agency. He was well suited to a role as mediator, making both sides of an issue feel as though they were well represented. David's work afforded the family many opportunities, including trips he'd earned through his sales. Over the years, the family enjoyed many travel adventures, including a memorable 6000-mile adventure to the West Coast. Summers found the family taking an annual vacation to Spider or Glen Lake near Traverse City. Throughout his life, David had the good fortune of traveling to all 50 states and 49 countries worldwide. David also dearly loved playing his trombone, often accompanied by Carolyn "Cookie" on piano or keyboards, for church services, hymn sings, and other gatherings, attended over the decades by thousands of listeners.
Always willing to give of his time and talents in service of his community, David was very active in many organizations. He served on many boards, including West YMCA, Grand Rapids Christian High School, Christian Counseling Center, Mission India, Christian School International, Mel Trotter, and Wedgewood Christian Services.
Sadly, Carolyn was diagnosed with Leukemia and died on October 11, 2001. In time, David was set up on a blind date with Bea Grotenhuis. The couple dated and married on December 27, 2003, at Calvary Church. After honeymooning by train to Vancouver, they moved to Raybrook. For many years, they enjoyed traveling and being active in their community.
A man who enjoyed the simple pleasures in life, David liked to eat at Bylsma Pancake House and later Red Lobster. He was proud of the German Mauser rifle he took off a German prisoner, his 1928 Ashland and 1937 Dodge, and his roll-up desk. By far, his favorite activity was spending time with his loved ones, especially his grandkids, whom he adored. He loved attending their activities and encouraged each of their endeavors.
David Byron Zylstra, age 96, of Grand Rapids, went to be with his Lord and Savior on December 30, 2020, while in the company of his wife, Bea. Preceded in death by his first wife, Carolyn "Cookie" Zylstra; Dave is survived by his wife, Beatrice "Bea" (Grotenhuis) Zylstra; children, Judith (Dr. Ronald) Warners, Mark (Urszula) Zylstra, Randall (Beth) Zylstra, Scott Zylstra, and Marlene Wood, Timothy Zylstra and Jeffrey (Renee) Zylstra; Bea's children, Judy (Dana) Sommers and W. Frank (Tami) Grotenhuis; 20 grandchildren; 16 great-grandchildren; and brother, James (Marilyn) Zylstra.
Due to COVID attendance restrictions, a private family funeral will take place. Burial in Washington Park Memorial Gardens. Contributions in his memory may be made to either Mission India, Mel Trotter Ministries, or West Side Christian School. Please visit www.heritagelifestory.com to read Dave's life story, submit a favorite memory, photo or to sign the guestbook online.
©2021 LifeStoryNet, LLC
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(once better quality vid comes out, I will post it).
While it mixes AU royalty and terrorists/political tension, it does seem much more IRIS than Goong, and as someone who was obsessed with the former (to put it mildly), good news for me! But then I loved IRIS, Swallow the Sun, Lobbyist etc, so I just go for those types of dramas.
Oh, and forget it, I am not calling it The King 2 Hearts. I am tired of title changes with this baby, especially since there is no point to the latest one except for making Lee Seung Gi's character sound like either a Time Lord or a serial killer.