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# JSONViewer
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## Why?
I made this because I didn't want to share my JSON over the internet using common linting websites.
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\section{Introduction}
In the series of our articles \cite{HabKhaPo}-\cite{KhaUMJ18}, we observed that such important objects for integrable nonlinear equations as the Lax pair and the recursion operator can be effectively obtained directly from the linearized equation. To do this, we construct an ordinary differential (or difference) equation that is compatible with the linearized equation for each solution of the nonlinear equation under consideration. We call this ODE a generalized invariant manifold. The aim of the present article is to show that the concept of invariant manifold provides an effective tool for constructing exact solutions for nonlinear equations. As a basic object we take the well known Volterra chain \cite{Volterra}
\begin{equation}\label{volterra}
\dot{u}_n=u_n(u_{n+1}-u_{n-1}),
\end{equation}
where the desired function $u=u_n(t)$ depends on the real $t$ and the integer $n$, and the dot above the letter means the derivative with respect to time $t$.
Volterra chain is one of the well studied models of the integrability theory \cite{Manakov}, \cite{KacvanMoerbeke}, \cite{Novikovbook}. Its finite-gap solutions are constructed in \cite{Veselov}, \cite{Ver}, \cite{Solomon} by using the discrete version of the algebro-geometric approach (see \cite{DubrovinMatveevNovikov}, \cite{Krichever77}). Various classes of exact self-similar solutions to the chain are found and their applications in the theory of two-dimensional quantum gravitation are discussed in \cite{ItsKitaevFokas}, \cite{FokasItsKitaev}. Alternative methods for studying explicit solutions of the integrable lattices are recently suggested in \cite{AdlerShabat1}, \cite{AdlerShabat2}, \cite{Nijhoff1}.
Let's recall some of the necessary definitions that we are going to use below. For a given equation of the Volterra type
\begin{equation}\label{volterratype}
\dot{u}_n=f(u_{n+1}, u_n,u_{n-1})
\end{equation}
the corresponding linearized equation is evaluated as follows
\begin{equation}\label{volterratypelin}
\dot{U}_n=\frac{\partial f}{\partial u_{n+1}}U_{n+1}+ \frac{\partial f}{\partial u_{n}}U_{n}+\frac{\partial f}{\partial u_{n-1}}U_{n-1}.
\end{equation}
According to the scheme proposed in \cite{HabKhaPo}, we add to the pair of equations (\ref{volterratype}), (\ref{volterratypelin}) an ordinary difference equation of the form
\begin{equation}\label{volterratypeGIM}
U_{n+k}=G(U_{n+k-1},U_{n+k-2},...U_{n-k_1}, [u_n]),\quad k\geq1,\quad k_1\geq0
\end{equation}
compatible with the linearized equation (\ref{volterratypelin}) for each solution of the equation (\ref{volterratype}). The notation $[u_n]$ indicates that function $G$ might depend on the variable $u_n$ and its several shifts $u_{n\pm1}$, $u_{n\pm2}$, \dots . We call the surface on the space of the dynamical variables, which is determined by the equation (\ref {volterratypeGIM}), a generalized invariant manifold. Actually function $G$ satisfies the following determining equation
\begin{equation}\label{determiningGIM}
\frac{dG}{dt}-D^k_n\dot{U}_n=0 \quad \mbox{mod}\,((1.2),(1.3),(1.4)).
\end{equation}
Let us explain the scheme of searching $G$ from relation (\ref{determiningGIM}). First, we must exclude in (\ref{determiningGIM}) all time-derivatives using equations (\ref {volterratype}), (\ref {volterratypelin}), and then we exclude the variables $ U_ {n + m} $ for $ m \geq k $, as well as for $ m \leq -k_1-1 $ using the relation
(\ref {volterratypeGIM}). It is possible since we request that the equation (\ref{volterratypeGIM}) can be rewritten in the form solved with respect to the variable $U_{n-k_1}$. After these manipulations we arrive at an equation which should hold identically with respect to the independent dynamical variables
$$U_{n+k-1}, U_{n+k-2}, ...U_{n-k_1};u_n,u_{n\pm1}, u_{n\pm2}, \dots .$$
Thus, we get an overdetermined system of equations for unknown function $ G $, which is usually effectively solved. Having found $G$ we get a triple of equations (\ref{volterratype}), (\ref{volterratypelin}) and (\ref{volterratypeGIM}) showing that the dynamics due to equations (\ref{volterratype}), (\ref{volterratypelin}) admits a compatible reduction (\ref{volterratypeGIM}). An intriguing fact is that for an appropriately chosen function $G$ the consistency condition of the equations (\ref{volterratypelin}) and (\ref{volterratypeGIM}) implies (\ref{volterratype}). Actually in such a case a pair of the equations (\ref{volterratypelin}), (\ref{volterratypeGIM}) provides a Lax pair for (\ref{volterratype}) (generally nonlinear!). The purpose of the article is to show that this nonlinear Lax pair can be used for constructing explicit solutions of the equation (\ref{volterratype}).
Since the invariant manifold deals with two sets of dynamical variables defined by the equation in question and the linearized equation, it has two orders. Sometimes the problem of determining orders causes problems. The corresponding relationship between these two orders for the case of equations of the KdV type was established in \cite{ZhangZY}.
Let us briefly discuss the content of the article. In section 2 for the Volterra chain we evaluate the generalized invariant manifold of the order two, depending on two constant parameters. At the end of the section it is shown that system (\ref{Laxt}), (\ref{Laxn}) provides a nonlinear Lax pair for the Volterra chain. In \S 3 we assumed that this Lax pair admits a polynomial in the parameter $\lambda$ solution and observed that this assumption puts a severe restriction on the solution of the nonlinear lattice in the form of the overdetermined system of difference and differential equations. The idea is to solve this overdetermined set of equations and find the desired particular solutions for the nonlinear chain. A scheme of realization of the idea is illustrated in \S4 by taking the Volterra chain as an example. Here for $m=1$, using an invariant manifold, we obtained a pair of scalar ordinary equations compatible with each other. One of which is differential, and the other is difference. Then we reduced the differential equation to an equation which is solved in terms of the Weierstrass $\wp$ function. It is worth to note that simultaneously the difference equation turns into the addition theorem for $\wp$. In generic case the problem of constructing a solution $u_n(t)$ is reduced to a system of ordinary differential equations (\ref{dubrt3}), (\ref{dubrt4}) with some additional constraint given by a system of $m$ algebraic equations (\ref{Rj}). These equations are shown in enlarged form for $m=2$ (see (\ref{dubrt_meq2}), (\ref{Rj1_meq2}), (\ref{Rj2_meq2})).
\section{Evaluation of the invariant manifold for the Volterra chain.}
In this section we concentrate on the Volterra chain (\ref{volterra}). First we find its linearization which obviously looks like
\begin{equation}\label{volterralin}
\dot{U}_n=u_n(U_{n+1}-U_{n-1})+(u_{n+1}-u_{n-1})U_n.
\end{equation}
The linearized equation needs some simplification, since it contains three extra variables $u_n, u_{n+1}, u_{n-1}$ which are interpreted here as some functional parameters. Let's change the variables in the latter by setting $U_n=u_n(P_{n+1}-P_{n-1})$. This relation is nothing else but the linearization of the formula $\log u_n=p_{n+1}-p_{n-1}$ relating the Volterra chain with the equation $\dot{p}_n=e^{p_{n+1}-p_{n-1}}$. In terms of the new variables the linearized equation looks much simpler
\begin{equation}\label{volterralin2}
\dot{P}_n=u_n(P_{n+1}-P_{n-1}).
\end{equation}
We look for an invariant manifold for the equation (\ref{volterralin2}) in the following form
\begin{equation}\label{volterralinGIM2}
P_{n+1}=F(P_{n},P_{n-1},u_n).
\end{equation}
To do this, we must solve the defining equation
\begin{equation}\label{determiningGIM2}
\frac{d}{dt}F(P_{n},P_{n-1},u_n)-D_n(u_n(P_{n+1}-P_{n-1}))=0 \, \mbox{mod}\,((1.1),(2.2),(2.3)).
\end{equation}
In addition to (\ref{volterralinGIM2}), in our further computations, we also need a function that, in a sense, is inverse to (\ref{volterralinGIM2})
\begin{equation}\label{volterralinGIM2-}
P_{n-1}=G(P_{n+1},P_{n},u_n)
\end{equation}
assuming that function $G$ is single-valued. Then, an equation of the form
\begin{equation}\label{volterralinGIM2-2}
P_{n-2}=G(P_{n},P_{n-1},u_{n-1})
\end{equation}
also defines an invariant manifold for the equation (\ref{volterralin2}) if the condition
\begin{equation}\label{determiningGIM2-}
\frac{d}{dt}G(P_{n},P_{n-1},u_{n-1})-D^{-2}_n(u_n(P_{n+1}-P_{n-1}))=0 \, \mbox{mod}\,((1.1),(2.2),(2.6))
\end{equation}
is satisfied.
Let's rewrite equations (\ref{determiningGIM2}) and (\ref{determiningGIM2-}) in a more detailed form
\begin{equation}\label{determiningGIM2df}
\frac{dF}{dP_{n}}P_{n,t}+\frac{dF}{dP_{n-1}}P_{n-1,t}+\frac{dF}{du_{n}}u_{n,t}-u_{n+1}(P_{n+2}-P_{n})=0,
\end{equation}
\begin{equation}\label{determiningGIM2-2df}
\frac{dG}{dP_{n}}P_{n,t}+\frac{dG}{dP_{n-1}}P_{n-1,t}+\frac{dG}{du_{n-1}}u_{n-1,t}-u_{n-2}(P_{n-1}-P_{n-3})=0,
\end{equation}
$\mbox{mod} \, ((1.1),(2.2),(2.3),(2.6))$.
Replace the variables $u_{n,t}$, $u_{n-1,t}$, $P_{n,t}$, $P_{n-1,t}$, $P_{n+2}$ and $P_{n-3}$ in equations (\ref{determiningGIM2df}) and (\ref{determiningGIM2-2df}) by virtue of the equations (\ref{volterra}), (\ref{volterralin2}), (\ref{volterralinGIM2}) and (\ref{volterralinGIM2-2}). Then we will get
\begin{eqnarray}\label{determiningGIM2df2}
&u_n(F-P_{n-1})\frac{dF}{dP_{n}}-u_{n-1}(G-P_{n})\frac{dF}{dP_{n-1}}\nonumber\\
&\qquad +u_{n}(u_{n+1}-u_{n-1})\frac{dF}{du_{n}}-u_{n+1}(D_n(F)-P_n)=0,
\end{eqnarray}
\begin{eqnarray}\label{determiningGIM2-2df2}
u_n(F-P_{n-1})\frac{dG}{dP_{n}}-u_{n-1}(G-P_{n})\frac{dG}{dP_{n-1}}\nonumber\\
\qquad +u_{n-1}(u_{n}-u_{n-2})\frac{dG}{du_{n-1}}+u_{n-2}(D^{-1}_n(G)-P_{n-1})=0,
\end{eqnarray}
where
\begin{eqnarray*}
&D_n(F(P_n,P_{n-1},u_n))=F(F(P_n,P_{n-1},u_n),P_n,u_{n+1}),\\
&D^{-1}_n(G(P_{n},P_{n-1},u_{n-1}))=G(P_{n-1},G(P_{n},P_{n-1},u_{n-1}),u_{n-2}).
\end{eqnarray*}
We differentiate relations (\ref{determiningGIM2df2}) and (\ref{determiningGIM2-2df2}) twice with respect to $u_{n+1}$ and $u_{n-2}$, respectively. As a result, we obtain
\begin{eqnarray}\label{determiningGIM2df2_up1}
u_{n+1}\frac{\partial^2}{\partial u^2_{n+1}}D_n(F)+2\frac{\partial}{\partial u_{n+1}}D_n(F)=0,
\end{eqnarray}
\begin{eqnarray}\label{determiningGIM2-2df2_um2}
u_{n-2}\frac{\partial^2}{\partial u^2_{n-2}}D^{-1}_n(G)+2\frac{\partial}{\partial u_{n-2}}D^{-1}_n(G)=0.
\end{eqnarray}
We apply the shift operator $D^{-1}_n$ to both sides of the equality (\ref{determiningGIM2df2_up1}) and, respectively, the operator $D_n$ to both sides of (\ref{determiningGIM2-2df2_um2}):
\begin{eqnarray}\label{determiningGIM2df2_u}
u_{n}\frac{\partial^2}{\partial u^2_{n}}F+2\frac{\partial}{\partial u_{n}}F=0,
\end{eqnarray}
\begin{eqnarray}\label{determiningGIM2-2df2_um1}
u_{n-1}\frac{\partial^2}{\partial u^2_{n-1}}G+2\frac{\partial}{\partial u_{n-1}}G=0.
\end{eqnarray}
The resulting equations are easily solved
\begin{eqnarray}\label{GIM_F}
F(P_n,P_{n-1},u_n)=F_1(P_n,P_{n-1})+\frac{1}{u_n}F_2(P_n,P_{n-1}),
\end{eqnarray}
\begin{eqnarray}\label{GIM_G}
G(P_n,P_{n-1},u_{n-1})=G_1(P_n,P_{n-1})+\frac{1}{u_{n-1}}G_2(P_n,P_{n-1}).
\end{eqnarray}
We substitute the found expressions into the equations (\ref{determiningGIM2df2}) and (\ref{determiningGIM2-2df2}). Now, since the dependence of the functions $F$ and $G$ on the variables $u_n$ and $u_{n-1}$ is already determined, we can compare in (\ref{determiningGIM2df2}) and (\ref{determiningGIM2-2df2}) the coefficients in front of the powers of the independent variables $u_{n+1}$, $u_{n-1}$ and $u_{n-2}$, $u_{n}$, respectively. As a result, we get a set of equations
\begin{eqnarray}
&\left(P_n-G_1(P_n,P_{n-1})\right)\frac{\partial F_1(P_n,P_{n-1})}{\partial P_{n-1}}=0,\label{GIM_F_eq1}\\
&u_n\left(D_n(F_1(P_n,P_{n-1}))-P_n\right)+F_2(P_n,P_{n-1})=0,\label{GIM_F_eq2}\\
&\left(P_n-G_1(P_n,P_{n-1})\right)\frac{\partial F_2(P_n,P_{n-1})}{\partial P_{n-1}}+F_2(P_n,P_{n-1})=0,\label{GIM_F_eq3}
\end{eqnarray}
\begin{eqnarray}
&\left(F_1(P_n,P_{n-1})-P_{n-1}\right)\frac{\partial G_1(P_n,P_{n-1})}{\partial P_{n}}=0,\label{GIM_G_eq1}\\
&u_{n-1}\left(D^{-1}_n(G_1(P_n,P_{n-1}))-P_{n-1}\right)+G_2(P_n,P_{n-1})=0,\label{GIM_G_eq2}\\
&\left(F_1(P_n,P_{n-1})-P_{n-1}\right)\frac{\partial G_2(P_n,P_{n-1})}{\partial P_{n}}-G_2(P_n,P_{n-1})=0.\label{GIM_G_eq3}
\end{eqnarray}
The equation (\ref{GIM_F_eq1}) confirms that there are two possibilities:
\begin{eqnarray*}
i) \quad &G_1(P_n,P_{n-1})=P_n,\\
ii) \quad &\frac{\partial F_1(P_n,P_{n-1})}{\partial P_{n-1}}=0.
\end{eqnarray*}
The first case leads to a trivial solution
\begin{eqnarray*}
F(P_n,P_{n-1},u_n)\equiv P_{n-1}, \quad G(P_n,P_{n+1},u_{n})\equiv P_{n+1}.
\end{eqnarray*}
Let's focus on $ ii) $, it gives right away
\begin{eqnarray*}
F_1(P_n,P_{n-1})=F_3(P_n).
\end{eqnarray*}
By virtue of the latter and due to the equation (\ref {GIM_G_eq1}) we obtain
\begin{eqnarray*}
G_1(P_n,P_{n-1})=G_3(P_{n-1}).
\end{eqnarray*}
We rewrite equations (\ref{GIM_F_eq2}) and (\ref{GIM_G_eq2}), taking into account the found functions $F_1(P_n,P_{n-1})$ and $G_1(P_n,P_{n-1})$
\begin{eqnarray}
&u_nF_3\left(F_3(P_n)+\frac{1}{u_n}F_2(P_n,P_{n-1})\right)-u_nP_n+F_2(P_n,P_{n-1})=0,\label{GIM_F_eq22}\\
&u_{n-1}G_3\left(G_3(P_{n-1})+\frac{1}{u_{n-1}}G_2(P_n,P_{n-1})\right)-u_{n-1}P_{n-1}\nonumber\\
& \qquad{} \qquad \qquad \qquad{} \qquad{}+G_2(P_n,P_{n-1})=0,\label{GIM_G_eq22}
\end{eqnarray}
We differentiate equation (\ref{GIM_F_eq22}) with respect to $u_n$ and equation (\ref{GIM_G_eq22}) with respect to $u_{n-1}$ twice, then we find that the functions $F_3(P_n)$ and $G_3(P_{n-1})$ have the form
\begin{eqnarray*}
&F_3(P_n)=C_2P_n+C_1,\\
&G_3(P_{n-1})=C_4P_{n-1}+C_3,
\end{eqnarray*}
where $C_i$, $i=1,2,3,4$ are arbitrary constants.
By substituting the found representations of the functions $F_3(P_n)$ and $G_3(P_{n-1})$ into equations (\ref{GIM_F_eq22}) and (\ref{GIM_G_eq22}) and comparing the coefficients before the variables $u_n$ and $u_{n-1}$, respectively, we find:
\begin{eqnarray*}
&C_2=-1,\quad C_4=-1.
\end{eqnarray*}
Now we substitute the above refinements into the equations (\ref {GIM_F_eq3}) and (\ref {GIM_G_eq3}), and then derive the equations for determining the functions $ F_2 (P_n, P_ {n-1}) $ and $ G_2 (P_n, P_{n-1}) $, which are easily solved:
\begin{eqnarray*}
&F_2(P_n,P_{n-1})=\frac{F_4(P_n)}{C_3-P_n-P_{n-1}},\\
&G_2(P_n,P_{n-1})=\frac{G_4(P_{n-1})}{C_1-P_n-P_{n-1}}.
\end{eqnarray*}
Summing up the above calculations, we can represent the functions $F(P_n,P_{n-1},u_n)$ and $G(P_n,P_{n-1},u_n)$ in the following form:
\begin{eqnarray}
&P_{n+1}=F(P_n,P_{n-1},u_n)=C_1-P_n+\frac{F_4(P_n)}{u_n\left(C_3-P_n-P_{n-1}\right)},\label{GIM_F_4}\\
&P_{n-2}=G(P_n,P_{n-1},u_{n-1})=C_3-P_{n-1}+\frac{G_4(P_{n-1})}{u_{n-1}\left(C_1-P_n-P_{n-1}\right)}.\label{GIM_G_4}
\end{eqnarray}
Let us apply the shift operator $D_n$ to both sides of (\ref{GIM_G_4}) and then express the variable $P_{n+1}$ from the obtained equation:
\begin{eqnarray}
P_{n+1}=F(P_n,P_{n-1},u_n)=C_1-P_n+\frac{G_4(P_n)}{u_n\left(C_3-P_n-P_{n-1}\right)}.\label{GIM_G_41}
\end{eqnarray}
By comparing equations (\ref{GIM_F_4}) and (\ref{GIM_G_41}) we observe that
\begin{eqnarray*}
G_4(P_n)=F_4(P_n).
\end{eqnarray*}
Thus, it remains to determine the only function $F_4(P_n)$.
It is easily observed that the Volterra equation and its linearization are invariant under the reflection transformation $n\rightarrow -n$, $t\rightarrow -t$, therefore it is natural to assume that formulas (\ref{volterralinGIM2}) and (\ref{volterralinGIM2-2}) are related to each other by the reflection transformation as well. Then it follows from formulas (\ref{GIM_F_4}) and (\ref{GIM_G_4}) that $C_1=C_3$.
Since the equation (\ref{volterralin2}) is invariant under the shift transformation we can remove the parameter $C_1$ by changing
$P_n=\frac{C_1}{2}+\tilde{P}_n$. Moreover, we are only interested in parameters that cannot be removed, so we put $C_1=0$.
We return to the equation (\ref{determiningGIM2df2}) and specify it using the obtained formulas:
\begin{eqnarray}\label{EQ_last}
\left[F_4\left(-P_n-\frac{F_4(P_n)}{u_n\left(P_n+P_{n-1}\right)}\right)-F_4(P_n)\right]u^2_n-\frac{F_4(P_n)F'_4(P_n)u_n}{P_n+P_{n-1}}\nonumber\\
\quad -\frac{(F_4(P_n))^2\left[F'_4(P_n)(P_n+P_{n-1})+F_4(P_{n-1})-F_4(P_n)\right]}{(P_n+P_{n-1})^4}=0.
\end{eqnarray}
Let us rewrite equation (\ref{EQ_last}) in a short form
\begin{eqnarray}\label{intermediate}
F_4(-x-\delta y)-F_4(x)=c\delta^2+d\delta,
\end{eqnarray}
where $x=P_n$, $\delta=\frac{1}{u_n}$ and so on. Then by taking $\delta=0$ we get that $F_4$ is an even function.
Differentiation of the equation with respect to $\delta$ three times implies (we can do that because equation (\ref{EQ_last}) is satisfied identically with respect to the variable $u_n$) $F_4'''=0$ which gives
\begin{eqnarray}\label{F4}
F_4(P_n)=C_5P^2_n+C_6.
\end{eqnarray}
Thus, an invariant manifold (\ref{volterralinGIM2}) is given by an equation of the form
\begin{eqnarray*}
P_{n+1}=-P_n+\frac{\lambda P_n^2+c}{u_n(P_{n}+P_{n-1})},
\end{eqnarray*}
where $\lambda=-C_5$, $c=-C_6$.
Let us summarize the reasonings and computations of this section as a statement.
{\bf Proposition 1}. A system of the equations
\begin{eqnarray}\label{Laxt}
\dot{P}_n=u_n(P_{n+1}-P_{n-1}),\\
u_n(P_{n+1}+P_n)(P_{n}+P_{n-1})=\lambda P_n^2+c
\label{Laxn}
\end{eqnarray}
is compatible if and only if the coefficient $u_n=u_n(t)$ solves the Volterra lattice (\ref{volterra}).
We have already proved above that for arbitrary solution $u_n(t)$ to the Volterra chain the system is compatible. The converse can be easily approved by a direct computation. We notice that in fact the system defines a nonlinear Lax pair with two arbitrary constant parameters $\lambda$ and $c$ for the Volterra chain. By applying the operator $D_n-1$ to the equation (\ref{Laxn}) we immediately obtain a linear equation (see \cite{HabKhaPo})
\begin{equation}\label{Recursion}
u_{n+1}(P_{n+2}+P_{n+1})-u_{n}(P_{n}+P_{n-1})=\lambda (P_{n+1}-P_{n})
\end{equation}
which also defines a generalized invariant manifold since it is compatible, as it is easily checked, with the equations (\ref{volterra}), (\ref{volterralin2}). Earlier in \cite{HabKhaTMP18}, we observed that equation (\ref{Recursion}) can be easily rewritten as the recursion operator for the Volterra chain.
\section{Separation of the variables.}
It is easily proved that nonlinear difference equation (\ref{Laxn}), where the constant parameter $c=c(\lambda)$ is in the form
\begin{equation}\label{a1}
c(\lambda)=-\lambda+c^{(0)} +c^{(1)}\lambda^{-1}+c^{(2)}\lambda^{-2}+\dots
\end{equation}
admits a solution given by the following formal asymptotic expansion:
\begin{equation}\label{a2}
P_n(\lambda)=1+\alpha_n^{(1)}\lambda^{-1}+\alpha_n^{(2)}\lambda^{-2}+\dots.
\end{equation}
Here the coefficients $\alpha_n^{(j)}$ are functions of a finite set of the dynamical variables $u_{n},\,u_{n\pm1},\,u_{n\pm2},\dots$ that are found successively from equation (\ref{Laxn}). Let us give exact representation for the first two coefficients:
\begin{eqnarray}\label{a3}
\alpha_n^{(1)}&=&2u_n-\frac{1}{2}c^{(0)},\\
\alpha_n^{(2)}&=&\frac{u_n}{2}(\alpha_{n+1}^{(1)}+2\alpha_n^{(1)}+\alpha_{n-1}^{(1)})-\frac{(\alpha_n^{(1)})^2+c^{(1)}}{4}.
\label{a4}
\end{eqnarray}
If we assume that formal series (\ref{a2}) is terminated at some natural $m$, such that $\alpha_n^{(j)}\equiv 0$ for $\forall j>m$, then we get a system of difference equations for the variables $\alpha_n^{(1)},\,\alpha_n^{(2)},\,\dots,\alpha_n^{(m)}, u_n$. This fact can be used for describing some particular solutions of the Volterra chain. For the terminated case we change slightly our ansatz (\ref{a1}), (\ref{a2}) by multiplying the given ones by powers of $\lambda$. We rewrite the sought function $P_n(\lambda)$ and function $c(\lambda)$ as polynomials on $\lambda$ and parametrize these polynomials by their roots (earlier this kind of parametrization was used, for instance, in \cite{DubrovinMatveevNovikov}, \cite {Calogero}). That allows us to make the mentioned system of difference equations more symmetrical. Therefore we can assume that in the equations (\ref{Laxt}), (\ref{Laxn}) the constant parameter $c$ and a solution $P_n$ are polynomials on the parameter $\lambda$ so that
\begin{equation}\label{parametrization}
P_{n}(\lambda)=\prod_{i=1}^{m}(\lambda-\gamma^i_n),\quad c(\lambda)=-\prod_{i=1}^{2m+1}(\lambda-e_i).
\end{equation}
By requesting that $P_n(\lambda)$ is a polynomial we impose to the Lax pair (\ref{Laxt}), (\ref{Laxn}) a constraint of the form
\begin{equation}\label{polynom}
\frac{\partial^{m+1}}{\partial \lambda^{m+1}}P_n(\lambda)=0.
\end{equation}
{\bf Proposition 2.} Let us suppose that a solution $u_n(t)$ to the Volterra chain (\ref{volterra}) satisfies condition (\ref{polynom})
for $\forall \, n\in \bf{Z}$, $t=0$. Then (\ref{polynom}) is preserved for $t>0$.
{\bf Proof}. Indeed, the relation (\ref{polynom}) is evidently compatible with equation (\ref{Laxt}).
We substitute the ansatz (\ref{parametrization}) into the equations (\ref{Laxt}), (\ref{Laxn}) and obtain
\begin{eqnarray}\label{paramLaxt}
\sum_{k=1}^{m}\dot{\gamma}^k_n\prod_{i\neq k}(\lambda-\gamma^i_n)&=&u_n\left(\prod_{i=1}^{m}(\lambda-\gamma^i_{n-1})-\prod_{i=1}^{m}(\lambda-\gamma^i_{n+1})\right),
\end{eqnarray}
\begin{eqnarray}
u_n\left(\prod_{i=1}^{m}(\lambda-\gamma^i_{n+1})+\prod_{i=1}^{m}(\lambda-\gamma^i_n)\right)\left(\prod_{i=1}^{m}(\lambda-\gamma^i_n)+\prod_{i=1}^{m}(\lambda-\gamma^i_{n-1})\right)\nonumber\\
=\lambda\prod_{i=1}^{m}(\lambda-\gamma^i_n)^2-\prod_{i=1}^{2m+1}(\lambda-e_i).\label{paramLaxn}
\end{eqnarray}
By comparing the coefficients in front of the power $\lambda^{2m}$ in (\ref{paramLaxn}) we derive a relation between the field variable $u_n$ and the roots of the polynomial $P_n(\lambda)$
\begin{equation}\label{un}
u_{n}=-\frac{1}{2}\sum_{i=1}^{m}\gamma^i_n+\frac{1}{4}\sum_{i=1}^{2m+1}e_i.
\end{equation}
By setting $\lambda=\gamma^j_n$ and $\lambda=\gamma^j_{n-1}$ for $j=1,2,...,m$ in the equation (\ref{paramLaxn}) we obtain a system of difference equations
\begin{equation}\label{dubrn}
u_n \prod_{i=1}^{m}(\gamma^j_n-\gamma^i_{n+1})\prod_{i=1}^{m}(\gamma^j_n-\gamma^i_{n-1})=-\prod_{i=1}^{2m+1}(\gamma^j_n-e_i),
\end{equation}
\begin{eqnarray}\label{dubrn2}
u_n\left(\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n+1})+\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})\right)\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})=\nonumber\\
\qquad \qquad \qquad\gamma^j_{n-1}\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})^2-\prod_{i=1}^{2m+1}(\gamma^j_{n-1}-e_i).
\end{eqnarray}
Similarly by taking $\lambda=\gamma^j_n$ in (\ref{paramLaxt}) we find
\begin{equation}\label{dubrt1}
\dot{\gamma}^j_n=\frac{u_n}{\prod_{i\neq j}(\gamma^j_n-\gamma^i_n)}
\left(\prod_{i=1}^{m}(\gamma^j_n-\gamma^i_{n-1})-\prod_{i=1}^{m}(\gamma^j_n-\gamma^i_{n+1})\right)
\end{equation}
by shifting $n\longmapsto n-1$ in (\ref{dubrt1}) we get
\begin{equation}\label{dubrt2}
\dot{\gamma}^j_{n-1}=\frac{u_{n-1}}{\prod_{i\neq j}(\gamma^j_{n-1}-\gamma^i_{n-1})}
\left(\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n-2})-\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})\right).
\end{equation}
Due to (\ref{dubrn}) we can rewrite (\ref{dubrt1}), (\ref{dubrt2}) as a closed system of ordinary differential equations of the order $2m$
\begin{equation}\label{dubrt3}
\dot{\gamma}^j_n=\frac{u_n}{\prod_{i\neq j}(\gamma^j_n-\gamma^i_n)}
\left(\prod_{i=1}^{m}(\gamma^j_n-\gamma^i_{n-1})+\frac{\prod_{i=1}^{2m+1}(\gamma^j_{n}-e_i)}{u_n\prod_{i=1}^{m}(\gamma^j_n-\gamma^i_{n-1})}\right),
\end{equation}
\begin{equation}\label{dubrt4}
\dot{\gamma}^j_{n-1}=\frac{-u_{n-1}}{\prod_{i\neq j}(\gamma^j_{n-1}-\gamma^i_{n-1})}
\left(\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})+\frac{\prod_{i=1}^{2m+1}(\gamma^j_{n-1}-e_i)}{u_{n-1}\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})}\right).
\end{equation}
Let us first exclude the variables $\gamma^i_{n+1}, \, i=1,2,\dots,m$ from the system of discrete equations (\ref{dubrn}), (\ref{dubrn2}). To this end we rewrite equation (\ref{dubrn}) in the form
\begin{equation}\label{Prodeqrj}
\prod_{i=1}^{m}(\gamma^j_{n}-\gamma^i_{n+1})=r^j, \, j=1,2,\dots,m,
\end{equation}
where
\begin{equation*}
r^j=-\frac{\prod_{i=1}^{2m+1}(\gamma^j_{n-1}-e_i)}{u_n\prod_{i=1}^{m}(\gamma^j_n-\gamma^i_{n-1})}.
\end{equation*}
After opening the parentheses in (\ref{Prodeqrj}) we arrive at a system of linear equations
\begin{equation}\label{lin_eq_gamma_rj}
\alpha^{(1)}(\gamma^j_n)^{m-1}+\alpha^{(2)}(\gamma^j_n)^{m-2}+\dots+\alpha^{(m)}=r^j-(\gamma^j_n)^{m},
\end{equation}
where $j=1,2,\dots,m$ and
\begin{eqnarray}\label{alpha_j}
\alpha^{(1)}=-\sum^{m}_{i=1}\gamma^{i}_{n+1},\nonumber\\
\alpha^{(2)}=-\sum_{i\neq j}\gamma^{i}_{n+1}\gamma^{j}_{n+1},\\
\dots\nonumber\\
\alpha^{(m)}=(-1)^m\gamma^{1}_{n+1}\gamma^{2}_{n+1}\dots\gamma^{m}_{n+1} \nonumber
\end{eqnarray}
are coefficients of the polynomial $P_{n+1}(\lambda)=\Pi^{i=1}_{m}\left(\lambda-\gamma^{j}_{n+1}\right)$. Equations (\ref{alpha_j}) give explicit representations of the coefficients $\alpha^j$ in terms of $\gamma^1_{n+1}, \gamma^2_{n+1},...\gamma^m_{n+1}$. On the other hand side by solving the linear system (\ref{lin_eq_gamma_rj}) we find representations for the same coefficients in terms of the variables $\gamma^{1}_{n},\, \gamma^{2}_{n},\, \dots, \,\gamma^{m}_{n};$ $\gamma^{1}_{n-1},\, \gamma^{2}_{n-1},\, \dots, \,\gamma^{m}_{n-1}:$
\begin{equation}\label{other}
\alpha^j=H^j(\gamma^{1}_{n},\, \gamma^{2}_{n},\, \dots, \,\gamma^{m}_{n}; \gamma^{1}_{n-1},\, \gamma^{2}_{n-1},\, \dots, \,\gamma^{m}_{n-1}).
\end{equation}
By comparing two representations (\ref{alpha_j}), (\ref{other}) we obtain an implicit formula determining dynamics on $n$ for the roots $\gamma^j$.
Now we concentrate on the system (\ref{dubrn2}). Evidently it can be represented in the form
\begin{equation}\label{Prodeqsj}
\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n+1})=s^j,
\end{equation}
where
\begin{equation*}
s^j=-\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})+\frac{\gamma^j_{n-1}\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})^2-\prod_{i=1}^{2m+1}(\gamma^j_{n-1}-e_i)}{u_n\prod_{i=1}^{m}(\gamma^j_{n-1}-\gamma^i_{n})}.
\end{equation*}
From (\ref{Prodeqsj}) we get
\begin{equation}\label{lin_eq_gamma_sj}
\alpha^{(1)}(\gamma^j_{n-1})^{m-1}+\alpha^{(2)}(\gamma^j_{n-1})^{m-2}+\dots+\alpha^{(m)}=s^j-(\gamma^j_{n-1})^{m}.
\end{equation}
By substituting the earlier found expressions (\ref{other}) for $\alpha^{(1)},\, \alpha^{(2)}\, \dots, \alpha^{(m)}$ into the system (\ref{lin_eq_gamma_sj}) we obtain exactly $m$ constraints relating variables $\gamma^{1}_{n},\, \gamma^{2}_{n},\, \dots, \,\gamma^{m}_{n}$ with the variables $\gamma^{1}_{n-1},\, \gamma^{2}_{n-1},\, \dots, \,\gamma^{m}_{n-1}:$
\begin{equation}\label{Rj}
R^j\left(\gamma^{1}_{n},\, \gamma^{2}_{n},\, \dots, \,\gamma^{m}_{n};\gamma^{1}_{n-1},\, \gamma^{2}_{n-1},\, \dots, \,\gamma^{m}_{n-1}\right)=0,
\end{equation}
$j=1,2,\dots,m.$
Now the problem is to find solution of the finite system of ordinary differential equations (\ref{dubrt3}), (\ref{dubrt4}), satisfying the additional constraint (\ref{Rj}).
Let us consider the system (\ref{dubrt3}), (\ref{dubrt4}) and constraint (\ref{Rj}) for the case $m=2$. For the simplicity we take
\begin{equation}\label{Pn_meq2}
P_n(\lambda)=(\lambda-\gamma_n)(\lambda-\delta_n).
\end{equation}
Then (\ref{un}) yeilds
\begin{equation*}
u_n=-\frac{1}{2}(\gamma_n+\delta_n)+\Sigma^{5}_{i=1}e_i.
\end{equation*}
Functions $\gamma_n=\gamma_n(t), \, \delta_n=\delta_n(t)$ satisfy the following system of differential equations
\begin{eqnarray}\label{dubrt_meq2}
\dot{\gamma}_{n-1}=\frac{-1}{\gamma_{n-1}-\delta_{n-1}}
\left(u_{n-1}(\gamma_{n-1}-\gamma_{n})(\gamma_{n-1}-\delta_{n})+\frac{\prod_{i=1}^{5}(\gamma_{n-1}-e_i)}{(\gamma_{n-1}-\gamma_{n})(\gamma_{n-1}-\delta_{n})}\right),\nonumber\\
\dot{\delta}_{n-1}=\frac{-1}{\delta_{n-1}-\gamma_{n-1}}
\left(u_{n-1}(\delta_{n-1}-\gamma_{n})(\delta_{n-1}-\delta_{n})+\frac{\prod_{i=1}^{5}(\delta_{n-1}-e_i)}{(\delta_{n-1}-\gamma_{n})(\delta_{n-1}-\delta_{n})}\right),\nonumber\\
\dot{\gamma}_n=\frac{1}{\gamma_n-\delta_n}
\left(u_n(\gamma_n-\gamma_{n-1})(\gamma_n-\delta_{n-1})+\frac{\prod_{i=1}^{5}(\gamma_{n}-e_i)}{(\gamma_n-\gamma_{n-1})(\gamma_n-\delta_{n-1})}\right),\\
\dot{\delta}_n=\frac{1}{\delta_n-\gamma_n}
\left(u_n(\delta_n-\gamma_{n-1})(\delta_n-\delta_{n-1})+\frac{\prod_{i=1}^{5}(\delta_{n}-e_i)}{(\delta_n-\gamma_{n-1})(\delta_n-\delta_{n-1})}\right).\nonumber
\end{eqnarray}
Constraints (\ref{Rj}) in this case take the form
\begin{eqnarray}\label{Rj1_meq2}
\frac{\prod_{i=1}^{5}(\gamma_{n-1}-e_i)}{(\gamma_{n-1}-\gamma_{n})^2(\gamma_{n-1}-\delta_{n})^2}+\frac{\prod_{i=1}^{5}(\gamma_{n}-e_i)}{(\gamma_{n-1}-\gamma_{n})^2(\gamma_n-\delta_{n})(\gamma_n-\delta_{n-1})}+\nonumber\\
\qquad \qquad \qquad \frac{\prod_{i=1}^{5}(\delta_{n}-e_i)}{(\gamma_{n-1}-\delta_{n})^2(\delta_{n}-\gamma_n)(\delta_n-\delta_{n-1})}=-2u_n+\gamma_{n-1},
\end{eqnarray}
\begin{eqnarray}\label{Rj2_meq2}
\frac{\prod_{i=1}^{5}(\delta_{n-1}-e_i)}{(\delta_{n-1}-\gamma_{n})^2(\delta_{n-1}-\delta_{n})^2}+\frac{\prod_{i=1}^{5}(\gamma_{n}-e_i)}{(\gamma_{n}-\delta_{n-1})^2(\gamma_n-\delta_{n})(\gamma_n-\gamma_{n-1})}+\nonumber\\
\qquad \qquad \qquad \frac{\prod_{i=1}^{5}(\delta_{n}-e_i)}{(\delta_{n}-\delta_{n-1})^2(\delta_{n}-\gamma_n)(\delta_n-\gamma_{n-1})}=-2u_n+\delta_{n-1}.
\end{eqnarray}
{\bf Proposition 3.} Solution of the system (\ref{dubrt_meq2}) satisfy the following constraint
\begin{eqnarray}\label{PrY}
\frac{\dot{\gamma}_{n}}{(\gamma_{n}-\gamma_{n-1})(\gamma_{n}-\delta_{n-1})}+\frac{\dot{\delta}_{n}}{(\delta_{n}-\gamma_{n-1})(\delta_{n}-\delta_{n-1})}-\nonumber\\
\quad \frac{\dot{\gamma}_{n-1}}{(\gamma_{n-1}-\gamma_{n})(\gamma_{n-1}-\delta_{n})}-\frac{\dot{\delta}_{n-1}}{(\delta_{n-1}-\gamma_{n})(\delta_{n-1}-\delta_{n})}-1=0.
\end{eqnarray}
Proposition 3 is easily proved by using constraints (\ref{Rj1_meq2}), (\ref{Rj2_meq2}) and system (\ref{dubrt_meq2}).
Dynamics on $n$ of the functions $\gamma$, $\delta$ is determined by the system of discrete equations
\begin{eqnarray}\label{Pr_sump1}
\gamma_{n+1}+\delta_{n+1}=\gamma_{n}+\delta_{n}-\nonumber\\
\frac{1}{u_n(\delta_{n}-\gamma_{n})}\left(\frac{\prod_{i=1}^{5}(\gamma_{n}-e_i)}{(\gamma_n-\gamma_{n-1})(\gamma_n-\delta_{n-1})}-\frac{\prod_{i=1}^{5}(\delta_{n}-e_i)}{(\delta_n-\gamma_{n-1})(\delta_n-\delta_{n-1})}\right),
\end{eqnarray}
\begin{eqnarray}\label{Pr_prp1}
\gamma_{n+1}\delta_{n+1}=\gamma_{n}\delta_{n}-\nonumber\\
\frac{1}{u_n(\delta_{n}-\gamma_{n})}\left(\frac{\delta_{n}\prod_{i=1}^{5}(\gamma_{n}-e_i)}{(\gamma_n-\gamma_{n-1})(\gamma_n-\delta_{n-1})}-\frac{\gamma_n\prod_{i=1}^{5}(\delta_{n}-e_i)}{(\delta_n-\gamma_{n-1})(\delta_n-\delta_{n-1})}\right).
\end{eqnarray}
We hope that the overdetermined system of equations (\ref{dubrt3}), (\ref{dubrt4}), (\ref{Rj}) can be used for constructing explicit solutions to the Volterra chain.
When $ m = 1 $, this is done below. However in generic case, the problem of solving these equations needs further investigation.
\section{Particular solutions to the Volterra chain.}
In this section, in a particular case we illustrate the application of the scheme above.
\subsection{Construction of soliton solutions.}
Let us concentrate on the simplest nontrivial case $m=1$ by taking
\begin{equation}\label{parametrization_meq1}
P_{n}(\lambda)=\lambda-\gamma_n,\quad c(\lambda)=-\prod_{i=1}^{3}(\lambda-e_i).
\end{equation}
Now a system of equations (\ref{paramLaxt}), (\ref{paramLaxn}), (\ref{un}) turns into
\begin{eqnarray}\label{paramLaxt_meq1}
\dot{\gamma}_n=u_n(\gamma_{n+1}-\gamma_{n-1}),
\end{eqnarray}
\begin{eqnarray}\label{paramLaxn_meq1}
u_n(2\lambda-\gamma_{n+1}-\gamma_{n})(2\lambda-\gamma_{n}-\gamma_{n-1})=\lambda(\lambda-\gamma_{n})^2- \prod_{i=1}^{3}(\lambda-e_i),
\end{eqnarray}
\begin{equation}\label{un_meq1}
u_{n}=-\frac{1}{2}\gamma_{n}+\frac{1}{4}\sum_{i=1}^{3}e_i.
\end{equation}
Similarly formulae (\ref{dubrn}), (\ref{dubrt1}) read as
\begin{equation}\label{dubrn_meq1}
(\gamma_{n}-\gamma_{n+1})(\gamma_{n}-\gamma_{n-1})=R(\gamma_{n}),
\end{equation}
\begin{eqnarray}\label{dubrt_meq1}
\dot{\gamma}_{n}=\left(-\frac{1}{2}\gamma_{n}+\frac{1}{4}\sum_{i=1}^{3}e_i\right)(\gamma_{n+1}-\gamma_{n-1}),
\end{eqnarray}
where
\begin{equation}
R(\gamma_{n})=-\frac{\prod_{i=1}^{3}(\gamma_{n}-e_i)}{-\frac{1}{2}\gamma_{n}+\frac{1}{4}\sum_{i=1}^{3}e_i}.
\end{equation}
{\bf Proposition 4.} The overdetermined system of the equations (\ref{dubrn_meq1}), (\ref{dubrt_meq1}) is compatible.
The proposition is proved by a direct computation.
{\bf Corollary of Proposition 4}. A common solution of (\ref{dubrn_meq1}), (\ref{dubrt_meq1}) exists, it produces due to (\ref{un_meq1}) a particular solution to the Volterra chain.
Obviously for the fixed value of $n$ the variables $\gamma_{n}$, $\gamma_{n-1}$ are found from the following system of the ordinary differential equations:
\begin{eqnarray}\nonumber
\dot{\gamma}_{n}=\left(-\frac{1}{2}\gamma_{n}+\frac{1}{4}\sum_{i=1}^{3}e_i\right)\left(\gamma_{n-1}-\gamma_{n}-\frac{R(\gamma_{n})}{\gamma_{n-1}-\gamma_{n}}\right),
\end{eqnarray}
\begin{eqnarray}\nonumber
\dot{\gamma}_{n-1}=\left(-\frac{1}{2}\gamma_{n}+\frac{1}{4}\sum_{i=1}^{3}e_i\right)\left(\gamma_{n}-\gamma_{n-1}-\frac{R(\gamma_{n-1})}{\gamma_{n-1}-\gamma_{n}}\right).
\end{eqnarray}
The discrete equation (\ref{dubrn_meq1}) specifies the dependence on the discrete variable $n$. The order of (\ref{dubrn_meq1}) is easily reduced since it admits an integral of motion:
\begin{equation}\label{dubrn2_meq1}
u_n(\gamma_{n}+\gamma_{n+1})(\gamma_{n}+\gamma_{n-1})=e_1e_2e_3
\end{equation}
that evidently follows from (\ref{paramLaxn_meq1}) with $\lambda=0$.
Actually (\ref{dubrn_meq1}) provides an example of the discrete integrable map admitting a symmetry (\ref{dubrt_meq1}) (about discrete maps see \cite{Veselov2}, \cite{Nijhoff}, \cite{Gubbiotti} and references therein). After some elementary transformations we get
\begin{eqnarray}\label{eq1_Gamma1}
\gamma_{n+1}=&-\frac{\gamma^2_{n}-(e_1+e_2+e_3)\gamma_{n}+e_1e_2+e_1e_3+e_2e_3}{2\gamma_{n}-e_1-e_2-e_3}\nonumber\\
&-\frac{S}{2\gamma_{n}-e_1-e_2-e_3},
\end{eqnarray}
\begin{eqnarray}\label{eq1_Gammat}
\gamma_{n,t}=\frac{S}{2},
\end{eqnarray}
where
\begin{eqnarray}\nonumber
S=\sqrt{(\gamma^2_{n}-e_1e_2-e_1e_3-e_2e_3)^2+4e_1e_2e_3(2\gamma_{n}-e_1-e_2-e_3)}.
\end{eqnarray}
It is easily checked that the overdetermined system (\ref{eq1_Gamma1}), (\ref{eq1_Gammat}) is consistent. Our goal now is to find a solution of the system and then due to the formula (\ref{un_meq1}) construct the corresponding solution of the Volterra chain.
Let us first consider the degenerate case when $e_3=e_2$. We fix the branch of the root $S$ assuming that for $\gamma\mapsto\infty$ its value is $\gamma^2$ and then rewrite the formulas (\ref{eq1_Gamma1}), (\ref{eq1_Gammat}) as follows
\begin{eqnarray}\label{gammat_solitoncase}
\gamma_{n,t}=\frac{\varepsilon(n)}{2}(e2-\gamma_{n})\sqrt{\gamma^2_{n}+2e_2\gamma_{n}+e_2^2-4e_1e_2},
\end{eqnarray}
\begin{eqnarray}\label{gammanp1_solitoncase}
\gamma_{n+1}=&-\frac{\gamma^2_{n}-(2e_2+e_1)\gamma_{n}+e_2^2+2e_1e_2}{2\gamma_{n}-e_1-2e_2}\nonumber\\
&+\frac{\varepsilon(n)(\gamma_{n}-e2)\sqrt{\gamma^2_{n}+2e_2\gamma_{n}+e_2^2-4e_1e_2}}{2\gamma_{n}-e_1-2e_2}.
\end{eqnarray}
It is convenient to return to the field variable $u_n(t)$ in the equations (\ref{gammat_solitoncase}), (\ref{gammanp1_solitoncase}). Due to (\ref{un_meq1}) we obtain:
\begin{eqnarray}\label{up1}
u_{n+1}&=\frac{8e_1u_n+16e_2u_n+e_1^2-4e_1e_2-16u_n^2}{32u_n}+\nonumber\\
& \, \frac{\varepsilon(n)(e_1-4u_n)\sqrt{16u_n^2-8e_1u_n-32e_2u_n+e_1^2-8e_1e_2+16e_2^2}}{32u_n},
\end{eqnarray}
\begin{eqnarray}\label{ut}
u_{n,t}=\frac{\varepsilon(n)(e_1-4u_n)\sqrt{16u_n^2-8e_1u_n-32e_2u_n+e_1^2-8e_1e_2+16e_2^2}}{16}.
\end{eqnarray}
We find in a standard way the solution of (\ref{ut}):
\begin{eqnarray}\label{u_ob}
u_n(t)=\frac{(e_2^2-w^2)\left(\frac{e_2-w}{e_2+w}e^{w(c(n)+\varepsilon(n)t)}-1\right)\left(\frac{(e_2+w)^2}{(e_2-w)^2}e^{w(c(n)+\varepsilon(n)t)}-1\right)}{4e_2\left(e^{w(c(n)+\varepsilon(n)t)}-1\right)\left(\frac{e_2+w}{e_2-w}e^{w(c(n)+\varepsilon(n)t)}-1\right)},
\end{eqnarray}
where $\omega=\pm \sqrt{e_2(e_2-e_1)}$, it is supposed that $e_2(e_2-e_1)>0$ and $c(n)$ is a function of n. Let us determine the explicit form of $c(n)$ by using the equation (\ref{up1}).
Since it depends on the choice of $\varepsilon(n)$ we consider separately all of the possible cases $\varepsilon=1$, $\varepsilon=-1$, $\varepsilon=(-1)^n$.
Assume that $\varepsilon=1$ and substitute (\ref{u_ob}) into equation (\ref{up1}). Then we get immediately that $c(n)$ solves one of the following equations
\begin{eqnarray*}
(e_2-w)e^{wc(n+1)}-(e_2+w)e^{wc(n)}=0, \label{eq1_cn}\\
(e_2+w)e^{wc(n+1)}-(e_2-w)e^{wc(n)}=0, \label{eq2_cn}
\end{eqnarray*}
which obviously imply:
\begin{eqnarray}
c(n)=\frac{1}{w} \left(n\,ln\left(\frac{e_2+w}{e_2-w}\right)+c_1+i\pi\right),\label{cn_11}\\
c(n)=\frac{1}{w} \left(n\,ln\left(\frac{e_2-w}{e_2+w}\right)+c_2+i\pi\right),\label{cn_12}
\end{eqnarray}
respectively. Here we request that $c_1$, $c_2$ are arbitrary constants, we put summand $i\pi$ in order to change the sign before the exponentials in (\ref{u_ob}). Direct computation convinces us that $u_n(t)$ defined in (\ref{u_ob}) solves the Volterra chain if $c(n)$ is given by (\ref{cn_11}):
\begin{eqnarray}\label{u_cn11}
u_n(t)=\frac{(e_2^2-w^2)\left(e^{(n-1)\theta+wt+c_1}+1\right)\left(e^{(n+2)\theta+wt+c_1}+1\right)}{4e_2\left(e^{n\theta+wt+c_1}+1\right)\left(e^{(n+1)\theta+wt+c_1}+1\right)},
\end{eqnarray}
where $\theta=ln\left(\frac{e_2+w}{e_2-w}\right)$.
Obviously these solutions coincide with Manakov's solutions found in \cite{Manakov} (see also \cite{Novikovbook}) up to notations. The cases $\varepsilon=-1,$ $\varepsilon=(-1)^{n}$ lead to the same solution (\ref{u_cn11}).
\subsection{Construction of elliptic solutions.}
Here we focus on a pair of compatible equations (\ref{eq1_Gamma1}) and (\ref{eq1_Gammat}). For the simplicity we introduce notations $\lambda_1=-e_1-e_2-e_3$, $\lambda_2=e_1e_2+e_1e_3+e_2e_3$ and $\lambda_3=-e_1e_2e_3$ and rewrite the equations as
\begin{eqnarray}\label{eq3_Gamma1}
\gamma_{n+1}=&-\frac{\gamma^2_n+\lambda_1\gamma_n+\lambda_2}{2\gamma_n+\lambda_1}-\frac{\sqrt{\gamma^4_n-2\lambda_2\gamma^2_n-8\lambda_3\gamma_n+\lambda^2_2-4\lambda_1\lambda_3}}{2\gamma_n+\lambda_1},
\end{eqnarray}
\begin{eqnarray}\label{eq3_Gammat}
\gamma_{n,t}=\frac{\sqrt{\gamma^4_n-2\lambda_2\gamma^2_n-8\lambda_3\gamma_n+\lambda^2_2-4\lambda_1\lambda_3}}{2}.
\end{eqnarray}
To solve the equation (\ref{eq3_Gammat}), we first convert the polynomial
\begin{equation}\label{G}
G(\gamma_n):=\frac{1}{4}\gamma_n^4-\frac{1}{2}\lambda_2\gamma^2_n+2\lambda_3\gamma_n+ \frac{\lambda^2_2-4\lambda_1\lambda_3}{4}=y_n^2
\end{equation}
into the Weierstrass normal form by applying the following fractionally rational transformation (for more details see the book \cite{Bateman})
\begin{equation}\label{transform}
\gamma_n= \frac{2\alpha_0\xi_n-\alpha_0 A_2-2A_1}{2\xi_n-A_2},\quad y_n=\frac{A_1\eta_n}{(\xi_n-\frac{1}{2}A_2)^2},
\end{equation}
where $A_1=-\frac{1}{4}\alpha_0^3+\frac{1}{4}\alpha_0\lambda_2+\frac{1}{2}\lambda_3$, $A_2=\frac{1}{4}\alpha_0^2-\frac{1}{12}\lambda_2$ and $\alpha_0$ is a root of the polynomial $G(\gamma_n)$.
In terms of the new variables $\xi_n$, $\eta_n$ we get an elliptic curve
\begin{equation}\label{eta}
4\xi_n^3-g_2\xi_n-g_3=\eta_n^2,
\end{equation}
where $g_2=\frac{1}{12}\lambda^2_2-\frac{1}{4}\lambda_1\lambda_3$, $g_3=-\frac{1}{216}\lambda^3_2-\frac{1}{16}\lambda_3^2+ \frac{1}{48}\lambda_1\lambda_2\lambda_3$.
As a result of the transformation (\ref{transform}) equations (\ref{eq3_Gamma1}) and (\ref{eq3_Gammat}) are reduced to the form
\begin{equation}\label{weier}
\xi_{n,t}=\sqrt{4\xi_n^3-g_2\xi_n-g_3}
\end{equation}
and
\begin{eqnarray}\label{xinp1}
\xi_{n+1}=\frac{\lambda_2}{12}+\frac{1}{4}\frac{3\lambda_1\lambda_3}{12\xi_n-\lambda_2}+\frac{1}{2}\frac{9\lambda_3^2}{(12\xi_n-\lambda_2)^2}\nonumber\\
\qquad \qquad \qquad -\frac{18\lambda_3\sqrt{4\xi_n^3-g_2\xi_n-g_3}}{(12\xi_n-\lambda_2)^2}.
\end{eqnarray}
Since the point transformation preserves the compatibility condition equations (\ref{weier}), (\ref{xinp1}) are also compatible. Now our goal is to construct common general solution to (\ref{weier}), which is obviously expressed in terms of the Weierstrass $\wp$ function
\begin{equation}\label{xin}
\xi_n(t)=\wp(t+c(n)).
\end{equation}
Actually we have to find the explicit form of the function $c(n)$ in the formula (\ref{xin}) such that it solves (\ref{xinp1}) as well.
{\bf Proposition 5.} Function $\xi_n(t)$ can be taken as
\begin{eqnarray}\label{xinwpeps1}
\xi_n(t)=\wp(t+v n+\beta).
\end{eqnarray}
{\bf Proof.} We rewrite formula (\ref{xinp1}) in the form
\begin{eqnarray}\label{addth_xi}
\xi_{n+1}=\frac{1}{4}\left[\frac{\sqrt{4\xi_n^3-g_2\xi_n-g_3}-\frac{\lambda_3}{4}}{\xi_n-\frac{\lambda_2}{12}}\right]^2-\xi_n-\frac{\lambda_2}{12}.
\end{eqnarray}
It is remarkable that (\ref{addth_xi}) looks very similar to the addition theorem for the Weierstrass $\wp$ function
\begin{eqnarray}\label{addth}
\wp(u+v)=\frac{1}{4}\left[\frac{\wp'(u)-\wp'(v)}{\wp(u)-\wp(v)}\right]^2-\wp(u)-\wp(v),
\end{eqnarray}
where $\wp'(u)$ is the derivative of the function $\wp(u)$ (see \cite{Bateman}).
We show that for an appropriate choice of the parameters these two equations coincide completely. In (\ref{addth}) we set $c(n)=vn+\beta$, $u=t+v n+\beta$, where $v$ is found from the equation $\wp(v)=\frac{\lambda_2}{12}$. It is easily verified that $\wp'(v)$ can be chosen in such a way $\wp'(v)=\sqrt{4\left(\frac{\lambda_2}{12}\right)^3-g_2\frac{\lambda_2}{12}-g_3}=\frac{\lambda_3}{4}$. Then actually we get coincidence of the equations (\ref{addth_xi}), (\ref{addth}) that implies that function (\ref{xinwpeps1}) solves equations (\ref{weier}) and (\ref{xinp1}). That completes the proof.
Now it remains to express $\gamma_n(t)$ through $\xi_n(t)$ and then write down a solution $u_n(t)$ of the Volterra chain.
For known $\xi_n(t)$ we can recover $\gamma_n(t)$ due to the M$\ddot{o}$bius transformation (\ref{transform})
\begin{eqnarray}
\gamma_n(t)=\frac{2\alpha_0\wp(z)-2A_1-\alpha_0A_2}{2\wp(z)-A_2},\label{transformMobius}
\end{eqnarray}
where $z=t+v n+\beta$. Then solution of the Volterra chain is given by (\ref{un_meq1}), i.e.
\begin{eqnarray}
u_n(t)=\frac{r_2\wp(z)+r_1}{\wp(z)+r_3},\label{solution_u}
\end{eqnarray}
where
\begin{eqnarray*}
&r_1=\frac{1}{2}A_1+\left(\frac{1}{4}\alpha_0+\frac{1}{8}\lambda_1\right)A_2,\\
&r_2=-\frac{1}{2}\alpha_0-\frac{1}{4}\lambda_1, \quad r_3=-\frac{1}{2}A_2.
\end{eqnarray*}
In order to transform (\ref{solution_u}) to an accepted form we express $u_n(t)$ in terms of the Weierstrass $\sigma$ function. According to the well known theorem \cite{Bateman} we have
\begin{eqnarray}
u_n(t)=C\frac{\sigma(z-\mu)\sigma(z+\mu)}{\sigma(z-\theta)\sigma(z+\theta)},\label{solution_usigma}
\end{eqnarray}
where $\mu$ and $\theta$ are found from the equations
\begin{eqnarray}
\wp(\mu)=-\frac{r_1}{r_2}, \quad \wp(\theta)=-r_3. \label{solution_muteta}
\end{eqnarray}
We find $C$ due to the fact that $\wp(z)$ has a pole at the point $z=0$:
\begin{eqnarray}
C=r_2\frac{\sigma^2(\theta)}{\sigma^2(\mu)}. \label{C}
\end{eqnarray}
Let us apply to (\ref{solution_usigma}) the point transformation $u\rightarrow r_2 u$, $t\rightarrow r_2 t$ preserving the Volterra chain and bring the found $u_n(t)$ to the form (cf. \cite{Veselov})
\begin{eqnarray}
u_n(t)=\frac{\sigma^2(\theta)}{\sigma^2(\mu)}\frac{\sigma(r_2t+vn+\beta-\mu)\sigma(r_2t+vn+\beta+\mu)}{\sigma(r_2t+vn+\beta-\theta)\sigma(r_2t+vn+\beta+\theta)}.\label{ves-solution_usigma}
\end{eqnarray}
\section*{Conclusions}
The article proposes an alternative way for constructing exact solutions for integrable models based on the concept of generalized invariant manifolds. We have illustrated the scheme with an example of the Volterra chain. The first step is to derive an appropriate generalized invariant manifold to the chain, containing at least two constant parameters that are not removed by a point transformation. Actually such a manifold determines a nonlinear Lax pair. Next we assume that this Lax pair has a solution polynomially depending on one of the parameters $\lambda$. We also require that the other parameter is a polynomial on $\lambda$, as well. It turned out that under these requirements the solution of the nonlinear Lax pair satisfies to an overdetermined system of differential and difference equations. In the case of small degree polynomials, in more details studied in the article, we arrive at a pair of consistent equations. One of the equations is differential, it defines an elliptic function and the other one is difference, it coincides with the addition theorem for that elliptic function. In the case of arbitrary $m$ system of equations (\ref{dubrt3}), (\ref{dubrt4}) with additional constraint (\ref{Rj}) are derived. However, the problem of solving these equations by the methods developed in \cite{DubrovinMatveevNovikov} (see also \cite{Veselov2}, \cite{Smirnov}, \cite{Bobenko}) needs further investigation.
\subsection*{Acknowledgements}
We thank M.V.Pavlov for his interest to the work.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,997 |
Q: How to prevent Android Studio 2.0 from running on same device With the new update of Android Studio, whenever I rebuild my app, it always launches it on the same emulator/device. In order for me to switch devices, I have to shutdown the emulator/disconnect the device, so that it brings up the device selection screen.
Is there any way around this?
A: For me I had to press the red stop button in order to stop the "instant run" mode which automatically runs your app on the same device.
Once I press the stop button I can run the app again and it will bring up the device chooser dialog.
A: Just do this:
Select this menu:
And then uncheck the "Use the same device.." option:
Uncheck thi option:
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,078 |
INDIA – Successful attack against paramilitary soldiers
On 4th of April the PLGA (People Liberation Guerrilla Army) conducted a successful action against the reaction! Four Indian paramilitary soldiers have been killed and others have been wounded. That took place in Kanker in the district of Chhattisgarh, when the soldiers were combing the forest with regard to the elections on 18th of April. The Indian media reported that there was an exchange of gunfire, but the rebels escaped.
Also in district of Chhattisgarh, near the village of Chameda, on the 5th of April, the 211st battalion of the PLGA had shot change and killed one police officer and another was seriously injured. The forces of reaction were conducting a search operation for guerrillas, when they were ambushed by the Maoist.
The CPI (maoist) calls for the boycott of the election which are going to take place in three phases, on 11th, 18th and 23rd of April!
#India #CPImaoist
People's War
PHILIPPINES - Celebration of 52 years Communist Party
INDIA - News on People's War
PHILIPPINES - New Year's statement from the Communist Party of the Philippines | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,963 |
_To Our Lady of Guadalupe, in the confidence that guided by her,
the people of the Christian Hemisphere will work together
to encounter her son, Jesus Christ_.
# CONTENTS
_Acknowledgments_
_Introduction_
_Part I:_ _Approaching an Apparition_
I An Apparition of Reconciliation and Hope
II The Image of the Mother
III Message of Truth and Love
IV A Multifaceted Love
_Part II:_ _Christ, a Life-Changing Event_
V Liberated by Love
VI A Call to Conversion
VII Mary and the Church
_Part III:_ _Unified in Dignity_
VIII The Gift of Vocation
IX The Face of the Hidden Christ
X The Christian Hemisphere
_Appendix A:_ _The Nican Mopohua_
_Appendix B:_ _Chronology of Guadalupan Events_
_Appendix C:_ _Prayers_
_Notes_
_Bibliography_
# Acknowledgments
The authors wish to acknowledge the contributions of the many people who contributed to the completion of this book. Among them are His Eminence Cardinal Norberto Rivera Carrera, Archbishop of Mexico, who so graciously allowed Msgr. Chávez to work on this book and a variety of other Guadalupe projects in the United States; Trace Murphy and the staff at Doubleday, who so patiently and carefully guided us through the process of writing and editing this book; Joe Tessitore, who has assisted us with many details of this project; the members of the Instituto Superior de Estudios Guadalupanos; and especially Maureen Hough, Jennifer Daigle, Andrew Walther, and Luis Guevara of the Knights of Columbus, whose work on so many aspects of this project is greatly appreciated.
# Introduction
# _Mother of the Civilization of Love_
## TWO NEW EVANGELISTS
The genesis of this book occurred on July 31, 2002, the day Pope John Paul II canonized Juan Diego Cuauhtlatoatzin in the Basilica of Our Lady of Guadalupe in Mexico City. We were both present that day in the basilica, but we had not yet met. One of us was participating in the liturgical event that he had worked to achieve for more than a decade as postulator of the Cause for Canonization of Juan Diego. The other had traveled to the basilica eighteen months earlier for his installation Mass as the head of the world's largest organization of Catholic laymen, the Knights of Columbus. Both of us were deeply touched by our experience that day in Mexico City, and both of us realized we had witnessed one of the most profound events in the Catholic Church during John Paul II's pontificate and indeed during our own lifetimes, an event that would give deep and lasting hope to the Catholic Church in North America.
This may strike many as an extraordinary claim; after all, John Paul II is now regarded universally as one of the greatest popes in the two-thousand-year history of the Catholic Church. As pope, he canonized and beatified hundreds of people, wrote numerous encyclicals on theological, moral, and social topics, and commissioned the _Catechism of the Catholic Church_ , the first definitive work of its kind in more than four hundred years. He brought interreligious dialogue to new and unexpected levels while guiding the Church into the new millennium with the focus of hope in Christ. Beyond the Church, he changed the political map of Europe and the very course of history by helping to liberate nations trapped behind the Iron Curtain during the Cold War and aiding in their cause for self-determination within the Soviet Union. Beyond Europe's borders, his concerns for the poor, the disadvantaged, and the war-torn brought a greater commitment to human rights and democracy, especially to Latin America and Africa. But in Mexico that day, as he knelt and prayed awhile before the image of Our Lady of Guadalupe after the ceremony, it was clear that he did not want to leave; when he rose to leave, he entrusted all people to the intercession of the newest saint in the Church. He had not only canonized a man of the past but also given our continent a saint for the future.
Yet, early in John Paul's pontificate, Our Lady of Guadalupe in Mexico was an important, perhaps even indispensable, presence. In an interesting way, John Paul II's first invitation to the basilica was not intended for him; the Latin American Bishops Conference had extended the invitation to his predecessor the month before, and it was only his predecessor's death that opened this opportunity to John Paul II. In a telling way, it was John Paul II's determined desire to pray at the Basilica of Our Lady of Guadalupe and to personally engage in the meetings regarding the future of the Church in America that caused him to accept the invitation his predecessor had declined. (Twenty-three years later, Mexico would see this same determination; shortly before his trip for Juan Diego's canonization, John Paul II met with his medical specialists, who advised against his making the trip. But at the end of the consultation, John Paul II thanked them for their concern and concluded the meeting by saying: "I will see you in Mexico.") Later, John Paul II would reflect on his first visit to Mexico, recalling that "to some degree, this pilgrimage inspired and shaped all of the succeeding years of my pontificate."
If John Paul's pilgrimage to Mexico shaped the rest of his life as the universal pastor of the Church, his choice to visit Mexico first and his words commending Juan Diego as an evangelist expressed a new importance and new understanding of the Church in the Americas. He recognized the Americas as a hemisphere with a unique, rich Catholic history, and thus as a hemisphere with a unique, rich place in the future of the Church, a hemisphere with great ability to respond to and benefit from a renewed living out of the Gospel of love seen in the witness of the saints. It was in this context that a few months later, the cause for Juan Diego's canonization was officially opened and the Church in the Americas was reexamined and given a new focus: the new evangelization.
## NEW SAINT, RENEWED DEVOTION
The story of St. Juan Diego is, of course, the story of Our Lady of Guadalupe. The event of his canonization cannot be understood apart from the events of her appearance. As with any apparition claim, every detail of the Guadalupan accounts must be examined: each word spoken, each miraculous or extraordinary event that deviates from the everyday, the sequence of events, the character of the people involved, their reactions to the event, their lives afterward, and especially any lingering miraculous effect. For this, we begin with Antonio Valeriano's _Nican Mopohua_ , an account of the Guadalupan apparitions in 1531, the earliest extant edition of which is currently housed in the New York Public Library. The historical record suggests that Valeriano may have derived the information in the _Nican Mopohua_ directly from Juan Diego himself, writing it down sometime before Juan Diego's death in 1548 and within two decades of the apparition. Besides this significant work, numerous historical records recall in varying degrees of detail the Guadalupan apparitions, the miraculous image, the church on Tepeyac hill, and Juan Diego's own life; some of the most substantial works include the _Nican Motecpana_ , the _Información de 1556_ , and the _Informaciones Jurídicas de 1666_. Other items composing the complex record of the Guadalupan event include written accounts, artwork, recorded oral testimonies, investigations, wills, and other works.
Because Juan Diego would be the first Mexican indigenous saint of that time and place, the canonization process demanded extensive research, requiring a grasp of both the history of colonization in New Spain and pre-Colonial culture and religion. Contemporary scholars, historians, and anthropologists specializing in the culture and history of Mexico's Indian people were consulted, and nearly four thousand documents related to Our Lady of Guadalupe were reviewed. Ultimately, the knowledge and insights from such research have revealed the profound relevance and symbolic richness of the apparitions and the miraculous image on Juan Diego's _tilma_ (a cloak-like garment), showing how the Guadalupan event conveys in the language and culture of the Indians a message of hope and love.
While the facts regarding Colonial Mexico cannot be changed, the perspective advocated by historians and even the public at large has changed. Unlike many scholars of the nineteenth and early twentieth centuries, contemporary biographers and historians often highlight the Spanish conquest and occupation of Mexico as a volatile period of spiritual repression, conflict, and violence. By bringing to light the complexity of this period, contemporary research has played an enormous role in helping us to better comprehend—and even test the veracity of—the Guadalupan apparitions. But unfortunately, while the idea of "conversion by the sword" is now familiar, some people may view the Guadalupan apparition and devotions as a mere by-product of colonization: as a strategic devotion fabricated by missionaries seeking to convert or pacify the Indians with a Christian-Aztec story, or as a subversive devotion adopted by Indians who were confused or sought to preserve elements of Aztec religion with a façade of Christianity. Undoubtedly, the Guadalupan devotions were a cause of concern and confusion at some times, but for us this should not be surprising, considering how even today, in the Information Age, we often encounter mixed reports even on less extraordinary events. While in this book we wish to do more than judge and debate about the Catholicity of Guadalupan devotees, nevertheless it is perhaps necessary to address some generalizations about the devotion that often sidetrack readers from the religious significance of the apparition's expression of the Gospel.
First, to write off the rise of Guadalupan devotion to manipulation and misunderstanding is not only simplistic but also historically incongruous regarding a politically and religiously complex situation. Among the missionaries, there was no unified front encouraging the apparitions, as many missionaries doubted and even tried to suppress the Indians' new devotion to the Guadalupan Virgin. Furthermore, while the missionaries desired conversions, their distrust of the Indians' Catholicity verged on the scrupulous, even by modern standards; these same missionaries, some of whom were sophisticated _letrados_ (theologians) in Spain, were known to hold off giving Indians the sacraments and to eliminate symbolic elements of sacraments that were too similar to Aztec rites solely in order to keep the distinctiveness of the Christian faith obvious. That is, although oversight may have occurred, purposeful theological contamination, deception, and obfuscation were largely out of character. Additionally, there was a whole range of converts among the Indians, including many who completely forsook their indigenous religious practices—but not culture—for a Christian way of life. What is more, their life as Christians went beyond practices or rites of belonging, such as baptism, to include catechesis.
Likewise, the rise of Guadalupan devotion cannot be explained as a devotion taken up to appease Christianizing government authorities; after all, at the time of the apparition, many of the Spanish authorities in Mexico were themselves incurring excommunication, caring less for Christian life and evangelization than for their greed-ridden pursuit of political gains. The fact is, while the _people_ in Mexico were involved in both political and religious changes, the _Guadalupan devotion_ was not used politically until the devotion arrived in Spain, when an admiral in the royal Spanish fleet took up the image of Our Lady of Guadalupe in the battle of Lepanto and attributed the subsequent victory to her intercession. Moreover, even though in later Mexican history the image and devotion were appropriated to serve various political and economic causes (notably during the 1810 War of Independence), the original meaning and message of Our Lady of Guadalupe transcends the flaws and purposes of those who have turned to her.
For some, Juan Diego may seem yet another individual "divinized" in answer to the universal human desire for heroes to look up to. This, too, reflects another trend of demythologizing the very men once lauded as great heroes. In this light, what is striking is how Juan Diego's role in the apparition—even as passed down in testimonies—is already rather humble. While he was called to a significant and meaningful task, he was not called to a "great" task by any measure of earthly grandeur. His role in the apparition itself—his first and greatest claim to renown—was not a call to conquer lands but a simple invitation to intercede on behalf of one person by communicating a request to another. In essence, it was to vouch for and trust in another person. This simple act is the kernel of meaning and truth that is served, and not obscured, by the grandeur of the divine visitations, the healings, and the miraculous tilma. It was by answering this simple invitation that Juan Diego set himself apart; his was a gesture of humility, communication, advocacy, and trust, a gesture that we perform in less miraculous ways and situations every day of our lives. It is one of the most fundamental gestures of our humanity and the foundation of any society that wishes to live beyond selfish utilitarianism.
## THE NEWNESS OF GUADALUPE
Consequently, although we believe the appearance of Our Lady of Guadalupe to be a historical fact, we do not think that it should be consigned only to the pages of history books. In fact, in a unique way, the full radicalness of Our Lady of Guadalupe's apparition can only be understood fully now, when Catholicism's most expressed model for society is a Civilization of Love and its greatest explication of human dignity is the Theology of the Body. For at the time, the violence institutionalized in Aztec religion was not the only place where harsh practices could be found: the European justice system employed in Colonial Mexico and many of the "standards" of holiness among Catholics often included severe punishments and harsh penitential practices that still make us uneasy, even if the practices were less violent and more theologically different from the Aztec human sacrifices. What is notable is that this harshness is not corroborated in the words of the Virgin. In fact, while other Marian apparitions (such as those at Fatima and Lourdes) included words of admonishment or even warnings, Our Lady of Guadalupe's only words of spiritual guidance are her gentle but persistent reminders to Juan Diego about love: a love that can be trusted, a love that gives dignity, a love that is personal. If we are to see in her words an answer to a spiritual problem, the spiritual problem it answers is a lack of love and a lack of understanding about love as relationship rather than as practice. The Guadalupan message is, in its originality, a spiritual education, an education in love.
Today, as life is often characterized by a lack of love and by misunderstandings and misgivings about love, her message is one to take to heart. For this reason, like John Paul II, we think that one of the greatest influences of Our Lady of Guadalupe upon the history of the Western Hemisphere may still be before us. In the sixteenth century, Our Lady of Guadalupe became an expression of hope and unity for millions throughout the Americas. We are convinced that Our Lady of Guadalupe's message is _today_ capable of being not only an expression but a true catalyst of hope and unity for millions more throughout North America and the world.
In the Christian sense, this hope and unity are spread through evangelization—that is, through helping one another to find in Jesus Christ the "adequate dimension" of our own life. A clear picture of how Christians in this hemisphere can approach this task spiritually can be seen in the triptych that Benedict XVI presented in Aparecida, Brazil, on his first apostolic journey to the Western Hemisphere, depicting St. Juan Diego "evangelizing with the Image of the Virgin Mary on his mantle and with the Bible in his hand" and inscribed below with the phrase "You shall be my witnesses." To evangelize in the future is to evangelize from and through these first witnesses of Christianity.
Historically in our continent, Mexico was not the only country to be changed by this Marian evangelization. As later missionaries left Mexico for the neighboring countries in the hemisphere, including to the lands of the future United States, their evangelization was defined by their devotion to Our Lady of Guadalupe. As John Paul II wrote in his apostolic exhortation _Ecclesia in America_ :
The appearance of Mary to the native Juan Diego on the hill of Tepeyac in 1531 had a decisive effect on evangelization. Its influence greatly overflows the boundaries of Mexico, spreading to the whole continent... [which] has recognized in the mestiza face of the Virgin of Tepeyac, "in Blessed Mary of Guadalupe, an impressive example of a perfectly enculturated evangelization."
For this reason, the pope continued, Our Lady of Guadalupe is venerated throughout the Western Hemisphere as "Queen of all America," and he encouraged that her December 12 feast day be celebrated not only in Mexico but throughout the hemisphere.
Just as Mary's enculturated evangelization overflowed Mexico's borders, so it overflows the confines of the era and the culture of the apparition. For this reason, she is not only the "Patroness of all America" but the "Star of the first and new evangelization" who will "guide the Church in America... so that the new evangelization may yield a splendid flowering of Christian life." Our Lady of Guadalupe is more than an event; she is a person. As part of her continuing witness to Christ, she continues to aid the men and women of the Western Hemisphere and lead them to a greater encounter with Christ.
While two oceans may delineate our hemisphere and define us as a single community, the solidarity of the Christian life proposed by Our Lady of Guadalupe brings us to a greater solidarity, a global solidarity, when she leads us to a greater encounter with "the uniqueness of Christ's real presence in the Eucharist." Through the Eucharist, believers of all nations and cultures find themselves on a path of communion. This communion finds its ultimate source and summit in the communion within the Holy Trinity. As John Paul II wrote:
Faced with a divided world which is in search of unity, we must proclaim with joy and firm faith that God is communion, Father, Son and Holy Spirit, unity in distinction, and that he calls all people to share in that same Trinitarian communion. We must proclaim that this communion is the magnificent plan of God the Father; that Jesus Christ, the Incarnate Lord, is the heart of this communion, and that the Holy Spirit works ceaselessly to create communion and to restore it when it is broken.
At Aparecida, Benedict XVI raised this conviction with even greater eloquence and at the same time emphasized its transformative power: "Only from the Eucharist will the civilization of love spring forth which will transform Latin America and the Caribbean, making them not only the Continent of Hope, but also the Continent of Love!"
To venerate Our Lady of Guadalupe as Patroness of the Americas and Star of the first and new evangelization is to venerate her precisely as a Eucharistic woman, a woman through whom Christ came to humanity, a woman who experienced a unique closeness with the Holy Trinity. By leading millions more to her Son, and especially to her Son's real presence in the Eucharist, she will guide the people of the Western Hemisphere to a greater unity whose source is itself Trinitarian communion. For her love surpasses herself, and leads us to the source of love, a Source which demands from us and enables us to love our neighbor without reservation, without hesitation, without borders. For this reason, Our Lady of Guadalupe should also be venerated under the title Mother of the Civilization of Love.
According to tradition, after approving the patronage of Our Lady of Guadalupe over New Spain in 1754, Pope Benedict XIV quoted Psalm 147, saying, "God has not done anything like this for any other nation." We may never understand the full uniqueness of this apparition. But through the devotion to Our Lady of Guadalupe, we can expect to see the beauty and power of this event in the transformation of our lives and blossoming of our communities and ultimately our continent.
# PART I
# Approaching an Apparition
# I.
_An Apparition of Reconciliation and Hope_
_I am the handmaid of the Lord_.
–WORDS OF MARY IN THE GOSPEL OF LUKE
## JUAN DIEGO
The Basilica of Our Lady of Guadalupe has witnessed many grand and reverent ceremonies through the centuries, but few like the Mass celebrated July 31, 2002 for the canonization of St. Juan Diego. Thousands of people attended the three-hour canonization Mass, while thousands more watched on screens set up just outside the basilica and throughout Mexico City. Entire boulevards were closed down to make room for pilgrims. Among those attending the Mass was Mexico's president, Vicente Fox, whose presence marked a historic occasion: the first time a Mexican chief executive attended a papal Mass. Priests read from the Bible in Spanish and in the indigenous language Náhuatl, and while a portrait of Juan Diego was carried to the altar, Indians in colorfully plumed headdresses danced up the main aisle of the basilica.
Twelve years earlier, Juan Diego's beatification had been the result of rigorous historical research and examination into his life and later testimonies about him. Through this research, evidence of early devotion to Juan Diego and recognition of his saintliness dating back to the sixteenth century was uncovered. With this, the approval of an immemorial cultus was granted and the requirements for beatification were met. For Juan Diego's canonization, however, something more was needed: a miracle. But as it happened, on the same day that John Paul II was celebrating Juan Diego's beatification Mass, that miracle happened.
On May 3, 1990, in Mexico City, nineteen-year-old Juan José Barragán suffered from severe depression and, wanting to commit suicide, he threw himself from the balcony of his apartment, striking his head on the concrete pavement thirty feet below, despite his mother's frantic attempts to hold onto him as she cried out to Juan Diego for help. The young man was rushed to the nearby hospital, where the doctor there noted his serious condition and suggested that the boy's mother pray to God. To this, the young man's mother replied that she already had prayed for Juan Diego's intercession. For three days, examination and intensive care continued, and physicians diagnosed a large basal fracture of the skull—a wound that normally would have killed at the moment of impact, and even now destroyed any hope of survival or repair. Given the mortal nature of the wounds, on May 6 all extraordinary medical support was ceased, and young Juan José's death was thought to be imminent. But that same day, Juan José sat up, began to eat, and within ten days was entirely recovered, with no debilitating side-effects, not even so much as a headache. In the scans, the doctors could see clear evidence of the life-threatening fracture, but to their surprise they noticed that the bone was mended, with the arteries and veins all in place. Astonished, they requested more tests by specialists for second opinions, only to have their original assessment confirmed. Impossible, unexplainable, it was declared a miracle.
As enormously as it changed Juan José's life, the miracle affirmed the life of another: an Indian convert born five centuries earlier at the height of the Aztec Empire, Juan Diego Cuauhtlatoatzin. Ultimately, it was this miracle that led to Juan Diego's canonization.
Juan Diego was born around the year 1474 in Cuautitlán of the Texcoco kingdom, part of the Triple Alliance with the Aztec Empire. Known by his indigenous name Cuauhtlatoatzin, meaning "eagle that speaks," he belonged to the Chichimecas, a people that had assumed Toltec culture whose wise men had reached the conception of only one God. As a _macehual_ , a middle-class commoner, he owned property through inheritance.
The first of many great changes in his life came around 1524, when the fifty-year-old Cuauhtlatoatzin and his wife requested baptism from one of the early Franciscan missionaries to Mexico and received their Christian names, Juan Diego and María Lucía. Together, they were one of the first Catholic married couples of the New World. Five years later, María Lucía died, leaving Juan Diego alone with his elderly uncle, Juan Bernardino, also a recent convert, in the town of Tulpetlac, near Mexico City.
Juan Diego's conversion had been made possible just ten years earlier, when Hernán Cortés and his men conquered the great Aztec Empire, ultimately laying waste to the Aztec capital of Tenochtitlán and its main temple, the Templo Mayor (Great Temple). Having lived in the town of Cuautitlán in the nearby kingdom of Texcoco and then in the town of Tulpetlac, Juan Diego was no doubt familiar with the Aztecs and their campaign for empire. He also would have been familiar with their religious practices, which demanded human sacrifices to sustain the Aztec gods and maintain the harmony of the cosmos. The introduction of Christianity to the New World came with the Spanish conquest of Mexico; missionaries were sent to teach the faith, and the Indians were discouraged from practicing their own religion. Human sacrifice was prohibited and temples were torn down. Despite these efforts, missionary activity in the New World met with only very modest success. It is this history of conquest and its aftermath, along with the cultural and religious heritage of the indigenous people, that constitutes a vital lens in interpreting the significance of the Guadalupan apparitions for the Colonial Indians. More than that, it shows how the apparitions at Guadalupe resolved some of the deep-seated problems posed by the Aztec religion—problems that were doubtless exacerbated in the Indians' encounter with Spanish colonialism.
After María Lucía's death, Juan Diego continued to grow in his faith; to the missionaries, who were accustomed to meager resources and unsuccessful efforts, Juan Diego's dedication to the Christian faith must have been a welcome surprise. Although there was no established church in the area, every Saturday and Sunday Juan Diego rose at dawn to walk nine miles to the nearest _doctrina_ (place of religious instruction) in Tlaltelolco, where he could attend Mass and receive instruction in the faith. At the time, Mexico City was a small island in Lake Texcoco, and so in order to attend these services in Tlaltelolco, Juan Diego would have to travel south from Tulpetlac, walk around the western side of Tepeyac hill, and then along a great causeway connecting Mexico City to the mainland.
## THE FIRST APPARITION
On one of his Saturday trips for catechesis, on December 9, 1531, when he arrived at the Tepeyac area, Juan Diego heard beautiful singing that seemed to be coming from the top of Tepeyac hill. The singing sounded like a chorus of birds, but more beautiful than the song of any birds Juan Diego had ever heard before. Juan Diego wondered as he looked eastward toward the top of the hill:
By any chance am I worthy, have I deserved what I hear? Perhaps I am only dreaming it? Perhaps I am only dozing? Where am I? Where do I find myself? Is it possible that I'm in the place our ancient ancestors, our grandparents, told us about: in the land of the flowers, in the land of corn, of our flesh, of our sustenance, perhaps in the land of heaven?
In these moments before Juan Diego encounters the first apparition of Our Lady of Guadalupe, signs of renewal are already present. In Christianity, the east, the direction of the rising sun, is often used to symbolize resurrection and renewal, themes especially evident in Juan Diego's initial words of wonderment. Drawing on images from his indigenous heritage, Juan Diego attempts to describe something of the indescribable mystery of heaven, referring to it as "the land of the flowers, the land of corn, of our flesh, of our sustenance." For the Indians, both flowers and corn held great religious and cultural significance. On one hand, flowers, like song, were evocative of the truth and were considered the only things that, as an Aztec sage once wrote, "will not come to an end;" on the other hand, corn was an essential food staple, relied upon heavily by the Aztecs and without which Aztec life would have suffered greatly.
Suddenly, the singing stopped, and a woman's voice called out to him: "Juantzin, Juan Diegotzin," the Náhuatl affectionate diminutive form of his Spanish baptismal name. Acknowledging the woman's affectionate greeting, Juan Diego ascended the hill and found himself before a beautiful woman adorned in clothing that "shone like the sun." She stood upon stones that seemed to send forth beams of light like precious jade and other jewels; the "earth seemed to shine with the brilliance of a rainbow," and the foliage had the brightness of turquoise and quetzal feathers. She asked Juan Diego where he was going, and Juan Diego replied that he was on his way to "your little house in Mexico, Tlaltelolco, to follow the things of God." Notably, he said this even before the Virgin introduced herself, thus underscoring Juan Diego's early awareness of the important relationship between Mary and the Church.
Then, speaking to Juan Diego in his native language Náhuatl and using Texcocan religious phrases, the woman introduced herself in an unmistakably clear way, saying:
I am the ever-perfect holy Mary, who has the honor to be the mother of the true God [ _téotl Dios_ ] by whom we all live [Ipalnemohuani], the Creator of people [Teyocoyani], the Lord of the near and far [Tloque Nahuaque], the Lord of heaven and earth [Ilhuicahua Tlaltipaque].
By using both the Náhuatl and Spanish words for God ( _téotl Dios_ ), the Virgin reaffirms the supremacy, oneness, and universality of God. In her humility, she speaks very little of herself, while referring to God by many titles; importantly, when she does speak of herself, she calls herself "mother" and the "ever-perfect holy Mary." This title identifies her as the Immaculate Conception, a title not officially recognized until Pope Pius IX approved it more than three centuries later, in 1854. By introducing herself in this way, Mary significantly underscores the humanity and divinity of her Son. As Christ's mother, Mary shows the humanity of her Son, since she is herself a human being; but as immaculate, she shows the divinity of her Son, who, as God, was singularly born of a sinless woman.
After introducing herself, the Virgin revealed the reason for her appearance:
I want very much that they build my sacred little house here, in which I will show Him, I will exalt Him upon making Him manifest, I will give Him to all people in all my personal love, Him that is my compassionate gaze, Him that is my help, Him that is my salvation. Because truly I am your compassionate Mother, yours and that of all the people that live together in this land, and also of all the other various lineages of men, those who love me, those who cry to me, those who seek me, those who trust in me.
The Virgin then explained to Juan Diego how she needed him to deliver her message to Friar Juan de Zumárraga, the head of the Church in Mexico City.
Within the context of European Catholicism, the first apparition makes poignantly clear the Virgin Mary's universal role as mother and her desire to bring all people closer to God through her loving intercession. Less obvious, though no less significant, is what the Virgin's request for the construction of a church would have meant to a learned Indian. For the indigenous, the temple was more than a religious building, and the establishment of a temple was more than a ceremonial religious occasion. So central was religion to indigenous culture that the temple was seen as the foundation of society. Historically, the construction of a new temple marked the inauguration of a new civilization. In fact, the Aztecs built a temple in the years immediately following their migration to the Valley of Mexico, and a common indigenous glyph, or pictogram, for a conquered people was the depiction of a temple toppling over, sometimes in flames. Thus, the Virgin's commission to Juan Diego was rich in meaning far beyond the construction of a building, and was made richer still by the fact that it had been given to an Indian.
## THE SECOND APPARITION
Juan Diego could hardly face a greater test than going to the head of the Church in Mexico, bishop-elect Friar Juan de Zumárraga. Friar Zumárraga, who had arrived in the New World no more than three years earlier, was an extremely prudent man who, like the other missionary friars, fought vigorously against the idolatry of the time; in fact, in a letter earlier in 1531, he declared that he had caused twenty thousand idols to be destroyed, and in 1529, his agents had been responsible for burning countless native codices, including those in the royal repository at Texcoco. He was particularly suspicious of supposed visions and apparitions, believing most of them to be forms of idolatrous Indian worship. Putting the bishop-elect's concerns in context, he and the other missionaries living in Mexico were confronted with a people and religion wholly strange to them. They feared that the old religion could interfere or undermine the Indians' understanding of and conversion to Christianity. Even so, Juan Diego went immediately to the friar's house, where he waited for a long time before Friar Zumárraga would see him.
Once admitted, Juan Diego told Friar Zumárraga of the apparition, but the bishop, while attentive, was skeptical of Juan Diego's story. Why would the mother of God appear to this recently converted Indian? Why would she request that a church be built on the flatland of Tepeyac hill, when the hill's peak had once held an ancient temple dedicated to the pagan goddess Coatlicue? It was a significant request, and the miraculous nature of an apparition was not to be taken lightly. Friar Zumárraga dismissed Juan Diego, telling him that he would listen more patiently to his story at another time.
Dejected by the response, Juan Diego returned to Tepeyac hill and, after recounting to the Virgin what had happened, pleaded with her to give the mission to someone more important than himself:
So I beg you... to have one of the nobles who are held in esteem, one who is known, respected, honored, have him carry on, take your venerable breath, your venerable word, so that he will be believed. Because I am really just a man from the country, I am the porter's rope, I am a back-frame, just a tail, a wing; I myself need to be led, carried on someone's back.... My Little Girl, my Littlest Daughter, my Lady, my Girl, please excuse me: I will afflict your face, your heart; I will fall into your anger, into your displeasure, my Lady Mistress.
Throughout the apparition account, these familiar and affectionate appellations reflect the indigenous form of address in which people might call one another by many titles; for example, consider how a younger boy of the nobility would greet his mother: "Oh my noble person, oh personage, oh Lady,... we salute your ladyship and rulership. How did you enjoy your sleep, and now how are you enjoying the day?"
The Virgin listened with tenderness but responded firmly:
Listen, my youngest son, know for sure that I have no lack of servants [and] messengers to whom I can give the task of carrying my breath, my word, so that they carry out my will. But it is necessary that you, personally, go and plead, that by your intercession my wish, my will, become a reality. And I beg you, my youngest son, and I strictly order you to go again tomorrow to see the bishop. And in my name, make him know, make him hear my wish, my will, so that he will bring into being, build my sacred house that I ask of him. And carefully tell him again how I, personally, the ever Virgin Holy Mary, I, who am the Mother of God, sent you as my messenger.
Certainly there were others more suitable for the task, in terms of both credibility and social status. And why was there a need for an intermediary at all? The Virgin herself could have appeared before the bishop. And yet the Virgin selected Juan Diego—a selection that reflects the words of the Virgin Mary in the Gospel, when she praises God who "has cast down the mighty from their thrones and lifted up the lowly." The Virgin's selection of a man of humble rank likewise resonates with Friar Zumárraga's own vocation as a Franciscan, an order valuing humility and renowned for its vows of poverty; in this spirit of humility, before coming to New Spain, Zumárraga had hoped to end his days living in a quiet, stable community, but was chosen instead for a prominent and demanding position in the New World. Juan Diego, too, seems to confirm the Virgin's selection; insisting that she find someone better for the task, he reveals himself as the perfect messenger, one who humbly withdraws in order to call attention to the message itself.
## THE THIRD APPARITION
When Juan Diego returned to the bishop the next day to deliver the Virgin's message, Friar Zumárraga questioned Juan Diego on many details of the apparition. This time, before sending Juan Diego away, Friar Zumárraga requested evidence that would confirm the truth of his story. Undaunted by this request, Juan Diego left, promising to return with a sign from the Virgin.
The bishop, disarmed by Juan Diego's confidence, sent two men to follow him to make sure that Juan Diego was not up to any tricks. The two men trailed Juan Diego for a good while but lost sight of him as he crossed the ravine near the bridge to Tepeyac. After a desperate and unsuccessful search, they returned to Friar Zumárraga's home and, infuriated with Juan Diego for having wasted their time, told Zumárraga that Juan Diego was a sorcerer and a fraud who deserved punishment to prevent him from lying again.
In the meantime, Juan Diego arrived at Tepeyac hill and found the Virgin there waiting for him. Kneeling down before her, he recounted his second meeting with Friar Zumárraga and told her of the bishop's request for a sign. Again with words of kindness, the Virgin thanked Juan Diego for his faithful service to her and assured him of the success of his mission, asking him to return the next day to receive a sign for him to take to Friar Zumárraga.
## THE FOURTH APPARITION
Upon Juan Diego's arrival home, however, his plans to return to the Virgin were quickly set aside. While he was away, his uncle Juan Bernardino had taken gravely ill. So the following day, instead of going to Tepeyac, Juan Diego spent his time finding and bringing a doctor to help his uncle, but to no avail; although the doctor ministered to Juan Bernardino, his efforts were too late and death became imminent.
Apart from his love for his uncle, this would have been devastating to Juan Diego because of the important role the uncle played in Indian culture. As Friar Sahagún, one of the early missionaries to the New World and a scholar of Indian culture, notes: "These natives were accustomed to leaving an uncle as guardian or tutor of their children, of their property, of their wife and of their whole house... as if it were his own." Additionally, being Juan Diego's elder, Juan Bernardino occupied another essential and well-respected role in his nephew's life and in the community at large. With the absence of writing, knowledge was primarily passed from one generation to the next by oral tradition, through the accurate, word-for-word recitation of discourses from the _huehuetlatolli_ , the "speech of the elders." Describing the importance of such speech, one indigenous man explained that the words of the _huehuetlatolli_ were "handed down to you... carefully folded away, stored up in your entrails, in your throat." It was through the traditions and wisdom passed down by the community elders that the contemporary indigenous world was guided and given meaning. Thus, community elders and the _huehuetlatolli_ , far more than just sources of advice and education, constituted the very fabric from which indigenous identity was formed.
Although both Juan Diego and Juan Bernardino were dedicated to their new Christian faith, there is no reason to think that the special role accorded to the elders in indigenous culture would not have retained a prominent place within Juan Diego's worldview. The death of his uncle would have signaled something more than just the passing of a close family member, difficult to bear as that would have been; it also could have been seen as the irrevocable loss of a part of Juan Diego's own identity. To a certain degree, the fear and uncertainty confronting Juan Diego were experienced by many other Indian communities and families as well, some of which were decimated by disease and uprooted from their traditional religious practices. And yet, even at this moment, Juan Bernardino showed the strength of his own faith and his trust in the faith of his nephew. He begged Juan Diego to bring a priest to hear his confession and prepare him for death. So the following day, December 12, Juan Diego wrapped himself in a tilma to protect his body from the cold and hurried off toward the _doctrina_ at Tlaltelolco.
As he approached Tepeyac hill, Juan Diego remembered his promised appointment with the Virgin. However, aware of his uncle's condition, he did not want to delay his journey, and so he avoided his usual path in the hope of evading the Virgin. Yet as he rounded the hill he saw the Virgin descend from the top of the hill to greet him. Concerned, she inquired: "My youngest son, what's going on? Where are you going? Where are you headed?"
Juan Diego, at once surprised, confused, fearful, and embarrassed, told the Virgin of his uncle's illness and of his new errand, and expressed something of the hopelessness he was then experiencing, saying, "Because in reality for this [death] we were born, we who came to await the task of our death." Still, even in his distress, he remained committed to his mission. He promised: "Afterwards I will return here again to go carry your venerable breath, your venerable word, Lady, my little girl. Forgive me, be patient with me a little longer, because I am not deceiving you with this... tomorrow without fail I will come in all haste."
The Virgin listened to Juan Diego's plea, and when he had finished she spoke to him:
Listen, put it into your heart, my youngest son, that what frightened you, what afflicted you, is nothing; do not let it disturb your face, your heart; do not fear this sickness nor any other sickness, nor any sharp and hurtful thing. Am I not here, I who have the honor to be your Mother? Are you not in my shadow and under my protection? Am I not the source of your joy? Are you not in the hollow of my mantle, in the crossing of my arms? Do you need something more?
In this passage, the Virgin's words not only have important associations with motherhood but also have imperial associations as well. Specifically, her words bear a special resemblance to the words addressed to the Aztec emperor upon his succession to the throne:
Perhaps at some time they [your people] may seek a mother, a father [protection]; but they will also weep before you, place their tears, their indigence, their penury... Also perhaps the tranquility, the joy with which they will receive from you, that will be gathered in you because you are their mother, their shelter, because you love them deeply, you help them, you are their lady.
The Virgin is mother and queen, and she reveals herself to Juan Diego in both of these capacities.
Interestingly, we see in this another sign of the cultural accuracy of the _Nican Mopohua_ ; although it was common to address persons using many different relationship titles, "inferiors never called superiors by name and rarely even referred openly to any relationship that might exist between them, whereas superiors could do both (though sparingly)." In the _Nican Mopohua_ , nowhere does Juan Diego address the Virgin by the name of Mary; nor does he address her as a mother. In contrast, the Virgin's first words call him by name, and from the beginning, she calls him "my dear son" and breaks the silence about their relationship by calling herself by the title Juan Diego does not address her by: mother.
## MORE THAN FLOWERS
At this moment, Our Lady of Guadalupe began to reveal herself to the world—not through Juan Diego himself but through his uncle. Following her words of consolation to Juan Diego, the Virgin assured Juan Diego of his uncle's recovery, saying, "Don't grieve your uncle's illness, because he will not die of it for now; you may be certain that he is already healed." In fact, as Juan Diego would later learn, at that very moment she was also appearing to Juan Bernardino. Juan Diego trusted the Virgin completely and again implored her for a sign that he could take as proof to Friar Zumárraga.
The Virgin instructed Juan Diego to go to the top of Tepeyac hill, where he would now find a variety of flowers for him to cut, gather, and bring back to her so that she could then arrange them in his tilma. Obediently, Juan Diego climbed up the hill and was amazed to find—in the arid winter environment, and in a rocky place where usually only thistles, mesquites, cacti, and thorns grew—a garden brimming with dew-covered flowers of the sweetest scent. Juan Diego quickly gathered them up in his tilma and took them back down to where the Virgin was waiting. The Virgin, taking the flowers from Juan Diego, arranged them in his tilma and said to him:
My youngest son, these different kinds of flowers are the proof, the sign that you will take to the bishop. You will tell him for me that in them he is to see my wish and that therefore he is to carry out my wish, my will; and you, you who are my messenger, in you I place my absolute trust.
Upon hearing this, Juan Diego set out once again for the bishop's house, reassured by the sign he carried and enjoying the beautiful fragrance of the flowers in his tilma.
Perhaps it is in this moment, as the Virgin stoops to rearrange the flowers in Juan Diego's tilma, that we are given the most poetically poignant expression of what the apparitions at Guadalupe would have meant to the Indian people. In her appearances on Tepeyac, the Virgin takes what is good and true in the Indian culture and rearranges it in such a way that these same elements are brought to the fulfillment of truth. In the Indian culture, flowers and song (which, you will recall, Juan Diego heard just before the first apparition) were symbols of truth—more specifically, the truth that, though somehow intuited by reason, is never comprehensively grasped. Thus the Virgin's sign of flowers, which had to undo the lie told to Friar Zumárraga by the false servants, possesses a double meaning: more than a sign for the bishop that is impossible to explain away as a mere trick by Juan Diego, it is also for the indigenous people a sign of truth.
With these flowers in his tilma, Juan Diego arrived at Friar Zumárraga's residence, but the doorman and servants refused to allow him to enter, pretending not to hear his request. Nevertheless, as Juan Diego continued to wait, the servants grew curious about what he carried in his tilma and approached him. Juan Diego, afraid he could not protect the flowers from their grasping hands, opened his tilma just enough for the servants to see some of the flowers. As the servants reached down into Juan Diego's tilma, the flowers suddenly appeared as if painted or embroidered on the tilma's surface.
Amazed, the servants took Juan Diego to see Friar Zumárraga, and Juan Diego, kneeling before the bishop, told him what the Virgin had said. Then Juan Diego unfolded his tilma, letting the flowers fall to the floor, only to reveal upon his tilma's rough surface an image of the Virgin Mary. In amazement, those present knelt down, overwhelmed with emotion. Friar Zumárraga likewise knelt in tears, praying for the Virgin's forgiveness for not having attended to her wish. Then Friar Zumárraga untied the tilma from around Juan Diego's neck, took it immediately into his private chapel, and welcomed Juan Diego to spend the rest of the day in his home.
The following day, Friar Zumárraga, guided by Juan Diego, went to see where the Virgin wished to have her chapel built. And in this place of craggy rocks, thorns, and spiny cacti—a place whose barren landscape was reminiscent only of death and the futility of life—people from the city and nearby towns immediately came and began construction of the Virgin's chapel.
In the account of the Guadalupan apparitions and miracles, there are many significant moments of reconciliation. In the image itself, one sees a perfect harmony of cultures and their respective symbols that convey the same truth. But for the Indians and laymen, the impression of the Virgin's image on the tilma and the acceptance of Juan Diego's tilma into the chapel are perhaps the most significant moments. In the Indian culture, the tilma reflected social status. A peasant's tilma would be plain and undecorated, while a tilma with color or decoration was reserved for noblemen and people of high social rank. But it would have held powerful cultural meaning as well. The tilma also represented protection, nourishment, matrimony, and consecration—all elements that would be important as the Guadalupan legacy unfolded. For the Indians, the Virgin, by placing her image on Juan Diego's tilma, gives a new and elevated dignity to the common person and especially the Indian. Moreover, this dignity is recognized by the bishop when, as the head of the Church in Mexico, he publicly and personally accepts the tilma into his own private chapel and welcomes Juan Diego into his home. At this moment, all of Juan Diego's roles that had previously impeded his total participation in the Church after the conquest—as an Indian, a convert, a layman, and a man of limited social significance—are welcomed as having an important and decisive place in the Church and its mission of evangelization.
## A NAME FOR THE VIRGIN
After fulfilling his duty, Juan Diego begged to be excused so that he could return to his uncle, who had been, when he saw him last, seriously ill and near to death. The bishop agreed and sent several men to accompany Juan Diego, ordering them to return with Juan Bernardino if he was in good health. When they arrived at the town of Tulpetlac, they were astonished to find Juan Bernardino completely recovered; he, on the other hand, was just as astonished to find his nephew so highly honored by the accompaniment of persons sent by Friar Zumárraga. Juan Diego then explained to his uncle where he had been, only to learn that Juan Bernardino already knew: the Virgin—exactly as Juan Diego had described her—had come to Juan Bernardino too. She had healed him, instructed him to show himself to the bishop, and told him everything that his nephew was doing for her.
What is more, the Virgin revealed to Juan Bernardino something even more important—her name. Henceforth, she was to be known as "the Perfect Virgin HOLY MARY OF GUADALUPE."
It is significant that the Virgin chose to disclose her full name not to Juan Diego but instead to his elderly uncle Juan Bernardino. Now there are two witnesses to the apparitions. While pointing to the veracity of Juan Diego's account, it also underscores the role of family relationships in learning about the faith and the value of spiritual solidarity. Even before the moment when Juan Bernardino tells his nephew of the Virgin's appearance, there is already a history of mutual trust and sharing in their relationship together as Christians.
Yet this moment especially speaks of Juan Bernardino in his combined role as community elder and Christian witness. In many of the biblical accounts of Christ's miraculous healings and those later performed by his apostles, the faith of the healer is integral, but so is the faith of those being healed. Christ would often say to those whom he healed: "Your faith has healed you." As already suggested, owing to his status as a community elder—a status presumably damaged following his conversion to Christianity—Juan Bernardino represented the indigenous community, both its collective knowledge and its identity. Thus, while his illness and imminent death paralleled the condition of many, so also did his recovery foretell a spiritual recovery and renewal. Specifically, the Virgin gives Juan Bernardino her name with two complementary effects. The first is a restoration of Juan Bernardino in his role as community elder, now as a witness of hope with new wisdom to share. The second is the rooting of her name in the collective knowledge of the Indian people, thus giving them a means to seek her intercession and to be spiritually healed in the hope of her promises. This is true renewal: a renewal of the individual in society and a renewal of culture in hope.
Both the Virgin's name and Juan Diego's name are significant within this context, pointing to the need for reconciliation between peoples of different cultures and especially to the importance of inculturation in achieving this reconciliation. While several scholars have argued that the name Guadalupe is of Náhuatl origin—a mistake that began with Luis Becerra Tanco in 1675, but was subsequently shown inaccurate—the fact is that the Virgin chose a name known by the Spaniards. The true origins of the Virgin's name run deeper still, once again bringing together elements of the New World and Spain. (The name "Mary," of course, is originally Hebrew, not Spanish.) In Spain, there was a river named Guadalupe that ran through Extremadura, Spain; the name itself was of Arabic origin and meant "river of black gravel." As legend has it, in the thirteenth century, after a statue of the Black Madonna was found on the banks of the Guadalupe River, the Royal Monastery of Santa María de Guadalupe was built in the Virgin's honor. But the historical record shows that the Spaniards did not give the Mexican Virgin the name "Guadalupe." She chose it—and in doing so she assumed a name that reflected her mission as the one that carries or brings the living water, Jesus Christ.
While it is significant that the Virgin chose a layman as her messenger, thereby underscoring the importance of lay ministry within the Church, it is especially significant that she chose Juan Diego Cuauhtlatoatzin, "eagle that speaks." In Aztec culture, the eagle played an important symbolic role, both as the herald of the Aztec civilization and as the symbol of their patron deity, the sun god. According to Aztec mythology, at some time in the fourteenth century, the Aztecs migrated south to the Valley of Mexico, where an eagle sitting atop a nopal cactus revealed the site of their future capital city, Tenochtitlan ("place of the nopal cactus rock"). But far more than recalling the beginnings of the Aztec civilization, eagles also played an important symbolic role in the contemporary Aztec world, specifically in Aztec religious sacrifice. Revering the eagle as a symbol of the sun, the Aztecs would place the hearts of sacrificial victims in a _cuauhxicalli_ , or "eagle gourd vessel," sometimes shaped like the head of an eagle; it was from these eagle vessels that the Aztecs believed the sun would be nourished. Now, at the Virgin's request, Juan Diego Cuauhtlatoatzin is designated as the messenger of a new civilization. This new civilization, however, is not one in which the lives of the gods are sustained by the sacrifice of human lives for food, but rather one in which all people are called to the God who in Christ is life-giving food for them.
## A HOME FOR THE IMAGE
On December 26, 1531, the chapel in the Virgin's honor was completed. Intended to be as much a home for the image on Juan Diego's tilma as it was a place for prayer, the chapel was built out of adobe, whitewashed, and roofed with straw in just two weeks. To dedicate the chapel, Juan Diego, Friar Zumárraga, and villagers from Cuauhtitlán processed to the foot of Tepeyac hill and placed the tilma over the chapel's altar. Housed in this new chapel, called the Hermitage, the tilma and its image attracted attention throughout New Spain. Antonio Valeriano concluded the _Nican Mopohua_ 's apparition account by noting that "absolutely everyone, the entire city, without exception... came to acknowledge [the image] as something divine. They came to offer her their prayers [and] they marveled at the miraculous way it had appeared."
Juan Diego, too, became an important figure in the Virgin's new shrine. Many who came to the shrine identified in the Virgin's messenger a beautiful expression of holiness that they wished to imitate so that, as some of the Indians put it, "we also could obtain the eternal joys of Heaven." In his homily for Juan Diego's canonization Mass, John Paul II recalled this early recognition of Juan Diego's holiness in the developing Mexican Church. Concluding his homily, he prayed:
Blessed Juan Diego, a good, Christian Indian, whom simple people have always considered a saint!... We entrust to you our lay brothers and sisters so that, feeling the call to holiness, they may imbue every area of social life with the spirit of the Gospel...
Beloved Juan Diego, "the Eagle that speaks"! Show us the way that leads to the "Dark Virgin" of Tepeyac, that she may receive us in the depths of her heart, for she is the loving, compassionate Mother who guides us to the true God. Amen.
Nearly five centuries after the apparitions, Juan Diego remains an example for us today, especially for the new evangelization. In his role in the apparition and in his life afterward, he is a model of faith, of devotion, of sacrifice, and of the role of _every_ believer to transform culture—"to imbue every area of social life with the spirit of the Gospel."
# II.
_The Image of the Mother_
_He who falls at the feet of Christ's mother most certainly shows honor to her Son. There is no God but one, He who is known and adored in the Trinity_.
—ST. JOHN DAMASCENE, ON HOLY IMAGES
## A MYSTERY FOR SCIENCE
On the morning of November 14, 1921, Luciano Perez Carpio, an employee of the private ministry of the presidency, entered the Basilica of Our Lady of Guadalupe with soldiers disguised as civilians protecting him on either side. Leaning down, he placed an arrangement of flowers at the base of the Virgin's image. Moments later, a bomb hidden inside the flowers detonated. The explosion was of such a magnitude that it ruined the basilica's altar, the candelabra, and the bronze crucifix set atop the altar, and it even shattered windows of neighboring homes within a one-kilometer radius. But just inches away, Juan Diego's tilma and its glass covering remained perfectly intact. While persecution of Catholics was prevalent in Mexico during the anticlerical regimes in the nineteenth and twentieth centuries, this attack reached a new level; yet even such an orchestrated effort as this could not undo the miracle, but rather only further underscored its supernatural origins.
More than a century earlier, in 1785, the tilma had already proven resistant to ordinarily devastating occurrences when nitric acid was spilled on it during a routine cleaning of the frame. As one eyewitness of the accident testified:
The spilling of a great deal of acid occurred while the side of the frame was being cleaned..., enough to destroy the whole of the surface being cleaned.... I have personally seen, on those occasions when the glass case has been open... that the [nitric acid] left a somewhat dull mark where it was spilled, though the painting is without any damage.
Remarkably, Juan Diego's tilma was barely damaged, having nothing to show from the incident except a dull mark where the acid was spilled, visible on the right side when looking at the image. But as for the unfortunate silversmith, the eyewitness testimony explained, "I also know that the silversmith responsible for the accident was so upset that it was believed he would become seriously ill... for everyone knows that nitric acid is so strong that it will destroy iron if it comes in contact with it."
The tilma's preservation throughout the centuries has become a mainstay in any consideration of its miraculous properties. Perhaps even more remarkable than these dramatic incidents of preservation is how the tilma withstood equally harmful conditions on a day-to-day basis for the first 116 years of its history. During this time, the tilma was displayed without any type of covering. Additionally, the image's lack of priming underneath the coloration is evident even today since the colors permeate the fabric all the way through and are visible from the back, producing a rough mirror image on the opposite side of the tilma. Unprotected, the tilma was particularly vulnerable to deterioration caused by a naturally corrosive substance, saltpeter, carried in the air from the lake, as well as by the blackening effect of dust and incense in the Hermitage and the "hands-on" devotion of pilgrims who would often kiss and touch the image. In addition, small pieces were even cut from the tilma in order to be venerated as relics, thus further exposing the edges of the tilma to fraying and deterioration.
Toward the end of the eighteenth century, José Ignacio Bartolache, a natural scientist and medical doctor, became interested in the tilma for its miraculous preservation. Intrigued by how the image had remained in such a remarkable state for two and a half centuries, he designed an experiment to test how long a tilma such as Juan Diego's would typically last in the inhospitable climate of the Tepeyac region. He commissioned several tilmas to be made of natural fibers, and instructed the most skilled Indian copyists to replicate the original image as faithfully as possible on each of the replica tilmas. Two replicas were then placed in the same area as the original image in order to expose them to the same environmental conditions. Unlike the original tilma, the replicas could not withstand the humidity and saltpeter characteristic of the Tepeyac hill climate. Before a short period had passed, the replica tilmas were discolored and falling to pieces.
Many other scientific examinations of the tilma have been commissioned and continue to be done today. Yet, even as scientific knowledge has progressed and more studies on the tilma have been performed, the tilma's mystery has only deepened. Beginning with the first-ever examination of the tilma in 1666, we have learned more ways that the tilma is miraculous than we have answers explaining the miracle. In the mid-seventeenth century, with the increasing popularity of Our Lady of Guadalupe in Mexico, many desired to celebrate the Virgin with official Mass and church services, which in turn required a formal investigation into the image and apparitions. While in 1556 a study was commissioned by Bishop Alonso Montúfar, Zumárraga's successor, this did not include a study of the tilma; in the 1666 investigation, however, recorded in the _Informaciones Jurídicas de 1666_ , interviews were conducted so as to learn of the rich Guadalupan oral tradition and the tilma itself was officially examined for the first time. Though the official Mass and liturgies for Our Lady of Guadalupe were not sanctioned until 1754, this investigation is important as the first formal inquiry into the apparitions and miraculous image.
To get permission to examine the tilma, a petition was formulated and addressed to Viceroy Marqués de Mancera. With the viceroy's approval granted, on March 13, 1666, the image was lowered from above the Hermitage's altar; watching over this solemn event were some of the highest authorities in New Spain. First the image was given to a team of art specialists, then to a group of respected chemists, both with the goal of establishing how the image came to be on the tilma.
The team of artists consisted of seven professional painters and art instructors, most of whom had practiced or taught painting for more than twenty years. However, despite their experience in the field, they could not explain how an image so beautiful could be painted upon a surface so rough. In a joint and unanimous statement, the artists marveled at the technique employed for the realization of the image and at its remarkable state of preservation:
[I]t is humanly impossible that any artist could paint and work something so beautiful, clean and well-formed on a fabric which is as rough as is the tilma or ayate... [T]here cannot be a painter, as skillful as he may be or as good as there have been in this New Spain, who could succeed perfectly to imitate the color, nor determine if such a painting is in tempera or oil, because it appears to be both, but it is not what it appears, because only Our God knows the secret of this painting, of its durability and preservation, of the permanence of its beautiful colors and the gold of its stars.
The seven art specialists concluded that the only reasonable explanation for the image—for its beauty, delicacy, and preservation—was God.
With the artists' study complete, on March 28, three chemists were also given access to the image to conduct their own study. After analyzing the environment around the Virgin's Hermitage, the chemists, too, were astounded. Given the location of the Hermitage—an area that was humid and filled with saltpeter—they noted that the tilma should have been destroyed by these environmental elements many years earlier. After all, the saltpeter in the air ruined even the Hermitage's stone ornamentation, and would have done the same with the silver adorning the inside of the Hermitage "if it were not for the very frequent care it receive[d],... since it is far less resistant than stone." Nevertheless, the image remained remarkably well preserved. Touching the image, the chemists were puzzled by the fact that, though the surface of the image was soft and gentle like silk, "on the backside, it is rough and hard." How could such a delicate and detailed painting, with such soft and gentle features, they wondered, be done on a surface with such a coarse weave? The chemists concluded their report in the same way as did the artists, saying: "Our limited intelligence cannot account for it."
Since 1666, as technology has developed, subsequent studies of the tilma have been numerous and diverse, underscoring the mystery of the tilma in new areas. With the rise of photography, Alfonso Marcue (1929), Carlos Salinas (1951), and ophthalmologist Dr. Javier Torroella Bueno (1956) took close-up photographs of the eyes of Our Lady of Guadalupe, and reported that the image's coloration not only depicts her pupils but also depicts the types of images one would see reflected in the eyes of a living human being—in this case, the reflected images of people. A few years later Dr. Rafael Torija Lavoignet (1958) and Dr. Charles Wahlig (1962) studied the Purkinje-Sanson effect in the image on the tilma. More recently, in 1981, Dr. José Aste Tönsmann likewise studied this same effect in the Virgin's eyes, and wrote of his findings: "The presence of the images in both of the eyes of the Virgin of Guadalupe constitutes, without doubt, one of the most forceful proofs... of the difficulty of obtaining a natural explanation for its creation."
Over the years, these studies have continued to deepen our understanding of the unnatural, inexplicable, and miraculous qualities of the tilma. Even as science has advanced and the complicated theories of earlier examiners have proved inadequate, it has become increasingly clear that there is no natural explanation for the phenomenon of the image of Our Lady of Guadalupe on Juan Diego's tilma. Moreover, with advancements made in our understanding of the history, religion, and culture of the New World Indians, it is becoming increasingly clear that the richness of the tilma is not exhausted by its physical properties, but extends to the profound symbolic relevance of its message.
Though people marvel that the tilma has survived corrosive air, an accident with acid, and even a bombing, as with any miracle, we should look beyond its miraculous elements and survival. Without God and the Virgin, the miracle of the tilma becomes unintelligible, since this miracle is not a mere event but an action—an ongoing action that we today are also gifted in bearing witness to. And like any action, this action has an agent and a purpose. By surveying the history of the tilma, we get a better idea not only of what science can tell us about the image but at the same time a better sense of its deeper spiritual message. This is not to suggest that every facet of the tilma and its history is imbued with meaning or is theologically explicable, or even that we can ever fully grasp its mystery. Rather, the implication is simpler: it is an invitation to reflect on what historical circumstances and science have revealed about the tilma in light of the ever deepening mystery and message to which they point.
## CONTEMPLATING THE IMAGE
As we will hear more about in the following chapters, the Virgin's miraculous image spoke profoundly to the Indian people of the New World in a way the missionaries never could. In his account of the history of the Guadalupan apparitions, the missionary Fernando de Alva Ixtlilxóchitl recalled:
[The Indians], submerged in profound darkness, still loved and served false little gods, clay figurines and images of our enemy the devil, in spite of having heard about the faith. But when they heard that the Holy Mother of Our Lord Jesus Christ had appeared, and since they saw and admired her most perfect Image, which has no human art, their eyes were opened as if suddenly day had dawned for them.
Appearing in this way, the Virgin herself affirms and continues a tradition already well established in the Church: that of artistic patronage and development of talent as a means to achieving or inculcating spiritual growth. While serving an important pedagogical function as a way to teach basic articles of the faith, art was also seen as having a deeper, spiritual significance as a means to contemplation and conversion. Yet in the early Church it was precisely this efficacy and purpose of art that was questioned. Some, popularly called iconoclasts, argued that to venerate images of Mary and the saints was idolatrous; instead of leading to God, they claimed that veneration of icons focused merely on the creature. But in A.D. 787, the historic Second Council of Nicaea upheld the value of holy images against the iconoclasts, stating: "For by so much more frequently as [the saints] are seen in artistic representation, by so much more readily are men lifted up to the memory of their prototypes, and to a longing after them; and to these should be given due salutation and honorable reverence." More than seven hundred years later, Our Lady of Guadalupe issued her own statement on the matter; even today, her holy image stands as a singular example of the power of holy art to bring about spiritual growth and conversion.
Artistically, the image of Our Lady of Guadalupe achieves a harmony of two seemingly incompatible styles. On one hand, the use of shading and color to produce depth and lifelikeness in her face, figure, and clothing is reminiscent of European techniques and styles. On the other hand, the two-dimensional gold designs over her tunic, which do not follow the folds of the tunic, are more similar to what we find in ancient Indian codices. In this chapter, we will consider the image apart from these gold designs found on the Virgin's tunic, an important feature of the image that will be discussed in the following chapter. Here, we will look at the three-dimensional features of the image—which of course include the Virgin herself—in view of the special way that these features convey important Christian concepts in an understandable and completely relevant way for the Indians of the New World.
In the image, we can easily recognize Mary as the woman described in the Book of Revelation—the woman "clothed with the sun, with the moon under her feet" (Rev. 12:1). But to understand how her Child's presence is evoked, we have to look at the dark purplish ribbon tied high above her waist. Typically, Indian women would wear this belt just at the waist, unless a woman was pregnant, in which case the belt would be worn higher up, as in the case of Our Lady of Guadalupe. (Interestingly, there is a nice parallel in the Spanish language, where the word for pregnant is _encinta_ , which literally means "adorned with ribbon.") In this way, the Virgin is a lady of Advent, of hope, patiently awaiting the birth of her Son.
Her pregnancy, as a symbol of birth and renewal, takes on greater meaning in that she is pregnant with God himself. In this, the clouds surrounding her would have elicited respect, and possibly even indicated something supernatural. It was written that when the Aztec emperor Moctezuma first greeted Cortés, he said: "I have been expecting this for some days, days in which my heart was looking at those places from where you have come _from among the fog and from among the clouds_ , a place unknown to all." Numerous scholars have debated what these words tell us about what Moctezuma actually believed about Cortés—for instance, did he really believe Cortés to be a god, or are these words mere mockery or simple etiquette? For us, however, what is important is not what these words actually meant in that context, but rather what they tell us about how the Indians spoke of supernatural things: "from among the fog and from among the clouds"— _mixtitlan ayauhtitlan_ in Náhuatl. These are also the same words reportedly spoken to the first twelve missionaries to the New World when they met both the Aztec priests and nobles. Here in the Virgin's image, she is surrounded by clouds, not because she herself is a god, but because she is with God as his mother.
A related kind of renewal is evoked as well: the renewal of the Indian civilization. Beneath her feet is an angel, bald but with the countenance of a child, thus evoking both wisdom and youthful innocence. His wings are those of an eagle, decorated by a rainbow of blue-green, white, and red. It is important to recall the eagle's role both as herald of the Aztec civilization and as the symbolic conveyor of the Aztecs' sacrificial offerings to their gods. Echoing these roles, this eagle-angel transports in his hands a new sacrifice, Christ present in the Virgin's womb. Christ the Redeemer comes to free the Indians from their perceived need for ritual sacrifice.
The clothing the Virgin wears tells us more about who she is; her mantle is adorned with stars. Some who have studied the stars on the Virgin's mantle have reported a remarkable coincidence between this pattern of stars and the constellations that appeared in the sky above Mexico City on the morning of December 12, 1531, the day the Virgin revealed herself on the tilma. The Virgin's mantle is of a rich blue-green color, which for the Aztecs was a color that had significant imperial associations. Traditionally, only the Aztec emperor wore a blue mantle, which typically was festooned with emeralds, thus symbolizing the heavens. Decorated in gold flowers, her earthy pink tunic evoked the earth, the land. At her feet, these two pieces of clothing, her tunic and mantle, are held by the eagle-angel, thus indicating the Virgin's reign over the whole cosmos; in her, the sky and earth are joined together. This revelation was not merely theoretical but profoundly incorporated into the Indians' understanding of their own lives. In a 1995 interview, the Indians of Zozocolco, Veracruz, while preparing to celebrate the feast of Our Lady of Guadalupe, explained:
With the harmony of the angel, who holds up the Heavens and the Earth, a new life will come forth. This is what we received from our elders, our grandparents, that our lives do not end, but rather that they have a new meaning.... This is what we celebrate today... the arrival of this sign of unity, of harmony, of new life.
These words, passed down through a centuries-old oral tradition, give us insight into ancient perspectives on the image and its meaning for the Indian people, who were able to see it in the context of their own lives, thus assuaging fears and imbuing hope.
Importantly for the Aztecs, this new harmony is illustrated specifically with depictions of the sun and the moon, the first set behind the Virgin, bathing her in a peaceful light, the second at her feet. These solar bodies were objects of great importance and fear for the Aztecs, who associated the sun and moon with gods and believed these gods to be in constant conflict with one another. Fearing that the sun would somehow perish in this cosmic battle, the Aztecs sustained their god with human sacrifices intended ultimately to achieve cosmological harmony and the preservation of life on earth. The Virgin's image speaks specifically to this fear, wherein the sun and moon are shown no longer as gods but as objects under her governance. While eclipses were often believed to be bad omens and a sign of the sun's defeat, in her image the Virgin eclipses the sun, but not in a menacing way; in doing so, she brings focus to her own womb, wherein is kept the true God. Interestingly, December 12, 1531, the day of the Virgin's last appearance, was the winter solstice, thus inaugurating the time of year when the sun increasingly conquers the darkness and the days become longer.
The moon set beneath the Virgin's feet has a superadded significance. While indicating her governance over the moon, and thus assuaging fear of cosmic collapse, the Virgin at the same time visually roots herself in the very origins of the Mexican civilization, which derives its name from a combination of three Náhuatl words: _meztli_ , "moon;" _xic_ (tli), "navel;" and _co_ , "in." Taken together, these three words mean "in the center of the moon." In the image, the Virgin literally stands in the center of the moon. However, though rooting herself in these origins, the Virgin at the same time suggests to the Indian people a radically new spiritual conception of the universe.
While her imperial-colored clothing and cosmic surroundings indicate that the Virgin is a heavenly queen, her posture indicates that there is someone greater than she, someone to whom she humbly prays. For us, the Virgin's clasped hands immediately indicate prayer; this would have been the same for the Spaniards at the time as well. But even more, for the Indians, her entire body would have indicated the Virgin was praying. For them, prayer was expressed not only in words or song but in solemn dance. In the image, the Virgin can be seen as in motion, with her weight on one foot and the other knee bent, in the dance-step position. This for the Indians was the highest form of prayer. As one missionary at the time explained, the Indians' religious ceremonies were elaborate events, with "many roses and green and bright things, and with chants solemn in style, and with dances... of great feeling and importance, without disagreeing in tone or step, since this was their main prayer." The Aztec emperor would also participate, uniting himself to his people in song and dance for their gods.
In this we can see a deeper meaning in the bright colors and beautiful birdsong that Juan Diego observed before his first encounter with the Virgin. With song and the transformation of Tepeyac hill into a place of brightness and light—the ground like jade and the foliage like the bright feathers of the quetzal bird—the Virgin prepares the hill as a place of worship and sacred celebration. And in this we are at the same time shown the true source of the Virgin's queenship; in her prayerful dance, lowly posture, and downcast eyes, she is queen precisely as handmaid, precisely through her humble openness to God.
The tilt of her head and direction of her gaze have another important meaning. Though for us the idiomatic phrase "to look sideways at" has a negative connotation and is often associated with indifference, in Náhuatl it was the reverse. To look sideways at someone or something was actually more complementary, meaning "thinking of he who is looked upon, not forgetting who is looked upon." In this case, the side of her face is turned toward the viewer. Even today, this was one of the things pointed out in the 1995 interview with the Indians of Zozocolco: "This Woman is important because She stands before the Sun, steps on the Moon and dresses herself with the Stars, but her countenance tells us that there is someone greater than She, because She is looking down, as a sign of respect."
One of the aspects of Our Lady of Guadalupe given most attention is the color of her skin. Neither white like the Spaniards nor dark like the Indians, the Virgin is a mestiza—a combination of the two. Specifically, the mestizos were people of mixed blood, having both European and Indian ancestry. In this way, she identifies herself completely with the people of the New World, since the first appearance of mestizo children was in the New World. At the same time, she does not exclude the Spaniards; they are a part of her as well. In this way, the Virgin borrows from the Spaniards and the Indians, reaffirming both in their uniqueness but at the same time representing an important link between them: she is their mother.
As their mother, she leads them to her Son. If we look closely at the Virgin's image, we can see that she is wearing a brooch with a bare cross. The Indians would have seen similar crosses worn by the New World missionaries, who wore bare crosses for fear that the Indians would mistake any depiction of the Crucifixion as an affirmation of their own human sacrifices. Yet the Virgin does much more than identify with the missionaries: while pointing to Christ's Crucifixion, the bare cross also points to his victory over death in the resurrection. With plagues drastically reducing the Indian population, and their sufferings compounded by the mistreatment they received at the hands of some of the Spanish officials and settlers, the Virgin's cross stands as an acknowledgment of their sufferings.
## A CULTURE OF THE IMAGE
In her miraculous image, the Guadalupan Virgin bequeathed something both enduring and physically concrete. Why she chose to perform a second miracle is a message in itself. After all, the flowers in wintertime, growing atop the barren Tepeyac hill, would likely have been miraculous enough to inspire a chapel, making the second miracle not only unexpected but completely unnecessary. Still, the Virgin chose a sign that achieved much more. Like the flowers, Juan Diego's tilma is composed of an organic, highly corruptible material—agave fibers—and yet, unlike the flowers, the tilma has neither fallen into obscurity nor suffered from the processes of normal decomposition. Rather, the tilma exists as a synthesis of the earthly and the heavenly whose enduring nature belies an enduring message and request, one that far exceeds the original purpose of the flowers.
The message contained in the Virgin's image, while certainly of universal import, is one whose presentation is geared specifically toward the Indian people. Drawing upon pictographs, myth, symbols, and ways of thinking already familiar to the Náhuatl mind, the Guadalupan Virgin provides us with an extraordinary model of enculturation, as Benedict XVI wrote:
Our Lady of Guadalupe is in many respects an image of the relationship between Christianity and the religions of the world: all of these streams flow together into it, are purified and renewed, but are not destroyed. It is also an image of the relationship of the truth of Jesus Christ to the truths of those religions: the truth does not destroy; it purifies and unites.
In her miraculous image, the Guadalupan Virgin borrows potentially good and fruitful elements from the Indian culture—those "seeds of the Word" that, until unearthed, lie dormant in every civilization. In this way, she reaffirms these elements, while at the same time purifying them and bestowing upon them the fullness of Christ. Thus, far from being outright rejected or condemned, the basic truths recognized in the Indian culture are confirmed and perfected in the Virgin's miraculous image, where they are pictorially united to the truth of Christ.
In any authentic Christian evangelization, the message is always truth—specifically, the truth of Christ—and enculturation speaks to how that truth is conveyed in a cultural "language" each recipient can understand. At the first Pentecost, the apostles were sent out from the upper chamber to evangelize the world. We are told that Mary was with them, praying in the cenacle. Biblically, no passage suggests that she too spoke in tongues. Still, we know that she has done so historically, through her various apparitions. Among these, none more beautifully points to the intimate link between love and communication than the apparitions at Guadalupe. Here, in the apparition, the Virgin Mary spoke in Náhuatl. Additionally, in looking at the image, the Indians could see reflected their own language and culture, and a codex with profound meaning.
The fact that this great and lasting miracle came as both an image and a codex was one way that this evangelization truly showed itself to be enculturated according to the New World culture rather than an imposition of European culture. In many ways, European culture could be understood as a "culture of the word," heavily influenced by its dependence upon written language—from the philosophy and literature of the ancient classical world to the study of sacred scripture and the spiritual writings of the Church Fathers. That culture was greatly advanced by medieval theologians such as St. Thomas Aquinas and during the Renaissance with the invention of the printing press some decades before the apparition. Yet the missionaries in New Spain faced a people who were entirely different in this regard. Communicating and recording ancient truths through their pictographs, the Indians of the New World lived in a culture more accurately described as a "culture of the image."
Today, we have seen a significant shift in our own culture, from a culture of the word to a new and very different culture of the image, suggested and reinforced by the prevalence and popularity of new technologies. As with any cultural shift, new problems and possibilities arise. Speaking of the special problems posed today, the late Cardinal Alfonso López Trujillo wrote, "The complex theme of language has to be taken up: 'Christ himself asks us to proclaim the Good news using a language that brings the Gospel closer and closer to today's new cultural realities.'" Undoubtedly, the theme of language and dialogue is one of central importance in Christianity, not only historically, as an indispensable means to sharing the faith, but more deeply as an expression of its very origin and sublime prototype: the communion of Trinitarian love. Today, this dialogue has been opened up to a variety of new forums. And yet if it is to be effective, it must remain personal, tailored to the unique values, language, and mediums of expression present in each culture. What is more, it must always manifest itself, as John Paul II once said, in the form of "an act of love which has its roots in God himself."
Enculturation requires dialogue. It requires a level of trust and understanding that makes it easy for the recipient to grasp the fundamental truths being taught. Paul VI spoke of the requirements for dialogue in this way:
Since the world cannot be saved from the outside, we must first of all identify ourselves with those to whom we would bring the Christian message—like the Word of God who Himself became a man. Next we must forego all privilege and the use of unintelligible language, and adopt the way of life of ordinary people in all that is human and honorable. Indeed, we must adopt the way of life of the most humble people, if we wish to be listened to and understood. Then, before speaking, we must take great care to listen not only to what men say, but more especially to what they have it in their hearts to say. Only then will we understand them and respect them, and even, as far as possible, agree with them.
Furthermore, if we want to be men's pastors, fathers and teachers, we must also behave as their brothers. Dialogue thrives on friendship, and most especially on service. All this we must remember and strive to put into practice on the example and precept of Christ.
One of the most extraordinary examples of this enculturated, personal dialogue is that of the image of Our Lady of Guadalupe. First made manifest to a people within a culture of the image, Our Lady of Guadalupe is becoming increasingly popular today, in a global community now transitioning to a new culture of the image. Thus, while she may come to us differently—in different times and circumstances—this ancient image of Our Lady remains as contemporary as when it first appeared nearly five hundred years ago. And more than any other dialogue, the conversions and hope that she inspires make clear that the gift of her image was and continues to be an act of love rooted in God.
# III.
_Message of Truth and Love_
_I am the Way, the Truth, and the Life_.
–GOSPEL OF JOHN 14:6
## THE TILMA: A NÁHUATL CODEX
While in Mexico for the installation Mass as the Supreme Knight of the Knights of Columbus, I purchased an oil painting depicting the Guadalupan apparition. Painted sometime between 1675 and 1700, it depicts in vibrant colors Our Lady of Guadalupe appearing to Juan Diego, accompanied by two angels, one gazing at the Virgin, the other laying a reassuring hand on Juan Diego's shoulder. The blue veil and stars, the Virgin's positioning, even the folds of her clothing and the bending of her knee are just as depicted in the image of the Virgin on Juan Diego's tilma. But upon examining the Guadalupan Virgin's image further, it became clear that something significant was very different in the later painting of the Virgin. The vinelike gold design over the Virgin's tunic lacked the most important, unduplicated flower—the four-petaled jasmine flower, representing God.
Rather than a unique "error" in copying, the anonymous painter of the picture was just another in a long history of artists who viewed, or at least painted, the floral design as mere "decoration"—something extraneous, undefined, and thus perfect for artistic license. Often, instead of the spacious, logical, even repetitive design, the Colonial-era artists would paint diverse, even radically ornate, baroque floral patterns. But this "error" is also understandable, since only recently has Náhuatl and Guadalupan scholarship uncovered the reason for the flowers to take the particular shape and position they have on the tilma: each flower resembles the pictographic writing, or glyphs, found in Náhuatl codices. That is, the flowers are more than flowers; they are symbols, words, and concepts. In this, the tilma becomes more than an image; it is a codex, conveying in the Náhuatl language and culture of the Indians the most fundamental elements of the Christian message: the relationship between God and man, a relationship of love and truth.
As in most civilizations, the indigenous manuscripts played a significant role in communicating information on topics of communal importance, including religious ceremonies, the gods, and special dates and feast days. Yet unlike European manuscripts written using alphabetic language, Náhuatl codices were primarily pictorial, comprising images and glyphs. Often the text was written in a circle or a square on the page, unlike the parallel lines in European text. Also, unlike bound books, these codices were more like folded stacks of long pages. When taken out to be read, the codices were unfolded flat on the floor, where their pages could be viewed from all sides. Most of the codices from central and southern Mexico we have today are composed of paper made primarily from the soft interior of tree bark, and a few from the maguey plant, the same plant family from which Juan Diego's agave tilma is made. Visually, the idea of the tilma and the Virgin's image being a codex is accentuated by the fact that the floral symbols or glyphs do not follow the folds of the Virgin's tunic, as it would if the pattern were merely a design on the tunic, but rather the floral glyphs overlie the tunic area, leaving the glyph lines undistorted and entirely visible.
## BASIC COMPONENTS
The floral design overlying the Virgin's tunic consists of three kinds of flowers: a four-petaled jasmine (appearing once), an eight-petaled flower (appearing eight times), and a flower cluster (appearing nine times). The meaning of these glyph-flowers is derived from their correspondence to ancient Náhuatl glyphs and from their relationship to the rest of the tilma image, described in the previous chapter. Additionally, because of the way these arabesque flowers are composed, they can resemble several different glyphs at the same time according to the direction they are viewed (just as the letter _p_ can resemble a _d_ when viewed upside down). Importantly, these multiple meanings of the glyphs are not simply common or generic glyph associations from other Náhuatl codices; rather, they are unique to the tilma, and thus imply a unique and multifaceted message and meaning.
## JASMINE FLOWER
The four-petaled jasmine flower (the flower absent in the seventeenth-century painting described at the beginning of the chapter) is the only one of its kind found on the Virgin's tunic. While basic in design, it is both central to the image of the Virgin and central to the identity of the Virgin and her Child.
For the Indians, the design of this four-petaled jasmine flower had many interrelated meanings in their religious thought. Cosmologically, it symbolized the four directions (north, south, east, and west), covering the whole universe. The design represents what the indigenous called the _Nahui Ollin_ , meaning "always in movement." Its fifth point, in the center, symbolizes the Fifth Sun, and is therefore the sun flower that represented in the theology of the _tlamatinime_ (wise men) of the Toltecs the only living and true god, whom they called _Ometéotl_. The Texcocans, of which Juan Diego was one, were heirs to Toltec thought and culture. Some of the titles attributed to this "unknown deity" were _Ipalnemohuani_ ("Him for whom one lives"), _Teyocoyani_ ("Creator of people"), _Tloque Nahuaque_ ("Owner of the near and close") and _Ilhuicahua Tlaltipaque_ ("Lord of heaven and earth"). However, in the Toltec mentality, _Ometéotl_ resided in the highest part of the heavens where no human being could have access to him and where this divinity would not have to concern himself with insignificant human beings.
In the apparition account, Our Lady of Guadalupe speaks of her Son using these titles (Him for whom one lives, Creator of people, Owner of the near and close, Lord of heaven and earth). And yet, it is absolutely clear that she is speaking of Christ. In the image of Our Lady of Guadalupe, this takes shape as a magnificent inculturated evangelization through the positioning of the jasmine flower on the womb of the image, just below her pregnancy belt, thus identifying her Child as divine. In this, the symbol of the four-petal jasmine shows the Indians that the omnipotent God is reachable by any human being; and not only is he interested in them but he delivers himself to them: it is wondrous that this omnipotent God, the deeply rooted God, now comes to find and deliver himself to mankind through his mother.
The fathers of the Second Vatican Council spoke of the "seeds of the Word"—the glimpses of the truth about God—that can be found in various cultures. This can be seen in the limited understanding the Indians had of this "unknown God," reflected in the four titles referenced above expressing certain truths about the omnipotent creator. In a special way, this harks back to St. Paul's address at the Areopagus, when he spoke to the Athenians about their worship of the "unknown God," whom they detected but could not understand. As St. Paul explained, this God is revealed fully in Christ as the one who created man and the natural order "so that people might seek God, even perhaps grope for him and find him, though indeed he is not far from any one of us. For 'in him we live and move and have our being,' as even some of your poets have said, 'for we too are his offspring.'"
Of course, there are many differences between the God of Jesus Christ and Ometéotl, the god of the Toltecs. The image on the tilma addresses a number of these differences. One of the fundamental differences addressed by the jasmine's placement over the Virgin's womb is a difference of love, presence, and care. For these Indians, despite Ometéotl's authority over the whole world, the god was believed to be indifferent to the affairs of the world; though sustaining all things in existence, he was inaccessible. Here, the flower over the Virgin's womb speaks to one of the radical concepts brought by Christianity, specifically by the belief in Christ's incarnation: the omnipotent God of the earth whom these Indians sought is near to mankind, and cares for them so deeply that he comes in a vulnerable and loving relationship, through his mother.
## EIGHT-PETALED FLOWER
Besides the implicit promise of birth in pregnancy, in this eight-petaled flower a different kind of beginning is expressed: the beginning of a new age, indicated by the harmony among the cosmic spheres. The eight-petaled flower, appearing eight times, symbolizes the planet Venus, also known as either the "morning star" or the "evening star," depending on whether it rises in the sky at sunrise or sunset. Among the Indians, the calendar system was very complex and was entrusted to their priests educated in astronomy and the cosmological rhythms. In particular, these priests followed three calendars of varying lengths: a ritual calendar, a solar calendar, and a calendar following the movements of the planet Venus. Over several centuries, without adequate compensation for leap years, the Indians' calendar system and feast days got off, so much so that eventually the feast days were being celebrated many months away from the corresponding seasons—for example, the rituals for use during the drought season were performed long before the droughts typically occurred. On the tilma, not only are the sun and moon "subdued" and eclipsed by the Virgin and now in harmony with each other, but the two orbs governing the calendars are in harmony as well. The sun and the planet Venus are present on the tilma, but neither is dominating the other, suggesting a harmony in the calendars.
Like many elements of the apparition, this peaceful timeliness reflects the birth of Christ. The Roman emperor at the time of Christ's birth, the Gospel of Luke tells us, was Caesar Augustus, who established the Pax Romana (Roman peace). During this time, an age of peace was welcomed into the empire and most of the continent. Thus, Christ was born at a time "when the whole world was at peace." Likewise, coming historically ten years following the conquest, "when the arrows and shields were put aside, when there was peace in all towns," Our Lady of Guadalupe is shown to be pregnant with Christ at an interval of peace.
## FLOWER CLUSTER
The most intricate design on the Virgin's tunic is the cluster of flowers. Importantly, this cluster begins to address the universal and timeless questions: Who am I? How do I relate to God? How does this affect how I relate to other persons and the world? Appearing nine times, each time with slight variations, this cluster has three main parts: a triangular blossom, a curving stem with leaves, and small flowers attached to the outside of the blossom. Comparing this flower cluster with symbols found in other Aztec codices, we uncover similarities between the designs of the pictographs and the design of the parts of the flower. The blossom, with its triangular shape and bumpy slopes, resembles the glyph for "hill" ( _tepec_ ) and the hill-like temple, while the curving stem corresponds to the glyph for "river."
In Náhuatl, it was standard to combine glyphs to create a new word or thought, especially to designate items that are unique, such as personal names or place names. In this case, the combination of glyphs for "hill" and "water" was an established combination: the _altepetl_ (hill + water) was a communal concept, ranging in meaning from a village to the larger concept of nation or civilization.
Historically for the Aztecs, with religion so deeply imbedded in the everyday life of the people and state, this larger concept of nation or civilization was tied to the temple. Tenochtitlan, the Aztecs' capital city, was itself believed to be the fulfillment of a promise made to the Aztecs by their patron deity, Huitzilopochtli, while they were still a subjugated people living in the land north of the Valley of Mexico. According to legend, the Aztecs were told to leave in search of a sign—an eagle set atop a nopal cactus. After much wandering, the Aztecs finally found the promised sign on an island in a lake. Here, on this island, they founded their capital—Tenochtitlan, "place of the cactus on a rock"—from which they forged an empire, coming to dominate the whole Valley of Mexico. Inaugurating their new nation and civilization, the Aztecs built a temple, which later was expanded upon numerous times over its long history. More than just a place of worship, the Templo Mayor transcended its sacrificial function and held greater importance in the identity of the Aztec Empire and civilization. Located in the center of the city, marked by the intersection of four converging causeways, the Templo Mayor was encircled by a wall, which separated the sacred precinct from the rest of the city. Inside the sacred precinct were palaces and numerous pyramids and other religious buildings. Though set within the enclosed space of the sacred precinct, the Templo Mayor nevertheless was connected with the rest of the city as a symbol of its sustaining and originating center, and indeed as a center of the cosmos.
With this history in mind, when we look at the reason for this particular combination of glyphs—the hill and river—on the Virgin's tunic, we can understand why these geographic features carried such great significance. "Mountains and water symbolized natural forces considered necessary for the life of the community." A hill represented the land's protection, a place of origin, and sustenance, while the river represented life itself. Similarly, hills were associated with temples or "sacred hills," both in writing and in reality, in part because of the obvious resemblance of a pyramid and a hill, making the spiritual sustenance and protection of the temple present as well. Only with these—spiritual sustenance, physical sustenance, and life—could a civilization exist and proceed for generations. Moreover, this combination of hill and water was also in part due to the Indians' belief that the mountains held water within them; one day, they feared, the mountains might break open, flooding the land.
## PROPOSING A NEW CIVILIZATION
While these are the basic glyphs composing the flower cluster, these are not the only glyphs that the flower cluster resembles. In the codices, the _altepetl_ glyphs often were written to reflect the distinguishing character of the city. In this case, upon fuller examination, it becomes clearer not only that a civilization is inscribed as a message, but also what kind of civilization.
The triangular blossom is not always pointing upward, but is situated differently, for instance, the flower-duster at the Virgin's feet or the one just above her hands. In this, the tilma-codex resembles other Náhuatl codices, which were sometimes written to be looked at from various angles. Looking at the tilma glyphs in this way, we can find another meaning in the flower-duster. When we view it upside-down, as shown in the picture, we see that the triangular blossom and the curving stem come to resemble a heart and its arteries. This depiction of a heart is similar to those found in indigenous codices, which often show the heart with the attached arteries, as found in the depictions of ritual sacrifice in the _Florentine Codex_ , for example. Yet, while obviously signaling the idea of sacrifice, the heart-flower does so in an entirely new way. Unlike the Aztecs' own ritual sacrifices, this sacrificial heart is shown to be a divine heart, a heart through which divine blood flows, indicating the sacrifice and thus love of God. With the heart's artery attached to the Virgin's celestial mantle, God is shown as the true Giver of life. Rather than being sustained by the Aztecs' ritual sacrifice, he is shown as the one who sustains his creation through the gift of his own divine life-giving blood.
In an obvious way, this civilization is characterized by truth in that the glyphs representing the _altepetl_ are composed in such a way that they resemble a flower—with a blossom, a stem, buds, and leaves. We have already mentioned how the Indians metaphorically associated truth with flowers and song. The image of the flower had a superadded relevance if we also recall the importance the Indians placed on tradition and the ancient word. As Miguel León-Portilla noted, the Náhuatl word for "truth," _neltiliztli_ , derives "from the same radical 'root,' _tla-nél-huatl_ , from which, in turn, comes _nelhuáyotl_ ," meaning "base" or "foundation." The flower could only survive if its roots were firmly planted in the soil; without this rootedness, the flower would die. Similarly with truth; to be true meant to be rooted or to have firm foundations. Here, the flower stem of each of the nine _altepetl_ flowers is rooted visually in the Virgin's mantle, that is, rooted in the heavens. The emphasis on truth—and the health of this divinely rooted truth—is seen as well in how smaller flowers are sprouting around the top of the heart-flower.
The relationship of this civilization to truth has a personal dimension as well. Looking at the interior of the hill-flower, we can see the outlined features of a face, with two squinting eyes, a large nose, and a long, smiling mouth. For the Indians, in the same way that flowers and song were a metaphor for truth, the human heart and face taken together were a metaphor for the human individual. More than that, they were specifically related to the mature and enlightened person. Since for the Indians the whole process of education was geared toward giving a "face" (personality) to the human being, believed to be born faceless, as well as humanizing his "heart" (will), the face and the heart brought together in the hill-flower refer to one both wise and mature, one with a purified heart. And as if to point out how such wisdom and maturity are to be nurtured and sustained in the human person, the main artery connects the face-heart (human person) to the divine, to God. Biblically, this has an important parallel in the prophetic books of Ezekiel, found in the Old Testament, when the Lord speaks to Ezekiel, saying: "I will give them [the Israelites] a new heart and put a new spirit within them; I will remove the stony heart from their bodies, and replace it with a natural heart,... they shall be my people and I will be their God." The opposite of a stony heart is a humanized heart, depicted literally on the tilma as a heart with a human face.
The _altepetl_ civilization, while rooted in the divine, is drawn over the Virgin's earth-colored tunic. In this way, the connectedness of this heart-civilization to the earth is made clear. While it is a civilization with special ties to God, it is likewise a civilization on the earth, for the earth—for the here and now. More specifically, looking again at the triangular hill part of the flower cluster, we discover that its design is not a generic hill; rather, according to the Náhuatl writing, the execution of the "hill" glyph exhibits identifying characteristics—in this case, including its topographical uniqueness. With its noticeably pointed top, this hill symbolizes Tepeyac hill, whose name in fact meant "hill of the nose" or "pointed hill." This civilization of the heart, then, far from showing itself as an abstract or indistinct concept, is linked to a real place with real people, beginning symbolically at Tepeyac hill.
The link between this heart-civilization and God is provided by the Virgin, a link pictorially shown in the connection between the heart-flower and the Virgin's mantle. Of the nine heart-shaped flower clusters, one is placed just above the Virgin's clasped hands. In the oral tradition of the Indians of Zozocolco, the Virgin's intercessory role is seen in the heart design in this way: "Our elders offered hearts to God that there may be harmony in life. This Lady says that, without tearing them out, we should place them in her hands so that she may then present them to the true God." This resonates beautifully with how the Indians' ancient oral tradition spoke of the emperor and the protection his hands were said to provide:
You make yourself just like the variety of fruit trees do; you rise with gracefulness, with gentleness. Next to you different birds suck, the hummingbird, the zaquan, the quecholli, the tzinitzan, the quetzal. In your hands they take shelter from the heat, they protect themselves from the sun.
Here, the large cluster of flowers is rooted in the Virgin's mantle, indicating that the Virgin takes this heart-civilization into her own maternal care. Through her, the Indians' prayers to her are offered to God. Her compassion is more than the offering of her own heart: it is the acceptance of our hearts, ourselves, within her—it is the promise of protection and continued life.
In the preceding chapter, we considered the image of Our Lady of Guadalupe in its basic composition—in the figure and face of the Virgin, and in her clothing and positioning. And with just that, already a profound message was discerned, a new conception of the universe gleaned. Now, adding to this a consideration of the two-dimensional floral glyphs on the Virgin's tunic, the message of Our Lady of Guadalupe takes on further insistence through a new expression. Not only is it a message about God but, insofar as it was a message about God—about his closeness and personal love—it was a message for them. It was a message about God, about his love for them. And insofar as it was a message for them, it is a message for us today as well. This universal message speaks to our basic human desire for God; it is God's promise to us, made through his mother, that we are never left without help. And like any promise, it always has two sides: God promises to help us, which at the same time requires that we let him. Love—even God's love—always seeks to be returned. This is the proper relationship between God and his people.
# IV.
_A Multifaceted Love_
_The key to every hope is found in love, solely in authentic love, because love is rooted in God_.
–BENEDICT XVI, "ADDRESS TO YOUNG PEOPLE"
## THE EVERLASTING, EVER-LOVING GOD
Each day and in every age and culture, people speak of many different kinds of love: love for a spouse, for family, friends, neighbors. Sometimes we speak of love and do not even realize it. For example, we often speak of charity for the poor, the stranger, the disadvantaged, without realizing that this, too, is a unique form of love ( _charity_ comes from the Latin word _caritas_ , meaning "love"). Even though these loves are different, they are alike in that love is valued most when it is unconditional and flourishing. A love less than this is unsatisfying, and could seem expendable. But what does unconditional mean in this context? What is truly demanded here? And is it reasonable to demand it? It was just these types of universal, human questions about love that inspired Benedict XVI to write his first encyclical on love. And in many ways the title alone, taken from the Bible, begins to explain the Christian significance of the tilma's message and, at the same time, to identify the attractiveness of the Gospel: _Deus caritas est_ (God is love). From him, we can learn how our love grows to become a higher, more perfect love:
It is part of love's growth towards higher levels and inward purification that it now seeks to become _definitive_ , and it does so in a twofold sense: both in the sense of _exclusivity_ (this particular person alone) and in the sense of being " _for ever_."... [L]ove looks to the eternal. [Emphasis added]
These are the two aspects of a pure, unconditional love that we seek: it is personal, and it is lasting. In a unique way, this is expressed in matrimonial vows, when the bride and groom pledge to be faithful in love to that specific person until death. But more than that, this unconditional love is shown in God's love for us—in his eternal love for each of us personally, as individuals. Importantly, God does not merely give love; he _is_ love. As such, he is the ultimate model of unconditional love. There is no condition that ends love; his love refuses to be stunted.
As mentioned earlier, the four-petaled jasmine on the Virgin's tunic spoke to a significant difficulty in the Toltec religion, namely, the Indians' belief that their supreme deity was completely inaccessible to them. Thus in the image of Our Lady of Guadalupe a love that is close and personal is evoked, but also important was the permanence of this love. The Aztecs believed that temporality was governed by the gods, and yet temporality was also a constant source of fear: Would the gods hold them in existence? After all, the four ages that had preceded theirs had ended in catastrophe, usually by a violent natural disaster. In fact, many of the Aztecs' sacrificial practices were coupled with fear. This was especially true at the end of every 104-year span, called a _huehuetilítli_. In these years, the Aztecs believed, one set of years ended and another, with the cooperation of the gods, might begin through the ritual of the New Fire ceremony, during which women and children were kept in their houses for fear that they would transform into wild animals and devour people. But the success of the ceremony was never certain, and each time the Aztecs feared the possibility that the gods would abandon them and end their world. These were gods who could not persevere in care for man. In this, man was more persevering, and the burden fell on all the Indian people.
The two elements of a purified, higher love—a love both personal and eternal—were lacking. Even more, these two elements were not being preached effectively, either in word or in example, by the Spanish missionaries and colonists. But in the Guadalupan apparition, these characteristics of the highest, purest love are shown symbolically in the enduring image on the tilma and expressed really in the Virgin's declaration of her motherhood and her role in the apparitions: her affectionate names for Juan Diego; her concern for his well-being; her healing of Juan Bernardino; her understanding of Zumárraga's doubt; her encouragement of Juan Diego even when he failed; her valuing of Juan Diego even when he saw himself as lowly and unworthy; her interest in his plans; her mestiza face, which made her truly one of the people of Mexico; her persistence in meeting Juan Diego even when he avoided her; even her sacrificial role in requesting a church, a lasting place of prayer, for her Son.
This last point is most significant: the Virgin comes not for herself but for God, on an errand of love. In his third epistle in the New Testament, St. John declares the depths of God's love, writing: "God so loved the world, that he sent his only son." In the apparition, coming in a way that expresses the particularity of the indigenous peoples and their cultures, the particularity of God's love comes through, so much so that we can apply it to each people and each person: God so loved the Aztecs, the Texcocans, the Tlaxcalans, the Spaniards, the New Yorkers, the Puerto Ricans, and so on, "that he sent his only son." In this way the event expresses not only the Virgin's love but God's love as well. Sending the Virgin is a personal touch.
## A CIVILIZATION OF TRUTH AND LOVE
We have heard significant voices express the need to improve our society through brotherly love. But that we speak of a "civilization of love" with any amount of familiarity is largely due to the writings of Pope John Paul II and more recently Pope Benedict XVI. Few have contributed so much to our understanding of the truth about the person and the truth about love, underscoring the real possibility—and necessity—of building a civilization upon love.
John Paul II's proclamation of a civilization of love first reached our ears in 1979, the day after his first visit to the Basilica of Our Lady of Guadalupe, when he postulated a specific truth for the foundation of our communication with each other: "The truth we owe to human beings," he said, "is first and foremost a truth about themselves." Cardinal Ratzinger, several years before becoming Pope Benedict XVI, likewise expressed the vital need for truth about the person particularly in regard to love: "Truth is love, and if love were to turn against truth, it would be mutilating itself." On the tilma, a suggestion of this can be seen in how the flower (symbolizing truth) is at the same time a heart with a face (the combination symbolizing the individual).
Of course, there are many truths about the human person that we can speak about. Many of the most popular explanations today define man by his interactions with the created world, including economics, politics, psychology and sociology. But Christ reveals the human person in light of the human person's interactions with the Creator. "The universe in which we live has its source in God and was created by him.... Consequently, his creation is dear to him, for it was willed by him and 'made' by him.... [T]his God loves man." While mankind resembles the rest of creation in being created by God, only man is called to a higher level of relationship to the Creator, a relationship of personal love with God, enabling each person to address God not only as "God the Creator" but, through Christ, as "Father."
Certainly, being a creator and a father are related: both produce something that did not exist before. But parenthood implies a correspondence, a continuity, a similarity—the similarity of being made "in the image and likeness" of another. And in this continuity we find in ourselves the "dignity which... brings demands" and our vocation to love. So important is this understanding of ourselves that soon after returning from Mexico, John Paul II devoted nearly four years to a series of weekly addresses that proclaimed this truth—what is now known as the Wednesday Catechesis on the Theology of the Body.
When we make the sign of the cross, praying, "In the name of the Father, and of the Son, and of the Holy Spirit," we identify the Trinity, the supreme model of a loving communion of persons, a communion of mutual love and truth. To be made in the image of God suggests something about the human person as a reflection of the Trinity: we, too, are created for a loving communion of people. We, too, are called to imitate in our lives and everyday dealings with others the same Trinitarian love and truth we are reflections of. This is the foundation of the civilization of love: that to be made in the image of God is not simply to be _fashioned_ as such, but to _function_ as an image of God—that is, to be ontologically destined for and capable of a life of loving communion with others.
Our connection to God goes beyond the genetic identity of our personhood. That is, God's fatherhood creates us for loving communion with others and at the same time gives us the capacity for loving communion with others. Like the heart-flower and the heart with a face on the _tilma_ , which are rooted in the divine, both the root of our personhood and the source of our love are not human but divine:
[God] has loved us first and he continues to do so; we too, then, can respond with love. God does not demand of us a feeling which we ourselves are incapable of producing. He loves us, he makes us see and experience his love, and since he has "loved us first," love can also blossom as a response within us.
God's loving relationship with us and our loving relationship with others are not separate relationships but connected. Christianity is relational: our relationship with Christ does not close in on itself, but shines forth in our lives, always demanding to be concretely revealed in our love of others. Through us, Christ's divine presence is no longer kept within the Church—as the divine water under the Templo Mayor—but is able to flow within and out from us.
In guiding our relations with others, what higher truth is there than the truth that, as made in the image and likeness of God, each person shares in a unique dignity? This is fundamental to our humanity, setting us apart from creation and demanding that we relate to one another in love. "Love contains the acknowledgment of the personal dignity of the other, and of his or her absolute uniqueness." Although this dignity ultimately comes from God, this is not a hidden or purely spiritual reality. Every person can—and naturally does—recognize some dignity that all people hold in common.
This is the "good news" of the Gospel that is too good to forget: that "Jesus Christ brought us a message that has emphasized the absolute value of life and of the human person, who comes from God and is called to live in communion with God." What Christ makes clear—in his incarnation, in coming to man fully as a man, and in his death—is that "each of us is the result of a thought of God; each of us is willed, each of us is loved." For this reason, to live as a Christian is not simply to follow rules of "being good" but to be united to a Person. As Benedict XVI said, "Christian ethics is not born from a system of commandments, but rather is the consequence of our friendship with Christ. This friendship influences life: if it is true, it incarnates and fulfills itself in love for neighbor." Our relationship with Christ is deeply positive, and rather than the source of prohibitions it is the source of love that compels us to return love with love and to treat others with compassion, generosity, and justice.
Even more, there is a kind of education through Christ, by which we each grow in love and become more like Christ, more like the ultimate child of God. As Christians, we understand that this becomes part of our identity: we bear the image of God and are also the face of Christ to those around us. "The Christian has the face of Christ imprinted in his heart in an indelible fashion. He is not only _alter Christus_ [another Christ], but _ipse Christus_ [Christ Himself]." Through love, we begin to see the face of Christ in others, and we ourselves through love are called to be the presence of Christ to others. In this education, Mary accepts a prominent part. For as the mother of Christ, she has an unprecedentedly close participation in God's Trinitarian love; even as she recognized the Messiah in her child, so she can "enlighten our vision, so that we can recognize Christ's face in the face of every human person, the heart of peace."
## A NEW SACRIFICE
In looking at the civilization of love in the heart-flower, we cannot overlook the idea of sacrifice simply because we associate the word with the violent and bloody sacrifices of the Aztecs. This love of God for man not only enables us to love each other but also gave us the supreme sacrificial love in Jesus Christ. By his life and death, he gave mankind his heart, in the most concrete way, on the Cross. In Christ we understand sacrifice in a new way: "This insistence on _sacrifice_ —a ' _making sacred_ ' [from the Latin _sacrum facere_ ]—expresses all the existential depth implied in the transformation of our human reality as taken up by Christ."
This act liberating mankind from sin is more than a mere declaration of dignity and worth. Likewise, this act is more than a historical event with timeless spiritual consequences. Christ's death is our salvation; his sacrifice was the only sacrifice that could truly enable man to return to communion with God. Every sacrament we celebrate depends upon the truth of Christ's sacrifice, and enables us to participate in it to some degree. In the Gospels, Christ's sacrifice is described physically after his death, when the Roman soldier pierces Christ's side with a lance and "immediately blood and water flowed out." As St. John Chrysostom reflects, "The water was a symbol of baptism and the blood [was a symbol] of the holy Eucharist." The water of baptism and the Eucharist are not only symbols of Christ's sacrifice and our participation therein but also a constant source.
A civilization of love engages reality. Like the barren Tepeyac that grew flowers out of the Virgin's love for man and attentiveness to Zumárraga's desire for truth, a civilization of love is the seedbed for truth. And like the hill-flower on the tilma, with its smaller flowers growing from the large flower, truth begets truth. When Benedict XVI spoke to educators in the United States, he said: "Truth means more than knowledge: knowing the truth leads us to discover the good. Truth speaks to the individual in his or her entirety, inviting us to respond with our whole being." This is the other aspect of being made in the image and likeness of God: as beings capable of love we are also beings capable of seeing the truth – another important attribute of God who is love. Our relationship with God and our development of a well-formed conscience thus purifies and elevates truth to a different level with new importance.
Seeing the truth—the truth about God, about his relationship with us, about who we are—can be overwhelming. Before the image of Our Lady of Guadalupe, Friar Zumárraga fell to his knees in penitence and awe. Juan Diego, too, when he encountered the Virgin, prostrated himself in her presence. Truth inspires awe; it inspires reverence. But the complexity of truth must never be used to obscure the good. Importantly, neither Friar Zumárraga nor Juan Diego was stunned into inaction; rather, both were inspired to act. Christianity and the Guadalupan event in particular give us a model of responding to truth and reality. Before the truth about the human person, his or her dignity and capacity to love, the only appropriate response is a gift of ourselves. This is a gift not of what we have but of who we are. In the following chapters, we will discuss how the event at Guadalupe highlights exactly this: who we are, and what this means about how we give. Precisely because it is a gift of ourselves, it is a gift that can be given wherever and whenever we are. In this, love truly becomes a universal presence. When we interact with God's creation—man and other beings alike—we interact with the fruits of love, the fruits of God's love. Through God's love, we are invited to imitate God's own creativity; we are invited to personally participate in his own being as Lord and Creator. And just as God's creative action is one of the ultimate expressions of his love, so with us: in all we do, love is the measure, the purpose, and the motivation.
# PART II
# Christ, a Life-Changing Event
# V.
_Liberated by Love_
_The radical freedom of man thus lies at the deepest level: the level of openness to God by conversion of heart, for it is man's heart that theroots of every form of subjection, every violation of freedom, are found. Finally for the Christian, freedom does not come from man himself; it is manifested in obedience to the will of God and in fidelity to his love_.
-JOHN PAUL II, MESSAGE FOR THE WORLD DAY OF PEACE, 1981
## A NEW AND TROUBLED WORLD
Shortly before becoming pope, Karol Wojtyla celebrated Mass at his homeland's much beloved Marian shrine, Czestochowa, in Poland. In his homily he reflected on the unique paths by which we engage some of the most complex and important elements of our world, saying: "To understand man, one must delve into the depth of the mystery; to understand a nation, one must come to its shrine." Few shrines express this so clearly as Mexico's Basilica of Our Lady of Guadalupe. In fact, the history of Mexico becomes incomprehensible without understanding its people's devotion to Our Lady of Guadalupe. Without this devotion, the face of the continent would be drastically different from what it is today. For Mexico of the sixteenth century, the apparition of Our Lady of Guadalupe was a dramatic invitation "to understand man" by "delving into the depth of the mystery" at a time when the Christian understanding of man—his freedom, love, and dignity—was being obscured by the most unchristian systems.
Two years before the apparition of Our Lady of Guadalupe, Friar Zumárraga penned a secret letter to Charles V, the king of Spain, a letter forbidden by the authorities from leaving Mexico's shores. For several months, the Spanish civil authorities in Mexico had feared such a letter being delivered, and did everything in their power to prevent it from being sent: they patrolled the roads, inspected ships from deck to ballast, and forbade anyone to accept letters from any friar unless the authorities read it first. So to send his letter, Zumárraga himself and another friar took the letter to the port of Veracruz at great personal risk; after all, the last friars who had tried to deliver Zumárraga's message had been robbed of not only the letter but all their possessions as well. Followed every step of the way by government agents, Zumárraga and his companion narrowly avoided having the letter confiscated, and when they reached the port, the letter was kept safe only because of the ingenuity of a Basque sailor who hid it in a cake of wax placed within a barrel, where it floated beside the ship with the other buoys. So although their pursuers arrived and searched the ship, the letter left the port undetected and a few weeks later reached the hands of Charles V.
The letter now resides in the Archives of Simancas, and it gives us insight into why the civil authorities during the Spanish occupation of Mexico went to such extremes to prevent Friar Zumárraga from contacting the king. It described in great detail the inhabitants' sufferings inflicted by members of the civil governing body in Mexico, the First Audience. Natives were enslaved, children sold, property confiscated, women abducted, contracts broken, dissenters knifed, workers swindled, and clergy threatened as the First Audience under Nuño de Guzman's leadership constructed a divisive, profit-seeking regime heedless of the human cost. "If God does not provide the remedy from His Hand," Zumárraga wrote despairingly to Charles V, "the land is about to be completely lost."
Considering the situation, it is no wonder the early missionary efforts failed. As the missionary Friar Mendieta recalled, "When the Spaniards again arrived to their lands, [the Indians] never ceased receiving the moral principles with enormous love and benevolence, until the point where they were shocked [by the First Audience's behavior] and learned their lesson." Besides the temporal damage to the people's lives, Mendieta lamented that the spiritual fallout caused by the many atrocities committed by First Audience officials had "completely prevented the salvation of an infinite number of persons."
Additionally, while considered some of the most learned and holy priests of their homelands, the friars themselves were unprepared to evangelize in the new language and culture, causing Friar Sahagún to lament their ignorance of Indian culture, likening it to a doctor hoping to cure a patient without knowing the illness. Facing the Aztec practices, they considered the religion diabolical for its violent sacrificial rites, and they, too, resorted to violent displays in tearing down the idols and statues. Such abrasive and reactionary tactics likewise proved ineffectual.
Although we often recognize Our Lady of Guadalupe's apparition as a defining moment in the life of St. Juan Diego, it was also a critical turning point for the Indians, the Spaniards, and Zumárraga himself, whose constant conflicts had led him to despair about the future of the land. History's first quantifiable indication of a massive, radical change in lives beyond Juan Diego's own was a flood of conversions after the apparitions—inspiring John Paul II to call Our Lady of Guadalupe the first evangelist to Mexico. Writing in 1537 in his _History of the Indians of New Spain_ , Friar Toribio de Benevente, whom the Indians called "Motolinia" (poor one), took great pains in calculating the number of baptisms from the figures submitted by each missionary and province. He concluded: "In my opinion and truthfully, there must have been baptized in the time I mention—a matter of fifteen years—more than nine million Indians." In fact, so great was the number of Indians seeking baptism that friars began abbreviating the baptismal rites, and in 1537 a council of missionaries gathered for the first of several meetings to discuss the challenges and needs of this new land. Many historians have attributed these many conversions to the effects of the Guadalupan apparitions. What is absolutely certain is that the continent for which Zumárraga had all but given up hope in 1530 was almost universally converted to Christianity in a matter of decades, and that Guadalupe is the single highest-profile Catholic event in the Americas in the sixteenth century.
## A NON-EUROPEAN CONVERSION
From the apparition account and the tilma, we can see how the event communicates the Gospel message and Christian concepts through Indians' culture and language. But to understand why such a message could inspire such conversions, and why it was communicated through the person of Mary by such an extraordinary medium as an apparition, we must look into more than cultural and linguistic analysis.
As radical as the large-scale conversion of Indians was for the Church in Mexico, it was also radical in the context of the sixteenth-century Europe, where the Church was suffering religious turmoil and failed religious reconciliation. Just months before Columbus encountered the New World, Spain had won the Reconquista, a series of wars regaining the land of Spain from Muslim rule, and Muslims and Jews in Spanish lands were forced to choose between conversion and expulsion. Similarly, two years before Cortés encountered the polytheistic Aztecs, the Catholic Church itself faced major division when Martin Luther wrote the Ninety-five Theses, marking the beginning of the Protestant movement, which drew millions of people from Catholicism. Evangelization in Europe was volatile and problematic.
In the twentieth century, several popular commentaries on the Guadalupan event compare the religious turmoil of the two continents. These commentaries often interpret the Mexican conversions as a type of healing within the Church, observing that at the same time an estimated nine million Europeans _left_ the Catholic Church for Protestantism, nine million Indians in the Americas converted _to_ the Catholic Church. Of course, this has its shortcomings: the conversion of one person can never make up for another person falling away, since each person's relationship with Christ is unique and irreplaceable. But this comparison is valuable for another reason: it points to the experience of the Guadalupan event and the subsequent conversions as something remarkably different from the European experience, where the king's religious belief held great sway over his people.
The Holy Roman Empire, ruled by Charles V at the time of the apparition, had been made possible through the conversion of whole countries and peoples of Europe. And in that history, often the hallmark of a country's conversion was that it was initiated by its leaders in a kind of top-down model. For example, after reverting to paganism once the Romans departed and the Anglo-Saxons invaded, England marked its new conversion to Christianity from the baptism of King Ethelbert of Kent in 597, which "had such an effect in deciding the minds of his wavering countrymen that as many as 10,000 are said to have followed his example within a few months." Likewise, France marks its conversion to Christianity from the baptism of King Clovis in 496, Poland from the baptism of King Mieszko in 966, and so on. Furthermore, expectations for this top-down model continued in Europe with the rise of Protestantism. For example, Lutheranism spread systematically through Germany according to the conversions of the county princes, and the effects of this were visible even into the twentieth century, as southern Germany was more Catholic than the Protestant north. Perhaps the most explicit example of this top-down model occurred just three years after the apparition, with the birth of Anglicanism in England. Rejecting the Catholic teaching on marriage and divorce, Henry VIII codified his rejection of the Catholic Church by writing the Act of Supremacy, a law establishing a new church, the Church of England, and naming himself—and all his heirs—as its leader. Even more, it commanded that all English citizens attend these Anglican services, demanding obedience to the king not only in political matters but in religious matters as well. This caused a delicate situation, to say the least, since anyone who rejected his religious mandates effectively was accused of treason, and many who resisted were punished, even put to death.
Thus for the missionaries and conquistadors, seeking out the Indian leaders was not only a savvy political move but a religious move as well. In this context, it would have been expected that the conversion of Mexico would soon follow the death of Emperor Moctezuma in 1520, the defeat of the Aztecs in 1521, the baptism of Moctezuma's sister Papantzin in Tlaltelolco in 1524, or even the baptism of Moctezuma's son. The account of this latter baptism, as Pardo notes, barely veils the obvious symbolism of his baptism as a result of political conquest. First, the event is recounted less for his request for baptism than for his dramatic shaking during the exorcism, making it seem as if "the devil was going out of him." Additionally, a sizeable part of the account of the event is devoted to the church later built upon the site of his home; the church was dedicated to St. Hippolytus, whose feast day became a major celebration throughout the land precisely because it was on his feast day that the Aztec Empire fell. In this, we can see an example of how evangelization and conversion had lost, under a cloud of political significance, their personal impetus and meaning. Consequently, though the Spanish had conquered Mexico politically and militarily, toppling the Aztec religious infrastructure, conversions were few and evangelization unsuccessful.
Resistance came as well from the Aztec worldview itself, in which the weight of religious authority rested not on the emperor per se but on tradition itself. This is partly why, after the Aztecs perceived weakness in Moctezuma's dealings with the Spaniards, who did not respect their temples and religion, Moctezuma's power was severely compromised; his dethronement, while not literal, was at least symbolically shown when he was stoned by his own people. This is also why, later, when the Aztec priests were challenged by the Christian missionaries to defend the Aztec religion, the priests appealed not to any traditional metaphysical argument but to tradition itself, saying in a 1524 debate: "It is a new word, this one you tell them, and because of it we are distressed, because of it we are extremely frightened. Indeed... our fathers... did not speak in this way.... And now are we the ones who will [perhaps] destroy the ancient law?" For the Aztecs, the wisdom of the past ages rooted the present and thereby gave meaning and direction to the contemporary world. If deprived of this rootedness, the Aztecs feared, their civilization, like a flower detached from its roots, would wither and collapse. And unlike their emperor or any other religious or political persons of authority, their traditions could be neither dethroned nor converted.
The drama of Cortés's campaign of conquest was more than the collapse of Mesoamerican military, social, economic, and political structures. By prohibiting sacrifices and other religious practices, he deprived their gods of their necessary sustenance, and the Indians anticipated an apocalyptic end of the world. Instead, they saw that, though the human sacrifices to the gods had ceased, the cycle of life around them continued. The result was increasing doubt among the Indians about their worldview. But doubt is not enough for conversion, especially in the face of the violence of those who practiced the religion of their conquerors. As the Aztec priests told the Franciscans:
_What are we to do then_ ,
_We who are small men and mortals;_
_If we die, let us die;_
_If we perish, let us perish;_
_The truth is that the gods also died_.
The collective depression from this crisis of faith was so great that some of the natives committed suicide.
Even as the Indians' way of life suffered this great religious and cultural crisis, they endured another profound assault when their very humanity and thus their rights were called into question. Was it Christian to conquer a land, appropriate other people's property, and even enslave the former property owners? Could they truly justify the subjugation of another people? Such doubts were voiced strongly by many missionaries as well as by some of the more conscientious lay Spaniards. Others, however, were eager to justify the invasion and appropriation by questioning the natives' innate capability to reason. If the natives were unable to demonstrate their humanity, they argued, then they would have no rights of ownership and their property could be seized. But if the debates were waged in Salamanca's university lecture halls and in the Spanish Court, the battleground was Mexico. Thus, the Indians needed liberation from the many forces impeding living a free life: their religion, their human sacrifices, their depression, the cruelties dealt and received, and the animosity that quickly developed between the conquered and the conquerors.
Speaking of freedom, Benedict XVI once wrote: "Freedom presupposes that in fundamental decisions, every person and every generation is a new beginning.... [N]ew generations can draw upon the moral treasury of the whole of humanity. But they can also reject it, because it can never be self-evident in the same way as material inventions." In a particular way, the situation of the Spanish discovery and occupation of Mexico marked a radically clear beginning when the "moral treasury of the whole of humanity" was at stake.
Although the Aztec political and religious authority were intertwined, Mexico would require a completely different method of conversion than countries in Europe. Mexico demanded a reconsideration of what evangelization is and what being a Christian means. Just as the failed attempts at governing the Indians raised new questions about what it means to be human, so the failed missionary attempts raised questions about the meaning of evangelization and being a Christian. The situation in Mexico demanded a reconsideration of conversion as a relationship with Christ, proposed though Christian witness, rather than as a transfer of religious loyalties tied to political authority. To answer these challenges, the missionaries could not simply follow the traditional European pattern.
Even as the old Christian culture was passing away, this conversion required the creation of a new Christian culture, precipitated neither by new law nor by new political leadership, but by a revelation of love. And rooted in love, this new culture would express some of the greatest concepts of Christianity in a new and renewed manner. It was a new kind of evangelization, an evangelization that was peer-to-peer rather than top-down, and as such it was marked by methods substantially different from the methods that had made Europe Catholic.
The recent precipitous decline of Christianity in Europe is significantly tied to its historical entanglement with that continent's monarchies. The Church at times became identified with the political power of the state, and as Europeans chose different forms of government, they also chose different forms of religion or none at all. It is for this reason that, discussing the legacy of the Church in light of the Roman emperor Constantine, Cardinal Ratzinger wrote in the 1980s, "In the long run, neither embrace nor ghetto can solve for Christians the problem of the modern world." What is needed is not a ghetto nor an embrace, but an engagement in which Christians preach primarily through the example of their own lives, of the great freedom that is brought by saying "yes" to Jesus Christ, and in this way informing the conscience of the society in which they live.
## MARY: A NEW EVANGELIST
By her apparitions, Our Lady of Guadalupe has replaced the top-down model of conversion with an evangelization rooted in a personalism independent of political and social hierarchy. Two of the titles by which she is often addressed in prayers—Morenita and Tonantzin—point toward two radical aspects of her relationship with the New World people that communicated this new evangelization based on personalizing Christ's love in a new way.
The name Morenita—a Spanish word meaning "the dark woman"—refers to the Virgin's darker skin tone, as depicted in the tilma's image. At the time, and especially the longer the Spaniards spent in the New World, race, ethnicity, and origin dominated the scale of social classes; Spaniards held the positions of power both socially and politically, while the Indians, as part of the conquered people, often held an inferior position. As the Morenita, the Virgin is identifiable as a mestiza. And while mestizos would later encompass most of the country, at the time these mestizo children were despised as products of the conquest and rape, and were left, as Bishop Vasco de Quiroga explained, to search among the animal stalls for food left for pigs and dogs.
Interestingly, since this was fairly soon after Cortés' conquest of Mexico, this was at a time when the number of mestizos in Mexico was still small—and the oldest mestizos in Mexico City would have been still quite young. As a mestiza, she is identified not only with the ostracized and rejected but also with the most vulnerable group, the children. Through identifying herself with them, her face becomes the face of the future.
A different facet of the universality of conversion of heart—a call to Christ not limited or prioritized by status or social position—is emphasized as well when in the apparition account the Virgin refuses Juan Diego's request that she send someone else—someone of higher social recognition—as a messenger to Zumárraga. At the Virgin's insistence, Juan Diego is instructed to seek out Zumárraga, a Spaniard known and chosen personally by the King and who, though good and humble, had an elite service in the Church and society as Mexico City's bishop. By the Virgin's request, Juan Diego is given a vital role in manifesting and promoting the solidarity that comes from a unified desire for God—a desire that transcends social hierarchy. Just as the mestiza Virgin, appearing with the face of the lowly, comes as the catalyst for conversion, so Juan Diego, a member of the conquered people, is enlisted in this significant gesture of heaven's interest in even the lowest. The mestiza Virgin, and to a certain extent humble Juan Diego, express the universality—the _catholicism_ —of God's desire for persons to turn to him, a universal desire that transcends the distances created by societies even as God's love sends missionaries to traverse oceans to make the message known.
The Indian Juan Diego and the mestiza Virgin, together occupying two lowly rungs of the New World's social hierarchy, become the catalysts for conversion, thereby introducing a new model of evangelization. Rather than the European pattern of conversion, moving from the top down in society, there is an obvious reversal. Though both the old world and new world models of evangelization had at their core the missionary activity of the Catholic Church, whose priests and bishops brought sacramental life to the previously unbaptized, the catalysts that opened each of these continets to Christianity could not have been more different. In most case in Europe, the missionary activity of the church gained ground through the good graces of a prince or king. In Mexico, the miracle the bishop had seen as necessary for the conversion of the people to the Catholic faith came not through the good graces of a prince but through the obedient actions of a simple Indian who followed the request of Our Lady, herself a mestiza woman. Instead of a conversion of a nation's people, it is a conversion of persons who together forged their mestiza nation.
The other title often attributed to Our Lady of Guadalupe, Tonantzin—a Náhuatl word meaning "our dear mother" or, colloquially, "our mama." According to Sahagún, this title was quite controversial, as some missionaries feared and believed that the Indians addressed Our Lady of Guadalupe in this way simply because they saw her as a manifestation of their Aztec goddess. Nevertheless, this title in its fundamental meaning simply means "our dear mother ( _tonantzin_ ). And the continued use of this title points to something very important about how the message of the Virgin was received into the lives of Mexican Christians: it was not enough to speak of the Virgin only as someone else's mother—the mother of God or the mother of Juan Diego. Her personal motherhood for each person, her motherhood to the Indians, and to all of us, was important, and the name "Tonantzin" recognized a relationship that no one should or would want to be denied in either earthly or spiritual life.
To come as a mother—mother of God, mother of Juan Diego, mother of all people—is to come precisely in the most personal capacity. Importantly, in the apparition account, the Virgin's motherhood is a continual role that is defined not by the fact of giving birth but by a relationship of love and care evidenced in her interactions with Juan Diego and Juan Bernardino, and later in her image.
At a time when the humanity of the Indians was still contested, the Virgin's mestiza face and her declaration of motherhood were a truly great expression of the Indians' humanity. When she declared she was the mother of Juan Diego, Mary removed any doubt: the Indians were indeed children of God and entitled to respect. In her mestiza face, the Indians recognized the Virgin not as a European but as a person of the New World, sharing in their distinct cultural identity and in their unique physical traits. For the Spanish, too, there is a profound message: these children were loved to such a degree that the mother of God took on their appearance, becoming family. As "our dear Mother," and specifically as "our dear mestiza mother," the relationship between the peoples of Mexico is delineated as a relationship of family.
Although this inculturation had a liberating aspect, it was not at its heart liberation through politics. It helped free the Church from the distrust of the Indians, and the Indians from the hopelessness of their own situation. In some ways, the reconstruction of the Indians' society can be seen in the historical accounts of the conversions after the apparition. Instead of conversions being inspired from the top down, that is, from political dignitaries to the people, the Guadalupan event allowed the evangelization of the New World to spread through personal exchanges between the Indians, especially within the family, working as a powerful complement to the work of the Spanish missionaries. As one missionary at the time recalled:
In the beginning, [the Indians] started going [to receive baptism] 200 at a time, then 300 at a time, always growing and multiplying, until they reached thousands; some from two days journey, others from three, others from four, and some from farther away. This caused great admiration in those who saw it. Grown people brought their children to be baptized, and the young baptized brought their parents; the husband brought his wife, and the wife, her husband.
Throughout Mexico's history, at various times the Church has been protected by the government and other times persecuted by the government. Although at a later time the Church in Mexico would be closely identified with the state, these early conversions appear to have occurred not as a result of Spanish civil authorities but in spite of the actions of many of them, including members of the First Audience. Coming thus as a mother brings a new dimension to the debates about conversion, the human person, and the building of society. The most universal element of being human—coming from a mother—is made the catalyst for solidarity. And the most fundamental element of society—the family—becomes the place where this society is created.
## A NEW KNIGHTHOOD
In the New World, many of those either officially knighted or other nobles of authority carried a devotion to Mary with them, including Cortés, who had a devotion to the Spanish Our Lady of Extremadura. In this, they reflected a long history of European knighthood, in which devotion to Mary constituted a strong part of the knight's identity as a Christian called to be a defender of Christianity. Even the knight's devotion to his lady was often seen as a suggestive parallel of his devotion to Our Lady, Mary. Consequently, St. Joseph was sometimes seen as the first knight, the first entrusted by God with the task of defending Christ and Mary, making him the model for the knight. (This view of St. Joseph was ratified further in 1870 when he was named patron and defender of the entire Church.) As defender of the faith, a knight was at the same time called to be a defender of those virtues and values upheld and embodied in Christianity, especially justice; as the thirteenth-century knight Geoffroi de Charny of France tells us, a knight fought "to defend and uphold the faith or out of pity for men and women who cannot defend their own rights."
In the New World, there was some degree of expectation by the Spanish Crown that this behavior, this defense of the faith and the disenfranchised, would—or should—continue. Beginning with the primary financier of Columbus's first voyage, Queen Isabel, the Spanish rulers sought to protect the rights of the natives, whom they considered "vassals" of Spain. In 1499, when a shipment of Indians arrived in Spain, Queen Isabel intervened directly, issuing a public proclamation demanding that the Indians be returned to their homeland in Hispaniola (today the Dominican Republic). After Isabel died, her husband, King Ferdinand, commissioned a group of scholars and theologians in the hopes of reaching some solution to the many problems facing the native people. As a result of this commission, Ferdinand promulgated the Laws of Burgos in 1512, enacting thirty-five laws intended to protect the Indians from maltreatment and to promote evangelization and missionary work among the natives, detailing many goods to be afforded to the Indians. Yet, an ocean away from the authorities, it became clear that legislation alone was insufficient. In practice, the rights upheld in the laws were often obscured and their enforcement lacking; the knightly defense of justice and of the poor was abandoned throughout Mexico by the very men entrusted with it, even to the point of Charles V naming Zumárraga, rather than one of the authorities or men of power, the "Defender of the Indians." The situation had reversed; the knights, nobles, and men-at-arms no longer protected the defenseless, but instead fought for personal gain. Into this situation, Mary herself enters, choosing a _macehual_ , a commoner and "a poor one," to help reestablish not only the humanity of the Indians but also the true vocation of the knight.
How could Spanish knights in the New World maintain a true devotion to the Virgin Mary and at the same time oppress those whom she now called children? For as Christians, we are called to see the face of Christ in others. And yet in this situation, while devotion to Mary was strong among the Spaniards, they failed to see or treat their Indian neighbors as Christ. After the apparition, when the Spanish and the Indians looked at the face of their beloved Mary, they saw the face of the other—their neighbors, their enemies, their parishioners. To reject each other was to reject Mary. To reject Our Lady of Guadalupe as an Indian daughter is to reject she who also appears as a Spanish daughter. In Our Lady of Guadalupe, each becomes the other, bringing a new visual manifestation of Christ's command: "Love your neighbor as yourself."
## LIBERATED CHURCH VS. LIBERATION THEOLOGY
The conversion of the Aztec Empire after the apparition of Our Lady of Guadalupe presents an extraordinary contrast to the other massive conversion in the history of Christianity: that following Emperor Constantine in the fourth-century Roman Empire. Speaking of Mary's role in the New World evangelization, Cardinal Ratzinger observed:
It has always been the Mother who reached people in a missionary situation and made Christ accessible to them. That is especially true of Latin America. Here, to some extent, Christianity arrived by way of Spanish swords, with deadly heralds. In Mexico, at first absolutely nothing could be done about missionary work—until the occurrence of that phenomenon at Guadalupe, and then the Son was suddenly near by way of his Mother.
Up to the time of Constantine, the spread of Christianity was originally motivated through the personal testimony of individual Christians, but suffered under the capricious whims of some of Rome's early emperors, beginning with Emperor Nero (who reigned around the time of the martyrdoms of apostles Peter and Paul). With the conversion of Emperor Constantine, Christianity gained not only a new protector but a new level of protection in law. In A.D. 313 Emperor Constantine issued the Edict of Milan, which gave official legal recognition to the Church, granted Christians freedom of worship, and restored Church properties that previously had been confiscated. In addition, in A.D. 325 Constantine convened the Council of Nicaea, which clarified the Church's teaching on Christ as both human and divine, and from which we now have the Nicene Creed. With the freedoms secured under Constantine, Christianity flourished, and by the end of the fourth century, it was declared the official religion of the Roman Empire.
In a way, there is a direct parallel between Constantine and Juan Diego: both were catalysts for the spread of Christianity in their respective times and cultures. But unlike the evangelization under Constantine, the conversion of a hemisphere begun by the Virgin of Guadalupe through her messenger Juan Diego and with the approval and implementation of the bishop was not born of political power. It was initiated first among the common people (family members Juan Diego and Juan Bernardino) and then spread by the Church throughout Mexico.
Appearing to both Juan Diego and Juan Bernardino, Our Lady of Guadalupe gives the message, based on the expression of Christ's love, to a family—which John Paul II so often called the first "school of love"—thereby accentuating the importance of the family's role in creating a civilization of love. Furthermore, through the faith and love of Juan Diego and Juan Bernardino, the loving message of Our Lady of Guadalupe immediately incorporates them into the universal family, the Church, for whom and through whom her message gains its full fulfillment and true meaning. From the family narrowly understood, as in the case of Juan Diego and his uncle, to the larger "human family" in Mexico—Indian, mestizo, and Spanish—all were united in her loving message and in a short time united also in their faith, nourished by the sacraments of the Catholic Church.
To evangelize effectively, the Church has always adapted to the situations in which it finds itself. In late antique and early medieval Europe, this often meant creating an alliance with the indigenous power structure of missionary lands, since the political hierarchy held great sway over the beliefs of its people. This "embrace," as Pope Benedict has pointed out, was not a perfect solution. With the advent of the Protestant reformation in the 16th century, the limits of this model became abundantly clear. Finding the opposite extreme in New Spain, a Church reduced to near "ghetto status," Our Lady of Guadalupe presented a perfectly inculturated message of conversion that began the process of transcending a need for an alliance with political power and drew the Church into the mainstream of the life and society of the New World.
Thus although today some see liberation as a political solution to a variety of spiritual and ethical problems, the message of Our Lady of Guadalupe could not be more different: her solution was not political but spiritual. And her solution, although articulated clearly in the Mexican situation, is one that is relevant in all ages, as Cardinal Ratzinger noted:
The Church cannot choose the times in which she will live. After Constantine, she was obliged to find a mode of coexistence with the world other than that necessitated by the persecutions of the preceding age. But it bespeaks a foolish romanticism to bemoan the change that occurred with Constantine while we ourselves fall at the feet of the world from which we profess our desire to liberate the Church. The struggle between _imperium_ and _sacerdotium_ in the Middle Ages, the dispute about the "enlightened" concept of state churches at the beginning of the modern age, were attempts to come to terms with the difficult problems created in its various epochs by a world that had become Christian.
The protections and limitations of close alliances between religious and civil authority were keenly visible in Zumárraga's life. On one hand, he had been named for the bishopric by Charles V himself, and sailed to the New World as the first bishop of Mexico City alongside the first Spanish civil authority, the First Audience. The First Audience, on the other hand, would use its authority to inhibit the missionaries and restrict the work of the Church. Additionally, not only was Zumárraga officiating in religious matters, but also Charles V had appointed him the "Defender of the Indians," situating him in the middle of the debates and conflicts over the mistreatment of the Indians. This was exacerbated by the fact that although he had been granted the title, his actual authority was challenged and constrained when the First Audience threatened to kill any Indian who ventured to Zumárraga for protection.
In contrast to this conflict between Church and state, the apparition account and the oral tradition passed down from the event are strikingly clear that this was a purely religious event. Juan Diego was a _macehual_ commoner uninvolved in the governing of the land; Zumárraga was invoked as the bishop. The Virgin tells Juan Bernardino of Juan Diego's visit even before the bishop approves. Already, it has become simultaneously a religious and a family matter. Even the accounts of the celebratory procession for the completed chapel relate how all the people—"absolutely everyone, the entire city, without exception"—came to the chapel and marveled at the image.
In a poignant way, this change toward a freed Church and a freer people is shown in how the practice of vigils was adopted by the later viceroys in New Spain. In the Middle Ages, the tradition of knighthood included praying a vigil in a chapel the eve before one's knighting ceremony, during which one would pray all night for courage, wisdom, and holiness. Continuing this tradition, newly appointed viceroys in New Spain would spend the night at Guadalupe before entering Mexico City. And at the end of their term as viceroy, many would keep another vigil with the Virgin before returning to Spain. Motivating this practice was a desire for a beneficent coexistence between the Church and the state, between the practice of faith and civil commitments and roles; the one should neither hinder nor obscure the other, but rather inform and serve.
Christian social doctrine requires continuous renewal of social structures to provide greater respect for the dignity of persons. For this reason, speaking of the role of love in a civilization, Benedict XVI noted:
Love— _caritas_ —will always prove necessary, even in the most just society. There is no ordering of the State so just that it can eliminate the need for a service of love. Whoever wants to eliminate love is preparing to eliminate man as such. There will always be suffering which cries out for consolation and help. There will always be loneliness. There will always be situations of material need where help in the form of concrete love of neighbour is indispensable. The State which would provide everything, absorbing everything into itself, would ultimately become a mere bureaucracy incapable of guaranteeing the very thing which the suffering person—every person—needs: namely, loving personal concern.
Peace, John Paul II constantly reminded us, requires freedom, justice, love, and truth. But it is only when each person recognizes and lives according to the dignity of each person that forgiveness seems reasonable, that justice is possible, and that peace is made real. For as Pope John Paul II recalled, "peace is not essentially about structures but about people." A civilization of love is strengthened with laws, but cannot be built solely with them. Fundamentally, it is only the encounter of love that can promise to renew both social structures and individuals. This was the origin of the change in Juan Diego's life, and this was the promise millions of converts saw and experienced through Our Lady of Guadalupe when they described their new faith in Christ.
Importantly, this message of love and liberation from the trials of life was not seen as the only or most significant message. After all, "the Church desires the good of man in all his dimensions, first of all as a member of the city of God, and then as a member of the earthly city." When this order is reversed, both are threatened. Unfortunately, this is sometimes a danger facing Christians today, when Christianity is reduced to a program of charity without an adequate spiritual dimension. As Cardinal Ratzinger noted:
The danger of some theologies is that they insist on... the exclusively earthly standpoint of secularist liberation programs. They do not and cannot see that from a Christian point of view, "liberation" is above all and primarily liberation from the radical slavery which the "world" does not notice, which it actually denies, namely, the radical slavery of sin.
Only by recognizing the profound reality of good and evil can reality and progress be not only measured but judged. When theology is harnessed only to drive earthly projects of "liberation," without acknowledging the spiritual, such liberation sacrifices what is truly fulfilling. Programs based on this kind of theology, called "liberation theology," begin with the view that "all reality is political," and thus reduce liberation itself to a strictly political concept that can only be achieved through strictly political means and action. In short, it endorses the view that "nothing lies outside political commitment. Everything has a political color." In effect, theology and our relationship with God are likewise reduced to the politically tangible, and any theology that is not "practical" or essentially "political" is misconstrued as "idealist" and divorced from reality. More than that, it condemns any theology that does not embrace this radical political dimension as a "vehicle for the oppressors' maintenance of power."
A decade before liberation theology entered Church circles, this type of flawed view of the Church as essentially a temporal power had led to the persecution of Catholics in Mexico during the 1920s. During this time, the Mexican government, under President Plutarco Elías Calles, decreed and enforced severe penalties on clergy and laymen alike, leading to the execution of many. Nevertheless, Mexican Catholics held on to their faith—and to their devotion to Our Lady of Guadalupe. Despite the hardships during this time, one of the first dual-language editions of the _Nican Mopohua_ was edited and published (1926), photographer Alfonso Marcue discovered and studied the reflections in the Virgin's eye (1929), and an examination of Tepeyac's flora and fauna was conducted in relation to the apparition roses (1923). It was during this time that the Basilica of Our Lady of Guadalupe was bombed (a few months after another bomb threat had been foiled), which caused such damage within the church but failed to destroy the main target, the tilma; the event inspired six thousand Catholics to gather in prayer at the cathedral that night, and caused the Mexican president, Álvaro Obregón, to visit the site himself, although his reputation for anti-Catholicism set off a protest. After these bombing fiascos, a different approach was taken. The government attempted to bring Guadalupe under its control in order to harness what it saw as Catholicism's "power." In another example of an attempt at conversion from the top down, the government founded a rival religious organization, the nationalist "Mexican Catholic Church." In addition, the leading nationalist labor organization, the Confederación Regional Obrera Mexicana (CROM), founded an underling group called the Knights of Guadalupe as an alternative to independent Catholic organizations such as the Knights of Columbus. The Knights of Guadalupe even went so far as to request that the Basilica of Our Lady of Guadalupe be turned over to the control of this nationalist organization in the hopes that it might become the seat or "Vatican City" for the new nationalist church. However, despite the pressures to convert to this non-apostolic church, few joined, and even the choice of name (Knights of _Guadalupe_ ) could not hide the order's allegiance to the nationalist church. In 1925, when the Knights of Guadalupe and the rogue "patriarch" of the newly organized nationalist church entered the Soledad Church near the basilica and removed the priest while he celebrated Mass, so great was the outrage that the parishioners protested long after President Calles closed the church.
(Here, for the sake of clarity, it must be noted that there is currently another existing order, also called the Knights of Guadalupe—Caballeros de Guadalupe—but this organization is in communion with the Catholic Church and bears no past or present relationship to the nationalist order of the 1920s.)
President Calles, the nationalist church, and the Knights of Guadalupe failed to understand the fundamental nature of both Christianity and the Guadalupe message. Even the Mexican president before Calles, Álvaro Obregón, two years after the failed bombing of the basilica, showed a greater understanding of the Guadalupan event: "The Virgin of Guadalupe always has been regarded as Mexico's Queen; as such she merits our gratitude and respect." But even this falls short. As described in the _Nican Motecpana_ :
She not only came to show herself as the queen of heaven, our precious mother of Guadalupe, in order to help the natives in their mundane miseries, but actually because she wanted to give them her light and her help, so that finally they would know the true and only God and through him see and know life in heaven.
The significance of Our Lady of Guadalupe is reducible neither to her value as "Mexico's Queen" nor to her value as an intercessor, as one to whom the Mexican people can look for material help and support. Rather, her value lies more precisely in how she never comes alone, in how she always brings her Son. It is true that in her actions, words, and image, Our Lady of Guadalupe speaks to the need for change, but the change she specifically requests is a new church on Tepeyac, and the purpose for this church, she tells us, is to present her Son. While the Church offers programs geared toward charitable service and the improvement of society, she is not reducible to these services and programs. The Church, like Our Lady of Guadalupe, never comes alone, but always with Christ; he is her true measure and gift to the world.
Like the message of Our Lady of Guadalupe, the message of the Church is to be preached in all times and to all people. She has stood in the center of the hemisphere, through times of close Church-state cooperation, and through times of anticlerical persecution of the Church. But her message of building a civilization of love has never changed. Like Juan Diego, we are called to spread the Gospel, to build the civilization of love his "dear mother," our _"tonantzin"_ represents. And we are called to do it at all times, in all political circumstances, and in all countries on the continent of baptized Christians.
## GUADALUPAN EVANGELIZATION
IN THE UNITED STATES
In Mexico, it may be easy to see the pervasiveness of the Guadalupan apparition in the Catholicism and devotional spirit of its people. In the United States, we need only look in our Capitol building to find a hint of how Our Lady of Guadalupe transformed our land through the faith of Catholic evangelists.
Many years before the United States had acquired its current breadth of land from the Atlantic to the Pacific oceans, Eusebio Kino arrived on his first missionary expedition to California, bringing with him an image of Our Lady of Guadalupe. As he exited the ship, a group of Indians came forward with their hunting weapons. Not knowing what to expect, Fr. Kino laid the image on the ground before them. Upon seeing the image, however, the Indians dispersed, only to return again with pearls and other precious treasures from their homes, which they strewed upon the image. Significantly, Our Lady of Guadalupe is treated not as a miracle but as a presence, as an evangelist. The missionaries trusted in the efficacy of her presence as an evangelist; she is an evangelizing mother to whom the missionaries continued to look in order to carry on the tradition of evangelization first initiated on Tepeyac. Understandably, after his encounter with the Indians, Fr. Kino established several missionary churches along the California coast for them, calling the first settlement in La Paz after Our Lady of Guadalupe.
While the codex of the tilma's image and its glyphs would not have been intelligible to the Indians to whom Fr. Kino first showed the image, the message of civilization of love and truth became truly evident in Fr. Kino's work. His care for the Indians extended beyond religious practices as he attended to their civilization itself, bringing about extensive improvements in the Indians' living conditions by better connecting them to other villages and building schools, roads, rancheros, and stockyards. In Fr. Kino's work as a priest and social pioneer, we find an exemplar of the type of evangelization inspired and enabled by Our Lady of Guadalupe. So great was Fr. Kino's presence that in 1965 the State of Arizona sent a statue of Fr. Kino—complete with crucifix, compass, and rosary—to the United States Capitol's Statuary Hall as one of two heroic persons who gave so much to the people in that state.
# VI.
_A Call to Conversion_
_"What does Mary mean to you, personally?" "An expression of the closeness of God... And the older I am, the more the Mother of God is important to me and close to me."_
–JOSEPH CARDINAL RATZINGER
## THE MIRACULOUS FACE OF FAITH
In the New World, the growing popularity of Our Lady of Guadalupe drew suspicion, so much so that, even after Archbishop Alonso Montúfar delivered a homily extolling the Guadalupan devotion, the Franciscan Provincial of Mexico, Friar Bustamante, fired back with his own harsh critique. He condemned the Indians' fascination with the image of Our Lady of Guadalupe on St. Juan Diego's tilma, suggesting that "the first person to claim that the image was capable of miracles should have been given one hundred lashes of the whip." Following this controversy, in September of 1556, Archbishop Montufar ordered an investigation into the Guadalupan devotion. To some degree, an important concern in this was the extent to which the Guadalupan devotion inculcated truly Christian beliefs and practices, and especially the extent to which it pointed beyond the Virgin to Christ. The danger to be avoided was perhaps expressed best by the great twentieth-century Mexican muralist Diego Rivera, when he said: "I don't believe in God... But I do believe in the Virgin of Guadalupe." Already the very location of the Virgin's shrine alarmed some of the missionaries who questioned how Tepeyac hill itself might influence belief, since nearby had once been an Aztec temple to the goddess Cuatlicue. Although many Catholic churches in Europe and in Italy, in particular, were built over pagan temples, in Mexico this particular spot drew suspicion, even leading some to distance their own devotion to the Spanish Virgin of Guadalupe (Our Lady of Extramadura) from the Indians' devotion to the Mexican Virgin of Guadalupe by insisting that the Virgin's title be changed to an Indian name, Tepeaquilla or Tepeaca (after Tepeyac). Nevertheless, by the end of Montúfar's investigation, the testimonies given ultimately vindicated devotion to Our Lady of Guadalupe and the popularity of her miraculous image, reaffirming it as a true expression of the Christian faith and an encouragement of the Gospel message. But how?
Listening to the Gospels read at Mass, we hear of many miracles performed by Jesus during his earthly ministry. Some are extraordinary miracles like raising the dead to life and healing the severely sick, and we can easily see them as expressions of God's grandeur and power. But they are also always linked to faith. As John Paul II explains, "Faith precedes the miracle and indeed is a condition for its accomplishment. Faith is also an effect of the miracle, because it engenders faith in the souls of those who are its recipients or witnesses." But, as John Paul II also points out, first and foremost, miracles are a mark of Christ's sonship, because "all that he does, even in working miracles, is done in close union with the Father."
The miracles are therefore "for man." In harmony with the redemptive finality of his mission, they are works of Jesus which reestablished the good where evil had lurked, producing disorder and confusion.... No other motive than love for humanity, merciful love, explains the "mighty deeds and signs" of the Son of Man.
Our Lady of Guadalupe's message of love and Christian personalism spoke to reestablishing a moral order that had been corrupted. But we cannot respond to this Christian personalism by interpretation alone. We live according to this Christian personalism only if we ourselves are continuously transformed. For this reason, like Christ's miracles during his public ministry, the miraculous occurrences at Tepeyac are "closely linked to the call of faith,"—not only because they are "miraculous" or because they are "religious" occurrences, but because they lead us to the source of our transformation, to her Son who redeems. In this, Our Lady of Guadalupe's apparition becomes a participation in Christ's sonship and Christ's mission. Through a gesture of closeness, words of compassion, and an invitation to come to Christ in our difficulties, Our Lady of Guadalupe makes a declaration of faith.
Of course, the faith of those engaged in the miracle, like Juan Diego, differs from the faith of those who approach the miracle afterward. There is no guarantee that the miracles will be received with faith or inspire faith. Even some of Christ's apostles who had witnessed his miracles betrayed him (Judas Iscariot), denied him (Peter), and doubted him (Thomas). Christ himself noted in one parable that for some, even if a person should return from the dead and give a warning to his brothers, his brothers would not change their ways, because they had already rejected the words of Moses and the prophets. This parable is interesting in that it suggests that there is a certain way of approaching God through the miraculous: "'Blessed are the pure in heart, for they shall see God.' The organ for seeing God is the purified heart." The miraculous is not a force, but a presentation, a communication, which can be ignored, misconstrued, or accepted only in part. It presupposes an openness of heart—an openness to transformation and to truth. This is in part "the task of Marian piety": "to awaken the heart and purify it in faith." And through it, we too become blessed to see God in each other, and to recognize his work in our lives and in each other's lives.
## TO JESUS THROUGH MARY
Ingrid Betancourt, an important political and social activist in Colombia, was kidnapped by Marxist guerillas while campaigning as a presidential candidate in 2002. As a rising political figure, her capture became an international cause of concern, especially in Europe, because of her dual citizenship with France. On July 2, 2008, after six years of failed negotiations and efforts for release, she and fourteen other hostages were liberated from their captors in the jungles of Colombia in a risky but ultimately bloodless military operation. Asked afterward about her trials, she spoke about her faith and her prayers to Mary. "When I thought of the Virgin, I thought of the Virgin of Guadalupe. I always felt she was very close to me. I know she is close now and is helping us and will help all those who continue in captivity in Colombia. She will bring them out, you'll see; she will do this miracle for us." For Betancourt, faith was not only a support during her trials but a foundation for life. "Without faith there is no hope, without hope there is no strength, no fortitude to continue fighting. Faith is everything; it's what gives meaning to life, especially faith in Christ."
Betancourt's rescue was not literally or scientifically miraculous in the sense of being absolutely impossible, nor would bringing out the rest of the captives be. However, as Betancourt suggests, obstacles do not have to be marked by scientific impossibility in order to be serious impediments. Human nature, sin, ignorance, habit, and fear can all seriously work against the hallmarks of a civilization of love and peace: truth, freedom, and justice. What is miraculous, what is worthy of our wonder (from _mirari_ , to wonder), is a change of heart.
This is one of the great beauties of conversion. Recalling the profound joy of the baptized Indians and of those preparing to receive the sacrament, one missionary wrote:
To see the fervent desire which these new converts brought to their baptism was truly something to notice and marvel at. One does not read about greater things in the primitive Church. And one does not know what to marvel at most, seeing these new people coming or seeing how God brought them.
This, too, is the true marvel about the apparition: the many changes of heart that happened in the face of the miraculous. Some of these changes are very real and practical, like Zumárraga accepting Juan Diego to stay in his house after he had previously sent Juan Diego away twice before; others are changes in the face of mystery, like Zumárraga's servants refusing Juan Diego admittance, then letting him in after the roses transform in Juan Diego's tilma. The most human changes are more subtle actions of following and trusting even when the path is not entirely understood; Juan Diego considers himself unworthy to approach Zumárraga, and yet at the Virgin's insistence he does. But why? He is still, after all, "a man from the country;" his tilma had not yet been imprinted with an image that would bear the signs of his dignity as a child of God. But he trusts, and this trust gives him determination and directs his determination. He goes because the Virgin has helped him to understand the honor of his sonship. He goes because he trusts the Virgin, who leads him and teaches him to follow a new relationship. This is miraculous, not in the sense that it is impossible, but in the sense that, through the Virgin of Guadalupe, Juan Diego, too, experiences a conversion of the heart.
When John Paul II placed his devotion to Mary as central to his pontificate, choosing the expression _Totus Tuus_ ("Entirely Yours") as his apostolic motto, he must have had Mary's role in inspiring conversion and leading others to Christ in mind. The expression was taken from one of St. Louis de Montfort's prayers to Mary, and in a personal way it became not only an expression of John Paul II's devotion to Mary but a result of his deeper understanding of the purpose of Marian devotion.
_Totus Tuus_. This phrase is not only an expression of piety, or simply an expression of devotion. It is more. During the Second World War, while I was employed as a factory worker, I came to be attracted to Marian devotion. At first, it had seemed to me that I should distance myself a bit from the Marian devotion of my childhood, in order to focus more on Christ. Thanks to Saint Louis of Montfort, I came to understand that true devotion to the Mother of God is actually Christocentric, indeed, it is very profoundly rooted in the Mystery of the Blessed Trinity, and the mysteries of the Incarnation and Redemption.
As it did for the young John Paul II, devotion to Mary leads one more closely into the mysteries of Christ and his mission of salvation in the world. Historically, it was only through Mary that God took on human flesh and became one with us. And today, like Juan Diego centuries before, we can be brought into closer unity with God by following Our Lady of Guadalupe's message of trust and love.
In the working document written in preparation for the Synod of Bishops for America, one of the subjects proposed for consideration was the role of Mary in the health and evangelization of the church in the Americas: The "Virgin Mary will be a model of conversion, communion and solidarity for the Church in America, so that the saving activity of her Son may reach all on the continent." Looking at the Gospel accounts mentioning Mary's life, we find significant moments when Mary's life, words, and interiority give a clear, concrete model. If the communion of saints truly matters, it is not only possible to detect a Marian model but a uniquely Guadalupan Marian model of conversion, communion, and solidarity.
After all, Our Lady of Guadalupe, Our Lady of Extremadura, Our Lady of Fatima, and countless other titles Mary is given throughout history and throughout the world are ultimately the same: each is Mary. There is no—and can be no—contradiction with the Virgin Mary spoken of in the Gospel. Mary is a woman for every age, and yet how she expresses herself can be different. In every age, she continues to participate in Christ's revelation of himself through miracles, just as she did in the first miracle of Christ recorded in the Gospels—the miracle of changing water into wine at the wedding at Cana. She continues to be aware of our situation and to propose to us that, as once she did at Cana, we "do whatever he tells us." What is unique among Marian apparitions is how each speaks to specific persons, in a specific time and place, who have unique problems, language, and preconceptions.
Consequently, it is possible—and for us in the Americas, we have a special invitation—to reflect on and follow the example of Our Lady of Guadalupe in our efforts toward personal conversion, communion, and solidarity. The subjects addressed in particular during the apparition—Mary's motherhood, charity, witness, intercession, trust, and truth—form the basis of the Guadalupan model of conversion, communion, and solidarity, and become the foundation of Guadalupan spirituality.
## MARIAN CLOSENESS TO GOD
In the previous chapter, we considered Mary as the first evangelist, and thus her involvement in "making converts." Although we can perhaps better understand Mary as a model evangelist, we do not ordinarily think of Mary as a model of communion, solidarity, or especially conversion. We often associate conversion only with converting from one religion to another religion, like the Indians converting from Aztec beliefs to Christianity. But conversion, as Benedict XVI said, is much broader and more continual than this:
What does "to be converted" actually mean? It means seeking God, moving with God, docilely following the teachings of his Son, Jesus Christ; to be converted is not a work for self-fulfillment because the human being is not the architect of his own eternal destiny. We did not make ourselves.... Conversion consists in freely and lovingly accepting to depend in all things on God, our true creator, to depend on love. This is not dependence but freedom.
Conversion is not a mere declaration of beliefs or a mark of membership. It is a wholehearted adherence to a Person, a continual "yes" to the transformative power of God's love. Its requirement is a simple gesture, a simple "yes," and in this way it is something we can all do; even more, it is something we are all called to. But importantly we are not left without a model; because conversion as a constant "yes" to God is something that Mary herself did so well, so perfectly, and in such a clear manner, it is possible for us to live our lives guided by her example. And for us today, in a new time and situation, Our Lady of Guadalupe again presents a model of seeking God, following Christ, depending on God, and depending on love.
It was precisely this universality of conversion that, on the eighth centenary of St. Francis' conversion, Benedict XVI underscored, saying: "Speaking of conversion means going to the heart of the Christian message, and at the same time to the roots of human existence." Of course, there are many ways we can describe the heart of the Christian message, but this is how Benedict XVI described it to a group of Catholic youth:
Here then we have reached the heart of the Christian message: Christ is the response to your questions and problems; in him every honest aspiration of the human being is strengthened. Christ, however, is demanding and shuns half measures. He knows he can count on your generosity and coherence; for this reason he expects a lot of you. Follow him faithfully and, in order to encounter him, love his Church, feel responsible, do not avoid being courageous protagonists, each in his own context.
To be converted thus means not pursuing one's own personal success—that is something ephemeral—but giving up all human security, treading in the Lord's footsteps with simplicity and trust so that Jesus may become for each one, as Blessed Teresa of Calcutta liked to say, "my All in all".
Often times, God's love is present to us in ways that we can neither anticipate nor comprehend. And yet, regardless of our intellectual limitations, preconceptions, and sometimes flawed expectations, the mystery of God's love is not, to borrow a phrase from Winston Churchill, "a riddle, wrapped in a mystery, inside an enigma." It is something so immense and overwhelmingly great that we can only experience and know it one part at a time. Like Juan Diego, who could not immediately understand the significance of the Virgin's miraculous apparitions and her declaration of motherhood and so tried to avoid her in his time of greatest need, we can be overwhelmed by the mystery of divine love, not immediately able to recognize its practical and profound significance. Importantly, however, the more we open ourselves up to it, the more it will make sense in our lives.
Speaking of the spiritual life, Fr. Benedict Groeschel once noted that spiritual intimacy requires three things: openness, self-giving, and vulnerability. The first two—openness and self-giving—are easy to accept in our spiritual life, precisely because we see their importance in our relationships with family and friends. Openness and self-giving are often hailed as the cornerstones of a good marriage, and most erroneous views and treatments of the marriage bond often return to some basic rejection of these two necessities. However, vulnerability is less often discussed, and for many it is the real crux of spiritual difficulties. To truly love completely is to love with complete openness and self-giving, even with the knowledge that one may be changed, one may even be hurt. It is the vulnerability Christians see in Christ, "obedient unto death, even to death on a cross."
Frankly, vulnerability cannot be explained as a good in a secular society. Today, we often equate vulnerability with weakness and inability and autonomy with strength and true happiness. As the German philosopher Friedrich Nietzsche wrote, happiness is "the feeling that power _increases—_ that resistance is overcome, not contentment, but more power; not peace at any price, but war; not virtue, but efficiency." To the contrary, in Christ men and women are called to be people of the Beatitudes. The paradox of Christianity is that in Christ "power and service went together."
If to serve is to become vulnerable through a gift of self, to serve through God's love is to become vulnerable to God's love and to experience service not as a loss but as a powerful fulfillment. It is the vulnerability seen in Mary's "yes" to God. It is the vulnerability asked of Juan Diego, to change his direction and his plans and to risk the ridicule of Zumárraga's servants, the humiliation of enduring Zumárraga's doubt, the fear and disappointment of failing, the frustration of being entrusted with a task believed to be too large for his shoulders. It is a vulnerability that leads us to the edge of what is easily desirable or possible through our efforts. It is a vulnerability that demands faith.
In Mary, we learn how closely God comes to us in this way. And in Mary, we learn how to be open, giving, and vulnerable to God. We learn to trust God. Millions of people around the world remember John Paul II for saying, wherever he visited in the world, "Do not be afraid! Open wide the doors to Christ!" In Benedict XVI's first homily as pope, he explained why we should not be afraid, why we should have the courage to welcome Christ in our lives: "Do not be afraid of Christ! He takes nothing away, and he gives you everything. When we give ourselves to him, we receive a hundredfold in return. Yes, open, open wide the doors to Christ—and you will find true life." This was the hopeful message of the Virgin in telling Juan Diego that his uncle had been healed. This was the courage of Juan Diego in not returning to this uncle, believing in the Virgin's word that Juan Bernardino had been healed.
## HUMANITY RENEWED
It is significant that Mary appeared to Juan Diego after his formal conversion to Christianity—his baptism—rather than before. Juan Diego's conversion to Christianity marked a definitive change in his life. With the Indian culture's emphasis on tradition and the wisdom of the ages, his conversion must have been very difficult. But even more, it marked a change of heart, a change in what he desired, since even after his baptism, Juan Diego pursued the faith through not only the sacraments and prayer, but also through catechesis. That is, Mary comes to a Catholic in order to encourage him to become a better Catholic. She comes to a Catholic and administers to his—and his community's—deepest needs. She comes to a Catholic and engages him in active Catholic life.
In the hagiographical writings and verbal testimonies about Juan Diego's life, often people focused on the facts of his life (marriage, location, age, etc.) and his sanctity. Juan Diego's flaws or spiritual difficulties before or after the apparition are largely or completely overshadowed by his record of holy living after the apparition. Even so, in the apparition account we can still glimpse Juan Diego's spiritual struggles to pursue God in truth and love. For example, just as the Spanish occupiers challenged the humanity of Indians like himself through exploitation, so his language in the apparition account expresses in a more subtle way how his own evaluation of himself deteriorated toward a utilitarian view of human worth and purpose: "Because I am really just a man from the country, I'm the porter's rope, I'm a back frame, just a tail, a wing; I myself need to be led, carried on someone's back."
Juan Diego's expression of humility reflects the common phrases and patterns of courteous speech for the Indians, placing him among the common people. But his mission was more. His mission was to intercede on behalf of the Virgin. Only a person can be a witness for another person. Only a person can show persistence, patience, humility, and dedication. Only a person can trust and inspire trust. Only a person with free will can choose virtue. Only a person can show desire. The Virgin's reply delineates the difference and also elevates him beyond the instrumentality of being only "one of the people," beyond identifying him by how he functions: "Listen my youngest son, know for sure that I have no lack of servants, of messengers, to whom I can give the task of carrying my breath, my word, so that they carry out my will; but it is necessary that you, personally, go and plead, that by your intercession, my wish, my will, become a reality."
When we come to the end of our abilities, when we approach something that our knowledge cannot explain, when we face something that we alone cannot surmount or a problem that man cannot fix, when we face something larger than ourselves, that is when we ask ourselves who we are. This happens when we face our death or the death of a loved one, serious health issues, situations "beyond our control." In a civilization of love, many of these still cannot be "fixed," but they can be approached satisfyingly when they are answered by love. It requires a trust in God's decisions, that God does not call us to something beyond us, only to abandon us without personally seeing to our help. Sometimes he calls us to face something beyond us in order to encourage—even demand—that we search him out, just as Christ did, depending on God and depending on love. Even while suffering on the cross, Christ turned to the Father, saying, "why have you abandoned me?," which was in fact a prayer, a recitation of one of the ancient psalms. And even just before his death, there still was the trust in the Father that made Christ entrust himself to the Father: "into your hands I commend my spirit."
The miracle of the apparition is a reminder of in whom and in what we place our trust—in God and his infallible love. Speaking of this love, Benedict XVI once wrote:
Let us... be overtaken by the reconciliation that God has given us in Christ, by God's 'crazy' love for us: No one and nothing could ever separate us from his love. With this certainty we live. And this certainty gives us the strength to live concretely the faith that works in love."
It is God's "crazy" love that makes him present himself to us, to be near to us; it was by this same "crazy" love that he sent his son and over fifteen centuries later that he sent his mother to Tepeyac. It was precisely trust in this love that was the trust at stake in Zumárraga's despairing letter to King Charles. This was the certainty received through Mary's apparition. Moreover, it is our certainty in God's love and the transformative power of his love that is the basis of our trust in others. It was out of his love for us that he created us and created all that is around us.
## CIVILIZATION OF TRANSFORMED PERSONS
Reflecting on the canonization of Juan Diego, John Paul II wrote:
But to return to Guadalupe, in 2002 I was privileged to celebrate the canonization of Juan Diego in this shrine. It was a wonderful opportunity to offer thanks to God. Juan Diego, having embraced Christianity without surrendering his indigenous identity, discovered the profound truth about the new humanity, in which all are called to be children of God in Christ. "I bless you, Father, Lord of heaven and earth, for although you have hidden these things from the wise and the learned you have revealed them to mere children.... " (Matt.11:25). And in this mystery, Mary had a particular role.
The call to conversion is central to what it means to live a Christian life. Yet what does it mean to say that a culture itself is called to conversion? Culture is not definitive; it cannot direct itself independent of people, but it takes on the character of those who live in a community and give it life. Conversion begins with individual people and, through the lives of these individuals, culture is transformed. Whether in the form of personal prayer, small or great acts of charitable service, the daily self-renunciations and sacrifices in our family lives, or integrity and competence in our work, we are called to be mediators between God and society. In this way culture is not only transformed, but converted.
In his book _Coworkers of the Truth_ , Joseph Ratzinger speaks of this kind of transformation, both in oneself and in the world:
Those who would be Christians must be "transformed" ever again. Our natural disposition, indeed, finds us always ready to assert ourselves to pay like with like, to put ourselves at the center. Those who want to find God need, again and again, that inner conversion, that new direction.... Yet the truth is that what is invisible is greater and much more valuable than anything visible. One single soul... is worth more than the entire visible universe.... _Metanoeite_ : change your attitude, so that God may dwell in you and, through you, in the world.
Thinking of the change in Juan Diego's life following the apparitions—in how he continued to grow in Christian practice and understanding, dedicating himself to a life of service and never neglecting the opportunity to share the Virgin's message with others—it is important to keep in mind that this change has a source. When we are led to him, Christ not only changes our lives, but he also presents the possibility of positively changing the lives of others. Through God's love for us, we are given the possibility of living out the two greatest commandments: loving God and loving our neighbor. Someone once said to Mother Teresa that not for a million dollars would he touch a leper. Much to his surprise, Mother Teresa responded: "Neither would I. If it were a case of money, I would not even do it for two million. On the other hand, I do it gladly for love of God." This is true conversion of the heart. Adhering to God in such a love, all else must fall beneath that love, becoming an expression of that love.
Some would call Mother Teresa unreasonable for being willing to touch a leper for the love of God, that is, to touch a leper in faith. But reason itself demands an enormous amount of trust, the fundamental trust that what has been is an indicator of what will be, that there is some order and logic to the world. This need for faith becomes all the more obvious when considering the actions and intentions of others, since a person's love for another is one value that is not a catalogue of actions. It is possible—and even easy—to do "works of charity" lovelessly. Love cannot be seen in the way that actions can be. That is why understanding a person's love requires faith. That is why faith must also be a gift, a gift from God. Faith is the gift that enables us to see and to trust his love.
This also shows the conversion of the heart in its most concrete, everyday expression. In Christ, building up a civilization of love is not an abstract good or hypothetical possibility. It is being built already. It is concrete. It is personal. It is found in the individual witness offered by all people truly touched by God. By converting to Christ, reality can reach its full potential. Even suffering finds its beauty in Christ. If we ourselves live close to Christ, we can bring Christ closer to other people. Personal witness is indispensable, as Pope Benedict XVI has made clear: "Only through men who have been touched by God can God come near to men."
Through devotion to Mary, "people can have a direct experience of Christianity as the religion of trust, of certainty." The Guadalupan event made that clear. Before the miracle of Our Lady's appearances, human efforts had failed; Christianity appeared to be neither a religion of trust nor of certainty but rather the religion of the conqueror, a religion with deadly heralds. Even with the great efforts and examples of some of the early missionaries, Christianity could not be offered as a good—not with so many witnesses against it. The fallibility of human works and the limitations of the world had become too obvious, and even human reason could not tell Friar Zumárraga of the truth of what he had heard from the Virgin's messenger. And yet, through Our Lady of Guadalupe, all of this was ultimately changed; revealing herself as a mother, suddenly trust was possible, certainty within reach.
The entire story of Guadalupe is one of transformation: from a continent of bloodletting and human sacrifice, to a continent where the mother who watched her Son pierced granted a reprieve to its inhabitants. It is the story of a barren hillside surprisingly covered in flowers, of a coarse tilma, imprinted with the beautiful image not made by human hands, of a bishop's heart softened, and of the beginning of millions of conversions, each of which represented an immortal victory for the civilization of love. It falls to us, to continue this sequence of conversions and through our conversion of self to become the codex that can bring conversion to those around us by our witness.
# VII.
_Mary and the Church_
_Here, at the feet of Mary, ever anew we "learn the Church," entrusted by Christ to the Apostles and to all of us. The mystery of Mary is linked inseparably to the mystery of the Church_.
-JOHN PAUL II, ADDRESS AT THE MARIAN SHRINE AT CZESTOCHOWA
## CHURCH ON TEPEYAC
One of the most interesting details of the apparition is the Virgin's request: to build a church on Tepeyac. With other churches in the general area, including the _doctrina_ that Juan Diego was accustomed to visit for instruction, building another church on Tepeyac may seem superfluous from a pragmatic standpoint. Would this chapel, uniquely requested, have a unique purpose? After all, although the tilma is now housed in the Basilica of Our Lady of Guadalupe, in the apparition account the tilma's image is unknown and unexpected at the time of Mary's first request for a church. Furthermore, why would _Mary_ request a church at all?
One of the main indications about Mary's personal role regarding the church's purpose is her declaration that it be a place of intercession: "There truly will I hear their cry, their sadness, to remedy, to cure all their various troubles, their miseries, their pains."
In August 1736, a painful and deadly plague of typhus broke out throughout the population of New Spain. Thousands died in the streets and in the hospitals of Mexico City. Within months, the city's nine hospitals were filled, along with nine abandoned buildings converted into makeshift infirmaries. Victims gathered in the city's plaza and in any large buildings that could be found. Soon existing cemeteries were filled, and outlying parts of the city were transformed into cemeteries that were also filled. The archbishop of Mexico paid to open pharmacies throughout the city to distribute free medicine. Despite these efforts, the plague raged on into the next year with its original intensity. During this time, Our Lady of Guadalupe remained a center of devotion. During nine days of a solemn novena to the Guadalupan Virgin, an unbroken line of pilgrims stretched from Mexico City to the Virgin's shrine at Tepeyac. A council of civic leaders was organized and vowed to have Our Lady of Guadalupe declared Patroness of the City of Mexico. At ten o'clock on the morning of April 27, 1737, the city swore its solemn allegiance to Our Lady of Guadalupe. On the eve of this consecration, only three burials took place in the city's cemetery of San Lázaro, where previously forty to fifty burials had been taking place each day.
This show of devotion to the Guadalupan Virgin, while astounding, is not unique to this occasion. Historically, whenever disasters struck, the people of Mexico City turned to her; nearly three hundred years later, in the swine flu epidemic in Mexico City in 2009, Cardinal Norberto Rivera Carrera led the citizens of that city in prayers seeking the protection of Our Lady of Guadalupe. Even in personal trials, illnesses, or injuries of family members, Our Lady of Guadalupe was almost always invoked, at least in prayer, if not in pilgrimage. And today, millions of people visit the Basilica of Our Lady of Guadalupe in Mexico each year, praying for intentions and attending Mass. That people come to her, even with simple personal troubles, as once she encouraged the grieving Juan Diego to do, points to the individuality in Mary's relationship to each of us, the individuality expressible only as a relationship between mother and child.
## CHRIST PRESENT
One of the greatest of Christ's promises recorded in the Gospels is this: "I am with you always, until the end of the age." From the human perspective, it is also one of the most enigmatic. Speaking these words after his resurrection, he had already proven in a tangible way that his love for his apostles was so strong—despite their faults, doubts, and denials—that he returned to them in his resurrected life. But Christ promises his presence not only until his ascension but until the end of the age. He promises enduring and personal affection not only to the apostles meeting him on the mountaintop in Galilee but to all subsequent generations, causing us to ask: how is he still with us today?
The answer to this question is made clearer in the context of what was said. Meeting with the apostles as he had requested, Jesus spoke of a time beyond his resurrected life on earth:
All power in heaven and on earth has been given to me. Go, therefore, and make disciples of all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, until the end of the age.
That is, Christ is made present by his power over heaven and earth, by our evangelization, by our baptism, and by our observance of his commandments. Aside from his power over heaven and earth, the rest are actions between persons and within the Church.
At its heart, the Guadalupan event points to Christ's presence precisely in this way. In the previous chapter, we looked at the apparition's Christocentric element, with Mary as a model of conversion to Christ and as a mother through whom Christians are led to a greater union with Christ. Much of this focused on her acceptance of God's will at the Annunciation, in the moment when she said "yes" to God, conceiving Christ, and thus becoming Christ's mother. However, Mary's motherhood also opens us up to a new dimension of encountering Christ, encountering Christ in the Church. The stated purpose for the Tepeyac chapel clearly speaks to this continuing presence of Christ.
I want very much that they build my sacred little house here, in which I will show Him, I will exalt Him on making Him manifest, I will give Him to all people in all my personal love, Him that is my compassionate gaze, Him that is my help, Him that is my salvation.
Only here is the full purpose for the church—and our full place in this church—made clear: it is a place where Christ is made present, a meeting place for God and man.
In the original Náhuatl, the personhood of what is given at the church is made more apparent in these last lines. Often, it is translated simply as "my compassionate gaze, my help, my salvation," but in the Náhuatl there is the affix _te_ , indicating a person. In this way, this compassionate gaze, help, and salvation are personified, or, more specifically, are a person. Christ is made present in a way that is personal, just as each person is loved particularly in a way that is lasting. This makes both the request for a church and the entire apparition Christocentric, centered on Christ.
For us today, just as for Juan Diego and the people of the New World, the church is not a place of remembrance, like a memorial, but a place where Christ is constantly present as a person. That continuing presence is the true and lasting miracle, enabled through Christ's power as God and made known to us through the witness of others in the life of Christ's body, the Church. By requesting this chapel, the Virgin encourages Juan Diego in a sharing of the faith. She relates to us as individuals seeking Christ but also as members of the Body of Christ, the Church.
Mary's motherhood is not the only motherhood given to us. Rather, both Mary and the Church are mothers in a spiritual sense, and the motherhood of each complements the other's. As John Paul II explained: "The two mothers, the Church and Mary, are both essential to Christian life. It could be said that the one is a more objective motherhood and the other more interior." While the motherhood of the Church is inextricably linked to the preaching of God's word and the sacraments, "Mary's motherhood," John Paul II continued, "is expressed in all the areas where grace is distributed, particularly within the framework of personal relations. They are two inseparable forms of motherhood: indeed both enable us to recognize the same divine love which seeks to share itself with mankind."
## UNDER MARY'S ROOF
This personal presentation of Christ is indicated in the second way the Virgin is identified with the church in the apparition account: It is her church and—what is more—her _home_. And Mary is identified not only with the church on Tepeyac but even with the doctrina in Mexico City: "I am going as far as your little house in Mexico, Tlaltelolco, to follow the things of God that are given to us, that are taught to us by those who are the images of the Lord, Our Lord, our priests."
The first conversation between Juan Diego and the Virgin, in which the Virgin asks Juan Diego where he is going, establishes more than where Juan Diego is headed. It is an opportunity for Juan Diego to express the continuity of the Church and of its teaching. Our Lady already appears in the context of Christ and the Church. She already is associated with the clergy, the "images of God," who teach. She already participates in the education of her children. In this context, the entire apparition, the entire mission entrusted to Juan Diego, brings Juan Diego to fuller participation in the Church's "yes" to God.
Giving birth to Christ began with an extraordinary "yes" to God's will. More extraordinary is the "yes" to Christ's life throughout his childhood and mission. Speaking of Christ's life, Benedict XVI highlights Christ's time in Mary's home as a formative period in Christ's life. Christ spent most of his life in Mary's home, and Mary was present at the wedding at Cana, the beginning of Christ's public ministry. Christ constantly calls himself the Son—and as Son he has both a father (God the Father) and a mother (Mary). Benedict XVI concludes that something in Christ's childhood and upbringing enabled him to uphold childhood as a model of our relationship with God:
"Truly I say to you, unless you turn and become like children, you will never enter the kingdom of heaven" (Mt 18:3). This means that Jesus does not regard "being a child" as a transient phase of human life that is a consequence of man's biological fate and thus is completely laid aside. Rather, it is in "being a child" that the very essence of what it is to be a man is realized, so much so that one who has lost the essence of childhood is himself lost.... If being a child remained so precious in Jesus' eyes and he saw this as the purest mode of human existence, then he must have had very happy memories of his own childhood days.
So close is this association between spiritual childhood and the favor of God that Benedict XVI sees the New Covenant established by Christ beginning "not in the Temple or on the holy mountain, but in the simple dwelling of the Virgin, in the house of the worker, in one of the forgotten places in 'Galilee of the Gentiles,' a town from which no one expected anything good to come."
Consequently, the Church continually returns to the home of Mary to discover, as Christ discovered, the calling to unite to God through the free and loving acceptance of his will and love.
What does Jesus learn from his mother? He learns to say Yes, _fiat_. Not just any Yes, but a Yes that goes ever farther, without getting weary. Everything that you desire, my God... This is the Catholic prayer that Jesus learned from his human mother, from the _Catholica Mater_ who was in the world before him and who was inspired by God to be the first to speak this word of the new and eternal covenant.
It is this prayer and outlook that each Christian learns from Mary; this too is how the Church follows the model of Mary. Mary's home is closely tied to the future of the Church. "The universal Church can grow and flourish only if she is aware that her hidden roots are kept safe in the atmosphere of Nazareth."
## MARY'S MOTHERHOOD AND WOMANHOOD
Just as Mary's motherhood of Christ went beyond birth, so with Juan Diego—and us. Her motherhood is not mere spiritual "fact," like a name on a birth certificate. Rather, she comes to Juan Diego making present her continued role as mother.
In the apparition account, one of the most striking moments is when Mary calls herself God's mother and Juan Diego's mother: _inantzin_ (dear mother) and _nohuacanantzin_ (compassionate mother). Both words have the same ending (- _tzin_ , dear). This ending was also used when she called to Juan Diego in the beginning: "Juantzin, Juan Diegotzin." This word ending was used to express affection, reverence, and intimacy. It is often loosely translated as "dear." However, it was significant whether this affectionate form was given to someone else or whether one called oneself by this endearing form. While calling another by this respectful form could indicate one's own feelings toward or relationship with the other person, to call oneself by this form indicates a degree of acceptance of this familiarity as a joy, an honor. In this case, the Virgin calling herself God's dear mother and Juan Diego's dear mother revealed a relationship that was both preexisting and honorable.
Of course, Mary was Christ's mother in a way different from her motherhood for us. She was Christ's biological mother: he was conceived within her, she carried him in her womb, and she gave birth to him. It is a relationship none of us has with her. But spiritually, she is the mother of all of us, the brothers and sisters of Christ. And this motherhood, too, was given by God, in Christ's words from the Cross when he entrusted Mary and the apostle John to each other:
When Jesus saw his mother and the disciple there whom he loved, he said to his mother, "Woman, behold, your son." Then he said to the disciple, "Behold, your mother." And from that hour the disciple took her into his home.
Christ did not ask for anything specific; he does not tell John, "Let her stay with you." He gives a relationship. He gives his mother a son. He gives this son a mother.
Even more, he entrusts his mother not only to John but to "the disciple there whom he loved." Through Christ's love for each of us, and through our discipleship to Christ, we too are both entrusted to Mary and entrusted with Mary: "Mary's motherhood, which becomes man's inheritance, is a gift: a gift which Christ himself makes _personally_ to every individual." This is seen, too, in the Guadalupan apparitions; not only is the Virgin's spiritual motherhood expressed to another "John" ("Juan" Diego), but the Virgin's words express how this is a relationship that already exists. She is already his mother. She already holds Juan Diego in her arms. Mary's motherhood affects each of us individually, as she intercedes and cares for us. Just as John, out of care for this relationship, accepted his new mother into his home, just as Our Lady of Guadalupe entered the home of Juan Bernardino and requested of Juan Diego a home, so each of us is called to invite her into our homes and our hearts.
Together and inseperably, Mary and the Church aid the person in continual conversion to Christ, helping us to show our faith in action, work for love and persevere through hope in Christ. In this task, Mary's womanhood bears special significance:
Mary is the believing other whom God calls. As such, she represents the creation, which is called to respond to God, and the freedom of the creature, which does not lose its integrity in love but attains completion therein. Mary thus represents saved and liberated man, but she does so precisely as a woman.
For Catholics, Mary provides the example of the perfect woman. Not only was she sinless, but also she displayed an incredible sense of interiority, a capacity to listen, and above all a trust in God that allowed her to say yes to his will, even when what was asked required heroic trust.
We can imagine that the Virgin Mary, as the wife of a carpenter, had to work very hard to help sustain her family and to raise Jesus. Yet even as a woman of great activity, Mary was also a woman of advent, of prayerful waiting for the Word. Mary's womanhood is significant since she is in a very real way Christ's mother, but it is also important in terms of the unique moral and spiritual values embodied therein. Mary exemplifies values integral to the affective, personal, and interpersonal life of the human person. Values such as attention, openness, receptivity, waiting, and patience—often called "ethic of care" values—are all deeply characteristic of Mary in her relation to God and to others. Through these values, Mary showed above all a trust in God that allowed her to say "yes" to his will, welcoming God into her womb and into her life. Precisely through her womanhood, she made her whole life "a response of God's love."
Of course, we must remember that in the New World, the face of mature Christians was not only European, but also male. In the apparition of the Virgin, the Church in the Americas is given anew the protoevangelist, Mary, who precisely as Virgin and mother represented saved and liberated humankind in a loving way. As in the life and person of Mary, action and prayer are shown to be integrally united in the single gift of self to God, as an action stemming from prayerful union with him. It is Mary's continued response to God's love, from Nazareth to Tepeyac, which inspires Juan Diego to the seek the same sensitivity to the will of God when he accepts God's plan for him—a plan he had not envisioned personally, but was able to recognize when proposed to him lovingly as a "response of God's love."
## THE CHURCH: BRIDE AND MOTHER
Within the Church, Mary's motherhood takes on a concrete character of care. Importantly, however, as a woman espoused to God, Mary does more than point toward the Church. She is in fact a model for the Church, the Church that is the Bride of Christ, and she is a model for us precisely in our participation in the Church. While a model for each individual believer, Mary's motherhood is also a model for the Church. In fact, as John Paul II wrote in his apostolic letter on the dignity of women, "Unless one looks to the Mother of God, it is impossible to understand the mystery of the Church, her reality, her essential vitality." In a fundamental way, the Church finds a model in Mary, who in her "yes" to God became that place where God and humanity met. The Church is called to an analogous vocation, bringing Christ in the sacraments to all people and, at the same time, giving birth through her preaching and the sacraments to a new community of believers.
Speaking of the Church, there are two terms most often used. One is _populus Dei_ , meaning the "people of God;" the other is _ecclesia_ , meaning "church." Together they describe two different aspects of the Church's relationship to Christ. As Cardinal Ratzinger explains:
In contrast to the masculine, activistic-sociological _populus Dei_ (people of God) approach, Church— _ecclesia_ —is feminine.... Church is more than "people," more than structure and action: the Church contains the living mystery of maternity and of the bridal love that makes maternity possible. There can be ecclesial piety, love for the Church, only if this mystery exists. When the Church is no longer seen in any but a masculine, structural, purely theoretical way, what is most authentically ecclesial about _ecclesia_ has been ignored.
The Church as _ecclesia_ points to the way the Church relates to Christ in a feminine manner. Several times in the Gospels, Christ spoke in parables of himself as the bridegroom with his bride. This bride is the Church. The Church incorporates into her own self-understanding the mystery of Mary, "the listening handmaid who—liberated in grace—speaks her _Fiat_ and, in so doing, becomes bride and thus body." Without the values of femininity and an ethic of care, as embodied in the person of Mary, the proper relationship between God and the Church is threatened. To lose sight of the Church as the patient bride of Christ risks reducing the relationship between human and divine to an exclusively human dialogue. Coming to the Indians and missionaries as a woman, Our Lady of Guadalupe awakens them to the femininity of the Church. Just as Mary's receptivity to God enables her to bring us to Christ, so the Church's continual receptivity to Christ enables the Church to bring Christ to all peoples. The image and codex of Our Lady of Guadalupe, which carried the symbol of divinity in the four-petaled flower at its center, brought to the Indians in a feminine form the message and truth of Christ. In the same way, our _ecclesia_ , our Church with its feminine aspects, brings Christ to our life both through its teaching and through its sacraments.
This understanding of the Church as _ecclesia_ , as feminine, does not preclude action, does not preclude the masculine. On the contrary, it is the very presupposition upon which the Church's charitable action is based. The choice is not between God and works of charity but rather between God and charity on one hand and mere action on the other. In the same way that the human person finds fulfillment only through a duality of both the so-called masculine and feminine, so too with the Church, who in her charitable work with others actively offers God to the world. This activity is not sought in and of itself, as a mere social service or program; neither is it proselytism. It is, rather, one of the many forms in which we open ourselves up to God's free gift of love to us, in our own free gift of love to others.
## MARY, THE CHURCH, AND THE SACRAMENTS
To speak of the motherhood of the Church in terms of her preaching and the sacraments may sound abstract and removed from the familiar reality confronting us in our individual church communities; it is not. At baptism, we are each individually born into the Body of Christ. Each person receives his or her name, is anointed individually, and is welcomed personally—by family, by godparents, and by the priest—into a living human and spiritual family.
The reality of the Church manifesting Christ's presence to us in the sacraments would have been particularly clear for those converting after the apparition. As the Náhuatl scholar Fr. Mario Rojas Sánchez points out, the Náhuatl word for "sacrament" itself suggested this reality of meeting Christ through the sacraments. While in English, as in Latin, the word _sacrament_ derives from the word for "holy" or "sacred," the Náhuatl word created for "sacrament" speaks more of the person. It is composed of three words: _celilia_ (to receive) + _tla_ (thing, object) + _te_ (someone, a person). Thus, the Náhuatl word for "sacrament" was _tetlaceliliztli_ , meaning to "receive something which is also a Someone."
For the missionaries of Mexico after the apparition, maintaining this personal element in baptism was one of their great concerns. In one of the accounts of the many baptisms, this personal element was stressed, noting that the missionaries placed chrism on each person being baptized. And despite the added work, as one priest explained, he "felt in his heart something more joyful in baptizing them than others."
This desire for Christ leads to a desire to live a Christian life, because only a life that has truth and love as its basis can bring closeness with Christ. The persistance of this desire was exemplified by the many Indians who had to travel great distances to be baptized, crossing over gorges and traveling through rough, mountainous terrain. Yet getting to a church or monastery where they could be baptized was not their only difficulty. Because there were so many Indians who wanted to receive the sacrament of baptism, often they would have to wait months before being baptized. In the convents of Guacachula and Tlaxcala, close to two thousand Indians were counted waiting in the courtyards, begging any missionary they saw to baptize them. This became such a problem that the friars of New Spain wrote to the pope in order to request permission to perform an abbreviated form of the baptismal ceremony. The Franciscan missionary Friar Toribio recorded that in a short space of time close to nine million Indians received the sacrament of baptism. Friar Juan de Torquemada, in his book _Monarquía Indiana_ , informs us that "many thousands were baptized in one day." In receiving baptism, the inhabitants of Mexico not only received Mary as their mother but also underwent through the process of the sacrament a spiritual rebirth. Their rebirth through Christ gave them a distinctive mother, both in Our Lady of Guadalupe and in the institution itself.
In addition to learning catechism and accepting the sacrament of baptism, the Indians often sought the sacrament of confession. As Friar Mendieta recalled:
It happened that on the roads, hills and wilderness, one thousand, two thousand Indian men and women would follow the [friars] just to confess, leaving their homes and lands [completely unattended]. Many of them were pregnant women, some giving birth on the way, and almost all of them carrying their children on their backs. Others were old men and women who could hardly stand with their canes and [some were] even blind.... Many would take their wife and children and their small amount of food as if they were moving to another place. They might stay one or two months waiting for a confessor, or a place to confess.
In this testimony, it is easy to see the hardships and persistence. But in comparison to the forms of confession practiced by the Aztecs, one may also see the way in which Christianity transformed the Indians' understanding of repentance—and forgiveness—as part of a continual conversion to Christ.
For the Aztecs, the goddess who tempted them to sin was the same goddess who could forgive them. Such forgiveness could only be granted at certain times, and the Aztec priest had to consult sources to be sure it was a proper time. Only once could a given sin be confessed and excused. Repeat offenses of the same sin were believed unpardonable, which led many to wait until very late in life before they would confess serious offenses. As Sahagún explained, only old men would confess great sins such as adultery, "from which fact we may deduce that they probably had greatly sinned in their youth but did not confess until they were old, but that they would continue sinning while young." Yet once they converted to Christianity, the sacrament of confession became a significant and much desired element in the Aztecs' life in the Church.
Many aspects of Indian religous ritual had direct—if obviously different—analogs to Catholicism. Thus Our Lady of Guadalupe proposed a new way of life on Tepeyac. She offered to the Indians the fulfillment of their own culture, replacing violence with love as the solution to the problems of everyday life. Ever the loving mother, Our Lady requested the Tepeyac church be built at the bottom of the hill—the humble root, a place easily accessible to the passerby. As Fr. Peter John Cameron observes, "Our Lady of Guadalupe comes to us even in the midst of our idolatries," and there her presence as mother provides guidance, protection, and a home in which we are invited to enter into a new life, with Christ our Redeemer as our brother.
# PART III
# Unified in Dignity
# VIII.
_The Gift of Vocation_
_In the Church there is a diversity of ministry,
but a oneness of mission_.
–PAUL VI, APOSTOLICAM ACTUOSITATEM
## THE VOCATION OF JUAN DIEGO
One of the striking facts about the first church built on Tepeyac is that it is almost impossible to separate the church's history from the life of Juan Diego himself. Shortly after the people of the town built the chapel in December 1531, Juan Diego requested permission from Zumárraga to live there, and a small room was attached to the chapel's east side. There, he soon found himself in the hub of Christian life. When he was not at Mass, deep in prayer, or sharing the apparition story with pilgrims, Juan Diego cared for the Virgin's chapel. One task he took upon himself was sweeping the floors, a great honor in Indian religious ritual, as the sixteenth-century missionary-historian Friar Mendieta wrote: "They have great reverence to all the temples and to all the things consecrated to God; even the principal elders take pride in sweeping the churches, maintaining the custom of their ancestors when they were pagans that, by sweeping the temples, they showed their devotion, even the important persons." In this way, Juan Diego made of his life what had been an honor for his ancestors.
With Juan Diego's close connections to the Church and his reputation for eagerly speaking of God and Mary, it is understandable that some of the early paintings of Juan Diego depict him wearing the garb of the first missionaries to Mexico, the Franciscan brown habit, instead of Indian clothing. For anyone familiar with the true situation of Indians regarding holy orders, this becomes rather extraordinary. Around the time of the apparition, Indians were barred from entering the priesthood; in fact, just four years before the apparition, three Indians had expressed a desire to become priests, but later word came back forbidding it. Not until 1539 did the bishops accept literate Inidans and mestizos into minor orders. Thus to depict Juan Diego in the Franciscan habit was in some ways an extraordinary expression of his extraordinary life. It placed his dedication to Christian life and evangelization, lived out within walking distance of his old home, on the level of the missionary friars who had given up everything and left their home continent in order to preach the Gospel and be a concrete witness of Christian life.
In a striking way, this points to the "oneness of mission" shared by all those in the Church: the call by God to holiness and to evangelization. It is our "universal vocation." However, although it may be poignant to depict Juan Diego in the Franciscan habit, not only is it historically inaccurate but it also overlooks the distinctiveness of Juan Diego's life and the "diversity of ministry" that is so vital to the Church—and to our world. The universal call to holiness and evangelization is lived out differently in each of our lives in what it is helpful to think of as complementary vocations—a vocation to the lay life, to the priesthood, to religious or consecrated life. In each of these, the person works toward holiness and the sharing of the Gospel in a distinct way defined by the unique opportunities of his or her way of life. Juan Diego sought Christ's love, and radiated Christ's love, precisely as a layman and a mediator between the old world and new.
This "oneness of mission" extends beyond the church walls, but sometimes it can be difficult to see how this is true. Consequently, lay men and women often view their lay vocation as the "default" vocation, and thus as less important. After all, it does not require seminary training or ordination, and except at marriage, there is rarely a definitive moment when the lay life is chosen or actively accepted. Living in the world, lay persons often engage in work and social interactions that have little or no overt connection to religious devotional practices. While the lay vocation is not a "religious" vocation in the clerical or monastic sense, religion still plays a vital part in it.
While in Rome attending the second session of the Second Vatican Council, Bishop Karol Wojtyla—who later became John Paul II—described the lay vocation in this way:
If someone were to ask me what the role of the laity in the Church is, I [would] answer that it consists in completing the work of Christ, the Son of God, in the world and with the world's help. It consists in regaining the world, in all of its facets and manifestations, for the Eternal Father. On the road to this, however, lies an even higher aim: to regain man himself, in his humanity, for the Eternal Father.
In their work, marriages, families, social life, and social interactions, lay men and women make their greatest transformations of society.
This role of lay witness is apparent in the tilma, an item that has some of the most beautiful uses in the indigenous culture—and also some of the most mundane ones. The tilma was an article of clothing bearing the indication of one's social status. It was also indispensable as a tool used to transport food and other goods, connecting it to the very life, work, and sustenance of the indigenous people. Used in the marriage ceremony, when the man's tilma was tied to the dress ( _huipil_ ) of the woman, it symbolized the person in the bond of marriage. In Juan Diego's tilma, these various facets of everyday life are affirmed in their proper dignity as forums for witnessing to and connecting with God. In a special way, Juan Diego recognized the possibility of living a spiritually fulfilling life in the world, in whatever situation one is placed, when he encouraged Juan Bernardino to remain at his home in Tulpetlac so that his uncle could continue the type of life they had lived together. This is what Benedict XVI called "the truly great thing in Christianity, which does not dispense one from small, daily things but must not be concealed by them either," namely, "this ability to come into contact with God." Witnessing to God even in the world, lay Christians offer God to the world. Transforming the world, lay Christians offer the world to God.
## EUCHARISTIC PARISH
Speaking of the codex of the image, we can see how an _altepetl_ -like flower represents a civilization. Joining this with the call to build a church, we see how this could have been understood as a new beginning, the establishment of a new civilization. The building of the church is not only symbolic in the indigenous historical and cultural context, where temples played a central role. In a very real way, the building of a church—specifically, the local church and its parish community—truly is the foundation and source for the building of a civilization of love.
In the codex, what is remarkable about this interpretation is that each of the flower clusters has unique elements that distinguish it from the others, and yet they all follow a certain model, continuing to maintain the _altepetl_ shape. Some of the flowers have more little flowers on the top of the hill, others more leaves or more buds on the stems; some are positioned on the garment, while others are positioned on the Virgin's sleeves. That is, they bear the same resemblance not only to _a_ civilization but to the _same_ civilization with the same divine source, without losing their uniqueness and independence.
The elements symbolized by the tilma's codex come into a concrete synthesis in the parish. In the parish church, individuals (hearts with a face) are educated in Christ; coming to know Christ, they learn to recognize the face of Christ in themselves and in one another. In the parish, Christ's sacrifice is made truly present in the Eucharist; through receiving the Eucharist, the parish is transformed from being a collection of persons to a unity of persons, "a family of families"—relatives through the blood of Christ. Finally, through the Eucharist, the parish family receives the grace to look beyond its borders.
In the codex, all of these civilization glyphs are located around the central glyph, the four-petal flower indicative of divinity. Likewise, in the Church at large, people of all civilizations center on Christ. Today, this "catholic" (in the sense of universal) reality of the Church is made particularly apparent in the diversity within our parishes.
In his message for the eighty-fifth World Migration Day, John Paul II addressed the difficulties individuals and families face when they leave their homeland to settle elsewhere. He spoke of many needs migrants have, but in particular he spoke of the church: "The parish is the place where all the members of the community come together and interact. It makes visible... God's plan to call all people to the covenant established in Christ, without any exception or exclusion." Called to become, individually, a spiritual center of worship for everyone, the parish is the place where the Church reaches out to the world. In the tangible witness offered by its parishioners—in their prayerful intercession, listening ears, open hearts, and willing gift of support and help—the parish manifests in concrete terms the universality and unifying power of God's love.
The church Our Lady asked to be built on Tepeyac was not simply to be another indigenous building. It was to be a place where she, like a magnet, could draw people to the presence of her Son in the Eucharist. The mission of building a church physically on Tepeyac hill, and to a greater extent the evangelization of his people, was Juan Diego's lasting contribution. The man sometimes painted in Franciscan robes was the New World corollary to St. Francis himself, who was instructed by Christ to "rebuild My church." While St. Francis rebuilt a small chapel in Assisi and had as a greater mission the role of messenger in the reevangelization of Christians in the Old World, Juan Diego was the messenger of first-time evangelization to those unfamiliar with Christ in the New World.
While the task of establishing the church was completed when Juan Diego successfully gained Zumárraga's approval, the mission of making the chapel a vibrant center of Catholic life filled Juan Diego's lifetime. Living at the hermitage with the image, Juan Diego found himself with many pilgrims coming to see the image and to hear of Mary's apparition. But many of these, we are told, came with their problems. Just as Mary asked Juan Diego to intercede for her, to bring her desire to the spiritual father within the Church on earth, Friar Zumárraga, so many pilgrims came to Juan Diego to intercede for them to our spiritual mother in heaven. And for hundreds of years those who have come to ask the favor of Our Lady of Guadalupe have received at the foot of her image her greatest gift, her Son in the Eucharist.
As at the Hermitage or the Basilica of Our Lady of Guadalupe, the parish too is ultimately a spiritual community founded upon Christ's unifying love offered in the Eucharist. As Pope Benedict XVI writes in his first encyclical, "'Worship' itself, Eucharistic communion, includes the reality both of being loved and of loving others in turn. A Eucharist which does not pass over into the concrete practice of love is intrinsically fragmented." Coming to the Eucharist, each Catholic receives Christ—and the grace to live more fully the life of Christ.
If the Church is to be the center and source of a civilization, it depends on the parish, the "family of families," that extends beyond itself to touch the lives of others. For this reason, too, shortly before becoming pope, Karol Wojtyla declared the need for families to perform service together, especially within the parish. "It is not sufficient merely to visit the sick, this should be made into a 'family' custom—but within the scope of the larger family of families which is the parish." Works of charity, unity, and truth are purified and find their true expression in the parish. In its people, the civilization of love is rooted in faith, and in their concrete community, the parish.
## UNIVERSAL VOCATION
In 1666, another great investigation into the Guadalupan events was conducted, in which numerous Indian elders recounted what they knew of Juan Diego and the apparition. Many of those testifying were very old, putting them within a generation of the apparition, and the details they recounted about his life, practices, and demeanor were passed down to them largely by word of mouth, making what was remembered and told a gauge of what truly struck the early Indian witnesses in the first place: the distinctiveness of his Christian life. In the many texts describing Juan Diego's life and holiness, the accounts of his practices and attitude toward others express the vitally important fact that his relationship with Christ dramatically changed the way he lived. One particularly moving testimony was given by Marcos Pacheco, who gave an account of the event as told to him by his aunt. He recalled how she was so impressed with Juan Diego's holiness that she would say to her nephew, "May God make you like Juan Diego and his uncle." Greeting the Mexican people during his 2002 trip to Mexico for the canonization of Juan Diego, John Paul II echoed these words, saying: "May God make us like Juan Diego!" But why look to Juan Diego's life at all?
Quite frankly, to ignore the saints is to ignore the work of God's love and to deny a vital part of our relationship to Christ as our Brother. To be a child of God means that not only do we have a fraternal relationship with Christ, but we have a fraternal relationship with all other children of God, both those living now and those who "have gone before us marked with the sign of faith." To accept the reality of the communion of saints—that there is a special bond between the saints in heaven and us on earth, enabled by God's love and our love of God—is to accept the power of God's love to extend beyond an individual human person.
Our Lady of Guadalupe's presence makes manifest in a dramatic way how death does not end our vocation to witness to the saving work of Christ. Learning about the saints, we open our eyes to the manifold ways God makes his love apparent in our lives, as well as to the manifold ways we respond to God's love. That is, looking at the lives of the saints, we are inspired and informed of the true nature of holiness and the various forms that it can take. Pope Benedict XVI writes of the saints:
Their human and spiritual experience shows that holiness is not a luxury, it is not a privilege for the few, an impossible goal for an ordinary person; it is actually the common destiny of all men called to be children of God, the universal vocation of all the baptized.
The fact is, the Church does not canonize a person simply because the Virgin Mary appears to him or her. The Church has recognized Juan Diego as a saint because of his holiness after the apparition. His relationship with Christ continued to be the foremost desire in his life. How Juan Diego responded to this call to holiness not only affected him spiritually but also changed his life, and it changed the lives of those around him as well. This type of witness—this following of Christ's call to holiness—can change the face of our civilization.
## THE NEW EVANGELIZATION
As an indigenous layman, Juan Diego was a new kind of witness for the evangelization in the New World. In a way, the full importance of Juan Diego's unique life after the apparition can be understood in a new dimension, in terms of his vocation as a layman, a sort of lay pioneer in a place where the majority of his people were not Catholic and none of the Indians was a priest or missionary. With the Church in Mexico so young and in such a precarious position, those lay converts needed then, as we do now, reminders of the importance of the laity in serving Christ in union with the Church hierarchy. Juan Diego's lived witness after the apparition was an inspiration to generations, as is made clear by the 1666 investigation. In the same way, each of us is called to live a life of heroic witness, whatever our circumstances, evangelizing those with whom we come in contact. Our role in the new evangelization is to make present and relevant the message of the Gospel—the message of faith, hope, and charity—in a culture that has often become deaf to its message. This requires our consistent living of the Gospel message in our parishes, in our homes, and in our public lives.
On the feast of Our Lady of Guadalupe in the Jubilee Year 2000, Cardinal Ratzinger gave an address to catechists entitled "The New Evangelization: Building the Civilization of Love" in which he said:
God cannot be made known with words alone. One does not really know a person if one knows about this person secondhandedly. To proclaim God is to introduce to the relation with God: to teach how to pray. Prayer is faith in action. And only by experiencing life with God does the evidence of his existence appear.
It is impossible to introduce others to someone one does not know, and it is impossible to know Christ without at the same time loving him. Juan Diego was an effective messenger because his prayerfulness enabled him to call others to God and to show them how to recognize God's presence in their lives. His closeness to Our Lady and her Son provided such a strong witness that his intercession was frequently sought, and is still sought today.
Closeness to Christ is not always easy. The parish is not just a collection of individuals and individual families, but a living body. Of course, the reality of this community is not always clear or apparent in our parish, since "parish life is especially related to the strengths and weaknesses and needs of the families that make it up." Like a family, one parish or another may become fragmented. Lack of concern, cooperation, or involvement—any ailment that can plague a family—can also affect a parish. Often, when individuals or families are affected by these, the parish is also affected, and this may lead to further discouragement in participation and interest in parish life. We may not care for every aspect of a particular parish experience: we may not like the music or the church architecture, or perhaps the homilies leave us untouched; maybe we even feel isolated or out of place. But we must remember two things. First, the Eucharist is the central and unifying element that supersedes any cosmetic issues we may see. Second, we should remember that we have an integral role to play in the parish. Its vitality depends in no small measure on the vitality breathed into it by each parishioner.
Wherever we find them, human faults and imperfections are nothing compared to God's gift of himself to us. It is this gift—the Eucharist—that truly binds us together and infuses all other parish activities with meaning.
Speaking of the future of the Church, Cardinal Ratzinger once said that "we don't have such urgent need" for reformers. What the Church really needs, he added, are "people who are inwardly seized by Christianity, who experience it as joy and hope, who have thus become lovers. And these we call saints."
To be "inwardly seized by Christianity," however, requires that we live with a well-formed conscience and an awareness of our vocation as Christians, which in turn require that we be authentic in our treatment of others—that we treat them with the charity and dignity due every human person. And the truth of their own dignity needs to be spoken often. This dignity should be universally defended in every culture, however diverse and however seemingly distant from God.
At the Basilica of Our Lady of Guadalupe, John Paul II suggested that the future of our country and our continent depends on this:
We must rouse the consciences of men and women with the Gospel, in order to highlight their sublime vocation as children of God. This will inspire them to build a better America. As a matter of urgency, we must stir up a new springtime of holiness on the Continent so that action and contemplation will go hand in hand.
In the new evangelization, no pope has laid such dramatic expectations of evangelization for a continent at the feet of just a few individuals, such as the first Franciscan missionary friars; but at the feet of the laity as a whole, they have. In words addressed to bishops from the United States, John Paul II made this imperative clear, declaring that "now is above all the hour of the lay faithful," and that the success of this new evangelization depends in large part upon the laity "being true leaven in every corner of life." The newness of the "new evangelization" is how it is "new in its ardor, methods and expression."
This is the importance of the Virgin's chosen description of God, expressed in the words of the Indians and in the image's codex. This was in part the distinctiveness of Juan Diego: he lived a distinctively Christian life in a new way, literally living near the ruins of his people's previous religious aspirations. Not only is Juan Diego a reminder of the transformative power of witness in the evangelization of people who had never met Christ, but he is a model for us today in the new evangelization of those who have met Christ but don't know him as they should.
# IX.
_The Face of the Hidden Christ_
_May the Continent of Hope also be the Continent of Life! This is our cry: life with dignity for all!... Dear brothers and sisters, the time has come to banish once and for all from the Continent every attack against life_.
–JOHN PAUL II, HOMILY AT THE BASILICA OF OUR LADY OF GUADALUPE
## NEW LIFE
For us today, a pilgrimage of children to the Guadalupan basilica may seem cute or quaint. For the children walking to the Virgin's Hermitage in 1544, however, it was perhaps the most concrete expression of how Christianity had changed their civilization. In that year, Mexico suffered one of its greatest plagues and one of the longest droughts in its history. Historically for the Aztecs, a drought not only devastated crops but also could be devastating to families. While adults were sacrificed to the god of war, children were sacrificed to the god of rain, Tlaloc. For example, an excavation of the Templo Mayor revealed a fifteenth-century burial site dating from a drought with the remains of at least forty-two young children who were sacrificed. Additionally, because it was considered good if the victims cried at the sacrifice, great care was taken over choosing the victims, and the bodies of the chosen children show a range of disabilities and prior painful sicknesses ranging from toothaches to bone diseases. But in 1544, we find a serious and dramatic manifestation of change from a culture of death to a culture of life, when instead of a sacrifice of children, a pilgrimage of children ages six and seven processed to the adobe Guadalupe Hermitage to pray for relief—a great and early testament of trust in the Virgin of Guadalupe as the Mother of Hope and the Mother of Life.
## THE EXPECTANT VIRGIN
All Marian apparitions are extraordinary and unique, and the Guadalupan apparition is no exception. There is the extraordinary relevance of the Virgin's image and codex to the Mesoamericans. There is the enduring image on the tilma that is a kind of continuous apparition. But one of the most significant aspects of the Guadalupan image is its depiction of the Blessed Virgin as a pregnant woman and Christ as neither a young boy nor a grown man but a helpless child hidden and alive within the womb.
In this, the image of Our Lady of Guadalupe may be understood as an image of the Visitation, introducing the people of the New World to Christ in a way similar to his introduction to the people of ancient Israel. Reading the Gospel of Luke, we find the first revelations of Christ's identity months before his birth in Bethlehem, while he was still a child in his mother's womb. When Mary (pregnant with Christ) visits her cousin Elizabeth (pregnant with St. John the Baptist), Elizabeth recognizes the specialness of Mary's pregnancy, saying, "Blessed are you among women, and blessed is the fruit of your womb." Elizabeth goes further, identifying what makes Mary and Christ uniquely blessed: "Who am I that the mother of my Lord should visit me?" Just as Mary is addressed as a mother before the birth of her Son, Christ is recognized as Lord before his birth. And just as Mary is identified by her relationship to Christ, so Elizabeth identifies Christ by his relationship to herself and to all of us.
Importantly, both Christ and Mary are acknowledged in these vital roles at a time when Christ is most helpless and vulnerable. And the next time Christ is most vulnerable, at his crucifixion, Mary is named our mother. In these moments of extreme vulnerability as a child and as a suffering man near death, Christ experiences and reveals a profound part of what it means to be human. Hans Urs von Balthasar writes:
The developing human being... is intrinsically ordered to "being _with_ " other persons, so much so that he awakens to self-consciousness only through other human beings, normally through his mother. In the mother's smile, it dawns on him that there is a world into which he is accepted and in which he is welcome, and it is in this primordial experience that he becomes aware of himself for the first time.
What is obvious physically about the human person is that the body is made for life: lungs are created to inhale the air that is so necessary for life, while the heart pumps the blood that is so necessary for life. What is profound is how our awareness of our ultimate purpose in life depends on other people. "Being with" others leads us to understand ourselves in relation to each other, and through other people we can come to understand the truths about ourselves as individuals and as a society. We are made aware that each of us was created out of love, for love. If we are cut off from love, or if we cut ourselves off from love, our life becomes unintelligible. God's calling us to love—our vocation to love—is made apparent by the love and presence of other persons.
Our dependence on others and their dependence on us may effectively reveal love. Using pregnancy as an example, Benedict XVI describes how our "being with" others also necessitates a "being for" others: "This _being-with_ compels the being of the other—that is, the mother—to become a _being-for_." And after birth, children become aware of their mother's care, her "being for" them. From the earliest stages of infancy, every mother and father watches their child naturally begin to discern for himself or herself what is around, but especially what exists "for" him or her, what persons or objects communicate some element of care or happiness. As the child grows, the child learns to discern the difference between a relationship with an object (a relationship of utility) and a relationship with a person—a relationship between two freedoms. Uncoincidentally, as children learn the limits of their physical freedom, they also learn the moral limits to their freedom: "I must live my freedom not out of competition but in a spirit of mutual support."
It is easy to see how a child must exist as "being with" another person in order to survive, and thus how a child depends on the "being with" and "being for" of the parents. This observation becomes especially clear looking at an unborn child. Yet important to keep in mind is the fact that "the child in the mother's womb is simply a very graphic depiction of the essence of human existence in general." The human person's need for others applies as much to the physically mature as it does to a child. Although a person can be isolated socially and distanced geographically from other people, no person can survive for long without entrusting himself or herself to others. Absolute autonomy cannot exist.
Thus, in a world where absolute autonomy is construed as freedom, and freedom as happiness, it is no surprise that people often see their attempts at happiness thwarted. The problem is not in equating happiness and freedom, but in equating happiness and freedom with autonomy. Absolute autonomy is more than just impossible; it contradicts the very nature of what it means to be human. Rather, as Thomas Merton once explained, true happiness is achieved through freedom, while freedom reaches its highest expression in a profound sharing of ourselves with others:
A happiness that is sought for ourselves alone can never be found: for a happiness that is diminished by being shared is not big enough to make us happy.... True happiness is found in unselfish love, a love which increases in proportion as it is shared. There is no end to the sharing of love, and, therefore, the potential happiness of such love is without limit. Infinite sharing is the law of God's inner life. He has made the sharing of ourselves the law of our own being, so that it is in loving others that we best love ourselves.
"Being for" others betters not only their circumstances and person but also our own. It helps us to better become what we are. While true love is always selfless love, it is not thankless love: with it, we work to realize our own dignity as made in the image of the Trinity, the dignity of being placed first and foremost within a communion of persons, a communion of love, the dignity of being something God called "very good" from the beginning of creation. Even more, as John Paul II notes, "while each day of creation concludes with the observation: 'God saw that it was good,' after the creation of man on the sixth day, it is said that 'God saw everything that he had made and behold, it was very good.'" Humanity's goodness extends beyond itself.
## HOPE IN ACTION
Granted, it can be difficult to see this dignity. A culture of life is more than a privilege for prosperous times. Even in the face of dramatic, decimating death, such as Mexico experienced at various times after the apparition, people are still able to build a culture of life. A culture of life respects life for what it is, not for what hardship and suffering frame it to be. A culture of life does not deny the reality of death. A culture of life deals with death as a serious but not ultimate principle. In fact, one of the most strikingly human elements conveyed in the apparition account is Juan Diego's grasp of the reality of death. As a man in his fifties, he certainly knew many people who had died, including his own wife, María Lucía. He had likewise lived during the smallpox epidemic that killed millions of Indians. And then, in a windfall of life-changing events, his uncle Juan Bernardino became ill with a swift and even visually grotesque plague. In his words to the Virgin, we glimpse how powerfully this impending death colored Juan Diego's view of life: "because in reality for this [death] we were born, we who came to await the task of our death." This statement, perhaps more than any other in Juan Diego's dialogue with the Virgin, resonates with contemporary views of the person; and the Virgin's response—"Am I not here, I who have the honor to be your mother"—tests our understanding of the person and points us toward hope.
The truth is that "God is not satisfied with finding his creation good; he wants it also to find itself good." Each of us makes the dignity of God's creation known. While the dignity of the other inspires love, through love we open ourselves up to the innumerable riches of that dignity; through love we show this dignity to others. By loving others, we lead others to recognize their dignity and to live according to it. In his homily at the beginning of his papacy, Benedict XVI spoke of this active role in helping others to see this truth about themselves:
The purpose of our lives is to reveal God to men. And only where God is seen does life truly begin. Only when we meet the living God in Christ do we know what life is. We are not some casual and meaningless product of evolution. Each of us is the result of a thought of God. Each of us is willed, each of us is loved, each of us is necessary.
The fact that Benedict XVI's first two encyclicals have dealt with love and hope reflects how these words continue to be the measure and guide of his life as pope. These words, too, can be a measure and guide for us in our lives if we ask ourselves each day, "Am I living in a way that makes Christ's presence apparent? Am I 'being with' and 'being for' people in such a way that they are encouraged to recognize and live up to the beautiful dignity that is theirs?"
This revelation of the dignity of the human person was one of the great gifts of the Guadalupan event for both the Indians and the Spaniards. On one hand, the Indians suffered from a profound depreciation of their dignity through their own culture. Having no knowledge of the Gospel, the culture of the natives of Mexico had taken even valid concepts to terryfiying conclusions. Men, women, and children had their hearts torn out above a sacrificial altar, and wars were waged to acquire more victims so that, as one Aztec chief explained, "each time and whenever... our god wishes to eat and feast, we may go... as one who goes to the market to buy something to eat." Although pregnancy and birth were largely honored, an Aztec "lecture" given to a newborn child shows clearly how violence was perceived as the means to most ends—no matter how noble.
My very beloved child, very tender... this house where you have been born is just a nest, a shelter... your own land is another, it is promised elsewhere, which is the field where wars are made, where battles are engaged: you are sent for it. Your role and job is war, your role is to give the sun the blood of your enemy to drink and to give the earth... the bodies of your enemies to eat. Your own land, your inheritance, and your father are the house of the Sun.... Hopefully you will deserve and be worthy of dying in this place and receiving in it a flowered death.
While the Indians had an understanding of the importance of sacrifice—and of its redemptive power—the manner in which they expressed this—through human sacrifice—was, of course, not one which was in keeping with human dignity. And after the conquest of Mexico, the Indians' dignity as persons was denied outright in the maltreatment they received from many of the Europeans. While affecting the Indians, these injustices at the same time impoverished the Spaniards, who reduced themselves to mere "things," moved not by truth or goodness but by the whims of greed and corrupt passion.
In the context of such abuse, the Virgin's response to Juan Diego's morbidness—her declaration of her motherhood—can seem a surprising counterargument to make. On the contrary, however, by speaking of her protection, her care, her nourishment, her constant being with and for us, the Virgin awakens Juan Diego to his own humanity. Through love, the Virgin puts death in perspective, thereby revealing the Marian dimension of the distinctiveness of Christian hope. Spiritually and metaphorically, there is a sense that this life is all a pregnancy. We are all waiting to be born out of this life, this life of transformation, of maturation, of becoming what we are meant to become, of becoming more aware of and sensitive to God's love. The more we become aware of and sensitive to God's love, the more we become aware of and sensitive to God's love for others, and thus our love for others grows as well. As Benedict XVI writes:
Hope in a Christian sense is always hope for others as well. It is an active hope, in which we struggle to prevent things moving towards the "perverse end." It is an active hope also in the sense that we keep the world open to God. Only in this way does it continue to be a truly human hope.
In the same way that our love of God reaches out to others, the Virgin's love of God and of each individual person reaches out to everyone. This movement of love to embrace the whole and to never exhaust itself provides another consolation and reason to trust in the Virgin's motherhood. Juan Diego can trust Our Lady of Guadalupe specifically because her motherhood is not closed in on itself but is open to all; she is a mother who rejects no one.
For this reason, too, the church on Tepeyac was requested not as a private place of prayer for Juan Diego but as a place of prayer for all, a "little home" open to all so that every person has a place to encounter Christ, not only for the Virgin's love of Christ, but specifically for her love for us: "Because truly I am honored to be your compassionate Mother... [T]here truly will I hear their cry, their sadness, in order to remedy, to cure all their various troubles, their miseries, their pains." In a very real sense, "prayer is hope in action," and the Church is a place where through the Eucharist and through prayer we encounter Christ, the ultimate transformative hope that saves.
## HOPE FOR OTHERS
What is more, the Church is a place where through prayer "we undergo those purifications by which we become open to God and are prepared for the service of our fellow human beings... and thus we become ministers of hope for others." That is, prayer leads us to charity, to the pursuit of "all serious and upright human conduct [which] is hope in action."
This "hope in action" is ultimately shown in our response to one another. As Paul VI wrote, "charity must be as it were an active hope for what others can become with the help of our fraternal support. The mark of its genuineness is found in a joyful simplicity, whereby all strive to understand what each one has at heart." Responding in this way can be effective only if our response is guided by the truth: the truth about who the person is and the reality of the situation. In hope, we perceive God as the beneficent giver that he is, and so in hope we at the same time perceive others as the recipients of God's graces and gifts; reflecting his merciful and generous giving to each person, we mercifully and generously give ourselves. Addressing members at a conference on the aftermath of abortion and divorce, Benedict XVI explained this, saying, "The Gospel of love and life is also always the _Gospel of mercy_ , which is addressed to the actual person and sinner that we are, to help us up after any fall and to [help us] recover from any injury." We truly see another person only if we appreciate that person in all his or her uniqueness and dignity; it is to understand not only what a person has done—for we have all sinned—but what a person is and by whom that person is loved.
In the Guadalupan event, this ethic of active hope and care for the person is shown especially in how the Virgin acted toward Juan Diego when he was faced with his uncle's death. Rather than expressing anger or frustration that Juan Diego disregarded her instructions, the Virgin met him where he was and showed compassionate concern for his situation. She was interested in him personally, asking: "My youngest son, what's going on? Where are you going? Where are you headed?" There are moments when difficulty obscures the truth and thus obscures the right course of action. There are moments when as a culture we realize that subtle motives and fears have made inhuman practices common and even forced custom itself down a path that few had envisioned or desired. These are moments when we must ask ourselves and each other, "Where are you headed?" These are questions that must be asked because the answers may surprise us.
Although God and Our Lady meet us where we are, our society and peers often do not—even to the point of creating a great silence about the facts of abortion and its aftermath. Often it even appears that too much time is wasted condemning the person, with so little time spent caring. With this, we are reminded of how our ways are unfortunately not always God's ways:
Christ is not a healer in the manner of the world. In order to heal us, he does not remain outside the suffering that is experienced; he eases it by coming to dwell within the one stricken by illness, to bear it and live it with him. Christ's presence comes to break the isolation which pain induces. Man no longer bears his burden alone: as a suffering member of Christ, he is conformed to Christ in his self-offering to the Father, and he participates, in him, in the coming to birth of the new creation.
We too must meet people as God does: where they are, rather than where we would like them to be. By reaffirming all persons, we must be the voice and helping hand of truth and love. Financially, love costs us nothing, but gives a part of ourselves. If we truly recognize the face of Christ in the poor and disadvantaged, in the unborn, in the worried faces of mothers and fathers who, like Christ in Gethsemane, feel abandoned and betrayed, we realize that we have more to offer than advice. Finding ourselves already a part of a community—our neighborhood, workplace, school, or parish—we realize that we already have a relationship with this parent or this peer or this child. We have to reach out when difficulties in life—whether circumstantial hardship or degradation—obscure the truth about the human person, about our brothers and sisters entrusted to our care.
## CARRYING CHRIST
When all we see around us are the weaknesses of human love, it is easy to lose hope and to believe that perhaps our desires for love and happiness exceed what is possible for us. To think like this, however, would be one of the greatest self-deceptions, for it would be to betray our deepest aspirations and to live as if God did not exist. While this is a very human temptation, it is especially a temptation for our time. Looking over the course of human history, we have seen some of the worst in ourselves. But at the same time, we find profound revelations of faith, hope, and love. We find that whenever people open themselves up to the reality of God, our greatest failures are less decisive: they are no longer the only reality, nor the most important part of reality. Human love in light of divine love can reach its true potential and goal: the perfection of love, hence the perfection of man.
Giving ourselves to others in "active hope" shares a part of ourselves with them, and at the same time it is an opportunity for them to share with us. We share our talents, our joys, our time, and our attention, but we also share our sufferings. We share in the sufferings of others. When Our Lady of Guadalupe first introduced herself to Juan Diego, she told him how she was the mother of God, using the Náhuatl word _inantzin_ , but when she referred specifically to her role as mother to him and to us, she called herself _nohuacantzin_ ; here, there is one important addition, _nohuac_ , meaning "compassionate." Etymologically, the English word _compassion_ derives from the Latin _compassio_ , meaning "to suffer together" ( _com_ meaning "together" and _passio_ from the verb "to suffer"). This compassion, this "suffering with" others, is possible through love and supremely modeled in Christ. Christ suffered and Christ liberated. His liberation was wrought precisely through his suffering with us. Like Christ himself, who welcomed our sufferings into his very body, the Virgin suffers with us as well. It is a fundamental way that we express our nearness to Christ in our attentiveness to the concerns, sufferings, and pain of others. It is how we carry the triumphant Christ to others, and at the same time how we receive the suffering Christ from them.
To see the reality of divine love requires something from God and from us: it requires his love, which is always given, and our reception of that love. It requires that we first trust in his love, that we give ourselves over to this love, so that in time our words and thoughts and hearts may be made one with the words and thoughts and heart of the Psalmist:
_O Lord, hear my voice when I call;_
_Have mercy and answer_.
_Of you my heart has spoken: "Seek his face."_
_It is your face, O Lord, that I seek;_
_Hide not your face_.
...
_Do not abandon or forsake me_ ,
_O God my help!_
_Though father and mother forsake me_ ,
_The Lord will receive me_.
...
_Hope in him, hold firm and take heart_.
_Hope in the Lord!"_ 30
Hope in God's love calls us out of ourselves in order to seek the face of God; it opens us to care about the day-to-day realities of our lives. We hope in God, in the fact that his love is something real for us today. More than that, hope in God gives us renewed energy and hope in others; it lets us see them, failures and all, but still love them, still believe in the dignity they have and the perfection to which they are called. It helps us carry Christ to them, to see Christ in them, and to receive Christ from them.
# X.
_The Christian Hemisphere_
_Not only in Central and South America, but in North America as
well, the Virgin of Guadalupe is venerated as Queen of all America_.
–JOHN PAUL II, ECCLESIA IN AMERICA
## THE CHURCH IN AMERICA
When the Knights of Columbus sponsored a tour of a relic of Juan Diego's tilma in 2003, Our Lady of Guadalupe's role as mother of all in the Americas became very clear. In city after city, crowds of tens of thousands of people came to venerate Our Lady of Guadalupe. What was most striking was not the number of people but the number of nationalities and ethnicities represented in each gathering. From Denver to Dallas and from La Crosse to Los Angeles, people of Mexican origin and people with no connection to Mexico whatsoever prayed side by side to the Virgin of Guadalupe.
During this tour, as the outpouring of devotion to Our Lady from the laity showed, nowhere is the laity's mission more eagerly embraced than in the American hemisphere. Here, the faith was transmitted—recently—from person to person and from family member to family member. Here, a layman, Juan Diego, became the single most important evangelist of the New World, and perhaps in the history of Christianity. But here and now there is one last question that must be answered: What does faith in this God give us?
This question, posed by Benedict XVI to the bishops at Aparecida in his first apostolic visit to the Western Hemisphere, is the perennial question about the distinctiveness of Christian life. For the Indians of Colonial Mexico, the answer seemed unintelligible, as the preached goodness of Christian life was obscured by encounters with sinful Christians. For the Indians and the Spaniards, and for us in the Western Hemisphere, we can find the answer clearly in Our Lady of Guadalupe, as she exemplifies the transformative power of God's love—a love that made her his mother, that fueled her sinless life, that brought her, transformed again, to make his love present at Tepeyac, transforming Juan Diego into a messenger, a man of church and charity, and eventually into a saint. To the bishops of the Americas gathered in Aparecida, Benedict XVI expressed this gift of Christian transformation in this way:
Faith in this God... gives us a family, the universal family of God in the Catholic Church. Faith releases us from the isolation of the "I," because it leads us to communion: the encounter with God is, in itself and as such, an encounter with our brothers and sisters, an act of convocation, of unification, of responsibility towards the other and towards others. In this sense, the preferential option for the poor is implicit in the Christological faith in the God who became poor for us, so as to enrich us with his poverty.
Today, we must keep this question constantly foremost, for only if our faith informs our actions, only if we live a distinctively Christian life, can our actions clearly speak the answer to others in a universal language and give the world the infallible hope it needs, a hope that cannot be found either in an individual alone or in humanity as a whole.
For this reason, a decade ago the bishops of the hemisphere met at the Synod of Bishops for America and invited all of us living in the hemisphere to rethink who we are as _Americans_ :
We believe that we are one community; and, although America comprises many nations, cultures and languages, there is so much that links us together and so many ways in which each of us affects the lives of our neighbors.
That meeting itself was an excellent model of cooperation among the bishops of the Americas. But the challenge to the Church of the Americas was a challenge to all the baptized. As Pope John Paul II wrote in _Ecclesia in America_ , "The renewal of the Church in America will not be possible without the active presence of the laity. Therefore, they are largely responsible for the future of the Church." The question is what Catholics—lay and clergy alike—can do to advance the promise of _Ecclesia in America_. For this promise is made increasingly urgent with each passing day.
The fact is, this promise for renewal of the Church in America, articulated in _Ecclesia in America_ , is based upon the reality that our unity in the sacramental life of the Church transcends every national border and joins us in a way that must have both profound and practical consequences. Speaking of the Church, Cardinal Ratzinger noted that our unity and thus our vitality come through the uniqueness of our unity and vitality as a Church:
The Church is not an apparatus; she is not simply an institution; neither is she only one of the usual sociological entities. She is a person. She is a Woman. She is a Mother. She is alive. The Marian understanding of the Church is the strongest and most decisive contrast with the concept of a purely organizational and bureaucratic Church. We cannot make the Church; we must _be_ the Church. It is only in the measure in which faith, above and beyond doing, forms our being, that we _are_ Church and the Church is in us. And it is only in being Marian that we become Church. Also at the beginning, the Church was not made, but born. She was born when the _fiat_ emerged from the soul of Mary. This is the most profound desire of the Council: that the Church is reawakened in our souls. Mary shows us the way.
Precisely because Mary is the archetypal believer, she is also the archetype of the Church. In the Americas, we have the particular example in the figure of Our Lady of Guadalupe, providing a model for us as both believers and as the Church in America.
Although during these centuries she has come to symbolize many things, today in light of _Ecclesia in America_ hers is a message of unity—she is the spiritual mother we all share, perfectly enculturated, a symbol of the "catholic" aspect of a Church where all are full members and all are welcome as equal heirs to the kingdom of God. As John Paul II wrote in _Ecclesia in America_ :
The appearance of Mary to the native Juan Diego on the hill of Tepeyac in 1531 had a decisive effect on evangelization. Its influence greatly overflows the boundaries of Mexico, spreading to the whole Continent. America, which historically has been, and still is, a melting-pot of peoples, has recognized in the _mestiza_ face of the Virgin of Tepeyac, "in Blessed Mary of Guadalupe, an impressive example of a perfectly enculturated evangelization." Consequently, not only in Central and South America, but in North America as well, the Virgin of Guadalupe is venerated as Queen of all America.
The unity of the Christians of this hemisphere—with Guadalupe at its center—continued throughout the twentieth century as well. It was at the Basilica of Our Lady of Guadalupe that the flags of twenty nations were sent in tribute in 1941, and where forty-five members of the American Hierarchy, led by Archbishop Cantwell of Los Angeles, pledged "devotion and fealty" to the "Virgin of the Americas." In 1961, the basilica was also the site of a congress held during the international Marian year, and where in 1962 President John F. Kennedy insisted on attending Mass during his visit to Mexico. Just as Our Lady of Guadalupe has been the center and motivation for such historic hemispheric unity, so she should remain as we discuss issues that affect the Americas today. Her patronage of churches and families throughout the hemisphere goes back hundreds of years, and yet today she affords us a new starting point to reconsider our cultures, our methods of evangelization, our life as a country, a continent, and a Church.
Our collective American continent is one on which we can find devotion to Our Lady of Guadalupe everywhere. In the northernmost American country, Canada, she has a prominent place in the Shrine of St. Anne du Beaupré in Quebec. In the southernmost American country, Argentina, she has a basilica in her honor near the town of Santa Fe. It is not too much to say that in every country of the Americas, Our Lady of Guadalupe is mother not only in her title, Empress of the Americas, but by adoption as well. She is the spiritual glue that holds together a continent that, though diverse in language and culture, shares in common—in every country—the experience of baptism.
In this, Our Lady of Guadalupe—a woman of enculturation—can play a vital role in shaping our new culture. Our Lady of Guadalupe presented to the Spanish and Indians alike a model of enculturation and cultural dialogue. Neither exclusively Spanish nor exclusively Indian, she appeared as a mestiza—a woman of both cultures, who transcended any cultural divide and called the Europeans and Native Americans alike to her Son. She came to two cultures that had recently been at war, and called them to overcome their differences and join a common cause: the evangelization of the Americas and, in this way, the building of a new civilization.
Our future as the "continent of baptized Christians" requires that we—whatever our ethnic or national backgrounds—likewise respond to this message and overcome our differences in order to build a hemisphere that can truly be called a civilization of love. As Pope Benedict XVI said in his first encyclical, _Deus caritas est_ : "To say that we love God becomes a lie if we are closed to our neighbor or hate him altogether." This is the message of the mestiza Virgin of Guadalupe as well, but at a more profound level. We are more than neighbors; we are all children of God, regardless of ethnic or national heritage.
When Our Lady of Guadalupe appeared to Juan Diego, there was no shortage of distrust between the Native Americans and the Spaniards. Similarly today, there is no shortage of distrust based on different cultures or nationalities. Perhaps one of the biggest hurdles to overcome is the fear by many in the United States of Hispanic immigration. Of all people, Catholics should have no trouble remembering that the same fears were harbored against the Irish and Italian immigrants of the nineteenth and early twentieth centuries. Few today would contest the contributions made by those immigrants to the United States, who not only assimilated but breathed dynamic life into the Catholic Church and helped to make it the largest single religious denomination in the United States.
For if our continental history is a shared one, so too is our future. David Rieff pointed out in the _New York Times Magazine_ in December 2006 just how interwoven our collective future as "American" Catholics is, noting: "Nationally, [in the United States] Hispanics account for 39 percent of the Catholic population... since 1960 they have accounted for 71 percent of new Catholics in the United States." At a time when Church attendance is faltering across Europe, it is stronger in this hemisphere and strongest in Latin American countries and in those places in the United States that Hispanic immigrants call home.
For Catholics in the United States, immigration from Latin America brings a unique benefit. Hispanic immigration brings with it the promise of a revitalization of our parishes through a rich tradition of spiritual devotion. It is up to the Catholics already in the United States to provide a rich spiritual environment that will feed the needs of these new arrivals. As these immigrants breathe new life into parish communities, it is our job to help them assimilate into our parishes and communities, as our parents and grandparents did, and to help them to live their faith with support from all Catholics.
## ACROSS THE BORDERS
A crucial part of this future is the cooperation between Catholics in the United States and Mexico and, by extension, the rest of the continent. From the halls of government to the parish pews, the willingness and ability of Catholics to respond to the "perfectly enculturated" call of the archetypal American Christian, Our Lady of Guadalupe, and to build bridges between those in the United States and those in Mexico and throughout the hemisphere will shape our future. This is a special responsibility of Catholics in the United States—especially leaders in business and finance. As John Paul II wrote to Christian workers, "The culture of workers... must remain a culture of solidarity"—a solidarity of equality, a solidarity that restores to each person the dignity of their work by recognizing the dignity of the human person. Likewise, the movement toward greater transparency in Mexican government and politics gives Catholics in the United States as well as those in Mexico an unprecedented opportunity for cooperation in economic and social reform.
This would not be the first time that Catholics of the United States have taken a leadership role to help their neighbors to the south. During the 1920s Mexican persecution, as thousands of Mexicans died with the words " _Viva Cristo Rey, viva la Virgen de Guadalupe_ " on their lips, those north of the border did not ignore their plight, and voices such as the Knights of Columbus helped Catholics in the United States successfully influence their government to take an active interest in the persecution. Moreover, when up to a million Mexican refugees fled north, American Catholics opened their arms to those displaced by the violence. Seminaries were built so that young Mexicans could study for the priesthood in the safety of the United States, and Mexican exiles, from archbishops to humble rancheros, received aid, here and in Mexico, from their fellow Catholics north of the border. And when the Cristero struggle ended in 1929, it was the active involvement and cooperation of Catholics on both sides of the border that had made that peace—and rebuilding of the Church—possible.
That unity between Catholics of the hemisphere had been prepared for hundreds of years. This unity is evident from the earliest colonial periods in the bringing of the Guadalupan devotion to the northern reaches of New Spain by the missionaries of what is now the southwestern United States—men such as Fr. Eusebio Kino and Blessed Junipero Serra in the seventeenth and eighteenth centuries. Meanwhile, men such as Friar Margil de Jesús brought the devotion to Central America, and others such as Count Luis Enríquez de Gúzman carried the image farther south, to Peru and beyond.
Our hemisphere is indeed a microcosm of the globalization process occurring worldwide. What happens in America will have a profound effect on the Church and the world, and what happens between the United States and Mexico will shape the future of our hemisphere. Catholics in both countries have every reason to work for a day when such close neighbors are even closer friends.
## UNITY WITHIN THE CHURCH
One model for the type of cooperation that can occur between the Catholics of the United States and Mexico is that seen in and fostered by the Knights of Columbus. Founded in 1882 in New Haven, Connecticut, the order rapidly expanded throughout the hemisphere. Canadian councils were founded in 1897, and Mexican councils in 1905. Since then, Knights from both sides of the border have accumulated more than a century of cooperation in many different forms. More recently, it has meant the support of seminarians from the United States to study in Mexico and learn Mexican culture, the support of Mexican seminarians to minister to the needs of Mexican immigrants in the United States, and the support of the Catholic Legal Immigration Network. Today, local Knights of Columbus councils along the border actively work together in Mexico and the United States on social, spiritual, and charitable projects. Increasing such cooperation is a high priority of the Knights of Columbus and it should be as well for other Catholic organizations in the United States. But all of this collaborative action is inspired by a belief that our unity as persons is stronger than any differences a border can raise, and our mission as Catholics is stronger than nationalism.
Beyond our common human family as men and women, we recognize a deeper bond of kinship as children of God and brothers and sisters of Christ, a bond that specifically orients us to the service of others. As Christians, and specifically as Catholics, we have a right to encourage in each other and expect in our dealings with each other a common measure of faith, hope, and love. In most of the countries of our hemisphere between 70 and 95 percent of the population is Catholic. In the United States, Catholics today represent one in every four people. If we, as Catholics, view one another as brothers and sisters in faith, as we have done before, and if we share that vision with the rest of our country, we truly have the opportunity to shape not just the future of the Church but the future of our own country, our continent, and our hemisphere.
It is precisely in this ecclesial childhood that we find in Our Lady of Guadalupe a true guide. Shortly before becoming pope, John Paul II spoke of the power of a mother's example of love: "This fundamental love, love for a human being who is conceived, who is carried beneath the heart of a mother, who is to be born: this is a fundamental love! The entire existence of the nation depends on it." How much more perfect is the example of Our Lady of Guadalupe to us! For regardless of our own upbringing or current family situation, Our Lady of Guadalupe comes reaffirming the goodness of our childhood before God and leading us by her example of compassion, by her affection and interest, by her persistence to guide and make easier Juan Diego's decision to follow the will of God, by her own trust in us to join her in her own missions and intercessions for God. Hans Urs Von Balthasar made this point as well when he wrote:
[Mary] is not only the maternal figure of the Church that gives birth to all the other members of the Body after it has borne the Head (Rev 12:17), and remains their Mother; she is also the archetypical member of the Church..., who [i.e., the Church] in all her members participates, through Christ and by virtue of word and sacrament, in the grace of being a child in the bosom of the Father.... [A]ll forms of the following of Christ within the Marian Church by carrying Christ's Cross with him, all priestly functions of the hierarchy and of the laity, are in the end ordered to this highest grace of childhood.
Thus if all forms of Christian living find a model in Mary, and if we speak of Mary as the archetypical believer and the archetype of the Church, it is possible to speak of her as the archetype of Christian culture, the Christian culture enacted by believers but not exclusively for believers, a culture that is the intersection of the Church and community.
Perhaps more than any other region in the Western Hemisphere, and even the world, North America is known for our diverse people, for being on the forefront of trends and creators of state-of-the-art technologies. Without a doubt, our versatile and often shifting culture requires new methods of evangelization, methods often proposed by these very changes. But also it requires us to ask ourselves what the face of Christian culture in the future should look like. For this, we can choose no better guide than Our Lady of Guadalupe when we search for those "seeds of the Word" in our culture and purify them into something supportive of all that is best in man and supportive of man's search for God.
In this, we do well to remember the words of Paul VI: "The civilization of a people is measured by its sensitivity in the face of suffering and its capacity to relieve it!" We do well to remember also the words of Benedict XVI to young people: "It is up to you and to your hearts to ensure that progress is transformed into a greater good for all. And the way of good—as you know—has a name: it is called love." As we consider our civilization and our place in our communities and families, we should remember that the generosity and the "way of love" in our culture and civilization cannot be summed up by the "generosity" of impersonal systems of aid, however efficient. If for this fact alone, the personal presence of Our Lady of Guadalupe presents a strong model of the importance of maintaining the personal element in charity. It is only through this personal element that aid takes on a magnitude beyond the problem itself, and becomes the face and foundation of love.
This is especially true for us as Christians. "It is therefore natural that those who truly want to be a companion of Jesus really share in his love for the poor. For us, the option for the poor is not ideological but is born from the Gospel." Just as we cannot ignore the poor on our spiritual journey to Christ, so we cannot ignore our faith as we labor for the poor. Catholics on both sides of the border should take the initiative to promote a Catholic solution to the problems of poverty and to promote economic and educational opportunities for the poor of the region. In Our Lady of Guadalupe, we have a model of "the preferential option for the poor" in a woman who came to conquered and conqueror alike in order to show to all—even the poorest and those without hope—the way to true resurrection. In her, we hear a voice expressing the true means to overcoming difficulties.
And if Our Lady of Guadalupe's message is one that has universal meaning for those in the Americas, then Juan Diego's example is also very important. On the continent where Juan Diego—a layman—was a key element in the evangelization of all those who inhabit it, we should carefully consider and follow his example. He found himself between two worlds. He was an Indian bringing this perfectly enculturated message both to his own people and to the Spanish. And he brought Our Lady of Guadalupe, and her message, to both groups consistently.
We too often find ourselves between worlds: between home and work, between our private and public lives. And yet, like Juan Diego, we are called to be witnesses of the truth—even when faced with disbelief. We must be undeterred as we bring the message of the mother of our continent and her Son to all those we encounter. We must witness by our lives, so that our example will be like a living tilma, and will provide a map to all we encounter of a civilization where love triumphs and motivates all action.
Our continent is one drawn together by shared history. On a historical level, all of its countries—to some degree—are nations of immigrants and of Native Americans. On a spiritual level, all of these countries share a common heritage of Christianity and baptism. And on a personal level, every person on this continent shares a mother: Our Lady of Guadalupe. From Canada to Argentina, all of us who live in the Americas are called, like Juan Diego, to bridge the divides of cultures, religion, and factions of any kind, by presenting to all the message of Our Lady of Guadalupe, the message of the mother of the civilization of love. What unites us as a Christian family, as children of a mother who has watched over us for nearly five hundred years, is far greater than anything that divides us.
But in this, no unity can be as great as the unity of a life with Christ, our Source. In Our Lady of Guadalupe, we find new signs of this great unity in the call for the chapel that would become the center of the parish and sacramental life for the people of not just Mexico, but of the Americas and indeed the whole world. As we prepare for the future, we do well to remember Benedict's words at Aparecida: "It is only from the Eucharist that the Civilization of Love will come forth." Thus Our Lady of Guadalupe offers us the promise that the Continent of Hope may one day blossom into a Civilization of Love.
# APPENDIX A
# _The Nican Mopohua_
_The_ Nican Mopohua _was probably written down sometime before Juan Diego's death in 1548 by the mestizo Antonio Valeriano (1520–1605), a student of the renowned missionary-scholar Friar Bernardino de Sahagún. The oldest copy of the_ Nican Mopohua _, dating from the mid-sixteenth century and containing about one-third of the apparition account, resides in the New York Public Library. The account reflects the popular style of speaking used by the Indians at the time of the apparitions. The paper on which the account is written contains watermarks, an indication of the paper's European origin, and it is written using Spanish lettering in vogue at the time. 1_
_Textually, the account contains a mixture of both Spanish and Náhuatl idioms and expressions characteristic of the time. 2 Notably, the indigenous elements appear in how the narrative and dialogue between Juan Diego and the Virgin reflect the high forms of indigenous speech, the_ huehuetlahtolli _, the speech of the elders. As the Náhuatl scholar Miguel León-Portilla notes, this suggests that the author was familiar with ancient Indian rhetoric and songs. Náhuatl rhetoric in the account includes the frequent use of poetic phrases (such as "your face, your heart" to mean "your person"), the unique expressions of honor through both titles of rank and diminutive titles of affection (such as "My Mistress, my Lady, my Queen, my littlest Daughter, my little Girl"), and multiple phrases to compound an idea, creating a cascade of images (such as when Juan Diego approaches the beautified Tepeyac_ _hill, saying, "By any chance am I worthy, have I deserved what I hear? Perhaps I am only dreaming it? Perhaps I'm only dozing? Where am I? Where do I find myself?"). 3 Although this form of speech may seem archaic to our ears, it has been retained in translation to give the reader the chance to experience naturally the rhetorical ebb and flow._
_This translation was greatly informed by the translation done by members of the Instituto Superior de Estudios Guadalupanos under the supervision of Msgr. Eduardo Chávez_.
Here is told and set down in order how a short time ago the Perfect Virgin Holy Mary Mother of God, our Queen, miraculously appeared on the Tepeyac, "nose of the hill," widely known as Guadalupe. First she caused herself to be seen by an Indian named Juan Diego, poor but worthy of respect; and then her precious and beloved image appeared before the recently named bishop, Don Fray Juan de Zumárraga.
1Ten years after the conquest of the water, mountain and city of Mexico, when the arrows and shields were put aside, when there was peace in all the towns, their waters and their mountains. 2Just as it budded, faith now grows green, now opens its corolla, the knowledge of the Giver of life, the true God.
3Then, in the year 1531, a few days into the month of December, it happened that there was an Indian, a _macehual_ , a poor man of the people; 4his name was Juan Diego, and he lived in Cuauhtitlán, as they call it, 5and in all the things of God he belonged to Tlatelolco.
6It was Saturday, not yet dawn, when he was coming in pursuit of God and His commandments. 7And as he drew near the little hill called Tepeyac, it was beginning to dawn. 8There he heard singing on the little hill, like the song of many precious birds. When their voices would stop, it was as if the hill were answering them. Extremely soft and delightful, their songs exceeded those of the _coyoltototl_ and the _tzinitzcan_ and other precious songbirds.
9Juan Diego stopped to look. He said to himself, "By any chance am I worthy, have I deserved what I hear? Perhaps I am only dreaming it? Perhaps I'm only dozing? 10Where am I? Where do I find myself? Is it possible that I am in the place our ancient ancestors, our grandparents, told us about: in the land of the flowers, in the land of corn, of our flesh, of our sustenance, perhaps in the land of heaven?"
11He was looking up toward the top of the hill, toward the direction from which the sun rises, toward where the precious heavenly song was coming from.
12And then when the singing suddenly stopped, when it could no longer be heard, he heard someone calling him from the top of the hill, someone was saying to him: "Dear Juan, dearest Juan Diego."
13Then he dared to go to where the voice was coming from, his heart was not disturbed and he felt extremely happy and contented, he started to climb to the top of the little hill to go see where they were calling him from. 14When he reached the top of the hill, he beheld a Maiden standing there. 15She called to him to come close to her.
16And when he reached where she was, he was filled with admiration for the way her perfect grandeur exceeded all imagination: 17her clothing was shining like the sun, as if it were sending out waves of light. 18And the stones, the crag on which she stood, seemed to be giving out rays 19like precious jades, like jewels they [the stones] gleamed. 20The earth seemed to shine with the brilliance of a rainbow in the mist. 21And the mesquites, prickly pear, and the other little plants that are generally up there seemed like quetzal feathers. Their foliage looked like turquoise. And their trunks, their thorns, their prickles, were shining like gold.
22He prostrated himself in her presence and listened to her venerable breath, her venerable words, which were extremely affable, extremely noble, as if from someone who was drawing him toward her and loved him. 23She said to him: "Listen my son, my youngest son, Juanito, where are you going?" 24And he answered her: "My Lady, my Queen, my Little Girl, I am going as far as your little house in Mexico Tlatelolco, to follow the things of God that are to us given, that are taught to us by our priests, those who are the images of the Lord, Our Lord."
25Then she spoke with him, she revealed her precious will; 26she said to him: "Know, know for sure my dearest and youngest son, that I am truly the ever perfect Holy Virgin Mary, who has the honor to be the Mother of the one true God for whom we all live, the Creator of people, the Lord of all around us and of what is close to us, the Lord of Heaven, the Lord of Earth.
"I want very much that they build my sacred little house here, 27in which I will show Him, I will exalt Him upon making Him manifest, 28I will give Him to all people in all my personal love, Him that is my compassionate gaze, Him that is my help, Him that is my salvation.
29"Because truly I am honored to be your compassionate mother, 30yours and that of all the people that live together in this land, 31and also of all the other various lineages of men; those who love me, those who cry to me, those who seek me, those who trust in me. 32Because there [at my sacred house] truly will I hear their cry, their sadness, in order to remedy, to cure all their various troubles, their miseries, their pains.
33"And to bring about what my compassionate and merciful gaze would achieve, go to the palace of the Bishop of Mexico, and tell him how I have sent you, so that you may reveal to him how I very much want him to build me a house here, to erect my temple on the plain; tell him everything, all you have seen and marveled at, and what you have heard. 34And know for sure that I will appreciate it very much and reward it, 35that because of it I will enrich you, I will glorify you; 36and because of it you will deserve very much how I will reward your fatigue, your service in going to petition the matter for which I am sending you. 37Now, my dearest son, you have heard my breath, my word; go, do what you are responsible for doing."
38And immediately he prostrated himself in her presence, and he said to her: "My Lady, My Little Girl, now I will go to make your venerable breath, your venerable word, a reality; for now, I leave you, I, your humble servant."
39Then he came down the hill to put her errand into action; he returned to the path and went straight to Mexico City. 40When he reached the center of the city, he went directly to the palace of the Bishop, the Governing Priest, who had just recently arrived; his name was Don Fray Juan de Zumárraga, a Franciscan Priest.
41And as soon as he got there, he tried to see him [the Bishop], he begged his [the Bishop's] servants, his helpers, to go and tell him that he needed to see him. 42After a long time, when finally the Reverend Bishop ordered that he [Juan Diego] enter, they [the Bishop's servants] came to call him. 43And as soon as he [Juan Diego] entered, first he knelt before him [the Bishop], he prostrated himself, then he revealed to him, he told him of the precious breath, the precious word of the Queen of Heaven, her message, and he also told him everything that made him marvel, what he saw, what he heard.
44But the Bishop, having heard his whole story, his message, as if he didn't particularly believe it to be true, 45answered him, he said to him: "My son, you will come again. At that time I will still hear you calmly, I will look at it carefully from the very beginning, I will consider the reason why you have come, what is your will, what is your wish."
46He [Juan Diego] left; he left sad because the errand entrusted to him was not immediately accepted. 47Then he returned, at the end of the day, he went straight from there to the top of the little hill, 48 and he arrived before Her, the Queen of Heaven: there, exactly where she had appeared to him the first time, she was waiting for him.
49As soon as he saw her, he prostrated himself before her, he threw himself to the ground, and he said to her: 50"My Mistress, my Lady, my Queen, my littlest Daughter, my little Girl, I went to where you sent me to carry out your venerable breath, your venerable word. Although I entered with difficulty to the place where the Governing Priest is, I saw him, and before him I placed your venerable breath, your venerable word, as you ordered me to do. 51He received me kindly and listened with attention, but, from the way he answered me, it's as if his heart didn't recognize it, he doesn't think it's true. 52He said to me: 'You will come again, and at that time I will still hear you calmly, I will look at it carefully from the very beginning, I will consider the reason why you have come, what is your will, what is your wish. 53We shall see,' the way he answered me; it's as though your venerable divine house that you want them to build here, that maybe I just made it up, or maybe that it doesn't come from your venerable lips.
54"So I beg you, my Lady, my Queen, my little Girl, to have one of the nobles who are held in esteem, one who is known, respected, honored, have him carry on, take your venerable breath, your venerable word, so that he will be believed. 55Because I am really just a man from the country, I'm the porter's rope, I'm a back frame, just a tail, a wing; I myself need to be led, carried on someone's back; there, where you sent me, it is not my place to go or to stay, my little Girl, my littlest Daughter, my Lady, my Girl. 56Please, excuse me, I will afflict your face, your heart; I will fall into your anger, your displeasure, my Lady Mistress."
57The Perfect Virgin, worthy of honor and veneration, answered him: 58"Listen my youngest son, know for sure that I have no lack of servants, of messengers, to whom I can give the task of carrying my breath, my word, so that they carry out my will; 59but it is necessary that you, personally, go and plead, that by your intercession, my wish, my will, become a reality. 60And I beg you, my youngest son, and I strictly order you, to go again tomorrow to see the Bishop. 61And in my name, make him know, make him hear my wish, my will, so that he will bring into being, he will build, my sacred house that I ask of him. 62And carefully tell him again how I, personally, the ever Virgin Holy Mary, I, who am the Mother of God, sent you as my messenger."
63For his part, Juan Diego responded and said to her: "My Lady, my Queen, my Little Girl, let me not anguish you or grieve your face, your heart; truly with gladness I will go to carry out your venerable breath, your venerable word; I absolutely will not fail to do it, nor does the road trouble me. 64I will go now, to carry out your will, but maybe I won't be heard and, if heard, maybe I won't be believed. 65But truly, tomorrow afternoon, when the sun goes down, I will come to return to your venerable breath, to your venerable word, what the Governing Priest answers to me. 66Now I respectfully say goodbye to you, my youngest Daughter, my young Girl, Lady, my Little Girl, rest a little more." 67And then he went to his house to rest.
68On the following day, Sunday, while it was still nighttime, everything was still dark, he went to Tlatelolco directly from his house, he came to learn about divine things and to be counted in roll call; then he went to see the Governing Priest.
69And around ten he was ready, he had been to Mass and was counted in the roll, and everyone had left. 70But he, Juan Diego, then went to the palace, the Reverend Bishop's house. 71And as soon as he arrived, he went through the whole struggle to see him and, after much effort, he saw him again. 72He knelt at his feet, he wept, he became sad as he spoke to him, as he revealed to him the venerable breath, the venerable word, of the Queen of Heaven. 73He hoped the errand would be believed, the will of the Perfect Virgin, to make for Her, to build for Her, Her sacred little house, where She had said, where She wanted it.
74And the Governing Bishop asked him many, many things, he interrogated him, in order to be certain about where he had seen Her, what She was like. He told absolutely everything to the Reverend Bishop. 75And although he told him absolutely everything that he had seen, that he had marveled at, that it appeared perfectly clear that She was the Perfect Virgin, the Kind, Wondrous Mother of Our Savior, Our Lord Jesus Christ; 76nevertheless his wish was not fulfilled. 77The Bishop said that not only on his [Juan Diego's] word would his petition be carried out, would what he requested happen, 78but that some other sign was very necessary if he were to believe how the Queen of Heaven, personally, was sending him [Juan Diego] as Her messenger.
79As soon as Juan Diego heard that, he said to the Bishop: 80"Señor Governor, think about what the sign you ask for will be, because then I will go ask for it of the Queen of Heaven who sent me." 81And as the Bishop saw that he [Juan Diego] was in agreement, that he did not hesitate or doubt in the slightest, he dismissed him. 82And as soon as he [Juan Diego] had left, the Bishop ordered some of his own household staff, in whom he had absolute trust, to go and follow him, to carefully observe where he went, whom he saw, and with whom he spoke. 83And this they did. And Juan Diego went straight along, following the path. 84But those who followed him, where the ravine opens, near Tepeyac, on the wooden bridge, came to lose him. And although they searched for him everywhere, they didn't see him anywhere.
85And so they turned back, not just because they had made terrible fools of themselves, but also because he had frustrated their attempt, he [Juan Diego] had made them angry. 86So they went to tell the Reverend Bishop, they put into his head that he shouldn't believe him [Juan Diego], they told him how he was only telling him lies, that he was only making up what he came to tell him, or that he was only dreaming or imagining what he was telling him, what he was asking of him. 87So they decided that if he came again, if he returned, they would grab him right there, and punish him severely, so that he would never again tell lies or get people all riled up.
88Meanwhile, Juan Diego was with the Most Holy Virgin, telling Her the response he brought from the Reverend Bishop: 89and, when She heard it, She said to him: 90"That's fine, my little son, you will come back here tomorrow so that you may take the Bishop the sign he has asked you for; 91with that he will believe you, and he will no longer have any doubts about all this, nor will he be suspicious of you; 92and know, my little son, that I will reward the care, the work and the fatigue that you have put into this for me; 93so go now; I will be waiting for you here tomorrow."
94And on the following day, Monday, when Juan Diego was to take some sign in order to be believed, he did not return. 95Because when he arrived at his house, the sickness had struck an uncle of his, named Juan Bernardino, who had become very ill. 96He [Juan Diego] went to get the doctor, who treated him, but it was too late; he was dying. 97And when night fell, his uncle begged him that, when it was still the early hours of the morning, when it was still dark, he [Juan Diego] go to Tlatelolco to call one of the priests to come and hear his [Juan Bernardino's] confession, to get him ready, 98because it was in his heart that it was truly now time, that now he would die, because he would no longer get up, he would no longer get well.
99And Tuesday, when it was still very dark, he left from there, from his house, to go to Tlatelolco to call a priest, 100and when he reached the side of the little hill, at the foot of Tepeyácac, the end of the mountain range, where the road comes out, towards where the sun sets, where he had always gone before, he said: 101"If I follow the road straight ahead, I don't want this Noble Lady to see me because, for sure, just like before, She'll stop me so I can take the sign to the Governing Priest for Her, as She ordered me to do. 102First we must get rid of our first affliction; first I must quickly call the priest since my poor uncle anxiously awaits him." 103He immediately went around the hill, climbed up the middle, crossing over it, and emerged towards where the sun rises; so he could quickly arrive in Mexico City, so that the Queen of Heaven would not stop him. 104He thought that where he made the turn the one who sees everywhere perfectly would not see him.
105But he saw how She was coming down from up on the hill, and that from there she had been looking at him, from where she saw him before. 106She came to meet him beside the hill, she came to block his way; she said to him: 107"My youngest son, what's going on? Where are you going? Where are you headed?" 108And he, perhaps he grieved a little, or perhaps he became ashamed? Or perhaps he became afraid of the situation, became fearful? 109He prostrated himself before Her, he greeted Her, he said to Her: 110"My little Maiden, my youngest Daughter, my Little Girl, I hope you are happy; how are you this morning? Does your beloved little body feel well, my Lady, my Girl?
111"Though it grieves me, I will cause your face and your heart anguish: I must tell you, my little Girl, that one of your servants, my uncle, is very ill. 112A terrible sickness has taken hold of him; he will surely die from it soon. 113And now I shall go quickly to your little house in Mexico, to call one of the ones beloved of Our Lord, one of our priests, so that he will go to hear his [my uncle's] confession and prepare him. 114Because in reality for this we were born, we who came to await the task of our death.
115"But, while I am going to do this, afterwards I will return here again to go carry your venerable breath, your venerable word, Lady, my Little Girl. 116Forgive me, be patient with me a little longer, because I am not deceiving you with this, my youngest Daughter, my Little Girl, tomorrow without fail I will come in all haste."
117As soon as She heard Juan Diego's words, the Merciful Perfect Virgin answered him: 118"Listen, put it into your heart, my youngest son, that what frightened you, what afflicted you is nothing; do not let it disturb your face, your heart; do not fear this sickness or any other sickness, nor any sharp or hurtful thing. 119Am I not here, I who have the honor to be your mother? Are you not in my shadow and under my protection? Am I not the source of your joy? Are you not in the hollow of my mantle, in the crossing of my arms? Do you need anything more?
120"Let nothing else worry you, disturb you; don't grieve over your uncle's illness, because he will not die of it for now, you may be certain that he is already healed." 121(And at that very moment his uncle was healed, as he later found out). 122And Juan Diego, when he heard the venerable breath, the venerable word, of the Queen of Heaven, he was greatly comforted by it, his heart became peaceful; 123and he begged her to send him immediately as messenger to see the Governing Bishop, to take him Her sign, for proof, so that he [the Bishop] would believe.
124And the Queen of Heaven ordered him then to go to the top of the little hill, where he had seen her before. 125She said to him: "Go up, my youngest son, to the top of the hill, to where you saw me and I told you what to do; 126there you will see spread out several kinds of flowers: cut them, gather them, put them all together: then come right down; bring them here, into my presence." 127And then Juan Diego climbed the little hill, 128and when he reached the top, he marveled at how many flowers were spread out there, their blossoms were open, flowers of every kind, lovely and beautiful, like those of Castille, when it was not yet their season 129because it was when the frost was worst. 130The flowers were giving off an extremely soft fragrance, like precious pearls, as if filled with the night's dew. 131Right away he began to cut them, gathered them all and put them in the hollow of his tilma. 132The top of the little hill was certainly not a place in which any flowers grew, because it was rocky, there were burs, thorny plants, prickly pear, and an abundance of mesquite bushes. 133And though some small grasses might grow, it was then the month of December, in which the ice eats everything up and destroys it.
134And immediately he came back down, he came to bring the Heavenly Maiden the different kinds of flowers which he had gone up to cut. 135And when She saw them, She took them with her venerable hands; 136 then She put them back in the hollow of Juan Diego's tilma and said to him: 137"My youngest son, these different kinds of flowers are the proof, the sign that you will take to the Bishop; 138you will tell him from me that in them he is to see my wish and that therefore he is to carry out my wish, my will; 139and you, you who are my messenger, in you I place my absolute trust. 140And I strictly order you that only alone, in the Bishop's presence, will you open your tilma and show him what you are carrying; 141and you will tell him everything exactly, you will tell him that I ordered you to climb to the top of the little hill to cut the flowers, and everything you saw and admired; 142so that you can convince the Governing Priest, so that he will then do what is entrusted to him, to build my little sacred house that I have asked for."
143And as soon as the Heavenly Queen gave him Her orders, he returned to the path, he went straight to Mexico, and now he went happily, 144his heart was tranquil now, because it was going to come out fine, the flowers would see to that. 145Along the way, he was very careful of what was in the hollow of his tilma, lest he lose something. 146As he went, he enjoyed the fragrance of the different kinds of exquisite flowers.
147When he arrived at the Bishop's house, the doorkeeper and the Governing Priest's other servants went to meet him. 148He begged them to tell him that he wanted to see him, but none of them was willing; they didn't want to listen to him, or perhaps because it was still very dark. 149Or maybe because they knew him by now, and all he did was bother and inconvenience them. 150And their companions [the other servants] had already told them about him, the ones who lost him when they were following him. 151For a long, long time he [Juan Diego] waited for his request to be granted. 152And when they [the servants] saw that he was simply standing there for a very long time, with his head down, doing nothing, in case he should be called, and how he was carrying something in the hollow of his tilma; then they came close to him to see what it was he was bringing and thus to satisfy their curiosity.
153And when Juan Diego saw that there was no way he could hide from them what he was carrying, and that therefore they would harass him, push him, or perhaps beat him, he gave them a little peek and they saw that it was flowers. 154And when they saw that they [the flowers] were all fine, different flowers, like those from Castille, and that it wasn't the season for them to be blooming, they admired them [the flowers] greatly, how fresh they were, with their buds open, how good they smelled, beautiful. 155And they wanted to grab them and pull a few out. 156They dared to try to take them three times, but there was no way they could do it, 157because when they tried, they couldn't see them [the flowers] anymore, instead they looked painted or embroidered or sewn into the tilma.
158They [the servants] went immediately to tell the Governing Bishop what they had seen, 159and how the lowly Indian who had come the other times wanted to see him, and that he had been waiting a very long time there for permission, because he wanted to see him [the Bishop]. 160And the Governing Bishop, as soon as he heard this, already had it in his heart that that was the sign to convince him, so he would carry out the work that the humble man had asked of him. 161He immediately ordered that they [the servants] let him in to see him. 162And, having entered, he [Juan Diego] prostrated himself in his [the Bishop's] presence, as he had done before. 163And again he [Juan Diego] told him about all he had seen, what he had admired, and his message. 164He said to him: "My Lord, Governor, I have truly done it, I carried out your orders; 165I went to tell the Lady, my Mistress, the Heavenly Maiden, Saint Mary, the Beloved Mother of God, that you asked for a sign in order to believe me, so that you would make her sacred little house, there where She asked that you build it; 166and I also told Her I had given you my word to come and bring you some sign, some proof of Her venerable will, as you told me to do. 167And She listened well to your venerable breath, your venerable word, and was pleased to receive your request for the sign, the proof, so that Her beloved will can be done, can be carried out. 168And now, when it was still nighttime, She ordered me to come again to see you; 169and I asked Her for Her sign so that I would be believed, as She said she would give to me, and immediately She kept her promise. 170And She sent me to the top of the little hill where I had seen Her before, to cut some different flowers there, like those from Castille. 171And when I had cut them, I took them down to Her below; 172and with Her venerable hands she took them. 173Then, again, She put them in the hollow of my tilma, 174so that I would come to bring them to you, so that I would deliver them to you personally. 175Although I knew well that the top of the hill isn't a place where flowers grow, because it's just rocks, burs, thorny plants, wild prickly pear and mesquite bushes, I didn't doubt because of that; I didn't hesitate because of that. 176When I reached the top of the little hill, I saw that it was now the Flowered Land [paradise]. 177There had sprung forth various flowers, like Castillian roses, the finest that there are, full of dew, splendid; so I went to cut them. 178And She told me that I should give them to you from Her, and that in this way I would prove it; so that you would see the sign you requested in order to carry out Her venerable will, 179and so that it would be clear that my word, my message, is the truth. 180Here they are; please receive them."
181And then he opened his white tilma, in the hollow of which were the flowers. 182And all the different flowers, like those from Castille, fell to the floor. 183Then and there his tilma became the sign, there suddenly appeared the Beloved Image of the Perfect Virgin Saint Mary, Mother of God, in the form and figure in which it is now, 184where it is preserved in her beloved little house, in her sacred little house in Tepeyac, which is called Guadalupe. 185And as soon as the Governing Bishop and all those who were there saw it, they knelt, they were full of awe, 186they stood up to see it, they were moved, their hearts were troubled, their hearts as well as their minds were raised. 187And the Governing Bishop, in tears, with sadness, begged Her, he asked Her forgiveness for not having carried out Her venerable will, Her venerable breath, Her venerable word.
188And the Bishop got up, and untied Juan Diego's garment, his tilma, from his neck where it was tied, 189on which appeared the venerable sign of the Heavenly Queen. 190And then he took it and placed it in his private chapel. 191And Juan Diego still stayed for the day in the Bishop's house, who still kept him there. 192And on the next day he [the Bishop] said to him: "Come, let's go so you can show me where it is that the venerable will of the Queen of Heaven wants Her chapel built." 193Immediately the order was given to make it, to build it. 194And Juan Diego, as soon as he showed where the Lady of Heaven had ordered that Her sacred little house be built, asked for permission to leave. 195He wanted to go home in order to see his uncle, Juan Bernardino, who was very ill when he left him, when he had gone to call on one of the priests in Tlatelolco to hear his confession and prepare him, the one whom the Queen of Heaven had said was already cured. 196But they didn't let him go alone, instead people went with him to his house. 197And when they arrived they saw that his venerable uncle was healthy, absolutely nothing pained him. 198And he, for his part, greatly admired the way in which his nephew was so accompanied and honored. 199He asked his nephew why this was happening, that they so honored him; 200and he [Juan Diego] told him [Juan Bernardino] how, when he left to go call on a priest to hear his confession, to prepare him, there in Tepeyácac the Lady of Heaven appeared to him. 201And She sent him to Mexico City to see the Governing Bishop, so that there he would build Her house in Tepeyácac.
202And She told him not to worry, because his uncle was already cured, and this very much consoled him. 203His uncle told him it was true, that She healed him at that exact moment, 204and he saw Her in exactly the same way She had appeared to his nephew. 205And She told him that she was also sending him to Mexico City to see the Bishop; 206and that also, when he went to see him, he should reveal absolutely everything to him, he should tell him what he had seen 207and the wonderful way in which She had healed him, 208 and that he should properly name Her Beloved Image thus: THE PERFECT VIRGIN, SAINT MARY OF GUADALUPE.
209And right away they took Juan Bernardino into the presence of the Governing Bishop, so that he could come to speak to him, to give him his testimony. 210And together with his nephew Juan Diego, the Bishop lodged them in his house for a few days, 211while the sacred little house of the Heavenly Maiden was built there in Tepeyac, where She revealed Herself to Juan Diego. 212And after some time, the Reverend Bishop moved the beloved Image of the Heavenly Maiden to the main church. 213He [the Bishop] took it from his palace, from his chapel where it had been, so that everyone could see and admire Her precious Image. 214And absolutely everyone, the entire city, without exception, trembled when they went to behold, to admire Her precious Image. 215They came to acknowledge it as something divine. 216They came to offer Her their prayers. 217They marveled at the miraculous way it had appeared 218since absolutely no one on Earth had painted Her beloved Image.
# APPENDIX B
# _Chronology of Guadalupan Events_
1474: An Indian named Cuauhtlatoatzin ("eagle that speaks") is born in Cuautitlán.
1476: Juan de Zumárraga is born in Spain.
1492: Christopher Columbus discovers the Americas, when he makes landfall on an island he calls San Salvador.
1517: Martin Luther writes his Ninety-Five Theses, commencing the Protestant Reformation.
1517: Francisco Hernández de Córdoba discovers Mexico.
1519-1521: Hernán Cortes lands in Mexico and conquers the capital city of the Aztecs.
1522: The first missionaries, including Pedro de Gante, arrive in Mexico.
1524: Official missionary activity begins with the arrival of twelve missionaries in Mexico City.
1525: The Indian Cuauhtlatoatzin is baptized and receives the Christian name of Juan Diego (John James).
1526: Dominican missionaries arrive in Mexico.
1528: Friar Juan de Zumárraga arrives in the New World.
1528: The first civil government, called the First Audience, arrives in New Spain, headed by President Nuño de Guzmán.
1529: Juan Diego's wife, María Lucía, dies.
1529, August 27: Problems arise between the First Audience officials and the evangelizing missionaries.
1530: There is a plot to assassinate bishop-elect Juan de Zumárraga, but he escapes harm.
1531: A series of natural events, including earthquakes, the appearance of Halley's comet, and a solar eclipse, leads the Indians to believe the world is about to end.
1531, December 9-12: During the winter solstice, Our Lady of Guadalupe appears to a Juan Diego Cuauhtlatoatzin, and asks him to be her messenger. The _tilma_ is presented to bishop-elect Juan de Zumárraga.
1531: The first chapel to Our Lady of Guadalupe of Tepeyac is built, and on December 26, the tilma with Our Lady of Guadalupe's image is carried in procession to this first chapel.
1531: The _Pregón del Atabal_ is composed, pairing pre-Conquest Aztec melodies with new words celebrating the procession of the _tilma_ to the chapel on the Tepeyac.
1537, June 9: Pope Paul III issues the papal bull _Sublimis Deus_ , which declares that Indians are able to receive the sacraments, encourages their catechesis, and defends their humanity.
1537: A _junta eclesiástica_ is convened to consider modifications of the baptismal ceremony which had been proposed and practiced to accommodate the unusually large number of baptisms. These discussions would continue for a couple decades.
1541: The Franciscan friar Toribio de Benavente, an early historian of New Spain, writes that some nine million Indians had converted to Christianity.
1544, May 15: The uncle of Juan Diego, Juan Bernardino, dies.
1545: During a great drought and plague, a pilgrimage of young children goes to the Guadalupan shrine.
c1545-1548: The _Nican Mopohua_ , an account of the apparitions of Our Lady of Guadalupe, is written down by a mestizo named Antonio Valeriano.
1548: Both Juan Diego and Bishop Juan de Zumárraga die in the same year.
Mid- to late sixteenth-century: Three of the most important extent manuscripts are written.
-The earliest extent manuscript of the _Nican Mopohua_ is written; the manuscript now resides in the New York Public Library;
-The _Códice 1548_ or _Codex Escalada_ is composed on deerskin, depicting the two of the apparitions of Our Lady of Guadalupe at the Tepeyac and Juan Diego wearing the tilma with the image on it; this manuscript also contains the date of Juan Diego's death, his name, a brief inscription in Náhuatl, and signatures of significant persons including Antonio Valeriano (author of the _Nican Mopohua_ ) and Sahagún;
-The _Codex Saville_ is written, a pictorial calendar in which a depiction of Our Lady of Guadalupe is place in the position representing the year 1531.
1555: In the Provincial Council, archbishop of Mexico, Alonso de Montúfar, formulates canons that indirectly approve the apparitions.
1555-1556 - The Chapel of the Tepeyac is put on the "Uppsala map" (named for the city—Uppsala—where the map is presently located.
1556: Archbishop Montúfar orders an investigation into the Guadalupan devotion during which several testimonies are taken that ultimately affirm the devotion as true expression of Christian faith and practices.
1556 - Archbishop Montúfar begins the construction of the second church in honor of Our Lady of Guadalupe.
1556: A chapel is built next to Juan Diego's house in Cuauhtitlán and another is built in Tulpetlac
c.1559: The daughter of Juan Martín García gives a detailed testimony about Juan Diego and his wife, María Lucía, including such details as where they were married and where they lived.
1562: A census, now located in the Basilica Museum, is conducted by Martín de Aranguren and speaks of the Virgin of Guadalupe.
1564: An image of Our Lady of Guadalupe is carried on the first formal expedition to the Philippine Islands.
1567: The new church begun by Archbishop Montúfar is completed.
1568: Bernal Díaz del Castillo, in his work _Verdadera Historia del Conquista de la Nueva España_ , twice mentions the Sanctuary of Our Lady of Guadalupe and notes that many miracles took place there.
1568: The pirate Miles Philips describes the great devotion of the Spaniards and Indians to Our Lady of Guadalupe.
1568: Friar Bernardino of Sahagún writes of the growing popularity of devotion to Our Lady Guadalupe on Tepeyac.
1570: Archbishop Montúfar sends King Philip II of Spain a copy of the image of Our Lady of Guadalupe done in oil paints.
1571: Admiral Giovanni Andrea Doria carries a copy of the image of Our Lady of Guadalupe aboard his ship during the battle of Lepanto and later credits the Virgin of Guadalupe with the victory over the Ottoman Empire forces.
1573: The historian Juan de Tovar, who transcribed the story of the apparitions from an earlier source, probably that written by Juan González, Zumárraga's translator, writes the "Primitive Relation."
1576: Pope Gregory XIII extends indulgences and blessings to the chapel at Tepeyac.
1582: Two important documents in the _File of Chimalhuacán Chalco_ , an exvoto (a sign of gratitude for a favor) and a sonnet, describe the apparitions of Guadalupe.
1589: In his _Treatise on the History of the Indies_ , Suárez de Peralta speaks of the apparition of Our Lady of Guadalupe.
1590: The _Nican Motecpana_ is written, providing an account of the apparitions and the virtuous life of Juan Diego.
1590: A sixteenth-century drawing that captures the apparition of Our Lady of Guadalupe to Juan Diego is completed.
1606: The first copy of the _tilma_ , dated and signed by Baltasar de Echave, is made.
1615: The artist Johannes Stradanus creates a copper engraving of the apparition of Our Lady of Guadalupe and the miracles attributed to her intercession.
1622: A publication from _Publicación de Diego Garrido_ captures the image of Our Lady of Guadalupe.
1647: The image of Our Lady of Guadalupe on the _tilma_ is covered with glass for the first time.
1648: For the 100th anniversary of Juan Diego's death, the priest Miguel Sánchez publishes _Imagen de la Virgen María, Madre de Dios de Guadalupe_ , a work recounting in Spanish the apparition story.
1649: Luis Lasso de la Vega publishes _Huei–Tlamahuicoltica_ , telling the story of the apparitions in Náhuatl and including earlier Náhuatl sources.
1650: The construction of the Indians' parish is completed, and the chapel is now used as a sacristy.
1666, February 18-March 22: A formal inquiry and investigation are conducted by the Church in order to inquire into the apparitions at Guadalupe and the miraculous tilma. The Vatican latter confirms the quality of investigation, designating them on the level of an Apostolic Visitation.
1666: The Chapel of the Cerrito is built at the highest point on the Tepeyac.
1667: Pope Clement IX institutes the feast of Our Lady of Guadalupe on December 12
1689: Carlos de Sigüenza y Góngora writes _Piedad Heroyca de don Fernando Cortés_ , in which he speaks of the apparitions of Guadalupe.
1695: The first stone of the new sanctuary is laid, and the sanctuary is solemnly dedicated in 1709.
1723: Another formal investigation ordered by Archbishop Lanziego y Eguilaz is conducted.
1737: The Most Holy Mary of Guadalupe is chosen as the patroness of Mexico City.
1746: The patronage of Our Lady of Guadalupe is accepted for all of New Spain, which includes the regions from northern California to El Salvador.
1746: Pope Benedict XIV approves the building of the Our Lady of Guadalupe College.
1746: The knight Boturini Benaducci promotes the solemn and official coronation of the image.
1754: Pope Benedict XIV approves the Virgin's patronage of New Spain and grants a Mass and Office proper to the celebration of her feast on December 12.
1756: The famous painter Miguel Cabrera publishes his study of the _tilma_ and image in the book _Maravilla Americana_.
1757: The Virgin of Guadalupe is named patroness of the city of Ponce in Puerto Rico.
1757: Pope Benedict XV allows King Ferdinand VII to use the offices and Masses of Our Lady of Guadalupe in the Spanish territories.
1767: When the Society of Jesus is expelled from the Spanish territory, the Jesuits carry the image with them around the world.
1787: Dr. Jose Ignacio Bartolache conducts an experiment on the tilma's miraculous preservation and commissions a group of artist to examine the Virgin's image.
1795: During a routine cleaning of the image's frame, acid is accidentally poured on the image of Our Lady of Guadalupe, yet miraculously the image is not damaged, except for a minor stain.
1810: Fr. Miguel Hidalgo y Costilla, leader of Mexico's movement for independence, takes the image of Guadalupe as his flag.
1821: Agustín de Iturbide puts the Mexican nation in the hands of Our Lady of Guadalupe and proclaims her Patroness and Empress of Mexico.
1895: Many of the bishops from throughout the Americas attend the pontifically authorized coronation of Our Lady of Guadalupe.
1899: The First Plenary Council of Latin America takes place in Rome and recognizes the special protection of Our Lady of Guadalupe.
1900: Pope Leo XII proclaims that the offices and Masses of Our Lady of Guadalupe are to be celebrated in perpetuity.
1904: Pope Pius X elevates the Church of Our Lady of Guadalupe to a minor basilica.
1910: Pope Pius X declares Our Lady of Guadalupe Patroness of Latin America.
1911: A church is built on the site of Juan Bernardino's home.
1921, November 14: A bomb placed beneath the image explodes, causing a great deal of damage within the basilica, but the _tilma_ is unharmed.
1926-1929: The Cristeros, fighting against Mexico's anticlerical government, adopt the battle cry: "Viva Cristo Rey, viva la Virgen de Guadalupe!" ("Long live Christ the King and Our Lady of Guadalupe!"). The North American episcopate and the Knights of Columbus support the persecuted Catholic Church in Mexico.
1926: For the first time, the feast of Our Lady of Guadalupe is celebrated at the Basilica without any priests participating, due to government restrictions on religion.
1928: A copy of the image is crowned in Santa Fe, Argentina.
1929: Photographer Alfonso Marcue makes the first documented discovery of an apparent reflection of a man's head in the right eye of the Virgin.
1931: In celebration of the 400 anniversary of the apparitions, the infants baptized in the Archdiocese of Guadalajara this year are given the name Guadalupe or José Guadalupe.
1933: The day Our Lady of Guadalupe was first proclaimed Patroness of Latin America is commemorated in St. Peters Basilica in Rome.
1935: Pope Pius XI names Our Lady of Guadalupe Patroness of the Philippines.
1938: The president of the Holy Name Society in California declares Our Lady of Guadalupe to be the Queen of the New World, who should be honored by all Catholics in the United States and Canada.
1941: Delegates representing twenty countries gather at the Basilica of Our Lady of Guadalupe to pray for peace. Among those attending is Archbishop John J. Cantwell of Los Angeles, who leads a delegation of American clergy to Mexico City and petitions that Our Lady of Guadalupe be named Patroness of the United States. The archbishop of Mexico City, Luis María Martínez, gives a small piece of the _tilma_ to Archbishop Cantwell.
1945: Pope Pius XII declares that the Virgin of Guadalupe is the Queen of Mexico and Empress of the Americas and upholds the divine origins of her miraculous image.
1946: Pope Pius XII declares Our Lady of Guadalupe Patroness of the Americas.
1951: Carlos Salinas examines the image and finds the reflection of a man's head in the right eye of the image of Our Lady.
1956 - Dr. Javier Torroella Bueno, an ophthalmologist, examines the eyes of the Virgin on the _tilma_ and confirms the existence of a reflection in her eyes.
1958: Dr. Rafael Torija-Lavoignet publishes his study of the Purkinje-Sanson effect as exhibited in the Guadalupan image.
1961: Pope John XXIII prays to Our Lady of Guadalupe as the Mother of the Americas and calls her the mother of and teacher of the faith to all people in the Americas.
1962: Studying a photograph of the image enlarged twenty-five times, Dr. Charles Wahlig, O.D., announces the discovery of two images reflected in the eyes of the Virgin.
1962: During a diplomatic visit to Mexico, President John F. Kennedy and the first lady Jacqueline Kennedy attend Mass at the Basilica of Our Lady of Guadalupe. 1966: Pope Paul VI sends a golden rose to the basilica of Our Lady of Guadalupe.
1975: The glass covering the image is removed so another ophthalmologist, Dr. Enrique Graue, can examine the image.
1976: The new basilica of Our Lady of Guadalupe, located four miles from central Mexico City, is dedicated.
1979: Pope John Paul II celebrates Mass in the sanctuary of Our Lady of Guadalupe during his first international pilgrimage.
1979: Dr. José Aste Tönsmann finds at least four human figures reflected in both eyes of the Virgin.
1981: The process of Juan Diego's canonization is officially opened.
1988: The liturgical celebration of Our Lady of Guadalupe on December 12 is raised to the status of a feast in all dioceses in the United States.
1990, May 3-6: Jose Barragan Silva fractures his skull and sustains life-threatening injuries in a fall from a balcony, but is healed through the intercession of Juan Diego. This miracle would become the miracle that would further Juan Diego's cause of canonization.
1990, May 6: Juan Diego is declared Blessed by Pope John Paul II at the Vatican. Later that year, Pope John Paul II returns to the Basilica in Mexico City to perform the beatification ceremony of Juan Diego.
1992: Pope John Paul II dedicates a chapel in honor of Our Lady of Guadalupe in St. Peter's Basilica.
1999: Pope John Paul II proclaims Our Lady of Guadalupe as Patroness of the whole American continent.
1999: The tilma is examined as part of the investigation process for Juan Diego's canonization.
2002, July 31: Juan Diego Cuauhtlatoatzin is canonized by Pope John Paul II at the Basilica of Our Lady of Gaudalupe in Mexico City.
2003: A relic of the _tilma_ tours the United States. The pilgrimage is organized by the Apostolate for Holy Relics and is sponsored by the Knights of Columbus and Holy Cross Family Ministries. The relic is then enshrined in the Cathedral of Our Lady of the Angels in Los Angeles.
2003: After the Juan Diego's canonization, Archbishop Norberto Rivera Carrera founds the Guadalupan Studies Institute to bring together scholars to further study the Guadalupan event.
2007, May 13: On his first apostolic journey, while in Brazil, Pope Benedict XVI underscores the continuing significance of Our Lady of Guadalupe by addressing the Bishops of Latin America and the Caribbean with the words she once spoke to Juan Diego centuries before.
# APPENDIX C
# _Prayers_
### PRAYER TO OUR LADY OF GUADALUPE
Given by John Paul II on his first apostolic visit to Mexico in 1979
_O Immaculate Virgin_ ,
_Mother of the true God and Mother of the Church!_
_You, who from this place reveal_
_your clemency and your compassion_
_to all who seek your protection_ ,
_hear the prayer that we address to you with filial trust_ ,
_and present it to your Son Jesus, our sole Redeemer_.
_Mother of Mercy, Teacher of hidden and silent sacrifice_ ,
_to you, who come to meet us sinners_ ,
_we dedicate on this day all our being and all our love_.
_We also dedicate to you our life, our work_ ,
_our joys, our infirmities and our sorrows_.
_Grant peace, justice and prosperity to our peoples;_
_for we entrust to your care all that we have and all that we are_ ,
_our Lady and Mother_.
_We wish to be entirely yours and to walk with you_
_along the way of complete faithfulness to Jesus Christ in His Church:_
_hold us always with your loving hand_.
_Lady of Guadalupe, Mother of the Americas_ ,
_we pray to you for all the Bishops, that they may lead the faithful along paths_
_of intense Christian life, of love and humble service to God and souls_.
_Contemplate this immense harvest, and intercede with the Lord_
_that He may instill a hunger for holiness in all the People of God_ ,
_and grant abundant vocations to the priesthood and religious life_ ,
_and that they be strong in the faith and zealous dispensers of God's ysteries_.
_Grant to our homes_
_the grace of loving and respecting life since its beginnings_ ,
_with the same love with which you conceived in your womb_
_the life of the Son of God_.
_Blessed Virgin Mary, Mother of Precious Love, protect our families_ ,
_so that they may always remain united, and bless the upbringing of our children_.
_You, our hope, look upon us with compassion_ ,
_teach us to go continually to Jesus and, if we fall, help us to rise again_ ,
_to return to Him, by confessing our faults_
_and sins in the Sacrament of Penance_ ,
_which brings peace to the soul_.
_We beg you to grant us a great love for all the holy Sacraments_ ,
_which are, as they were, the signs that your Son left us on earth_.
_Thus, Most Holy Mother, with the peace of God in our conscience_ ,
_with our hearts free from evil and hatred_ ,
_we will be able to bring to all true joy and true peace_ ,
_which come to us from your Son, our Lord Jesus Christ_ ,
_who with God the Father and the Holy Spirit_ ,
_lives and reigns forever and ever. Amen_.
### PRAYER TO ST. JUAN DIEGO
From John Paul II's homily at the canonization of St. Juan Diego at the Basilica of Our Lady of Guadalupe in 2002
_Blessed Juan Diego, a good, Christian Indian_ ,
_whom simple people have always considered a saint!_
_We ask you to accompany the Church..._
_so that she may be more evangelizing and more missionary each day_.
_Encourage the Bishops, support the priests_ ,
_inspire new and holy vocations_ ,
_help all those who give their lives_
_to the cause of Christ and the spread of his Kingdom_.
_Happy Juan Diego, true and faithful man!_
_We entrust to you our lay brothers and sisters_
_so that, feeling the call to holiness_ ,
_they may imbue every area of social life_
_with the spirit of the Gospel_.
_Bless families, strengthen spouses in their marriage_ ,
_sustain the efforts of parents_
_to give their children a Christian upbringing_.
_Look with favor upon the pain of those who are suffering_
_in body or in spirit, on those afflicted by poverty_ ,
_loneliness, marginalization or ignorance_.
_May all people, civic leaders and ordinary citizens_ ,
_always act in accordance with the demands of justice_
_and with respect for the dignity of each person_ ,
_so that in this way peace may be reinforced_.
_Beloved Juan Diego, "the talking eagle"!_
_Show us the way that leads to the "Dark Virgin" of Tepeyac_ ,
_that she may receive us in the depths of her heart_ ,
_for she is the loving, compassionate Mother_
_who guides us to the true God. Amen_.
### CONSECRATION OF THE FAMILY
TO OUR LADY OF GUADALUPE
_Holy Mary, Mother of God_
_you fled into Egypt with your husband Joseph_
_to save your child from a death defended by unjust laws_ ,
_seeking protection in a foreign land_.
_Holy Mary, Mother to Juan Diego_ ,
_on the Tepeyac hill you came not as an exile_
_but as a mother offering protection and peace_
_to a new family in this new land, bringing with you such beauty_
_that could only compare with his dreams of the heavenly_.
_Come now to the Tepeyac of our family_.
_To you we consecrate not only ourselves as individuals_ ,
_but we consecrate our shared life – our unity of love as a family_.
_Our family has shared many joys_ ,
_and you have known them all_.
_Continue to watch over us and strengthen us_
_in our shared life and love_.
_When one of us faces difficulty_ ,
_help us all to be supportive_.
_When division seems inevitable_ ,
_help us all to be patient and to grow together_.
_When we are lost in ourselves and our own cares_ ,
_distracted by demands_ ,
_when anger rises and unkind words lash out_ ,
_help us all to forgive and to seek forgiveness_ ,
_and to see our good in the good of our family_.
_Holy Mary, come to the Tepeyac of our hearts_ ,
_make such a presence of Christ there_
_that we may no longer roam the empty places_
_where difficulties are unanswered_ ,
_but find instead in you a way to Christ_ ,
_and find in Christ the way of love_ ,
_transforming even hard times into opportunities for love_.
_Help our family to create a place of love_
_for the unborn children with us_ ,
_for the sick and the elderly among us_ ,
_for the persons dying before us_.
_And when comfort seems inexpressible_
_and help seems beyond our abilities_ ,
_let your words and your hands guide us_.
_To those who seem far from your healing presence_ ,
_let our family radiate your love of Christ_.
_To those whose families have been broken by hardship_ ,
_who can no longer see the obscured or severed bond of love_ ,
_let our family be a refuge, an extension of your mantle_.
_To those who cannot see the face of Christ in their lives_ ,
_let our family's life in Christ, our domestic church_ ,
_become a place for meeting God_.
_Holy Mary, Mother at Guadalupe_ ,
_We, your children, trust you with the safety of our family_ ,
_Save us from tolerating evil, encourage our defense of the good_ ,
_that our family may become a thing of beauty_
_in building a civilization truly worthy of God's family_.
### PRAYER FOR LIFE
From John Paul II 1995 encyclical "The Gospel of Life" (Evangelium Vitae)
_O Mary_ ,
_bright dawn of the new world_ ,
_Mother of the living_ ,
_to you do we entrust the cause of life_.
_Look down, O Mother_ ,
_upon the vast numbers_
_of babies not allowed to be born_ ,
_of the poor whose lives are made difficult_ ,
_of men and women_
_who are victims of brutal violence_ ,
_of the elderly and the sick killed_
_by indifference or out of misguided mercy_.
_Grant that all who believe in your Son_
_may proclaim the Gospel of life_
_with honesty and love_
_to the people of our time_.
_Obtain for them the grace_
_to accept that Gospel_
_as a gift ever new_ ,
_the joy of celebrating it with gratitude_
_throughout their lives_
_and the courage to bear witness to it_
_resolutely, in order to build_ ,
_together with all people of good will_ ,
_the civilization of truth and love_ ,
_to the praise and glory of God_ ,
_the Creator and lover of life_.
### PLEDGE TO OUR LADY OF GUADALUPE
_Our Lady of Guadalupe_ ,
_by saying "yes" to God's gift of life_ ,
_you brought life to the world_.
_May I, like you, always be prepared_
_to accept the gift of new life_.
_You who told us that you would be our Mother_ ,
_always keep me close to your motherly heart_.
_May all the sons and daughters of this great land_
_yet to be born always be welcomed and protected_.
_O Virgin Mary, you who are_
_the Immaculate tabernacle of the Sacramental Jesus_ ,
_today I consecrate myself entirely to you_ ,
_and so I promise you that I will defend human life_
_from conception until the moment_
_when our Lord calls each person to His presence_.
### MAGNIFICAT
Also known as the Canticle of Mary, spoken by Mary at the Visitation in the Gospel
_My soul proclaims the greatness of the Lord;_
_my spirit rejoices in God my savior_
_for he has looked upon his lowly servant_.
_From this day all generations will call me blessed:_
_the Almighty has done great things for me_ ,
_and holy is his Name_.
_He has mercy on those who fear him
in every generation_.
_He has shown the strength of his arm_ ,
_he has scattered the proud in their conceit_.
_He has cast down the mighty from their thrones_ ,
_and has lifted up the lowly_.
_He has filled the hungry with good things,
and the rich he has sent away empty_.
_He has come to the help of his servant Israel_
_for he has remembered his promise of mercy_ ,
_the promise he made to our fathers_ ,
_to Abraham and his children for ever_.
### PRAYER FOR A CONTINENT OF HOPE AND LOVE
John Paul II's prayer from his homily given at the Basilica of Our Lady of Guadalupe in Mexico City in 1999
_O Mother! You know the paths followed_
_by the first evangelizers of the New World_ ,
_from Guanahani Island and Hispaniola_
_to the Amazon forests and the Andean peaks_ ,
_reaching to Tierra del Fuego in the south_
_and to the Great Lakes and mountains of the north_.
_Accompany the Church working in the nations of America_ ,
_so that she may always preach the Gospel_
_and renew her missionary spirit_.
_Encourage all who devote their lives_
_to the cause of Jesus and the spread of his kingdom_.
_O gentle Lady of Tepeyac, to you we present_
_this countless multitude of the faithful_
_praying to God in America_.
_You who have penetrated their hearts_ ,
_visit and comfort the homes, the parishes_ ,
_and the dioceses of the whole continent_.
_Grant that Christian families_
_may exemplarily raise their children_
_in the Church's faith and in love of the Gospel_ ,
_so that they will be the seed of apostolic vocations_.
_Turn your gaze today upon young people_
_and encourage them to walk with Jesus Christ_.
_O Lady and Mother of America!_
_Strengthen the faith of our lay brothers and sisters_ ,
_so that in all areas of social, professional, cultural and political life_
_they may act in accord with the truth and the new law_
_which Jesus brought to humanity_.
_Look with mercy on the distress of those suffering_
_from hunger, loneliness, rejection or ignorance_.
_Make us recognize them as your favorite children_
_and give us the fervent charity to help them in their needs_.
_Holy Virgin of Guadalupe, Queen of Peace!_
_Save the nations and peoples of this continent_.
_Teach everyone, political leaders and citizens_ ,
_to live in true freedom and to act_
_according to the requirements of justice_
_and respect for human rights_ ,
_so that peace may thus be established once and for all_.
_To you, O Lady of Guadalupe_ ,
_Mother of Jesus and our Mother_ ,
_belong all the love, honor, glory and endless praise_
_of your American sons and daughters!_
# NOTES
### INTRODUCTION
1. Cardinale, "First Stop Puebla."
2. John Paul II, _Rise, Let Us Be on Our Way_ , 55.
3. John Paul II, Homily at the Basilica of Our Lady of Guadalupe, January 27, 1979, §2.
4. León-Portilla addresses this authorship and dating in _Tonantzin Guadalupe_.
5. For example, _Información de 1556_.
6. Cf. Pardo, _The Origins of Mexican Catholicism_ , especially Chapter I on conversion and baptism, 20–48.
7. For example, on March 7, 1530, Friar Zumárraga excommunicated several members of the governing body, the First Audience, for torturing and killing a Crown priest, Cristóbal de Angulo, and García de Llerena, a servant to Cortés. Cf. García Icazbalceta, _Fray Juan de Zumárraga_ , 54–61.
8. John Paul II, General Audience, February 14, 1979, §6.
9. Benedict XVI, Address at the Inaugural Session of the Fifth General Conference of the Bishops of Latin America, §6.
10. John Paul II, _Ecclesia in America_ , §11.
11. Ibid.
12. Ibid.
13. Ibid., §12.
14. Ibid., §33.
15. Benedict XVI, Address at the Inaugural Session of the Fifth General Conference of the Bishops of Latin America, §4.
### CHAPTER 1
1. Luke 1:38.
2. "Actas e Informes Médicos y Técnicos sobre el Caso del Joven Juan José Barragán," in the Archive for the Cause of the Canonization of Saint Juan Diego, in the Congregation for the Causes of Saints, Holy See.
3. Cf. Sigüenza y Góngora, _Piedad Heróica de D. Fernando Cortés_ , 31.
4. Specifically, the _macehuales_ were "free commoners." See Owensby, _Empire of Law and Indian Justice in Colonial México_ , 16. Some of the early works about Juan Diego include: Alva Ixtlilxóchitl, "Nican Motecpana;" Becerra Tanco, _Felicidad de México_ ; Sigüenza y Góngora, _Piedad Heróica de D. Fernando Cortés_ ; and Florencia, _Estrella del Norte de México_. Early testimonies about Juan Diego can be found in "Informaciones Guadalupanas de 1666 y 1723," in de la Torre Villar and Navarro de Anda, eds., _Testimonios Históricos Guadalupanos_. Finally, for some contemporary sources, see Chávez, _Juan Diego: La Santidad de un Indio Humilde_ and Chávez, _Juan Diego: Una Vida de Santidad Que Marcó la Historia_.
5. _Informaciones Jurídicas de 1666_ , f. 158r.
6. For an extended account of Cortés's conquest of the Aztec Empire, see Levy, _Conquistador_ ; also, for an excellent account specifically on Cortés's siege of Mexico City, see Hanson, _Carnage and Culture_ , 170–93. For more on the practice of human sacrifice in the Aztec culture, see Carrasco, _City of Sacrifice_.
7. Throughout this chapter, the account of the apparition is based on Valeriano, _Nican Mopohua_. Full translation included in Appendix A.
8. Valeriano, _Nican Mopohua_ , 9–10.
9. _Cantares Mexicanos_ , fol. 16v.
10. In the Aztec civilization, corn was of such great importance that it took on profound religious significance; the Aztecs had several gods of corn to whom they made numerous offerings and for whom they performed human sacrifices. In _Patterns in Comparative Religion_ , Mircea Eliade describes one rite when, just as the corn seed began to sprout, the Aztecs would go out to the fields "to find the god of the maize," "a shoot which they brought back to the house and offered food, exactly as they would a god." He continues: "In the evening, [the sprout] was brought to the temple of Chicome-coatl, goddess of sustenance, where a group of young girls were gathered, each carrying a bundle of seven ears of maize saved from the last crop, wrapped in red paper and sprinkled with sap. The name given to the bundle, _chicomolotl_ (the sevenfold ear), was also the name of the goddess of the maize. The girls were of three different ages, very young, adolescent, and grown up—symbolizing, no doubt, the stages in the life of the maize—and their arms and legs were covered in red feathers, red being the colour of the maize divinities. This ceremony, intended simply to honour the goddess and obtain her magic blessing upon the newly-germinating crop, did not involve any sacrifice. But three months later, when the crop was ripe, a girl representing the goddess of the new maize, Xilonen, was beheaded." See Eliade, _Patterns in Comparative Religion_ , 343.
11. Valeriano, _Nican Mopohua_ , 12.
12. Ibid., 17.
13. Ibid., 18–21.
14. Ibid., 24.
15. Ibid., 26.
16. Ibid., 26–31.
17. See, for instance, the analysis of the _Codex Mendoza_ in Carrasco, _City of Sacrifice_ , 25.
18. For more information on Friar Zumárraga, see García Icazbalceta, _Fray Juan de Zumárraga_.
19. Letter from June 12, 1531. Regarding the destruction of the codices, see Von Hagen, _The Aztec and Maya Papermakers_ , 31–32.
20. Valeriano, _Nican Mopohua_ , 54–56.
21. Quoted in Lockhart, _Nahuas and Spaniards_ , 5–6; cf. Karttunen and Lockhart, _The Art of Náhuatl Speech_.
22. Valeriano, _Nican Mopohua_ , 57–62.
23. Luke 1:52, the Magnificat.
24. Mendieta, _Historia Eclesiástica Indiana_ , 514.
25. Sahagún, _Historia General_ , 546.
26. Sahagún, _Florentine Codex_ , VI, 35.
27. The plagues that ravaged Mexico throughout the sixteenth century were catastrophic, beginning with the smallpox epidemic of 1519–20, when between five million and eight million people perished, and perhaps culminating—though by no means ending—with the epidemic of cocoliztli of 1545–48. This later epidemic, responsible for the death of Juan Diego's uncle in 1545, killed between five million and fifteen million people, or up to 80 percent of the native population of Mexico. As pointed out by Acuña-Soto in "Megadrought and Megadeath in 16th Century Mexico": "In absolute and relative terms the 1545 epidemic was one of the worst demographic catastrophes in human history, approaching even the Black Death of bubonic plague, which killed approximately 25 million in western Europe from 1347 to 1351 or about 50% of the regional population."
28. Valeriano, _Nican Mopohua_ , 107.
29. Ibid., 114.
30. Ibid., 115–16.
31. Ibid., 118–19.
32. _Testimonios de la Antigua Palabra_ , 161.
33. Lockhart, _Nahuas and Spaniards_ , 6.
34. Valeriano, _Nican Mopohua_ , 120.
35. Ibid., 137–39.
36. In order to come near the distinctiveness of the Nahuatl concepts, it is important to keep in mind the _difrasismos_ (the "disguises"): in different texts written in Nahuatl in the sixteenth-century we find certain linguistic forms that have definitive discursive contexts. These forms have been given several names: metaphors, couplets, _difrasismos_ , or paired phrases. The word difrasismo was coined by Father Angel Maria Garibay, one of the first scholars of Náhuatl language and culture in Mexico. See especially Garibay K., _History of Náhuatl Literature_ , 926. The main characteristic is the juxtaposition of two or even three words that create a meaning that is not the sum of its parts, but rather produces a third meaning.
Later we shall see the difrasismo "face-heart," which the Indians used to refer to the human person or individual, as well as the difrasismo " _in alt, in tepetl_ ," meaning nation or civilization; equally present in Indian speech and writing is the difrasismo " _in xóchitl, in cuícatl_ ," meaning "flower and song." As León-Portilla notes in _Aztec Thought and Culture_ , 75, beyond its literal meaning, "the phrase is a metaphor for poetry or a poem." And poetry—"flower and song"—was prized by the Indians as "the only truth on earth." Thus the above identification of flower and song with truth.
37. Anawalt, _Indian Clothing Before Cortés_ , 27.
38. Valeriano, _Nican Mopohua_ , 208.
39. Luke 18:42; Matthew 9:22.
40. To this day, countless historical anthropologists and specialists in the Náhuatl language have yet to reach a consensus regarding the origin of the Virgin's name. Around the time of the first official inquiry of 1666, Luis Becerra Tanco, one of the most influential commentators on the Guadalupan event, argued in his work _Felicidad de México_ that Juan Bernardino, a native Náhuatl-speaker, would not have understood the Spanish "Guadalupe," because the sounds for "g" and "d" do not occur in the Náhuatl language, a language that was at the time still used by a vast majority of the Indians. Becerra Tanco then suggested two alternative Náhuatl names that sounded similar to the Spanish "Guadalupe": _Tequatlanopeuh_ , "she whose origins were in the rocky summit," and _Tequantlaxopeuh_ , "she who banishes those who devoured us." See Becerra Tanco, _Felicidad de México_ , 21. Becerra Tanco's argument has over the years inspired countless scholars to propose alternative indigenous names that, as these scholars contend, were more likely to have been used by the Guadalupan Virgin and Juan Bernardino. Yet Becerra Tanco and others fail to take into account three important facts. First, at the time of the apparition, Juan Bernardino and Juan Diego would have been familiar with the "g" and "d" sounds, which were necessary in the pronunciation of their baptismal names received in 1525. Second, prior to the 1675 publication of Becerra Tanco's _Felicidad de México_ , there is no historical evidence indicating that the Virgin was called by any of the names proposed by Becerra Tanco or later Náhuatl scholars, including the earliest account of the event written in Náhuatl. Third, there is a wealth of historical documents, written by the early Spaniards and the Franciscan friars, contending that the Virgin's title be changed to _Tepeaquilla_ or _Tepeaca_ , thus indicating that the original name was "Guadalupe" and not an Indian name, for had it been a Náhuatl name, there would have been no controversy. See, for instance, the 1556 testimony of the Franciscan Friar Alonso de Santiago in "Información de 1556 ordenada realizar por Alonso de Montúfar," in de la Torre Villar and Navarro de Anda, (eds.), _Testimonios Históricos Guadalupanos_ , 61–62.
41. The name "Guadalupe" has been analyzed by a number of authors, most of whom agree that the first part of the name means "river," and as for the second part the name, some authors propose that it derives from a Latin word "lupus" meaning "wolf." However, it is little probable that a word would be partly derived from Arabic and partly from Latin, and agree with the analysis that it come from _lub_ , meaning in Arabic "black lava," "black gravel," "black stones," or as "hidden" (that is, staying in the dark or darkness). This interpretation points to the very bottom of the river, that which is "hidden" or "in blackness" – that is, the riverbed which carries the water and moves it about. (Cf. Guerrero, _The Nican Mopohua_ , T. I, 92).
42. Crémoux, _Pélerinages et Miracles_ , 10–12.
43. An early chronicle on the Aztec migration and discovery of the eagle-sign recounts: "Seeing that everything / was filled with mystery, / [the Aztecs] went on, to seek / the omen of the eagle / and wandering from place to place / they saw the cactus and on it the eagle, / with its wings spread out to the rays of the sun, / enjoying its warmth and the cool of the morning, / and in its claws it held an elegant bird / with precious and resplendent feathers. /... [T]hey began to weep and give vent to their feelings, / and to make displays and grimace and tremble, as a sign of their joy and happiness, and as an expression of thanks, saying: 'How have we deserved such a blessing? Who made us worthy of such grace / and greatness and excellence? / Now we have seen what we have sought, / and we have found our city and place. / Let us give thanks to the Lord of creation / and to our god Huitzilopochtli.' / [T]hen, the next day, the priest / Cuauhtloquetzqui said to all of the tribe, / 'My children, we should be grateful to our god / and thank him for the blessing he has given us. / Let us all go and build at the place of the cactus / a small temple where our god may rest." See Durán, _Historia de las Indias de Nueva España_ , cited in Matos Moctezuma, "Templo Mayor: History and Interpretation," in Broda, Carrasco, and Matos Moctezuma, eds., _The Great Temple of Tenochtitlan_ , 29–30. Also, for the above-cited information on the eagle in Aztec culture and religion, as well as the etymological information on Tenochtitaln, see Miller and Taube, _An Illustrated Dictionary_ , 83.
44. The name of this first chapel, commonly called in Spanish the "hermita," can be confusing. This is in part due to the fact that this building had a particular unique history and unique uses as a shrine to Our Lady, as a chapel where Mass was celebrated, as a pilgrimage site, and as the place where Juan Diego lived. Throughout the book, we tend to refer to the "hermita" as the hermitage, following the tradition of several Guadalupan scholars.
45. Valeriano, _Nican Mopohua_ , 214–18.
46. Alva Ixtlilxóchitl, "Nican Motecpana," 305.
47. John Paul II, Homily for the Canonization of St. Juan Diego, §5.
48. Ibid.
### CHAPTER 2
1. Damascene, _On Holy Images_ , 48.
2. Cf. González Fernández et al, _El Encuentro de la Virgen de Guadalupe_.
3. Andrade, "Testimonio de Manuel Ignacio Andrade," ff. 19v–20v.
4. Ibid.
5. Molina, _Química Aplicada al Manto de la Virgen de Guadalupe_ , 3.
6. Cf. Bartolache y Posadas, "Manifiesto Satisfactorio" in de la Torre and Navarro de Anda, eds., _Testimonios Históricos Guadalupanos_. Bartolache's study consisted of two parts, one being the experiment described above. The other part was an examination of the tilma by several expert painters. After examining the image, the painters gave testimony as to their findings. When asked, "Taking into account the rules of your faculty, and without any personal passion or desire, do you consider this Holy Image as having been painted miraculously?" the artists responded: "Yes, inasmuch as what is considered substantial and [original], in our Holy Image, but not inasmuch as certain touch-ups and details which without a doubt, seem to have been done later by impudent hands" (ibid., 648).
7. Molina, _Química Aplicada al Manto de la Virgen de Guadalupe_ , 3.
8. The team of artists included Juan Salguero, an art teacher for more than thirty years; Thomas Conrado, an art teacher for more than eight years; Sebastián López de Ávalos, a painter for more than thirty years; Nicolás de Fuenlabrada, a painter for over twenty years; Nicolás de Angulo, an art teacher and painter for more than twenty years; Juan Sánchez, an art teacher and painter for more than fifteen years; and Alonso de Zárate, an art teacher and painter for over fourteen years. See ibid.
9. _Informaciones Jurídicas de 1666_ , ff. 138v–140r.
10. The chemists on the team included Dr. Lucas de Cárdenas Soto, Dr. Jerónimo Ortiz, and Dr. Juan Melgarejo. See Molina, _Química aplicada al manto de la Virgen de Guadalupe_ , 3.
11. _Informaciones Jurídicas de 1666_ , f. 185r–185v. As the _Informaciones Jurídicas de 1666_ notes, the Hermitage was built on the ground bordering the south side of the lake which extended almost to the very edge of the Hermitage near the main entrance. Even at this time, the chemists saw evidence of the humidity of the sacristy and the church. Not only did the Hermitage suffer from the breezes constantly blowing from the lake, but during the rainy season, waters would rise up from the lake and reach the Hermitage, thus making the area around the Hermitage very wet and allowing the ground's humidity to seep through the foundation from underneath. See ibid. ff. 182v–183r.
12. Ibid., f. 187r.
13. Ibid.
14. Aste Tönsmann, _El Secreto de Sus Ojos_ , 48.
15. Alva Ixtlilxóchitl, "Nican Motecpana," 307.
16. Second Council of Nicaea, "Decree of the Holy, Great, Ecumenical Synod, the Second of Nicaea."
17. Sahagún, _Florentine Codex_ , XII, f. 25r.
18. See _Coloquios y Doctrina Cristiana con que los Doce Frailes de San Francisco Enviados por el Papa Adriano VI y por el Emperador Carlos V_ , f.34r and f.35r, respectively.
19. For example, Hernández Illescas and Fernando Ojeda Llanes. See also Hernández Illescas, Rojas Sánchez, and Salazar, _La Virgen de Guadalupe y las Estrellas_.
20. The complete text and its judicial ratification are in the Sacred Congregation for the Causes of the Saints, Archive for the Cause of the Canonization of Juan Diego.
21. Cf. especially Carrasco, _City of Sacrifice: the Aztec Empire and the Role of Violence in Civilization_.
22. The difference between the day of the winter solstice in 1531 and now is due to the change from the Julian calendar to the Gregorian calendar (used today); while the Julian calendar was largely satisfactory, it failed to take into account the need for adding an extra day (leap day) every fourth year, causing a ten-day difference between the calendars.
23. González et al., _El Encuentro de la Virgen de Guadalupe y Juan Diego_ , 213.
24. Mendieta, _Historia Eclesiástica Indiana_ , 99.
25. Soustelle and O'Brian, _Daily Life of the Aztecs_ , 46–47.
26. _Testimonios de la Antigua Palabra_ , 77.
27. The complete text and its judicial ratification are in the Sacred Congregation for the Causes of the Saints, Archive for the Cause of the Canonization of Juan Diego.
28. This is also seen in the early catechetical artwork, depicting the crucifixion only symbolically, showing the instruments of the crucifixion—the nails, the crown of thorns, the spear – instead of showing the body of Christ himself.
29. Ochoterena, Análisis de Unas Fibras del Ayate de Juan Diego o Icono de Nuestra Señora de Guadalupe.
30. Benedict XVI, _On the Way to Jesus Christ_ , 74.
31. López Trujillo, Address at the Opening of the International Theological Pastoral Congress Fourth World Meeting of Families; cf. _Documento Final de Santo Domingo_ (IV Conferencia Episcopal Latinoamericano, 1992), 30.
32. John Paul II, Address at Vigyan Bhavan.
33. Paul VI, _Ecclesiam Suam_ , §87.
### CHAPTER 3
1. Fr. Mario Rojas developed the concept of the indigenous glyphs in a way that can easily be compared with those that the Virgin of Guadalupe had on her dress. Cf. Rojas Sánchez, _Guadalupe: Símbolo y Evangelización_.
2. Boone, _Stories in Red and Black_ , 23.
3. For example, the island city of Tenochtitlan itself was designed according to it, divided into four quadrants by four roads, all connecting to the central temple, the Templo Mayor. For more on the symbolism of the Templo Mayor in relation to the Aztec religion and cosmology, see Matos Moctezuma, "Symbolism of the Templo Mayor."
4. During his reign, which ended with his death in 1472, Nezahualcóyotl, the great poet, philosopher, and king, erected a temple directly opposite the temple of Huitzilopochtli, which was previously built in recognition of Tenochtitlan's political supremacy. Unlike the temples of Tenochtitlan, this new temple was dedicated to Tloque Nahuaque, "lord of the close and near, invisible as the night and intangible as the wind," and held no images or idols. See León-Portilla, _Fifteen Poets of the Aztec World_ , 76.
5. As suggested, the four-petal flower over the Virgin's womb is in the position of the _Nahui Ollin_ , meaning constant movement. The _Nahui Ollin_ is represented in several Indian codices, such as the _Codex Ríos_ and the _Telleriano Remensis_ manuscript, and in certain artistic and archeological works, including some found in the Templo Mayor as well as the so-called "Sun Stone," depicting the Aztec calendar. In Rojas Sánchez's _Guadalupe: Símbolo y evangelización_ (p. 151-154 and Plates No. 55), the four-petal flower is related to Indian cosmology and solar symbols. In this, some of the important Indian cosmological concepts to keep in mind are: _Tonatih_ (the Sun itself), _Nahui Ollin_ (four movements), _Chalchiuhmichihuacan_ (where the master of the jade fish lives; life), and _Omeyocan_ (place of duality, of god). Significantly, each of these concepts is expressed through the four-petal flower. And thus, the flower on the Virgin's womb, representing Christ, draws on each of these concepts, thereby expressing to the Indians a God who is singular, omnipotent, eternal, and in constant movement. That is, a God who is the Lord of life, of the heavens, and of the earth.
To better understand the symbolic richness and synthesis of these concepts, see León-Portilla, _La Filosofía Náhuatl_ , particularly chapters II and III. Here, León-Portilla considers the following passage from _Manuscript 1558_ , in which the Indians identify the Sun with the Nahui Ollin and connect these with the preservation of life: "This Sun, named four movement, this is the Sun on which we live now." Analyzing this passage, León-Portilla writes: "Just as the text states, this can also be seen at the marvelous Sun stone, where the central figure represents the face of _Tonatiuh_ (Sun), inside the sign of 4 movement (nahui ollin) from _Tonalámatl_. Given that this fifth Sun makes its entrance into the Nahuatl cosmological thought, the idea of movement becomes extremely important in the image of the world and its destiny" ( _La Filosofía Náhuatl_ , 8).
Notably, the idea of a single deity was not foreign to some of the Indians, specifically the Indian sages or _tlamatinime_. As León-Portilla notes: "In sharp contrast with the popular worship of the sun god _Huitzilopochtli_ , the _tlamatinime_ kept the ancient belief in a single god who exists beyond all heavenly levels." Called by many names, this god, while absolutely one, was at the same time seen as having two aspects or faces, one masculine and the other feminine, hence his popular title Ometéotl, meaning "the god of duality." See León-Portilla, _Visión de los vencidos_. See also López Austin, "The Mexicas and Their Cosmos," 26.
6. Valeriano, _Nican Mopohua_ , 26.
7. Second Vatican Council, _Ad Gentes_ , §11.
8. Acts 17:27–28.
9. For more information on Aztec cosmology and Ometéotl, see León-Portilla, _Aztec Thought and Culture_ , 62–103.
10. For more information on the Aztec calendar system, see Aguilar-Moreno, _Handbook to Life in the Aztec World_ , 290–98.
11. Cf. Hassig, _Time, History and Belief in Aztec and Colonial Mexico_ , 7–17.
12. Luke 2; traditional "Christmas Proclamation" from the _Roman Martyrology_.
13. Valeriano, _Nican Mopohua_ , 1.
14. As suggested, for the natives, the difrasismo " _in atl in tepetl_ " or "water and mountain" means "village," "city," or "civilization." The mountain—the stone, the wood, the cavern, etc.—provided protection, while the water was associated with life. For the natives, the two realities together signified civilization. Cf. Luis Becerra Tanco, _Felicidad de México_ , 1979 translation; León-Portilla, _Tonantzin Guadalupe_ , 92-93; Guerrero, _El Nican Mopohua_ , T. I, 109.
15. León-Portilla, _Fifteen Poets of the Aztec World_ , 148.
16. Ibid., 147–48.
17. For more on the symbolism of the Templo Mayor in relation to Aztec cosmology, see Matos Moctezuma, "Symbolism of the Templo Mayor," 185–209.
18. Broda, "The Provenience of the Offerings: Tribute and Cosmovisión," 230–31.
19. Sahagún, quoted and discussed in ibid.
20. León-Portilla, _Aztec Thought and Culture_ , 8.
21. The Náhuatl sage, as León-Portilla explains, "was called _te-ix-tlamach-tiani_ , 'teacher of people's faces.'" Additionally, describing the task of the Aztec educator, one Aztec poet wrote: "He makes wise the countenances of others; / he contributes to their assuming a face; / he leads them to develop it.... / Thanks to him, people humanize their will." See León-Portilla, _Aztec Thought and Culture_ , 115. Cf. Sahagún, _Códice Matritense de la Real Academia_ , VIII, f. 118v.
22. León-Portilla, _Aztec Thought and Culture_ , 142–43.
23. Ezekiel 11:19–20.
24. The complete text and its judicial ratification are in the Sacred Congregation for the Causes of the Saints, Archive for the Cause of the Canonization of Juan Diego.
25. _Testimonios de la Antigua Palabra_ , 145.
### CHAPTER 4
1. Benedict XVI, Address to the Local Population and the Young People.
2. Words from 1 John 4:8; quoted in Benedict XVI, _Deus Caritas Est_ , §1.
3. Ibid., §6.
4. Carrasco, _City of Sacrifice_ , 96. Carrasco writes: "[I]n this fearful night, women were closed up in granaries to avoid their transformation into fierce beasts who would eat men, pregnant women put on masks of maguey leaves, and children were punched and nudged awake to avoid being turned into mice while asleep." Also, as is relayed in the _Florentine Codex_ , on that night, the people of Mexico "became filled with dread that the sun would be destroyed forever. All would be ended, there would evermore be night. Nevermore would the sun come forth. Night would prevail forever, and the demons of darkness would descend, to eat men." See Sahagún, _Florentine Codex_ , 7:27, quoted in Carrasco, _City of Sacrifice_ , 97. Finally, León-Portilla cites ancient annals that provide mythological information on the previous four creations or "suns" and also gives more specific information about the anticipated apocalypse: "Under this sun [the fifth sun or present age] there will be earthquakes and hunger, and then our end shall come." See _Annals of Cuauhtitlán_ and _Leyenda de los Soles_ , cited in León-Portilla, _Aztec Thought and Culture_ , 39.
5. John 3:16.
6. John Paul II, Address at the Puebla Conference, Mexico, 1979.
7. Benedict XVI (Joseph Cardinal Ratzinger), _God and the World_ , 181.
8. Benedict XVI, _Deus Caritas Est_ , §9.
9. Genesis 1:27.
10. John Paul II, _Christifideles Laici_ , §17.
11. For more on John Paul II's Theology of the Body, see Anderson and Grenados _, Called to Love: Approaching John Paul II's Theology of the Body_.
12. Benedict XVI, _Deus Caritas Est_ , §17.
13. John Paul II, _Letter to Families_ , February 2, 1994, §19.
14. John Paul II, Address to Participants in the International Games for Disabled Persons, §7.
15. Benedict XVI, Homily for the Inauguration of his Pontificate.
16. Benedict XVI, General Audience, November 23, 2008.
17. Benedict XVI, Interview with Deutsche Welle.
18. Saraiva Martins, "The Face of Christ in the Face of the Church," §2.
19. Pope Benedict XVI, Message for the Celebration of the World Day of Peace, January 1, 2007.
20. Benedict XVI, _Sacramentum Caritatis_ , §70.
21. John 19:34.
22. Chrysostom, "Catecheses," 3, 13–19.
23. Benedict XVI, Meeting with Catholic Educators.
### CHAPTER 5
1. John Paul II, Message for the World Day of Peace 1981, §11.
2. John Paul II (Karol Wojtyla), Homily at Jasna Góra (Czestochowa), 1978. Quoted in Boniecki, _The Making of the Pope of the Millennium_ , 739.
3. Mendieta, _Historia Eclesiástica Indiana_ , 311–12. "And fearing that the friars would send notice of their tyrannies to the King and to his advisors, they placed the potential stagecoach under watch as well as interrupted all of the roads and trails so that word could not get through. And so they decreed that no one take a letter from religious men without the authorities reading it first. And subsequently they ordered inspections of the ships upsetting everything down to the ballast, looking for letters from the friars. And not content with this, for one or another reason, they wished to forestall, at the expense of the honor of the innocents, that if any letter arrived from them it should be discredited. To this effect, since they themselves were the witnesses and secretaries, they wrote their reports slandering the Holy Bishop and the friars with ugly and unimaginable comments."
4. For a fuller account, see García Icazbalceta, _Fray Juan de Zumárraga_ , 36.
5. Zumárraga, _Letter from Friar Juan de Zumárraga to the King of Spain_.
6. Ibid., f. 314 v.
7. Mendieta, _Historia Eclesiástica Indiana_ , 53.
8. Ibid., 49.
9. Sahagún, _Historia General_ , 17.
10. This began scarcely one year after the twelve Franciscans arrived, beginning in Texcoco and later continuing in Mexico, Tlaxcala, and Huejotzingo. Motolinia provides us with interesting detailed news of this first "battle" against the idols by the Franciscans under the cover of night with the aid of some of their young catechism students: "Idolatry was as well established as before until, on the first day of the year 1525, which in that year fell on a Sunday, in _Texcoco_ , the location of most and greatest of the _teocallis_ or temples, and those most filled with idols and which were very well serviced by priests or ministers, on that very night three monks, from ten at night until dawn, scared and drove away all those who were in the houses and halls of the devils; and that day, after mass, they preached to them forbidding them to murder and ordering them, on behalf of God and the King, not to do that again otherwise they would be punished according to how God commanded such to be punished. This was the first battle against the devil, and then in Mexico and its surroundings, and in _Coauthiclan_ [Cuautitlan]." Motolinia, _History of the Indians of New Spain_ , 22.
11. John Paul II, _Letter to Men and Women Religious of Latin America_. Also quoted in John Paul II, _Message for the IV World Day of the Sick_ , October 11, 1995, which was held at the Basilica of Our Lady of Guadalupe in Mexico.
12. Motolinia took great pains in calculating the number of baptisms, acquiring figures from missionaries and provinces. See his _History of the Indians of New Spain_ , 131–33.
13. Macpherson, "St. Ethelbert." As the eighth-century English historian St. Bede the Venerable described the situation: "When he [King Ethelbert], among the rest, believed and was baptized,... greater numbers began daily to flock together to hear the Word, and, forsaking their heathen rites, to have fellowship, through faith, in the unity of Christ's Holy Church. It is told that the king, while he rejoiced at their conversion and their faith, yet compelled none to embrace Christianity, but only showed more affection to the believers, as to his fellow citizens in the kingdom of Heaven. For he had learned from those who had instructed him and guided him to salvation, that the service of Christ ought to be voluntary, not by compulsion." Bede, _Historia Ecclesiastica Anglorum_ , Book I, Chapter 26.
14. This especially became apparent in the Peace of Augsburg in 1555, based upon the principle of _cujus regio, ejus religio_ (whose region, his religion), which declared that the religion of the ruling prince dictated the religion permitted in his realm, effectively forcing religious segregation.
15. Motolinia, _History of the Indians of New Spain_ , Bk. 2, Ch. III, 131.
16. Ibid. Cf. Pardo, _The Origins of Mexican Catholicism_ , 23–24, and 170–71 n. 10.
17. Hanson, _Carnage and Culture_ , 175.
18. Sahagún, _Florentine Codex_ , VI, 35.
19. Cf. León-Portilla, _Aztec Thought and Culture_.
20. _Coloquio y Doctrina Cristiana_ , f. 36r. Also see León-Portilla, _El Reverso de la Conquista_ , 25.
21. Cf. León-Portilla, _Visión de los Vencidos_. Also see León-Portilla, _El Reverso de la Conquista_.
22. Benedict XVI, _Spe Salvi_ , §24.
23. Benedict XVI (Joseph Cardinal Ratzinger), _Principles of Catholic Theology_ , 391.
24. González Fernandez, et al., _El Encuentro de la Virgen de Guadalupe y Juan Diego_ , 204.
25. "Carta de Vasco de Quiroga al Consejo de Indias" in _Documentos Inéditos_ , Torres de Mendoza, T. XIII, 421, in Cuevas, _Historia de la Iglesia en México_ , T. I, 312.
26. Sahagún, "Sobre Supersticiones," in his _Historia General de las Cosas de Nueva España_. Available in de la Torre Villar and Navarro de Anda, eds., _Testimonios Históricos Guadalupanos_ , 142–44.
27. Valeriano, _Nican Mopohua_ , 26: "Sancta Maria in inantzin in huel nelli Teotl Dios" (Holy Mary, mother of the true God).
28. Mendieta, _Historia Eclesiástica Indiana_ , 276.
29. Charny, _The Book of Chivalry of Geoffroi de Charny_ , 177.
30. For more on the Spanish monarchy and its policies toward the Indians of the New World, see especially Owensby, _Empire of Law and Indian Justice in Colonial Mexico_.
31. Similarly, in 1503, Queen Isabel issued a decree ordering that the Indians be treated "as the free people they are, and not as serfs" (ibid., 134).
32. Another important figure in the debates regarding the humanity and rights of the Indians is the missionary Bartolomé de las Casas. Las Casas first came to the New World with his father in 1502, settling on Hispaniola, where he was given a _repartimiento_. However, in 1514, after a dramatic conversion, las Casas gave up his own _repartimiento_ and became an adamant defender of the Indians. Later, he joined the Dominican Order, and dedicated much of his time to denouncing both in his preaching and scholarly writing the unjust treatment of the Indians, admitting, however, that "no account, no matter how lengthy, how long it took to write, nor how conscientiously it was compiled, could possibly do justice to the full horror of the atrocities committed." Bartolomé de las Casas, _A Short Account of the Destruction of the Indies_ , 43. For more information on De las Casas and New World rights, see Vickery, _Bartolomé de las Casas: Great Prophet of the Americas_.
33. Mark 12:31.
34. Benedict XVI (Joseph Cardinal Ratzinger), _God and the World_ , 300.
35. Benedict XVI (Joseph Cardinal Ratzinger), _Principles of Catholic Theology_ , 391.
36. "His uncle told him it was true,... and he saw her in exactly the same way she had appeared to his nephew. And she told him that she was also sending him to Mexico City to see the Bishop." Valeriano, _Nican Mopohua_ , 203–5.
37. Leies, _Mother for a New World_ , 231–35.
38. Benedict XVI, _Deus Caritas Est_ , §28b.
39. John Paul II, _Message for the World Day of Peace_ , January 1, 2003, §9.
40. Congregation for the Doctrine of the Faith and Joseph Cardinal Ratzinger, _Instruction on Christian Freedom and Liberation_ , §63.
41. Benedict XVI (Joseph Cardinal Ratzinger), _The Ratzinger Report_ , 173.
42. Ibid, 176.
43. Ibid.
44. De la Vega, _El Gran Acontecimiento_. Regarding the study of Tepeyac hill, see Letter from Guillermo Gándara (director of the Mexico City Herbario) to Fr. Jesús García Gutiérrez (secretary of the Academia de la Historia Guadalupana), February 19, 1924. The study by Ign. Carlos F. de Landero was conducted September 15, 1923; quoted in Benítez, _El Misterio de la Virgen de Guadalupe_ , 204.
45. According to the _New York Times_ , June 2, 1921, even prior to the successful bombing, another bombing attempt was made: "Three persons disguised as beggars, one of them said to be carrying a dynamite bomb, were arrested last night at the village of Guadalupe, near here, charged with attempting to break up religious services being held in the cathedral there under the auspices of the Knights of Columbus. They were later released upon orders from Celestino Gasca, Governor of the Federal District, who declared there was no evidence against them." Incidentally, Celestino Gasca was also a leader of the CROM. "Vera Cruz Radicals in 24-Hour Strike."
46. In fact, all churches in Mexico were asked to hold prayer services. See _Brownsville Herald_ , "Catholics Make Solemn Atonement for Desecration," November 19, 1921. Regarding Obregon's visit, see "Two Goddesses," _New York Times_ , November 20, 1921.
47. Meyer, _La Cristiada: El conflicto entre la iglesia y el estado_ , 148–51.
48. Ibid., 148.
49. "A group of Separatists, known as the Knights of Guadalupe, has requested the Mexican Government to give the Basilica of Guadalupe to the National or Separatist Church as the seat of the Mexican Catholic religion. The knights say their intention is to make Guadalupe the Vatican of Mexico." "One Dead, Three Hurt in Mexico City Riot," _New York Times_ , February 24, 1925.
50. Ibid.; "Calles Closes Church." _New York Times_ , March 15, 1925.
51. Álvaro Obregón, quoted in "Papal Notes," _Time_ , September 1, 1924.
52. Alva Ixtlilxóchitl, "Nican Motecpana," 307.
53. Leies, _Mother for a New World_ , 266.
54. Suzanne Silvercruys, "Eusebio F. Kino."
### CHAPTER 6
1. Benedict XVI (Joseph Cardinal Ratzinger) and Seewald, _God and the World_ , 296.
2. Regarding the _Información_ , quoted in Johnson, _The Virgin of Guadalupe_ , 47.
3. Onis, "Mexican Pilgrims Flock to the Shrine of Our Lady of Guadalupe."
4. Cf. _Información de 1556_ and the _Letter of the Hieronymite Brother Diego de Santa María_.
5. John Paul II, General Audience, December 16, 1987, §1.
6. John Paul II, General Audience, December 9, 1987, §3.
7. Ibid., §1.
8. John Paul II, General Audience, December 16, 1987, §1.
9. John Paul II, General Audience, December 16, 1987, §1.
10. Luke 16:19-31.
11. Benedict XVI, "Thoughts on the Place of Marian Doctrine and Piety in Faith and Theology as a Whole," in Benedict XVI (Joseph Cardinal Ratzinger) and Von Balthasar, _Mary: the Church at the Source_ , 36.
12. Ibid.
13. Betancourt, Interview on Vatican Radio. Quoted in "Betancourt Trusting Our Lady for End to Conflict."
14. Ibid.
15. Mendieta, _Historia Eclesiástica Indiana_ , 277.
16. Valeriano, _Nican Mopohua_ , 55.
17. John Paul II, _Crossing the Threshold of Hope_ , 212-13.
18. Synod of Bishops, _Encounter with the Living Jesus Christ: The Way to Conversion, Communion and Solidarity in America_ , §14.
19. John 2:1-11. Quotation from John 2:5. New American Bible translation.
20. Benedict XVI, General Audience, February 21, 2007.
21. Benedict XVI, Homily at the Lower Basilica of St. Francis.
22. Benedict XVI, General Audience, February 21, 2007.
23. Churchill, Radio Broadcast, London, October 1, 1939. In Churchill, _Never Give In_ , 199.
24. Groeschel, _Spiritual Passages_ , 54-55.
25. Philippians 2:8.
26. Nietzsche, _Antichrist_ , §2.
27. John Paul I, Address to the Roman Clergy.
28. John Paul II, _Homily for the Inauguration of his Pontificate_ , §5.
29. Benedict XVI, _Homily for the Inauguration of his Pontificate_.
30. Valeriano, _Nican Mopohua_ , 55.
31. Ibid., 58-59.
32. Mark 15:34; see also Psalm 22, verse 1.
33. Luke 23:46.
34. Benedict XVI, General Audience, November 26, 2008.
35. John Paul II, _Rise, Let Us Be On Our Way_ , 56.
36. Benedict XVI, _Co-Workers of the Truth_ , 390-91.
37. Mother Teresa and González-Balado, _Mother Teresa_ , 35.
38. Thornton and Varenne, _The Essential Pope Benedict XVI_ , 335.
39. Benedict XVI, _God and the World_ , 299.
### CHAPTER 7
1. John Paul II, Address at Czestochowa.
2. Valeriano, _Nican Mopohua_ , 32.
3. Leies, _Mother for a New World_ , 188–93.
4. Matthew 28:20.
5. Matthew 28:16–20.
6. Valeriano, _Nican Mopohua_ , 26–28.
7. Most authors translate verses 27 and 28 of the _Nican Mopohua_ as: "in order to show all my love, compassion, aid, and defense in [the temple]." However, in Nahuatl, different words—and thus different concepts—can be created by joining them or bringing them together in single words, enabling Náhuatl to express rich, multilayered concepts with few words. Breaking down the Nahuatl version of the Virgin's words, these layers are made visible: "Notlazotlaliz" means "my love." "Noicnoitaliz" means "my compassion" and can also be literally translated as "my compassionate glance." However, the Nahuatl text in the _Nican Mopohua_ says: "Notetlazotlaliz," "noteicnoitaliz": this - _te_ \- does not refer to "something," which means that it does not refer to the action that she is carrying out, but to "someone," a person. Thus, she is not talking about herself but of someone else. Therefore, the literal translation of this is "Notetlazotlaliz," "my love-person;" "Noteicnoitaliz," "my compassionate glance-person;" "Notepalehuiliz," "my aid-person;" "Note- manahuiliz," "my salvation-person." This would be hard to pronounce in many languages, including English and Spanish, but from a theological point of view, it has a very profound meaning, given that it is obvious that Saint Mary of Guadalupe is talking with her Son. That is why the most faithful translation, taking into account all the richness and implications, is: "He who is my love;" "He who is my compassionate glance;" "He who is my aid;" "He who is my salvation." It is also interesting to note that the translators of Nican Mopohua had not paid attention to the syllable - _te_ -, which we can clearly observe in the original text, because in their own philological notes about the usage of the syllable - _te_ \- they mention that it makes reference to "someone." It seems, though, that they might have done this in order make the reading of the text in Spanish easier, given the difficulty translating these concepts literally. The result is that these content-rich words that join the syllable _-te-_ end up being translated more simply and clearly in Spanish as "my love" or "my compassionate glance," which, even if not completely incorrect, do not express the richness of this expression. José Luis Guerrero was a pioneer in capturing the deep theological significance of taking into account the syllable - _te_ -, which attaches itself to the word. Even though this makes the translation more literal and thus more difficult to articulate in common Spanish, it helps one capture the depth and richness of the words of Saint Mary of Guadalupe, given that she is making reference to "someone," who evidently is her Son Jesus Christ. Cf. Guerrero, _El Nican Mopohua: Un Intento de Exégesis_ , 171–77.
8. John Paul II, General Audience, April 13, 1997, §5.
9. Ibid.
10. Valeriano, _Nican Mopohua_ , 24.
11. Ibid.
12. Benedict XVI, _The God of Jesus Christ_ , 69–84.
13. Ibid., 78.
14. Von Balthasar, quoted in Benedict XVI, _The God of Jesus Christ_ , 75.
15. Benedict XVI, _The God of Jesus Christ_ , 77.
16. For more information about the uses and interpretations of the ending _-tzin_ as an expression of courtesy and reverence toward another person, see Ortiz de Montellano, _Nican Mopohua_ , 275. Also cf. Guerrero, _El Nican Mopohua: Un Intento de Exégesis_ , 136, 142–43. For more information about _-tzin_ as acknowledging honor in one's own situation, see Guerrero, _El Nican Mopohua: Un Intento de Exégesis_ , 321. Cf. also Chávez, _La Verdad de Guadalupe_ , 318.
17. John 19:26–27.
18. John Paul II, _Redemptoris Mater_ , §45.
19. Quoted by John Paul II, Homily at Mass, Accra, Ghana, §2: "You have shown your faith in action, worked for love and persevered through hope, in your Lord Jesus Christ."
20. Benedict XVI, in Benedict XVI (Joseph Cardinal Ratzinger) and Von Balthasar, _Mary: The Church at the Source_ , 31.
21. Cf. John Paul II, _Mulieris Dignitatem_ , §2-5, as well as John Paul II, _Letter to Women_ , §10.
22. John Paul II, Homily at Mass, Suva.
23. John Paul II, _Mulieris Dignitatem_ , §22.
24. Benedict XVI, in Benedict XVI and Von Balthasar, _Mary: The Church at the Source_ , 25.
25. Ibid., 25.
26. Quoted in González Fernandez, _Guadalupe: Pulso y Corazón de un Pueblo_ , 187–88.
27. Mendieta, _Historia Eclesiástica Indiana_ , 278.
28. Wainwright and Tucker, eds., _The Oxford History of Christian Worship_ , 631–48.
29. Benavente, _History of the Indians of New Spain_ , 22.
30. Torquemada, _Monarquía Indiana_ , III:140.
31. Mendieta, _Historia Eclesiástica Indiana_ , 282–83.
32. Sahagún, _Historia General_ , 38.
33. Cameron, Homily.
### CHAPTER 8
1. Paul VI, _Apostolicam Actuositatem_ , §2.
2. Mendieta, _Historia Eclesiástica Indiana_ , 429.
3. For example, one such artistic representation, dated to the sixteenth century, was presented in 1983 by the commission for the beatification of Juan Diego. Done on wood by an unknown Indian artist, this portrait depicts Juan Diego crowned with a halo and wearing the Franciscan habit. Also presented as evidence for Juan Diego's beatification was a sculpture of Juan Diego, again done by an anonymous Indian artist, portraying Juan Diego as a Franciscan, with a missionary staff.
4. Toribio de Benavente discussed this in _Memoriales_ and in _History of the Indians of New Spain_. See Pardo, _Origins of Mexican Catholicism_ , 49.
5. Pardo, _The Origins of Mexican Catholicism_ , 52.
6. Paul VI, _Apostolicam Actuositatem_ , §2.
7. Benedict XVI, General Audience, August 20, 2008.
8. John Paul II (Karol Wojtyla), Vatican Radio, November 25, 1963. Quoted in Boniecki, _The Making of the Pope of the Millennium_ , 221.
9. Benedict XVI, Address to the Bishops of Switzerland, November 9, 2006.
10. John Paul II, Messaggio per la Giornata Mondiale del Migrante 1992, §6.
11. Benedict XVI, _Deus Caritas Est_ , §14.
12. John Paul II (Karol Wojtyla), quoted in Boniecki, _The Making of the Pope of the Millennium_ , 196.
13. "Testimonio de Marcos Pacheco," in _Informaciones Jurídicas de 1666_ , f. 15r–15v, facsimile and translation available in Chávez, _La Virgen de Guadalupe y Juan Diego_.
14. John Paul II, Address for the Arrival Ceremony in Mexico, July 30, 2002, §3.
15. "Eucharistic Prayer I," in _The Sacramentary_ , 546.
16. Benedict XVI, General Audience, August 20, 2008.
17. Of course, all Catholics are in some measure "witnesses" to the faith, participating in the evangelization through their actions if not their words as well; likewise, the margin of Indians who did convert became witnesses in this same capacity. However, Juan Diego still stood out in recognition by others, in that he was hailed as a good Christian, an exemplary witness of the life of faith.
18. Benedict XVI (Joseph Cardinal Ratzinger), "The New Evangelization: Building the Civilization of Love."
19. John Paul II, Address to a Spanish-Speaking Parish, in _The Pope Speaks to the American Church_ , 222, §3.
20. Benedict XVI (Joseph Cardinal Ratzinger) and Peter Seewald, _Salt of the Earth_ , 265.
21. John Paul II, Homily at the Basilica of Our Lady of Guadalupe, January 23, 1999, §8.
22. John Paul II, Ad Limina Address to the Bishops of Indianapolis, Chicago and Milwaukee; John Paul II, _Ecclesia in Oceania_ , §19.
23. 23. John Paul II, Address at the Puebla Conference. Quoted in John Paul II, _Ecclesia in America_ , §66.
### CHAPTER 9
1. John Paul II, Homily at the Basilica of Our Lady of Guadalupe, January 23, 1999, §8.
2. Acuña-Soto, et al., "Megadrought and Megadeath in 16th Century Mexico."
3. López Luján, _The Offerings of the Templo Mayor of Tenochtitlan_ , 151. See also Juan Alberton Román Berrelleza, "Offering 48 of the Templo Mayor: A Case of Child Sacrifice," in Boone, ed., _The Aztec Templo Mayor_.
4. This pilgrimage is recounted in the _Nican Motecpana_ and in Miguel Sánchez's _Imagen de la Virgen_ (1648). Both available in de la Vega, ed., _The Story of Guadalupe_ , 94–97 and 142–43, respectively.
5. Luke 1:39–56; quotation from Luke 1:42. _New American Bible translation_.
6. Luke 1:43. _New American Bible translation_.
7. Benedict XVI (Joseph Cardinal Ratzinger) and Von Balthasar, _Mary: The Church at the Source_ , 102–3.
8. Benedict XVI (Joseph Cardinal Ratzinger), "Truth and Freedom."
9. Ibid.
10. Ibid.
11. Merton, _No Man Is an Island_ , 3.
12. Genesis 1:31.
13. John Paul II, General Audience, April 9, 1986, §4.
14. Valeriano, _Nican Mopohua_ , 114.
15. Ibid., 119.
16. Weil, _Waiting for God_ , 65.
17. Benedict XVI, Homily for the Inauguration of his Pontificate.
18. For more information on the role of sacrifice in Aztec cosmology, see León-Portilla, _Aztec Thought and Culture_ , especially 25–61.
19. Durán, _Historia de las Indias_ , I, 242. Quoted also in León-Portilla, _Aztec Thought and Culture_ , 163–64.
20. Sahagún, _Historia General_ , 384–85. Quoted in Chávez, _Our Lady of Guadalupe and Saint Juan Diego_ , 55–56.
21. Benedict XVI, _Spe Salvi_ , §34.
22. Valeriano, _Nican Mopohua_ , 26–32.
23. Benedict XVI, Address at the Conclusion of the Meeting with the Bishops of Switzerland.
24. Benedict XVI, _Spe Salvi_ , §34.
25. Ibid., §35.
26. Paul VI, _Evangelica Testificatio_ , §39.
27. Benedict XVI, Address to an International Congress, "Oil on the Wounds."
28. Valeriano, _Nican Mopohua_ , 107.
29. Benedict XVI, Homily at the Basilica of Notre-Dame du Rosaire.
30. Psalm 27.
### CHAPTER 10
1. John Paul II, _Ecclesia in America_ , §11.
2. Benedict XVI, Address at the Inaugural Session of the Fifth General Conference of the Bishops of Latin America, §3.
3. Synod of Bishops, Message of the Special Assembly for America, §4.
4. John Paul II, _Ecclesia in America_ , §44, quoting from _Propositio_ 54.
5. Benedict XVI, "Die Ecclesiologie des Zweiten Vatikanums."
6. John Paul II, _Ecclesia in America_ , §11.
7. Leies, _Mother for a New World_ , 323–29.
8. Benedict XVI, Address at the Inaugural Session of the Fifth General Conference of the Bishops of Latin America, §4.
9. Benedict XVI, _Deus Caritas Est_ , §16.
10. David Rieff, "Nuevo Catholics."
11. John Paul II, Letter to World Movement of Christian Workers.
12. Quoted in Boniecki, _The Making of the Pope of the Millennium_ , 720.
13. Von Balthasar, _Unless You Become Like This Child_ , 73.
14. Paul VI, Homily for Beatification, November 1, 1975.
15. Benedict XVI, Address to the Fathers of the General Congregation of the Society of Jesus.
16. Benedict XVI, Address at the Inaugural Session of the Fifth General Conference of the Bishops of Latin America, §4.
### APPENDIX A
1. Burrus, _The Oldest Copy of the Nican Mopohua_ , 3–4. Observations on the Spanish and Náhuatl textual similarities first made by Fr. Mario Rojas Sánchez.
2. Ibid.
3. León-Portilla, _Tonantzin Guadalupe_ , 87.
4. Translation provided by the Instituto Superior de Estudios Guadalupanos.
### APPENDIX C
1. John Paul II, Prayer.
2. John Paul II, Homily for the Canonization of St. Juan Diego Cuauhtlatoatzin, §5.
3. John Paul II, _Evangelium Vitae_ , §105.
4. Luke 1:46–55.
5. John Paul II, Homily at the Basilica of Our Lady of Guadalupe, January 23, 1999, §9.
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Von Balthasar, Hans Urs. _Unless You Become Like This Child_. Translated by Erasmo Leiva-Merikakis. San Francisco: Ignatius Press,1991.
Wainwright, Geoffrey, and Karen Westerfield Tucker, eds. _The Oxford History of Christian Worship_. New York: Oxford University Press, 2006.
Weil, Simone. _Waiting for God_. Trans. Emma Craufurd. New York, NY: HarperCollins Publishers, 2001.
Wood, Michael. _Conquistadors_. Berkeley: University of California Press, 2000.
Zavala, Silvio. _La Filosofía Política en la Conquista de América_. Mexico City: FCE, 1947.
———. _Repaso Histórico de la Bula Sublimis Deus de Paulo III_ , _en Defensa de los Indios_. Mexico City: Universidad Iberoamericana, Departamento de Historia, 1991.
———. _El Servicio Personal de los Indios en la Nueva Espana_. Mexico City: Colegio Nacional, 1984–95.
Zumárraga, Juan de. Letter from Friar Juan de Zumarraga to the King of Spain. August 27, 1529. In Archivo de Simancas, Bibl. Miss., II, 229, carta 13. Muñoz collection, T. 78.
Copyright © 2009 by Carl A. Anderson
All rights reserved.
Published in the United States by Doubleday Religion, an imprint of the Crown Publishing Group, a division of Random House, Inc., New York.
www.crownpublishing.com
The images , , , , , , and are courtesy of the Archdiocese of Mexico
DOUBLEDAY and the DD colophon are registered trademarks of Random House, Inc.
Library of Congress Cataloging-in-Publication Data
Anderson, Carl A.
Our Lady of Guadalupe: Mother of the civilization of love / Carl A. Anderson, Eduardo
Chávez.—1st ed.
1. Guadalupe, Our Lady of. I. Chávez, Eduardo. II. Title.
BT660G8A63 2009
232.91′7097253—dc22 2009012120
eISBN: 978-0-307-58949-1
v3.0
| {
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Trump-era State Department staffer arrested, charged over alleged role in Capitol riot: Report
Over the last two months, the FBI has been working hard to track down and arrest individuals believed to have committed crimes as part of the violent riot that disrupted proceedings at the U.S. Capitol building on Jan. 6. Now, the bureau is making big moves on that front.
Politico reports that Federico Klein, who worked as a low-level staffer at the State Department during the Trump administration, was arrested Thursday in Virginia for his alleged role in the unrest.
The 42-year-old is now facing several felony charges, including assault on police officers, interfering with police during civil disorder, and obstructing official proceedings, among other less serious offenses.
Politico reports that Klein's arrest marks "the first known instance of an appointee of President Donald Trump facing criminal prosecution in connection" with the Capitol riot.
Trump-era aide arrested
According to Politico, Klein served as a "tech analyst" for then-candidate Trump's 2016 campaign, and after Trump was elected president, he joined the State Department as a low-level appointee.
Per official records, Klein worked within the Bureau of Western Hemisphere Affairs, including at the Office of Brazilian and Southern Cone Affairs, prior to transferring to the office that handles Freedom of Information Act (FOIA) requests.
He is believed to be a former U.S. Marine and was said to have held Top Secret security clearance at the time of the riot. In fact, he was still a federal employee at that point, as he didn't resign from his position until the day after President Joe Biden's inauguration, Politico noted.
More than 300 charged
According to a "Statement of Facts" from the FBI, Klein was spotted by investigators in video footage of the Capitol attack, and his picture was posted on a "wanted" bulletin that led to multiple tips.
The agent who wrote the Statement of Facts said Klein was observed, in both police bodycam footage as well as "open source" videos of the riot posted online, "resisting officers, attempting to take items from officers, and assaulting officers with a riot shield."
More specifically, Klein stands accused of having "physically and verbally engaged" with police in a manner that prevented them from dispersing the crowd, of stealing a riot shield from a police officer, of using that shield as both a weapon against officers and a wedge to force open a door, and of encouraging other riot participants to act unlawfully.
According to NBC News, Klein is now one of more than 300 individuals who are facing charges related to the unrest at the Capitol building.
If convicted, Klein is facing upwards of 20 years in prison, Politico notes. | {
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I love me some good full coverage foundation but when summer time comes around, I'm all about BB cushion.
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Best believe when Influenster advised me they would be sending me Laneige BB Cushion, I was super excited to test it out.
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Personally I'm only familiar with their skin care products, which I love!
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Laneige is available in Sephora.
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It has SPF 50+ and it advises you to apply 15 minutes before sun exposure.
Reapply at least every 2 hours if necessary.
Comes with a refill of the BB Cushion.
The product is detrmatologist tested.
Have you tried this product yet? Let me know your experience about it.
**Disclaimer: I received this product complimentary through Influenster. All opinions are my own.
yes it is! so far I love it.
That looks an amazing product. Thanks for sharing.
Its great to know that Laneige is available in Sephora. Though this product is new to me but I have tried some Korean product and I can tell they have some really good formulas when it comes to skin care products.
yes I love their skin care products too.
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I personally enjoy it, I love light weight makeup during summer.
I have not tried this BB product but reading your review on the basis of your own experience I am quite curious to try.
I enjoy it, I've been wearing it all summer so far.
Love that it's a foundation that has spf!
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I love this! I am also a beauty blogger but I have not heard of this brand before, as you mentioned it is a foreign brand. I love the packaging…I have been known to purchase makeup just because of the packaging. For some reason it is pretty important to me. I love it! I may have to try this brand! Great post!
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This looks perfect for summer and easy to reapply. Who wants thick foundation when you're out and ready to swim?
I love how this foundation shines and glows, especially on you. I have to pick this up and try it.
A Laneige loyalist here. Will try to have a look in the store for that item.
I'm going to try out more of their product.
I know I love it too.
I cannot remember where I have seen this product before.
you've properly have seen their name because their more known for face care products.
That's a great review. Thanks for sharing!!
Looks like a must try product. Will look for this here in Japan. Thanks for the review!
I love the fact that it looks natural. thanks for recommending .
How did this last throughout the day?
It did last for me but I only used during the day not the whole day.
yea I love it for the summer! | {
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Now 57, Lorin has been meditating since age 18, when he signed up to be part of a research project on the physiology of meditation. He was a control subject, and received no instructions whatsoever – they wanted him to just sit in a totally dark, soundproofed room in the lab for two hours a day for several weeks, and measure his brain waves. With no instructions, and never having heard of meditation, Lorin just paid attention to the total silence and darkness, and spontaneously entered entered a state of intense alertness. A few months later, someone handed him a little book describing 112 meditation practices. Lorin realized that he had experienced some of them while sitting in the lab.
This experience made it clear that meditation is a spontaneous and natural human experience, and that there are many doorways into meditation. After the experiment was over, people began asking Lorin to teach simple meditation practices as part of scientific studies and then university classes. Remember, this was 1968. One thing led to another, and soon Lorin was running his own Experimental College, which went by the name, Esalen at Irvine Experiential Workshops. He invited teachers from Esalen to come to Irvine and offer workshops in meditation, Tai Chi, Structural Integration Movement Awareness, Yoga, Dance Meditation, Art Meditation, and Gestalt Body Awareness.
Lorin was born in 1949 and grew up in Southern California, in little beach towns such as Ventura, Malibu and Dana Point, which in the 1950's, 60's and 70's were middle-class and unpretentious. Both his parents were surfers and members of the San Onofre Surfing Club from the 1940's on, and took him into the ocean before he could walk.
With his wife, Camille Maurine, Lorin is the author of Meditation Secrets for Women and Meditation 24/7: Practices to Enlighten Every Moment of the Day. He is also the author of Meditation Made Easy, Breath Taking, and Whole Body Meditations, and The Radiance Sutras.
Lorin received his Ph.D. in Social Science from the University of California at Irvine in 1987. His doctoral dissertation was about the language meditators come up with to describe their experiences. In other words, the maps they make to navigate their inner worlds. His Master's Degree work focused on the hazards of meditation and the crisis points in a meditator's development.
Both his Master's and Doctoral research were based on an 8-year period in which he sought out meditators of all types: Zen, Christian, Buddhist, Vipassana, Kundalini, TM, Sikh, Hindu, Tibetan, Jewish, Kaballah, Wicca, Native American, Theosophist, Arcana, Agni Yoga, Hatha Yoga, Raja Yoga, Bhakti Yoga, Brain Wave Biofeedback, Autogenic Training, Neurolinguistic Programming, Ericksonian Hypnosis, Gestalt, Charltte Selver Sensory Awareness, Feldenkrais Awareness Through Movement, Shamanism, and others. He made a special effort to seek out people who were meditating on their own, with no teacher and no tradition, just making it up. Lorin asked all these people, "What are you experiencing now?" and took notes as they talked for hours and described their sensations, images, feelings, and auditory perceptions. Listening to all these different kinds of meditators describe their experiences helped Lorin develop his models of individual perception, the way that each individual has her or his own unique style of approaching meditation.
At the same time (1975 to 1989 especially), as word got around that there was a meditation teacher on the loose who would just listen to you, all kinds of meditators started coming to Lorin to share their experiences. They were seeking a "meditation therapist," actually, who could help them with the problems they were having in adapting their inner world to the outer world. During this time he worked with many hundreds of meditators of all traditions who were having difficulties, doubts, or health problems that they suspected were related to their meditation practice. Listening to all these meditators talk about the difficulties was an incredibly rich learning experience for Lorin, because it let him map out the challenges on the path. This led to the development of a system of diagnosing meditation injuries and problems.
During the 1960's and 1970's Lorin worked as a research assistant on some of the physiological studies on meditation, then in the 1980's he became involved in research on the subjective experience of meditation.
Dr. Roche has been exploring, researching and teaching meditation since 1968. In 1975 stopped teaching the Transcendental Meditation Technique and started developing an approach, which he calls Instinctive Meditation, that works with the fine structure of individual uniqueness, rather than imposing a standardized, one-size-fits-all approach. He draws on insights into how people learn gained from the best of Western science and Eastern Yoga. This integrative approach results in straightforward ways for people to learn many different meditation techniques.
Instinctive Meditation tends to feel more like an innate skill that you are remembering than a technique that you are learning. Lorin's work is aimed at activating an individual's internal guidance systems and bringing forth your instinctive knowing, so that you can safely practice meditation without being dependent upon Gurus, systems or external authorities.
Lorin says, "Meditation can feel like it comes as a gift from the outside – from the lineage of monks, gurus and sages over the millennia. You become their servant and seek to kill off anything in you that does not fit into the aura of a meditation ashram, where the ideal is to be celibate and detached from it all. I have done this type of meditation and loved it. But it is for people who don't mind being dominated by dead Asian males.
"Meditation can feel like it comes from the inside, from your own inner knowing, your inner self, your instincts and body. Instinctive Meditation is based on what works for Westerners, and is for people who embrace the adventure of making it up for themselves." | {
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Jarry named Ephesus/AHL Graduate of the Month | TheAHL.com
June 22, 2020 9:50 PM a minute read
Picture: Scott ThomasSPRINGFIELD, Mass. … The American Hockey League introduced in the present day that Tristan Jarry of the Pittsburgh Penguins has been chosen because the Ephesus/AHL Graduate of the Month for December.Jarry was Eight-1-Zero in 9 appearances for Pittsburgh in December, posting a 1.54 goals-against common, a .947 save share and three shutouts.A second-round choose by the Penguins within the 2013 NHL Draft, Jarry spent elements of 4 seasons with Pittsburgh's high growth affiliate in Wilkes-Barre/Scranton, compiling a document of 77-48-15 with a 2.55 GAA, a .915 save share and 9 shutouts in 141 AHL video games. Jarry was an AHL All-Star in 2017 and was a recipient of the Harry "Hap" Holmes Memorial Award for crew goaltending in 2016-17. On Nov. 14, 2018, Jarry grew to become the 14th goaltender in AHL historical past to attain a aim when his 170-foot shot discovered an empty internet in a recreation towards Springfield.The Nationwide Hockey League's "Second Star" of the Month for December, Jarry is 13-6-1 for Pittsburgh this season and leads the NHL in goals-against common (1.99), save share (.935) and shutouts (three, tied). The 24-year-old native of Surrey, B.C., has a document of 27-14-Four (2.48, .918) in 49 profession NHL contests.At present in use in 10 AHL arenas, Eaton's Ephesus LED sports activities lighting product line supplies optimum lighting that illuminates extra uniformly on the taking part in floor and presents an improved stage for gamers and followers. The system is simple to put in, requires little to no upkeep for years and presents the bottom complete working prices in comparison with different conventional sports activities lighting techniques. For extra info, go to www.eaton.com/ephesus.
Carlson named Ephesus/AHL Graduate of the Month | TheAHL.com
Sports activities Motion Images/AHLSPRINGFIELD, Mass. … The American Hockey League introduced at this time that John Carlson of the Washington Capitals has been chosen because the Ephesus/AHL Graduate of the Month for October.Carlson was named the First Star o...
Marchand named Ephesus/AHL Graduate of the Month | TheAHL.com
SPRINGFIELD, Mass. … The American Hockey League introduced right this moment that Brad Marchand of the Boston Bruins has been chosen because the Ephesus/AHL Graduate of the Month for November.Marchand recorded 11 targets and 11 assists for 22 factors in 14 vid...
Rask named Ephesus/AHL Graduate of the Month | TheAHL.com
JustSports Pictures/AHLSPRINGFIELD, Mass. … The American Hockey League introduced in the present day that Tuukka Rask of the Boston Bruins has been chosen because the Ephesus/AHL Graduate of the Month for Might.Rask had a document of Eight-1 throughout Might,...
Giordano named Ephesus/AHL Graduate of the Month | TheAHL.com
Photograph: Michelle BishopSPRINGFIELD, Mass. … The American Hockey League introduced in the present day that Mark Giordano of the Calgary Flames has been chosen because the Ephesus/AHL Graduate of the Month for June.On the NHL Awards ceremony in Las Vegas on...
Tristan Jarry, Penguins Beat Oilers 5-2
EDMONTON — Tristan Jarry made 26 saves and Jared McCann scored the eventual winner because the Pittsburgh Penguins emerged with a 5-2 victory over the Edmonton Oilers on Friday night time. Pittsburgh Penguins goaltender Tristan Jarry (Kim Klement-USA TODAY Sp...
Nedeljkovic named AHL's high goaltender for 2018-19 | TheAHL.com
SPRINGFIELD, Mass. … The American Hockey League introduced as we speak that Alex Nedeljkovic of the Charlotte Checkers is the winner of the Aldege "Baz" Bastien Memorial Award because the AHL's excellent goaltender for the 2018-19 season.The award is voted on ... | {
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{"url":"https:\/\/economics.stackexchange.com\/questions\/10461\/given-the-supply-and-demand-function-what-is-the-difference","text":"# Given the supply and demand function what is the difference? [closed]\n\nGiven some unspecified demand and supply functions.\n\nI need to calculate quantity and price after taxation, but what is the difference between having a tax of 20% of total revenue and 20% of profits ($\\pi$)? I mean how to calculate this tax in those two cases? And what precisely should be changed when calculating Q and P after tax of those two types is imposed on.\n\nWhether tax of 20% of total revenue creates simply consumer price on a level of $1.2(\\text{producerprice})$ or is it something different? In case of profits is it $0.8(\\text{supplyfunction})$? **hope this question now satisfies required format **\n\n\u2022 It remains always a homework question even after your edit. You can ask your homework questions but you should show what you have tried and tell us where you stuck. Unless, nobody will give an answer once they understand that it is a homework question without effort made. Feb 1, 2016 at 14:30\n\u2022 I was trying to follow sites policy on homework questions i.e. to make it more general. What if some questions are short thus you cannot be stuck at any point as the question is obviously too short? Feb 2, 2016 at 10:40\n\u2022 No better? I do not understand this strange attitude towards question of this kind, frankly you have no idea whether it's a homework question or not. It may be simply a question from a book i am reading right know. Am i right? Feb 2, 2016 at 10:44\n\u2022 @mkopkowski It is not strange at all. For this kind of short questions, If you don't show where you stuck in terms of reasoning or calculation, we assume that you did not make any effort. Do edit your question in this way, with words or calculations to tell us where you stuck. If you are not giving any effort, nobody will make any effort. Feb 2, 2016 at 13:31\n\u2022 I think i have added the point where i am stuck Feb 2, 2016 at 13:38\n\nDeriving those functions, you can find $p^*$ and $Q^*$, which, multiplying both, should give you equilibrium profits. Now, if you take that tax rate (20%, I reckon?), you can calculate taxes over profits and the tax-free price.\nTo find the equilibrium price $p^*$, you need to have those two functions in a system and solve for $p$. Then, just replace it on both functions and you should have equilibrium quantities.\nNevertheless, to calculate taxes, you need to work around with the system. That will give you total revenues, cost and tax free. Applying 20% leads to net earnings, because you do not have any production costs associated with the problem - something commonly done in Economics. That or consider a variable $c$ of costs.","date":"2022-05-16 21:33:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5532345771789551, \"perplexity\": 554.3282139678096}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662512249.16\/warc\/CC-MAIN-20220516204516-20220516234516-00764.warc.gz\"}"} | null | null |
\section{Introduction}
Discretisation methods supporting meshes with general, possibly non standard, element shapes have experienced a vigorous growth over the last few years.
In the context of solid-mechanics, this feature can be useful for several reasons including, e.g., improved robustness to mesh distortion and fracture, local mesh refinement, or the use of hanging nodes for contact and interface problems.
A non-exahustive list of contributions in the context of elasticity problems includes \cite{Hughes.Cottrell.ea:05,Tabarraei.Sukumar:06,Beirao-Da-Veiga:10,Beirao-da-Veiga.Brezzi.ea:13*1,Droniou.Lamichhane:15,Gain.Talischi.ea:14,Di-Pietro.Lemaire:15,Di-Pietro.Ern:15,Botti.Di-Pietro.ea:17,Artioli.Beirao-da-Veiga.ea:17,Koyama.Kikuchi:17,Cockburn.Fu:18,Sevilla.Giacomini.ea:18,Caceres.Gatica.ea:19}; see also references therein.
For large three-dimensional simulations, or whenever one cannot expect the exact solution to be smooth, low-order methods are often privileged in order to reduce the number of unknowns.
It is well-known, however, that low-order Finite Element (FE) approximations are in some cases unsatisfactory:
affine conforming FE methods are not robust in the quasi-incompressible limit owing to their inability to represent non-trivial divergence-free displacement fields;
nonconforming (Crouzeix--Raviart) FE methods, on the other hand, yield unstable discretisations unless appropriate measures are taken; see, e.g., the discussions in \cite{Brenner.Sung:92,Hansbo.Larson:03}.
The underlying reason for this lack of stability is the non-fulfillment of a discrete counterpart of Korn's inequality owing to a poor control of rigid-body motions at mesh faces.
For similar reasons, the stability of Hybrid High-Order (HHO) methods for linear elasticity requires the use of polynomials of degree $k\ge 1$ as unknowns; see \cite[Lemma 4]{Di-Pietro.Ern:15}.
As a matter of fact, as we show in Section \ref{sec:comparison.hho} below, the stability and consistency requirements on the local HHO stabilisation term are incompatible when $k=0$, that is, when piecewise constant polynomials on the mesh and its skeleton are used as discrete unknowns.
In this paper we highlight a modification of the HHO method which recovers stability for $k=0$.
The proposed fix consists in adding a novel term which penalises in a least square sense the jumps of the local affine displacement reconstruction.
This modification is inspired by the Korn inequality on broken polynomial spaces proved in Lemma \ref{lem:korn} below, which appears to be a novel extension of similar results to general polyhedral meshes.
The proof combines the techniques of \cite[Lemma 2.2]{Brenner:03} with the recent results of \cite{Di-Pietro.Ern:12} and \cite{Di-Pietro.Droniou:17} concerning, respectively, the node-averaging operator and local inverse inequalities on polyhedral meshes.
In the context of Crouzeix--Raviart FE approximations of linear elasticity problems on standard meshes, similar jump penalisation terms have been considered in \cite{Hansbo.Larson:03}.
The resulting method has several appealing features:
it is valid in two and three space dimensions, paving the way to unified implementations;
it hinges on a reduced number of unknowns ($15$ for a tetrahedron, $21$ for a hexahedron and, for more general polyhedral shapes, $3$ unknowns per face plus $3$ unknowns inside the element);
it is robust in the quasi-incompressible limit;
it admits a formulation in terms of conservative numerical tractions, which enables its integration in existing Finite Volume simulators (a particularly relevant feature in the context of industrial applications).
We carry out a complete convergence analysis based on the abstract framework of \cite{Di-Pietro.Droniou:18} for methods in fully discrete formulation.
Specifically, we show that the energy and $L^2$-norms of the error converge, respectively, as $h$ and $h^2$ (with $h$ denoting, as usual, the meshsize).
As for the original HHO method of \cite{Di-Pietro.Ern:15}, the error estimates are additionally shown to be robust in the quasi-incompressible limit.
Key to this result is the fact that the gradient of the local displacement reconstruction satisfies a suitable commutation property with the $L^2$-orthogonal projector.
The theoretical results are supported by a thorough numerical investigation, including two- and three-dimensional test cases, as well as a comparison with the original HHO method of \cite{Di-Pietro.Ern:15} on a test case mimicking a mode 1 fracture.
The rest of the paper is organised as follows.
In Section \ref{sec:continuous.setting} we formulate the continuous problem along with the assumptions on the problem data.
In Section \ref{sec:discrete.setting} we establish the discrete setting: after briefly recalling the notion of regular polyhedral mesh, we introduce local and broken polynomial spaces and projectors thereon, and we prove a discrete counterpart of Korn's first inequality on broken polynomial spaces.
In Section \ref{sec:discretisation} we introduce the space of discrete unknowns, define a local affine displacement reconstruction, formulate the discrete bilinear form, discuss the differences with respect to the original HHO bilinear form of \cite{Di-Pietro.Ern:15}, and state the discrete problem.
Section \ref{sec:convergence} addresses the convergence analysis of the method in the energy- and $L^2$-norms, while Section \ref{sec:numerical.tests} contains an exhaustive panel of two- and three-dimensional numerical tests.
Finally, in Section \ref{sec:flux} we show that the method satisfies local balances with equilibrated tractions, for which an explicit expression is provided.
\section{Continuous setting}\label{sec:continuous.setting}
Consider a body which, in its reference configuration, occupies a given region of space $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$.
In what follows, it is assumed that $\Omega$ is a bounded connected open polygonal (if $d=2$) or polyhedral (if $d=3$) set that does not have cracks, i.e., it lies on one side of its boundary $\partial\Omega$.
We are interested in finding the displacement field $\vec{u}:\Omega\to\mathbb{R}^d$ of the body when it is subjected to a given force per unit volume $\vec{f}:\Omega\to\mathbb{R}^d$.
We work in what follows under the small deformation assumption which implies, in particular, that the strain tensor $\tens{\varepsilon}$ is given by the symmetric part of the gradient of the displacement field, i.e., $\tens{\varepsilon}=\tens{\nabla}_{\rm s}\vec{u}$ where, for any vector-valued function $\vec{z}=(z_i)_{1\le i\le d}$ smooth enough, we have set $\vec{\nabla}\vec{z}=(\partial_jz_i)_{1\le i,j\le d}$ and $\tens{\nabla}_{\rm s}\vec{z}\coloneq\frac12\left(\vec{\nabla}\vec{z}+\vec{\nabla}\vec{z}^\top\right)$.
We further assume, for the sake of simplicity, that the body is clamped along its boundary $\partial\Omega$.
Other standard boundary conditions can be considered up to minor modifications.
The displacement field is obtained by solving the following linear elasticity problem, which expresses the equilibrium between internal stresses and external loads:
Find $\vec{u}:\Omega\to\mathbb{R}^d$ such that
\begin{subequations}\label{eq:strong}
\begin{alignat}{2}
-\vec{\nabla}{\cdot}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u})) &= \vec{f}&\qquad&\text{in $\Omega$},\label{eq:strong:pde}
\\
\vec{u} &= \vec{0}&\qquad&\text{on $\partial\Omega$},\label{eq:strong:bc}
\end{alignat}
\end{subequations}
where, denoting by $\mathbb{R}_{\rm sym}^{d\times d}$ the set of symmetric real-valued $d\times d$ matrices, the mapping $\tens{\sigma}:\mathbb{R}_{\rm sym}^{d\times d}\to\mathbb{R}_{\rm sym}^{d\times d}$ represents the strain-stress law.
For isotropic homogeneous materials, the strain-stress law is such that, for any $\tens{\tau}\in\mathbb{R}_{\rm sym}^{d\times d}$,
\begin{equation}\label{eq:sigma}
\tens{\sigma}(\tens{\tau}) = 2\mu\tens{\tau} + \lambda\tr(\tens{\tau})\Id,
\end{equation}
where $\tr(\tens{\tau})\coloneq\sum_{i=1}^d\tau_{ii}$ is the trace operator and $\Id$ the $d\times d$ identity matrix.
The real numbers $\mu$ and $\lambda$, which correspond to the Lam\'e coefficients when $d=3$, are assumed such that, for a real number $\alpha>0$,
\begin{equation}\label{eq:lambda.mu.bounds}
2\mu - d\lambda^-\ge\alpha,
\end{equation}
where $\lambda^-\coloneq\frac12\left(|\lambda|-\lambda\right)$ denotes the negative part of $\lambda$.
In what follows, $\mu$, $\lambda$, the related bound \eqref{eq:lambda.mu.bounds}, and $\vec{f}$ will be collectively referred to as the problem data.
For any open bounded set $X\subset\Omega$, we denote by $({\cdot},{\cdot})_X$ the usual inner product of the space of scalar-valued, square-integrable functions $L^2(X;\mathbb{R})$, by $\norm[X]{{\cdot}}$ the corresponding norm, and we adopt the convention that the subscript is omitted whenever $X=\Omega$.
The same notation is used for the spaces of vector- and tensor-valued square-integrable functions $L^2(X;\mathbb{R}^d)$ and $L^2(X;\mathbb{R}^{d\times d})$, respectively.
With this notation, a classical weak formulation of problem \eqref{eq:strong} reads:
Find $\vec{u}\in H_0^1(\Omega;\mathbb{R}^d)$ such that
\begin{equation}\label{eq:weak}
(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u}),\tens{\nabla}_{\rm s}\vec{v}) = (\vec{f},\vec{v})
\qquad\forall\vec{v}\in H_0^1(\Omega;\mathbb{R}^d),
\end{equation}
where $H_0^1(\Omega;\mathbb{R}^d)$ classically denotes the space of vector-valued functions that are square-integrable along with all their partial derivatives, and whose traces on $\partial\Omega$ vanish.
\section{Discrete setting}\label{sec:discrete.setting}
\subsection{Mesh}
Throughout the rest of the paper, we will use for the sake of simplicity the three-dimensional nomenclature also when $d=2$, i.e., we will speak of polyhedra and faces rather than polygons and edges.
We consider here meshes corresponding to couples $\Mh\coloneq(\Th,\Fh)$, where $\Th$ is a finite collection of polyhedral elements $T$ such that $h\coloneq\max_{T\in\Th}h_T>0$ with $h_T$ denoting the diameter of $T$, while $\Fh$ is a finite collection of planar faces $F$. It is assumed henceforth that the mesh $\Mh$ matches the geometrical requirements detailed in \cite[Definition 7.2]{Droniou.Eymard.ea:18}; see also~\cite[Section 2]{Di-Pietro.Tittarelli:18}.
This covers, essentially, any reasonable partition of $\Omega$ into polyhedral sets, not necessarily convex or even star-shaped.
For every mesh element $T\in\Th$, we denote by $\Fh[T]$ the subset of $\Fh$ containing the faces that lie on the boundary $\partial T$ of $T$.
Symmetrically, for every face $F\in\Fh$, we denote by $\Th[F]$ the subset of $\Th$ containing the (one or two) mesh elements that share $F$.
For any mesh element $T\in\Th$ and each face $F\in\Fh[T]$, $\vec{n}_{TF}$ is the constant unit normal
vector to $F$ pointing out of $T$.
Boundary faces lying on $\partial\Omega$ and internal faces contained in
$\Omega$ are collected in the sets $\Fh^{{\rm b}}$ and $\Fh^{{\rm i}}$, respectively.
For any $F\in\Fh^{{\rm i}}$, we denote by $T_1$ and $T_2$ the elements of $\Th$ such that $F\subset\partial T_1\cap\partial T_2$.
The numbering of $T_1$ and $T_2$ is assumed arbitrary but fixed, and we set $\vec{n}_F\coloneq\vec{n}_{T_1F}$.
Our focus is on the $h$-convergence analysis, so we consider a sequence of refined meshes that is regular in the sense of~\cite[Definition~3]{Di-Pietro.Tittarelli:18}. This implies, in particular, that the diameter $h_T$ of a mesh element $T\in\Th$ is comparable to the diameter $h_F$ of each face $F\in\Fh[T]$ uniformly in $h$, and that the number of faces in $\Fh[T]$ is bounded above by an integer $N_\partial$ independent of $h$.
\subsection{Local and broken spaces and projectors}
In order to alleviate the exposition, throughout the rest of the paper we use the abridged notation $a\lesssim b$ for the inequality $a\le Cb$ with real number $C>0$ independent of the meshsize, possibly on the problem data, and, for local inequalities, on the mesh element or face.
We also write $a\simeq b$ for $a\lesssim b$ and $b \lesssim a$.
The dependencies of the hidden constant are further specified whenever needed.
Let $X$ denote a mesh element or face.
For a given integer $l\ge 0$, we denote by $\Poly{l}(X;\mathbb{R})$ the space spanned by the restriction to $X$ of $d$-variate, real-valued polynomials of total degree $\le l$.
The corresponding spaces of vector- and tensor-valued functions are respectively denoted by $\Poly{l}(X;\mathbb{R}^d)$ and $\Poly{l}(X;\mathbb{R}^{d\times d})$.
A similar notation is used also for the vector and tensor versions of the broken spaces introduced in what follows.
At the global level, we denote by $\Poly{l}(\Th;\mathbb{R})$ the space of broken polynomials on $\Th$ whose restriction to every mesh element $T\in\Th$ lies in $\Poly{l}(T;\mathbb{R})$, i.e.,
$$
\Poly{l}(\Th;\mathbb{R})\coloneq\left\{v_h\in L^2(\Omega;\mathbb{R})\; : \; v_{h|T}\in\Poly{l}(T;\mathbb{R})\quad\forall T\in\Th\right\}.
$$
We also introduce the broken Sobolev spaces
$$
H^s(\Th;\mathbb{R})\coloneq\left\{
v\in L^2(\Omega;\mathbb{R})\; : \; v_{|T}\in H^s(T;\mathbb{R})\quad\forall T\in\Th
\right\},
$$
which will be used in the error estimates to express the regularity requirements on the exact solution.
On $H^s(\Th;\mathbb{R})$, we define the broken seminorm
$$
\seminorm[H^s(\Th;\mathbb{R})]{v}\coloneq\left(
\sum_{T\in\Th}\seminorm[H^s(T;\mathbb{R})]{v}^2
\right)^{\frac12}.
$$
Again denoting by $X$ a mesh element or face, the local $L^2$-orthogonal projector $\lproj[X]{0}:L^2(X;\mathbb{R})\to\Poly{0}(X;\mathbb{R})$ maps every $v\in L^2(X;\mathbb{R})$ onto the constant function equal to its mean value inside $T$, that is,
\begin{equation}\label{eq:lproj}
\lproj[X]{0}v\coloneq\frac{1}{\meas{X}}\int_X v,
\end{equation}
with $\meas{X}$ denoting the Hausdorff measure of $X$.
The vector and tensor versions of the $L^2$-projector, both denoted by $\vlproj[X]{0}$, are obtained applying $\lproj[X]{0}$ component-wise.
From \cite[Lemmas 3.4 and 3.6]{Di-Pietro.Droniou:17}, it can be deduced that, for any mesh element $T\in\Th$ and any function $v\in H^1(T;\mathbb{R})$, the following approximation properties hold:
\begin{equation}\label{eq:lproj:approx}
\norm[L^2(T;\mathbb{R})]{v-\lproj[T]{0}v}
+ h_T^{\frac12}\norm[L^2(\partial T;\mathbb{R})]{v-\lproj[T]{0}v}
\lesssim h_T\seminorm[H^1(T;\mathbb{R})]{v},
\end{equation}
where $\partial T$ denotes the boundary of $T$ and the hidden constant is independent of $h$, $T$, and $v$.
The global $L^2$-orthogonal projector $\lproj{0}:L^2(\Omega;\mathbb{R})\to\Poly{0}(\Th;\mathbb{R})$ is such that, for any $v\in L^2(\Omega;\mathbb{R}^d)$,
\begin{equation}\label{eq:lproj.h}
(\lproj{0} v)_{|T}\coloneq\lproj[T]{0} v_{|T}\qquad\forall T\in\Th.
\end{equation}
The vector and tensor versions, both denoted by $\vlproj{0}$, are obtained applying $\lproj{0}$ component-wise.
We will also need the elliptic projector $\eproj[T]{1}:H^1(T;\mathbb{R})\to\Poly{1}(T;\mathbb{R})$ such that, for all $v\in H^1(T;\mathbb{R})$,
\begin{equation}\label{eq:eproj}
\vec{\nabla}\eproj[T]{1}v = \vlproj[T]{0}(\vec{\nabla} v)\mbox{ and }
\frac{1}{\meas{T}}\int_T\eproj[T]{1}v = \frac{1}{\meas{T}}\int_T v.
\end{equation}
The first relation makes sense since $\vec{\nabla}\Poly{1}(T;\mathbb{R})=\Poly{0}(T;\mathbb{R}^d)$, and it defines $\eproj[T]{1}v$ up to a constant, which is then fixed by the second relation.
Also in this case, the vector version $\veproj[T]{1}$ of the projector is obtained applying the scalar version component-wise.
The following approximation properties for the elliptic projector are a special case of \cite[Theorems 1.1 and 1.2]{Di-Pietro.Droniou:17*1}:
For all $T\in\Th$ and all $v\in H^2(T;\mathbb{R})$,
\begin{equation}\label{eq:eproj:approx}
\norm[L^2(T;\mathbb{R})]{v-\eproj[T]{1}v}
+ h_T^{\frac12}\norm[L^2(\partial T;\mathbb{R})]{v-\eproj[T]{1}v}
\lesssim h_T^2\seminorm[H^2(T;\mathbb{R})]{v},
\end{equation}
where the hidden constant is independent of $h$, $T$, and $v$.
For further use, we also define the global elliptic projector $\eproj{1}:H^1(\Th;\mathbb{R})\to\Poly{1}(\Th;\mathbb{R})$ such that, for any $v\in H^1(\Th;\mathbb{R})$,
$$
(\eproj{1} v)_{|T}\coloneq\eproj[T]{1} v_{|T}\qquad\forall T\in\Th.
$$
The vector version $\veproj{1}$ of the global elliptic projector is obtained applying $\eproj{1}$ component-wise.
\subsection{Discrete Korn inequality on broken polynomial spaces}
The stability of our method hinges on a discrete counterpart of Korn's inequality in discrete polynomial spaces stating that the $H^1$-seminorm of a vector-valued broken polynomial function is controlled by a suitably defined strain norm.
The goal of this section is to prove this inequality.
Let us start with some preliminary results.
Recalling that, for any $F\in\Fh^{{\rm i}}$, we have denoted by $T_1$ and $T_2$ the elements sharing $F$ and assumed that the ordering is arbitrary but fixed, we introduce the jump operator such that, for any function $v$ smooth enough to admit a (possibly two-valued) trace on $F$,
\begin{subequations}\label{eq:jump}
\begin{equation}
\jump{v}\coloneq (v_{|T_1})_{|F} - (v_{|T_2})_{|F}.
\end{equation}
This operator is extended to boundary faces $F\in\Fh^{{\rm b}}$ by setting
\begin{equation}
\jump{v}\coloneq v_{|F}.
\end{equation}
\end{subequations}
When applied to vector-valued functions, the jump operator acts componentwise.
Let now $\fTh$ denote a matching simplicial submesh of $\Mh$ in the sense of \cite[Definition 4.2]{Di-Pietro.Tittarelli:18}, and let $\fFh$ be the corresponding set of simplicial faces.
Given an integer $l\ge 1$, we define the node-averaging operator $\Iav[l]:\Poly{l}(\Th;\mathbb{R})\to\Poly{l}(\Th;\mathbb{R})\cap H_0^1(\Omega)$ such that, for any function $v_h\in\Poly{l}(\Th;\mathbb{R})$ and any Lagrange node $V$ of $\fTh$, denoting by $\fTh[V]$ the set of simplices sharing $V$,
$$
(\Iav[l] v_h)(V)\coloneq\begin{cases}
\dfrac{1}{\card(\fTh[V])}\displaystyle\sum_{\tau\in\fTh[V]} (v_h)_{|\tau}(V) & \text{if $V\in\Omega$},
\\
0 & \text{if $V\in\partial\Omega$}.
\end{cases}
$$
The vector-version, denoted by $\vIav[l]$, acts component-wise.
Adapting the reasoning of \cite[Section 5.5.2]{Di-Pietro.Ern:12} (based in turn on \cite{Karakashian.Pascal:03}), we infer that it holds, for all $T\in\Th$,
\begin{equation}\label{eq:est.Iav.l2}
\norm[T]{v_h-\Iav[l] v_h}^2\lesssim\sum_{F\in\Fh[\mathcal{V},T]}h_F\norm[F]{\jump{v_h}}^2,
\end{equation}
where $\Fh[\mathcal{V},T]$ denotes the set of faces whose closure has nonempty intersection with the closure of $T$ and the hidden constant is independent of $h$, $T$, and $v_h$.
Combining this result with an inverse inequality (see \cite[Remark A.2]{Di-Pietro.Droniou:17}) we obtain, with hidden constants as before,
$$
\begin{aligned}
\seminorm[H^1(\Th;\mathbb{R})]{v_h-\Iav[l] v_h}^2
&\lesssim\sum_{T\in\Th} h_T^{-2}\norm[T]{v_h-\Iav[l] v_h}^2
\\
&\lesssim\sum_{T\in\Th} h_T^{-2}\sum_{F\in\Fh[\mathcal{V},T]}h_F\norm[F]{\jump{v_h}}^2
\\
&\lesssim\sum_{F\in\Fh}\sum_{T\in\Th[\mathcal{V},F]}h_F^{-1}\norm[F]{\jump{v_h}}^2,
\end{aligned}
$$
where we have used \eqref{eq:est.Iav.l2} to pass to the second line while, to pass to the third line, we have invoked the mesh regularity to write $h_F h_T^{-2}\lesssim h_F^{-1}$ and we have exchanged the order of the sums after introducing the notation $\Th[\mathcal{V},F]$ for the set of mesh elements whose closure has nonzero intersection with the closure of $F$.
Using again mesh regularity to infer that $\card(\Th[\mathcal{V},F])$ is bounded uniformly in $h$, we arrive at
\begin{equation}\label{eq:est.Iav.h1}
\seminorm[H^1(\Th;\mathbb{R})]{v_h-\Iav[l] v_h}^2\lesssim\sum_{F\in\Fh}h_F^{-1}\norm[F]{\jump{v_h}}^2.
\end{equation}
We are now ready to prove the discrete Korn inequality.
\begin{lemma}[Discrete Korn inequality]\label{lem:korn}
Let an integer $l\ge1$ be fixed and set, for all $\vec{v}_h\in\Poly{l}(\Th;\mathbb{R}^d)$,
\begin{equation}\label{eq:norm.dG}
\norm[\tens{\varepsilon},h]{\vec{v}_h}\coloneq\left(
\norm{\tens{\nabla}_{{\rm s},h}\vec{v}_h}^2
+ \seminorm[{\rm j},h]{\vec{v}_h}^2
\right)^{\frac12}
\mbox{ and }
\seminorm[\mathrm{j},h]{\vec{v}_h}\coloneq\left(
\sum_{F\in\Fh}h_F^{-1}\norm[F]{\jump{\vec{v}_h}}^2
\right)^{\frac12},
\end{equation}
where $\tens{\nabla}_{{\rm s},h}:H^1(\Th;\mathbb{R}^d)\to L^2(\Omega;\mathbb{R}_{\rm sym}^{d\times d})$ is the broken symmetric gradient such that $(\tens{\nabla}_{{\rm s},h}\vec{v})_{|T}=\tens{\nabla}_{\rm s}\vec{v}_{|T}$ for any $T\in\Th$.
Then, for all $\vec{v}_h\in\Poly{l}(\Th;\mathbb{R}^d)$, it holds with hidden constant depending only on $\Omega$, $d$, and the mesh regularity parameter:
\begin{equation}\label{eq:korn}
\seminorm[H^1(\Th;\mathbb{R}^d)]{\vec{v}_h}\lesssim\norm[\tens{\varepsilon},h]{\vec{v}_h}.
\end{equation}
\end{lemma}
\begin{proof}
The proof adapts the arguments of \cite[Lemma 2.2]{Brenner:03}.
We can write
$$
\begin{aligned}
\seminorm[H^1(\Th;\mathbb{R}^d)]{\vec{v}_h}^2
&\lesssim\seminorm[H^1(\Omega;\mathbb{R}^d)]{\vIav[l]\vec{v}_h}^2 + \seminorm[H^1(\Th;\mathbb{R}^d)]{\vec{v}_h-\vIav[l]\vec{v}_h}^2
\\
&\lesssim\norm{\tens{\nabla}_{\rm s}\vIav[l]\vec{v}_h}^2 + \seminorm[\mathrm{j},h]{\vec{v}_h}^2
\\
&\lesssim\norm{\tens{\nabla}_{{\rm s},h}\vec{v}_h}^2 + \norm{\tens{\nabla}_{{\rm s},h}(\vIav[l]\vec{v}_h-\vec{v}_h)}^2 + \seminorm[\mathrm{j},h]{\vec{v}_h}^2
\\
&\lesssim\norm{\tens{\nabla}_{{\rm s},h}\vec{v}_h}^2 + \seminorm[\mathrm{j},h]{\vec{v}_h}^2
=\norm[\tens{\varepsilon},h]{\vec{v}_h}^2,
\end{aligned}
$$
where we have inserted $\pm\vIav[l]\vec{v}_h$ into the seminorm and used a triangle inequality in the first line,
we have applied the first Korn inequality in $H^1(\Omega;\mathbb{R}^d)$ to the first term and invoked \eqref{eq:est.Iav.h1} for the second term after recalling the definition \eqref{eq:norm.dG} of the jump seminorm in the second line,
we have inserted $\pm\tens{\nabla}_{{\rm s},h}\vIav[l]\vec{v}_h$ and used a triangle inequality to pass to the third line,
we have invoked again \eqref{eq:est.Iav.h1} to estimate the second term in the right-hand side to pass to the fourth line, and we have used the definition \eqref{eq:norm.dG} of the strain norm to conclude.
\end{proof}
\begin{remark}[Korn--Poincar\'e inequality]
Combining the discrete Poincar\'e inequality resulting from \cite[Theorem 6.1]{Di-Pietro.Ern:10} (see also \cite[Theorem 5.3 and Corollary 5.4]{Di-Pietro.Ern:12}) with \eqref{eq:korn}, we infer that it holds, for all $\vec{v}_h\in\Poly{l}(\Th;\mathbb{R}^d)$,
\begin{equation}\label{eq:korn-poincare}
\norm{\vec{v}_h}\lesssim\norm[\tens{\varepsilon},h]{\vec{v}_h},
\end{equation}
with hidden constant independent of $h$ and $\vec{v}_h$.
\end{remark}
\section{Discretisation}\label{sec:discretisation}
\subsection{Discrete space}
Given a mesh $\Mh=(\Th,\Fh)$, we define the following space of discrete unknowns:
$$
\vUh\coloneq\left\{
\uvec{v}_h=( (\vec{v}_T)_{T\in\Th},(\vec{v}_F)_{F\in\Fh} )\; : \;
\vec{v}_T\in\Poly{0}(T;\mathbb{R}^d)\quad\forall T\in\Th\mbox{ and }
\vec{v}_F\in\Poly{0}(F;\mathbb{R}^d)\quad\forall F\in\Fh
\right\}.
$$
For all $\uvec{v}_h\in\vUh$, we denote by $\vec{v}_h\in\Poly{0}(\Th;\mathbb{R}^d)$ the piecewise constant function obtained by patching element-based unknowns, that is,
\begin{equation}\label{eq:vh}
(\vec{v}_h)_{|T}\coloneq\vec{v}_T\qquad\forall T\in\Th.
\end{equation}
The restrictions of $\vUh$ and $\uvec{v}_h\in\vUh$ to a generic mesh element $T\in\Th$ are respectively denoted by $\vUT$ and $\uvec{v}_T=(\vec{v}_T,(\vec{v}_F)_{F\in\Fh[T]})$.
The vector of discrete variables corresponding to a smooth function on $\Omega$ is obtained via the global interpolation operator $\vIh:H^1(\Omega;\mathbb{R}^d)\to\vUh$ such that, for all $\vec{v}\in H^1(\Omega;\mathbb{R}^d)$,
$$
\vIh\vec{v}\coloneq( (\vlproj[T]{0}\vec{v}_{|T})_{T\in\Th}, (\vlproj[F]{0}\vec{v}_{|F})_{F\in\Fh} ).
$$
Its restriction to a generic mesh element $T\in\Th$ is the local interpolator $\vIT:H^1(T;\mathbb{R}^d)\to\vUT$ such that, for all $\vec{v}\in H^1(T;\mathbb{R}^d)$,
\begin{equation}\label{eq:vIT}
\vIT\vec{v} = (\vlproj[T]{0}\vec{v}, (\vlproj[F]{0}\vec{v}_{|F})_{F\in\Fh[T]}).
\end{equation}
The displacement is sought in the following subspace of $\vUh$ that strongly incorporates the homogeneous Dirichlet boundary condition:
$$
\vUhD\coloneq\left\{
\uvec{v}_h\in\vUh\; : \;\vec{v}_F=\vec{0}\quad\forall F\in\Fh^{{\rm b}}
\right\}.
$$
\subsection{Displacement reconstruction}
Let a mesh element $T\in\Th$ be fixed.
We define the local displacement reconstruction operator $\vpT:\vUT\to\Poly{1}(T;\mathbb{R}^d)$ such that, for all $\uvec{v}_T\in\vUT$,
\begin{equation}\label{eq:vpT}
\vec{\nabla}\vpT\uvec{v}_T = \sum_{F\in\Fh[T]}\frac{\meas{F}}{\meas{T}}(\vec{v}_F-\vec{v}_T)\otimes\vec{n}_{TF}
\mbox{ and }
\frac{1}{\meas{T}}\int_T\vpT\uvec{v}_T=\vec{v}_T.
\end{equation}
\begin{remark}[Explicit expression for the displacement reconstruction operator]
From \eqref{eq:vpT}, one can infer the following explicit expression for the displacement reconstruction operator: For all $\vec{x}\in T$,
\begin{equation}\label{eq:vpT:bis}
\vpT(\vec{x}) = \vec{v}_T
+ \sum_{F\in\Fh[T]}\frac{\meas{F}}{\meas{T}}(\vec{x}-\overline{\vec{x}}_T){\cdot}\vec{n}_{TF}~(\vec{v}_F-\vec{v}_T),
\end{equation}
where $\overline{\vec{x}}_T\coloneq\frac{1}{\meas{T}}\int_T\vec{x}$ denotes the centroid of $T$.
\end{remark}
\begin{proposition}[Commutation properties for the displacement reconstruction]
It holds, for all $\vec{v}\in H^1(T;\mathbb{R}^d)$,
\begin{equation}\label{eq:vpT:commutation}
\vec{\nabla}(\vpT\vIT\vec{v})=\tlproj[T]{0}(\vec{\nabla}\vec{v})\mbox{ and }
\vpT(\vIT\vec{v}) = \veproj[T]{1}\vec{v}.
\end{equation}
\end{proposition}
\begin{proof}
Let $\vec{v}\in H^1(T;\mathbb{R}^d)$.
Recalling the definition \eqref{eq:vIT} of the local interpolator, we have that
$$
\begin{aligned}
\vec{\nabla}\vpT\vIT\vec{v}
&= \sum_{F\in\Fh[T]}\frac{\meas{F}}{\meas{T}}(\vlproj[F]{0}\vec{v}-\vlproj[T]{0}\vec{v})\otimes\vec{n}_{TF}
\\
&= \frac{1}{\meas{T}}\sum_{F\in\Fh[T]}\int_F\vec{v}\otimes\vec{n}_{TF}
- \frac{1}{\meas{T}}\sum_{F\in\Fh[T]}\int_F\vlproj[T]{0}\vec{v}\otimes\vec{n}_{TF}
\\
&= \frac{1}{\meas{T}}\int_T\vec{\nabla}\vec{v}
- \cancel{\frac{1}{\meas{T}}\int_T\vec{\nabla}\vlproj[T]{0}\vec{v}},
\end{aligned}
$$
where we have used the definition \eqref{eq:vpT} of the local displacement reconstruction with $\uvec{v}_T=\vIT\vec{v}$ in the first line,
the definition \eqref{eq:lproj} of the $L^2$-orthogonal projector $\vlproj[F]{0}$ along with the fact that $\vlproj[T]{0}\vec{v}\otimes\vec{n}_{TF}$ is constant over $F$ to pass to the second line,
the Stokes theorem to pass to the third line and the fact that $\vlproj[T]{0}\vec{v}$ is constant inside $T$ to cancel the second term therein.
This proves the first relation in \eqref{eq:vpT:commutation}.
The second relation in \eqref{eq:vpT:commutation} immediately follows accounting for the first and recalling the definition \eqref{eq:eproj} of the elliptic projector after observing that the second relation in \eqref{eq:vpT} gives
\[
\pushQED{\qed}
\frac{1}{\meas{T}}\int_T\vpT\vIT\vec{v}
= \vlproj[T]{0}\vec{v}
= \frac{1}{\meas{T}}\int_T\vec{v}.\qedhere
\]
\end{proof}
To close this section, we define the global displacement reconstruction operator $\vph:\vUh\to\Poly{1}(\Th;\mathbb{R}^d)$ obtained by patching the local reconstructions:
For all $\uvec{v}_h\in\vUh$,
\begin{equation}\label{eq:vph}
(\vph\uvec{v}_h)_{|T}\coloneq\vpT\uvec{v}_T\qquad\forall T\in\Th.
\end{equation}
\subsection{Discrete bilinear form}
We define the bilinear form $\mathrm{a}_h:\vUh\times\vUh\to\mathbb{R}$ such that, for all $\uvec{w}_h,\uvec{v}_h\in\vUh$,
\begin{equation}\label{eq:ah}
\mathrm{a}_h(\uvec{w}_h,\uvec{v}_h)
\coloneq
(\tens{\sigma}(\tens{\nabla}_{{\rm s},h}\vph\uvec{w}_h),\tens{\nabla}_{{\rm s},h}\vph\uvec{v}_h)
+ (2\mu)~\mathrm{j}_h(\vph[1]\uvec{w}_h,\vph[1]\uvec{v}_h)
+ (2\mu)~\mathrm{s}_h(\uvec{w}_h,\uvec{v}_h).
\end{equation}
In the above expression, $\mathrm{j}_h: H^1(\Th;\mathbb{R}^d)\times H^1(\Th;\mathbb{R}^d)\to\mathbb{R}$ is the jump penalisation bilinear form such that, for all $\vec{w},\vec{v}\in H^1(\Th;\mathbb{R}^d)$,
$$
\mathrm{j}_h(\vec{w},\vec{v})
\coloneq\sum_{F\in\Fh}h_F^{-1}(\jump{\vec{w}},\jump{\vec{v}})_F,
$$
while $\mathrm{s}_h:\vUh\times\vUh\to\mathbb{R}$ is a stabilisation bilinear form defined from local contributions as follows:
\begin{equation}\label{eq:sh}
\mathrm{s}_h(\uvec{w}_h,\uvec{v}_h)\coloneq\sum_{T\in\Th}\mathrm{s}_T(\uvec{w}_T,\uvec{v}_T)\mbox{ with }
\mathrm{s}_T(\uvec{w}_T,\uvec{v}_T)
\coloneq \sum_{F\in\Fh[T]}\frac{\meas{F}}{h_F}\vec{\delta}_{TF}\uvec{w}_T{\cdot}\vec{\delta}_{TF}\uvec{v}_T
\mbox{ for all $T\in\Th$}.
\end{equation}
In the above expression, for all $T\in\Th$ and all $F\in\Fh[T]$, we have introduced the boundary difference operator $\vec{\delta}_{TF}:\vUT\to\mathbb{R}^d$ is such that, for any $\uvec{v}_T\in\vUT$,
\begin{equation}\label{eq:dTF}
\vec{\delta}_{TF}\uvec{v}_T\coloneq\vlproj[F]{0}\vpT\uvec{v}_T-\vec{v}_F.
\end{equation}
It can be proved that the stabilisation bilinear form enjoys the following consistency property:
For all $\vec{w}\in H^1(\Omega;\mathbb{R}^d)\cap H^2(\Th;\mathbb{R}^d)$,
\begin{equation}\label{eq:sh:consistency}
\mathrm{s}_h(\vIh\vec{w},\vIh\vec{w})^{\frac12}\lesssim h\seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{w}},
\end{equation}
with hidden constant independent of both $h$ and $\vec{w}$.
\subsection{Comparison with the original HHO method and role of the jump penalisation term}\label{sec:comparison.hho}
Compared with the original HHO bilinear form defined by \cite[Eqs. (24)--(26) and (38)]{Di-Pietro.Ern:15} and written for $k=0$, the bilinear form \eqref{eq:ah} includes a novel jump penalisation contribution inspired by the discrete Korn inequality of Lemma \ref{lem:korn}.
This term is needed for stability which, for HHO discretisations of the linear elasticity problem, cannot be achieved through local stabilisation terms for $k=0$.
As a matter of fact, following the ideas of \cite[Section 4.3.1.4]{Di-Pietro.Tittarelli:18}, stability would require the use in \eqref{eq:sh} of a family of local symmetric, positive semidefinite stabilisation bilinear forms $\left\{\mathrm{s}_T\; : \; T\in\Th\right\}$ satisfying the following properties:
\begin{compactenum}[(i)]
\item \emph{Local stability and boundedness.} For all $T\in\Th$ and all $\uvec{v}_T\in\vUT$, with hidden constants independent of $h$, $T$, and $\uvec{v}_T$,
\begin{equation}\label{eq:sT:norm.equivalence}
\norm[T]{\tens{\nabla}_{\rm s}\vpT\uvec{v}_T}^2 + \mathrm{s}_T(\uvec{v}_T,\uvec{v}_T)
\simeq
\sum_{F\in\Fh[T]}h_F^{-1}\norm[F]{\vec{v}_F-\vec{v}_T}^2.
\end{equation}
\item \emph{Polynomial consistency.} For all $\vec{w}\in\Poly{k+1}(T;\mathbb{R}^d)$,
\begin{equation}\label{eq:sT:polynomial.consistency}
\mathrm{s}_T(\vIT\vec{w},\uvec{v}_T)=0\qquad\forall\uvec{v}_T\in\vUT.
\end{equation}
\end{compactenum}
Actually, as noticed in \cite[Chapter 7]{Di-Pietro.Droniou:19}, properties \eqref{eq:sT:norm.equivalence} and \eqref{eq:sT:polynomial.consistency} are incompatible.
To see it, assume \eqref{eq:sT:polynomial.consistency}, consider a rigid-body motion $\vec{v}_{{\rm rbm}}$, that is, a function over $\overline{T}$ for which there exist a vector $\vec{t}_{\vec{v}}\in\mathbb{R}^d$ and a skew-symmetric matrix $\tens{R}_{\vec{v}}\in\mathbb{R}^{d\times d}$ such that, for any $\vec{x}\in\overline{T}$, $\vec{v}_{{\rm rbm}}(\vec{x}) = \vec{t}_{\vec{v}} + \tens{R}_{\vec{v}}\vec{x}$.
Take now $\uvec{v}_T=\vIT\vec{v}_{{\rm rbm}}$.
Since $\vec{v}_{{\rm rbm}}\in\Poly{1}(T;\mathbb{R}^d)$, the first relation in \eqref{eq:vpT:commutation} shows that $\vec{\nabla}\vpT\uvec{v}_T=\vlproj[T]{0}(\vec{\nabla}\vec{v}_{{\rm rbm}})=\vec{\nabla}\vec{v}_{{\rm rbm}}=\tens{R}_{\vec{v}}$ so that, in particular, $\vec{\nabla}\vpT\uvec{v}_T$ is skew-symmetric. Hence, $\tens{\nabla}_{\rm s}\vpT\uvec{v}_T=\tens{0}$.
Moreover, by \eqref{eq:sT:polynomial.consistency}, $\mathrm{s}_T(\uvec{v}_T,\uvec{v}_T)=\mathrm{s}_T(\vIT[0]\vec{v}_{{\rm rbm}},\uvec{v}_T)=0$, again because $\vec{v}_{\rm rbm}\in\Poly{1}(T;\mathbb{R}^d)$.
Hence, the left-hand side of \eqref{eq:sT:norm.equivalence} vanishes for all $\uvec{v}_T=\vIT\vec{v}_{{\rm rbm}}$ with $\vec{v}_{{\rm rbm}}$ rigid-body motion.
It is, however, easy to construct a rigid-body motion $\vec{v}_{{\rm rbm}}$ such that the right-hand side does not vanish, which shows that \eqref{eq:sT:norm.equivalence} cannot hold.
For this reason, the assumption that the discrete unknowns are at least piecewise affine is required in the original HHO method; see \cite[Section 4]{Di-Pietro.Ern:15}.
Notice that the choice of $\mathrm{s}_T$ in \eqref{eq:sh} retains the polynomial consistency property \eqref{eq:sT:polynomial.consistency}, which is crucial to prove \eqref{eq:sh:consistency}.
We next discuss how the stability property modifies for $k=0$.
To this end, recalling the definitions \eqref{eq:norm.dG} of the double-bar strain norm $\norm[\tens{\varepsilon},h]{{\cdot}}$ and \eqref{eq:sh} of the stabilisation bilinear form we introduce the triple-bar strain norm such that, for any $\uvec{v}_h\in\vUh$,
\begin{equation}\label{eq:tnorm.strain.h}
\tnorm[\tens{\varepsilon},h]{\uvec{v}_h}
\coloneq\left(
\norm[\tens{\varepsilon},h]{\vph\uvec{v}_h}^2 + \seminorm[\mathrm{s},h]{\uvec{v}_h}^2
\right)^{\frac12}\mbox{ with }
\seminorm[\mathrm{s},h]{\uvec{v}_h}\coloneq\mathrm{s}_h(\uvec{v}_h,\uvec{v}_h)^{\frac12}.
\end{equation}
\begin{lemma}[Global stability and boundedness]
For all $\uvec{v}_h\in\vUhD$ it holds
\begin{equation}\label{eq:global.norm.equivalence}
\norm{\tens{\nabla}_{{\rm s},h}\vph\uvec{v}_h}^2 + \seminorm[\mathrm{s},h]{\uvec{v}_h}^2
\lesssim
\sum_{T\in\Th}\sum_{F\in\Fh[T]}h_F^{-1}\norm[F]{\vec{v}_F-\vec{v}_T}^2
\lesssim
\tnorm[\tens{\varepsilon},h]{\uvec{v}_h}^2,
\end{equation}
with hidden constant independent of both $h$ and $\uvec{v}_h$.
\end{lemma}
\begin{proof}
It follows from \cite[Lemma 4]{Di-Pietro.Ern.ea:14} that
\begin{equation}\label{eq:global.norm.equivalence:1}
\norm{\vec{\nabla}_h\vph\uvec{v}_h}^2 + \seminorm[\mathrm{s},h]{\uvec{v}_h}^2
\simeq
\sum_{T\in\Th}\sum_{F\in\Fh[T]}h_F^{-1}\norm[F]{\vec{v}_F-\vec{v}_T}^2,
\end{equation}
where $\vec{\nabla}_h: H^1(\Th;\mathbb{R}^d)\to L^2(\Omega;\mathbb{R}^{d\times d})$ is the broken gradient such that $(\vec{\nabla}_h\vec{v})_{|T}=\vec{\nabla}\vec{v}_{|T}$ for any $T\in\Th$.
On the other hand, using the definition of the symmetric gradient for the first bound and Korn's inequality \eqref{eq:korn} for the second, we can write
\begin{equation}\label{eq:global.norm.equivalence:2}
\norm{\tens{\nabla}_{{\rm s},h}\vph\uvec{v}_h}^2
\lesssim\norm{\vec{\nabla}_h\vph\uvec{v}_h}^2
\lesssim\norm{\tens{\nabla}_{{\rm s},h}\vph\uvec{v}_h}^2 + \seminorm[\mathrm{j},h]{\vph\vec{v}_h}^2.
\end{equation}
Combining \eqref{eq:global.norm.equivalence:2} with \eqref{eq:global.norm.equivalence:1} yields the result.
\end{proof}
\subsection{Discrete problem}
The low-order scheme for the approximation of problem \eqref{eq:strong} reads:
Find $\uvec{u}_h\in\vUhD$ such that
\begin{equation}\label{eq:discrete}
\mathrm{a}_h(\uvec{u}_h,\uvec{v}_h)
= (\vec{f},\vec{v}_h)\qquad\forall\uvec{v}_h\in\vUhD.
\end{equation}
Using the coercivity of the bilinear form $\mathrm{a}_h$ proved in Lemma \ref{lem:ah} below together with the discrete Korn inequality \eqref{eq:korn-poincare}, we infer that the discrete problem is well-posed and the a priori estimate $\tnorm[\tens{\varepsilon},h]{\uvec{u}_h}\lesssim \alpha^{-\frac12}\norm{\vec{f}}$ holds for the discrete solution, with hidden constant independent of both $h$ and of the problem data, and triple-bar strain seminorm defined by \eqref{eq:tnorm.strain.h} below.
\begin{remark}[Static condensation for problem \eqref{eq:discrete}]\label{rem:static.condensation}
The jump stabilisation introduces a direct link among discrete unknowns attached to neighbouring mesh elements.
As a result, static condensation of element-based unknowns no longer appears to be an interesting option.
\end{remark}
\section{Convergence analysis}\label{sec:convergence}
In this section, after studying the properties of the discrete bilinear form $\mathrm{a}_h$, we prove a priori estimates for the error in the energy- and $L^2$-norms.
\subsection{Properties of the discrete bilinear form}
\begin{lemma}[Properties of $\mathrm{a}_h$]\label{lem:ah}
The bilinear form $\mathrm{a}_h$ enjoys the following properties:
\begin{compactenum}[(i)]
\item \emph{Stability and boundedness.} Recalling the definition \eqref{eq:tnorm.strain.h} of the triple-bar strain norm and the bound \eqref{eq:lambda.mu.bounds} on Lam\'e's coefficients, for all $\uvec{v}_h\in\vUh$ it holds
\begin{equation}\label{eq:ah:stability}
\alpha\tnorm[\tens{\varepsilon},h]{\uvec{v}_h}^2
\lesssim\norm[\mathrm{a},h]{\uvec{v}_h}^2
\lesssim\left(2\mu+d|\lambda|\right)\tnorm[\tens{\varepsilon},h]{\uvec{v}_h}^2
\mbox{ with }
\norm[\mathrm{a},h]{\uvec{v}_h}\coloneq\mathrm{a}_h(\uvec{v}_h,\uvec{v}_h)^{\frac12},
\end{equation}
where the hidden constants are independent of both $h$ and the problem data.
\item \emph{Consistency.} It holds for all $\vec{w}\in H_0^1(\Omega;\mathbb{R}^d)\cap H^2(\Th;\mathbb{R}^d)$ such that $\vec{\nabla}{\cdot}\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w})\in L^2(\Omega;\mathbb{R}^d)$,
\begin{equation}\label{eq:ah:consistency}
\tnorm[\tens{\varepsilon},h,*]{\Cerr{\vec{w}}{\cdot}}
\lesssim h\left(
2\mu\seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{w}}
+ \seminorm[H^1(\Th;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{w}}
\right),
\end{equation}
where the hidden constant is independent of $\vec{w}$, $h$, and of the problem data, the linear form
$\Cerr{\vec{w}}{{\cdot}}:\vUhD\to\mathbb{R}$ representing the consistency error is such that, for all
$\uvec{v}_h\in\vUhD$,
\begin{equation}\label{eq:Eh}
\Cerr{\vec{w}}{\uvec{v}_h}
\coloneq
-(\vec{\nabla}{\cdot}\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w}),\vec{v}_h)
- \mathrm{a}_h(\vIh\vec{w},\uvec{v}_h),
\end{equation}
and its dual norm is given by
$$
\tnorm[\tens{\varepsilon},h,*]{\Cerr{\vec{w}}{\cdot}}\coloneq
\sup_{\uvec{v}_h\in\vUhD,\tnorm[\tens{\varepsilon},h]{\uvec{v}_h}=1}\left|\Cerr{\vec{w}}{\uvec{v}_h}\right|.
$$
\end{compactenum}
\end{lemma}
\begin{proof}
(i) \emph{Stability and boundedness.}
Let $\uvec{v}_h\in\vUh$.
We recall the Frobenius product such that, for all $\tens{\tau},\tens{\eta}\in\mathbb{R}^{d\times d}$, $\tens{\tau}{:}\tens{\eta}\coloneq\sum_{i=1}^d\sum_{j=1}^d\tau_{ij}\eta_{ij}$ with corresponding norm $\norm[\rm F]{\tens{\tau}}\coloneq(\tens{\tau}{:}\tens{\tau})^{\frac12}$.
Writing \eqref{eq:ah} for $\uvec{w}_h=\uvec{v}_h$, using the assumption \eqref{eq:lambda.mu.bounds} on Lam\'e's parameters to infer that $\tens{\sigma}(\tens{\tau}){:}\tens{\tau}\ge\alpha\norm[\rm F]{\tens{\tau}}^2$ for any $\tens{\tau}\in\mathbb{R}_{\rm sym}^{d\times d}$, recalling the definitions \eqref{eq:norm.dG} and \eqref{eq:tnorm.strain.h} of the double- and triple-bar strain norms, and observing that $2\mu\ge\alpha$, the first inequality in \eqref{eq:ah:stability} follows.
The second inequality can be obtained in a similar way: write \eqref{eq:ah} for $\uvec{w}_h=\uvec{v}_h$, observe that $|\tens{\sigma}(\tens{\tau}){:}\tens{\tau}|\le(2\mu + d|\lambda|)\norm[\rm F]{\tens{\tau}}^2$ for any $\tens{\tau}\in\mathbb{R}_{\rm sym}^{d\times d}$, and use again \eqref{eq:norm.dG} and \eqref{eq:tnorm.strain.h}.
\medskip\\
(ii) \emph{Consistency.}
Let $\uvec{v}_h\in\vUhD$.
We reformulate the components of the consistency error.
Integrating by parts element by element, we infer that
$$
-(\vec{\nabla}{\cdot}\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w}),\vec{v}_h)
= \sum_{T\in\Th}\sum_{F\in\Fh[T]}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w})_{|T}\vec{n}_{TF},\vec{v}_F-\vec{v}_T)_F,
$$
where we have used the continuity of normal tractions across interfaces together with the fact that boundary unknowns are set to zero in $\vUhD$ to insert $\vec{v}_F$ into the right-hand side.
To reformulate the second term in \eqref{eq:Eh}, in the expression \eqref{eq:ah} of $\mathrm{a}_h$ we use the first property in \eqref{eq:vpT:commutation} together with the linearity of the strain-stress law $\tens{\sigma}$ to write, for all $T\in\Th$, $\tens{\sigma}(\tens{\nabla}_{\rm s}\vpT\vIT\vec{w})=\tens{\sigma}(\tlproj[T]{0}(\tens{\nabla}_{\rm s}\vec{w}))=\tlproj[T]{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w}))$ and obtain
$$
\mathrm{a}_h(\vIh\vec{w},\uvec{v}_h)
= \sum_{T\in\Th}(\tlproj[T]{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w})),\tens{\nabla}_{\rm s}\vpT\uvec{v}_T)_T
+ (2\mu)~\mathrm{j}_h(\vph\vIh\vec{w},\vph\uvec{v}_h)
+ (2\mu)~\mathrm{s}_h(\vIh\vec{w},\uvec{v}_h).
$$
After expanding, for all $T\in\Th$, $\tens{\nabla}_{\rm s}\vpT\uvec{v}_T$ according to its definition \eqref{eq:vpT}, we deduce that
$$
\mathrm{a}_h(\vIh\vec{w},\uvec{v}_h)
= \sum_{T\in\Th}\sum_{F\in\Fh[T]}(\tlproj[T]{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w}))\vec{n}_{TF},\vec{v}_F-\vec{v}_T)_F
+ (2\mu)~\mathrm{j}_h(\vph\vIh\vec{w},\vph\uvec{v}_h)
+ (2\mu)~\mathrm{s}_h(\vIh\vec{w},\uvec{v}_h).
$$
Plugging the above relations into the expression \eqref{eq:Eh} of the consistency error, passing to absolute values, using a generalised H\"older inequality with exponents $(2,\infty,2)$ along with $\norm[L^\infty(F;\mathbb{R}^d)]{\vec{n}_{TF}}\le 1$ and $h_F\le h_T$ for the first term in the right-hand side, and Cauchy--Schwarz inequalities for the remaining terms, we get
\begin{equation}\label{eq:consistency:basic}
\begin{aligned}
\left|\Cerr{\vec{w}}{\uvec{v}_h}\right|
&=
\underbrace{%
\left(
\sum_{T\in\Th}h_T\norm[\partial T]{\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w})_{|T}-\tlproj[T]{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w}))}^2
\right)^{\frac12}%
}_{\mathfrak{T}_1}
\left(
\sum_{T\in\Th}\sum_{F\in\Fh[T]}h_F^{-1}\norm[F]{\vec{v}_F-\vec{v}_T}^2
\right)^{\frac12}
\\
&\qquad
+ \underbrace{%
(2\mu)\seminorm[\mathrm{j},h]{\vph\vIh\vec{w}}%
}_{\mathfrak{T}_2}
~\seminorm[\mathrm{j},h]{\vph\uvec{v}_h}
+ \underbrace{%
(2\mu)\seminorm[\mathrm{s},h]{\vIh\vec{w}}%
}_{\mathfrak{T}_3}
~\seminorm[\mathrm{s},h]{\uvec{v}_h}
\\
&\lesssim
\left(\mathfrak{T}_1+\mathfrak{T}_2+\mathfrak{T}_3\right)\tnorm[\tens{\varepsilon},h]{\uvec{v}_h},
\end{aligned}
\end{equation}
where we have used the second inequality in \eqref{eq:global.norm.equivalence} together with the definition \eqref{eq:tnorm.strain.h} of the triple-bar strain norm to conclude.
Recalling the expression \eqref{eq:sigma} of the strain-stress law, we get, for any $T\in\Th$,
\begin{equation}\label{eq:est.sigma}
\begin{aligned}
h_T^{\frac12}\norm[\partial T]{\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w})_{|T}{-}\tlproj[T]{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{w}))}
&\le (2\mu)h_T^{\frac12}\norm[\partial T]{\tens{\nabla}_{\rm s}\vec{w}{-}(\tlproj[T]{0}\tens{\nabla}_{\rm s}\vec{w})}
+ h_T^{\frac12}\norm[\partial T]{\lambda\vec{\nabla}{\cdot}\vec{w}{-}\lproj[T]{0}(\lambda\vec{\nabla}{\cdot}\vec{w})}
\\
&\lesssim
h\left( (2\mu)\seminorm[H^2(T;\mathbb{R}^d)]{\vec{w}} + \seminorm[H^1(T;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{w}}\right),
\end{aligned}
\end{equation}
where we have used the approximation properties \eqref{eq:lproj:approx} of the $L^2$-orthogonal projector along with $h_T\le h$ to conclude.
Using the above estimate, we infer for the first term
\begin{equation}\label{eq:consistency:T1}
\mathfrak{T}_1\lesssim h\left(
(2\mu)\seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{w}}
+ \seminorm[H^1(\Th;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{w}}
\right).
\end{equation}
Moving to the second term, we start by observing that
$$
\begin{aligned}
\mathfrak{T}_2^2
&= (2\mu)^2~\seminorm[\mathrm{j},h]{\veproj{1}\vec{w}}^2
\\
&= (2\mu)^2\sum_{F\in\Fh}h_F^{-1}\norm[F]{\jump{\veproj{1}\vec{w}}}^2
\\
&= (2\mu)^2\sum_{F\in\Fh}h_F^{-1}\norm[F]{\jump{\veproj{1}\vec{w}-\vec{w}}}^2
\\
&\lesssim (2\mu)^2\sum_{F\in\Fh}\sum_{T\in\Th[F]}h_F^{-1}\norm[F]{\veproj[T]{1}\vec{w}-\vec{w}_{|T}}^2
\\
&\lesssim (2\mu)^2\sum_{T\in\Th}h_T^{-1}\norm[\partial T]{\veproj[T]{1}\vec{w}-\vec{w}_{|T}}^2
\end{aligned}
$$
where we have used, in this order, the second relation in \eqref{eq:vpT:commutation}, the definition \eqref{eq:norm.dG} of the jump seminorm, the fact that the jumps of $\vec{w}$ vanish across any $F\in\Fh$, the definition \eqref{eq:jump} of the jump operator together with the triangle inequality, and the relation
\begin{equation}
\label{eq:sumTF=sumFT}
\sum_{T\in\Th}\sum_{F\in\Fh[T]} \bullet =\sum_{F\in\Fh}\sum_{T\in\Th[F]} \bullet
\end{equation}
to exchange the sums over elements and faces.
Hence, using the approximation properties \eqref{eq:eproj:approx} of the elliptic projector, $h_T\le h$, and taking the square root, we arrive at
\begin{equation}\label{eq:consistency:T2}
\mathfrak{T}_2 \lesssim (2\mu) h \seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{w}}.
\end{equation}
For the third term, \eqref{eq:sh:consistency} readily gives
\begin{equation}\label{eq:consistency:T3}
\mathfrak{T}_3\lesssim (2\mu) h\seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{w}}.
\end{equation}
Plugging \eqref{eq:consistency:T1}, \eqref{eq:consistency:T2}, and \eqref{eq:consistency:T3} into \eqref{eq:consistency:basic} and passing to the supremum yields \eqref{eq:ah:consistency}.
\end{proof}
\subsection{Energy error estimate}
\begin{theorem}[Energy error estimate]\label{thm:err.est}
Let $\vec{u}\in H_0^1(\Omega;\mathbb{R}^d)$ denote the unique solution to \eqref{eq:weak}, for which we assume the additional regularity $\vec{u}\in H^2(\Th;\mathbb{R}^d)$.
For all $h\in{\cal H}$, let $\uvec{u}_h\in\vUhD$ denote the unique solution to \eqref{eq:discrete}.
Then,
\begin{equation}\label{eq:err.est}
\tnorm[\tens{\varepsilon},h]{\uvec{u}_h-\vIh\vec{u}}
\lesssim \alpha^{-1} h\left(
(2\mu)\seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{u}}
+ \seminorm[H^1(\Th;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{u}}
\right),
\end{equation}
with hidden constant independent of $h$, $\vec{u}$, and of the problem data.
\end{theorem}
\begin{proof}
Applying to the present setting the results of \cite[Theorem 10]{Di-Pietro.Droniou:18} gives the abstract estimate
$$
\tnorm[\tens{\varepsilon},h]{\uvec{u}_h-\vIh\vec{u}}
\le\alpha^{-1}\tnorm[\tens{\varepsilon},h,*]{\Cerr{\vec{u}}{\cdot}}.
$$
Using the assumed regularity for the exact solution to invoke \eqref{eq:ah:consistency}, \eqref{eq:err.est} follows.
\end{proof}
\begin{remark}[Robustness in the quasi-incompressible limit]\label{rem:robustness}
In the numerical approximation of linear elasticity problems, a key point consists in devising schemes that are robust in the quasi incompressible limit corresponding to $\frac{\lambda}{2\mu}\gg 1$ (which requires, in particular $\lambda\ge 0$).
From a mathematical perspective, this property is expressed by the fact that the error estimates are uniform in $\lambda$.
For $d=2$ and $\Omega$ convex, it is proved, e.g., in \cite[Lemma 2.2]{Brenner.Sung:92} that
\begin{equation}\label{eq:a-priori}
(2\mu)\norm[H^2(\Omega;\mathbb{R}^d)]{\vec{u}}
+ \norm[H^1(\Omega;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{u}}
\lesssim\norm{\vec{f}},
\end{equation}
with hidden constant possibly depending on $\Omega$ and $\mu$ but independent of $\lambda$.
This result can be extended to $d=3$ reasoning as in the above reference and accounting for the regularity estimates for the Stokes problem derived in \cite[Theorem 3]{Amrouche.Girault:91}.
Plugging \eqref{eq:a-priori} into \eqref{eq:err.est} and observing that, when $\lambda\ge 0$, we can take $\alpha=2\mu$ (cf. \eqref{eq:lambda.mu.bounds}), we can write, with hidden constant independent of both $h$ and $\lambda$,
\begin{equation}\label{eq:en.err.est.robust}
\tnorm[\tens{\varepsilon},h]{\uvec{u}_h-\vIh\vec{u}}\lesssim h \norm{\vec{f}},
\end{equation}
which shows that our error estimate \eqref{eq:err.est} is uniform in $\lambda$.
The key point to obtain robustness is the first commutation property in \eqref{eq:vpT:commutation}, which is used to estimate the term $\mathcal{T}_1$ in the proof of Lemma \ref{lem:ah}.
\end{remark}
\begin{remark}[Quasi-optimality of the error estimate]
It follows from the second inequality in \eqref{eq:ah:stability} that the bilinear form $\mathrm{a}_h$ is bounded with boundedness constant independent of $h$.
Hence, following \cite[Remark 11]{Di-Pietro.Droniou:18}, the error estimate \eqref{eq:err.est} is quasi-optimal.
\end{remark}
\begin{remark}[{Energy estimate in the $\norm[\mathrm{a},h]{{\cdot}}$-norm for $\lambda\ge 0$}]\label{rem:err.est:norm.a.h}
When $\lambda\ge0$, a consistency estimate in $h$ holds for $\norm[\mathrm{a},h,*]{\Cerr{\vec{w}}{\cdot}}$, the norm of the consistency error linear form dual to $\norm[\mathrm{a},h]{{\cdot}}$ (see \eqref{eq:ah:stability}).
To see it, observe that, from \eqref{eq:consistency:basic} together with $(2\mu)^{\frac12}\tnorm[\tens{\varepsilon},h]{\uvec{v}_h}\le\norm[\mathrm{a},h]{\uvec{v}_h}$ (a consequence of the assumption $\lambda\ge 0$), it follows $\left|\Cerr{\vec{w}}{\uvec{v}_h}\right|\lesssim\left(\mathfrak{T}_1+\mathfrak{T}_2+\mathfrak{T}_3\right)(2\mu)^{-\frac12}\norm[\mathrm{a},h]{\uvec{v}_h}$.
Hence, passing to the supremum over $\big\{\uvec{v}_h\in\vUhD\; : \;\norm[\mathrm{a},h]{\uvec{v}_h}=1\big\}$, we infer
$$
\norm[\mathrm{a},h,*]{\Cerr{\vec{w}}{\cdot}}
\lesssim h\left((2\mu)^{\frac12}\seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{w}}
+(2\mu)^{-\frac12}\seminorm[H^1(\Th;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{w}}\right).
$$
Invoking again \cite[Theorem 10]{Di-Pietro.Droniou:18}, this time with $\vUhD$ equipped with the $\norm[\mathrm{a},h]{{\cdot}}$-norm, it is inferred
$$
\norm[\mathrm{a},h]{\uvec{u}_h-\vIh\vec{u}}
\lesssim h\left(
(2\mu)^{\frac12}\seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{u}}
+ (2\mu)^{-\frac12}\seminorm[H^1(\Th;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{u}}
\right),
$$
with hidden constant having the same dependencies as in \eqref{eq:err.est}.
\end{remark}
\subsection{Improved $L^2$-error estimate}
Combining the discrete Korn--Poincar\'e inequality \eqref{eq:korn-poincare} with the error estimate \eqref{eq:err.est}, we can infer an estimate in $h$ for the $L^2$-norm of the displacement error $\norm{\vec{u}_h-\vlproj{0}\vec{u}}$, where we remind the reader that $\vec{u}_h$ is defined according to \eqref{eq:vh} as the broken polynomial obtained patching element unknowns, while $\vlproj{0}\vec{u}$ is the $L^2$-orthogonal projection of the exact solution on $\Poly{0}(\Th;\mathbb{R}^d)$.
It is well-known, however, that improved $L^2$-error estimates can be derived in the context of HHO methods when elliptic regularity holds.
In this section, we show that the same is true for the low-order method considered in this work.
For the sake of simplicity, we assume throughout this section that
$$
\lambda\ge 0.
$$
This assumption could be removed, but we keep it here to simplify the discussion and point out the robustness in the quasi-incompressible limit.
Recalling the discussion in Remark \ref{rem:robustness}, elliptic regularity for our problem entails that, for all $\vec{g}\in L^2(\Omega;\mathbb{R}^d)$, the unique solution of the (dual) problem:
Find $\vec{z}_{\vec{g}}\in H_0^1(\Omega;\mathbb{R}^d)$ such that
\begin{equation}\label{eq:dual}
(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}}),\tens{\nabla}_{\rm s}\vec{v}) = (\vec{g},\vec{v})
\qquad\forall\vec{v}\in H_0^1(\Omega;\mathbb{R}^d)
\end{equation}
satisfies the a priori estimate
\begin{equation}\label{eq:dual:a-priori}
(2\mu)~\norm[H^2(\Omega;\mathbb{R}^d)]{\vec{z}_{\vec{g}}}
+ \norm[H^1(\Omega;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{z}_{\vec{g}}}
\lesssim\norm{\vec{g}},
\end{equation}
with hidden constant only depending on $\Omega$ and $\mu$.
\begin{theorem}[Improved $L^2$-error estimate]\label{thm:l2.err.est}
Under the assumptions and notations of Theorem \ref{thm:err.est}, and further assuming $\lambda\ge 0$, elliptic regularity, and $\vec{f}\in H^1(\Th;\mathbb{R}^d)$, it holds that
\begin{equation}\label{eq:l2.err.est}
\norm{\vec{u}_h-\vlproj{0}\vec{u}}
\lesssim h^2\norm[H^1(\Th;\mathbb{R}^d)]{\vec{f}},
\end{equation}
where the hidden constant is independent of both $h$ and $\lambda$ (but possibly depends on $\mu$).
\end{theorem}
\begin{proof}
Inside the proof, hidden constants have the same dependencies as in \eqref{eq:l2.err.est}.
Applying the results of \cite[Theorem 13]{Di-Pietro.Droniou:18} to the present setting gives the basic estimate
\begin{equation}\label{eq:l2.basic.est}
\norm{\vec{u}_h-\vlproj{0}\vec{u}}
\le
\tnorm[\tens{\varepsilon},h]{\uvec{u}_h-\vIh\vec{u}}
{\sup_{\vec{g}\in L^2(\Omega;\mathbb{R}^d),\norm{\vec{g}}\le 1}\tnorm[\tens{\varepsilon},h,*]{\Cerr{\vec{z}_{\vec{g}}}{\cdot}}}
+ {\sup_{\vec{g}\in L^2(\Omega;\mathbb{R}^d),\norm{\vec{g}}\le 1}\Cerr{\vec{u}}{\vIh\vec{z}_{\vec{g}}}}.
\end{equation}
We proceed to bound the addends in the right-hand side, denoted for the sake of brevity $\mathfrak{T}_1$ and $\mathfrak{T}_2$.
\medskip\\
(i) \emph{Estimate of $\mathfrak{T}_1$.}
Since $\vec{z}_{\vec{g}}\in H_0^1(\Omega;\mathbb{R}^d)\cap H^2(\Omega;\mathbb{R}^d)$, the consistency estimate \eqref{eq:ah:consistency} followed by the elliptic regularity bound \eqref{eq:dual:a-priori} yield, for any $\vec{g}\in L^2(\Omega;\mathbb{R}^d)$,
$$
\tnorm[\tens{\varepsilon},h,*]{\Cerr{\vec{z}_{\vec{g}}}{\cdot}}
\lesssim h\left(
(2\mu)\seminorm[H^2(\Th;\mathbb{R}^d)]{\vec{z}_{\vec{g}}}
+ \seminorm[H^1(\Th;\mathbb{R})]{\lambda\vec{\nabla}{\cdot}\vec{z}_{\vec{g}}}
\right)
\lesssim h\norm{\vec{g}}.
$$
Combined with the energy error estimate \eqref{eq:en.err.est.robust}, this yields
\begin{equation}\label{eq:l2.err.est:T1}
\mathfrak{T}_1\lesssim h^2\norm{\vec{f}}.
\end{equation}
\\
(ii) \emph{Estimate of $\mathfrak{T}_2$.}
Recalling the expression \eqref{eq:Eh} of the consistency error, expanding the bilinear form $\mathrm{a}_h$ according to its definition \eqref{eq:ah} with $\uvec{w}_h=\vIh\vec{u}$ and $\uvec{v}_h=\vIh\vec{z}_{\vec{g}}$, and invoking \eqref{eq:vpT:commutation} to replace $\vph\vIh$ with $\veproj{1}$ and $\tens{\nabla}_{{\rm s},h}\vph\vIh$ with $\tlproj{0}\tens{\nabla}_{\rm s}$, we can write
\begin{multline}\label{eq:l2.err.est:T2:1}
\Cerr{\vec{u}}{\vIh\vec{z}_{\vec{g}}}
= (-\vec{\nabla}{\cdot}\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u}),\vlproj{0}\vec{z}_{\vec{g}})
- (\tlproj{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u})),\tlproj{0}\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}})
\\
-(2\mu)~\mathrm{j}_h(\veproj{1}\vec{u},\veproj{1}\vec{z}_{\vec{g}})
-(2\mu)~\mathrm{s}_h(\vIh\vec{u},\vIh\vec{z}_{\vec{g}}).
\end{multline}
We have that
$$
\begin{aligned}
-(\vec{\nabla}{\cdot}\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u}),\vlproj{0}\vec{z}_{\vec{g}})
=
(\vec{f},\vlproj{0}\vec{z}_{\vec{g}})
&=
(\vlproj{0}\vec{f},\vec{z}_{\vec{g}})
\\
&=
(\vlproj{0}\vec{f}-\vec{f},\vec{z}_{\vec{g}})
+ (\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u}),\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}})
\\
&=
(\vlproj{0}\vec{f}-\vec{f},\vec{z}_{\vec{g}}-\vlproj{0}\vec{z}_{\vec{g}})
+ (\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u}),\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}}),
\end{aligned}
$$
where we have used the fact that \eqref{eq:strong:pde} holds almost everywhere in $\Omega$ to replace $-\vec{\nabla}{\cdot}\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u})$ with $\vec{f}$ along with the definitions \eqref{eq:lproj.h} and \eqref{eq:lproj} of the global and local $L^2$-orthogonal projectors in the first line,
we have added the quantity $(\vec{f},\vec{z}_{\vec{g}})-(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u}),\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}})=0$ (see \eqref{eq:weak}) in the second line,
while, to pass to the third line, we have used the fact that, by definition of $\vlproj{0}$, the function $(\vlproj{0}\vec{f}-\vec{f})$ is $L^2(\Omega;\mathbb{R}^d)$-orthogonal to $\Poly{0}(\Th;\mathbb{R}^d)$ to insert $\vlproj{0}\vec{z}_{\vec{g}}$ into the first term.
The Cauchy--Schwarz inequality and the approximation property \eqref{eq:lproj:approx} of the $L^2$-orthogonal projector inside each mesh element yield for the first term in the right-hand side
\begin{equation}\label{eq:T2:1}
\left|
(\vlproj{0}\vec{f}-\vec{f},\vec{z}_{\vec{g}}-\vlproj{0}\vec{z}_{\vec{g}})
\right|
\lesssim h^2\seminorm[H^1(\Th;\mathbb{R}^d)]{\vec{f}}\seminorm[H^1(\Omega;\mathbb{R}^d)]{\vec{z}_{\vec{g}}}
\lesssim h^2\seminorm[H^1(\Th;\mathbb{R}^d)]{\vec{f}}\norm{\vec{g}},
\end{equation}
where we have used a standard estimate on $\seminorm[H^1(\Omega;\mathbb{R}^d)]{\vec{z}_{\vec{g}}}$ obtained letting $\vec{v}=\vec{z}_{\vec{g}}$ in \eqref{eq:dual} and using the Cauchy--Schwarz and Korn inequalities to bound the right-hand side.
On the other hand, using the definitions \eqref{eq:lproj.h} and \eqref{eq:lproj} of the global and local $L^2$-orthogonal projectors, we can write
\begin{equation}\label{eq:T2:2}
\begin{aligned}
&\left|
\big(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u}),\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}}
\big)
- \big(
\tlproj{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u})),\tlproj{0}\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}}
\big)
\right|
\\
&\qquad
=\left|
\big(
\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u})-\vlproj{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u})),\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}}-\tlproj{0}(\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}})
\big)
\right|
\\
&\qquad
\le
\norm{\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u})-\tlproj{0}(\tens{\sigma}(\tens{\nabla}_{\rm s}\vec{u}))}
~\norm{\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}}-\tlproj{0}(\tens{\nabla}_{\rm s}\vec{z}_{\vec{g}})}
\\
&\qquad
\lesssim h^2\left(
(2\mu)\seminorm[H^2(\Omega;\mathbb{R}^d)]{\vec{u}} + \seminorm[H^1(\Omega;\mathbb{R}^d)]{\lambda\vec{\nabla}{\cdot}\vec{u}}
\right)\seminorm[H^2(\Omega;\mathbb{R}^d)]{\vec{z}_{\vec{g}}}
\\
&\qquad
\lesssim h^2\norm{\vec{f}}~\norm{\vec{g}},
\end{aligned}
\end{equation}
where we have used a Cauchy--Schwarz inequality to pass to the third line,
\eqref{eq:est.sigma} with $\vec{w}=\vec{u}$ together with the approximation property \eqref{eq:lproj:approx} of the $L^2$-orthogonal projector to pass to the fourth line,
and the elliptic regularity bound \eqref{eq:dual:a-priori} to conclude.
Finally, using Cauchy--Schwarz inequalities, we can write
\begin{equation}\label{eq:T2:3+4}
\begin{aligned}
(2\mu)~\mathrm{j}_h(\veproj{1}\vec{u},\veproj{1}\vec{z}_{\vec{g}})
+(2\mu)~\mathrm{s}_h(\vIh\vec{u},\vIh\vec{z}_{\vec{g}})
&\lesssim
(2\mu)~\seminorm[\mathrm{j},h]{\veproj{1}\vec{u}}
\seminorm[\mathrm{j},h]{\veproj{1}\vec{z}_{\vec{g}}}
+ (2\mu)~\seminorm[\mathrm{s},h]{\vIh\vec{u}}
\seminorm[\mathrm{s},h]{\vIh\vec{z}_{\vec{g}}}
\\
&\lesssim
(2\mu)h^2\seminorm[H^2(\Omega;\mathbb{R}^d)]{\vec{u}}
\seminorm[H^2(\Omega;\mathbb{R}^d)]{\vec{z}_{\vec{g}}}
\lesssim
h^2\norm{\vec{f}}~\norm{\vec{g}},
\end{aligned}
\end{equation}
where, to pass to the second line, we have used \eqref{eq:sh:consistency} for the terms involving $\mathrm{s}_h$ and we have proceeded as in the estimate of $\mathfrak{T}_2$ in the proof of point (ii) of Lemma \ref{lem:ah} for the terms involving $\mathrm{j}_h$ while, to conclude, we have invoked \eqref{eq:dual:a-priori}.
Taking absolute values in \eqref{eq:l2.err.est:T2:1} and using the estimates \eqref{eq:T2:1}, \eqref{eq:T2:2}, \eqref{eq:T2:3+4} yields $\left|\Cerr{\vec{u}}{\vIh\vec{z}_{\vec{g}}}\right|\lesssim h^2\norm[H^1(\Th;\mathbb{R}^d)]{\vec{f}}\norm{\vec{g}}$.
Hence, passing to the supremum, we obtain
\begin{equation}\label{eq:l2.err.est:T2}
\mathfrak{T}_2
\lesssim h^2\norm[H^1(\Th;\mathbb{R}^d)]{\vec{f}}.
\end{equation}
Plugging \eqref{eq:l2.err.est:T1} and \eqref{eq:l2.err.est:T2} into \eqref{eq:l2.basic.est} concludes the proof.
\end{proof}
\section{Numerical tests}\label{sec:numerical.tests}
In what follows we verify, through numerical examples, the results stated in the previous section.
\subsection{Two-dimensional quasi-incompressible case}\label{sec:numerical.tests:brenner}
The first test case is inspired by \cite{Brenner:93}: we solve on the unit square $\Omega=(0,1)^2$ the homogeneous Dirichlet problem corresponding to the exact solution such that
$$
\renewcommand{\arraystretch}{1.2}
\vec{u}(\vec{x}) =
\begin{pmatrix}
(\cos(2\pi x_1)-1)\sin(2\pi x_2)+\frac{1}{1+\lambda}\sin(\pi x_1)\sin(\pi x_2)
\\
(1-\cos(2\pi x_2))\sin(2\pi x_1)+\frac{1}{1+\lambda}\sin(\pi x_1)\sin(\pi x_2)
\end{pmatrix}.
$$
The corresponding forcing term is
$$
\renewcommand{\arraystretch}{1.2}
\vec{f}(\vec{x}) =
\begin{pmatrix}
-\mu\left[
4\sin(2\pi x_2)\left(1-2\cos(2\pi x_1)\right)
-\frac{2}{1+\lambda}\sin(\pi x_1)\sin(\pi x_2)
\right]
- \frac{\lambda+\mu}{1+\lambda}\cos(\pi(x_1+x_2))
\\
-\mu\left[
4\sin(2\pi x_1)\left(2\cos(2\pi x_2)-1\right)
-\frac{2}{1+\lambda}\sin(\pi x_1)\sin(\pi x_2)
\right]
- \frac{\lambda+\mu}{1+\lambda}\cos(\pi(x_1+x_2))
\end{pmatrix}.
$$
We take $\mu=1$ and, in order to assess the robustness of the method in the quasi-incompressible limit, we let $\lambda$ vary in $\{1,10^3,10^6\}$.
For the numerical solution, we consider structured and unstructured triangular, Cartesian orthogonal, and deformed quadrangular mesh families; see Figure \ref{fig:numerical.results:brenner:meshes}.
The solutions corresponding to $\lambda=1$ and $\lambda=10^6$ on the finest Cartesian orthogonal mesh are represented in Figure \ref{fig:numerical.results:brenner:solution}, where we have plotted the components of the global displacement reconstruction obtained from the discrete solution according to \eqref{eq:vph}.
The numerical results are collected in Tables \ref{tab:numerical:results:brenner:tria}--\ref{tab:numerical:results:brenner:distQuad}, where the following quantities are monitored:
$\Ndofs$, the number of degrees of freedom;
$\Nnz$, the number of non-zero entries in the problem matrix;
$\tnorm[\mathrm{a},h]{\uvec{u}_h-\vIh\vec{u}}$, the energy-norm of the error;
$\norm{\vec{u}_h-\vlproj{0}\vec{u}}$, the $L^2$-norm of the error estimated in Theorem \ref{thm:l2.err.est}.
Notice that, in view of Remark \ref{rem:err.est:norm.a.h}, in this and in the following numerical tests the energy error is measured using the $\norm[\mathrm{a},h]{{\cdot}}$-norm, whose computation can be done using the already assembled problem matrix.
We additionally display the Estimated Order of Convergence (EOC) which, denoting by $e_i$ an error measure on the $i$th mesh refinement with meshsize $h_i$, is computed as
$$
{\rm EOC} = \frac{\log e_i - \log e_{i+1}}{\log h_i - \log h_{i+1}}.
$$
In all the cases, the asymptotic EOC match the ones predicted by the theory, that is, $1$ for the energy-norm of the error and $2$ for the $L^2$-norm.
The results additionally highlight the robustness of the method in the quasi-incompressible limit (see Remark \ref{rem:robustness}) and with respect to the mesh, showing errors of comparable magnitude irrespectively of the value of $\lambda$ and of the selected mesh family.
\begin{figure}\centering
\begin{minipage}{0.35\textwidth}\centering
\includegraphics[height=4.25cm]{Tri}
\subcaption{Structured triangular mesh}
\end{minipage}
\begin{minipage}{0.35\textwidth}\centering
\includegraphics[height=4.25cm]{distTri}
\subcaption{Unstructured triangular mesh}
\end{minipage}
\vspace{0.25cm} \\
\begin{minipage}{0.35\textwidth}\centering
\includegraphics[height=4.25cm]{Quad}
\subcaption{Cartesian orthogonal mesh}
\end{minipage}
\begin{minipage}{0.35\textwidth}\centering
\includegraphics[height=4.25cm]{distQuad}
\subcaption{Distorted quadrangular mesh}
\end{minipage}
\caption{Meshes for the numerical test of Section \ref{sec:numerical.tests:brenner}.\label{fig:numerical.results:brenner:meshes}}
\end{figure}
\begin{figure}\centering
\begin{minipage}{0.40\textwidth}\centering
\includegraphics[width=5cm]{brenner_l1_square_10_128_u1}
\subcaption{$\lambda=1$, $u_1$}
\end{minipage}
\begin{minipage}{0.40\textwidth}\centering
\includegraphics[width=5cm]{brenner_l1_square_10_128_u2}
\subcaption{$\lambda=1$, $u_2$}
\end{minipage}
\vspace{0.25cm} \\
\begin{minipage}{0.40\textwidth}\centering
\includegraphics[width=5cm]{brenner_l1e6_square_10_128_u1}
\subcaption{$\lambda=\pgfmathprintnumber{1e6}$, $u_1$}
\end{minipage}
\begin{minipage}{0.40\textwidth}\centering
\includegraphics[width=5cm]{brenner_l1e6_square_10_128_u2}
\subcaption{$\lambda=\pgfmathprintnumber{1e6}$, $u_2$}
\end{minipage}
\caption{Numerical solution for the test of Section \ref{sec:numerical.tests:brenner} on the $128\times128$ Cartesian orthogonal mesh.\label{fig:numerical.results:brenner:solution}}
\end{figure}
\begin{table}\centering
\caption{Numerical results for the test of Section \ref{sec:numerical.tests:brenner}, structured triangular mesh family.\label{tab:numerical:results:brenner:tria}}
\small
\begin{tabular}{cccccc}
\toprule
$\Ndofs$
& $\Nnz$
& $\norm[\mathrm{a},h]{\uvec{u}_h-\vIh\vec{u}}$
& EOC
& $\norm{\vec{u}_h-\vlproj{0}\vec{u}}$
& EOC \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+00}) $ } \\
\midrule
144 & 3680 & 3.82e+00 & -- & 2.08e-01 & -- \\
608 & 17856 & 1.96e+00 & 0.97 & 6.97e-02 & 1.58 \\
2496 & 78080 & 9.64e-01 & 1.02 & 1.87e-02 & 1.90 \\
10112 & 326016 & 4.84e-01 & 1.00 & 4.74e-03 & 1.98 \\
40704 & 1331840 & 2.43e-01 & 1.00 & 1.19e-03 & 1.99 \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+03}) $ } \\
\midrule
144 & 3680 & 5.09e+00 & -- & 2.05e-01 & -- \\
608 & 17856 & 1.95e+00 & 1.38 & 7.15e-02 & 1.52 \\
2496 & 78080 & 9.15e-01 & 1.09 & 2.00e-02 & 1.84 \\
10112 & 326016 & 4.52e-01 & 1.02 & 5.18e-03 & 1.95 \\
40704 & 1331840 & 2.25e-01 & 1.00 & 1.31e-03 & 1.98 \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+06}) $ } \\
\midrule
144 & 3680 & 1.10e+02 & -- & 2.05e-01 & -- \\
608 & 17856 & 1.48e+01 & 2.90 & 7.15e-02 & 1.52 \\
2496 & 78080 & 2.07e+00 & 2.83 & 2.00e-02 & 1.84 \\
10112 & 326016 & 5.08e-01 & 2.03 & 5.19e-03 & 1.95 \\
40704 & 1331840 & 2.27e-01 & 1.16 & 1.31e-03 & 1.98 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}\centering
\caption{Numerical results for the test of Section \ref{sec:numerical.tests:brenner}, unstructured triangular mesh family.\label{tab:numerical:results:brenner:distTria}}
\small
\begin{tabular}{cccccc}
\toprule
$\Ndofs$
& $\Nnz$
& $\norm[\mathrm{a},h]{\uvec{u}_h-\vIh\vec{u}}$
& EOC
& $\norm{\vec{u}_h-\vlproj{0}\vec{u}}$
& EOC \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+00}) $ } \\
\midrule
234 & 6572 & 3.00e+00 & -- & 1.38e-01 & -- \\
978 & 30012 & 1.60e+00 & 0.90 & 4.11e-02 & 1.75 \\
3986 & 127372 & 8.15e-01 & 0.98 & 9.37e-03 & 2.13 \\
15542 & 505828 & 4.27e-01 & 0.93 & 2.61e-03 & 1.85 \\
63584 & 2089920 & 2.12e-01 & 1.01 & 6.65e-04 & 1.97 \\
249238 & 8228988 & 1.08e-01 & 0.97 & 1.71e-04 & 1.96 \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+03}) $ } \\
\midrule
234 & 6572 & 3.57e+00 & -- & 1.45e-01 & -- \\
978 & 30012 & 1.60e+00 & 1.15 & 4.52e-02 & 1.68 \\
3986 & 127372 & 8.00e-01 & 1.00 & 1.07e-02 & 2.07 \\
15542 & 505828 & 4.18e-01 & 0.94 & 2.99e-03 & 1.85 \\
63584 & 2089920 & 2.08e-01 & 1.01 & 7.63e-04 & 1.97 \\
249238 & 8228988 & 1.06e-01 & 0.97 & 1.97e-04 & 1.96 \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+06}) $ } \\
\midrule
234 & 6572 & 6.17e+01 & -- & 1.45e-01 & -- \\
978 & 30012 & 7.55e+00 & 3.03 & 4.52e-02 & 1.68 \\
3986 & 127372 & 1.14e+00 & 2.72 & 1.07e-02 & 2.07 \\
15542 & 505828 & 4.33e-01 & 1.40 & 2.99e-03 & 1.85 \\
63584 & 2089920 & 2.08e-01 & 1.06 & 7.63e-04 & 1.97 \\
249238 & 8228988 & 1.06e-01 & 0.98 & 1.97e-04 & 1.96 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}\centering
\caption{Numerical results for the test of Section \ref{sec:numerical.tests:brenner}, Cartesian orthogonal mesh family.\label{tab:numerical:results:brenner:quad}}
\small
\begin{tabular}{cccccc}
\toprule
$\Ndofs$
& $\Nnz$
& $\norm[\mathrm{a},h]{\uvec{u}_h-\vIh\vec{u}}$
& EOC
& $\norm{\vec{u}_h-\vlproj{0}\vec{u}}$
& EOC \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+00}) $ } \\
\midrule
80 & 2768 & 3.13e+00 & -- & 1.55e-01 & -- \\
352 & 15856 & 1.84e+00 & 0.77 & 4.08e-02 & 1.93 \\
1472 & 73904 & 1.09e+00 & 0.75 & 1.04e-02 & 1.98 \\
6016 & 317488 & 5.89e-01 & 0.89 & 2.89e-03 & 1.84 \\
24320 & 1314608 & 3.02e-01 & 0.97 & 7.73e-04 & 1.90 \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+03}) $ } \\
\midrule
80 & 2768 & 3.08e+00 & -- & 1.64e-01 & -- \\
352 & 15856 & 1.81e+00 & 0.77 & 4.72e-02 & 1.80 \\
1472 & 73904 & 1.08e+00 & 0.75 & 1.37e-02 & 1.78 \\
6016 & 317488 & 5.81e-01 & 0.89 & 3.96e-03 & 1.79 \\
24320 & 1314608 & 2.97e-01 & 0.97 & 1.06e-03 & 1.90 \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+06}) $ } \\
\midrule
80 & 2768 & 3.08e+00 & -- & 1.64e-01 & -- \\
352 & 15856 & 1.81e+00 & 0.77 & 4.72e-02 & 1.80 \\
1472 & 73904 & 1.08e+00 & 0.75 & 1.37e-02 & 1.78 \\
6016 & 317488 & 5.81e-01 & 0.89 & 3.96e-03 & 1.79 \\
24320 & 1314608 & 2.97e-01 & 0.97 & 1.06e-03 & 1.90 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}\centering
\caption{Numerical results for the test of Section \ref{sec:numerical.tests:brenner}, distorted quadrangular mesh family.\label{tab:numerical:results:brenner:distQuad}}
\small
\begin{tabular}{cccccc}
\toprule
$\Ndofs$
& $\Nnz$
& $\norm[\mathrm{a},h]{\uvec{u}_h-\vIh\vec{u}}$
& EOC
& $\norm{\vec{u}_h-\vlproj{0}\vec{u}}$
& EOC \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+00}) $ } \\
\midrule
80 & 2768 & 3.51e+00 & -- & 1.89e-01 & -- \\
352 & 15856 & 1.91e+00 & 0.88 & 5.45e-02 & 1.79 \\
1472 & 73904 & 1.08e+00 & 0.82 & 1.34e-02 & 2.03 \\
6016 & 317488 & 5.83e-01 & 0.89 & 3.52e-03 & 1.93 \\
24320 & 1314608 & 2.97e-01 & 0.97 & 9.18e-04 & 1.94 \\
97792 & 5348656 & 1.49e-01 & 0.99 & 2.33e-04 & 1.98 \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+03}) $ } \\
\midrule
80 & 2768 & 3.44e+00 & -- & 1.96e-01 & -- \\
352 & 15856 & 1.87e+00 & 0.88 & 5.89e-02 & 1.73 \\
1472 & 73904 & 1.07e+00 & 0.81 & 1.63e-02 & 1.85 \\
6016 & 317488 & 5.74e-01 & 0.89 & 4.48e-03 & 1.86 \\
24320 & 1314608 & 2.92e-01 & 0.97 & 1.18e-03 & 1.93 \\
97792 & 5348656 & 1.47e-01 & 0.99 & 3.00e-04 & 1.97 \\
\midrule
\multicolumn{6}{c}{$ (\mu,\lambda) = (\pgfmathprintnumber{1.00e+00},\pgfmathprintnumber{1.00e+06}) $ } \\
\midrule
80 & 2768 & 9.12e+00 & -- & 1.96e-01 & -- \\
352 & 15856 & 2.27e+00 & 2.00 & 5.89e-02 & 1.73 \\
1472 & 73904 & 1.08e+00 & 1.07 & 1.63e-02 & 1.85 \\
6016 & 317488 & 5.74e-01 & 0.91 & 4.48e-03 & 1.86 \\
24320 & 1314608 & 2.92e-01 & 0.97 & 1.18e-03 & 1.93 \\
97792 & 5348656 & 1.47e-01 & 0.99 & 3.00e-04 & 1.97 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Two-dimensional singular case}\label{sec:numerical.tests:singular}
We next consider the solution of \cite[Section 5.1]{Ainsworth.Senior:97} which, in polar coordinates $(r,\theta)$, reads
$$
\renewcommand{\arraystretch}{1.2}
\vec{u}(r,\theta) = \begin{pmatrix}
\frac{1}{2G} r^L\left[ (\kappa - Q(L+1))\cos(L \theta) - L \cos((L-2)\theta)\right]
\\
\frac{1}{2G} r^L\left[ (\kappa + Q(L+1))\sin(L \theta) + L \sin((L-2)\theta)\right]
\end{pmatrix},
$$
where the various parameters take the following numerical values: $\mu=\pgfmathprintnumber{0.65}$, %
$\lambda=\pgfmathprintnumber{0.975}$, %
$G=\frac{5}{13}$, %
$\kappa=\frac{9}{5}$, %
$L=\pgfmathprintnumber[precision=14]{0.5444837367825}$, %
$Q=\pgfmathprintnumber[precision=14]{0.5430755788367}$.
The forcing term in this case is equal to zero, while the Dirichlet boundary condition is inferred from the exact solution.
\begin{figure}\centering
\begin{tikzpicture}[scale=2]
\draw[->] (-2.1,0) -- (2.1,0) node[below]{$x$};
\draw[->] (0,-2.1) -- (0,2.1) node[left]{$y$};
\draw[very thick] (1.414213562373,0) node[above,anchor=north west] {$(\sqrt{2},0)$} %
-- (0,1.414213562373) node[right,anchor=south west] {$(0,\sqrt{2})$} %
-- (-0.707106781186548,0.707106781186548) node[left] {$(-\frac{\sqrt{2}}2,\frac{\sqrt{2}}2)$} %
-- (0,0) node[right,anchor=north west] {$(0,0)$} %
-- (-0.707106781186548,-0.707106781186548) node[left] {$(-\frac{\sqrt{2}}2,-\frac{\sqrt{2}}2)$}
-- (0,-1.414213562373) node[right,anchor=north west] {$(0,-\sqrt{2})$} %
-- (1.414213562373,0);
\end{tikzpicture}
\caption{Domain for the test case of Section \ref{sec:numerical.tests:singular}.\label{fig:numerical.tests:singular:domain}}
\begin{minipage}{0.45\textwidth}\centering
\includegraphics[width=0.85\textwidth]{singular_u1}
\subcaption{$u_1$}
\end{minipage}
\begin{minipage}{0.45\textwidth}\centering
\includegraphics[width=0.85\textwidth]{singular_u2}
\subcaption{$u_2$}
\end{minipage}
\caption{Numerical solution for the test of Section \ref{sec:numerical.tests:singular}.\label{fig:numerical.tests:singular:solution}}
\end{figure}
The domain $\Omega$ is illustrated in Figure \ref{fig:numerical.tests:singular:domain}, while the solution on the finest computational mesh considered here is depicted in Figure \ref{fig:numerical.tests:singular:solution}.
This test case is representative of real-life situations corresponding to a mode 1 fracture in a plain strain problem.
The solution exhibits a singularity in the origin, which prevents the method from attaining the full orders of convergence predicted for smooth solutions.
For the numerical resolution, we consider a sequence of refined structured quadrangular meshes.
The numerical results collected in the top half of Table \ref{tab:numerical.tests:singular} show an asymptotic EOC in the energy-norm of about $0.54$, while the asymptotic EOC in the $L^2$-norm is about $1.31$.
For the sake of completeness, we show, in the bottom half of Table \ref{tab:numerical.tests:singular}, a comparison with the original HHO method of \cite{Di-Pietro.Ern:15} with $k=1$.
Also in this case, the EOC are limited by the regularity of the solution, and coincide with those observed for the method studied in this work.
As expected, the number of unknowns on a given mesh is larger for the method of \cite{Di-Pietro.Ern:15} compared to the method proposed here, despite the fact that static condensation is applied in the former case.
It has to be noticed, however, that the reduction in the number of unknowns is balanced by the increased number of nonzero entries in the matrix, due to both the absence of static condensation (see Remark \ref{rem:static.condensation}) and the presence of the jump penalisation term.
This phenomenon is specific to the two-dimensional case: in dimension $d=3$, the matrix corresponding to the method of \cite{Di-Pietro.Ern:15} with $k=1$ is generally more dense; see, e.g., Table \ref{tab:numerical.tests:3d}.
The errors in the energy norm appear to be smaller for the HHO method of \cite{Di-Pietro.Ern:15}, but this is in part due to the fact that the natural energy norm associated with the corresponding bilinear form does not contain the norm of the jumps.
\begin{table}\centering
\caption{Numerical results for the test of Section \ref{sec:numerical.tests:singular} and comparison with the high-order method of \cite{Di-Pietro.Ern:15} with $k=1$.
For the latter, the energy norm is the one associated to the corresponding bilinear form without jump stabilisation.\label{tab:numerical.tests:singular}}
\small
\begin{tabular}{cccccc}
\toprule
$\Ndofs$
& $\Nnz$
& $\norm[\mathrm{a},h]{\uvec{u}_h-\vIh\vec{u}}$
& EOC
& $\norm{\vec{u}_h-\vlproj{0}\vec{u}}$
& EOC \\
\midrule
\multicolumn{6}{c}{Present work}- \\
\midrule
256 & 10616 & 7.65e-01 & -- & 7.51e-02 & -- \\
1088 & 52728 & 5.63e-01 & 0.44 & 3.34e-02 & 1.17 \\
4480 & 232568 & 3.97e-01 & 0.50 & 1.40e-02 & 1.25 \\
18176 & 974712 & 2.76e-01 & 0.53 & 5.72e-03 & 1.29 \\
73216 & 3988856 & 1.90e-01 & 0.54 & 2.31e-03 & 1.31 \\
293888 & 16136568 & 1.31e-01 & 0.54 & 9.29e-04 & 1.31 \\
\midrule
\multicolumn{6}{c}{HHO method of \cite{Di-Pietro.Ern:15}, $k=1$} \\
\midrule
320 & 7584 & 1.07e-01 & -- & 9.40e-03 & -- \\
1408 & 36512 & 7.32e-02 & 0.55 & 3.64e-03 & 1.37 \\
5888 & 158880 & 5.01e-02 & 0.55 & 1.41e-03 & 1.36 \\
24064 & 661664 & 3.43e-02 & 0.55 & 5.52e-04 & 1.36 \\
97280 & 2699424 & 2.35e-02 & 0.54 & 2.17e-04 & 1.35 \\
391168 & 10903712 & 1.61e-02 & 0.54 & 8.57e-05 & 1.34 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Three-dimensional compressible case}\label{sec:numerical.tests:3d}
To test the performance of the method in three space dimensions, we solve on the unit cube domain $\Omega=(0,1)^3$ the homogeneous Dirichlet problem corresponding to the exact solution $\vec{u}=(u_i)_{1\le i\le d}$ such that
$$
u_i(\vec{x})=\sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3)\qquad\forall 1\le i\le 3.
$$
The corresponding forcing term is
$$
\renewcommand{\arraystretch}{1.2}
\begin{aligned}
\vec{f}(\vec{x})
&=\mu\begin{pmatrix}
2\sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3) - \sin(\pi x_2)\cos(\pi(x_3+x_1)) - \sin(\pi x_3)\cos(\pi(x_1+x_2))
\\
2\sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3) - \sin(\pi x_3)\cos(\pi(x_1+x_2)) - \sin(\pi x_1)\cos(\pi(x_2+x_3))
\\
2\sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3) - \sin(\pi x_1)\cos(\pi(x_2+x_3)) - \sin(\pi x_2)\cos(\pi(x_3+x_1))
\end{pmatrix}
\\
&\quad
+ \lambda\begin{pmatrix}
\sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3) - \cos(\pi x_1)\sin(\pi(x_2+x_3))
\\
\sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3) - \cos(\pi x_2)\sin(\pi(x_3+x_1))
\\
\sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3) - \cos(\pi x_3)\sin(\pi(x_1+x_2))
\end{pmatrix}.
\end{aligned}
$$
For the numerical solution, we take $\mu=\lambda=1$.
Table \ref{tab:numerical.tests:3d} collects the numerical results on Cartesian orthogonal and unstructured simplicial
mesh families.
The monitored quantities are the same as for the other test cases to which we add, for the sake of comparison, the number of unknowns and of nonzero matrix entries for the method of \cite{Di-Pietro.Ern:15} with $k=1$.
For both mesh families, the asymptotic EOC for both the energy- and the $L^2$-norms of the error agree with the ones predicted.
Specifically, on the simplicial mesh family an EOC close to 1 is attained starting from the third mesh refinement in the energy norm, whereas an EOC close to 2 is already observed starting from the second mesh refinement; on the Cartesian orthogonal mesh family, on the other hand, the orders of convergence take longer to settle to the corresponding asymptotic values, likely because the first computational meshes are very coarse.
\begin{table}\centering
\caption{Numerical results for the test of Section \ref{sec:numerical.tests:3d}.
The number of degrees of freedom and of nonzero matrix entries for the method of \cite{Di-Pietro.Ern:15} are also included for comparison (except for the last mesh refinement).
\label{tab:numerical.tests:3d}}
\small
\begin{tabular}{cccccccc}
\toprule
\multicolumn{2}{c}{$\Ndofs$}
& \multicolumn{2}{c}{$\Nnz$}
& \multirow{2}{*}{$\norm[\mathrm{a},h]{\uvec{u}_h-\vIh\vec{u}}$}
& \multirow{2}{*}{EOC}
& \multirow{2}{*}{$\norm{\vec{u}_h-\vlproj{0}\vec{u}}$}
& \multirow{2}{*}{EOC}
\\
$k=0$ & ($k=1$) & $k=0$ & ($k=1$) \\
\midrule
\multicolumn{8}{c}{Cartesian orthogonal mesh sequence} \\
\midrule
60 & (108) & 2772 & (4860) & 2.42e+00 & -- & 1.76e-01 & -- \\
624 & (1296) & 70128 & (97200) & 2.07e+00 & 0.23 & 1.01e-01 & 0.81 \\
5568 & (12096) & 831024 & (1057536) & 1.31e+00 & 0.65 & 4.09e-02 & 1.30 \\
46848 & (103680) & 7879824 & (9673344) & 7.19e-01 & 0.87 & 1.27e-02 & 1.68 \\
384000 & (857088) & 68277456 & (82425600) & 3.71e-01 & 0.95 & 3.46e-03 & 1.88 \\
3108864 & -- & 567808848 & -- & 1.87e-01 & 0.98 & 8.95e-04 & 1.95 \\
\midrule
\multicolumn{8}{c}{Unstructured simplicial mesh sequence} \\
\midrule
1584 & (3024) & 107136 & (167184) & 1.38e+00 & -- & 4.70e-02 & -- \\
13248 & (25920) & 1008288 & (1539648) & 7.61e-01 & 0.85 & 1.64e-02 & 1.52 \\
108288 & (214272) & 8676288 & (13125888) & 3.96e-01 & 0.94 & 4.39e-03 & 1.91 \\
875520 & (1741824) & 71860608 & (108241920) & 2.02e-01 & 0.97 & 1.14e-03 & 1.95 \\
7041024 & -- & 584706816 & -- & 1.02e-01 & 0.99 & 2.89e-04 & 1.98 \\
\bottomrule
\end{tabular}
\end{table}
\section{Local balances and continuity of numerical tractions}\label{sec:flux}
In this section we show that our method satisfies local force balances with equilibrated face tractions.
This property can be exploited, e.g., to derive a posteriori error estimates by flux equilibration, and it makes the proposed method suitable for integration into existing Finite Volume codes.
\begin{lemma}[Traction formulation of the discrete bilinear form]\label{lem:ah:flux}
We have the following reformulation of the discrete bilinear form $\mathrm{a}_h$ defined by \eqref{eq:ah}:
For all $\uvec{w}_h,\uvec{v}_h\in\vUhD$,
\begin{equation}\label{eq:ah:flux}
\mathrm{a}_h(\uvec{w}_h,\uvec{v}_h)
= \sum_{T\in\Th}\sum_{F\in\Fh[T]}\meas{F}\vec{\Phi}_{TF}(\uvec{w}_h){\cdot}(\vec{v}_T-\vec{v}_F),
\end{equation}
where, for all $T\in\Th$ and all $F\in\Fh[T]$, we have introduced the numerical traction $\vec{\Phi}_{TF}:\vUT\to\Poly{0}(F;\mathbb{R}^d)$ such that
$$
\vec{\Phi}_{TF}(\uvec{w}_h)
\coloneq -\tens{\sigma}(\tens{\nabla}_{\rm s}\vpT\uvec{w}_T)\vec{n}_{TF}
+ (2\mu)~\vec{\Phi}_{\mathrm{j},TF}(\uvec{w}_h)
+ (2\mu)~\vec{\Phi}_{\mathrm{s},TF}(\uvec{w}_T),
$$
with jump penalisation and stabilisation contributions respectively defined as
$$
\begin{aligned}
\vec{\Phi}_{\mathrm{j},TF}(\uvec{w}_h)
&\coloneq
\frac{\epsilon_{TF}}{h_F\meas{F}}\int_F\jump{\vph\uvec{w}_h}
+ \sum_{G\in\Fh[T]}
\frac{\epsilon_{TG}}{h_G\meas{T}}
(\overline{\vec{x}}_G-\overline{\vec{x}}_T){\cdot}\vec{n}_{TG}
\int_G\jump[G]{\vph\uvec{w}_h},
\\
\vec{\Phi}_{\mathrm{s},TF}(\uvec{w}_T)
&\coloneq
\frac{1}{h_F}\vec{\delta}_{TF}\uvec{w}_T
+ \sum_{G\in\Fh[T]}\frac{\meas{G}}{h_G\meas{T}}(\overline{\vec{x}}_T-\overline{\vec{x}}_G){\cdot}\vec{n}_{TF}~\vec{\delta}_{TG}\uvec{w}_T,
\end{aligned}
$$
where, for any $X$ mesh element or face, we have denoted by $\overline{\vec{x}}_X\coloneq\frac{1}{\meas{X}}\int_X\vec{x}$ its centroid and, for any $T\in\Th$ and any $F\in\Fh[T]$, $\epsilon_{TF}\coloneq\vec{n}_{TF}{\cdot}\vec{n}_F$ defines the orientation of $F$ relative to $T$.
\end{lemma}
\begin{proof}
We proceed to reformulate the three terms in the right-hand side of \eqref{eq:ah} in order to highlight the corresponding contribution to the numerical traction.
For the consistency term, we can write
$$
\begin{aligned}
(\tens{\sigma}(\tens{\nabla}_{{\rm s},h}\vph\uvec{w}_h),\tens{\nabla}_{{\rm s},h}\vph\uvec{v}_h)
&=
\sum_{T\in\Th}\meas{T}\tens{\sigma}(\tens{\nabla}_{\rm s}\vpT\uvec{w}_T){:}\tens{\nabla}_{\rm s}\vpT\uvec{v}_T
=
\sum_{T\in\Th}\meas{T}\tens{\sigma}(\tens{\nabla}_{\rm s}\vpT\uvec{w}_T){:}\vec{\nabla}\vpT\uvec{v}_T
\\
&=
\sum_{T\in\Th}\meas{T}\tens{\sigma}(\tens{\nabla}_{\rm s}\vpT\uvec{w}_T){:}\left(
\sum_{F\in\Fh[T]}\frac{\meas{F}}{\meas{T}}(\vec{v}_F-\vec{v}_T)\otimes\vec{n}_{TF}
\right)
\\
&=
-\sum_{T\in\Th}\sum_{F\in\Fh[T]}
\meas{F}\tens{\sigma}(\tens{\nabla}_{\rm s}\vpT\uvec{w}_T)\vec{n}_{TF}{\cdot} (\vec{v}_T-\vec{v}_F),
\end{aligned}
$$
where we have used the fact that, for any $T\in\Th$, both $\tens{\sigma}(\tens{\nabla}_{\rm s}\vpT\uvec{w}_T)$ and $\tens{\nabla}_{\rm s}\vpT\uvec{v}_T$ are constant inside $T$ along with the fact that $\tens{\sigma}(\tens{\nabla}_{\rm s}\vpT\uvec{w}_T)$ is symmetric to replace $\tens{\nabla}_{\rm s}$ with $\vec{\nabla}$ in the first line,
the first relation in \eqref{eq:vpT} to pass to the second line,
and we have rearranged the products and sums to conclude.
For the jump penalisation term, we can start by observing that
$$
\begin{aligned}
\mathrm{j}_h(\uvec{w}_h,\uvec{v}_h)
&=
\sum_{F\in\Fh}\frac{1}{h_F}\left(\jump{\vph\uvec{w}_h},\jump{\vph\uvec{v}_h}\right)_F
\\
&=
\sum_{F\in\Fh}\sum_{T\in\Th[F]}\frac{\epsilon_{TF}}{h_F}\left(\jump{\vph\uvec{w}_h},\vpT\uvec{v}_T\right)_F
=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\frac{\epsilon_{TF}}{h_F}\left(\jump{\vph\uvec{w}_h},\vpT\uvec{v}_T\right)_F,
\end{aligned}
$$
where we have used the definition of the jump operator to pass to the second line and exchanged the sums over elements and faces according to \eqref{eq:sumTF=sumFT} to conclude.
Using the explicit expression \eqref{eq:vpT:bis} of the local displacement reconstruction, we can go on writing
$$
\begin{aligned}
\mathrm{j}_h(\uvec{w}_h,\uvec{v}_h)
&=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\frac{\epsilon_{TF}}{h_F}\left(
\jump{\vph\uvec{w}_h},\vec{v}_T + \sum_{G\in\Fh[T]}\frac{\meas{G}}{\meas{T}}(\vec{x}-\overline{\vec{x}}_T){\cdot}\vec{n}_{TF}(\vec{v}_G-\vec{v}_T)
\right)_F
\\
&=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\frac{\epsilon_{TF}}{h_F}\left(\jump{\vph\uvec{w}_h},\vec{v}_T-\vec{v}_F\right)_F
\\
&\qquad
+ \sum_{T\in\Th}\sum_{F\in\Fh[T]}\sum_{G\in\Fh[T]}\frac{\epsilon_{TF}\meas{G}}{h_F\meas{T}}\left(
\jump{\vph\uvec{w}_h}(\vec{x}-\overline{\vec{x}}_T){\cdot}\vec{n}_{TF},\vec{v}_G-\vec{v}_T
\right)_F
\\
&=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\meas{F}\left(\frac{\epsilon_{TF}}{h_F\meas{F}}\int_F\jump{\vph\uvec{w}_h}\right){\cdot}(\vec{v}_T-\vec{v}_F)
\\
&\qquad
- \sum_{T\in\Th}\sum_{G\in\Fh[T]}\meas{G}\left(
\sum_{F\in\Fh[T]}\frac{\epsilon_{TF}}{h_F\meas{T}}\int_F\jump{\vph\uvec{w}_h}(\vec{x}-\overline{\vec{x}}_T){\cdot}\vec{n}_{TF}
\right){\cdot}(\vec{v}_T-\vec{v}_G)
\\
&=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\meas{F}\vec{\Phi}_{\mathrm{j},TF}(\uvec{w}_h){\cdot}(\vec{v}_T-\vec{v}_F),
\end{aligned}
$$
where, to insert $\vec{v}_F$ into the first term in the second line, we have used the fact that $\jump{\vph\uvec{w}_h}$ is single-valued at interfaces together with $\vec{v}_F=\vec{0}$ on boundary faces,
to pass to the third line we have used the fact that the discrete unknowns in $\uvec{v}_h$ are constant over mesh elements to take them out of the integrals over faces
while, to conclude, we have observed that $(\vec{x}-\overline{\vec{x}}_T){\cdot}\vec{n}_{TF}=(\overline{\vec{x}}_F-\overline{\vec{x}}_T){\cdot}\vec{n}_{TF}$ for all $\vec{x}\in F$ and we have used the definition of $\vec{\Phi}_{\mathrm{j},TF}(\uvec{w}_h)$ after switching the names of the mute variables $F$ and $G$ in the second term of the third line.
Moving to the stabilisation term, we can write
$$
\begin{aligned}
\mathrm{s}_h(\uvec{w}_h,\uvec{v}_h)
&=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\frac{\meas{F}}{h_F}\vec{\delta}_{TF}\uvec{w}_T{\cdot}\vec{\delta}_{TF}\uvec{v}_T
\\
&=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\frac{\meas{F}}{h_F}\vec{\delta}_{TF}\uvec{w}_T{\cdot}\left(
\vec{v}_T - \vec{v}_F + \sum_{G\in\Fh[T]}\frac{\meas{G}}{\meas{T}}(\vec{v}_G-\vec{v}_T)(\overline{\vec{x}}_F-\overline{\vec{x}}_T){\cdot}\vec{n}_{TG}
\right)
\\
&=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\frac{\meas{F}}{h_F}\vec{\delta}_{TF}\uvec{w}_T{\cdot}(\vec{v}_T - \vec{v}_F)
\\
&\qquad
+ \sum_{T\in\Th}\sum_{G\in\Fh[T]}\meas{G}\left(
\sum_{F\in\Fh[T]}\frac{\meas{F}}{h_F\meas{T}}(\overline{\vec{x}}_T-\overline{\vec{x}}_F){\cdot}\vec{n}_{TG}~\vec{\delta}_{TF}\uvec{w}_T
\right){\cdot}(\vec{v}_T-\vec{v}_G)
\\
&=
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\meas{F}\vec{\Phi}_{\mathrm{s},TF}(\uvec{w}_T)
{\cdot}(\vec{v}_T-\vec{v}_F),
\end{aligned}
$$
where we have used the definition \eqref{eq:dTF} of the boundary difference operator together with the explicit expression \eqref{eq:vpT:bis} of the local displacement reconstruction to pass to the second line,
we have rearranged the terms to pass to the third line,
and we have used the definition of $\vec{\Phi}_{\mathrm{s},TF}(\uvec{w}_T)$ after switching the names of the mute variables $F$ and $G$ in the second term of the third line to conclude.
\end{proof}
\begin{corollary}[Local balances and equilibrated tractions]
Under the assumptions and notations of Lemma \ref{lem:ah:flux}, we have that $\uvec{u}_h\in\vUhD$ solves the discrete problem \eqref{eq:discrete} if and only if:
For all $T\in\Th$ the following balance holds
\begin{equation}\label{eq:balance}
\sum_{F\in\Fh[T]}\meas{F}\vec{\Phi}_{TF}(\uvec{u}_h) = \int_T\vec{f},
\end{equation}
and, for any interface $F\in\Fh^{{\rm i}}$ shared by the mesh elements $T_1$ and $T_2$, it holds that
\begin{equation}\label{eq:equilibrated.tractions}
\vec{\Phi}_{T_1F}(\uvec{u}_h) + \vec{\Phi}_{T_2F}(\uvec{u}_h) = \vec{0}.
\end{equation}
\end{corollary}
\begin{proof}
Plugging the flux reformulation \eqref{eq:ah:flux} of the bilinear form $\mathrm{a}_h$ into the discrete problem \eqref{eq:discrete}, and recalling \eqref{eq:vh}, we infer that it is equivalent to:
Find $\uvec{u}_h\in\vUhD$ such that
\begin{equation}\label{eq:discrete:flux}
\sum_{T\in\Th}\sum_{F\in\Fh[T]}\meas{F}\vec{\Phi}_{TF}(\uvec{u}_h){\cdot}(\vec{v}_T-\vec{v}_F)
= \sum_{T\in\Th}\int_T\vec{f}{\cdot}\vec{v}_T
\qquad\forall\uvec{v}_h\in\vUhD.
\end{equation}
Taking, for a given mesh element $T\in\Th$, $\uvec{v}_h$ such that $\vec{v}_{T'}=\vec{0}$ for all $T'\in\Th\setminus\{T\}$, $\vec{v}_F=\vec{0}$ for all $F\in\Fh$, and letting $\vec{v}_T$ span $\Poly{0}(T;\mathbb{R}^d)$, \eqref{eq:discrete:flux} reduces to \eqref{eq:balance}.
Similarly, given an interface $F\in\Fh^{{\rm i}}$ shared by the mesh elements $T_1$ and $T_2$, taking in \eqref{eq:discrete:flux} $\uvec{v}_h$ such that $\vec{v}_T=\vec{0}$ for all $T\in\Th$, $\vec{v}_{F'}=\vec{0}$ for all $F'\in\Fh\setminus\{F\}$, and letting $\vec{v}_F$ span $\Poly{0}(F;\mathbb{R}^d)$, \eqref{eq:discrete:flux} reduces to \eqref{eq:equilibrated.tractions} after recalling that the numerical tractions are constant over $F$.
\end{proof}
\section*{Acknowledgements}
The work of the second author was partially supported by \emph{Agence Nationale de la Recherche} grants HHOMM (ANR-15-CE40-0005) and fast4hho (ANR-17-CE23-0019).
\bibliographystyle{plain}
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Q: Union of the graphs of derivatives of a function. For a function $ f: \mathbb{R} \to \mathbb{R} $, write Graph($f$) $:= \{(x,f(x)):x\in \mathbb{R}\} $.
For $ f \in C^{\infty}(\mathbb{R}) $, define $ A(f) := \bigcup\limits_{n=0}^{\infty} \textrm{Graph}(f^{(n)}) $.
It is clear that $ g = f^{(n)} $ implies $ A(g) \subset A(f) $. The question is, is the converse also true? That is, does $ A(g) \subset A(f) $ imply $ g = f^{(n)} $ for some $ n \geq 0 $?
A: Let $g(x)=e^{- \frac 1 {x^2}}$ for $x \neq 0$, $g(0)=0$. Then $g \in C^{\infty} (\mathbb R)$ and all derivatives of g vanish at 0. Let $f(x)=g(x)$ for $x \leq 0$ and $f(x)=g'(x)$ for $x>0$. For any n and x $(x,f^{(n)} (x))=(x,g^{(n)} (x))$ or $(x,f^{(n)} (x))=(x,g^{(n+1)} (x))$ so $A(f) \subset A(g)$. However, there is no n such that $f=g^{(n)}$. (Sorry, I switched f and g!)
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{"url":"https:\/\/testbook.com\/question-answer\/at-a-sluice-gate-across-a-rectangular-channel-the--5ecd6172f60d5d76e0aa7739","text":"# At a sluice gate across a rectangular channel, the upstream flow conditions are: depth of 2.0 m; velocity of flow of 1.25 m\/sec. The flow conditions at the vena contracta just downstream of the gate can be taken as: depth of 0.44 m; velocity of flow of 5.68 m\/sec. What is the total thrust on the gate on its upstream face (to the nearest 10 units)?\n\nFree Practice With Testbook Mock Tests\n\n1. 770 kgf\n\n2. 800 kgf\n\n3. 825 kgf\n\n4. 870 kgf\n\n## Solution:\n\nConcept:\n\nThe thrust \u2018F\u2019 is given by,\n\n$$F = \\frac{{VB{{\\left( {{y_1} - {y_2}} \\right)}^3}}}{{2\\;\\left( {{y_1} + {y_2}} \\right)}}$$\n\nCalculation:\n\nGiven, y1 = 2 m, V1 = 1.25 m\/s, y2 = 0.44 m, V2 = 5.68 m\/s\n\n$$F = \\frac{{VB{{\\left( {{y_1} - {y_2}} \\right)}^3}}}{{2\\;\\left( {{y_1} + {y_2}} \\right)}}$$\n\nFor unit width i.e. b = 1 m\n\n$$F = \\frac{{9.81 \\times 1 \\times {{\\left( {2 - 0.44} \\right)}^3}}}{{2\\left( {2 + 0.44} \\right)}} = 7.595\\;kN\/m\\;$$\n\n\u2234\u00a0$$F = \\frac{{7.595 \\times {{10}^3}}}{{9.81}} = 774\\;Kgf\/m$$\n\n\u2234 Option 1 is more appropriate.","date":"2021-08-05 05:37:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7626301050186157, \"perplexity\": 3512.1052381457343}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046155322.12\/warc\/CC-MAIN-20210805032134-20210805062134-00103.warc.gz\"}"} | null | null |
Porto dos Milagres é uma telenovela brasileira produzida e exibida pela TV Globo de 5 de fevereiro a 29 de setembro de 2001, em 203 capítulos. Substituiu Laços de Família e foi substituída por O Clone, sendo a 60ª "novela das oito" exibida pela emissora.
Escrita por Aguinaldo Silva e Ricardo Linhares com a colaboração de Nelson Nadotti, Filipe Miguez, Glória Barreto e Maria Elisa Berredo, foi inspirada nos romances Mar Morto e A Descoberta da América pelos Turcos, ambos do escritor brasileiro Jorge Amado, que faleceu enquanto a novela foi exibida. A direção foi de Fabrício Mamberti e Luciano Sabino, sob a direção-geral de Roberto Naar e Marcos Paulo, também diretor de núcleo.
Contou com as participações de Marcos Palmeira, Flávia Alessandra, Antônio Fagundes, Cássia Kis, Luíza Tomé, Leonardo Brício, Camila Pitanga e Arlete Salles.
Algumas controvérsias envolvendo a produção aconteceram durante e depois de sua exibição, como o núcleo reduzido de atores negros apesar da trama ser ambientada na Bahia, além de membros de Igreja Católica criticarem a TV Globo pelo culto ao candomblé dentro da novela. Políticos brasileiros também expressaram descontentamento com a trama, devido a enredos serem muito parecidos com o que acontecia na vida real, com personagens representando casos verídicos de escândalos. Apesar disso, Porto dos Milagres foi um sucesso de audiência, conquistando uma média de 44 pontos em geral e sendo geralmente bem recebida pela crítica especializada, mas que apontou que a dupla de autores estava defasada ao escrever novamente um folhetim de realismo fantástico.
Antecedentes
Na virada do milênio, o horário de novelas das oito estava obtendo recordes de audiência exibindo novelas como Terra Nostra e Laços de Família, depois do relativo fracasso de Suave Veneno, de autoria de Aguinaldo Silva. Para substituir Laços de Família, foi escolhida Segredos do Mar, de autoria de Silva e Ricardo Linhares, depois renomeada para Porto dos Milagres, para manter a boa audiência que a novela estava rendendo à emissora. Silva, a respeito de manter os bons índices da novela anterior, comentou: "Não podemos ficar preocupados com essa herança. Novela quando acaba é como jornal de ontem. No começo, há um estranhamento, mas a saída é fazer algo bem diferente", afirmando também que o segredo para não deixar a "bola cair" é marcar logo a diferença. Marcos Paulo, diretor-geral de Porto dos Milagres, afirmou também que "não dá para comparar. São produtos completamente diferentes. E estamos apostando nessa diferença. A nossa expectativa é de, no mínimo, entrar no mesmo barco".
Produção
Inspiração
Para escrever a trama, Aguinaldo Silva citou como inspiração dois livros de Jorge Amado, Mar Morto e A Descoberta da América pelos Turcos: "Eu queria adaptar 'Mar Morto' havia muito tempo. Li o livro quando era adolescente, tinha uma ideia de como era aquela história de amor desenfreado. Quando o reli, percebi que ele tinha uma linguagem poética meio datada e que sua história não daria uma novela. Então tive que fazer uma adaptação muito livre", afirmou ele. Mar Morto estava prevista para ir ao ar como uma minissérie em 1995, mas foi cancelada pela TV Globo antes de estrear, pois a emissora queria transformar o folhetim em novela no ano seguinte, mas isso não se concretizou. O autor também comentou que também se inspirou em uma história contada por um amigo pecuarista de Barretos, cidade do interior do estado de São Paulo. Quando este contou a história a Aguinaldo, achou apropriada para uma novela. Logo depois, Marluce Dias da Silva, diretora geral da TV Globo na época, chamou-o para continuar a adaptação de Mar Morto, e ele acrescentou esta história à trama.
Silva disse também que a autonomia de adaptar livros às novelas deu oportunidade de inserir temas políticos: "A trama mais forte de Porto dos Milagres era a política. Foi o fato de eu ter transformado o personagem do Antonio Fagundes, o Félix, em prefeito e candidato a governador, casado com uma mulher ambiciosa, a Adma, vivida pela Cássia Kiss, que deu o tom da novela". Ele complementou dizendo que isso era uma ferrenha crítica aos métodos que alguns políticos utilizam no Brasil. Assim, a "trama política prevaleceu sobre a romântica. Confesso que não tinha muita paciência com a trama romântica. Achava que aquela relação do pescador com a moça de classe média era uma história dos anos 1930, 1940. Hoje o romantismo não caminha mais por aí", comentou o autor. Outra inspiração para os autores foi Macbeth, de William Shakespeare, para a trama em que Félix Guerrero (Antônio Fagundes), que ouve de uma cigana a previsão de que se tornaria rei, com as mãos sujas de sangue, tendo uma grande mulher por trás, Adma (Cássia Kis).
Filmagens
Em outubro de 2000, Porto dos Milagres entrou em sua fase de pré-produção e o elenco participou de um workshop, assistindo a um vídeo no qual o próprio Jorge Amado fala sobre sua obra Mar Morto. No mês seguinte, Antônio Fagundes e Cássia Kis, juntamente com a equipe, embarcaram para Sevilha, Espanha, para gravar as primeiras cenas da novela, e no final do mesmo mês, as gravações no Brasil começaram na ilha de Comandatuba, na Bahia. Para as gravações no litoral, o elenco era deslocado a cada mês para a Bahia. "Lá tem o cheiro do mar, que não podemos reproduzir em estúdio. Também não podíamos gravar tudo fora, como Tropicaliente [gravada em Fortaleza, pois é grande o núcleo da cidade", disse o diretor geral Marcos Paulo.
A imprensa chamou a novela de a "maior superprodução da história dramatúrgica brasileira", com custos de mais de US$ 80 mil por capítulo. Foi também especulado que o custo total da novela poderia chegar a US$ 8 milhões, mas que esse valor poderia ultrapassar os US$ 11 milhões, dependendo da aceitação do público e, consequentemente, do número de capítulos. Esse custo elevado foi atribuído ao uso de efeitos especiais na trama, utilizados nas cenas de barco em alto-mar – como, por exemplo, quando, no início da trama, o personagem Guma (Marcos Palmeira) enfrenta uma tempestade – foram realizadas em parceria com a empresa norte-americana de Los Angeles Digital Domain, responsável por alguns dos efeitos de filmes como Titanic e O Segredo do Abismo.
Escolha do elenco
Originalmente, os irmãos Félix e Bartolomeu não seriam gêmeos, mas devido à desistência de um dos atores, Antônio Fagundes ficou encarregado de dar vida aos dois irmãos. Inicialmente, Arlete Salles viveria a personagem Rita, uma mulher simples e batalhadora, boa mãe, pilar da família, e Joana Fomm ficaria com o papel da aristocrata e arrogante Augusta Eugênia. Para o livro Autores, Histórias da Teledramaturgia, Aguinaldo Silva declarou que o diretor Marcos Paulo propôs a troca, fazendo Fomm e Salles interpretarem personagens diferentes do que estavam acostumadas a fazer na televisão. A princípio, o ator Francisco Cuoco interpretaria o senador Victório Vianna, mas o personagem acabou sendo interpretado por Lima Duarte. Pitágoras Williams Mackenzie, o deputado que Ary Fontoura interpretou na novela A Indomada (1997) seria apenas uma participação, mas acabou permanecendo como personagem fixo. Para interpretar Olímpia, a madrasta de Adma e Amapola na reta final da novela, foi inicialmente escalada a atriz Eva Wilma, mas logo depois foi substituída por Débora Duarte.
Cenografia
Duas cidades cenográficas de dez mil metros quadrados foram construídas para formar a fictícia cidade de Porto dos Milagres. A primeira delas, com na Ilha de Comandatuba, representava a parte baixa da cidade, onde ficam o cais, o nicho de Iemanjá, a capelinha, o bar Farol das Estrelas, o Mercado Municipal, as casas de Guma e dos outros pescadores. A estátua de Iemanjá que aparece na trama foi feita pela equipe de artesãos dos Estúdios Globo. A construção da cidade levou cerca de dois meses e o desenho foi feito em tempo recorde, duas semanas. Foi preciso construir depois o vazamento da cidade, bem como a parte alta da mesma nos estúdios, no Rio de Janeiro, ampliando o cenário em . Assim, várias casas construídas na cidade da Ilha de Comandatuba tiveram que ser reproduzidas também no Rio de Janeiro - como o bar Farol das Estrelas, que tem duas entradas: uma na Bahia e outra no Rio de Janeiro - para que uma parte se funda à outra. Várias cenas se passaram no Hotel Transamérica, situado na ilha. A cidade cenográfica foi mantida durante anos pelo hotel, e podia ser visitada pelos hóspedes, mas foi demolida depois.
Para o palacete dos Guerrero foi construído um minarete de dezesseis metros de altura, que corresponde a uma construção de cerca de cinco andares. De qualquer ponto da cidade, é possível ver a mansão. Além disso, a casa tem influências mouras e alguns leves toques espanhóis. Os mosaicos da casa foram feitos com pastilhas de verdade, ao invés das cenográficas. A decoração na primeira fase da novela é mais "masculina". Para a segunda fase, foi dado um toque mais "feminino", como se Adma (Cássia Kis) tivesse feito as modificações. Outras construções foram feitas na cidade baiana de Canavieiras, principalmente na ruela que serviu de locação para as cenas do bordel de Dona Coló (Glória Menezes). Com a casa de Augusta Eugênia (Arlete Salles) e sua família, a cenografia tratou de acompanhar a decadência da família. Na primeira fase, ela é brilhante, ornado com quadros, cortinas e móveis novos. Na segunda, sua casa dá mostras do empobrecimento dos personagens: as cortinas levaram um banho de chá para parecerem mais velhas e os móveis estão envelhecidos. A residência de Amapola (Zezé Polessa) apresenta uma arquitetura baseada na cidade grande e a decoração é mais moderna. Já o núcleo dos pescadores é simples e não tem alvenaria até o teto.
Figurino
O figurinista Lessa de Lacerda foi o encarregado de compor o visual dos personagens, seguindo a linha dos autores e diretores da novela. "Em novela de Aguinaldo e Ricardo, a gente pode dar passos acima dos usuais. É um universo muito rico para ficar apenas em figurinos realistas. Nós vamos além", disse ele. Todas as roupas sofreram a influência do local, com a "temperatura da Bahia". Podem ser detalhes sutis, como os brincos da personagem de Flávia Alessandra, ou roupas extremas e que chamam atenção para a moda da cidade. Lessa também destacou as personagens de Camila Pitanga, descrevendo-a como florida, sensual e que está sempre criando roupas, amarrando panos, compondo decotes, e de Flávia Alessandra, que é mais moderna e, embora use roupas leves, coloridas, de materiais modernos como microfibra e sintéticos, também faz uso dos biquínis, cangas e faz um intercâmbio entre a cidade grande e Porto dos Milagres.
O personagem de Marcos Palmeira segue a linha dos pescadores e, além de sua guia no pescoço, tem uma tatuagem de Iemanjá no braço esquerdo. Como a personagem de Zezé Polessa adorava gastar e comprar roupas de grife, Lessa fez uso exagerado das cores: turquesa, azul claro, verdes, amarelos, rosa, "praticamente um arco-íris". Para as personagens de Júlia Lemmertz e Mônica Carvalho, que interpretavam irmãs, o figurino marcou os contrastes de suas personalidades: a primeira usava roupas comportadas e a segunda, ganhou um tom sensual porém dissimulada, com flores com bordados, mas roupas justas que deixavam à mostra suas formas. Leontina (Louise Cardoso), tia de Lívia, levava "um figurino déjà vu, meio Blanche du Bois".
Enredo
Em Porto dos Milagres, pequena cidade localizada no litoral da Bahia, chegam Félix Guerrero (Antônio Fagundes) e Adma (Cássia Kis), um casal de golpistas que, após uma cigana profetizar que Félix, ao atravessar o mar, tornar-se-á rei, fogem da Espanha. Lá, vive seu irmão gêmeo, Bartolomeu (Antônio Fagundes), que após ter sofrido um golpe do irmão no passado, recepciona ele e a cunhada mal. Apesar disso, ele construiu um império na cidade, e Adma mata o cunhado com um forte raticida sem Félix saber, e ele então assume os negócios de seu irmão. No entanto, Bartolomeu havia se envolvido com a prostituta Arlete (Letícia Sabatella), que descobre estar esperando um filho dele. Arlete, após o nascimento do menino, vai com este à mansão Guerrero e é recepcionada por Adma; desejando eliminar os herdeiros do cunhado, Adma manda o capataz Eriberto (José de Abreu) livrar-se deles, mais uma vez sem contar a Félix. Eriberto os leva para alto-mar, mas Arlete o engana durante uma distração e coloca o menino em um cesto, que coloca na água, e mata-se afogada ao pular no mar.
Guiado por Iemanjá, o cesto é levado pelas ondas até perto do barco do pescador Frederico (Maurício Mattar), que estava fazendo o difícil parto de sua esposa Eulália (Cristiana Oliveira), cujo filho nasce morto. Entretanto, Frederico ouve um choro de criança e encontra no mar o cesto com o bebê de Arlete. Acreditando que o menino é uma bênção de Iemanjá, o pescador resgata-o e mostra-o a Eulália como se fosse o filho deles. Enfraquecida pelo parto, Eulália batiza o menino como Gumercindo e morre em seguida. Frederico então passa a criar o menino como seu filho. Pouco tempo depois, durante uma noite de tempestade, Frederico desaparece no mar. Seu irmão Francisco (Tonico Pereira) e sua mulher Rita (Joana Fomm) assumem a criação do sobrinho. Laura (Carolina Kasting) é a irmã mais nova da doce Leontina (Louise Cardoso) e da prepotente Augusta Eugênia (Arlete Salles). Renegando sua família, a jovem se casa com o pescador Leôncio (Tuca Andrada), com quem tem uma filha, Lívia. Inconformada com o casamento da irmã com um simples pescador, Augusta denuncia Leôncio à polícia por contrabando, sem saber que Laura estava no barco. A polícia aborda Leôncio para prendê-lo e o casal morre, depois que um tiro dado pelos policiais causa a explosão do motor do barco. A família então assume a criação da sobrinha. Rosa Palmeirão (Luiza Tomé), irmã de Arlete, está prestes a se casar com Otacílio (Eduardo Galvão), e no dia de seu casamento com este, mata o coronel Jurandir de Freitas (Reginaldo Faria) que violentara a sua irmã mais nova. Rosa é condenada a 20 anos de prisão, e diz que ao sair da prisão dará uma reviravolta em sua vida.
Passados vinte anos, Lívia (Flávia Alessandra) deixa a cidade do Rio de Janeiro e vai com o seu namorado Alexandre (Leonardo Brício), filho de Félix e Adma, para Porto dos Milagres. Já na cidade, ela conhece Gumercindo (Marcos Palmeira), mais conhecido como Guma, que se tornou um homem de caráter e um líder respeitado na cidade baixa. Os dois se apaixonam, tendo que passarem por poucas e boas para concretizar esse amor, como a hostilidade de Alexandre, que não se conforma em perder Lívia para um pescador, e a ambição de sua tia Augusta Eugênia, que quer ver sua sobrinha casada com o herdeiro de Félix, e ainda ter que lidar com as armações da bela e sedutora Esmeralda (Camila Pitanga), moça apaixonada pelo pescador. Já Rosa Palmeirão deixa a cadeia disposta a descobrir o paradeiro do filho de Arlete, e se instala na cidade abrindo um bordel, conhecido por todos como Centro Noturno de Lazer. Otacílio, com quem ela pretendia casar, se casou com Amapola, com quem teve dois filhos. Rosa pensa que Félix é o culpado pelo desaparecimento de sua irmã e seu sobrinho e torna-se amante dele, mas se apaixona por ele no processo.
Exibição
Porto dos Milagres foi a 60a telenovela a ser exibida no horário das oito pela TV Globo, estreando em 5 de fevereiro de 2001. Seu último capítulo foi exibido em 28 de setembro do mesmo ano, uma sexta-feira, com uma reprise indo ao ar no dia seguinte.
Em 2005, a emissora planejou reexibir a telenovela no Vale a Pena Ver de Novo, porém teve que desistir após o veto do Ministério da Justiça para o horário. Em seu lugar entrou a reprise de Força de um Desejo. No ano seguinte a TV Globo tentou novamente reexibir a novela, mas devido ao mesmo fato, Porto dos Milagres deu lugar à reexibição de A Viagem.
Foi reexibida na íntegra pelo Canal Viva, de 11 de fevereiro a 4 de outubro de 2019, substituindo a segunda reexibição de Vale Tudo e sendo substituída por Cabocla, às 15h30. Além disso, Porto dos Milagres foi vendida para diversos países, entre eles Portugal onde foi exibida pela SIC, México pela MVS Televisión, Angola e Moçambique pela Globo On.
Outras mídias
No dia 1 de fevereiro de 2021, Porto dos Milagres foi disponibilizada na plataforma digital Globoplay.
Elenco
Participações especiais
Música
Volume 1
A primeira trilha sonora da telenovela foi lançada em 2001 pela Som Livre. A capa trouxe apenas o logo da novela.
Volume 2
A segunda trilha sonora da novela foi também lançada em 2001 pela Som Livre. Flávia Alessandra ilustrou a capa do álbum.
Repercussão
Audiência
A novela estreou com uma média de 47 pontos, com picos de 50, superando assim a sua antecessora, Laços de Família, que havia estreado com média de 45 pontos, medida pelo Instituto Brasileiro de Opinião Pública e Estatística (IBOPE). Durante suas três primeiras semanas de exibição, Porto dos Milagres competiu diretamente com a novela mexicana Esmeralda, exibida pelo SBT, durante meia hora. Durante o tempo em que ficaram simultaneamente no ar, de 5 a 20 de fevereiro, a novela registrou 38 pontos no IBOPE, considerado pouco para os padrões da TV Globo, contra 18 da novela mexicana. Em 26 de julho, com a prisão de Rodrigo (Kadu Moliterno) na ocasião de seu casamento com a professora Dulce (Paloma Duarte), a novela marcou o maior índice de audiência até então: 59 pontos. Desde o início da trama ela não havia marcado mais que 53 pontos até aquele momento. O último capítulo de Porto dos Milagres, exibido em 28 de setembro de 2001, marcou uma média de 61 pontos, e picos de 65, números não vistos desde 1999 com o último capítulo de Torre de Babel, que obteve a mesma média. A trama registrou uma média geral de 44 pontos, com 62,5% de televisores ligados na novela ao horário de sua exibição.
Apesar da audiência elevada, foi notado pela imprensa especializada que a novela não foi um sucesso de repercussão com o público. Kadu Moliterno, intérprete de Rodrigo, defendeu que "O que acontece com 'Porto' é por ser uma novela voltada para o entretenimento. Não é uma novela de fofocas". Ele acredita que a antecessora, Laços de Família, tinha um maior efeito junto ao público por tratar de assuntos do cotidiano. "Então, as pessoas discutiam a questão social e de saúde no dia seguinte. Já agora, as pessoas se divertem na hora e depois acabou", conclui o ator.
Avaliação em retrospecto
Amelia Gonzalez dO Globo fez uma avaliação positiva à trama, dizendo que sua sonoridade foi "uma das marcas mais felizes" de seu primeiro capítulo através do som da castanhola embalando as cenas do casal Adma e Félix na Espanha, além da percussão baiana que se mesclou bem à fotografia de um pescador jogando sua rede no mar para mostrar que a dupla chegava ao Brasil. A autora também complementou dizendo "elenco e cenários não deixam a desejar", elogiando a atuação de Arlete Salles como Augusta Eugênia, e dizendo que a novela "crava um gol" quando "dá de presente ao público uma dose dupla de Antônio Fagundes. O ator, que interpreta os irmãos gêmeos Bartolomeu e Félix, mostra que a maturidade só tem lhe acrescentado talento. Não deu para desgrudar os olhos da tela quando Bartolomeu viu Félix".
Patricia d'Abreu do Jornal do Brasil também escreveu uma crítica positiva na estreia da novela, elogiando os desempenhos de Fagundes, Salles, Cássia Kis e José de Abreu, além de classificar a atuação de Natália Thimberg como a "boa surpresa da noite": "Quase sempre na pele de damas - a terrível governanta Juliana de A Sucessora tinha ares mais madamescos que serviçais - Natália emprestou seu estilo elegante de interpretar à submissão da sábia empregada Ondina e a mistura funcionou", comentou ela. Finalizou sua crítica dizendo que ao todo, a estreia da novela parecia ter uma "trinca poderosa" - "atores experientes em primeiro plano, direção cuidadosa e um autor que domina com criatividade seu tema. Ingredientes que fazem Greenville, Asa Branca ou Porto dos Milagres não precisarem de Sevilha". Leila Reis do Estadão escreveu uma crítica negativa à trama, dizendo que era "uma colagem de velhas histórias" de Aguinaldo Silva, e que "assim como as crianças, que quanto mais veem um desenho mais querem revê-lo, o telespectador - bem ao contrário do destemido Guma (Marcos Palmeira) - parece sentir-se mais à vontade nadando em águas conhecidas. E como a regra é não arriscar, as redes seguem reeditando o que já foi testado no ibope".
Controvérsias
Ausência de atores negros
Porto dos Milagres sofreu grandes críticas do movimento negro da Bahia, pelo número reduzido de atores negros no elenco e pela ausência de protagonistas da mesma etnia. O fato deu-se pela novela ser ambientada na Bahia, o estado com maior população negra do Brasil. Na época, Tony Tornado chegou a comentar que "A minha briga no momento é por causa dessa novela das oito [Porto dos Milagres], que é uma Bahia branca. Vi dois ou três casais negros, só. Não é possível que Jorge Amado tenha escrito assim. Não pode ser que Porto dos Milagres seja tão branco quando 90% da Bahia é negra", acrescentando que "o Marquinhos Palmeira deve estar tomando laser para ficar preto e ser baiano".
Em 2006, o secretário de Direitos Humanos do Rio de Janeiro, coronel Jorge da Silva, comentou sobre o caso: "Em Porto dos Milagres, a maioria dos atores eram brancos e a novela parecia acontecer no Paraná, enquanto a história original, escrita por Jorge Amado, era toda ambientada na Bahia, com personagens negros e mulatos. O autor, na época, justificou como licença poética, então porque não usam esta liberdade de criação para retratar um casal de negros como patrões de imigrantes italianos nas novelas?", questionou.
Temas religiosos
Os temas de religião africana causaram controvérsias entre a comunidade cristã. Wilson Victoriano Ferreira da Silva, diretor espiritual da Renovação Carismática da Diocese de Jundiaí, acreditou que os problemas começavam pelo título da novela. "Iemanjá é uma divindade pagã e não pode ser comparada com Nossa Senhora, mãe de Deus. Não aceitamos essa identificação. Um deus pagão não pode fazer milagres", comentou ele. Já a visão do babalorixá Salvador Maria é contrária; ele assegura que a novela é inofensiva, e retrata com riqueza de detalhes o candomblé. Segundo ele, que não tinha costume de assistir novelas, a história teve uma boa repercussão entre os espiritualistas. "Não acredito que uma história contada na televisão influencie tanto assim as pessoas. Quem tem fé e religião não muda só por causa de uma ficção", disse. Para ele, Mãe Ricardina, personagem de Zezé Motta, mostrou que a religião não faz ligações com o mal. "O candomblé sofre preconceito e a novela está mostrando a verdade", enfatizou.
Por sua vez, Aguinaldo Silva tratou de discordar dos bispos metodistas, dizendo: "Usamos o candomblé como elemento dramático, não como propaganda religiosa. A novela se baseia em dois livros de Jorge Amado (A Descoberta da América pelos Turcos e Mar Morto), nos quais o candomblé tem papel de destaque. Aliás, eu sou católico". O co-autor da trama, Ricardo Linhares, diz que muito da religiosidade da novela é de tom farsesco. "Não defendemos nada. A novela traz situações absurdas que abordamos em tom cômico. Tomamos licenças poéticas com o culto de candomblé. Não o seguimos ao pé da letra", completou.
Política
À época de sua exibição, a direção da TV Globo recebeu algumas críticas pela paródia política na novela de algumas autoridades políticas, que demonstraram seu descontentamento em recados enviados aos dirigentes da emissora. Também na mesma época, um escândalo político referido como "escândalo do painel eletrônico" tornou-se conhecido no Brasil, em que envolveu o nome do senador do estado da Bahia Antônio Carlos Magalhães, filiado ao então Partido da Frente Liberal (PFL), em gravações de voz. Na trama, era exibido um enredo em que o senador baiano Victório Vianna (Lima Duarte) tem uma conversa gravada secretamente por Félix Guerrero (Antônio Fagundes), para o desgosto de Magalhães. Ricardo Linhares, autor da novela, disse que o personagem do senador não foi diretamente inspirado em Magalhães nem em nenhum outro político. "A inspiração vem certamente do escândalo no Senado e do quadro político atual", comentou na época.
No final da novela, um programa eleitoral foi exibido dentro da novela, para caracterizar a candidatura de Félix Guerrero à governador da Bahia e a de Guma à prefeitura de Porto dos Milagres. No entanto, deputados do Partido da Social Democracia Brasileira (PSDB) reclamaram dizendo que o programa prejudicava a imagem do governo. Enquanto Félix, que faz parte da situação, passa a imagem de uma pessoa inescrupulosa, Guma, seu adversário e mocinho da novela, estaria associado ao Partido dos Trabalhadores (PT); uma das semelhanças seria o nome fictício da legenda, Partido das Causas Trabalhistas. "Não há dúvida de que há uma indução para que o eleitor perceba que o Guma representa o que seria a oposição e o candidato a governador (Fagundes), o governo. Até com o mesmo impacto vocal do nome Guma parece com Lula. Isso vai para o inconsciente do eleitor e acaba sendo nocivo", acusou o vice-líder do PSDB na Câmara dos Deputados, Nárcio Rodrigues. Outro deputado aliado, o maranhense Sebastião Madeira, disse que o "horário eleitoral passa uma mensagem subliminar. O lado do governo é o corrupto. [...] Repare, até o nome do partido, PCT, lembra o PT. Então, a Globo quer que o Lula seja presidente?", indagou.
Ricardo Linhares reconheceu a semelhança, dizendo que Guma poderia ter uma semelhança com o partido, mas que não acreditava que isso influenciasse, e acrescentou ser apartidário. Luís Erlanger, diretor da Central Globo de Comunicação e um dos idealizadores da campanha fictícia, disse que a ideia da emissora foi chamar atenção para o fim da trama. "Esse tipo de ligação [associar as chamadas com as eleições de 2002] só quem fez, pelo visto foi o JB. Até porque, espontaneamente, ninguém se expôs descontentado", comentou. O diretor-geral da novela, Marcos Paulo, minimizou a controvérsia, dizendo: "Tudo não passa de uma sátira bem-humorada da política. A intenção era a de provocar um ruído como de fato aconteceu".
Acusação de plágio
Em 2004, as escritoras Adelaide Magalhães Veiga Ferreira e Ione de Moraes Bueno ajuizaram uma ação contra a TV Globo e os autores Aguinaldo Silva e Ricardo Linhares por um suposto plágio no enredo de Porto dos Milagres. De acordo com as escritoras, elas teriam enviado um manuscrito do livro Seara Santa para Nilton Travesso, diretor e produtor de novelas no canal, por intermédio de sua secretária antes de a novela ir ao ar, e dizem ter carta como prova, mas, pelo entendimento do juiz, as escritoras não tinham qualquer comprovante de recebimento ou resposta de tal correspondência. Assim, devido à falta de materialidade, o pedido de indenização das autoras foi indeferido pelo tribunal.
Mesmo que ficasse provado que Linhares e Silva tiveram acesso ao livro antes de escreverem a novela, o juiz entendeu que de qualquer forma não houve plágio de Seara Santa. O juiz amparou sua decisão em laudo pericial que diz: "Realizada a leitura das obras e feita a comparação entre elas, verificamos que as breves colidências entre estas, como especificado, não correspondem, isoladamente, em elementos passíveis de proteção autoral (que contém, por sua natureza, a prerrogativa de seu autor de exclusividade de utilização contra terceiros), não se configurando, consequentemente, na hipótese dos autos, o ilícito de plágio". Em sua defesa, os autores da novela disseram que a mesma fora baseada em obras específicas de Jorge Amado, permeada com realismo fantástico e "forte tempero baiano", já característico da dupla em seus trabalhos.
Prêmios e indicações
Bibliografia
Ligações externas
Programas de televisão do Brasil que estrearam em 2001
Programas de televisão do Brasil encerrados em 2001
Cidades fictícias
Telenovelas e séries baseadas em obras de Jorge Amado
Telenovelas ambientadas na Bahia
Telenovelas em português
Telenovelas da TV Globo da década de 2000
Telenovelas ambientadas na Espanha
Telenovelas exibidas no Viva | {
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Q: SQLSTATE[HY093]: Invalid parameter number: number of bound variables does not match number of tokens keeps showing I have been knocking my head over this for hours now but still i cant see where i m going wrong. Please refere to code below
try
{
$stmt1 = $db_con->prepare("SELECT * FROM users_profile WHERE email= :email ");
$stmt1->execute(array(":email"=>$email));
$count = $stmt1->rowCount();
if($count==0){
$stmt = $db_con->prepare("INSERT INTO users_profile(referencecode,name,surname,cellphone,email,address,region,password,datejoined,amount,status) VALUES(?,?,?,?,?,?,?,?,?,?,?)");
$stmt->bind_param("sssssssssss",$reference,$name,$surname,$cellphone,$email,$address,$region,$password,$joining_date,$amount,$status);
if($stmt->execute())
{
echo "registered";
}
else
{
echo "Query could not execute !";
}
}
else{
echo "1"; // not available
}
}
catch(PDOException $e){
echo $e->getMessage();
}
}
This is my table structure
Users_profile
I know the question has been asked a million times, but whatever solution i try it doesnt work for me. Maybe its something in my code i am oversighting.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,222 |
require 'spec_helper'
describe 'openvpn::config', type: :class do
on_supported_os.each do |os, facts|
context "on #{os}" do
let(:facts) do
facts
end
it { is_expected.to compile.with_all_deps }
case facts[:os]['family']
when 'Debian'
context 'on Debian based machines' do
it { is_expected.to contain_concat('/etc/default/openvpn') }
it { is_expected.to contain_concat__fragment('openvpn.default.header') }
context 'enabled autostart_all' do
let(:pre_condition) { 'class { "openvpn": autostart_all => true }' }
it {
is_expected.to contain_concat__fragment('openvpn.default.header').with(
'content' => %r{^AUTOSTART="all"}
)
}
end
context 'disabled autostart_all' do
let(:pre_condition) { 'class { "openvpn": autostart_all => false }' }
it {
is_expected.to contain_concat__fragment('openvpn.default.header').with(
'content' => %r{^AUTOSTART=""}
)
}
end
end
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,798 |
Die Liste der Stolpersteine in der Provinz Toledo enthält die Stolpersteine, die vom deutschen Künstler Gunter Demnig in der spanischen Provinz Toledo verlegt wurden. Die Provinz Toledo zählt zur Autonomen Gemeinschaft Kastilien-La Mancha. Stolpersteine erinnern an das Schicksal der Menschen, die von den Nationalsozialisten ermordet, deportiert, vertrieben oder in den Suizid getrieben wurden. Sie liegen im Regelfall vor dem letzten selbstgewählten Wohnsitz des Opfers.
Die kastilische Übersetzung des Begriffes Stolpersteine lautet: pedres que fan ensopegar. In Spanien werden sie jedoch zumeist piedras de la memoria (Erinnerungssteine) genannt. Die ersten zwei Verlegungen fanden am 10. Juni 2021 in Camuñas ab.
Verlegte Stolpersteine
In Camuñas wurden bis Januar 2022 drei Stolpersteine verlegt, vier sind geplant.
Verlegedatum
Die ersten beiden Verlegungen erfolgten am 10. Juni 2021 (für Pedro Gallego Romero und Noe Ortega Aranda).Anwesend war Patrocinio Yuste Aranda, die damals 106 Jahre alte Schwester von Emiliano Yuste Aranda. Die Verlegung von dessen Stolperstein wurde COVID-19-bedingt verschoben, um Verwandten aus Frankreich und England die Möglichkeit zur Anreise zu geben.
Weblinks
Stolpersteine.eu, Demnigs Website
Einzelnachweise
Provinz Toledo
Provinz Toledo | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 50 |
Шантіллі () — переписна місцевість (CDP) в США, в окрузі Ферфакс штату Вірджинія. Населення — осіб (2010).
Географія
Шантіллі розташоване за координатами (38.886765, -77.445004). За даними Бюро перепису населення США в 2010 році переписна місцевість мала площу 31,47 км², з яких 31,13 км² — суходіл та 0,34 км² — водойми.
Демографія
Згідно з переписом 2010 року, у переписній місцевості мешкало осіб у домогосподарствах у складі родин. Густота населення становила 732 особи/км². Було 7403 помешкання (235/км²).
Расовий склад населення:
До двох чи більше рас належало 4,3 %. Частка іспаномовних становила 15,9 % від усіх жителів.
За віковим діапазоном населення розподілялося таким чином: 27,8 % — особи молодші 18 років, 65,7 % — особи у віці 18—64 років, 6,5 % — особи у віці 65 років та старші. Медіана віку мешканця становила 36,1 року. На 100 осіб жіночої статі у переписній місцевості припадало 100,2 чоловіків; на 100 жінок у віці від 18 років та старших — 98,7 чоловіків також старших 18 років.
Середній дохід на одне домашнє господарство становив долари США (медіана — ), а середній дохід на одну сім'ю — доларів (медіана — ). Медіана доходів становила долари для чоловіків та доларів для жінок. За межею бідності перебувало 8,9 % осіб, у тому числі 13,8 % дітей у віці до 18 років та 6,6 % осіб у віці 65 років та старших.
Цивільне працевлаштоване населення становило осіб. Основні галузі зайнятості: науковці, спеціалісти, менеджери — 27,9 %, освіта, охорона здоров'я та соціальна допомога — 16,5 %, роздрібна торгівля — 9,3 %, публічна адміністрація — 8,3 %.
Примітки
Джерела
Переписні місцевості Вірджинії
Населені пункти округу Ферфакс (Вірджинія) | {
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} | 1,498 |
Pluralities of New Hampshire adults prefer someone else getting elected instead of incumbent Congresswomen Carol Shea-Porter and Ann Kuster. Senator Jeanne Shaheen remains popular and a plurality of Granite Staters support her re-election in November 2014. Senator Kelly Ayotte's favorability has declined since April.
Survey Center, UNH, "NH Congressional Delegation Facing Tough Re-Election, Shaheen Solid 8/01/13" (2013). All UNH Survey Center Polls. 192. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,533 |
\section{Introduction}
International Linear Collider (ILC) is a next-generation electron-positron linear collider.
International Large Detector (ILD) is one of two validated detector concepts for the ILC.
ILD is based on particle flow concept, which requires separation of each particle in jets.
Jet energy in the particle flow is measured with momentum of tracks and energy of neutral clusters.
Since momentum resolution of tracks is much better than energy resolution of clusters in general,
this gives better energy resolution of jets.
The particle flow requires a high granular calorimeter system to separate particles, especially in
electromagnetic calorimeter (ECAL). ILD ECAL has $5\times5$ mm$^2$ granularity, which is a major challenge
in both detector and electronics, and requires high cost. Two options are available for ILD ECAL:
one is silicon-tungsten ECAL (SiECAL) and the other is scintillator-tungsten ECAL (ScECAL).
SiECAL utilizes silicon diode pads for readout, sandwiched with tungsten absorber.
Silicon pads can be easily divided to desired area ($5\times5$ mm$^2$)
and the total cost mainly depends on the sensor area.
ScECAL utilizes strip scintillators with silicon-photomultipliers (SiPMs) for readout instead of silicon.
To reduce number of SiPM sensors which are the cost driver,
strip scintillators ($5\times45$ mm$^2$) placed perpendicularly to the neighbor layers have been adopted for the ScECAL design.
The total cost of ScECAL is around a half of SiECAL in the cost estimation in Detailed Baseline Design (DBD)~\cite{Behnke:2013lya} report.
The average cost of SiECAL and ScECAL is around 30\% of total cost of ILD
with the baseline design of inner radius of ECAL at 1800 mm from the center and 30 layers of sensors.
\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|l|l|}\hline
& SiECAL & ScECAL \\ \hline\hline
sensor & silicon & scintillator with SiPM \\ \hline
pixel size & $5\times5$ mm$^2$ & $5\times45$ mm$^2$ \\ \hline
thickness & 320 to 500 $\mu$m & 1 or 2 mm \\ \hline
MIP response & around 25 K pairs & $\mathcal{O}(10)$ photoelectrons \\
& in 320 $\mu$m thickness & (depending on detailed structure) \\ \hline
sensor gain & 1 & $\mathcal{O}(10^5)$ \\ \hline
sensor stability & stable & varied by temperature and \\
& & sensor overvoltage \\
& & periodic calibration required \\ \hline
gain in electronics & higher & lower \\ \hline
saturation & only in electronics & SiPM saturation, depending on \\
& & number of pixels of SiPM\\ \hline
assembly & easier & complicate, including wrapping \\
& & and placing tiny strips \\ \hline
cost & higher & lower \\ \hline
\end{tabular}
\caption{Comparison of SiECAL and ScECAL.}
\label{tbl:compsisc}
\end{center}
\end{table}
Table \ref{tbl:compsisc} shows comparison of characteristics of SiECAL and ScECAL.
Each has advantages and disadvantages. SiECAL has advantages on number of pairs,
sensor stability and granularity, though requirements on electronics are higher
(number of channels and gain of preamplifier) and sensor cost is higher.
ScECAL is a cheaper solution but has stronger requirements
on periodic sensor calibration, saturation correction and assembly.
To keep the DBD cost equal or lower with using SiECAL, we have two options:
(1) shrinking detector to smaller size, with reduced number of sensors,
(2) introducing hybrid ECAL, which is a combination of SiECAL and ScECAL.
The option (1) keeps robustness and simplicity of SiECAL but the detector
performance should be degraded at some extent because of worse particle
separation and worse momentum resolution of tracks due to the smaller detector.
The performance degradation should be carefully estimated since the degradation
should usually be compensated by more luminosity to obtain the similar impact of
physics results. More luminosity means more operation cost, which may be equal or
more to the reduction of detector cost.
With the option (2) performance degradation should be much smaller than option (1)
but more complexity is introduced. One consideration is that one of the HCAL option
of ILD is also scintillator-SiPM complex. If it is adopted, the total complexity of
calorimeter system is maintained at similar level compared to the simple SiECAL and
scintillator HCAL. In the hybrid ECAL, the SiECAL technology should be used in the inner
part of ECAL, which requires more granularity to separate particles and then expect
more ghost hits if we use strip ScECAL sensors.
\section{Optimization of hybrid ECAL}
Studies are ongoing to establish a optimal hybrid ECAL configuration in ILD.
We consider optimization with constraints that: (1) ILD of similar size to DBD version,
(2) ILD of similar cost to DBD and (3) combination of SiECAL and ScECAL.
Parameters of the optimization include (a) number of layers
and thicknesses of absorber, (b) pixel size of each layer, and
(c) order and fraction of SiECAL and ScECAL.
For (c), resolution of jet energy at various energies has been compared
with SiECAL, ScECAL and several order of hybrid ECAL in Fig.~\ref{fig:alternate}.
ILD full simulation and reconstruction software (ILCSoft~\cite{ILCsoftweb} v01-16-02 with Pandora PFA~\cite{Thomson:2009rp} v00-09-02)
with $q\bar{q}$ two jet events are used.
The three configurations of normal, single alternating and double alternating
give almost same jet energy resolution at every energy.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\linewidth]{alternate.eps}
\end{center}
\caption{Comparison of jet energy resolution among non-alternating (`HybridECAL', inner SiECAL and outer ScECAL),
single and double alternating configuration of hybrid ECAL, full SiECAL and ScECAL configuration.}
\label{fig:alternate}
\end{figure}
For (a), a comprehensive study is planned to understand response to longitudinal parameters
of sandwiched calorimeter. Longitudinal configuration of ILD ECAL is separated to two: inner and outer layer.
Current baseline adopts the configuration with thickness ratio of 1 by 2 in inner and outer layer,
separated by the half of radiation length with total 29 (inner 20 and outer 9) layers with additional
one layer before absorber. Total absorber thickness is set to 22.8 radiation length.
However, the thickness ratio, total number of layers and the border position of inner and outer layer
is practically not optimized. We can also consider three configuration of inner, middle, outer or
even fully variant thickness in each layer.
In the jet energy measurements with the particle flow algorithm, the longitudinal structure mainly affects
the intrinsic energy resolution of single particle, while the transverse structure is more important on the
clustering to separate contribution of charged particles.
Contribution of resolution of single particle in ECAL is not large in jet energy measurements since
HCAL energy resolution dominates the performance on the lower energy and the clustering dominates
on the higher energy.
However, ILC features non-jet measurements as well as jet physics, such as $\pi_0$ reconstruction from
$\tau$ decay, $H \to \gamma\gamma$ and non-pointing photons from new physics models.
These measurements heavily depend on the ECAL energy resolution and thus longitudinal structure is still important.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\linewidth]{photon.eps}
\end{center}
\caption{Comparison of energy resolution of single photons, by reducing number of layers in inner and outer region.}
\label{fig:photon}
\end{figure}
As the first step of optimization, we compared energy resolution of single photons with several longitudinal configuration,
shown in in Fig.~\ref{fig:photon}. In this figure, degradation of photon energy resolution is larger with reducing inner
layers than reducing outer layers.
A lot of analysis are still needed to optimize the hybrid ECAL, such as looking at hadron energy resolution,
confusion study with various transverse configuration, etc.
We plan to conclude one hybrid model to be compared with small detector after the optimization studies.
\section{Combined DAQ}
Hardware aspect is as important as the optimization aspects.
CALICE~\cite{CALICEweb} collaboration aims to combine all efforts of linear collider calorimeter development into a consistent manner.
For the calorimeter data acquisition (DAQ), many detector systems for ILC, including silicon and scintillator system, are based on
`ROC'-family integrated readout chip developed at OMEGA~\cite{OMEGAweb} group. The all ROC chips are based on a same basic design,
and have similar protocol for configuration and data readout.
Interface of ROC chips, timing control and data readout has been also developed for the all `ROC'-based
CALICE systems, but each detector system currently uses readout structure
which is not fully compatible each other by historical reasons.
To integrate full calorimeter system (and more), we need to develop a consistent
readout system. In hybrid ECAL system, we are trying to develop a combined DAQ system of
SiECAL and ScECAL, which can be a good start point to larger integration.
Our design is to use existing hardware and software in each system, with minimal interaction of them.
The interaction includes clock synchronization, common readout counting, run and stop controller software
working at higher level than each readout system, and combined data collection software.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{0.48\linewidth}
\includegraphics[width=1\linewidth]{combdaq-hardware.eps}
\end{minipage}
\begin{minipage}{0.48\linewidth}
\includegraphics[width=1\linewidth]{combdaq-timing.eps}
\end{minipage}
\end{center}
\caption{Block diagram and timing chart of combined DAQ.}
\label{fig:hardware}
\end{figure}
Figure \ref{fig:hardware} shows the block diagram and timing chart of the combined DAQ hardware.
`Spill' in the figures comes from testbeam control. The spill in the testbeam is assumed to be
different from ILC operation mode, that long spills (400 ms in the timing chart) come
several times a minute (ILC spill is only 1 ms long with 5 Hz operation).
To maximize data taking efficiency, the spill should be subdivided to several `readout cycle's, determined by
scintillator CCC (Clock and Control Card) by looking at `busy' signals from each scintillator module.
Scintillator busy is flagged from each SPIROC2 chip when the memory of the chip is full and
cleared when the readout has finished (silicon busy is not treated due to a technical reason).
The acquisition is stopped with busy flag, and the next readout cycle is started
after all busy flags are cleared. The acquisition period is shared from scintillator CCC
to silicon CCC by a level signal, provide the same readout cycle counting.
Scintillator CCC also creates a master clock to be synchronized with silicon CCC.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.48\linewidth]{combdaq-software.eps}
\end{center}
\caption{Software structure of combined DAQ.}
\label{fig:software}
\end{figure}
Figure \ref{fig:software} shows the software structure.
SiECAL and ScECAL have their own readout software, based on a system called Calicoes~\cite{Cornat:2014cha} in SiECAL
and a LabVIEW-based system in ScECAL. For the combined system, we adopted EUDAQ~\cite{EUDAQweb} framework,
which can be connected to each readout system via TCP socket connection.
We developed a EUDAQ producer, which talks to each readout system via socket to obtain raw data packets.
The received raw data is converted to LCIO format within the EUDAQ framework, to be stored in LCIO files.
Readout cycle is numbered and checked in the EUDAQ data collector to ensure that
the same event data of SiECAL and ScECAL is stored in the same LCIO event.
The EUDAQ run control is also used to provide start and stop signal to each system and assign run numbers.
The combined DAQ has been tested at CERN PS testbeam facility from November 26 to December 8, 2014.
A SiECAL layer was placed in front of a scintillator stuck, including three ScECAL layers and eleven HCAL layers.
The combined data taking run successfully, taking data with 7 GeV muons and 2-8 GeV pions.
The concurrent hits have been found between silicon and scintillator layers,
with consistent timing difference of electronics, proving successful synchronization of two systems.
Detailed analysis is ongoing.
\section{Summary}
Hybrid ECAL is an cost-effective option for ILD ECAL. We have started the optimization of hybrid ECAL
by looking at energy resolution of photons, hadrons and jets with various configuration.
The alternation of SiECAL and ScECAL layers gives similar performance to the combination of
SiECAL in inner part and ScECAL in outer part. Reducing number of layers have some impact on the
photon energy resolution, and the inner region is more important.
For the DAQ, we developed a combined SiECAL and ScECAL DAQ with minimal modification
of each readout framework.
We adopted EUDAQ as a higher level run control and data integration.
The testbeam at CERN was successful, obtaining concurrent hits at silicon and scintillator layers.
\section*{Acknowledgment}
We thank LLR, DESY, Shinshu and CERN colleagues for supplying sensors, DAQ and testbeam environment.
The computing resource for the optimization study was mainly provided from KEK computing center.
This work was supported by MEXT/JSPS KAKENHI Grant Numbers 23000002 and 23104007,
and by Kyushu University Interdisciplinary Programs in Education and Projects in Research Development.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,426 |
Harmen Liemburg (b. 1966, the Netherlands) is an Amsterdam-based graphic designer, printmaker, and journalist. He studied social geography and cartography at Utrecht University before graduating in 1998 with a degree in graphic design from the Gerrit Rietveld Academie (Amsterdam). First collaborating with graphic designer Richard Niessen and later working on his own, Liemburg produces and self-publishes his work and holds workshops and lectures worldwide.
Images: Harmen Liemburg and Gert Jan van Rooij (bottom left). | {
"redpajama_set_name": "RedPajamaC4"
} | 4,934 |
\section*{Acknowledgments}
We thank the SAFARI group members and Intel Labs for feedback and the stimulating intellectual environment. We acknowledge the generous gifts and support provided by our industrial partners: Intel, Google, Huawei, Microsoft, VMware, and the Semiconductor Research Corporation.
\balance
\setstretch{0.85}
\bibliographystyle{IEEEtranS}
\small
\section{Introduction} \label{sec:introduction}
Hidden Markov Models (HMMs) are useful for calculating the probability of a sequence of previously unknown (hidden) events (e.g., the weather condition) given observed events (e.g., clothing choice of a person)~\cite{eddy_what_2004}. To calculate the probability, HMMs use a graph structure where a sequence of nodes (i.e., states) are visited based on the series of observations with a certain probability associated with visiting a state from another. HMMs are very efficient in decoding the continuous and discrete series of events in many applications~\cite{mor_systematic_2021} such as speech recognition~\cite{mor_systematic_2021, mustafa_comparative_2019, mao_revisiting_2019, hamidi_interactive_2018, xue_novel_2018, li_hybrid_2013, patel_speech_2010}, text classification~\cite{nasim_sentiment_2020, kang_opinion_2018, zeinali_text-dependent_2017, ahmad_open-vocabulary_2016, vieira_t-hmm_2014}, gesture recognition~\cite{moreira_acoustic_2020, sinha_computer_2019, haid_inertial-based_2019, calin_gesture_2016, deo_-vehicle_2016, malysa_hidden_2016, nguyen-duc-thanh_two-stage_2012, shrivastava_hidden_2013}, and bioinformatics~\cite{liang_bayesian_2007, boufounos_basecalling_2004, narasimhan_bcftoolsroh_2016, yin_args-oap_2018, tamposis_semi-supervised_2019, zhang_fish_2014, eddy_profile_1998, huang_hardware_2017, wu_173_2020}. The graph structures (i.e., designs) of HMMs are typically tailored for each application, which defines the \emph{roles} and probabilities of the states and edges connecting these states, called \emph{transitions}. One important special design of HMMs is known as the \emph{profile Hidden Markov Model} (pHMM) design~\cite{eddy_profile_1998}, which is commonly adopted in bioinformatics~\cite{baldi_hidden_1994, bateman_pfam_2002, zhang_profile_2003, sgourakis_method_2005, friedrich_modelling_2006, steinegger_hh-suite3_2019, durbin_biological_1998, edgar_coach_2004, madera_profile_2008, eddy_accelerated_2011, wheeler_dfam_2012, firtina_hercules_2018, firtina_apollo_2020, lanyue_long_2020, yoon_hidden_2009}, malware detection~\cite{ali_profile_2022, sasidharan_prodroid_2021, liu_adversarial_2019, pranamulia_profile_2017, ravi_behavior-based_2013, attaluri_profile_2009} and pattern matching~\cite{riddell_reliable_2022, kazantzidis_profile_2018, saadi_framework_2016, ding_skeleton-based_2015, liu_characterizing_2015, liu_who_2009}.
Identifying differences between biological sequences (e.g., DNA sequences) is an essential step in bioinformatics applications to understand the effects of these differences (e.g., genetic variations and their relations to certain diseases). PHMMs enable efficient and accurate identification of differences by comparing sequences to a few graphs that represent a group of sequences rather than comparing many sequences to each other, which is computationally very costly and requires special hardware and software optimizations~\cite{alser_technology_2021, alser_going_2022, alser_accelerating_2020, singh_fpga-based_2021, alser_sneakysnake_2020, angizi_pim-aligner_2020, goenka_segalign_2020, senol_cali_genasm_2020, turakhia_darwin_2018, kim_grim-filter_2018, mansouri_ghiasi_genstore_2022, nag_gencache_2019, firtina_blend_2021, cali_segram_2022, kim_airlift_2021, kim_fastremap_2022}. Figure~\ref{fig:phmm} illustrates a \emph{traditional} design of pHMMs. A pHMM represents a single or many sequences with a graph structure using states and transitions. There are three types of states for each character of a sequence that a pHMM graph represents: insertion (\texttt{I}), match or mismatch (\texttt{M}), and deletion (\texttt{D}) states. Each state accounts for a certain difference or a match between a graph and an \emph{input sequence} at a particular position. For example, the \texttt{I} states recognize insertions in an input sequence that are missing from the pHMM graph at a position.
Many bioinformatics applications use pHMM graphs rather than directly comparing sequences to avoid the high cost of many sequence comparisons.
The applications that use pHMMs include protein family search~\cite{baldi_hidden_1994, bateman_pfam_2002, zhang_profile_2003, friedrich_modelling_2006, steinegger_hh-suite3_2019, soding_hhpred_2005, finn_pfam_2010, madera_comparison_2002}, multiple sequence alignment (MSA)~\cite{durbin_biological_1998, edgar_coach_2004, madera_profile_2008, attaluri_profile_2009, eddy_accelerated_2011, wheeler_dfam_2012, sgourakis_method_2005, steinegger_hh-suite3_2019, mulia_profile_2012, pei_promals_2007, edgar_satchmo_2003}, and error correction~\cite{firtina_hercules_2018, firtina_apollo_2020, lanyue_long_2020}.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/phmm.pdf}
\caption{Portion of an example pHMM design that represents a DNA sequence (\texttt{PHMM Sequence}).
Differences between \texttt{PHMM Sequence} and \texttt{Sequences \#1}, \texttt{\#2}, and \texttt{\#3} are highlighted by colors. Highlighted transitions and states identify each corresponding difference.}
\label{fig:phmm}
\end{figure}
To accurately model and compare DNA or protein sequences using pHMMs, assigning accurate probabilities to states and transitions is essential.
PHMMs allow updating these probabilities to fit the observed biological sequences to the pHMM graph accurately. Probabilities are adjusted during the \emph{training} step. The training step aims to maximize the probability of observing the input biological sequences in a given pHMM, also known as \emph{likelihood maximization}. There are several algorithms that perform such a maximization in pHMMs~\cite{baum_inequality_1972, scott_bayesian_2002, lewis_bayesian_2008, rezaei_generalized_2013}. The Baum-Welch algorithm~\cite{baum_inequality_1972} is commonly used to calculate the likelihood maximization~\cite{boussemart_comparing_2009} as it is highly accurate and scalable to real-size problems (e.g., large protein families)~\cite{lewis_bayesian_2008}. The next step is \emph{inference}, which aims to identify either 1)~the similarity of an input observation sequence to a pHMM graph or 2)~the sequence with the highest similarity to the pHMM graph, which is known as the \emph{consensus sequence} of the pHMM graph and used for error correction in biological sequences. Parts of the Baum-Welch algorithm can be used for calculating the similarity of an input sequence in the inference step.
Despite its advantages, the Baum-Welch algorithm is a computationally expensive method~\cite{lyngso_consensus_2002, kahsay_quasi-consensus-based_2005} due to the nature of its dynamic programming approach.
Several works~\cite{eddy_accelerated_2011, ren_fpga_2015, pietras_fpga_2017, yu_gpu-accelerated_2014, soiman_parallel_2014} aim to accelerate either the entire or smaller parts of the Baum-Welch algorithm for HMMs or pHMMs to mitigate the high computational costs. While these works can improve the performance for executing the Baum-Welch algorithm, they either 1)~provide software- or hardware-only solutions for a fixed pHMM-design or 2)~are completely oblivious to the pHMM design.
To identify the inefficiencies in using pHMMs with the Baum-Welch algorithm, we analyze the state-of-the-art implementations of three pHMM-based bioinformatics applications: 1)~error correction, 2)~protein family search, and 3)~multiple sequence alignment (Section~\ref{sec:motivation}). We make six key observations. 1)~The Baum-Welch algorithm is the main performance bottleneck in the pHMM applications as it constitutes at least around $50\%$ of the total execution time of these applications. 2)~SIMD-based approaches cannot fully vectorize the floating-point operations. 3)~Significant portion of floating-point operations is redundant in the training step due to a lack of a mechanism for reusing the same products. 4)~Existing strategies for filtering out the negligible states from the computation are costly despite their advantages. 5)~The spatial locality inherent in pHMMs cannot be exploited in generic HMM-based accelerators and applications as these accelerators and applications are oblivious to the design of HMMs. 6)~The Baum-Welch algorithm is the bottleneck even for the non-genomic application we evaluate. Unfortunately, software- or hardware-only solutions cannot solve these inefficiencies. These observations show that there is a pressing need for a flexible, high-performant, and energy-efficient hardware-software co-design to efficiently and effectively solve these inefficiencies in the Baum-Welch algorithm for pHMMs.
Our \textbf{goal} is to accelerate the Baum-Welch algorithm while eliminating the inefficiencies when executing the Baum-Welch algorithm for pHMMs. To this end, we propose \proposal, the \emph{first} flexible hardware-software co-designed acceleration framework that can significantly reduce the computational and energy overheads of the Baum-Welch algorithm for pHMMs. \proposal is built on four \textbf{key mechanisms}. First, \proposal is highly flexible that can use different pHMM designs with the ability to change certain parameter choices to enable the adoption of \proposal for many pHMM-based applications. This enables 1)~additional support for pHMM-based error correction~\cite{lanyue_long_2020, firtina_hercules_2018, firtina_apollo_2020} that traditional pHMM design cannot efficiently and accurately support~\cite{firtina_hercules_2018}. Second, \proposal exploits the spatial locality that pHMMs provide with the Baum-Welch algorithm by efficiently utilizing on-chip memories with memoizing techniques. Third, \proposal efficiently eliminates negligible computations with a hardware-based filter design. Fourth, \proposal avoids redundant floating-point operations by 1)~providing a mechanism for efficiently reusing the most common products of multiplications in lookup tables (LUTs) and 2)~identifying pipelining and broadcasting opportunities where certain computations are moved between multiple steps in the Baum-Welch algorithm without extra storage or computational overheads. Among these mechanisms, the fourth mechanism includes our software optimizations, while on-chip memory and hardware-based filter require a special and efficient hardware design.
To evaluate \proposal, we 1)~design a flexible hardware-software co-designed acceleration framework in an accelerator and 2)~implement the software optimizations for GPUs. We evaluate the performance and energy efficiency of \proposal for executing 1)~the Baum-Welch algorithm and 2)~several pHMM-based applications and compare \proposal to the corresponding CPU, GPU, and FPGA baselines. First, our extensive evaluations show that \proposal provides significant 1)~speedup for executing the Baum-Welch algorithm by \pbaumcpuc - \pbaumcpua (CPU), \pbaumgpud - \pbaumgpua (GPU), and \pbaumfpga (FPGA) and 2)~energy efficiency by \ebaumcpua (CPU) and \ebaumgpud-\ebaumgpua (GPU). Second, \proposal improves the overall runtime of the pHMM-based applications, error correction, protein family search, and MSA, by \perrgpud - \perrcpua, \pprofpga - \pprocpua, and \pmsafpga - \pmsacpua and reduces their overall energy consumption by \eerrcpua - \eerrgpua, \eprocpua, \emsacpua over their state-of-the-art CPU, GPU, and FPGA implementations, respectively. We make the following \textbf{key contributions}:
\begin{itemize}
\item We introduce \proposal, the \emph{first} flexible hardware-software co-designed framework to accelerate pHMMs. We show that our framework can be used at least for three bioinformatics applications: 1)~error correction, 2)~protein family search, and 3)~multiple sequence alignment.
\item We provide \proposal-GPU, the \emph{first} GPU implementation of the Baum-Welch algorithm for pHMMs, which includes our software optimizations.
\item We identify key inefficiencies in the state-of-the-art pHMM applications and provide key mechanisms with efficient hardware and software optimizations for significantly reducing the computational and energy overhead of the Baum-Welch algorithm for pHMMs.
\item We show that \proposal provides significant speedups and energy reductions for executing the Baum-Welch algorithm compared to the CPU, GPU, and FPGA implementations, while \proposal-GPU performs better than the state-of-the-art GPU implementation.
\item We provide the source code of our software optimizations, \proposal-GPU, as implemented in an error correction application. The source code is available at \url{https://github.com/CMU-SAFARI/ApHMM-GPU}.
\end{itemize}
\section{Background} \label{sec:background}
\subsection{Profile Hidden Markov Models (pHMMs)}\label{subsec:phmms}
\subsubsection{High-level Overview}\label{subsubsec:phmmoverview}
We explain the design of profile Hidden Markov Models (pHMMs). Figure~\ref{fig:phmm} shows the \emph{traditional} structure of pHMMs. To represent a biological sequence and account for differences between the represented sequence and other sequences, pHMMs have a constrained graph structure. Visiting nodes, called \emph{states}, via directed edges, called \emph{transitions}, are associated with certain probabilities to identify differences. To assign a probability for any modification at any sequence position, states are created for each character of the represented sequence. When visited, states may \emph{emit} one of the characters from the defined alphabet of the biological sequence (e.g., A, C, T, and G in DNA sequences) with a certain probability. Transitions preserve the correct order of the represented sequence while allowing insertions and deletions to the represented sequence.
To represent and compare biological sequences, pHMMs are used in three steps.
First, to represent a sequence, pHMM builds the states and transitions by iterating over each character of the sequence. Multiple sequences can also be represented with a single pHMM graph.
A typical pHMM graph includes insertion, match/mismatch, and deletion states for each character of the represented sequence. Connections between states have predefined patterns, as illustrated in Figure~\ref{fig:phmm}. Match states have connections to only match and deletion states of the next character and insertion state of the same character. Deletion states connect to match and deletion states of the next character. Insertion states connect to themselves with a loop and the match state of the next character. The flow from previous to next characters ensures the correct order of the represented sequence in a pHMM graph.
Second, the training step maximizes the similarity score of sequences that are similar to the sequence that the pHMM graph represents. To this end, the training step uses additional input sequences as observation to modify the probabilities of the pHMM. The Baum-Welch algorithm~\cite{baum_inequality_1972} is a highly accurate training algorithm for pHMMs.
Third, the inference step aims to either 1)~calculate the similarity score of an input sequence to the sequence represented by a pHMM or 2)~identify the consensus sequence that generates the best similarity score from a pHMM graph. 1)~Calculating the similarity score is useful for applications such as protein family search and MSA. This is because pHMM graphs can avoid making redundant comparisons between sequences by comparing a sequence to a single pHMM graph that represents multiple sequences. Parts of the Baum-Welch algorithm (i.e., the Forward and Backward calculations) can be used in this step for calculating the scores~\cite{eddy_accelerated_2011}. 2)~The goal of generating the consensus sequence is to identify the modifications that need to be applied to the represented sequence. These modifications enable error correction tools to identify and correct the errors in DNA sequences. Decoding algorithms such as the Viterbi decoding~\cite{viterbi_error_1967} are commonly used for inference from pHMMs~\cite{kern_predicting_2013, friedrich_modelling_2006}.
\subsubsection{Components of pHMMs} \label{subsubsec:phmmcomponents}
We formally define the pHMM graph structure and its components. We assume that pHMM is a graph, $G(V, A)$, the sequence that the pHMM represents is $S_{G}$, and the length of the sequence is $n_{S_{G}}$. To accurately represent a sequence, pHMMs use four components: 1)~states, 2)~transitions, 3)~emission, and 4)~transition probabilities. We represent the \emph{states} and \emph{transitions} as the members of the sets $V$ and $A$, respectively. First, for each character of sequence $S_{G}$ at position $t$, $S_{G}[t] \in S_{G}$, pHMMs include $3$ consecutive states, $v_{3t}$, $v_{3t+1}$, and $v_{3t+2} \in V$: 1)~match, 2)~insertion, and 3)~deletion states. Each of these states modifies the character $S_{G}[t]$, inserts additional characters after $S_{G}[t]$, or deletes $S_{G}[t]$.
Second, pHMM graphs include transitions from state $v_i$ to state $v_j$, $\alpha_{ij} \in A$, such that the condition $i \leq j$ always holds true to preserve the correct order of characters in $S_{G}$.
Third, to define how probable to observe a certain character when a state is visited, emission probabilities are assigned for each character in a state. These emission probabilities can account for matches and substitutions in match states when comparing a sequence to a pHMM graph. We represent the emission probability of character $c$ in state $v_{i}$ as $e_c(v_{i})$.
Fourth, to identify the series of states to visit, probabilities are assigned to transitions. We represent the transition probability of a character between states $v_{i}$ and $v_{j}$ as $\alpha_{ij}$. These four main components build up the entire pHMM graph to represent a sequence and calculate the similarity scores when compared to other sequences.
\subsubsection{Identifying the Modifications} \label{subsubsec:modifications}
Figure~\ref{fig:phmm} shows three types of modifications that pHMMs can identify, 1)~insertions, 2)~deletions, and 3)~substitutions when comparing the sequence a pHMM represents (i.e., \texttt{PHMM Sequence} in Figure~\ref{fig:phmm}) to other sequences. First, insertion states can identify the characters that are missing from the pHMM sequence at a certain position. For example, Sequence \#1 in Figure~\ref{fig:phmm} includes three additional \texttt{G} characters after \texttt{A}. To identify such insertions, the highlighted insertion state \texttt{I} can be taken three times after visiting the state with label \texttt{A}. Second, deletion states can identify the characters that are deleted from the sequences we compare with the pHMM sequence. Sequence \#2 in Figure~\ref{fig:phmm} provides significant similarity to the pHMM sequence only with a single character missing. To identify the missing character, the highlighted deletion state is visited as it corresponds to deleting the second character in the pHMM sequence, \texttt{C}. Third, match states can identify the characters in sequences different than the character at the same position of a pHMM sequence, which we call substitutions. The states in Figure~\ref{fig:phmm} with DNA letters are match states and show the characters they represent in the corresponding pHMM sequence. The last character of Sequence \#3 is different than the last character of the pHMM sequence in Figure~\ref{fig:phmm}. Such a substitution is identified by visiting the highlighted match state of the last character of the pHMM sequence.
\subsection{The Baum-Welch Algorithm}\label{subsec:BW_alg}
To maximize or calculate the similarity score of input observation sequences in a pHMM graph, the Baum-Welch algorithm~\cite{baum_inequality_1972} solves an \emph{expectation-maximization} problem~\cite{moon_expectation-maximization_1996, tavanaei_training_2018, lindberg_petro-elastic_2015, hubin_adaptive_2019}, where the \emph{expectation} step calculates the statistical values based on an input sequence to train the probabilities of pHMMs. To this end, the algorithm performs the expectation-maximization based on an observation sequence $S$ for the pHMM graph $G(V,A)$ in three steps: 1)~forward calculation, 2)~backward calculation, and 3)~parameter updates.
\subsubsection{Forward Calculation}
The goal of performing the forward calculation is to compute the probability of observing sequence $S$ when we compare $S$ and $S_G$ from their first characters to the last characters. Equation~\ref{eq:forward} shows the calculation of the forward value $F_{t}(i)$ of state $v_i$ for character $S[t]$. The forward value, $F_{t}(i)$, represents the likelihood of emitting the character $S[t]$ in state $v_{i}$ given that \emph{all} previous characters $S[1 \dots t-1]$ are emitted by following an \emph{unknown} path \emph{forward} that leads to state $v_{i}$. $F_{t}(i)$ is calculated for all states $v_i \in V$ and for all characters of $S$. Although $t$ represents the position of the character of $S$, we use the \emph{timestamp} term for $t$ for the remainder of this paper. To represent transition and emission probabilities, we use the $\alpha_{ji}$ and $e_{S[t]}(v_{i})$ notations as we define in Section~\ref{subsubsec:phmmcomponents}.
\begin{equation}
F_{t}(i) = \sum_{j \in V} F_{t-1}(j) \alpha_{ji} e_{S[t]}(v_{i}) \enspace i \in V, \enspace 1 < t \leq n_{S} \tag{1} \label{eq:forward}
\end{equation}
\subsubsection{Backward Calculation}
The goal of the backward calculation is to compute the probability of observing sequence $S$ when we compare $S$ and $S_G$ from their last characters to the first characters. Equation~\ref{eq:backward} shows the calculation of the backward value $B_{t}(i)$ of state $v_i$ for character $S[t]$. The backward value, $B_{t}(i)$, represents the likelihood of emitting $S[t]$ in state $v_{i}$ given that \emph{all} further characters $S[t+1 \dots n_{S}]$ are emitted by following an \emph{unknown} path \emph{backwards} (i.e., taking transitions in reverse order). $B_{t}(i)$ is calculated for all states $v_i \in V$ and for all characters of $S$.
\begin{equation}
B_{t}(i) = \sum_{j \in V} B_{t+1}(j) \alpha_{ij} e_{S[t+1]}(v_{j}) \enspace i \in V, ~1 \leq t < n_{S} \tag{2} \label{eq:backward}
\end{equation}
\subsubsection{Parameter Updates}\label{subsec:parameterupdate}
The Baum-Welch algorithm uses the values that the forward and backward calculations generate for the observation sequence $S$ to \emph{update} the emission and transition probabilities in $G(V, A)$. The parameter update procedure maximizes the similarity score of $S$ in $G(V, A)$. This procedure updates the parameters as shown in Equations~\ref{eq:transition} and ~\ref{eq:emission}. The special $[S[t] = X]$ notation in Equation~\ref{eq:emission} is a conditional variable such that the variable returns $1$ if the character $X$ matches with the character $[S[t]$, and returns $0$ otherwise.
\begin{equation}
\alpha^{*}_{ij} = \dfrac{\sum\limits_{t=1}^{n_{S}-1} \alpha_{ij}e_{S[t+1]}(v_{j})F_{t}(i)B_{t+1}(j)}
{\sum\limits_{t=1}^{n_{S}-1}\sum\limits_{x \in V} \alpha_{ix}e_{S[t+1]}(v_{x})F_{t}(i)B_{t+1}(x)} \quad \forall \alpha_{ij} \in A
\tag{3} \label{eq:transition}
\end{equation}
\begin{equation}
e^{*}_{X}(v_{i}) = \dfrac{\sum\limits_{t=1}^{n_{S}} F_{t}(i)B_t(i)[S[t] = X]}
{\sum\limits_{t=1}^{n_{S}} F_{t}(i)B_t(i)} \quad \forall X \in \Sigma, \forall i \in V \tag{4} \label{eq:emission}
\end{equation}
\subsection{Use Cases for Profile HMMs}
\subsubsection{Error Correction}\label{subsec:apollo}
The goal of error correction is to locate the erroneous parts in DNA or genome sequences to replace these parts with more reliable sequences~\cite{vaser_fast_2017, hu_nextpolish_2020, huang_neuralpolish_2021, walker_pilon_2014, zimin_genome_2020, chin_nonhybrid_2013} to enable more accurate genome analysis (e.g., read mapping and genome assembly). Apollo~\cite{firtina_apollo_2020} is a recent error correction algorithm that takes an assembly sequence and a set of reads as input to correct the errors in an assembly. Apollo constructs a pHMM graph for an assembly sequence to correct the errors in two steps: 1)~training and 2)~inference. First, to correct erroneous parts in an assembly, Apollo uses reads as observations to train the pHMM graph with the Baum-Welch algorithm. Second, Apollo uses the Viterbi algorithm~\cite{viterbi_error_1967} to identify the consensus sequence from the trained pHMM, which translates into the corrected assembly sequence. Apollo uses a slightly modified design of pHMMs to avoid certain limitations associated with traditional pHMMs when generating the consensus sequences\cite{lyngso_consensus_2002, kahsay_quasi-consensus-based_2005}. The modified design avoids loops in the insertion states and uses transitions to account for deletions instead of deletion states. These modifications allow the pHMM-based error correction applications~\cite{lanyue_long_2020, firtina_hercules_2018, firtina_apollo_2020} to construct more accurate consensus sequences from pHMMs.
\subsubsection{Protein Family Search}\label{subsubsec:bg_proteinfamilysearch}
Classifying protein sequences into families is widely used to analyze the potential functions of the proteins of interest~\cite{mulder_tools_2001, jeffryes_rapid_2018, seo_deepfam_2018, vicedomini_multiple_2022, turjanski_natural_2018, bileschi_using_2022}. Protein family search finds the family of the protein sequence in existing protein databases. A pHMM usually represents one protein family in the database to avoid searching for many individual sequences. The protein sequence can then be assigned to a protein family based on the similarity score of the protein when compared to a pHMM in a database. This approach is used to search protein sequences in the Pfam database~\cite{mistry_pfam_2021}, where the HMMER~\cite{eddy_accelerated_2011} software suite is used to build HMMs and assign query sequences to the best fitting Pfam family.
Similar to the Pfam database, HMMER's protein family search tool is integrated into the European Bioinformatics Institute (EBI) website as a web tool.
The same approach is also used in several other important applications, such as classifying many genomic sequences into potential viral families~\cite{skewes-cox_profile_2014}.
\subsubsection{Multiple Sequence Alignment}
Multiple sequence alignment (MSA) detects the differences between several biological sequences. Dynamic programming algorithms can optimally find differences between genomic sequences, but the complexity of these algorithms increases drastically with the number of sequences~\cite{just_computational_2001, wang_complexity_1994}. To mitigate these computational problems, heuristics algorithms are used to obtain an approximate yet computationally efficient solution for multiple alignment of genomic sequences. PHMM-based approaches provide an efficient solution for MSA~\cite{chowdhury_review_2017}. The pHMM approaches, such as \emph{hmmalign}~\cite{eddy_accelerated_2011}, assign likelihoods to all possible combinations of differences between sequences to calculate the pairwise similarity scores using forward and backward calculations or other optimization methods (e.g., particle swarm optimization~\cite{zhan_probpfp_2019}). PHMM-based MSA approaches are mainly useful to avoid making redundant comparisons as a sequence can be compared to a pHMM graph, similar to protein family search.
\section{Motivation and Goal} \label{sec:motivation}
\subsection{Sources of Inefficiencies}\label{subsec:inefficiencies}
To identify and understand the performance bottlenecks of state-of-the-art pHMM-based applications, we thoroughly analyze existing tools for the three use cases of pHMM: 1)~Apollo~\cite{firtina_apollo_2020} for error correction, 2)~hmmsearch~\cite{eddy_accelerated_2011} for protein family search, and 3)~hmmalign~\cite{eddy_accelerated_2011} for multiple sequence alignment (MSA). We make six key observations based on our profiling with Intel VTune~\cite{profiler2022intel} and gprof~\cite{graham_gprof_2004}.
\textbf{Observation 1: The Baum-Welch Algorithm is the Bottleneck.} Figure~\ref{fig:motivation} shows the percentage of the execution time of all three steps in the Baum-Welch algorithm for the three bioinformatics applications. We find that the Baum-Welch algorithm is overall the \emph{performance bottleneck} for all three applications as the algorithm constitutes from 45.76\% up to $98.57\%$ of the total CPU execution time. Our profiling shows that these applications are mainly compute-bound. Forward and Backward calculations are the common steps in all three applications, whereas Parameter Updates step is executed only for error correction. This is because protein family search and MSA use the Forward and Backward calculations mainly for scoring between a sequence and a pHMM graph as a part of inference. We do \emph{not} include the cost of training for these applications as it is executed either once or only a few times, such that the cost of training becomes insignificant compared to the frequently executed inference. However, the nature of error correction requires frequently performing both training and inference for every input sequence such that the cost of training is not negligible for this application. As a result, accelerating the entire Baum-Welch algorithm is key for accelerating the end-to-end performance of the applications.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/motivation.pdf}
\caption{Percentage of the total execution time for the three steps of the Baum-Welch algorithm}
\label{fig:motivation}
\end{figure}
\textbf{Observation 2: SIMD-based tools provide suboptimal vectorization.}
The Baum-Welch algorithm requires frequent floating-point multiplications and additions.
To resolve performance issues, hmmalign and hmmsearch use SIMD instructions.
We observe that these tools have poor SIMD utilization due to poor port utilization and poor vector capacity usage (below $50\%$).
These issues show that optimizations for floating-point operations provide limited computational benefits when executing the Baum-Welch algorithm.
\textbf{Observation 3: Significant portion of floating-point operations is redundant.} We observe that the same multiplications are repeatedly executed in the training step because certain floating-point values associated with transition and emission probabilities are mainly constant during training in error correction. Our profiling analysis with VTune shows that these redundant computations constitute around $22.7\%$ of the overall execution time when using the Baum-Welch algorithm for training in error correction.
\textbf{Observation 4: Filtering the states is costly despite its advantages.} The Baum-Welch algorithm requires performing many operations for a large number of states. These operations are repeated in many iterations, and the number of states can grow in each iteration. There are several approaches to keep the state space (i.e., number of states) near-constant to improve the performance or the space efficiency of the Baum-Welch algorithm~\cite{kirkpatrick_optimal_2012, miklos_linear_2005, grice_reduced_1997, wheeler_optimizing_2000, tarnas_reduced_1998, firtina_hercules_2018, firtina_apollo_2020}. A simple approach is to pick the \texttt{best-n} states that provide the highest scores at each iteration while the rest of the states are ignored in the next iteration, known as filtering~\cite{firtina_hercules_2018}. Figure~\ref{fig:motivation-filtersize} shows the relation between the filter size (i.e., the number of states picked as \texttt{best-n} states), runtime, and accuracy. Although the filtering approach is useful for reducing the runtime without significantly degrading the overall accuracy of the Baum-Welch algorithm, such an approach requires extra computations (e.g., sorting) to pick the \texttt{best-n} states. We find that such a filtering approach incurs non-negligible performance costs by constituting around $8.5\%$ of the overall execution time in the training step.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/motivation-filtersize.pdf}
\caption{Effect of the filter size on the runtime and the accuracy of the Baum-Welch algorithm}
\label{fig:motivation-filtersize}
\end{figure}
\textbf{Observation 5: HMM accelerators are suboptimal for accelerating pHMMs.}
Generic HMMs do not require any constraints on the connection between states (i.e., transitions) and the number of states. PHMMs are a special case for HMMs where transitions are predefined, and the number of states is determined based on the sequence that a pHMM graph represents. These design choices in HMMs and pHMMs affect the data dependency pattern when executing the Baum-Welch Algorithm. Figure~\ref{fig:motivation-datadependency} shows an example of the data dependency patterns in pHMMs and HMMs when executing the Baum-Welch algorithm. We observe that although HMMs and pHMMs provide similar temporal localities (e.g., only the values from the previous iteration are used), pHMMs provide better spatial localities with their constrained design. This observation suggests that HMM-based accelerators cannot fully exploit the spatial localities that pHMMs provide as they are oblivious to the design of pHMMs.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/motivation-datadependency.pdf}
\caption{Data dependency in pHMMs and HMMs}
\label{fig:motivation-datadependency}
\end{figure}
\textbf{Observation 6: Non-genomics pHMM-based applications suffer from the computational overhead of the Baum-Welch algorithm.} Among many non-genomics pHMM-based implementations~\cite{ali_profile_2022, sasidharan_prodroid_2021, liu_adversarial_2019, pranamulia_profile_2017, ravi_behavior-based_2013, attaluri_profile_2009, riddell_reliable_2022, kazantzidis_profile_2018, saadi_framework_2016, ding_skeleton-based_2015, liu_characterizing_2015, liu_who_2009}, we analyze the available CPU implementation of a recent pattern-matching application that uses pHMMs~\cite{riddell_reliable_2022}. Our initial analysis shows that almost the entire execution time ($98\%$) of this application is spent on the Forward calculation, and it takes significantly longer times to execute a relatively small dataset compared to the bioinformatics applications.
Many applications use either the entire or parts of the Baum-Welch algorithm for training the probabilities of HMMs and pHMMs~\cite{bateman_pfam_2002, zhang_profile_2003, sgourakis_method_2005, friedrich_modelling_2006, steinegger_hh-suite3_2019, attaluri_profile_2009, eddy_accelerated_2011, firtina_hercules_2018, firtina_apollo_2020, lanyue_long_2020, boufounos_basecalling_2004, chen_detecting_2016, ali_profile_2022, sasidharan_prodroid_2021, ravi_behavior-based_2013, riddell_reliable_2022, kazantzidis_profile_2018, ding_skeleton-based_2015}.
However, the Baum-Welch algorithm can result in significant performance overheads on these applications due to computational inefficiencies.
Solving the inefficiencies in the Baum-Welch algorithm are mainly important for services that frequently use these applications, such as the EBI website using HMMER for searching protein sequences in protein databases~\cite{madeira_embl-ebi_2019}. Based on the latest report in 2018, there have been more than 28 million HMMER queries on the EBI website within two years (2016-2017)~\cite{potter_hmmer_2018}. On average, these queries execute parts of the Baum-Welch algorithm more than 38,000 times daily. Such frequent usage leads to significant waste in compute cycles and energy due to the inefficiencies in the Baum-Welch algorithm.
\subsection{Goal}
Based on our observations, we find that we need to have a specialized, flexible, high-performant, and energy-efficient design to \circlednumber{$1$} support different pHMM designs with specialized compute units for each step in the Baum-Welch algorithm, \circlednumber{$2$} eliminate redundant operations by enabling efficient reuse of the common multiplication products \circlednumber{$3$} exploit spatio-temporal locality in an on-chip memory, and \circlednumber{$4$} perform efficient filtering. Such a design has the potential to significantly reduce the computational and energy overhead of the applications that use the Baum-Welch algorithm in pHMMs. Unfortunately, software- or hardware-only solutions cannot solve these inefficiencies easily. There is a pressing need to develop a hardware-software co-designed and flexible acceleration framework for several pHMM-based applications that use the Baum-Welch algorithm.
In this work, our \textbf{goal} is to computational and energy overhead of the pHMMs-based applications that use the Baum-Welch algorithm with a flexible, high-performance, energy-efficient hardware-software co-designed acceleration framework. To this end, we propose \proposal, the \emph{first} highly flexible, high-performant, and energy-efficient accelerator that can support different pHMM designs to accelerate wide-range pHMM-based applications.
\section{ApHMM Design} \label{sec:aphmm}
\subsection{Microarchitecture Overview}\label{subsec:uarchreview}
\proposal provides a \textbf{flexible}, high-performant, and energy-efficient hardware-software co-designed acceleration framework for calculating each step in the Baum-Welch algorithm. Figure~\ref{fig:aphmm-overview} shows the main flow of \proposal when executing the Baum-Welch algorithm for pHMMs. To exploit the massive parallelism that DNA and protein sequences provide, \proposal processes many sequences in parallel using multiple copies of hardware units called \emph{\proposal Cores}. Each \proposal Core aims to accelerate the Baum-Welch algorithm for pHMMs. An \proposal Core contains two main blocks: 1)~Control Block and 2)~Compute Block. Control block provides efficient on- and off-chip synchronization and communication with CPU, DRAM, and on-chip L2 and L1 memory. Compute block efficiently and effectively performs each step in the Baum-Welch algorithm: 1)~Forward calculation, 2)~Backward calculation, and 3)~Parameter Updates with respect to their corresponding equations in Section~\ref{subsec:BW_alg}.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=1\linewidth]{figures/aphmm_workflow.pdf}
\caption{Overview of \proposal}
\label{fig:aphmm-overview}
\end{figure}
\proposal starts when the CPU loads necessary data to memory and sends the parameters to \proposal \circlednumber{$1$}. \proposal uses the parameters to decide on the pHMM design (i.e., either traditional pHMM design or modified design for error correction) and steps to execute in the Baum-Welch algorithm. The parameters related to design are sent to Compute Block so that each Compute Block can efficiently make proper state connections \circlednumber{$2$}. For each character in the input sequence that we aim to calculate the similarity score, Compute Block performs 1)~Forward, 2)~Backward, 3)~and Parameter Updates steps. \proposal enables disabling the calculation of Backward and Parameter Updates steps if they are not needed for an application. \proposal iterates over the entire input sequence to fully perform the Forward calculation with respect to Equation~\ref{eq:forward}\circlednumber{$3$}. \proposal then re-iterates each character on the input sequence character-by-character to perform the Backward calculations for each timestamp $t$ with respect to Equation~\ref{eq:backward} (i.e., step-by-step) \circlednumberb{$4.1$}. \proposal updates emission \circlednumberb{$4.2$} and transition probabilities \circlednumberb{$4.3$} as the Backward values are calculated in each timestamp.
\subsection{Control Block}\label{subsec:control}
Control Section is responsible for managing the input and output flow of the compute section efficiently and correctly by issuing both memory requests and proper commands to Compute Block to configure for the next set of operations (e.g., the forward calculation for the next character of sequence $S$). Figure~\ref{fig:aphmm-overview} shows three main units in Control Block.
\textbf{Parameters.} Control Block contains the parameters of pHMM and the Baum-Welch algorithm. These parameters define 1)~pHMM design (i.e., either the traditional design or modified design for error correction) and 2)~steps to execute in the Baum-Welch algorithm as \proposal allows disabling the calculation of Backward or Parameter Updates steps.
\textbf{Data Control.} To ensure the correct, efficient, and synchronized data flow, \proposal uses Data Control to 1) arbitrate among the read and write clients and 2) pipeline the read and write requests to the memory and other units in the accelerator (e.g., Histogram Filter). Data control is the main memory management unit for issuing a read request to L1 memory to obtain 1)~each input sequence $S$, 2)~corresponding pHMM graph (i.e., $G(V,A)$), 3) corresponding parameters and coefficients from the previous \emph{timestamp} (e.g., Forward coefficients from timestamp $t-1$ as shown in Equation~\ref{eq:forward}). Data Control collects and controls the write requests from various clients to ensure data is synchronized.
\textbf{Histogram Filter.} The filtering approach is useful for eliminating negligible states from Forward and Backward calculations without significantly degrading the accuracy (Section~\ref{sec:motivation}). The \textbf{challenge} for truthfully implementing a simple filtering mechanism is to perform sorting in hardware, which is challenging to implement efficiently. Our \textbf{key idea} is to replace the sorting mechanism with a histogram-based filter to enable placing the values into different bins based on their values. This provides quick and approximate identification of non-negligible states (i.e., states with best values until the filter is full) based on the bins they are located. To enable such a binning mechanism, we employ a \textbf{flexible} \emph{histogram-based} filtering mechanism in the \proposal on-chip memory.
Figure~\ref{fig:filtering-histogram} shows the overall structure of our Histogram Filter. Our filtering places the states into bins that correspond to a memory block based on their Forward or Backward values from the current timestamp of the execution. The Histogram Filter divides the entire range of single-precision floating-point numbers into 16 equal parts (e.g., the range between two parts is 4.25E\textsuperscript{+37}), where each bin corresponds to a range of predefined threshold values. We empirically chose to use 16 blocks to ensure our filtering mechanism achieves the same minimum accuracy when the filter size is 500 in Figure~\ref{fig:motivation-filtersize}. The addresses of the states are assigned such that all the states in between the same two threshold values fall into the same memory space block.~Such an addressing mechanism enables \proposal to efficiently discard states that fall under the chosen threshold value in the next timestamp as their addresses are already known without requiring sorting. To build a \textbf{flexible} framework for many applications, the microarchitecture is configurable to vary these threshold values based on the application and the average sequence length.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=0.7\linewidth]{figures/aphmm_histogram.pdf}
\caption{Overall structure of a Histogram Filter}
\label{fig:filtering-histogram}
\end{figure}
\proposal allows disabling the filtering mechanism if the application does not require a filter operation to achieve more optimal computations. Figure~\ref{fig:filtering-result} shows the trade-off when using \proposal with and without filter with sequences of different lengths. We observe that enabling the filtering mechanism provides significantly higher performance.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/aphmm_filtering.pdf}
\caption{Effect of the Histogram Filter approach in \proposal for different sequence lengths}
\label{fig:filtering-result}
\end{figure}
\subsection{Compute Block}\label{subsec:computeblock}
Figure~\ref{fig:aphmm-compute} shows the overall structure of a Compute Block. Compute Block is responsible for performing core compute operations of each step in the Baum-Welch algorithm (Figure~\ref{fig:aphmm-overview}) based on the configuration set by the Control Block via Index Control\circlednumber{$1$}. A Compute Block contains two major units: 1)~a unit for calculating Forward (Equation~\ref{eq:forward}) and Backward (Equation~\ref{eq:backward}) values\circlednumber{$2$} and updating transition probabilities (Equation~\ref{eq:transition}) \circlednumberb{$3.1$}, and 2)~a unit for updating the emission probabilities (Equation~\ref{eq:emission}) \circlednumberb{$3.2$}. Each unit performs the corresponding calculations in the Baum-Welch algorithm.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=1\linewidth]{figures/aphmm_compute.pdf}
\caption{Overview of a Compute Block. Red arrows show on- and off-chip memory requests.}
\label{fig:aphmm-compute}
\end{figure}
\subsubsection{Forward and Backward Calculations}\label{subsec:fb}
Our goal is to calculate the forward and backward values for all states in a pHMM graph $G(V,A)$, as shown in Equations~\ref{eq:forward} and ~\ref{eq:backward}, respectively. To calculate the Forward or Backward value of a state $i$ at a timestamp $t$, \proposal uses Processing Engines (PEs). Since pHMMs may require processing hundreds to thousands of states to process at a time, \proposal includes many PEs and groups them PE Groups. Each PE is responsible for calculating the Forward and Backward values of a state $v_i$ per timestamp $t$. Our \textbf{key challenge} is to balance the utilization of the compute units with available memory bandwidth. We address and discuss this trade-off between the number of PEs and memory bandwidth in Section~\ref{sec:hw-config}. To achieve efficient calculation of the Forward and Backward values, PE performs two main operations.
First, PE uses the parallel four lanes in \emph{Dot Product Tree} and \emph{Accumulator} to perform multiple multiply and accumulation operations in parallel, where the final summation is calculated in the Reduction Tree. This design enables efficient multiplication and summation of values from previous timestamps (i.e., $F_{t-1}(j)$ or $B_{t+1}(j)$). Second, to avoid redundant multiplications of transition and emission probabilities, \textbf{the key idea} in PEs is to efficiently enable the reuse of the products of these common multiplications. To achieve this, our key mechanism stores these common products in lookup tables (LUTs) in each PE while enabling efficient retrievals of the common products. We store these products as these values can be preset (i.e., fixed) before the training step starts and frequently used during training while causing high computational overheads.
Our \textbf{key challenge} is to design minimal but effective LUTs to avoid area and energy overheads associated with LUTs without compromising the computational efficiency LUTs provide. To this end, we analyze error correction, protein family search, and multiple sequence alignment implementations. We observe that 1) redundant multiplications are frequent only during training and 2) the alphabet size of the biological sequence significantly determines the number of common products (i.e., 4 in DNA and 20 in proteins). Since error correction is mainly bottlenecked during the training step, we focus on the DNA alphabet and the pHMM design that error correction uses. We identify that each state uses 1) at most $4$ different emission probabilities (i.e., DNA letters) and 2) on average $7$ different transitions. This results in $28$ different combinations of emission and transition probabilities. To enable slightly better flexibility, we assume $9$ different transitions and include 36 entries in LUTs.
The \textbf{key benefit} is LUTs provide \proposal with a bandwidth reduction of up to $66\%$ per PE while avoiding redundant computations. \proposal is \textbf{flexible} such that it enables disabling the use of LUTs and instead performing the actual multiplication of transition and emission probabilities with the shared \texttt{TE MUL} unit in Figure~\ref{fig:aphmm-compute}.
\subsubsection{Updating the Transition Probabilities}\label{subsubsec:transition}
Our goal is to update the transition probabilities of all the states, as shown in Equation~\ref{eq:transition}. To achieve this, we design the \emph{Update Transition (UT)} compute unit and tightly couple it with PEs, as shown in Figure~\ref{fig:aphmm-compute}. Each UT efficiently calculates the denominator and numerator in Equation~\ref{eq:transition} for a state $v_i$. UTs include two key mechanisms.
First, to enable efficient broadcasting of common values between Backward calculation and Parameter Updates steps, \proposal connects PEs with UTs for updating transitions. Each PE in a PE Group is broadcasted with the \emph{same} previously calculated $F_{t}(i)$ or $B_{t+1}(j)$ values from the previous timestamp for calculating the $F_{t+1}(j)$ or $B_{t}(i)$ values, respectively. Incoming red arrows in Figure~\ref{fig:aphmm-compute} show the flow of these values in PEs and UTs. This \textbf{key design choice} exploits the broadcast opportunities available within the common multiplications in the Baum-Welch equations. \proposal Cores are designed to directly consume the broadcasted Backward values in multiple steps of the Baum-Welch algorithm in parallel to reduce the bandwidth and storage requirements. We exploit the broadcasting opportunities because we observe that Backward values do not need to be fully computed, and they can be consumed as they are broadcasted in the current timestamp. We update Emission and Transition probabilities step-by-step as Backward values are calculated, which is a hardware-software optimization that we call partial compute approach. \textbf{The key benefits} of our broadcasting and partial compute approach are 1)~decoupling hardware scaling from bandwidth requirements and 2)~reducing the bandwidth requirement by $4\times$ (i.e., 32 bits/cycle instead of 128 bits/cycle).
Second, to exploit the spatio-temporal locality in pHMMs, we utilize on-chip memory in UTs with memoization techniques that allow us to store the recent transition calculations. We observe from Equation~\ref{eq:transition} that transition update is calculated using the values of states connected to each other. Since the connections are predefined and provide spatial locality (Figure~\ref{fig:motivation-datadependency}), our \textbf{key idea} is to memoize the calculation of all the numerators from the same $i$ to different states by storing these numerators in the same memory space. This enables us to process the same state $i$ in different timestamps within the same PE Engine to reduce the data movement overhead within \proposal. To this end, we use an 8KB on-chip memory (Transition Scratchpad) to store and reuse the result of the numerator of Equation~\ref{eq:transition}. Since we store the numerators that contribute to all the transitions of a state $i$ within the same memory space, we perform the final division in Equation~\ref{eq:transition} by using the values in the Transition Scratchpad. We use an 8KB memory as this enables us to store 256 different numerators from any state $i$ to any other state $j$. We observe that pHMMs have 3-12 distinct transitions per state. Thus, 8KB storage enables us to operate on at least 20 different states within the same PE. \textbf{The memoization technique allows} 1) skipping redundant data movement and 2) reducing the bandwidth requirement by $2 \times$ per UT.
\subsubsection{Updating the Emission Probabilities}\label{subsec:emission}
Our goal is to update the emission probabilities of all the states, as shown in Equation~\ref{eq:emission}. To achieve this, we use the \emph{Update Emission (UE)} unit, as shown in Figure~\ref{fig:aphmm-compute}, which includes three smaller units: 1)~Calculate Emission Numerator, 2)~Calculate Emission Denominator, and 3)~Division \& Update Emission. UE performs the numerator and denominator computations in parallel as they are independent of each other, which includes a summation of the products $F_{t}(i)B_t(i)$.
These $F_{t}(i)$ and $B_t(i)$ values are used to update \emph{both} the transition and emission probabilities, as shown in Equation~\ref{eq:transition}.
To reduce redundant computations, our \textbf{key design} choice is to use the $F_{t}(i)$ and $B_t(i)$ values as broadcasted in the transition update step since these values are also used for updating the emission probabilities. Thus, we broadcast these values to UEs through \emph{Write Selectors}, as shown in Figure~\ref{fig:aphmm-compute}.
The \proposal Core writes and reads both the numerator and denominator values to L1 memory to update the emission probabilities. The results of the division operations and the posterior emission probabilities (i.e., $e^{*}_{X}(v_{i})$ in Equation~\ref{eq:emission}) are written back to L1 memory after processing each read sequence $S$. If we assume that the number of characters in an alphabet $\Sigma$ is $n_{\Sigma}$ (e.g., $n_{\Sigma} = 4$ for DNA letters), \proposal stores $n_{\Sigma}$ many different numerators for each state of the graph as emission probability may differ per character for each state. Our microarchitecture design is \textbf{flexible} such that it allows defining $n_{\Sigma}$ as a parameter.
\subsection{Data distribution and L1 Memory Layout}\label{subsec:datadistribution}
To implement genomic sequence execution in a limited cache environment, the sequences are divided into \emph{chunks} of sequence lengths ranging from 150 to 1,000 characters to represent both sequencing reads and almost all protein sequences, as these protein sequences are mostly smaller than 1,000 characters~\cite{brocchieri_protein_2005}. For longer sequences, a sequence may be chunked into small pieces while preserving the relative order between sequences. An analysis of a similar software-level optimization reveals that chunking does not degrade the accuracy of the training and inference steps~\cite{firtina_apollo_2020}.
\proposal partitions the L1 memory into four major sections: 1) chunked sequences that can be fed directly to the \proposal Core, 2) a pHMM graph, 3) parameters to calculate the Baum-Welch algorithm, and 4) other temporary results generated by the \proposal Core. Chunking the memories into blocks is not hard coded, and each section can use more space if needed. \proposal identifies the sections in memory blocks using additional 2 bits that label these four sections. Figure~\ref{fig:l1_data_distribution} shows the size of different Baum-Welch parameters that must be stored in memory based on the sequence length. It also captures the details for storing the data efficiently across the memory hierarchy. Since the entire genomic data set is traditionally large, it is typically stored in DRAM, and only smaller subsets of the entire data are fetched to the L2 and L1 memory. Similarly, \proposal stores the entire forward values in DRAM and fetches them into L2 and L1 memory when needed. \proposal uses the L1 memory of 128KB to support a larger spectrum of sequence lengths ranging between 150-1000 characters. We show the data distribution in L1 memory in Figure~\ref{fig:l1_data_distribution}. Our key observation from the data distribution is that the size of Baum-Welch parameters grows as the sequence length increases. Thus, the number of sequences that L1 memory can hold reduces with increased read length. This does not cause frequent data load from DRAM or the L2 memory as longer sequences occupy the \proposal Core usually for a longer duration, which compensates for the less number of read sequences stored in L1.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/storage_distribution.pdf}
\caption{Data distribution across memory hierarchy.}
\label{fig:l1_data_distribution}
\end{figure}
\subsection{System Mapping and Execution Flow}\label{subsec:sysmapping}
We show a system-level scale-up version of the \proposal Core in Figure~\ref{fig:system_mapping}. \proposal uses the L2-DMA table to load the data into the L2 memory and the L1-DMA table to write the corresponding data into the L1 memory per \proposal Core according to the data distribution, as described in Section~\ref{subsec:datadistribution}. \proposal enables Probs-DMA to load the transition probabilities from DRAM to the local memory when the LUTs are not utilized, as discussed in Section~\ref{subsec:fb}. In such a scenario, local memory inside the PE Engine is loaded with appropriate transition probability data to perform the multiplications without using LUTs.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=0.8\linewidth]{figures/system-integration.pdf}
\caption{System integration of the \proposal Core.}
\label{fig:system_mapping}
\end{figure}
We present the execution flow of the system with multi-\proposal Core in Figure~\ref{fig:exec_flow}. The operation starts with the host loading the data into DRAM and issuing DMA across various memory hierarchies through a global event control. Each \proposal Core can start asynchronously, and near the completion of all reads from L1, hardware sets a flag for fetching the next set of sequences from L2. Similarly, a counter-based signaling tells L2 to fetch the next set of sequences from DRAM. Once all reads are issued, \proposal sends a completion signal and releases the control back to the host.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/execution_flow.pdf}
\caption{Control and execution flow of \proposal Cores.}
\label{fig:exec_flow}
\end{figure}
\subsection{Hardware Configuration Choice}\label{sec:hw-config}
Our goal is to identify the ideal number of memory ports and processing elements (PE) for better scaling \proposal with many cores. We identify the number of memory ports and their dependency on the hardware scaling in four steps.
First, \proposal requires one input memory port for reading the input sequence to update the probabilities in a pHMM graph.
Second, updating the transition probabilities requires $3$ memory ports: 1)~reading the forward value from L1, 2)~reading the transition and 3)~emission probabilities if using the LUTs is disabled (Section~\ref{subsec:fb}). Since these ports are shared across each PE Engine, the number of PEs and memory bandwidth per port determines the utilization of these memory ports.
Third, \proposal requires $4$ memory ports to update the emission probabilities for 1)~calculating the numerator and 2)~denominator in Equation~\ref{eq:emission}, 3)~reading the forward from Write Selectors, and 4)~writing the output. These memory ports are \emph{independent} of the impact of the number of PEs in a single \proposal Core.
Fourth, \proposal does not require additional memory ports for each step in the Baum-Welch algorithm as a result of the broadcasting feature of \proposal (Section~\ref{subsubsec:transition}). Instead, computing these steps depends on the 1)~memory bandwidth per port, which determines the number of multiplications and accumulations in parallel in a PE, and 2)~number of processing engines (PEs).
We conclude that the overall requirement for the \proposal Core is $8$ memory ports with the same bandwidth per port.
In Figure~\ref{fig:hw_scaling}(a), we show the acceleration speedup while scaling \proposal with the number of PEs and bandwidth per memory port, where we keep the number of memory ports fixed to $8$. Based on Figure~\ref{fig:hw_scaling}(a), we observe that a linear trend of increase in acceleration is possible until the number of PEs reaches $64$, where the rate of acceleration starts reducing. We explore the reason for such a trend in Figure~\ref{fig:hw_scaling}(b). We find that the acceleration on the transition step starts settling down as the number of PEs grows due to memory port limitation that reduces parallel data read from memory per PE, eventually resulting in the underutilization of resources. We conclude that the acceleration trend we observe in Figure~\ref{fig:hw_scaling}(a) is mainly due to the scaling impact on the forward and backward calculation when the number of PEs is greater than $64$ where $8$ memory ports start becoming the bottleneck.
Choosing the memory bandwidth affects the number of PE Groups and PE Engines while keeping the number of PEs constant. Although our hardware can scale for higher bandwidth, we choose $16$ Bytes/cycle, which results in 4 PE Engines ($128$ bit$/32$ FP32), and 16 PE Groups ($64$PEs$/4=16$). This configuration enables us to choose a smaller Transition Scratchpad with the increased parallelism across PE Engines without significantly compromising performance.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/hw-config.pdf}
\caption{(a) Acceleration scaling with number of processing element. (b) Transition probability compute cycle acceleration with the increased number of PEs.}
\label{fig:hw_scaling}
\end{figure}
\subsubsection{Number of \proposal Cores}\label{subsec:aphmmcores}
We show our methodology for choosing the ideal number of \proposal-cores for accelerating the applications. Figure~\ref{fig:aphmm_cores} shows the speedup of three bioinformatics applications when using single, 2, 4, and 8 \proposal Cores. We incorporate the estimated off- and on-chip data movement overhead in our analysis. We observe that using 4 \proposal-cores provides the best speedup overall. This is because the applications provide smaller rooms for acceleration, and the data movement overhead starts becoming the bottleneck as we increase the number of cores. This observation suggests that there is still room for improving the performance of \proposal by placing \proposal inside or near the memory (e.g., high-bandwidth memories) to eliminate these data movement overheads. We use 4-core \proposal to achieve the best performance.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/aphmm_cores.pdf}
\caption{Normalized runtimes of multi-core \proposal compared to the single-core \proposal (\proposal-1).}
\label{fig:aphmm_cores}
\end{figure}
\section{Evaluation} \label{sec:results}
We evaluate our acceleration framework, \proposal, for three use cases:~1)~error correction, 2)~protein family search and 3)~multiple sequence alignment (MSA). We compare our results with the CPU, GPU, and FPGA implementations of the use cases.
\subsection{Evaluation Methodology}\label{subsec:evalmethod}
We use the configurations shown in Table~\ref{tab:ApHMM_parameters} to implement the \proposal design described in Section~\ref{sec:aphmm} in SystemVerilog. We carry out synthesis using Synopsys Design Compiler~\cite{noauthor_tool_nodate} in a typical 28nm process technology node at 1GHz clock frequency with tightly integrated on-chip memory (1GHz) to extract the logic area and power numbers. We develop an analytical model to extract performance and area numbers for a scale-up configuration of \proposal. We use 4 \proposal cores in our evaluation (Section~\ref{subsec:aphmmcores}). We account for 5\% extra cycles to compensate for arbitrating the across memory ports. These extra cycles estimate the cycles for synchronously loading data from DRAM to L2 memory of a single \proposal core and asynchronous accesses to DRAM when more data needs to be from DRAM for a core (e.g., Forward calculation may not fit the L2 memory).
\begin{table}[tbh]
\centering
\caption{Microarchitecture Configuration}\label{tab:ApHMM_parameters}
\resizebox{\columnwidth}{!}{
\begin{tabular}{@{}ll@{}}\toprule
Memory & Memory BW (Bytes/cycle): 16, Memory Ports (\#): 8\\
& L1 Cache Size: 128KB \\\cmidrule{1-2}
Processing & PEs (\#): 64, Multipliers per PE (\#): 4, Adders per PE (\#): 4\\
Engine & Memory per PE: 8, Update Transitions (\#): 64, Update Emissions (\#): 4 \\ \bottomrule
\end{tabular}}
\end{table}
We use the CUDA library~\cite{nickolls_scalable_2008} (version 11.6) to provide a GPU implementation of the software optimizations described in Section~\ref{sec:aphmm} for executing the Baum-Welch algorithm. Our GPU implementation, \textbf{\proposal-GPU}, uses the pHMM design designed for error correction, implements LUTs (Section~\ref{subsec:fb}) as a shared memory, and uses buffers to arbitrate between current and previous Forward/Backward calculations to reflect the software optimizations of \proposal in GPUs. We integrate our GPU implementation with a pHMM-based error correction tool, Apollo~\cite{firtina_apollo_2020}, to evaluate the GPU implementation. Our GPU implementation is the \emph{first} GPU implementation of the Baum-Welch algorithm for profile Hidden Markov models.
We use gprof~\cite{graham_gprof_2004} to profile the baseline CPU implementations of the use cases on the AMD EPYC 7742 processor (2.26GHz, 7nm process) with single- and multi-threaded settings. We use the CUDA library and \emph{nvidia-smi} to capture the runtime and power usage of \proposal-GPU on NVIDIA A100 and NVIDIA Titan V GPUs, respectively.
We compare \proposal with the CPU, GPU, and FPGA implementations of the Baum-Welch algorithm and use cases in terms of execution time and energy consumption. To evaluate the Baum-Welch algorithm, we execute the algorithm in Apollo~\cite{firtina_apollo_2020} and calculate the average execution time and energy consumption of a single execution of the Baum-Welch algorithm. To evaluate the end-to-end execution time and energy consumption of error correction, protein family search, and multiple sequence alignment, we use Apollo~\cite{firtina_apollo_2020}, hmmsearch~\cite{eddy_accelerated_2011}, and hmmalign~\cite{eddy_accelerated_2011}. We replace their implementation of the Baum-Welch algorithm with \proposal when collecting the results of the end-to-end executions of the use cases accelerated using \proposal. When available, we compare the use cases that we accelerate using \proposal to their corresponding CPU, GPU, and FPGA implementations. For the GPU implementations, we use both \proposal-GPU and HMM\_cuda~\cite{yu_gpu-accelerated_2014}. For the FPGA implementation, we use the FPGA Divide and Conquer (D\&C) accelerator proposed for the Baum-Welch algorithm~\cite{pietras_fpga_2017}. When evaluating the FPGA accelerator, we ignore the data movement overhead and estimate the acceleration based on the speedup as provided by the earlier work.
\subsubsection{Data Set}
To evaluate the error correction use case, we prepare the input data that Apollo requires: 1) assembly and 2) read mapping to the assembly. To construct the assembly and map reads to the assembly, we use reads from a real sample that includes overall 163,482 reads of Escherichia coli (E.coli) genome sequenced using PacBio sequencing technology. The accession code of this sample is SAMN06173305. Out of 163,482 reads, we randomly select 10,000 sequencing reads for our evaluation. We use minimap2~\cite{li_minimap2_2018} and miniasm~\cite{li_minimap_2016} to 1) find \emph{overlapping reads} and 2) construct the assembly from these overlapping reads, respectively. To find the read mappings to the assembly, we use minimap2 to map the same reads to the assembly that we generate using these reads. We provide these inputs to Apollo for correcting errors in the assembly we construct.
To evaluate the protein family search, we use the protein sequences from a commonly studied protein family, Mitochondrial carrier (PF00153), which includes 214,393 sequences with an average length of 94.2. We use these sequences to search for similar protein families from the entire Pfam database~\cite{mistry_pfam_2021} that includes 19,632 pHMMs. To achieve this, the hmmsearch~\cite{eddy_accelerated_2011} tool performs the Forward and Backward calculations to find similarities between pHMMs and sequences.
To evaluate multiple sequence alignment, we use 1,140,478 protein sequences from protein families Mitochondrial carrier (PF00153), Zinc finger (PF00096), bacterial binding protein-dependent transport systems (PF00528), and ATP-binding cassette transporter (PF00005). We align these sequences to the pHMM graph of the Mitochondrial carrier protein family. To achieve this, the hmmalign~\cite{eddy_accelerated_2011} tool performs the Forward and Backward calculations to find similarities between a single pHMM graph and sequences.
\subsection{Area and Power}\label{subsec:overhead}
Table~\ref{tab:ApHMM_area} shows the area breakup of the major modules in \proposal. For the area overhead, we find that the Update Transition (UT) units take up most of the total area ($77.98\%$). This is mainly because UTs consist of several complex units, such as a multiplexer, division pipeline, and local memory. For the power consumption, Control Block and PEs contribute to almost the entire power consumption ($86\%$) due to the frequent memory accesses these blocks make. Overall, the \proposal Core incurs an area overhead of 6.5mm\textsuperscript{2} in 28nm with a power cost of 0.509W.
\begin{table}[tbh]
\centering
\caption{Area and Power breakdown of \proposal}\label{tab:ApHMM_area}
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{@{}lrr@{}}\toprule
\textbf{Module Name} & \textbf{Area (mm$^2$)} & \textbf{Power (mW)} \\ \midrule
Control Block & 0.011 & 134.4 \\
64 Processing Engines (PEs) & 1.333 & 304.2 \\
64 Update Transitions (UTs) & 5.097 & 0.8 \\
4 Update Emissions (UEs) & 0.094 & 70.4 \\
\textbf{Overall} & 6.536 & 509.8 \\\cmidrule{1-3}
128KB L1-Memory & 0.632 & 100 \\ \bottomrule
\end{tabular}}
\end{table}
\subsection{Accelerating the Baum-Welch Algorithm}\label{subsec:baumwelch}
Figure~\ref{fig:baum_energy} shows the performance and energy improvements of \proposal for executing the Baum-Welch algorithm. Based on these results, we make six \emph{key} observations.
First, we observe that \proposal is \pbaumcpuc - \pbaumcpua, \pbaumgpud-\pbaumgpua, and \pbaumfpga faster than the CPU, GPU, and FPGA implementations of the Baum-Welch algorithm, respectively.
Second, \proposal reduces the energy consumption for calculating the Baum-Welch algorithm by \ebaumcpua and \ebaumgpud-\ebaumgpua compared to the single-threaded CPU implementation and GPU implementations, respectively. These speedups and reduction in energy consumption show the combined benefits of our software-hardware optimizations.
Third, the parameter update step is the most time-consuming step for the CPU and the GPU implementations, while \proposal takes the most time in the forward calculation step. The reason for such a trend shift is that \proposal reads and writes to L2 Cache and DRAM more frequently during the forward calculation than the other steps, as \proposal requires the forward calculation step to be fully completed and stored in the memory before moving to the next steps as we explain in Section~\ref{subsubsec:transition}.
Fourth, we observe that \proposal-GPU performs better than HMM\_cuda by \pbaumgpucomp on average. HMM\_cuda executes the Baum-Welch algorithm on any type of hidden Markov model without a special focus on pHMMs. As we develop our optimizations based on pHMMs, \proposal-GPU can take advantage of these optimizations for more efficient execution.
Fifth, both \proposal-GPU and HMM\_cuda provide better performance for the Forward calculation than \proposal. We believe that the GPU implementations are a better candidate for applications that execute only the Forward calculations as \proposal targets providing the best performance for the complete Baum-Welch algorithm.
Sixth, the GPU implementations provide a limited speedup over the multi-threaded CPU implementations mainly because of frequent access to the host for synchronization and sorting (e.g., the filtering mechanism). These required accesses from GPU to host can be minimized with a specialized hardware design, as we propose in \proposal for performing the filtering mechanism.
We conclude that \proposal provides substantial improvements, especially when we combine speedups and energy reductions for executing the complete Baum-Welch algorithm compared to the CPU and GPU implementations, which makes it a better candidate to accelerate the applications that use the Baum-Welch algorithm than the CPU, GPU, and FPGA implementations.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/baum_speedup-energy.pdf}
\caption{(a) Normalized speedups of each step in the Baum-Welch algorithm over single-threaded CPU (CPU-1). (b) Energy reductions compared to the CPU-1 implementation of the Baum-Welch algorithm and three pHMM-based applications.}
\label{fig:baum_energy}
\end{figure}
\subsection{Use Case 1: Error Correction}
Figures~\ref{fig:aphmm_applications} and \ref{fig:baum_energy} show the end-to-end execution time and energy reduction results for error correction, respectively. We make four key observations.
First, we observe that \proposal is \perrcpuc - \perrcpua, \perrgpud - \perrgpua, and \perrfpga faster than the CPU, GPU, and FPGA implementations of Apollo, respectively.
Second, \proposal reduces the energy consumption by \eerrcpua and \eerrgpud - \eerrgpua compared to the single-threaded CPU and GPU implementations. These two observations are in line with the observations we make in Section~\ref{subsec:baumwelch} as well as the motivation results we describe in Section~\ref{sec:motivation}: Apollo is mainly bounded by the Baum-Welch algorithm, and \proposal accelerates the Baum-Welch algorithm significantly, providing significant performance improvements and energy reductions for error correction.
We conclude that \proposal \emph{significantly} improves the energy efficiency and performance of the error correction mainly because the Baum-Welch algorithm constitutes a large portion of the entire use case.
\begin{figure}[tbhp!]
\centering
\includegraphics[width=\linewidth]{figures/use_cases.pdf}
\caption{Speedups over the single-threaded CPU implementations. In protein family search, we compare \proposal with each CPU thread separately.}
\label{fig:aphmm_applications}
\end{figure}
\subsection{Use Case 2: Protein Family Search}
Our goal is to evaluate the performance and energy consumption of \proposal for the protein family search use case, as shown in Figures~\ref{fig:aphmm_applications} and \ref{fig:baum_energy}, respectively. We make three key observations.
First, we observe that \proposal provides speedup by \pprocpuc - \pprocpua, and \pprofpga compared to the CPU and FPGA implementations.
Second, \proposal is \eprocpua more energy efficient than the single-threaded CPU implementation. The speedup ratio that \proposal provides is lower in protein family search than error correction because 1) \proposal accelerates a smaller portion of protein family search ($45.76\%$) than error correction ($98.57\%$), and 2) the protein alphabet size (20) is much larger than the DNA alphabet size (4), which increases the DRAM access overhead of \proposal by $12.5\%$. Due to the smaller portion that \proposal accelerates and increased memory accesses, it is expected that \proposal provides lower performance improvements and energy reductions compared to the error correction use case. Third, \proposal can provide better speedup compared to the multi-threaded CPU as a large portion of the parts that \proposal does not accelerate can still be executed in parallel using the same amount of threads, as shown in Figure~\ref{fig:aphmm_applications}.
We conclude that \proposal improves the performance and energy efficiency for protein family search, while there is a smaller room for acceleration compared to error correction.
\subsection{Use Case 3: Multiple Sequence Alignment}
Our goal is to evaluate the \proposal's end-to-end performance and energy consumption for multiple sequence alignment (MSA), as shown in Figures~\ref{fig:aphmm_applications} and \ref{fig:baum_energy}, respectively. We make three key observations.
First, we observe that \proposal performs \pmsacpua and \pmsafpga better than the CPU and FPGA implementations, while \proposal is \emsacpua more energy efficient than the CPU implementation of MSA. We note that the hmmalign tool does not provide the multi-threaded CPU implementation for MSA.
\proposal provides better speedup for MSA than protein family search because MSA performs more forward and backward calculations ($51.44\%$) than the protein search use case ($45.76\%$), as shown in Figure~\ref{fig:motivation}. Third, \proposal provides slightly better performance than the existing FPGA accelerator (FPGA D\&C) in all applications, even though we ignore the data movement overhead of FPGA D\&C, which suggests that \proposal may perform much better than FPGA D\&C in real systems.
We conclude that \proposal improves the performance and energy efficiency of the MSA use case better than protein family search.
\section{Related Work}\label{sec:relatedWork}
To our knowledge, this is the first work that provides a flexible and hardware-software co-designed acceleration framework to efficiently and effectively execute the complete Baum-Welch algorithm for pHMMs. In this section, we explain previous attempts to accelerate \emph{HMMs}. Previous works~\cite{ibrahim_reconfigurable_2016, soiman_parallel_2014, huang_hardware_2017, yu_gpu-accelerated_2014, pietras_fpga_2017, eddy_accelerated_2011, ren_fpga_2015, li_improved_2021, wertenbroek_acceleration_2019, banerjee_accelerating_2017, wu_high-throughput_2021, wu_173_2020, jiang_cudampf_2018, quirem_cuda_2011, derrien_hardware_2008, oliver_high_2007, oliver_integrating_2008} mainly focus on specific algorithms and designs of HMMs to accelerate the HMM-based applications. Several works~\cite{ibrahim_reconfigurable_2016, jiang_cudampf_2018, quirem_cuda_2011, derrien_hardware_2008, oliver_high_2007, oliver_integrating_2008} propose FPGA- or GPU-based accelerators for pHMMs to accelerate a different algorithm used in the inference step for pHMMs.
A group of previous works~\cite{huang_hardware_2017, soiman_parallel_2014, ren_fpga_2015, li_improved_2021} accelerates the Forward calculation based on the HMM designs different than pHMMs for FPGAs and supercomputers. HMM\_cuda~\cite{yu_gpu-accelerated_2014} uses GPUs to accelerate the Baum-Welch algorithm for any HMM design. \proposal differs from all of these works as it accelerates the entire Baum-Welch algorithm on pHMMs for more optimized performance, while these works are oblivious to the pHMM design when accelerating the Baum-Welch algorithm.
A related design choice to pHMMs is Pair HMMs. Pair HMMs are useful for identifying differences between DNA and protein sequences. To identify differences, Pair HMMs use states to represent a certain scoring function (e.g., affine gap penalty) or variation type (i.e., insertion, deletion, mismatch, or match) by typically using only one state for each score or difference.
This makes Pair HMMs a good candidate for generalizing pairwise sequence comparisons as they can compare pairs of sequences while being oblivious to any sequence. Unlike pHMMs, Pair HMMs are not built to represent sequences. Thus, Pair HMMs cannot 1)~compare a sequence to a group of sequences and 2)~perform error correction. Pair HMMs mainly target variant calling and sequence alignment problems in bioinformatics. There is a large body of work that accelerates Pair HMMs~\cite{huang_hardware_2017, li_improved_2021, ren_fpga_2015, wertenbroek_acceleration_2019, banerjee_accelerating_2017, wu_high-throughput_2021, wu_173_2020}. \proposal differs from these works as its hardware-software co-design is optimized for pHMMs.
\section{Conclusion} \label{sec:conclusion}
We propose \proposal, the \emph{first} hardware-software co-design framework that accelerates the execution of the entire Baum-Welch algorithm for pHMMs.
\proposal particularly accelerates the Baum-Welch algorithm as it is a common computational bottleneck for important bioinformatics applications. \proposal proposes several hardware-software optimizations to efficiently and effectively execute the Baum-Welch algorithm for pHMMs. The hardware-software co-design of \proposal provides significant performance improvements and energy reductions compared to CPU, GPU, and FPGAs, as \proposal minimizes redundant computations and data movement overhead for executing the Baum-Welch algorithm.
We hope that \proposal enables further future work by accelerating the remaining steps used with pHMMs (e.g., Viterbi decoding) based on the optimizations we provide in \proposal.
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\section{Introduction} For a given graph $H$, the Ramsey number $r(H)$ is defined to be the smallest integer $n$ such that for any two-coloring of the edges of the complete graph on $n$ vertices, $K_n$, we can find a monochromatic copy of $H$. In this paper we will consider $\textit{ordered Ramsey numbers}$, which are an analogue of Ramsey numbers for ordered graphs. The systematic study of ordered Ramsey numbers began with a 2014 paper by Conlon, Fox, Lee, and Sudakov \cite{conlon}.
An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. We say that an ordered graph $G$ on $N$ vertices \textit{contains} an ordered graph $H$ on $n$ vertices if there is a map $\phi: [n] \to [N]$ such that $\phi(i) < \phi(j)$ for $1 \leq i < j \leq n$ and such that if $(i,j) \in E(H)$, then $(\phi(i), \phi(j)) \in E(G)$. \cite{conlon} Thus the containment is order-preserving in the sense that given a copy of $H$ in $G$ the lowest ordered vertex (by the ordering of $G$) in the copy must correspond to the vertex labeled 1 in $H$ and so on. For example, if $H$ is the cycle on four vertices with labeling $\{1,2,3,4\}$ where $E(H)=\{(1,2),(2,3),(3,4),(4,1)\}$, then a possible monochromatic copy of $H$ in some larger graph $G$ could be on vertices $\{2,5,7,9\}$ with monochromatic edges $\{(2,5),(5,7),(7,9),(9,2)\}$.
Then we can define the $\textit{ordered Ramsey number}$ , $r_< (H)$, of an ordered graph $H$ to be smallest integer $n$ such that for any ordering and any two-coloring of $K_n$ we can find a monochromatic, order-preserving copy of $H$ contained in $K_n$. Recall that a \textit{coloring} by $m$ colors of the edges $E(G)$ of a graph $G$ is a map $c: E(G) \to [m]$. The first observation we can make about ordered Ramsey numbers is that for any ordering of a graph $H$, we clearly have $r(H) \leq r_<(H)$ where $r(H)$ is the usual Ramsey number of an unordered $H$ and $r_<(H)$ is the ordered Ramsey number. This gives us a trivial lower bound. Also observe that the trivial upper bound for the ordered Ramsey number of an ordered graph $H$ on $n$ vertices is the usual Ramsey number of $K_n$. This follows from the following lemma, whose truth is easy to see.
\begin{Lemma}
An ordered monochromatic complete graph on $n$ vertices necessarily contains an ordered copy of any ordered graph on $n$ vertices, regardless of its ordering.
\end{Lemma}
This is clear since for any ordered graph on $n$ vertices we can find vertices in $K_n$ with the ordering we want and we already know all the edges are monochromatic. Despite its simplicity, Lemma 1.1 will be useful in many of our proofs.
The paper by Conlon et al. proved a number of results for ordered Ramsey numbers of certain infinite families of graphs \cite{conlon}. Balko et al. established results for ordered Ramsey numbers on particular orderings of certain graph families such as paths, stars, and cycles \cite{balko}. Thus we will not investigate these graphs on four vertices in this paper. So far the only paper focusing on proving ordered Ramsey results for small graphs was by Chang in \cite{chang}. In that paper, Chang proved upper bounds for Ramsey numbers of 1-orderings for graphs on 4 vertices. A 1-ordering of a graph $H$ on $n$ vertices consists of a labeling of just one vertex with some integer from $\{1,...,n\}$. Then a copy of $H$ in some ordered complete graph just needs to preserve the ordering of this given vertex. Here we will focus on complete orderings of graphs on 4 vertices.
In this paper we will prove upper bounds on the Ramsey numbers for certain total orderings of graphs on four vertices. Specifically, in Section \ref{sec:K2} we will examine orderings of $K_2 \cup K_2$, in Section \ref{sec:K4e} we examine orderings of the diamond graph, and in Section \ref{sec:pan} we examine the 3-pan graph. In section \ref{sec:pendant} we extend our upper bound of the 3-pan graph to the infinite family of complete graphs with a pendant edge. Specifically, we will prove the following, where the definition of a ``complete with 1-pendant'' graph is given in Section \ref{sec:pendant}.
\begin{thm}
The ordered Ramsey number of the complete with 1-pendant graph on $n+1$ vertices is $R(n)+2n-1$.
\end{thm}
Note that upper bounds on some orderings immediately give the same upper bound on ``symmetric'' orderings. By ``symmetric'' orderings, we mean that if we had a graph with vertices $a,b,c,d$ labeled $1,2,3,4$, then an upper bound on this ordering of the graph would also apply to the ordering $4,3,2,1$ by just ``flipping'' the argument. We will not explicitly mention when this symmetry applies to our results, but it is possible to apply it to a number of our results.
\section{Ordered Ramsey Numbers of $K_2 \cup K_2$}\label{sec:K2}
The proofs for upper bounds on orderings of $K_2 \cup K_2$ will be relatively straightforward, but hopefully illustrative of techniques we will use on other graphs. Also, we will be able to exhibit constructions showing that our lower bounds are tight for some orderings of $K_2 \cup K_2$, thus completely determining the ordered Ramsey number for those orderings.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{2k2ver1.jpg}
\caption{Ordering $A$ of $K_2 \cup K_2$}
\end{figure}
We will first investigate the ordering of $K_2 \cup K_2$ given in Figure 1, which we will refer to as ordering $A$.
\pagebreak
\begin{Prop}
The ordered Ramsey number of $K_2 \cup K_2$ with ordering $A$ is 6.
\end{Prop}
\begin{proof} We will first show that $r_{<_A}(K_2 \cup K_2) \leq 6$. Recall that a \textit{coloring} by $m$ colors of the edges $E(G)$ of a graph $G$ is a surjective map $c:[m] \to E(G)$. So when we say ``color'' an edge, we mean choosing some color $\{1,...,m\}$ to assign to the edge, and here specifically we only deal with the case where $m=2$ and we have colors red and blue. Without loss of generality, color the edge between vertices 1 and 2 red. Then we know that all of the edges between vertices 3 through 6 must be blue, or else we would have an ordered copy of $K_2 \cup K_2$. But then we have a complete blue graph on four vertices, and thus by Lemma 1.1 we have a monochromatic copy of $K_2 \cup K_2$. So $r_{<_A}(K_2 \cup K_2) \leq 6$.
Now we will demonstrate that we can find a complete graph on 5 vertices with an edge coloring that does not produce a copy of $K_2 \cup K_2$. Color every edge from 1 to $\{2,3,4,5\}$ red. Then color the edge from 2 to 3 red. Color every other edge blue. See Figure 2. We claim that this graph does not have a monochromatic copy of $K_2 \cup K_2$. There is no red $K_2 \cup K_2$ since every red edge except $(2,3)$ originates from 1, and thus to find a copy of $K_2 \cup K_2$ with ordering $A$ we would need to find an edge with a higher lowest vertex than 1, which is only given by $(2,3)$, but there is no copy of $K_2 \cup K_2$ with $(2,3)$ either. So there is no red $K_2 \cup K_2$.
We can also see that there is no blue $K_2 \cup K_2$. Since every edge from 1 is red, a monochromatic copy of $K_2 \cup K_2$ would have to be within the subgraph induced by the vertices $\{2,3,4,5\}$. The only possible way to get a blue copy of $K_2 \cup K_2$ on these vertices is if we have a blue $(2,3)$ and a blue $(4,5)$, which is not the case. So $r_{<_A}(K_2 \cup K_2) > 5$, which proves our result $r_{<_A}(K_2 \cup K_2) = 6$.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{lowerbound2k2A.jpg}
\caption{Colored $K_5$ with no $K_2 \cup K_2$ having ordering $A$}
\end{figure}
Now we will examine the ordering on $K_2 \cup K_2$ given in Figure 3, which we will refer to as ordering $B$.
\begin{figure}[h]
\centering
\includegraphics[width=5cm]{2k2verb.jpg}
\caption{Ordering $B$ of $K_2 \cup K_2$}
\end{figure}
\begin{Prop}
The ordered Ramsey number of $K_2 \cup K_2$ with ordering $B$ is 5.
\end{Prop}
\begin{proof}
Note that we trivially have that $r_{<_B}(K_2 \cup K_2) \geq 5$ since the usual Ramsey number of $K_2 \cup K_2$ is 5 \cite{small}, and we noted in the introduction that $r(H) \leq r_<(H)$ for any ordering of $H$. So we only need to prove that $r_{<_B}(K_2 \cup K_2) \leq 5$.
Without loss of generality, color the edge $(1,3)$ red. This forces the edges $(2,4)$ and $(2,5)$ to both be blue. And since $(2,5)$ is blue, the edge $(1,4)$ is forced to be red, which then forces the edge $(3,5)$ to be blue. However, this gives us a blue copy of $K_2 \cup K_2$ with ordering $B$ having edges $(2,4)$ and $(3,5)$. So $r_{<_B}(K_2 \cup K_2) \leq 5$, which proves $r_{<_B}(K_2 \cup K_2) = 5$.
\end{proof}
The last ordering of $K_2 \cup K_2$ to consider is given in Figure 4, which we will refer to as ordering $C$.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{2k2verc.jpg}
\caption{Ordering $C$ of $K_2 \cup K_2$}
\end{figure}
\begin{Prop}
The ordered Ramsey number of $K_2 \cup K_2$ with ordering $C$ is 6.
\end{Prop}
\begin{proof}
First we will show that $r_{<_C}(K_2 \cup K_2) \leq 6$. Without loss of generality color edge $(1,4)$ red. Then this forces $(2,3)$ to be blue. Then $(2,3)$ being blue, forces edges $(1,5)$ and $(1,6)$ to be red. Then edge $(1,6)$ being red forces edges $(2,4)$ and $(2,5)$ to be blue. But now any coloring of $(3,4)$ will gives us a monochromatic copy of $K_2 \cup K_2$. If $(3,4)$ is red, then it forms a copy with $(1,6)$, while if $(3,4)$ is blue, then it forms a copy with $(2,5)$. Thus we have that $r_{<_C}(K_2 \cup K_2) \leq 6$.
Now we will show that $r_{<_C}(K_2 \cup K_2) > 5$ by exhibiting an ordered $K_5$ with no copy of $K_2 \cup K_2$ having ordering $C$. Color every edge from 1 red and color every edge from 5 red. Then color the triangle formed by $\{2,3,4\}$ blue. This clearly doesn't have a blue copy. To see that it doesn't have a red copy, note that since every red edge involves either 1 or 5, the only way to get a copy of $K_2 \cup K_2$ with ordering $C$ is if the copy involves the edge $(1,5)$, since the lowest and highest vertices in a copy must share an edge. But then all of the edges between $\{2,3,4\}$ are blue, so there is no red copy with edge $(1,5)$. Thus $r_{<_C}(K_2 \cup K_2) > 5$, so we get our result $r_{<_C}(K_2 \cup K_2) = 6$.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{lowerbound2k2C.jpg}
\caption{Colored $K_5$ with no $K_2 \cup K_2$ having ordering $C$}
\end{figure}
\section{Ordered Ramsey Numbers of the Diamond Graph} \label{sec:K4e}
The diamond graph can be considered as $K_4-e$, i.e. the complete graph on four vertices with an edge removed. The usual Ramsey number for the diamond graph is 10 \cite{small}, so this is the trivial lower bound on the ordered Ramsey number of the diamond graph for any ordering. Also recall that the trivial upper bound for a graph on four vertices is $R(4)=18$. In \cite{chang}, Chang obtained upper bounds between 13 and 17 for 1-orderings of $K_4-e$. He also demonstrated that the lower bound for ordering $A$ (see Figure 6) of $K_4-e$ is at least 12, but was thought to be higher since his program was able to find 25536 constructions of $K_4-e$ with ordering $A$ on 11 vertices. In Section \ref{sec:SAT}, we will see that the correct number is 15.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{diamondA.jpg}
\caption{Ordering $A$ of $K_4-e$}
\end{figure}
We will begin by proving that $r_{<_A}(K_4-e) \leq 17$. Our proof will rely on the fact that $R(K_3,K_4)=9$. \cite{small} Where we recall that $R(G,H)$ refers to the minimum integer $n$ such that for any edge-coloring of the complete graph on $n$ vertices we will get either a red $H$ or a blue $G$. So in this case, this means that for any edge coloring of $K_9$, we will either get a red $K_3$ or a blue $K_4$. Also recall the trivial fact that $R(K_4,K_3)=R(K_3,K_4)$.
\begin{Th}
The ordered Ramsey number of $K_4-e$ with ordering $A$ is bounded above by 17.
\end{Th}
\begin{proof}
Consider all of the edges from vertex 17 in the complete graph $K_{17}$. There are 16 such edges. Assume that $x\geq 9$ of them are the same color, which, without loss of generality, we can assume to be red. Then consider the set $X$ of the $x$ vertices connected to 17 by a red edge. Since $x \geq 9$, the subgraph of $K_{17}$ induced by these vertices contains either a red $K_3$ or a blue $K_4$. If there is a blue $K_4$, then we have a copy of $K_4-e$ with ordering $A$ by Lemma 1.1, so we can assume that we have a red $K_3$ instead. But then all three vertices in this red triangle also share a red edge to vertex 17, which implies that we have a red $K_4$. Thus we again would get a monochromatic copy of $K_4-e$ with ordering $A$. So there cannot be 9 or more edges of the same color form vertex 17.
Thus we can assume that there are exactly 8 red and 8 blue edges from vertex 17. Let the set of 8 vertices connected to 17 by a red edge be $X$ and let the set of vertices connected to 17 by a blue edge be $Y$. Vertex 1 is either in $X$ or $Y$; assume, without loss of generality, that $1 \in Y$. Then take the set $Z=\{1\} \cup X$. Then $|Z|=9$, so again we either have a red triangle or a blue $K_4$. We can assume again that there is no blue $K_4$, so there must be a red $K_3$ in $Z$. If the vertices of the red $K_3$ are in $X$, then we get a red copy of $K_4$ by considering the edges from vertex 17, so we're done. Thus vertex 1 must be in the red triangle. Let $p,q \in Y$ be the other two vertices in the red triangle. Then we know that vertex 1 has red edges to $p$ and $q$, and that $p$ and $q$ have a red edge between each other, and finally that $p$ and $q$ have red edges to vertex 17 since $p,q \in X$. Thus we get a red copy of $K_4-e$ with ordering $A$. Thus we have that $r_{<_A}(K_4-e) \leq 17$
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{reddiamondA.jpg}
\caption{Red copy of $K_4-e$ with $p,q \in X$}
\end{figure}
Now we will consider the ordering of the diamond graph given by Figure 8, which we will refer to as ordering $B$.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{diamondB.jpg}
\caption{Ordering $B$ of $K_4-e$}
\end{figure}
\begin{Th}
The ordered Ramsey number of $K_4-e$ with ordering $B$ is bounded above by 15.
\end{Th}
\begin{proof}
Consider the complete graph $K_n$ where $n$ is yet to be determined. Without loss of generality, assume that the edge $(1,2)$ is colored red. Now we will define four sets that partition the remaining vertices $\{3,4,...,n\}$. Let $RR$ be the set of vertices that have a red edge from 1 and a red edge from 2. Let $BB$ be the set of vertices that have a blue edge from 1 and a blue edge from 2. Let $RB$ be the set of vertices that have a red edge from 1 and a blue edge from 2. Finally, let $BR$ be the set of vertices that have a blue edge from 1 and a red edge from 2. Note that these four sets form a partition of all vertices $\{3,4,...,n\}$. Now assume that we do not have a monochromatic copy of $K_4-e$ with ordering $B$.
Clearly we have that $|RR| \leq 1$ since otherwise we would get a copy of $K_4-e$ with ordering $B$ since all the vertices in this set are necessarily greater than 1 or 2, so if 1 and 2 both had red edges to more than other vertex we would get a copy.
Now consider $RB$. We claim that $|RB| \leq 3$. To see this, assume that $|RB|=4$ and note that the vertices in $RB$ must have some total ordering. Without loss of generality, order them $3,4,5,6$. Now we know that out of the edges $(3,4),(3,5),(3,6)$ at least two of them must be the same color. Let these two edges of the same color be $(3,x)$ and $(3,y)$ with $x<y$. Then regardless of which color these two edges are, we get a monochromatic copy of $K_4-e$ with ordering $B$ since if they are red, then the vertices $\{1,3,x,y\}$ form a red copy, while if they are blue, then the vertices $\{2,3,x,y\}$ form a blue copy (see Figures 9 and 10). Thus we have that $|RB| \leq 3$. And a completely analogous argument shows that $|BR| \leq 3$.
So finally we need to consider $BB$. We claim that $|BB| \leq 5$. To see this, assume $|BB|=6$. The vertices in $BB$ are totally ordered, so without loss of generality, number them $3,4,5,6,7,8$. Note that of the edges $(3,4),(3,5),(3,6),(3,7),(3,8)$ only one of them can be blue since if we had two blue edges $(3,x)$ and $(3,y)$, then we'd get a copy of $K_4-e$ with ordering $B$ on vertices $\{1,3,x,y\}$.
First we assume that one vertex from $\{4,5,6,7,8\}$ does have a blue edge from 3. Let it be vertex $x$. Then consider the subgraph on the vertices $Q=\{4,5,6,7,8\} \setminus \{x\}$. Let $y$ be the lowest ordered vertex in $Q$. Then we know that $y$ has an edge to each of the other three vertices in $Q$ and thus at least two of these edges are the same color. If there are two blue edges, call them $(y,y')$ and $(y,y'')$, then we get a copy of $K_4-e$ with ordering $B$ on vertices $\{1,y,y',y''\}$ since $y < y'$ and $y < y''$. If there are two red edges, call them $(y,y')$ and $(y,y'')$, then we get a copy on the vertices $\{3,y,y',y''\}$ where we recall that 3 has red edges to every vertex except $x$, which is not in $Q$. So if there is a blue edge then we see that we get a copy of $K_4-e$ with ordering $B$ if $|BB| \leq 5$.
Now consider the case in which 3 has red edges to all of $\{4,5,6,7,8\}$. Then we can clearly see that we can use the same argument as the case in which we do have a blue edge since we still have at least four vertices to which 3 has a red edge. In fact we can just "forget" vertex 8. Then we can see that vertex 4 either has two red or two blue edges to $\{5,6,7\}$, so we will get a monochromatic copy of $K_4-e$ with ordering $B$ either using vertex 1 if it's a blue copy or vertex 3 if it's a red copy. So we have that $|BB| \leq 5$.
Thus we have that in order to avoid a monochromatic copy of $K_4-e$ with ordering $B$ we need $n$ to be less than or equal to $2+|RR|+|RB|+|BR|+|BB| \leq 2+1+3+3+5=14$. Any vertex we add to the graph will have to go in one of $RR$, $BB$, $RB$, or $BR$, which would thus give us a monochromatic $K_4-e$ with ordering $B$. So $r_{<_B}(K_4-e) \leq 15$.
\end{proof}
We will see in Section \ref{sec:SAT} that 12 is the correct Ramsey number.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{reddiamondB.jpg}
\caption{Red copy of $K_4-e$ with ordering $B$}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{bluediamondB.jpg}
\caption{Blue copy of $K_4-e$ with ordering $B$}
\end{figure}
The last ordering of the diamond graph we will consider is the one given in Figure 11, which we will refer to as ordering $C$.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{diamondC.jpg}
\caption{Ordering $C$ of $K_4-e$}
\end{figure}
In order to establish an upper bound on this ordering, we will first need two lemmas concerning the ordered graph on three vertices with edges $E=\{(1,2),(2,3)\}$\, i.e. the ordered path on 3 vertices. Denote this graph $P_3^<$.
The following two lemmas follow from Lemma 18 in \cite{balko}, but are given short proofs here for completeness.
\begin{Lemma}
Any edge coloring of the ordered complete graph on 5 vertices either contains a red copy of $P_3^<$ or a blue copy of $K_3$, i.e. $r_<(P_3^<, K_3) \leq 5$.
\end{Lemma}
\begin{proof}
Consider $K_5$ and consider the subgraph on vertices $\{1,2,3\}$. Then we know there must be some red edge, call it $e_1=(x_1,x_2)$. Now consider the subgraph on $\{x_2,4,5\}$, then there must also be some red edge on this subgraph. But we know that this edge cannot involve vertex $x_2$, or else we would get a red copy of $P_3^<$. Thus we must have that $(4,5)$ is red. Then this implies that $(1,4),(2,4),(3,4)$ must all be blue. But then this implies that $(1,2)$ must be red or else we'd get a blue $K_3$ on $\{1,2,4\}$ and also implies that $(2,3)$ must be red or we'd get a blue $K_3$ on $\{2,3,4\}$. But then we have that $(1,2)$ and $(2,3)$ are both red, so we get a red copy of $P_3^<$. Thus $r_<(P_3^<, K_3) \leq 5$.
\end{proof}
\begin{Lemma}
Any edge-coloring of the ordered complete graph on 7 vertices either contains a red $P_3^<$ or a blue $K_4$, i.e. $r(P_3^<,K_4) \leq 7$.
\end{Lemma}
\begin{proof}
Consider the complete ordered graph of $K_7.$ Consider vertex 4. Vertex 4 must have a blue edge connected to each vertex in the set $\{1,2,3\}$ or $\{5,6,7\}$ or else we would get a red copy of $P_3^{<}.$ Without loss of generality, assume vertex 4 has blue edges to the set $\{1,2,3\}.$ Now, among the edges $(1,2),(1,3)$ and $(2,3)$ at least one edge has to be red in order to avoid a blue copy of $K_4.$ There must also be a blue edge among the edges $(1,2),(1,3)$ and $(2,3)$ in order to avoid a red copy of $P_3^{<}.$ Assume the set $\{x_1, x_2, x_3\}$ represents the set of vertices $\{1,2,3\}$ in some order. Let the edge $(x_1,x_2)$ with $x_1 < x_2$ be red. Let the edge $(x_2,x_3)$ be blue. Now the vertex $x_2$ must have blue edges to each vertex in the set $\{5,6,7\}$ in order to avoid a red copy of $P_3^{<}.$ Now, among the edges $(5,6),(5,7)$ and $(6,7),$ one must be red in order to avoid a blue copy of $K_4.$ Assume the set $\{y_1,y_2,y_3\}$ represents the set $\{5,6,7\}$ in some order. Let the edge $(y_1,y_2)$ be red with $y_1 < y_2.$ Now, each vertex in the set $\{5,6,7\}$ must have at least one red edge to the blue triangle $(x_2,x_3,4)$ in order to avoid a blue copy of $K_4.$ Thus, creating a red copy of $P_3^{<}$ from a vertex in the triangle $(x_2,x_3,4)$ and the edge $(y_1,y_2).$ Therefore, $r(P_3^{<}, K_4) \leq 7.$
\end{proof}
\begin{Th}
The ordered Ramsey number of $K_4-e$ with ordering $C$ is bounded above by 14.
\end{Th}
\begin{proof}
Consider $K_{14}$. Consider the edges from vertex 14. Let $B$ be the set of vertices to which 14 has blue edges, and let $R$ be the set of vertices to which 14 has red edges. Clearly either $R$ or $B$ has size greater than or equal to 7. Assume, without loss of generality, that $|R| \geq 7$. Then since $|R|\geq 7$ we know by Lemma 3.4 that $R$ either has a red $P_3^<$ or a blue $K_4$. If we have a blue $K_4$, then we're done. But then if we have a red $P_3^<$, we get a red copy of $K_4-e$ with ordering $C$ since 14 is connected to all three vertices in the copy of $P_3^<$ by a red edge.
\end{proof}
We will see in Section \ref{sec:SAT} that 14 is indeed the correct Ramsey number.
\section{Ordered Ramsey Numbers of the 3-Pan Graph}\label{sec:pan}
All of the orderings of the 3-pan graph that we will consider will have a pendant edge from the triangle on vertices $\{2,3,4\}$ to vertex 1. We will consider the orderings we get from attaching vertex 1 to all three possible vertices of the triangle. Note that the usual Ramsey number of the 3-pan is 7, so that is our trivial lower bound. First we will investigate the ordering in which vertex 1 is attached to vertex 4, which we will refer to as ordering $A$.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{3panA.jpg}
\caption{Ordering $A$ of the 3-pan}
\end{figure}
\begin{Th}
The ordered Ramsey number of the 3-pan with ordering $A$ is bounded above by 10.
\end{Th}
\begin{proof}
Consider $K_{10}$. Vertex 10 must have at least 5 edges that are red or 5 edges that are blue to the other 9 vertices. Assume, without loss of generality, that there are 5 red edges. Let $R$ be the set of $\geq 5$ vertices with red edges to vertex 10. Since all of the vertices of $K_{10}$ are totally ordered, the vertices in $R$ are totally ordered. Remove the $|R|-4$ lowest vertices from $R$ so that we are left with the four vertices in $R$ with the highest ordering. Now if there is any red edge $e=(x_1,x_2)$ with $x_1 < x_2$ amongst these four vertices, then we get a red triangle on $\{x_1,x_2,10\}$, and since 10 has a red edge to at least one other vertex with ordering less than both $x_1$ and $x_2$, we get a red copy of the 3-pan with ordering $A$. But if we don't have a red edge amongst the four highest vertices in $R$, then we get a blue $K_4$, so by Lemma 1.1 we get a blue copy of the 3-pan with ordering $A$. Thus $r_{<_A}(\text{3-pan}) \leq 10$.
\end{proof}
Next we will consider the ordering we get by attaching vertex 1 to vertex 3. We will refer to this as ordering $B$ of the 3-pan.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{3panB.jpg}
\caption{Ordering $B$ of the 3-pan}
\end{figure}
\begin{Th}
The ordered Ramsey number of the 3-pan with ordering $B$ is bounded above by 14.
\end{Th}
\begin{proof}
Consider $K_{14}$. Now just consider the subgraph on vertices $\{9,10,11,12,13,14\}$. We know that any complete graph on 6 vertices contains a monochromatic triangle \cite{small}, so assume there is a red triangle $\{x,y,z\}$ with $x < y < z$ in this subgraph. Then in order to avoid a red copy of the 3-pan with ordering $B$, we must have that vertex $y$ has a blue edge to all of the vertices $\{1,2,3,4,5,6,7,8\}$.
Now consider the subgraph on the vertices $\{3,4,5,6,7,8\}$. Again we know that this subgraph must have a monochromatic triangle. If it were a blue monochromatic triangle, then we would get a blue copy of $K_4$ with the triangle and vertex $y$. So we must have a red triangle $\{a,b,c,\}$ with $a < b < c$ on these vertices. Now in order to avoud a red copy of the 3-pan with ordering $B$, we know that vertex $b$ has blue edges to vertices 1 and 2. But now we can take the subgraph on vertices $\{1,2,b,y\}$ and see that we get a blue copy of the 3-pan with ordering $B$. Thus $r_{<_B}(\text{3-pan}) \leq 14$.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{blue3panB.jpg}
\caption{Blue copy of the 3-pan with ordering $B$}
\end{figure}
Now we will consider the ordering of the 3-pan in which vertex 2 is attached to vertex 1, which we will refer to as ordering $C$ of the 3-pan.
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{3panC.jpg}
\caption{Ordering $C$ of the 3-pan}
\end{figure}
\begin{Th}
The ordered Ramsey number of the 3-pan with ordering $C$ is bounded above by 11.
\end{Th}
\begin{proof}
Consider $K_{11}$. Initialize a list $Q=\{2,3,4,5,6,7\}$. Then we know that the subgraph induced by the vertices in $Q$ must contain a monochromatic triangle. Remove the lowest vertex of this triangle from $Q$ and add vertex 8 to $Q$. Then the subgraph of vertices now in $Q$ must also contain a monochromatic triangle. Again remove its lowest vertex and now add in 9. Continue this process twice more, adding in vertices 10 and 11. Then we will get five monochromatic triangles all with a different lowest vertex. And we know that in order to avoid a copy of the 3-pan with ordering $C$ none of these triangles can have an edge of the same color as the triangle to any of the vertices in $K_{11}$ lower than the triangle's lowest vertex.
Since there are five monochromatic triangles, we know that three of them must be the same color. Assume, without loss of generality, that we have three red triangles. Then let their lowest vertices be $x, y, z$ with $x < y < z$. Then we know that $z$ has a blue edge to $y$ and $x$, we know that $y$ has a blue edge to $x$ and we know that $x$ has a blue edge to vertex 1. Thus we get a blue copy of the 3-pan with ordering $C$. Thus $r_{<_C}(\text{3-pan}) \leq 11$.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{blue3panC.jpg}
\caption{Blue copy of the 3-pan with ordering $C$}
\end{figure}
\section{Ordered Ramsey Number of $K_n$ with a Pendant Edge}\label{sec:pendant}
Finally in this section we will be able to extend our proof of the upper bound on ordering $C$ of the 3-pan to the infinite family of graphs with a copy of $K_{n-1}$ on the vertices $\{2,3,...,n\}$ and with an edge between vertices 1 and 2. We make the following definition
\begin{Def}
The $\textit{complete with 1-pendant}$ ordering of a graph on $n$ vertices consists of a complete subgraph on vertices $\{2,3,...,n\}$ and an edge between vertex 1 and vertex 2.
\end{Def}
For example, the complete with 1-pendant graph on 6 vertices is shown below in Figure 17
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{k5w1.jpg}
\caption{Complete with 1-pendant graph on 6 vertices}
\end{figure}
\begin{Prop} \label{main} Let $K_n^1$ be the complete with 1-pendant graph on $n+1$ vertices. Then we have that $r_< (K_m^1,K_n^1) \leq R(m,n) + m + n -1$ where $R(m,n)$ is the standard Ramsey number for the complete graph on $m$ vertices vs. the complete graph on $n$ vertices.
\end{Prop}
\begin{proof}
Start with $1+R(m,n)$ vertices. Then we have a set $V=\{2,3,...,1+R(m,n)\}$ which contains either a $K_m$ or $K_n.$ Remove the lowest ordered vertex, and replace it with $2+R(m,n).$ If we continue this process until we have added $(m+n-2)$ vertices, then we have $(m+n-1)$ copies of either $K_m$ or $K_n.$ Specifically, we have $n$ red copies of $K_m$ or $m$ blue copies of $K_n.$ Suppose we have $n$ red copies of $K_m.$ Then all of the copies have a unique lowest vertex. Now, we know that each copy's lowest vertex must have a blue edge to all other lower ordered vertices or else we would get a red copy of $K_m^1$ on $m+1$ vertices. But then if we arrange these $n$ lowest vertices along with vertex 1 in decreasing order, $v_n,v_{n-1},...,v_2,v_1,1,$ we know that each vertex has to have blue edges to all vertices after it, so we get a blue $K_{n+1}$, which inherently contains a $K_n^1.$ Thus, by Lemma 1.1, we get a blue copy of the complete with 1-pendant graph on $n+1$ vertices. Now instead suppose we have $n$ blue copies of $K_n.$ This proof follows the same idea for when we have $n$ copies of $K_m.$ Therefore, $r_{<}(K_m^1,K_n^1)\leq R(m,n) + m+n-1.$
\end{proof}
This bound is actually tight, which can be demonstrated by a construction that we found with David Conlon.
\begin{Prop}\label{prop:lb}
There exists an edge coloring of the complete graph on $R(m,n)+m+n-2$ vertices that does not contain a monochromatic copy of $K_m^1$ in the first color nor of $K_n^1$ in the second color.
\end{Prop}
\begin{proof}
Arbitrarily order the $R(m,n)+m+n-2$ vertices in the complete graph $G=K_{R(m,n)+m+n-2}.$ Take the highest ordered $R(m,n)-1$ vertices of $G.$ Then we know there is some way to color the edges among these vertices so that we do not get a red copy of a $K_m$ or a blue copy of a $K_n$. Color the edges this way. Call this subgraph of the highest ordered $R(m,n)-1$ vertices $Z.$ Now take the $n-1$ vertices before $Z$ in the ordering of $G$. Color all the edges among these vertices blue so that we get a blue $K_{n-1}.$ Call this subgraph $Y$. Now take the next $m-1$ vertices before $Y$ in the ordering of $G$. Color all the edges among these vertices red so that we get a red $K_{m-1}$. Call this subgraph $X$. Then the only vertex of $G$ that is not in $Z,Y,$ or $X$ is vertex $1$. Now color all the edges between $Z$ and $X$ blue. Color all the edges between $Y$ and $X$ blue. Color all the edges between $X$ to vertex $1$ red. Color the edges between $Z$ and $Y$ red. Color the edges between $Z$ to vertex $1$ red. Finally, color the edges between $Y$ to vertex $1$ blue. See Figure 1 below. Recall that $1< X < Y < Z$ where by $<$ we mean that all the vertices in one set have order less than all the vertices in the other set. We can see that this construction guarantees that whenever we get a red copy of $K_m$, there are no red edges from this $K_m$ to a lower vertex in $G.$ Similarly, whenever we get a blue copy of $K_n$, there are no blue edges from this $K_n$ to a lower vertex in $G.$ Thus, there is no red copy of a $K_m^1$ and no blue copy of a $K_n ^1$ on $m+n+1$ vertices.
\end{proof}
\begin{figure}[h]
\centering
\begin{tikzpicture} [scale = 1.3]
\draw[red, thick] (7,.15) -- (9.75,2.05);
\draw[red, thick] (7,.25) -- (9.7,2.15);
\draw[red, thick] (6.94,.35)-- (9.6,2.21);
\draw[red, thick] (10.3,2.1) -- (12.98,.15);
\draw[red, thick] (10.35,2.2) -- (12.99,.25);
\draw[red, thick] (10.3,2.4) -- (13.05,.35);
\draw[blue, thick] (12.97,-.23) -- (10.23,-2.1);
\draw[blue, thick] (13.1,-.24) -- (10.33,-2.15);
\draw[blue, thick] (13.2,-.3) -- (10.4,-2.25);
\draw[red, thick] (9.75,-2.2) -- (6.95,-.1);
\draw[red, thick] (9.7,-2.3) -- (6.9,-.2);
\draw[red, thick] (9.65,-2.4) -- (6.85,-.3);
\draw[blue, thick] (9.9,-1.95) -- (9.9,1.95);
\draw[blue, thick] (10,-1.95) -- (10,1.95);
\draw[blue, thick] (10.1,-1.95) -- (10.1,1.95);
\draw[blue, thick] (7,-.1) -- (13,-.1);
\draw[blue, thick] (7,0) -- (13,0);
\draw[blue, thick] (7,.1) -- (13,.1);
\draw[black] (6.7,0) circle (9.5pt) node[anchor=center] {$1$};
\draw[black] (10,2.3) circle (9.5pt) node[anchor=center] {Z} node[anchor=west] {\hspace{.5cm} $R(m,n)-1$};
\draw[blue] (13.3,0) circle (9.5pt) node[anchor=center] {Y} node[anchor=west] [black] {\hspace{.5cm} $n-1$};
\draw[red] (10,-2.3) circle (9.5pt) node[anchor=center] {X} node[anchor=west] [black] {\hspace{.5cm} $m-1$};
\end{tikzpicture}
\caption{Edge colorings between the sets $X$, $Y$, $Z$, and 1}
\end{figure}
\begin{Th}
We have $r_{<}(K_m^1,K_n^1) = R(m,n) + m+n-1$.
\end{Th}
\begin{proof}
The proof follows immediately from combining Propositions \ref{main} and .
\end{proof}
\section{SAT Solver Results}\label{sec:SAT}
We use the SAT solver Minisat to determine the ordered Ramsey numbers for all orderings of $K_4-e$, including the off-diagonal cases. We introduce for each edge $ij$ a boolean variable $x_{i,j}$. We interpret assigning $x_{i,j}$ to TRUE as coloring $ij$ red, and interpret assigning $x_{i,j}$ to FALSE as coloring $ij$ blue. Then we naturally express the condition ``$K_N$ has no monochromatic ordered $K_4-e$'' as a boolean formula in conjunctive normal form (CNF), which is to say, a conjunction of disjunctions (an ``\emph{and} of \emph{or}s'').
For example, the following CNF forbids, in each color, a monochromatic $K_4-e$ with the missing edge being between the lowest two vertices:
\[
\Phi_{0,1} = \bigwedge_{0\leq i<j<k<\ell<N} (x_{i,k} \vee x_{i,\ell} \vee x_{j,k} \vee x_{j,\ell} \vee x_{k,\ell} ) \wedge (\neg x_{i,k} \vee \neg x_{i,\ell} \vee \neg x_{j,k} \vee \neg x_{j,\ell} \vee \neg x_{k,\ell} )
\]
The formula $\Phi$ above is satisfiable for $N\leq 11$ and unsatisfiable for $N\geq 12$. Satisfying assignments correspond to colorings without any monochromatic ordered $K_4 - e$. Hence the Ramsey number is 12.
\begin{table}[hbt]
\centering
\begin{tabular}{c|c|c|c|c|c|c}
& 0-1 & 0-2 & 0-3 & 1-2 & 1-3 & 2-3 \\
\hline
0-1 & 12 & 14 & 14 & 13 & 14 & 13 \\
0-2 & & 14 & 15 & 14 & 15 & 14 \\
0-3 & & & 15 & 14 & 15 & 14 \\
1-2 & & & & 13 & 14 & 13 \\
1-3 & & & & & 14 & 14 \\
2-3 & & & & & & 12 \\
\end{tabular}
\caption{Table of $r_<(K_4 - e, K_4-e)$, with missing edge indicated}
\label{tab:my_label}
\end{table}
\section{Summary}
In this paper we were able to completely determine the ordered Ramsey numbers for every ordering of $K_2 \cup K_2$, for every ordering of $K_4 - e$, and for the complete with 1-pendant graph on any number of vertices (relative to the classical Ramsey number). The latter result is particularly interesting considering it is often difficult to prove exact results using Ramsey numbers that are not exactly known themselves, i.e. the fact that the ordered Ramsey number depends on $R(n)$.
An idea for future work would be to try to determine \textit{why} certain orderings of a graph give different ordered Ramsey numbers than others; the work in \cite{balko} on paths suggests that the more ``monotonic'' the ordering, the larger the Ramsey number will be.
\section{Acknowledgements}
The first author would like to thank Professor Conlon for reviewing preliminary results and introducing him to ordered Ramsey numbers. We would also like to thank him for his work on the construction in Theorem 5.3.
The last three authors recognize the majority contribution of the first author by breaking the tradition of alphabetical ordering of the authors.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,946 |
Plugins
=======
A plugin for differ allows to change how the differ works. By default the
differ will look at each path and see how it has changed. Depending on what
the user is comparing this could give false postives because if a file
contains counters then it will be quite likely that the files look like
they changed when the important content did not change.
Due to this we have a default plugin for iptables to provide an example.
.. literalinclude:: ../differ/plugins/strip_iptables_counters.py
plugins
+++++++
.. autoclass:: differ.plugins.Plugins
:members:
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,434 |
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One of these Asteroids is unlike the other! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,030 |
23 Jan 2019, 8:12 p.m.
Boulia Shire mayor and beef producer Rick Britton, who said Australia's agriculture sector lacked moral support from both sides of politics.
Producers say they have 'best interests' in animal welfare.
Queensland producer Rick Britton says there is no "moral support" for the rural sector as the fallout continues from allegations that animal activists paid for distressing live export images.
Animals Australia is currently facing allegations that it paid for footage of sheep suffering in the heat during a live export trip to the Middle East.
The video had been used to support calls to ban the live export trade, although the cash-for-footage allegations have since raised concerning questions about whether payment creates a market in animal cruelty.
Mr Britton, a beef producer and the mayor of Boulia Shire Council, said there was a lack of understanding among the general public for agricultural industries like the live export trade.
He said the public should be able to engage directly with producers to learn the truth about controversial subjects such as live export.
"If you're in Melbourne or Sydney or Brisbane or any of those places, and you really want to know what's going on in the rural industry, ask a farmer," he said.
Mr Britton said the live export saga had exposed a lack of "moral support" for Australian agriculture, stretching across both sides of politics.
Some cattle businesses were still feeling the pinch from the last live export ban in 2011, Mr Britton said.
That temporary ban created a supply glut in Australia that drove down cattle prices, he added.
"Everyone in the beef industry was affected, regardless of whether you were in live export or not.
"It had a massive financial impact right across Australia. From Darwin to Adelaide, from Perth to Melbourne. There was too much supply."
At the time of the 2011 ban, Mr Britton said his Goodwood cattle business had just come out of drought and was looking to capitalise on the live export market.
"In 2009 and 2010 - when we were buying replacement breeders - we invested in big Brahman cows to go to the live export trade as a good steady cash flow," he said.
Last year Mr Britton sent about 50 per cent of his stock for live export.
Northern Territory Live Exporters Association chairman David Warriner said producers and exporters were the "true custodians of animal welfare".
"While many tax exempt organisations claim to be the voice of animals, we are the ones actually looking after them each and every day, because our businesses and livelihoods depend on good animal welfare outcomes," he said.
Live export had a positive impact across all Australian cattle markets, Mr Warriner said.
"Having competition in the market for cattle during a time of drought is vital to the environmental and market sustainability of the northern production sector.
Mr Warriner said the Animals Australia allegations had to be investigated, and that both sides of politics needed to work together to get to the bottom of the saga. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,358 |
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The prizes are displayed on a box on the screen and winnings are added automatically. All in all, I loved being a World Champion through this scratchcard game. | {
"redpajama_set_name": "RedPajamaC4"
} | 335 |
{"url":"https:\/\/math.stackexchange.com\/questions\/3605280\/is-lim-n-to-inftyfx-n-f-lim-n-to-inftyx-n-always-true\/3605290","text":"# Is $\\lim_{n\\to \\infty}f(x_n)=f(\\lim_{n\\to \\infty}x_n)$ always true?\n\nI was studying about sequences of numbers and their limits. My book states the standard rules for algebra of limits involving sums, differences, products and quotients of convergent sequences.\n\nBut the author, while solving an example problem implicitly used the fact that $$\\lim_{n\\to \\infty}\\sqrt{\\frac{1}{n+1}} = \\frac{1}{\\sqrt{\\lim_{n\\to \\infty}(n)+1}}=0$$\n\nBut my question is can we generalise this result to be applicable to all types of functions...\n\nI found this answer for composite functions - limit of composite function underlying principles and special cases\n\nBut since the definitions of limits of number sequences and limit of a function are different, how do we formally prove the following result (this is not a composite function as such..)? $$\\lim_{n\\to \\infty}{f(x_n)}=f(\\lim_{n\\to \\infty}{x_n})\\ \\ where \\ \\ n\\in \\mathbb N$$ Here $$x_n$$ is a sequence of real numbers and its range is contained in the domain of $$f$$. Also, is this result applicable in all cases..if not...when can it be used..\n\nI am specifically looking for a formal proof of the statement..in those cases where it is applicable\n\n\u2022 It is true when $f$ is continuous. In fact, this holding for all sequences $(x_n)_n$ converging to $x$ is equivalent to the continuity of $f$ at $x$. Apr 1 '20 at 18:00\n\u2022 This is not true in general but it is true if you assume $f$ is continuous and that $\\lim_{n\\to \\infty} x_n$ is in the domain of $f$ Apr 1 '20 at 18:00\n\u2022 But how do we prove it using the definition of limit of a sequence? Apr 1 '20 at 18:01\n\u2022 @binarybitarray see my answer below using the definitions Apr 1 '20 at 18:05\n\nHere's a counter-example to show why you need continuity. Let $$f(x)$$ be $$0$$ if $$x$$ is rational, and $$1$$ otherwise. Let $$x_n = \\frac{\\pi}{n}$$. Then \\begin{align*} \\lim_{n\\to\\infty} f(x_n) &= \\lim_{n\\to\\infty} 1 = 1 \\\\ f\\left(\\lim_{n\\to\\infty}x_n\\right) &= f(0) = 0 \\end{align*}\n\n\u2022 Yes..this makes sense...Thanks a lot! Apr 1 '20 at 18:07\n\nThis is not always true. Consider the following function: $$$$f(x)= \\begin{cases} 1, \\quad x=0;\\\\~\\\\ x, \\quad x>0 \\end{cases}$$$$ Now if you take the sequence $$(x_{n})$$ such that $$x_{n}=1\/n$$, then clearly $$\\lim_{n\\to\\infty}~f(x_{n})=0$$, but $$f(\\lim_{n\\to\\infty}~x_{n})=1$$.\n\nThe condition you stated in the question title is actually a definition for continuity. That is, if that condition holds, then $$f$$ is continuous at $$x_{0}\\equiv\\lim_\\limits{n\\to\\infty}~x_{n}$$ (see Elementrary Analysis ed.2,K. A. Ross, Page 124).\n\nSuppose $$f$$ is continuous at $$x$$ where $$x=\\lim_{n\\to \\infty} x_n$$ (and assume $$x_n\\in D(f)$$ for all $$n\\in \\mathbb{N}$$). Let $$\\epsilon>0$$ and let $$\\delta>0$$ be such that whenever $$|y-x|<\\delta$$, $$|f(x)-f(y)|<\\epsilon$$ (this is the definition that $$f$$ is continuous as $$x$$). Now since $$x_n\\to x$$, there exists some $$N\\in \\mathbb{N}$$ such that for all $$n\\geq N$$, $$|x_n-x|<\\delta$$. Therefore $$|f(x_n)-f(x)|<\\epsilon$$ (for $$n\\geq N$$). But this is what it means to say that $$f(x_n)\\to f(x)$$\n\n\u2022 Yeah..I got it now..Thanks for your answer! Apr 1 '20 at 18:07\n\nNot true in general. For a simple example, let $$f(x)=\\begin{cases} x,\\,x\\ne1\\\\2,\\,x=1\\end{cases}$$. Now consider a sequence converging to $$1$$, like $$x_n=1+1\/n$$.\n\nThere's a limit point definition of continuous functions that you may want to take a look at. In particular, it's true when $$f$$ is continuous.","date":"2021-10-27 23:45:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 39, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9963182210922241, \"perplexity\": 102.68497804654581}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323588244.55\/warc\/CC-MAIN-20211027212831-20211028002831-00422.warc.gz\"}"} | null | null |
Wormaldia matagalpa är en nattsländeart som beskrevs av Oliver S. Flint Jr. 1995. Wormaldia matagalpa ingår i släktet Wormaldia och familjen stengömmenattsländor. Inga underarter finns listade i Catalogue of Life.
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matagalpa | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,447 |
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HALLOWEEN and THE BIBLE
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"Test all things; hold fast to what is good. Abstain from every form of evil." 1 Thessalonians 5:21–22
A Celebration of Evil
Halloween is a religious day, but it is not Christian. Tom Sanguinet, a former high priest in Wicca has said: "The modern holiday that we call Halloween has its origins in the full moon closest to November 1, the witches' new year. It is a time when the spirits (demons) are supposed to be at their peak power and revisiting planet earth… Halloween is purely and absolutely evil, and there is nothing we ever have, or will do, that would make it acceptable to the Lord Jesus."
Day of Death
Halloween has strong roots in paganism and witchcraft. It began as the Druid festival of Samhain. The Celts considered November 1st the day of death, because, in the Northern hemisphere, this was the beginning of winter, the leaves were falling, it was getting darker earlier, and temperatures were dropping. They believed that their sun god was losing strength and Samhain, the lord of death, was overpowering the sun god. The druids also taught that on 31 October, on the eve of the feast, Samhain assembled the spirits of all who had died during the previous year to return to their former home to visit the living.
Human and Animal Sacrifices
On Halloween, for thousands of years, druid priests have conducted diabolical worship ceremonies in which cats, horses, sheep, oxen, human beings and other offerings were rounded up, stuffed into wicca cages and burned to death. These human and animal sacrifices were apparently required to appease Samhain and keep the spirits from harming them.
To obtain these sacrifices, druid priests would go from house to house asking for fatted calves, black sheep and human beings. Those who gave were promised prosperity, and those who refused to give were threatened and cursed. This is the origin of "trick or treat."
The Jack-O-Lantern has its origin in the candle-lit pumpkin or skull, which served as a signal to mark those farms and homes that supported the druids' religion, and thus were seeking the "treat" when the terror of Halloween began. The World Book Encyclopedia says: "The apparently harmless lighted pumpkin face of the Jack-O-Lantern is an ancient symbol of a damned soul."
While people and animals were screaming in agony, being burnt to death, the druids and their followers would dress in costumes made of animal skins and heads. They would dance, chant and jump through the flames in the hope of warding off evil spirits.
One of the popular heroes of Halloween, Count Dracula, was also a real person. Dracula lived from 1431 to 1476. During his six year reign, Count Dracula massacred over 100,000 men, women and children in the most hideous ways. He devised a plan to rid his country of the burden of beggars, the handicapped, the sick and the aged by inviting them to a feast at one of his palaces. He fed them well and got them drunk. Then he asked them: "Do you want to be without cares, lacking nothing in the world?" When his guests yelled: "Yes!" Dracula ordered the palace boarded up and set on fire. No one escaped this original "house of horror."
"When you come into the land, which the Lord your God is giving you, you shall not learn to follow the abominations of those nations. There shall not be found among you anyone who makes his son or his daughter pass through the fire, or one who practices witchcraft, or a soothsayer, or one who interprets omens, or a sorcerer, or one who conjures spells, or a medium, or a spiritist, or one who calls up the dead. For all who do these things are an abomination to the Lord, and because of these abominations the Lord your God drives them out from before you. You shall be blameless before the Lord your God, for these nations which you will dispossess, listened to soothsayers and diviners; but as for you the Lord your God has not appointed such for you." Deuteronomy 18:9–14
"…teach my people the difference between the holy and the unholy, and cause them to discern between the unclean and the clean." Ezekiel 34:23
"My people are destroyed for lack of knowledge. Because you have rejected knowledge, I will also reject you from being priests for Me; because you have forgotten the Law of your God, I will also forget your children." Hosea 4:6
"Train up a child in the way he should go, then when he is old he will not depart from it." Proverbs 22:6
"That whoever causes one of these little ones who believe in Me to sin, it would be better for him if a millstone was hung around his neck, and he were drowned in the depths of the sea. Woe to the world because of offenses! For offenses must come, but woe to that man by whom the offense comes!" Matthew 18: 6–7
"Let love be without hypocrisy. Abhor what is evil. Cling to what is good." Romans 12:9
"Therefore, let us cast off the works of darkness, and let us put on the Armour of Light." Romans 13:12
"You cannot drink the cup of the Lord and the cup of demons; you cannot partake of the Lord's Table and of the table of demons." 1 Corinthians 10:21
"Do not be unequally yoked together with unbelievers. For what fellowship has righteousness with lawlessness? What communion has light with darkness? And what accord has Christ with Belial? For what part has a believer with an unbeliever? And what agreement has the temple of God with idols? …Therefore, come out from among them and be separate says the Lord. Do not touch what is unclean and I will receive you." 2 Corinthians 6:14–17
"And have no fellowship with the unfruitful works of darkness but rather expose them." Ephesians 5:11
"Finally, brethren, whatever things are true, whatever things are noble, whatever things are just, whatever things are pure, whatever things are lovely, whatever things are of good report, if there is any virtue and if there is anything praiseworthy - meditate on these things." Philippians 4:8
"Now the Spirit expressly says that in latter times some will depart from the Faith, giving heed to deceiving spirits and doctrines of demons, speaking lies and hypocrisy, having their own conscience seared with a hot iron." 1Timothy 4:1-2
"Therefore, submit to God. Resist the devil and he will flee from you." James 4:7
"Pure and undefiled religion before God and the Father is this: to visit orphans and widows in their trouble, and to keep oneself unspotted from the world." James 1:27
"Beloved, do not imitate what is evil, but what is good. He who does good is of God, but he who does evil has not seen God." 3 John 11
Participating in Paganism
Instead of participating in paganism, walking with Wicca, being in harmony with Halloween, having our children celebrate cruelty, and dabbling in a day of death, we should focus our family and congregation on celebrating Reformation Day this 31 October.
Reformation Day vs Halloween
It was on 31 October 1517, that Dr. Martin Luther nailed the 95 theses on the door of the Slosskirche (castle church) in Wittenberg, Germany. His bold challenge against the unBiblical practices of the medieval Roman papacy inspired the Protestant Reformation. All Bible believing churches should celebrate the greatest revival of Faith and freedom ever. The Reformation was one the most important turning points in world history. The energies that were released by the rediscovery of the Bible in the common tongue led to the most extraordinary spiritual Revival in history. The Reformation freed the Christians of Northern Europe from the decadence of Renaissance paganism and led to the greatest freedoms and scientific discoveries in history.
Every Bible believing Christian should celebrate the Reformation. No Christian should have part in celebrating the occultic Halloween.
We are in a spiritual world war. Cruelty to animals, vandalism and even murders occur with far greater frequency during Halloween. Every Halloween many thousands of animals, and even people, are sacrificed in satanic rituals worldwide, while millions of other people, including well meaning Christians, participate in Halloween celebrations. Halloween is a prime recruiting time for witches and Satanists. Many people have testified that they were introduced to the occult at a Halloween party. Halloween is very religious, but it is not Christian.
"Do not be overcome by evil, but overcome evil with good." Romans 12:21
On 31 October, take a stand against Halloween by mobilizing your family and congregation to celebrate the Reformation, and to engage in spiritual warfare, earnest prayer, praying the Psalms, and sharing the Gospel with our friends and neighbors, particularly those who may be unthinkingly participating in this occultic celebration of divination, necromancy, human sacrifice and cruelty to animals.
"Finally, my brethren, be strong in the Lord and in the power of His might. Put on the whole Armour of God, that you may be able to stand against the wiles of the devil. For we do not wrestle against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this age, against spiritual hosts of wickedness in the heavenly places. Therefore, take up the whole Armour of God, that you may be able to stand in the evil day, and having done all, to stand. Stand therefore, having girded your waist with truth, having put on the Breastplate of Righteousness, and having shod your feet with the preparation of the Gospel of Peace. Above all, taking the Shield of Faith with which you will be able to quench all the fiery darts of the wicked one. And take the Helmet of Salvation, and the Sword of the Spirit, which is the Word of God; praying always with all prayer and supplication in the Spirit, being watchful to this end with all perseverance and supplication for all the saints." Ephesians 6:10-18
Creation Worldview Ministries Designed by Centella Consulting | {
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Bindi is het vervolg op de debuutroman Echte mannen eten geen kaas (2008) van auteur Maria Mosterd. Het boek verscheen op 21 april 2009 bij Van Gennep in Amsterdam. De titel verwijst naar de rode stip op het voorhoofd van Indiase vrouwen, maar ook naar het hondje van Maria. Op 11 juni 2009 overhandigde Mosterd in het Internationaal Perscentrum Nieuwspoort het eerste exemplaar aan VVD-Kamerlid Fred Teeven.
In haar tweede boek beschrijft Mosterd hoe ze het eerste jaar na haar ontsnapping is losgekomen van haar afhankelijkheid van haar pooier. Ook wordt het behandeltraject beschreven: de behandeling in India, Borculo en Deventer, maar ook het feit dat er tijdens die behandelingen ook het nodige leed te verwerken was.
Over het waarheidsgehalte van onder meer Bindi schreef misdaadjournalist Hendrik Jan Korterink in zijn boek Echte mannen eten wél kaas (2010).
Non-fictieboek
Boek uit 2009
Werk van Maria Mosterd | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,596 |
Q: How to force UITableViewCell selection using selectRowAtIndexPath? I am trying to force selection(highlight) of a UITableViewCell using selectRowAtIndexPath..
For eg,
Lets say i want to always have the first cell in a table view to be highlighted,I apply the following logic in viewDidAppear
[self.tableViewPrompts selectRowAtIndexPath:[NSIndexPath indexPathForRow:0 inSection:0] animated:NO scrollPosition:0];
Works nicely.But when i try to apply the same logic elsewhere in the application the cell is not selected.I even tried deselecting other cells before calling the above method and reloading tableview before and after calling this method but none of the approaches seem to work!
I even used the tableview's delegate to forcibly make a call to the didSelectRowAtIndexPath.Event this does not work.
PS: The selection of the cell does not trigger any action all i am trying to achieve is highlighting the desired cell.
What am i doing wrong here?
Help is greatly appreciated!!
A:
The selection of the cell does not trigger any action all i am trying to achieve is highlighting the desired cell.
Why don't you use the cell's highlighted property?
UITableViewCell *cell = [self cellForRowAtIndexPath: [NSIndexPath indexPathForRow: 0 inSection: 0]];
cell.highlighted = YES;
Cant test it now, but it should work.....
edit
Better yet, of course: change your cellForRowAtIndexPath to do that
-(UITableViewCell) tableView:cellForRowAtIndexPath:
//...
if (indexPath.row == 0 && indexPath.section == 0) cell.highlighted = YES
else cell.highlighted = NO;
//...
return cell;
A: I'm puzzled why isn't this working in your case. Here is my method I use from several places inside my view controller including viewDidAppear and I see no real difference with what you're doing:
- (void)selectRow1
{
NSIndexPath* idx = [NSIndexPath indexPathForRow:1 inSection:0];
[_table selectRowAtIndexPath:idx animated:NO scrollPosition:UITableViewScrollPositionMiddle];
[self tableView:_table didSelectRowAtIndexPath:idx];
}
_table is my table property variable, and the call to didSelectRowAtIndexPath is made so the action of selection could happen but otherwise the only real difference with your code is the scrollPosition parameter.
A: Simply use setSelected to highlight cell:
[cell setSelected:YES];
I have created this extension for the whole purpose UITableView-Ext
| {
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Q: How to access columns that contains the argument in data frame and sort in decreasing order I am dealing with a data frame with column names, company name, division name all_production_2017, bad_production_2017...with many years back
Now I am writing a function that takes a company name and a year as arguments and summarize the company's production in that year. Then sort it by decreasing order in all_production_year
I have already converted the year to a string and filter the rows and columns required. But how can I sort it by a specific column? I do not know how to access that column name because the argument year is the suffix of that.
Here is a rough sketch of the structure of my data frame.
structure(list(company = c("DLT", "DLT", "DLT", "MSF", "MSF", "MSF"),
division = c("Marketing", "CHANG1", "CAHNG2", "MARKETING", "CHANG1M", "CHANG2M"),
all_production_2000 = c(15, 25, 25, 10, 25, 18),
good_production_2000 = c(10, 24, 10, 8, 10, 10),
bad_production_2000 = c(2, 1, 2, 1, 3, 5)))
with data from 2000 to 2017
I want to write a function that given a name of the company and a year.
It can filter out the company and the year relevant, and sort the all_production_thatyear, by decreasing order.
I have done so far.
ExportCompanyYear <- function(company.name, year){
year.string <- toString(year)
x <- filter(company.data, company == company.name) %>%
select(company, division, contains(year.string))
}
I just do not know how to sort by decreasing order because i do not know how to access the column name which contains the argument year.
A: Although it seems OP has provided a very simple sample data which contains data for only year 2000.
A solution approach could be:
1. Convert the list to data.frame
2. Use gather from tidyr to arrange dataframe in way where filter can be applied
ll <- structure(list(company = c("DLT", "DLT", "DLT", "MSF", "MSF", "MSF"),
division = c("Marketing", "CHANG1", "CAHNG2", "MARKETING", "CHANG1M",
"CHANG2M"), all_production_2000 = c(15, 25, 25, 10, 25, 18),
good_production_2000 = c(10, 24, 10, 8, 10, 10),
bad_production_2000 = c(2, 1, 2, 1, 3, 5)))
df <- as.data.frame(ll)
library(tidyr)
gather(df, key = "key", value = "value", -c("company", "division"))
#result:
# company division key value
#1 DLT Marketing all_production_2000 15
#2 DLT CHANG1 all_production_2000 25
#3 DLT CAHNG2 all_production_2000 25
#4 MSF MARKETING all_production_2000 10
#5 MSF CHANG1M all_production_2000 25
#6 MSF CHANG2M all_production_2000 18
#7 DLT Marketing good_production_2000 10
#8 DLT CHANG1 good_production_2000 24
#9 DLT CAHNG2 good_production_2000 10
#10 MSF MARKETING good_production_2000 8
#11 MSF CHANG1M good_production_2000 10
#12 MSF CHANG2M good_production_2000 10
#13 DLT Marketing bad_production_2000 2
#14 DLT CHANG1 bad_production_2000 1
#15 DLT CAHNG2 bad_production_2000 2
Now, filter can be applied easily on above data.frame.
A: You definitely need to reshape your data in such a way that year values could be passed as a parameter.
To create a reproducible example, I have added another year 2001 in the data.
df = data.frame(company = c("DLT", "DLT", "DLT", "MSF", "MSF", "MSF"), division = c("Marketing", "CHANG1", "CAHNG2", "MARKETING", "CHANG1M", "CHANG2M"), all_production_2000 = c(15, 25, 25, 10, 25, 18), good_production_2000 = c(10, 24, 10, 8, 10, 10), bad_production_2000 = c(2, 1, 2, 1, 3, 5),all_production_2001 = 2*c(15, 25, 25, 10, 25, 18), good_production_2001 = 2*c(10, 24, 10, 8, 10, 10), bad_production_2001 = 2*c(2, 1, 2, 1, 3, 5))
Now you can reshape the data using the reshape function in R.
Here, the variables "all_production","good_production","bad_production" are varying with time, and year values are changing for those variables.
So we specify v.names = c("all_production","good_production","bad_production").
df2 = reshape(df,direction="long",
v.names = c("all_production","good_production","bad_production"),
varying = names(df)[3:8],
idvar = c("company","division"),
timevar = "year",times = c(2000,2001))
For your data.frame you can specify times=2000:2017 and varying=3:ncol(df)
>df2
company division year all_production good_production bad_production
DLT.Marketing.2000 DLT Marketing 2000 15 2 10
DLT.CHANG1.2000 DLT CHANG1 2000 25 1 24
DLT.CAHNG2.2000 DLT CAHNG2 2000 25 2 10
MSF.MARKETING.2000 MSF MARKETING 2000 10 1 8
MSF.CHANG1M.2000 MSF CHANG1M 2000 25 3 10
MSF.CHANG2M.2000 MSF CHANG2M 2000 18 5 10
DLT.Marketing.2001 DLT Marketing 2001 30 4 20
DLT.CHANG1.2001 DLT CHANG1 2001 50 2 48
DLT.CAHNG2.2001 DLT CAHNG2 2001 50 4 20
MSF.MARKETING.2001 MSF MARKETING 2001 20 2 16
MSF.CHANG1M.2001 MSF CHANG1M 2001 50 6 20
MSF.CHANG2M.2001 MSF CHANG2M 2001 36 10 20
Now you can filter and sort like this:
library(dplyr)
somefunc<-function(company.name,yearval){
df2%>%filter(company==company.name,year==yearval)%>%arrange(-all_production)
}
>somefunc("DLT",2001)
company division year all_production good_production bad_production
1 DLT CHANG1 2001 50 2 48
2 DLT CAHNG2 2001 50 4 20
3 DLT Marketing 2001 30 4 20
| {
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Marité Cardenas is a researcher and has her own group at FORSKAREN, Malmö University, Malmö. Marité is a member of the steering committee for the Swedish Neutron Scattering Society (SNSS). She is an enthusiastic, result focused and experienced physical chemist with strong background on colloid and surface chemistry of biological systems. Marité´s main research interests cover relations between structure/function and composition of biological interfaces and biological colloids including diverse ap The group's current interest deals with the understanding of the key physicochemical properties of biointerfaces, and how they determine the interactions with biomolecules in solution. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,922 |
Tag / Biko Gayman
All-WACBA Men's & Women's Basketball Teams (Tuesday, March 31st, 2020)
Amber Wilson, Angelina Marazzi, Anna Maria College, Assumption College, Avery LaBarbera, Becker College, Biko Gayman, Bobby Perette, Brianna Capacchione, Cassidy Harrison, Catherine Sweeney, Chris Cardoso, Clark University, DeAnte Bruton, Fitchburg State, Garrett Stephenson, Hannah Favaloro, Holy Cross, Isabella Nerney, Jake Wisniewski, Joe Pridgen, Lauren Manis, Malik Brown, Mateus Ribeiro, Matt Morrow, Matthew Kelly, Mike Rapoza, Mya Mosley, Natasha Pacheco, Nichols College, Sierra Johnson, Spencer Vinson, Tyler Dion, Worcester State
By NoontimeSports.com | @NoontimeNation & @WACBAHoops
With another season of college basketball in the Worcester area is in the books, it is time to unveil our All-Worcester Area College Basketball Association (WACBA) Men's and Women's Basketball Teams.
Noontime Sports is excited to share our end of season All-WACBA Teams for the seventh consecutive season after being the host site for the organization's weekly honor rolls, which have appeared every Tuesday throughout the Worcester basketball season.
Fans of Worcester college basketball are encouraged to follow @WACBAHoops on Twitter for scores, news and more.
WACBA Men's Basketball Postseason Award
WACBA Women's Basketball Postseason Awards
All-WACBA Men's Basketball (First Team)
Mike Rapoza
Charlton, Mass.
North Potomac, Md.
Joe Pridgen
Winchendon, Mass.
DeAnte Bruton
New London, Conn.
Tyler Dion
Barre, Mass.
Garrett Stephenson
Townsend, Mass.
All-WACBA Men's Basketball (Second Team)
Bobby Perette
Marshfield, Mass.
Waldorf, Md.
Mateus Ribeiro
Biko Gayman
Fort Pierce, Fla.
Matt Morrow
Leicester, Mass.
Chris Cardoso
Jamaica Plain, Mass.
Jake Wisniewski
West Brookfield, Mass.
All-WACBA Women's Basketball (First Team)
Brianna Capacchione
Washington Township, N.J.
Hannah Favaloro
Blackstone, Mass.
Lauren Manis
Franklin, Mass.
Avery LaBarbera
Harrison, N.Y.
Mya Mosley
Worcester, Mass.
All-WACBA Women's Basketball (Second Team)
Sierra Johnson
Brockton, Mass.
Cassidy Harrison
Foxborough, Mass.
Natasha Pacheco
Fairfax, Va.
Angelina Marazzi
Manchester, N.H.
Isabella Nerney
Guilford, Conn.
Catherine Sweeney
Lowell, Mass.
Spencer Vinson
February 4, 2020 by Matt Noonan
WACBA Men's Basketball Honor Roll (Tuesday, February 4th, 2020)
Anna Maria College, Basketball, Becker College, Biko Gayman, Bobby Perette, Clark University, College Basketball, Erik Bjorn, Guney Kilitcioglu, Jake Wisniewski, Sam Dion, WACBA, Worcester Area College Basketball Association, Worcester State, WPI
Another week of college basketball in the Worcester area is in the books, which means it is time to unveil our ninth Worcester Area College Basketball Association (WACBA) Men's Honor Roll for the 2019-20 season.
Noontime Sports is proud to partner with WACBA for the seventh consecutive season to host the weekly honor roll, which will be seen every Tuesday throughout the current college basketball season. Each week, we will select a player and rookie of the week, as well as highlight various milestones, too.
Biko Gayman averaged 24 points, six rebounds, and four assists in two games last week for the Clark University men's basketball team. (PHOTO COURTESY: Clark University Athletics)
Men's Basketball Player of the Week: Biko Gayman (Clark University | Jr. | Fort Pierce, Fla.): Gayman averaged 24 points, six rebounds, and four assists in a pair of games last week for the Cougars. The junior guard began the week netting 27 points in a six-point setback to MIT before tallying 21 points against Springfield College. He registered seven assists, five rebounds, and one steal against the Engineers before recording seven rebounds against the Pride. Gayman has now netted 20 points or more in 11 contests this season.
Men's Rookie of the Week: Sam Dion (Worcester State | Fr. | Barre, Mass.): In a 1-1 week for the Lancers, Dion averaged 15 points, 3.5 rebounds, 2.5 assists, and three steals. Against Baruch, the first-year guard produced 19 points to go with four rebounds (three defensive caroms), four steals, and one block. Two days later against Bridgewater State, Dion registered 11 points on 4 of 6 shooting along with three defensive caroms and two assists.
Anna Maria College senior Bobby Perette netted his 1,000th career point last week against Colby-Sawyer. (PHOTO COURTESY: Anna Maria College Athletics)
Men's Basketball Milestone: Bobby Perette (Anna Maria College | Sr. | Marshfield, Mass.): Perette averaged 20 points and three rebounds in back-to-back games last week for the AMCATs. He eclipsed the 1,000-point mark in a one-point setback to Colby-Sawyer last Saturday to bump his total to 1,009 points in 87 contests. The Marshfield, Massachusetts native concluded the game with 19 points, five rebounds, two steals, and one assist. Prior to netting his 1,000th career point, the senior guard posted 21 points, one rebound, and one assist in a 95-82 setback to Emmanuel College.
Men's WACBA Honor Roll | Tuesday, February 4th, 2020
Guney Kilitcioglu (Becker College | Jr.| Istanbul, Turkey): Kilitcioglu had a standout week for the Hawks, averaging 8.5 points per game over two games on 55.6 percent shooting. The junior transfer also averaged 6.0 rebounds, 3.0 assists and 1.0 blocks per game.
Erik Bjorn (Worcester State | Jr. | Holden, Mass.): In his lone outing of the week against Bridgewater State, the junior forward recorded a double-double of 19 points and 13 rebounds to go with three assists, three steals, and two blocks.
Jake Wisniewski (WPI | Sr. |West Brookfield, Mass.): The senior forward posted a season-high (and game-high) 23 points on 10 of 16 shooting to help WPI cap its week with an 86-73 win over Wheaton College last Saturday. Wisniewski also recorded 13 rebounds (seven defensive caroms) and four assists in the win after tallying just four points and three rebounds in 17 minutes against Springfield College three days earlier.
January 28, 2020 February 3, 2020 by Matt Noonan
WACBA Men's Basketball Honor Roll (Tuesday, January 28th, 2020)
Anna Maria College, Assumption College, Basketball, Becker College, Biko Gayman, Bobby Perette, Clark University, College Basketball, Jake Wisniewski, Jamir Carr, John Lowther, Matthew Kelly, Tyler Dion, WACBA, Worcester Area College Basketball Association, Worcester State, WPI
Another week of college basketball in the Worcester area is in the books, which means it is time to unveil our eighth Worcester Area College Basketball Association (WACBA) Men's Honor Roll for the 2019-20 season.
Biko Gayman became the 28th player in Clark University men's basketball program history to reach 1,000 points last week. (PHOTO COURTESY: Clark University Athletics)
Men's Basketball Player of the Week: Biko Gayman (Clark University | Jr. | Fort Pierce, Fla.): Despite being a part of two setbacks last week against Coast Guard Academy and WPI, Gayman was a player to watch for the Cougars last week, averaging 20 points per game to go with four-and-a-half assists, three rebounds, and two steals. The junior guard began the week by netting 22 points on 9 of 19 shooting against the Bears to become the 28th member of the Clark men's basketball program to reach 1,000 points. Three days later against the Engineers of WPI, he tallied 18 points on 7 of 13 shooting along with four assists, three rebounds, and one steal. As of press deadline, Gayman is averaging a team-best 19.6 points per game.
John Lowther averaged 13.5 points and 7.5 rebounds in a 2-0 week for WPI men's basketball. (PHOTO COURTESY: WPI ATHLETICS)
Men's Rookie of the Week: John Lowther (WPI | Fr. | Hingham MA): Lowther shot an eye-popping 68.4% (13-of-19) from the floor as the Crimson and Gray tallied a pair of road victories last week. The freshman registered 15 points, six rebounds and four assists as the Engineers bested Emerson before narrowly missing a double-double with 12 points and nine rebounds in a road triumph as Clark. For the week, Lowther averaged 13.5 points and 7.5 rebounds.
Men's WACBA Honor Roll | Tuesday, January 28th, 2020
Bobby Perette (Anna Maria College | Sr. | Marshfield, Mass.): In a pair of wins for the AMCATs last week, the senior guard from Marshfield, Massachusetts averaged 25.5 points, three-and-a-half assists, and two-and-a-half rebounds. Against Rivier University, he matched classmate Mike Rapoza with a game and team-high 26 points on 10 of 17 shooting, including five three-pointers to go with three assists, three steals, and two rebounds. Three days later against Regis College, Perette scored 25 points on 8 of 19 shooting, including 7 of 12 from beyond the arc.
Matthew Kelly (Assumption College | Jr.|North Potomac, Md.): Kelly finished the week with 31 points in two games for the Hounds. He put up 17 points to go along with five rebounds, four assists, and three steals in a setback at Adelphi on Wednesday. He added 14 points, five assists, and three steals in a setback to New Haven at home on Saturday.
Jamir Carr (Becker College | Jr. | Philadelphia, Penn.): Carr averaged 13.5 points and 9.0 rebounds per game for the week over two games for the Hawks this past week, to go along with 1.5 assists and 1.5 steals. The junior forward helped spark Becker to a 72-67 win at Elms College last Wednesday before posting his first double-double of the season on Saturday of 18 points and 10 rebounds in a near-upset of New England College.
Tyler Dion (Worcester State | Sr. | Barre, Mass.): Tyler Dion enjoyed another successful week, averaging 18.5 points and 5.0 assists per game. In an overtime defeat against MCLA last Wednesday, Dion scored 13 points with seven assists while going 50 percent from long distance, which was followed up with a 24-point performance in a win over Fitchburg State. Dion went six-for-eight from three-point land in the win.
Jake Wisniewski (WPI | Sr. | West Brookfield, Mass.): Wisniewski was WPI's leading scorer as the Engineers notched a pair of road wins last week, extending their win streak to four, all away from Harrington Auditorium. The senior averaged 17.5 points on 56.5% shooting from the floor and 5.5 rebounds per game in the two contests. Wisniewski established a new career-high with 21 points in a rematch of last year's New England Women's and Men's Athletic Conference (NEWMAC) championship game with Emerson College before pouring in 14 points in a road win at Clark University. The returning NEWMAC Defensive Player of the Year also anchored a defensive effort that limited opponents to less than 40% (38.7%) shooting from the floor.
January 21, 2020 by Matt Noonan
WACBA Men's Basketball Honor Roll (Tuesday, January 21st, 2020)
Assumption College, Basketball, Becker College, Biko Gayman, Chris Cardoso, Clark University, Colin McNamara, College Basketball, Holy Cross, Joe Pridgen, Malik Brown, Mateus Ribeiro, Worcester Area College Basketball Association, Worcester Basketball, Worcester State, WPI
Another week of college basketball in the Worcester area is in the books, which means it is time to unveil our seventh Worcester Area College Basketball Association (WACBA) Men's Honor Roll for the 2019-20 season.
Worcester State's Chris Cardoso nearly averaged a double-double in three games last week for the Lancers. Cardoso concluded the week with 21.67 and 9.0 rebounds (PHOTO COURTESY: Worcester State University Athletics)
Men's Basketball Player of the Week: Chris Cardoso (Worcester State | Jr. | Jamaica Plain, Mass.): In a 3-0 week for the Lancers, the third-year guard averaged 21.67 points and nine rebounds to go along with 1.67 steals and one assist. Cardoso recorded back-to-back double-doubles against Framingham State and Westfield State while netting a season-high 28 points on 10 of 14 shooting last Tuesday which helped the Lancers secure their first of three-straight conference wins. The Jamaica Plain native scored 14 points against the Rams before adding 23 points this past Saturday in his team's 80-67 win over Salem State. Through 15 games this season, Cardoso is averaging 15.3 points and 6.9 rebounds.
Joe Pridgen recorded his first double-double this past weekend against American University. (PHOTO COURTESY: Holy Cross Athletics)
Men's Rookie of the Week: Joe Pridgen (Holy Cross | Fr. | Winchendon, Mass.): In a pair of games last week for the Holy Cross men's basketball team, the first-year forward averaged 22.5 points, 9.0 rebounds and 2.0 steals in a pair of games while hitting 58.8 percent of his field-goal attempts. He began the week by tallying 17 points, six rebounds, and two steals against Army before leading the Purple and White with 28 points, a career-high 12 rebounds and two steals against American University. The 28 points, 12-rebound effort was Pridgen's first double-double with the Crusaders.
Men's WACBA Honor Roll | Tuesday, January 21st, 2020
Malik Brown (Assumption College | RS-Jr. | Waldorf, Md.): Brown finished the week averaging a double-double of 12 points and 10.5 rebounds per game. He started the week with 11 points and nine rebounds in a 67-61 victory over Pace before recording 13 points and 12 rebounds in a 75-54 win over American International College this past Saturday. Additionally, he added seven assists and came up with four steals on the week.
Mateus Ribeiro (Becker College | Jr.| Sao Paulo, Brazil): Ribeiro led the Hawks to a 78-64 win over Mitchell College this past week, posting another great all-around performance with 18 points, seven rebounds and five assists while shooting 8-of-16 from the floor. As of press time, Ribeiro ranks tenth in the nation in minutes per game and total minutes.
Biko Gayman (Clark University | Jr. | Fort Pierce, Fla.): In a pair of games last week, Gayman averaged 23 points, 4.0 assists, 3.5 rebounds, and 2.0 steals. Against Springfield College last Wednesday, the third-year guard led the Cougars with 21 points on 10 of 17 shooting to go along with four assists, three steals, and one rebound. Three days later against Babson College, Gayman again led the Red and White with 25 points on 7 of 15 shooting while hitting 9 of 12 freebies. Gayman concluded the contest with six rebounds (all defensive caroms), four assists, and one steal, which helped the Cougars claim their first win over the Beavers in Worcester since January 25th, 2014. Heading into a new week, Gayman is seven points shy of becoming the 28th member of the Clark men's basketball program to reach 1,000 points.
Colin McNamara (WPI | Jr. | Arlington, MA): McNamara shot 50 percent from the floor and averaged 15 points, five rebounds, four assists and two steals at the Engineers went on the road and posted victories at Wheaton College and Coast Guard Academy.
December 10, 2019 January 6, 2020 by Matt Noonan
WACBA Men's Basketball Honor Roll (Tuesday, Dec. 10th, 2019)
Assumption College, Becker College, Biko Gayman, Clark University, College Basketball, Fitchburg State, Garrett Stephenson, Holy Cross, Joe Pridgen, Mateus Ribeiro, Matthew Kelly, Nicholas Tracy, Worcester Area College Basketball Association, Worcester Basketball, WPI
Another week of college basketball in the Worcester area is in the books, which means it is time to unveil our fourth Worcester Area College Basketball Association (WACBA) Men's Honor Roll for the 2019-20 season.
Noontime Sports is proud to sponsor with WACBA for the seventh consecutive season to host the weekly honor roll, which will be seen every Tuesday throughout the current college basketball season. Each week, we will select a player and rookie of the week, as well as highlight various milestones, too.
PROGRAM NOTE: This will be our final WACBA Honor Roll report until Tuesday, January 7th, 2020!
Becker College's Mateus Ribeiro averaged 19.0 points, 12.0 rebounds, and 2.5 assists in a pair of games for the Hawks last week. (PHOTO COURTESY: Becker College Athletics)
Men's Player of the Week: Mateus Ribeiro (Becker College | Jr. | Sao Paulo, Brazil): Ribeiro continued his outstanding play this past week, leading the Hawks to back-to-back wins over Clark University and Anna Maria College. The junior averaged 19.0 points, 12.0 rebounds and 2.5 assists per game, all while shooting 60 percent from the floor. Facing the Cougars this past Tuesday, Ribeiro scored 22 points with 15 rebounds – both career-highs – in a 69-54 victory. Then, against the AMCATs on Thursday, the junior scored 16 points with nine boards and five assists to spark a 66-56 win. Ribeiro played every minute of both contests, averaging 40 minutes per game for the week. Ribeiro was named New England Collegiate Conference (NECC) Player of the Week and Becker Athlete of the Week for the week ending December 8, 2019. At press time, Ribeiro was ranked seventh in the nation in double-doubles with five and fifth in the nation in total minutes.
Joe Pridgen averaged 18.5 points and 9.0 rebounds per game on the week while hitting 53.6 percent of his field-goal attempts last week for Holy Cross. (PHOTO COURTESY: Holy Cross Athletics)
Men's Rookie of the Week: Joe Pridgen (Holy Cross | Fr. | Winchendon, Mass.): Pridgen led the Crusaders in scoring in both games this past week. He netted career-highs of 21 points and nine rebounds in his team's win over Mercer before adding 16 points and matching his career-best with nine boards at San Diego. Pridgen averaged 18.5 points and 9.0 rebounds per game on the week while hitting 53.6 percent of his field-goal attempts.
Men's WACBA Honor Roll | Tuesday, December 10th, 2019
Matthew Kelly (Assumption College | Jr. | North Potomac, Md.): Kelly led the Hounds in their lone game of the week by tallying 27 points to go along with four rebounds and a game-leading six assists against Bentley University. He also picked up three steals while concluding the contest shooting 57.9% (11-19) from the field and 45.5% (5-11) from behind the arc.
Biko Gayman (Clark University | Jr. | Fort Pierce, Fla.): Gayman was an integral part of the Cougars' offense this past week. In a victory over Framingham State, the junior netted a season-high 28 points, moving his average for the week to 21.5 points and his points per game shooting percentage to 80%. Prior to his team's meeting against Framingham State, Gayman was able to produce a team-high 15 points against Becker College.
Nicholas Tracy (Fitchburg State | Sr. | Indianapolis, Ind.): Tracy averaged 12 points, 6.5 rebounds, one assist, 2.5 steals and 0.5 blocks per game over two contests for the Green and Gold. He collected eight points, six rebounds, one assist, four steals and one block in a 91-82 setback to Gordon College on Thursday before adding 16 points, seven caroms, one assist and one steal in an 80-75 loss to MIT on Saturday.
Garrett Stephenson (WPI | Jr. | Townsend, MA): Stephenson averaged 13 points and 6.5 rebounds during a 1-1 week for the Engineers. The junior paced the WPI with a game-best 21 points on 8-of-11 from the floor to go along with six rebounds and two assists in a 72-62 victory over previously undefeated Tufts last Tuesday. | {
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Ελληνικα Greek el
You are here: Home1 / News2 / Eurovision3 / Eurovision 20224 / Ukraine: The country is exempted from being required to film a Live-on-Tape...
Ukraine: The country is exempted from being required to film a Live-on-Tape performance!
March 30, 2022 /0 Comments/in Eurovision, Eurovision 2022, Ukraine/by Spyros Koronakis
Claudio Fasulo, the vice president of Rai 1, reveiled that Ukraine has been exempted from having to film a Live-on-Tape performance for Eurovision 2022.
The vice president of Rai 1, confirmed that Ukraine is excluded from the requirement of filming a live performance for the contest.
During the event of FIMI today, Mr. Fasulo explained:
Given the conditions we have waived the fact that the live-on-tape must be made with certain characteristics: eventually, we would use the video of their selection (Vidbir performance).
The Ukrainian delegation haw programmed to film their performance in Lviv in western Ukraine, however, the city became a target of Russian bombing the past weeks.
What is Live On tape?
According to what was announced, all participating broadcasters were asked to record a live performance of their participation in their country or somewhere else. This recording will be delivered before the contest and will take place in a studio. The recording will take place in real time (as it would in the Contest) without making changes to the vocals or any part of the show itself after the recording.
There will be freedom in the missions, to present as they consider their entries better, but instructions will be given which will ensure the fairness and integrity of the competition. There will be no audience and the recording should be unique and not be published before the event in May.
Delegations are allowed to use similar technical capabilities and dimensions that would be available on stage in Rotterdam, but are also free to choose a more limited production facility. Video recordings must not contain augmented or virtual reality, overlays, confetti, drone shots, water, color use or green screen.
It's the second year that Live-on-Tape videos are required by the participants. These videos will be used instead of the live performance on stage in Turin, in the event that a delegation is not able to travel in May, or a member of the delegation is diagnosed with COVID-19 before the first rehearsals take place.
Watch Kalush Orchestra's performance from the national final:
https://www.youtube.com/watch?v=UiEGVYOruLk
The European Broadcasting Union and Rai are currently looking after the participation of Ukraine in the Eurovision Song Contest 2022. Kalush Orchestra are going to perform sixth in the first semi-final.
Source: eurovoix
Stay tuned on Eurovisionfun for all the news regarding the Ukrainian participation at the Eurovision Song Contest 2022 in Turin, Italy!
Tags: Eurovision 2022 Ukraine, Kalush Orchestra, live on tape video, Stefania
https://eurovisionfun.com/wp-content/uploads/2022/03/ukraine-2022-kalush-orchestra-1.jpg 700 1400 Spyros Koronakis https://eurovisionfun.com/wp-content/uploads/2018/10/eurovision-fun-transparent-1-3.png Spyros Koronakis2022-03-30 01:20:252022-03-30 01:20:25Ukraine: The country is exempted from being required to film a Live-on-Tape performance!
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Eurovision 2022: What does the running order for the First Semi-Final reveal?... Switzerland: Watch Marius Bear's first Live Performance of "Boys Do... | {
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\section{Introduction}
Since the theoretical discovery of topological insulators (TIs) \cite{Kane_Mele_QSH_05,Kane_Mele_Z2_05}, topological materials have been the subject of intense theoretical and experimental investigation. Topological materials are characterized by their metallic surface (edge) states protected by the topology of the bulk wave function. A significant number of TIs have been found so far this decade, including topological crystalline insulators \cite{Liang_Fu_11,Hsieh_12,Tanaka_Ren_12,Dziawa_Kowalski_12}. Furthermore, closely related issues such as topological superconductors, \cite{Kitaev_01,Mourik_12,Rokhinson_12,Das_12,Yoshida_Sigrist_15,Daido_16} {and Weyl semimetals \cite{Murakami_07,Wan_11,Burkov_11,Weng_15,Xu_15,Lv_15} have also been the focus of intense interest, and topology has become a ubiquitous issue in condensed matter physics.
Recently, the notion of TIs has been extended to strongly correlated systems, such as topological Kondo insulators \cite{Dzero_10,Dzero_16}.
In particular, SmB$_6$ \cite{Neupane_13,Xu_14} has attracted much attention as a promising candidate for the topological Kondo insulator.
This compound has been known as a Kondo insulator for about 40 years, and it has been recently proposed that the saturation of its electrical resistivity at low temperatures is due to the topological surface current.
In addition to topological Kondo insulators, there are intriguing topological phenomena \cite{Hohenadler_11,Yoshida_Fujimoto_Kawakami_12,Tada_12,
Shitade_09,Chadov_10,Lin_10,Takimoto_11,Yan_12,Zhang_13,Lu_13,Hsieh_14,Kargarian_13,Weng_14}
under electron correlations such as topological Mott insulators \cite{Pesin_Balents_10,Yoshida_Peters_Fujimoto_Kawakami_14,Yoshida_Kawakami_16,Wu_16}, interaction-reduced classifications \cite{Fidkowski_10,Fidkowski_11,Turner_11,Ryu_Zhang_12,Yao_Ryu_13,Qi_13,Lu_Vishwanath_12,Levin_12,Fidkowski_13,Gu_14,Hsieh_Chang_14,
Isobe_Fu_15,Yoshida_Furusaki_15,Wang_Potter_Senthil_14,You_Cenke_14,Wang_Senthil_14,Morimoto_15,Yoshida_Daido_17,Yoshida_Danshita_17}, and the competition between long-range-ordered phases and TIs \cite{Mong_10,Essin_Gurarie_12,Okamoto_14,Guo_11,He_11,Yoshida_13,He_12,Yoshida_SSTI_13,
Rachel_10,Varney_10,Yamaji_11,Zheng_11,Griset_12,Wu_12,Yu_11}.
The strong correlation effects on topological states are particularly interesting because of nontrivial properties originating from the interplay between topology and correlation.
In this paper, we focus on the topological properties in long-range-ordered phases such as ferromagnetic (FM) and antiferromagnetic (AFM) phases.
The emergence of topological properties in the magnetic phases has already been studied. For example, {an antiferromagnetic topological insulator (AFTI) \cite{Mong_10} has been proposed for three-dimensional (3D) systems. In such AFTIs, the translation symmetry is broken and the unit cell is doubled.
The AFM order breaks time-reversal ($\Theta$) and primitive-lattice translational ($T_{1/2}$) symmetries but preserves their combined symmetry $S=T_{1/2} \Theta$. Under this symmetry, the system has a $Z_2$ topological number, which is related to the strong index of 3D topological insulators. Note, however, that this $Z_2$ number is allowed only for 3D systems, as confirmed by the periodic table \cite{ten-fold_way_08,ten-fold_way_09,ten-fold_way_10,Slager_13,Slager_17,Shiozaki_nonsymm_16}. We here attempt to solve this problem and demonstrate that we can realize AFTIs even in two-dimensional (2D) systems by taking into account a mirror symmetry.
Concerning nontrivial properties in FM phases, a spin-selective topological insulator (SSTI) was proposed for heavy-fermion systems \cite{Yoshida_SSTI_13}. This is a half-metallic FM phase \cite{Beach_Assaad_08,Peters_12,Howczak_Spalek_12,Peters_Kawakami_12,Denis_Rok_13} around quar]ter filling, where one spin sector acquires a gap while the other remains gapless, which is called a spin-selective gap. If the insulating sector has a nontrivial topological number, we have an SSTI, which is a unique TI embedded in a metallic phase.
Unfortunately, however, it has been shown that this phase emerges only in spin-conserving systems and is not generally compatible with the presence of spin-orbit coupling, which is necessary for topological properties.
Here, we will overcome this difficulty and propose a way to realize an SSTI for a spin-nonconserving system in the presence of spin-orbit coupling.
We explore the above-mentioned topological states in 2D magnetically ordered phases by using an effective model of the topological Kondo insulator for heavy-fermion systems, and demonstrate how they can emerge by using the Hartree-Fock (HF) approximation.
Our main idea for realizing such topological states in 2D magnetic phases is to take into account crystal symmetries, in particular, a mirror symmetry. We address two kinds of magnetic phases: a 2D AFM phase at half filling and a half-metallic FM phase around quarter filling. By taking into account the mirror symmetry, we elucidate the remarkable facts that a 2D AFM phase can have a topologically nontrivial structure specified by a mirror Chern number and that a half-metallic FM phase can have a topologically nontrivial structure specified by a Chern number. An important point is that these states can appear for spin-nonconserving systems.
The rest of this paper is organized as follows.
In Sect. \ref{sec:model}, we introduce the model and the method. We then
discuss the results for an AFTI at half filling in Sect. \ref{sec:afm}, and a half-metallic FM topological insulator near quarter filling in Sect. \ref{sec:fm}. In Sect. \ref{sec:green}, we briefly address the electron correlation effect on our topological phases. A brief summary is given in Sect. \ref{sec:summary}.
\section{Model and Method \label{sec:model}}
We explore the topological properties of the heavy-fermion systems by employing a 2D periodic Anderson model with nonlocal $d$-$f$ hybridization \cite{Dzero_10,Dzero_16,Legner_Sigrist_14,Legner_Sigrist_15}.
Specifically, we analyze the following topological periodic Anderson model showing band inversion at X points \cite{Takimoto_11,Tran_12,Lu_13,Alexandrov_13}due to next-nearest-neighbor (n.n.n.) hopping, which is important for describing SmB$_6$.
Here, note that $d$-electrons are conduction electrons.
In addition to the strong interaction $U_f$ between $f$-electrons, we also consider the interaction $U_d$ between $d$-electrons.
The Hamiltonian reads
\begin{subequations}
\label{eq:Hamiltonian}
\begin{eqnarray}
H&=&\sum_{\bm{k}}
\begin{pmatrix}
\bm{d}^{\dagger}_{\bm{k}} &\bm{f}^{\dagger}_{\bm{k}}
\end{pmatrix}
\begin{pmatrix}
\epsilon^d_{\bm{k}} &V_{\bm{k}} \\
V^{\dagger}_{\bm{k}}&\epsilon^f_{\bm{k}}
\end{pmatrix}
\begin{pmatrix}
\bm{d}_{\bm{k}} \\
\bm{f}_{\bm{k}}
\end{pmatrix}
\nonumber \\
&+&\sum_{j; \alpha ( =d,f)}U_{\alpha}n^{\alpha}_{j\uparrow}n^{\alpha}_{j{\downarrow}},
\end{eqnarray}
with
\begin{eqnarray}
\epsilon^d_{\bm{k}} &=& [-2t_d(\cos{k_x}+\cos{k_y})\nonumber \\
&&-4t_d^{\prime}\cos{k_x}\cos{k_y} ]\sigma_0 , \\
\epsilon_{\bm{k}}^f &=& [ \epsilon_f-2t_f(\cos{k_x}+\cos{k_y})\nonumber \\
&&-4t_f^{\prime}\cos{k_x}\cos{k_y}]\sigma_0, \\
V_{\bm{k}}&=& -2[ \sigma_x\sin{k_x} (V_1+V_2\cos{k_y}) \nonumber \\
&&+\sigma_y\sin{k_y} (V_1+V_2\cos{k_x}) ],
\end{eqnarray}
\end{subequations}
where $\epsilon^d_{\bm{k}}$($\epsilon^f_{\bm{k}}$) is the dispersion of $d$-($f$-) electrons, $V_{\bm{k}}$ is a Fourier component of the nonlocal $d$-$f$ hybridization, and $\bm{k}$ is a wave number.
The annihilation operators are defined as
$\bm{d}_{\bm{k}}=
\begin{pmatrix}
d_{\bm{k}\uparrow} &
d_{\bm{k}\downarrow}
\end{pmatrix}
^{T}$
and
$\bm{f}_{\bm{k}}=
\begin{pmatrix}
{f}_{\bm{k}\uparrow} &
{f}_{\bm{k}\downarrow}
\end{pmatrix}
^{T}$.
The basis function for this model is
$(d_{\uparrow}, d_{\downarrow}, f_{\uparrow}, f_{\downarrow})^T$,
where $\sigma_{i}$($i =0,x,y,z$) are the Pauli matrices for spins.
Here,
$t_d$, $t_f$, $t_d^{\prime}$, and $t_f^{\prime}$ are hopping parameters, $V_1$ and $V_2$ denote the $d$-$f$ hybridization, and
$\epsilon_f$ is the difference between the $d$- and $f$-electron energies.
We consider the above model on a 2D square lattice in the $x$-$y$ plane, which is a 2D version of the topological crystalline insulator in a 3D cubic lattice.
The system has inversion symmetry, so that the hybridization has odd parity, $V_{\bm{k}}=-V_{\bm{-k}}$.
In order to study the ground state of the model in Eq. (\ref{eq:Hamiltonian}), we employ the following HF approximation for the Coulomb term:
\begin{eqnarray}
n_{i \uparrow}^{\alpha}n_{i \downarrow}^{\alpha}
&\sim&n_{i \uparrow}^{\alpha} \langle n_{i \downarrow}^{\alpha} \rangle + \langle n_{i \uparrow}^{\alpha} \rangle n_{i \downarrow}^{\alpha} - \langle n_{i \uparrow}^{\alpha} \rangle \langle n_{i \downarrow}^{\alpha} \rangle,
\end{eqnarray}
where $n_{i\sigma}^{\alpha}$($\alpha =d,f$) is the number operator.
Here, $\langle \cdots \rangle$ denotes the expectation value at zero temperature.
We introduce the mirror operation $M_z$, which inverts the $z$-axis,
\begin{eqnarray}
M_z&=&i\, \tau_z \otimes \sigma_z ,
\end{eqnarray}
where $\tau_i$($i=0,x,y,z$) specifies $d$- and $f$-electrons.
In order to consider the magnetically ordered topological insulating states with a mirror symmetry, we introduce the corresponding topological number.
First, recall that the Chern number in multiband systems is given as
\begin{eqnarray}
C &=& \frac{1}{2\pi} \sum_{i }\int_{S} [\nabla_{\bm{k}}\times \mathcal{A}_i]_z dk_x dk_y,
\end{eqnarray}
where $\mathcal{A}_i(\bm{k})=-i\langle u_i(\bm{k})|\nabla_{\bm{k}}|u_i(\bm{k})\rangle$ is the $U(1)$ Berry connection,
where $|u_i(\bm{k}) \rangle$ is a Bloch state with occupied band index $i$, which is an eigenstate of ${\cal H}(\bm{k})$.
In the mirror-symmetric system, all the eigenstates are characterized by their mirror parities and divided into two subspaces as
\begin{eqnarray}
\mathcal{H}(\bm{k})=
\begin{pmatrix}
{\mathcal H}_{ M_z=+i}(\bm{k}) &0 \\
0&{\mathcal H}_{M_z=-i}(\bm{k})
\end{pmatrix}
.
\end{eqnarray}
The net Chern number $C$ and the mirror Chern number $C_m$ are defined by the Chern numbers $C_{\pm i}$ obtained in each mirror subspace,
namely,
\begin{subequations}
\begin{eqnarray}
C&=&C_{M_z=+i}+C_{M_z=-i} ,\\
C_m&=&(C_{M_z=+i}-C_{M_z=-i})/2 .
\end{eqnarray}
\end{subequations}
\section{Results}
We here separately discuss the obtained results for the magnetic phases at half filling and around quarter filling separately.
The values of the parameters we employ in the following are
$t_d=1$ (energy unit), $t_d^{\prime}=-0.5, t_f=-t_d/5, t_f^{\prime}=-t_d^{\prime}/5, U_d=2$. Unless otherwise noted we set $(V_1 ,V_2)=(0.1,-0.4)$.
The choice of these parameters will be explained below.
The magnetic properties of the system are studied by the HF method and the Chern number is calculated by the Fukui-Hatsugai method\cite{Fukui_Hatsugai_05}, which is efficient for numerical calculations.
\subsection{Antiferromagnetic phase\label{sec:afm}}
For a heavy-fermion system at half filling, there are some AFM phases and a Kondo insulating phase in the ordinary Doniac phase diagram. In contrast to a previous study \cite{Mong_10}, we here demonstrate that the AFTI can emerge in 2D mirror-symmetric systems.
The model we employ here is a topological mirror Kondo insulator introduced in Refs \citen{Legner_Sigrist_14,Legner_Sigrist_15}, and \citen{Rui-Xing_Zhang_16}, which is a mirror-symmetric extension of the topological Kondo insulator. This model was previously used to address a nonmagnetic Kondo insulating phase. The net Chern number is zero, $C=0$, because of the time-reversal symmetry, but there is still a possibility of having a non-zero mirror Chern number $C_m \neq 0$.
We elucidate below that the system can change from a paramagnetic phase to an AFM phase without breaking its mirror symmetry, thus leading to an AFTI. The AFM phase, where the magnetization is along the $z$-axis in our case, breaks a space translation symmetry $T_{1/2}$, and thus the period of the unit cell is doubled. Time-reversal symmetry is also broken by the magnetization, but we show that the net Chern number is zero by using the combined symmetry $S=\Theta T_{1/2}$ of the time-reversal $\Theta$ and primitive-lattice translation $T_{1/2}$.
We assume that the nesting vector is $\bm{Q}=(\pi,\pi)$ for the AFM phase, which is justified for $(t^{\prime}_d,t^{\prime}_f,V_2)=(0,0,0)$, see below.
The mean-field Hamiltonian is given by
\begin{subequations}
\begin{eqnarray}
{\cal H}^{mf}_{\bm{k}}&=&
\begin{pmatrix}
\epsilon^d_{\bm{k}} +h^d_{int} & V_{\bm{k}} \\
V_{\bm{k}}^{\dagger} &\epsilon^f_{\bm{k}} +h^f_{int}
\end{pmatrix}
,
\end{eqnarray}
with
\begin{eqnarray}
\epsilon^d_{\bm{k}} &=& [-2t_d(\cos{k_x}+\cos{k_y})\eta_x \nonumber \\
& &-4t_d^{\prime}\cos{k_x}\cos{k_y}\eta_0 ]\sigma_0 , \\
\epsilon^f_{\bm{k}} &=& [ \epsilon_f \eta_0 -2t_f(\cos{k_x}+\cos{k_y})\eta_x \nonumber \\
&&-4t_f^{\prime}\cos{k_x}\cos{k_y}\eta_0 ]\sigma_0 , \\
V_{\bm{k}}&=& -2 [ \sin{k_x}\cdot \sigma_x(V_1 \eta_x+V_2 \eta_0\cos{k_y}) \nonumber \\
&&+\sin{k_y}\cdot \sigma_y(V_1 \eta_x+V_2 \eta_0\cos{k_x}) ] ,
\end{eqnarray}
\begin{eqnarray}
h^{\alpha}_{int}= U_{\alpha}
\begin{pmatrix}
\langle n^{A\alpha}_{0\downarrow} \rangle &0 &0 &0 \\
0 &\langle n^{B\alpha}_{0\downarrow} \rangle &0 &0 \\
0 &0 &\langle n^{A\alpha}_{0\uparrow} \rangle \\
0 &0 &0 &\langle n^{B\alpha}_{0\uparrow} \rangle
\end{pmatrix}
,
\end{eqnarray}
\label{eqn:Hami}
\end{subequations}
where $\alpha=d,f$, and $\eta_{i}$($i=0,x,y,z$) are the Pauli matrices for sublattice indices. The basis function is
$(d_{\uparrow}^A,d_{\uparrow}^B, d_{\downarrow}^A,d_{\downarrow}^B, f_{\uparrow}^A, f_{\uparrow}^B,f_{\downarrow}^A, f_{\downarrow}^B)^T$.
The mirror operation in the sublattice is
\begin{eqnarray}
M_z&=&i\, \tau_z \otimes \sigma_z \otimes \eta_0 .
\end{eqnarray}
\subsubsection{ Case of $(t^{\prime}_d,t^{\prime}_f,V_2)=(0,0,0)$ }
\begin{figure}[h!]
\centering
\includegraphics[width=\hsize,clip,bb=64 453 539 743]{./68707fig1.eps}
\caption{
(Color) Magnetic and topological properties for a simplified model with only n.n. hopping and interaction, $(t_d^{\prime},t_f^{\prime},V_2)=(0,0,0)$ and $(U_d,V_1)=(2.0,0.1)$, obtained with the HF approximation:
(a) spin configuration of the AFM phase, (b) staggered magnetic moments, (c) indirect gap and direct gap, (d) Chern number for each sector.
In (b), the red dashed line denotes a spontaneous symmetry breaking (SSB) transition, the black dashed line denotes a topological phase transition (TPT), and there is a small hysteresis loop because of the first-order transition.
In (c), the indirect gap is the band gap between the conduction and valence bands $\mathrm{min}(E^{\mathrm{conduction}}_{\bm{k}}-E^{\mathrm{valence}}_{\bm{k}^{\prime}})$ ($\bm{k}$ is not necessarily equal to $\bm{k}^{\prime}$) while the direct gap is the band gap at wave number $\bm{k}$ $\mathrm{min}(E^{\mathrm{conduction}}_{\bm{k}}-E^{\mathrm{valence}}_{\bm{k}})$, where $\mathrm{indirect \, gap} \leq \mathrm{direct \, gap}$.
We have a TI for $U_f<1.42$, an AFTI for $1.42<U_f<2.72$, and an AF trivial insulator (AFI) for $2.72<U_f$.
Except when $U_f=2.72$, the system is an insulator because of the finite indirect gap.
}
\label{fig:U0_all}
\end{figure}
In order to see the essence of the results more clearly, we start with
a simplified model having only nearest-neighbor (n.n.) hopping and hybridization, i.e.,
$(t_d^{\prime},t_f^{\prime},V_2)=(0,0,0)$.
We first determine the easy axis of the magnetization by using second-order perturbation theory in the strong correlation limit.
As a result, we conclude that the $z$-direction is the easy axis.
The detail of the derivation is given in Appendix \ref{sec:spin}.
It turns out that the magnetic moments of $f$- and $d$-electrons align antiparallel at each site in the $z$-direction, as shown in Fig. \ref{fig:U0_all}(a).
The mean-field results for $(U_d,V_1)=(2,0.1)$ are summarized in Figs. \ref{fig:U0_all} (b)- \ref{fig:U0_all} (d).
Note that the results are not sensitive to the value of $V_1$.
We obtain an AFTI phase as shown in Fig. \ref{fig:U0_all}(b).
At $U_f=2.72$, there is a first-order magnetic phase transition, and the spin configuration for $U_f>2.72$ in Fig. \ref{fig:U0_all}(a), where $d$- and $f$-electrons align in the opposite directions, is in accordance with the second-order perturbation analysis.
We show the direct and indirect gaps in Fig. \ref{fig:U0_all}(c).
The former is important for determining the topological structure, and we confirm that there is indeed a finite direct gap in the AFM phase.
Note that the phase transition in Fig. \ref{fig:U0_all}(c) is a Lifshitz transition, where gap closing seems to occur at a single point.
However, this is an accidental phenomenon due to our choice of parameters.
It is seen that the AFTI phase extends between $U_f=1.42$ and $U_f=2.72$, where the mirror Chern number takes a value of $C_m=1$ in Fig. \ref{fig:U0_all}(d) with the finite magnetization in Fig. \ref{fig:U0_all}(b).
In the strong interaction region, there is a topological phase transition to a trivial phase at $U_f=2.72$, where both the direct and indirect gaps are closed. As seen from Fig. \ref{fig:U0_all}(d), the transition is accompanied by a change in the mirror Chern number from $C_m=1$ to $C_m=0$ while the net Chern number is zero.
This topological phase transition is triggered by the competition between the two types of gap. Namely, in the weak interaction region, the topologically nontrivial gap due to the nonlocal hybridization $V_1$ is dominant, while
in the strong interaction region, the topologically trivial gap with the AFM order is dominant. The competition between the two different states gives rise to a topological phase transition accompanied by gap closing (see Appendix \ref{sec:direct}).
\subsubsection{ Case of $(t^{\prime}_d,t^{\prime}_f,V_2)=(-0.5,0.1,-0.4)$ }
We now investigate the model with a specific choice of the parameters, $(t_d^{\prime},t_f^{\prime},V_2)=(-0.5,0.1,-0.4)$. Importantly, these parameters can describe band inversions for the X points in the 3D Brillouin zone (see Appendix \ref{sec:three}), leading to a strong topological insulator phase, as observed for SmB$_6$ via angle-resolved photoemission spectroscopy measurements \cite{Neupane_13,Xu_14}.
The results obtained for topological and magnetic properties at half filling are shown in Fig. \ref{fig:smb6_U2}(a).
A prominent feature in this model is that the system becomes metallic where the indirect gap is closed in the AFM phase even at half filling, as seen in Fig. \ref{fig:smb6_U2}(b).
Note, however, that the topological properties still remain intact in this region because the direct gap is not closed.
Namely, the Chern number is still well defined [Fig. \ref{fig:smb6_U2}(c)] in the region where the direct gap is open.
Thus, the topological properties remain even in a ``metal'', and such a metal adiabatically connected to a topological insulator is called a topological semimetal.
Note that this definition of a semimetal is standard in condensed matter but slightly different from that for Dirac/Weyl semimetals, which are zero-gap semiconductors by definition.
Finally, there are several topological phase transitions between different Chern numbers, as seen in Fig. \ref{fig:smb6_U2}(c).
This spin configuration is of the AFM-II type in Fig. \ref{fig:smb6_U2}(d).
Summarizing all these results, we arrive at the phase diagram shown in Fig. \ref{fig:phase-afm}.
The horizontal axis denotes the strength of the interaction $U_f$ and the vertical axis the strength of hybridization $V_1$.
There are two AFM phases in Fig. \ref{fig:phase-afm}.
The above analysis for the dashed blue line in Fig. \ref{fig:phase-afm} ($V_1=0.1,V_2=-0.4$) also applies to the region $|V_1|<|V_2|$ where the spin configuration is of the AFM-II type.
In the AFM phase for these parameters, a semimetallic AFM topological phase is realized, which we refer to as an AFM topological semimetal.
The mirror Chern number has various values in the phase diagram, which is due to the presence of n.n.n. hopping and hybridization, and is enriched by $U_d$.
The changes in the mirror Chern number are driven by the shift of the $f$-band.
In general, a complex band structure brings about various topological numbers (mirror Chern numbers), for example, see Refs. \citen{Yoshida_Daido_17} and \citen{Rui-Xing_Zhang_16}.
For reference, in Appendix \ref{sec:afti}, we show the phase diagram for $U_d=0$, which is much simpler than the one discussed above.
\begin{figure}[h!]
\centering
\includegraphics[width=\hsize,clip,bb= 76 300 546 626]{./68707fig2.eps}
\caption{
(Color) Magnetic and topological properties for the effective model of SmB$_6$, $(t_d^{\prime},t_f^{\prime},V_2)=(-0.5,0.1,-0.4)$ and $(U_d,V_1)=(2.0,0.1)$ at half filling:
(a) staggered magnetic moments, (b) indirect gap and direct gap, (c) Chern number for each sector, (d) spin configurations of the two AFM phases.
In (a), there is a small hysteresis loop because of the first-order transition.
In (b), the region where the indirect gap is closed is semimetallic, and the points where the direct gap is closed denote the topological phase transitions.
In (c), there are two Chern numbers for two mirror sectors, and the change in the Chern numbers signals the topological phase transition.
We have an AF topological semimetal (AFSM) for $1.8<U_f<3.25$ and an AF trivial semimetal (AFS) for $U_f<3.25$.
}
\label{fig:smb6_U2}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.8\hsize,clip,bb=87 368 458 660]{./68707fig3.eps}
\caption{
(Color) Phase diagram of mirror-symmetric AFTI at half filling for $U_d=2$ as a function of the interaction $U_f$ and hybridization $V_1$.
The white (dashed) line denotes the topological (insulator-metal) phase transition line and
the black dashed line separates two AFM phases, i.e., AFM-I and AFM-II.
We set $V_2=-0.4$.
In the metallic region, the indirect gap is closed; thus, there is a Fermi surface. However, the direct gap at the same wave
number $\bm{k}$ is not closed; thus, the Chern number is still well defined.
The white numbers are mirror Chern numbers.
The mirror Chern numbers are 4, 3, 2, 0, -1, and -2 in this figure.
}
\label{fig:phase-afm}
\end{figure}
Here some comments on the difference between the current results and the previous ones are in order. So far, topological properties with the AFM order have been studied in Refs. \citen{Okamoto_14,Guo_11,He_11,He_12,Yoshida_13}, focusing on the systems with spin $U(1)$ symmetry.
The Hamiltonian with spin $U(1)$ symmetry can be block-diagonalized for two spin sectors.
In such a case, the topology of the AFM phase is characterized by the spin Chern number.
In the presence of spin-orbit coupling, however, such $U(1)$ symmetry may disappear generally. Here, we stress that the AFTI in our analysis is more generic in the sense that our scenario does not require spin $U(1)$ symmetry. AFM systems respecting mirror symmetry with strong spin-orbit coupling are candidates for the AFTI proposed in this paper.
\subsection{Ferromagnetic phase\label{sec:fm}}
We now move on to an intriguing topological half-metallic state.
Around quarter filling in the Kondo lattice system, it has been known that a half-metallic FM phase dubbed a spin-selective Kondo insulator \cite{Beach_Assaad_08,Peters_12,Howczak_Spalek_12,Peters_Kawakami_12,Denis_Rok_13} appears, where a spin-selective gap opens, namely, one spin sector is metallic while the other is insulating.
This has been demonstrated for spin-conserving systems and has been extended later to a topological version referred to as a spin-selective topological insulator (SSTI) \cite{Yoshida_SSTI_13}, where the insulating sector has topologically nontrivial properties.
A crucial problem in the previous proposals is that all the results on the SSTI rely on spin $U(1)$ symmetry, which will disappear in the presence of spin-orbit coupling in general. Thus, one might naively think that the SSTI cannot appear in reality.
To overcome this difficulty, we here demonstrate that by using a mirror symmetry, such a topological half-metallic state can indeed exist in the 2D FM phase.
\begin{figure}[t]
\centering
\includegraphics[width=\hsize,clip,bb=118 273 541 655]{./68707fig4.eps}
\caption{
(Color) Magnetic and topological properties around quarter filling for $(U_d,U_f)=(0,4)$ and filling of 0.335:
(a) magnetization of $f$- ($d$-) electrons, where
the Chern number for the gapped sector is plotted in the inset.
(b) DOS at $U_f=0$ and $U_f=4$, (c) gap of insulating sector, (d) electron filling for each mirror sector, (e) spin configurations.
In (a), the red (blue) line represents $d$- ($f$-) electron magnetization and the black line the total magnetization.
For all $U_f$, the system has the same Chern number $C_{-i}=-3$ in the mirror sector $M_z=-i$.
In (b), we show the DOS for the sector of $M_z=+i$ ($M_z=-i$) by the blue (red) line,
where the chemical potential $\mu$ is indicated by the dashed black line.
In (c), the blue line is the gap of the mirror sector $M_z=-i$.
In the region of $U_f>2.3$, the sector $M_z=-i$ is an insulator.
We have a metallic state at $U_f=0$, while at $U_f=4$, we have the SSTI, where the sector $M_z=+i$ ($M_z=-i$) is a metal (an insulator).
}
\label{fig:smb6_U0}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\hsize,clip,bb=110 350 481 645]{./68707fig5.eps}
\caption{
(Color) Phase diagram of mirror-selective topological insulator around quarter filling for $U_d=0$.
The white dashed line denotes the insulator-metal phase transition line and
the black dashed line separates two FM phases, i.e., FM-I and FM-II.
In the half-metallic region, the sector $M_z=-i$ has a finite gap and the sector $M_z=+i$ is metallic.
The white numbers are mirror Chern numbers of the sector $M_z=-i$ in the whole region.
The blue dashed line represents the filling of 0.335.
}
\label{fig:phase-fm}
\end{figure}
In Figs. \ref{fig:smb6_U0} and \ref{fig:phase-fm}, we show the results obtained around quarter filling.
At a filling of 0.335 and $U_d=0$, a FM phase emerges, as seen in Fig. \ref{fig:smb6_U0}(a), where the magnetization has a hysteresis loop.
From the density of states (DOS) shown in Fig. \ref{fig:smb6_U0}(b), we find that the system is metallic at $U_f=0$, whereas the system is half-metallic at $U_f=4$ with the $M_z=+i$ sector being metallic while the $M_z=-i$ sector is insulating, as seen in Fig. \ref{fig:smb6_U0}(c).
This mirror-selective gap gives rise to the nontrivial topological number $C_{-i}=-3$ in Fig. \ref{fig:smb6_U0}(a), resulting in a mirror-selective topological insulator where the filling of the insulating sector is always half, as seen in Fig. \ref{fig:smb6_U0}(d).
This spin configuration in Fig. \ref{fig:smb6_U0}(e) is of the FM-I type.
All these results are put together in the phase diagram of Fig. \ref{fig:phase-fm}, shown as functions of the strength of the interaction $U_f$ and the filling in the system.
There are two FM phases having different types of spin configuration in Fig. \ref{fig:smb6_U0}(e) and the system has competition between two magnetic orders. We also study the case including the finite interaction $U_d$, as shown in Appendix \ref{sec:msti}. At $U_d=2$, there is no topological phase, in contrast to the above-mentioned case of $U_d=0$ in Fig. \ref{fig:phase-fm}, which shows a nontrivial topological phase in some parameter region.
Summarizing, we find the mirror-selective topological insulator in a half-metallic FM phase, which can emerge for spin nonconserving systems, in contrast to the previous proposals.
\subsection{Electron correlation effect\label{sec:green}}
So far, we have discussed the nontrivial topological states in the AFM phase and the half-metallic FM phase in the HF approximation.
One may ask what will happen if electron correlations are taken into account beyond the HF treatment.
Here, we argue that the topological properties obtained from the mean-field Hamiltonian can persist even if we consider electron correlations by, for example, dynamical mean-field theory, provided the Mott transition is absent according to Refs. \citen{Volovik_03,Gurarie_11,Zhong_Wang_12,Essin_Gurarie_11,Zhong_13}.
Recall that the Chern number of each mirror sector is given in terms of the Green's function as
\begin{eqnarray}
C_{\sigma}&=& \int \frac{d \omega d^2k}{24 \pi^2}{\rm Tr} [ \epsilon^{\mu \nu \rho}\, G_{\sigma}\partial_{\mu} G_{\sigma}^{-1}\, G_{\sigma}\partial_{\nu} G_{\sigma}^{-1}\, G_{\sigma}\partial_{\rho} G_{\sigma}^{-1}], \nonumber \\
\end{eqnarray}
where $\epsilon_{\mu \nu \rho}$ is a totally antisymmetric Levi-Civita tensor, and $(\partial_0,\partial_1,\partial_2)=(\partial_{\omega},\partial_{k_x},\partial_{k_y})$, $k=(\omega, \bm{k})$. Summation is assumed over repeated indices $\mu, \nu, \rho=0,1,2$.
$\sigma$ specifies the mirror parity and $G_{\sigma}$ is the full single-particle Green's function, which is related to the free Green's function $G_{\sigma 0}$ via $G^{-1}_{\sigma}(i\omega ,\bm{k})=G^{-1}_{\sigma 0}(i\omega ,\bm{k})-\Sigma _{\sigma}(i\omega ,\bm{k})$, where $\Sigma _{\sigma}(i\omega ,\bm{k})$ is the self-energy.
In the present treatment, $G_{\sigma}$ is a $4 \times 4$ ($2 \times 2$) matrix in the AFM (half-metallic FM) case.
According to Refs. \citen{Zhong_Wang_12} and \citen{Zhong_13}, the Chern number is determined by the topological Hamiltonian $h^{\rm eff}_{\sigma}(\bm{k})=-G^{-1}_{\sigma}(0,\bm{k})=-G^{-1}_{\sigma 0}(0,\bm{k})+\Sigma_{\sigma} (0,\bm{k})$.
This is because the Chern number does not change under the smooth deformation as
\begin{eqnarray}
G_{\sigma}(i\omega, \bm{k}, \lambda)=(1-\lambda)G_{\sigma}(i\omega, \bm{k})+\lambda [i\omega+G^{-1}_{\sigma}(0,\bm{k})]^{-1}, \nonumber \\
\end{eqnarray}
where $\lambda \in [0,1]$, provided ${\rm det}G_{\sigma} \neq 0$ and ${\rm det}G_{\sigma}^{-1} \neq 0$ are satisfied.
The cases of ${\rm det}G_{\sigma} = 0$ and ${\rm det}G_{\sigma}^{-1} = 0$ respectively correspond to the gap closing
or the emergence of Mott insulators with ${\rm Im}[\Sigma (0,\bm{k})]\rightarrow -\infty$.
Therefore, provided the Mott transition does not occur, the electron correlation effect on the AFM (half-metallic FM) phase can be treated with the renormalized band insulator, and thus the HF results may not be changed qualitatively, although the phase diagram should be modified quantitatively.
\section{Summary\label{sec:summary}}
We have explored two topological states in the AFM/FM phases by taking account of the mirror symmetry in heavy-fermion systems.
Concretely, in reference to topological crystalline insulators, we have proposed 2D topological crystalline insulating states in magnetic phases for interacting systems.
In particular, we have shown that in the AFM phase at half filling there is a topological state characterized by a mirror Chern number.
In the case of a SmB$_6$ film, an AFM topological semimetallic phase is expected.
We have also shown that in the half-metallic FM phase around quarter filling, the spin-selective topological insulating state characterized by a Chern number is realized.
In contrast to the previous studies, which assumed spin $U(1)$ symmetry to obtain such topological properties in the magnetic phases, our proposal is that these phases can be realized even in the absence of spin $U(1)$ symmetry by taking into account crystalline symmetry in magnetic phases. Generally, spin $U(1)$ symmetry is not preserved in the presence of spin-orbit coupling; thus, the present scenario without respecting spin $U(1)$ symmetry will provide a feasible platform to realize magnetic topological insulators for 2D systems.
In this paper, we have employed the HF approximation to address the above phases. We have discussed the correlation effects qualitatively and shown that the topological properties of these states may not change in the presence of correlation effects. Nevertheless, more elaborate calculations should be carried out to confirm this conclusion, which is now under consideration.
In addition, a 3D version of the mirror-selective topological insulator has been discussed \cite{Peters_18}. It might be interesting to study how our mirror-selective topological insulator extends to three dimensions by increasing the thickness of the layers.
\section*{Acknowledgments}
We thank R. Peters for the fruitful discussion.
This work was partly supported by JSPS KAKENHI Grant No. JP15H05855 and No. JP16K05501.
The numerical calculations were performed on the supercomputer at the Institute for Solid State Physics in the University of Tokyo, and SR16000 at Yukawa Institute for Theoretical Physics in Kyoto University.
| {
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In the months leading up to this year's Literary Writer's Conference, we'll be sitting down with some of our favorite agents to talk about their work, working with writers, and their LWC experience. This week we're chatting with Sonali Chanchani of Folio Literary Management. She has been with Folio since 2015, and is drawn to upmarket and literary fiction, nonfiction, essay collections, and memoirs.
What is it about LWC that excites you? What are you most looking forward to about this year's conference?
I've never been to LWC before, but I've heard wonderful things about it — in particular, that it brings together a smart, dedicated group of writers, invested in mastering their craft and committed to understanding the ins and outs of the industry. Those are exactly the kinds of writers I love working with and look forward to meeting!
The Opening Lines clinic seems particularly useful for writers. What goes on in this type of clinic? What can writers expect, and why is having a strong opening so important?
The Opening Lines clinic is an opportunity for writers to see how agents read their work. What are the opening lines that compel us to keep reading? What is it that makes these lines so compelling?
The simplest reason that first pages are so important: For readers browsing in a bookstore, if they're not immediately engaged by that opening, they'll put the book back on the shelf and move on to another title. So what are they looking for in those first few pages? What draws them in and what drives them away? And how should you be thinking about your opening lines to make sure they do the former?
We'll go through writers' first pages and talk through why we would (or would not) request more to help answer those questions.
Do you feel it's important for writers of color to connect with agents who have a similar background?
I think it's important for a writer of color to connect with someone who understands the value of their voice and story.
Sometimes, that means an agent from a similar background: someone uniquely poised to understand certain facets of their story because they've experienced it for themselves. That familiarity can elicit such a strong emotional response to the material — and in a business driven by passion, the depth of that connection is everything because it means an author's found someone who will give their all to championing their work.
That said, being of a particular background doesn't mean you have a monopoly on certain feelings or experiences. I find myself connecting with stories by and about people from all walks of life all the time. I also might feel passionately about a project precisely because it introduces me to a new experience or perspective.
Above all else, what you need in an agent is someone who believes in your voice and story, respects your perspective, and will give everything to fight for your work. Sometimes that agent is of the same background as you, sometimes they're not.
Do you feel that being a woman of color gives you a different perspective or set of intentions?
As someone from a marginalized background, I do have a particular passion for supporting underrepresented writers. I understand how powerful it is to see someone like you represented on the page or read a book that captures the experience of living on the margins — that gives shape to the feelings and sensations you've come to believe are all in your own head or you don't even realize you're experiencing until you see them articulated on the page.
So certainly, I'm determined to find and support those voices. But ultimately, I think my goals are the same as any other agent: to find great writers and help bring their work to the wider world. I think it's perhaps just about being keenly aware that many of those writers are being overlooked or passed over, despite the massive audience of readers desperate to see their work.
How does an event like this connect with your daily life as an agent? What are some of your essential takeaways?
I love these conferences because they're an opportunity to engage with the person behind the story or submission. It sometimes feels like so much of the job takes place over email, but meeting people in person to talk about their work or the business makes it feel personal and meaningful. Events like these are a great opportunity to revitalize relationships with other agents and editors and to start conversations with prospective clients. I always come away from them feeling so energized and excited. | {
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Iuj pensas ke labori estas perdi.
laboras sincere, tiom li akiras spirite.
proprieton per la sugestoj el la laboro.
estaĵo en la okuloj de Dio.
Mi preparis tiom da donacetoj, kiom da gastoj mi havis.
Li elektis librojn kiel eble plej facilajn.
One may think he loses something by having to work.
This is a mistaken notion.
then think of the spiritual gain in wholly mastering that job.
Man is, after all, an imperfect being in the eyes of God.
while humbly remaining mindful of his duty as God's servant. | {
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/algebra\/algebra-2-common-core\/chapter-4-quadratic-functions-and-equations-4-6-complete-the-square-practice-and-problem-solving-exercises-page-237\/32","text":"## Algebra 2 Common Core\n\n$\\dfrac{9}{4}$\nRECALL: To complete the square for $x^2 +mb$, just add $\\left(\\dfrac{m}{2}\\right)^2$. Using the rule above, the term that must be added to the given expression to complete the square is: With $m=-3$ \\begin{align*} &=\\left(\\frac{-3}{2}\\right)^2\\\\\\\\ &= \\frac{9}{4} \\end{align*}","date":"2020-10-25 19:52:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 1.0000046491622925, \"perplexity\": 414.08005509668124}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107889651.52\/warc\/CC-MAIN-20201025183844-20201025213844-00076.warc.gz\"}"} | null | null |
Bruce Springsteen Returns For Marathon Concert At United Center
Sep 01, 2016 James Currie Features, Music News, Reviews 0
By Christopher David
The Boss and his mighty ensemble arrived in Chicago again on Sunday as their tour commemorating landmark album The River winds to a close, and the band blazed through a three and a half hour set visiting every corner of their catalog.
Part of what makes Springsteen's live show and story-songs so universal and relevant is his ability to connect with his audience on so many different levels. River deep-cuts "I'm A Rocker" and "Cadillac Ranch" are pure, fun, party rock, "Rosalita" and "Tenth Avenue Freeze Out" give the band a chance to show off their chops even more than usual, and tracks like "Jack of All Trades" and "Death to My Hometown" offer catharsis through Springsteen's well-documented worldview of the Everyman, the hard-working American who watches—often helplessly—as the political and social tides of his country ebb and flow. This was perhaps more relevant here in Chicago than anywhere else on the tour as the band downshifted into a wrenching rendition of "American Skin," a song about the 1999 wrongful shooting death of Amadou Diallo by NYC police, followed by "Murder, Incorporated," one of Springsteen's most unsung rockers. Given the recent clashes between citizens and cops in our fair city, the context wasn't lost on the sold-out crowd.
Springsteen's penchant for storytelling is what he's probably best known for, and that was on full display as the band threw down some of the strongest tales in their canon: opener "New York City Serenade" complete with a string section, "The River," Born in the U.S.A. outtake and fan favorite "None But the Brave," and the epic "Backstreets" were all spot-on reminders of how deftly the man can set a scene.
And of course, the band. The E-Street Band is untouchable.
Let's just get that out of the way. I don't say it just to grab attention, be dramatic, anything like that. As rock n' roll bands go as of this moment in time, there is no band on earth that can match their level of precision, showmanship, swagger, or road-tested perfection. Try to find one. Actually, no, you know what? Don't waste your time.
Truly, what more can be said, at this point in history, about the E-Street Band? Rising from the streets of New Jersey in the mid-70s to take over the world, guitarists Steven Van Zandt and Nils Lofgren, pianist Roy Bittan, drummer Max Weinberg, and bassist Garry W. Tallent (along with guitarist/violinist Soozie Tyrell, keyboardist Charlie Giordano, and Jake Clemons—nephew to the late Clarence Clemons—on sax) are such a tight unit that it's almost impossible to measure, and Springsteen's talent as a bandleader always gets a chance to come to the fore, particularly when the 'sign request' portion of the show comes along, with the Boss often coming out into the crowd to retrieve signs requesting everything from the most beloved to the most obscure tunes in their canon.
Thanks to fan requests, the United Center received a rarely played "Mary's Place" from 2002's post-9/11 retort The Rising, the aforementioned "None But the Brave" (perhaps my favorite Springsteen song of all time, dating back to the early bootlegging days!), and a version of "Waitin' On A Sunny Day" that was made all the more fun by the little girl who requested it—the second Springsteen pulled her up on stage to sing a chorus, she grabbed the mic and commanded the 50,000+ crowd like she'd been doing it for years. I don't know who she was, but she's a rock star in the making.
Bruce Springsteen & the E-Street Band are wholly unique in the world of music – a band that exemplifies the best qualities of half a dozen genres, a band that can emotionally connect with a massive crowd at an effortless level, and, perhaps most importantly, a band that still, after over forty years, clearly loves playing together. Let's hope they're also a band that doesn't intend to stop touring any time soon.
For more on Bruce Springsteen & The E Street Band, click here
Bruce Springsteen & the E-Street Band – United Center, Chicago, August 28, 2016 (setlist)
New York City Serenade
Prove it All Night
My Love Will Not Let You Down
Sherry Darling
Mary's Place
Racing in the Street
None But the Brave
Hungry Heart
Out in the Street
Death to My Hometown
American Skin (41 Shots)
Murder, Incorporated
I'm A Rocker
Waitin' On A Sunny Day
Candy's Room
Backstreets
Rosalita (Come Out Tonight)
Tenth Avenue Freeze-Out
Shout (The Isley Brothers)
Bobby Jean
Prophets of Rage @ Hollywood Casino Amphitheatre (Tinley Park) Buddy Guy To Be Honored By The City of Chicago At Fifth Star Awards | {
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#include "ElemDDLConstraintAttr.h"
// #include "ElemDDLConstraintAttrDeferrable.h"
#include "ElemDDLConstraintAttrDroppable.h"
#include "ElemDDLConstraintAttrEnforced.h"
//
// End of File
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\section{Introduction}
{}{The} gravitational memory effect was first reported by Zel'dovich and Polnarev \cite{memory} in linearized gravity and further investigated by Christodoulou in full Einstein gravity \cite{Christodoulou:1991cr} (see also \cite{Braginsky:1986ia,1987Natur,Wiseman:1991ss,Thorne:1992sdb,Frauendiener} for further development and \cite{Lasky:2016knh,Nichols:2017rqr,Yang:2018ceq} for the realization with gravitational wave detectors). The memory effect is a relative displacement of nearby observers. It is therefore called {}{the} displacement memory effect. Recently, memory effect{}{s have obtained} renewed interest from {}{a} purely theoretical point of view. A fundamental connection between {}{the} displacement gravitational memory effect and Weinberg's soft graviton theorem \cite{Weinberg:1965nx} was discovered by Strominger and Zhiboedov \cite{Strominger:2014pwa}\footnote{See also the analogue in gauge theory \cite{Bieri:2013hqa,Pasterski:2015zua,Susskind:2015hpa,Mao:2017axa,Mao:2017wvx,Pate:2017vwa,Ball:2018prg,Afshar:2018sbq}}. The gravitational memory formula and the Fourier transformation of Weinberg's soft graviton formula are mathematically equivalent.
{}{A} recent investigation on soft graviton theorems \cite{Cachazo:2014fwa} shows that the universal property goes beyond Weinberg's pole formula and contains next-to-leading orders in the low-energy expansion. Inspired by the sub-leading soft graviton theorem, a new gravitational memory was proposed in \cite{Pasterski:2015tva}. This new gravitational memory effect is suggested to be a relative time delay between different orbiting light rays induced by radiative angular momentum flux. Accordingly it is called the spin memory.
The standard treatment of the memory effect \cite{Christodoulou:1991cr,Frauendiener} is based on a special choice of the topology of null infinity which is $S^2\times \mathbb{R}$. From the geometrical point of view, null infinity is not part of space-time but can be added to it by conformal compactification \cite{Penrose:1962ij,Penrose:1965am}. Hence, the topology of any asymptotically flat space-time can be always set to be the standard $S^2\times \mathbb{R}$ via changing the conformal factor during the compactification. However, asymptotically flat solutions may not be in their simplest form with the boundary topology being unit 2 sphere but {}{an} arbitrary 2 surface. Relating those solutions to the standard boundary topology will lose the simplicity both geometrically and algebraically. For instance, the well-known Robinson-Trautman waves \cite{Robinson:1960zzb} will not be truncated in the $\frac1r$ expansion with a unit 2 sphere boundary topology. It is extremely difficult, if not impossible, to see the geometrical property of this simple but very important exact solution with gravitational radiation. Therefore, it is definitely meaningful to extend the formula of the memory effect to the case of arbitrary 2 surface boundary topology. This is precisely what we will show in following pages.
Another purpose of this paper is to provide a new observational effect of the spin memory. In \cite{Pasterski:2015tva}, it is proposed that light rays orbiting in different directions acquire a relative delay which will induce a shift in the interference fringe. Alternatively, we propose to examine the spin memory effect by time-like free falling observers who are very close to null infinity. We find that {}{a} free falling observer, which is initially static, is forced to orbit by the gravitational radiation (see also \cite{Grishchuk:1989qa,Podolsky:2002sa,Podolsky:2010xh,Podolsky:2016mqg,Zhang:2017rno,Zhang:2017jma,Zhang:2018srn,Compere:2018ylh}). It receives {}{a} time delay due to massive objects in the space-time, \textit{e.g} massive stars or black holes, and gravitational radiation (see also \cite{Flanagan:2018yzh}). The former is the well-known Shapiro time delay \cite{Shapiro:1964uw} (see \cite{Visser:1998ua,Visser:1999fe} for recent development) and was observationally verified almost 40 years ago \cite{Reasenberg:1979ey}, while the {}{latter} is less stressed elsewhere. If a ring of freely falling observers who are initially static and synchronized can be set, the changes of the proper time of every observer on this ring will be different at later time. It can therefore memorize the waveform of the gravitational waves. Though {}{a} stationary massive object can cause {}{a} time delay {}{for a} single observer, the change {}{in} proper time of {}{each} observer {}{is the same}. Consequently, the ring of free falling observers will {}{detect} two memory effects: the displacement memory that will squash and stretch the shape of the ring and the time delay that will cause {}{a} difference in the proper time of nearby observers.
The plan of this paper is quite simple. In {}{the} next section, we will derive the formula of the displacement memory with arbitrary 2 surface boundary topology. Section 3 will present the displacement memory of Robinson-Trautman waves as a precise example. In section 4, we compute the time delay formula of free falling time-like observers. Then we will prove that conjugate points on a time-like geodesic are very far from each other in the asymptotic region in section 5. Some comments will be given in the discussion section. There are also two appendices providing useful information for the main text.
\section{Displacement memory effect}
By setting the boundary topology to be $S^2\times \mathbb{R}$, the null basis vector $n$ in the standard Newman-Penrose formalism \cite{Newman:1961qr} is tangent to null geodesic{}{s} with affine parameter $u$ on null infinity. The displacement memory effect in such case{}{s} is controlled by the time integration of the asymptotic shear of $n$, \textit{i.e.} $\lambda^0$. This is equivalent to the change of the asymptotic shear of $l$, \textit{i.e.} $\sigma^0$, at early time $u_i$ and late time $u_f$ \cite{Frauendiener}. However, according to Newman-Unti \cite{Newman:1962cia} (see also Appendix \ref{NU solution}), the most general asymptotically flat solution is derived with the boundary topology being arbitrary 2 surface including the well-known Robinson-Trautman metrics \cite{Robinson:1960zzb}. Actually, Robinson-Trautman is very special as it allows {}{a} shear-free null geodesic congruence, namely $\sigma=0$. From the standard formula, nothing will be memorized, but it is indeed an exact solution representing spherical radiation. Clearly, a general formula of displacement memory is of {}{urgent} need for answering this type of question. This is what we will setup in this section.
We will first check the memory formula for the solutions with arbitrary 2 surface null boundary topology by {}{mapping them to} the unit 2 sphere via Weyl transformations, {}{as} is well studied very recently by Barnich and Troessaert \cite{Barnich:2016lyg}. In Appendix \ref{Weyl}, relevant results are presented and we will follow the convention of \cite{Barnich:2016lyg}. We use $(u,z,\bar{z})$ coordinates and unprimed quantities for {}{the} unit 2 sphere case with boundary metric $ds^2=\frac{4}{(1+z\bar{z})^2}dzd\bar{z}$ while $(u',z',\bar{z}')$ coordinates\footnote{One should not confuse {}{this} with the notation in Appendix \ref{NU solution} where $(u,z,\bar{z})$ is used for a general solution with arbitrary 2 surface boundary.} and primed quantities are for the solution in {}{its} original form with arbitrary 2 surface boundary metric $ds^2=\frac{2}{P'\bar P'}dz'd\bar{z}'$. The two coordinates are connected in the following way\footnote{The full coordinate transformation is given in the form of {}{an} asymptotic expansion, but only the leading terms are involved {}{in} deriving the transformation law of \j{the} relevant fields.}
\begin{equation}
z'=z,\;\; u'=\int^u_0 dv \;\frac{P_s}{\sqrt{P'\bar P'}},\;\; P_s=\frac{1+ z \bar{z}}{\sqrt{2}}.
\end{equation}
Under such {}{a} coordinate transformation, $\sigma'^0$ is transformed as
\begin{equation}
\sigma_s^0=P_sP'^{-\frac32}\bar P'^{\frac12}\sigma'^0 - \partial_{\bar{z}} \left(P_s \sqrt{P'\bar P'} \partial_{\bar{z}} u' \right) + \sqrt{P'\bar P'} \partial_{\bar{z}} u' \partial_u \left(\sqrt{P'\bar P'} \partial_{\bar{z}} u'\right).
\end{equation}
$P',\;\bar P'$ and $\sigma'^0$ are scalar fields, hence
\begin{equation}
P'(u',z',\bar{z}')=P(u,z,\bar{z}),\;\bar P'(u',z',\bar{z}')=\bar P(u,z,\bar{z}),\;\;\sigma'^0(u',z',\bar{z}')=\sigma^0(u,z,\bar{z}).
\end{equation}
We can just drop the prime
\begin{equation}
\sigma_s^0=P_sP^{-\frac32}\bar P^{\frac12}\sigma^0 - \partial_{\bar{z}} \left(P_s \sqrt{P\bar P} \partial_{\bar{z}} u' \right) + \sqrt{P\bar P} \partial_{\bar{z}} u' \partial_u \left(\sqrt{P\bar P} \partial_{\bar{z}} u'\right).
\end{equation}
Since we have put the solution in $(u,z,\bar{z})$ coordinates with $S^2\times \mathbb{R}$ boundary topology, the standard displacement memory formula {}{works}. It is just the change of $\sigma_s^0$ at early time $u_i$ and late time $u_f$ in this coordinates.
Alternatively, in $(u',z',\bar{z}')$ coordinates, one can define
\begin{equation}
{\sigma'}_s^0=P_s P'(u',z',\bar{z}')^{-\frac32}\bar P'(u',z',\bar{z}')^{\frac12}\sigma'^0(u',z',\bar{z}') + P_s \partial_{\bar{z}'}^2\left[\int^{u'}_0 dv \;\sqrt{P'\bar P'}\right].
\end{equation}
This is the Weyl invariant part of $\sigma'^0(u',z',\bar{z}')$ \cite{Barnich:2016lyg}, namely it is unchanged as a function of their variables under Weyl transformation. This ${\sigma'}_s^0$ determines the memory effect in $(u',z',\bar{z}')$ coordinates. The displacement memory can be derived {}{from} the change of ${\sigma'}_s^0$ at early time $u_i'$ and late time $u_f'$ and is given by
\begin{multline}\label{dmemory}
\Delta {\sigma'}_s^0\mid_{u_i'}^{u_f'}=P_sP'(u'_f,z',\bar{z}')^{-\frac32}\bar P'(u'_f,z',\bar{z}')^{\frac12}\sigma'^0(u'_f,z',\bar{z}') \\
-P_sP'(u'_i,z',\bar{z}')^{-\frac32}\bar P'(u'_i,z',\bar{z}')^{\frac12}\sigma'^0(u'_i,z',\bar{z}')
+ P_s\partial_{\bar{z}'}^2\left[\int^{u'_f}_{u'_i} dv \;\sqrt{P'\bar P'}\right].
\end{multline}
We would like to comment more, from the geometric point of view, on the case of {}{an} arbitrary 2 surface. From the geodesic equation
\begin{equation}
\nabla_n n=-(\gamma+\bar \gamma)n_\mu + \bar\nu \bar{m}_\mu + \nu m_\mu,
\end{equation}
we find that $n$ will not be tangent to a null geodesic on null infinity when $P$ is $u$-dependent. ${\mu}^0$ and ${\lambda}^0$ are the asymptotic expansion and shear of {}{$n$ respectively \cite{Newman:1961qr}}. In the unit 2 sphere case, $n$ is tangent to null geodesic{}{s} with affine parameter $u$. The Weyl tensor $\Psi_{s3}^0$ and $\Psi_{s4}^0$ are completely determined by the asymptotic shear of the null geodesic congruence. We call them {}{the} \textit{news} functions as they indicate the existence of gravitational waves. However, in the case of arbitrary 2 surface, the asymptotic shear ${\lambda}^0$ will not only be controlled by gravitational waves, but will be also affected by the reference system. Nevertheless, we can define
\begin{equation}
{\lambda}_s^0=P_s^2\left(\frac{\lambda^0}{\bar P^2} + \frac{ \partial_z^2\sqrt{P\bar P}}{\sqrt{P\bar P}}\right),
\end{equation}
so that ${\lambda}_s^0$ measures only the gravitational wave contribution, while the second piece in the parentheses on the right hand side is purely the reference system effect.
{}{The special choice of boundary topology in the standard treatment of the memory effect represents physical space-times which {}{contain} isolated systems, {}{with} no geometrical or topological information from outside world \cite{Geroch:1977jn}. An arbitrary 2 surface boundary is certainly compatible with the condition that no geometrical information {}{is} coming from {}{the} outside world. However, topological information can not be avoided, for instance {}{a} null geodesic on {}{null} infinity may not be complete. Hence the Weyl transformations are usually singular in such case. Another interesting transformation that {}{involves} singularities are the so-called super-rotations \cite{Barnich:2009se,Barnich:2010eb,Barnich:2011ct}. The physical status of finite super-rotations {}{and the transition between such states is} related to the breaking of a cosmic string via quantum black hole pair nucleation \cite{Strominger:2016wns}. It is definitely of interest to investigate the physical consequence of super-rotations in memory effects elsewhere.}
\section{Robinson-Trautman waves}
In this short section, we are ready to clarify the puzzle about the memory effect of the Robinson-Trautman waves via the generalized displacement memory formula developed in {}{the} previous section. {}{The} Robinson-Trautman metric was originally derived in \cite{Robinson:1960zzb} to demonstrate a very simple kind of spherical radiation. Adapted to our notation, the metric is\footnote{To compare with the solution in \cite{Newman:2009}, $P$ should {}{be multiplied by a factor of $\frac12$.}}
\begin{equation}
ds^2=2\left(-r\partial_u \ln P + P^2\partial_z\partial_{\bar{z}}\ln P + \frac{\Psi_2^0}{r} \right)du^2+2dudr-2\frac{r^2}{P^2}dzd\bar{z},
\end{equation}
where $\Psi_2^0$ is a real constant and $P$ is a real arbitrary function of $(u,z,\bar{z})$ satisfying
\begin{equation}\label{RTequation}
3\Psi^0_2\partial_u P + P^3\partial_{\bar{z}}^2 P \partial_z^2 P - P^4\partial_z^2\partial_{\bar{z}}^2 P=0.
\end{equation}
In {}{the} NP formalism, the solution is given by
\begin{align}\label{RT}
&\Psi_0=\Psi_1=\sigma=\lambda=\tau=X^A=\omega=0,\nonumber\\
&\Psi_2=\frac{\Psi_2^0}{r^3},\;\;\;\;\mu_0=-P^2 \partial\xbar\partial \ln P\nonumber\\
&\Psi_3=\frac{P\partial_z \mu^0}{r^2},\;\;\;\;\Psi_4=\frac{-\partial_z(P^2\partial_z\partial_u\ln P)}{r}-\frac{P^2\partial_z\partial_{\bar{z}}\mu^0}{r^2},\nonumber\\
&\rho=-\frac{1}{r},\;\;\;\;\alpha=\frac{\partial_z P}{2r},\;\;\;\;\beta=-\frac{\p_{\bz} P}{2r},\;\;\mu=\frac{\mu_0}{r} - \frac{\Psi^0_2}{r^2},\\
&\gamma=-\frac12 \partial_u \ln P-\frac{\Psi^0_2}{2r^2},\;\;\nu=-P\partial_z\partial_u\ln P-\frac{P\partial_z \mu^0}{r},\nonumber\\
&U=r \partial_u \ln P + \mu^0-\frac{\Psi^0_2}{r},\;\;L^z=0,\;\;L^{\bar{z}}=\frac{P}{r}.\nonumber
\end{align}
As explained in {}{the} previous section, $n$ is not tangent to a null geodesic on {}{null} infinity in this case. Both gravitational radiation and the effect of {}{the} reference system will contribute to the asymptotic shear of the null congruence ${\lambda}^0$ that $n$ is tangent to. Their contributions happen to cancel, namely ${\lambda}^0=0$. The standard displacement memory formula does not apply in this situation. To eliminate the reference effect, one needs to use a ``good'' reference system with time coordinate $\tilde{u}=\int^u_0 dv \;\frac{P}{P_s}$. However, according to \eqref{dmemory}, the displacement memory effect from gravitational radiation can be obtained directly in the original coordinates. It is just
\begin{equation}
P_s\partial_{\bar{z}}^2\left(\int^{u_f}_{u_i} dv \;P\right).
\end{equation}
To the best of our knowledge, non-trivial explicit solutions of equation \eqref{RTequation} {}{rarely} exist. One may need to turn to numerical methods to find the exact value of displacement memory.
\section{Spin memory effect}
Recently, Pasterski, Strominger, and Zhiboedov discovered a new type of gravitational memory, the so-called spin memory effect \cite{Pasterski:2015tva}. They proposed that the observational effect of spin memory is the relative time delay of different light rays at very large radial distance $r_0$. In this section, we will provide a new observational effect by looking at time-like geodesics with affine parameter very close to null infinity.
The observer will be constrained {}{to} a fixed radial distance $r_0$ which is very far from the gravitational source, \textit{e.g. on the earth}. The $r=r_0$ hypersurface is time-like, its induced metric can be derived easily from the most general NU solutions in Appendix \ref{NU solution}. Up to relevant orders, the induced metric is given by
\begin{multline}\label{metric}
ds^2=\left[1+\frac{\Psi_2^0 + \xbar \Psi_2^0}{r}-\frac{\xbar\eth\Psi_1^0 + \eth\xbar \Psi_1^0}{r^2} + O(r^{-3})\right]du^2 \\
- 2 \left[\frac{\eth \xbar\sigma^0}{P_s} - \frac{2 \xbar\Psi_1^0}{3P_s r} + O(r^{-2})\right] du dz
- 2 \left[\frac{\xbar\eth \sigma^0}{P_s} - \frac{2 \Psi_1^0}{3P_s r} + O(r^{-2})\right] du d\bar{z} \\
- \left[2\frac{\xbar\sigma^0r}{P^2_s} - \frac{\xbar\Psi_0^0}{3P_s r} + O(r^{-2})\right]dz^2- \left[2\frac{\sigma^0r}{P^2_s} - \frac{\Psi_0^0}{3P_s r} + O(r^{-2})\right]d\bar{z}^2\\
-2\left[\frac{r^2}{P^2_s} + \frac{\sigma^0\xbar\sigma^0}{P^2_s} + O(r^{-2})\right]dzd\bar{z}.
\end{multline}
We will work in the unit 2 sphere case. {}{An arbitrary 2 surface can be mapped onto {}{the} unit 2 sphere by a Weyl transformation as discussed} in previous section (see more details in \cite{Barnich:2016lyg}).
Free falling observers on this hypersurface will travel along time-like geodesics. Supposing that {}{the} vector $V$ is tangent to a time-like geodesic, it should satisfy the geodesic equation
\begin{equation}
\xbar\nabla_V V=0,
\end{equation}
where $\xbar\nabla$ is the covariant derivative on this 3 dimensional hypersurface. Actually, $V$ is induced from a 4 dimensional vector $\tilde V$. When $r\rightarrow\infty$, a 4 dimensional time-like vector $\tilde V$ will either vanish or {}{be} proportional to the null basis $n$ (in {}{the} Newman-Penrose formalism) which is the generator of null infinity, because the light-cone will be squashed to a line on null infinity. Hence $V$ should have the following asymptotic behavior:
\begin{equation}
V^u=1 + \sum\limits_{a=1}^\infty\frac{V^u_a}{r^a},\;\;\;\;V^z=\sum\limits_{a=2}^\infty\frac{V^z_a}{r^{a}}.
\end{equation}
Then we need to solve the geodesic equation order by order. The solution is (up to relevant order):
\begin{equation}
\begin{split}
&V^u_1=-\frac{\Psi_2^0 + \xbar \Psi_2^0}{2}+V_{1I}^u(z,\bar{z})\\
&V^u_2=\frac16(\xbar\eth\Psi_1^0 + \eth \xbar\Psi_1^0) - \eth\xbar\sigma^0\xbar\eth\sigma^0 + \frac38(\Psi_2^0 + \xbar \Psi_2^0)^2 - \frac12V_{1I}^u(\Psi_2^0 + \xbar \Psi_2^0)+V_{2I}^u(z,\bar{z})\\
&V^z_2=-P_s \xbar\eth\sigma^0+V_{2I}^z(z,\bar{z}) ,\\
&V^z_3=P_s\left[2\eth\xbar\sigma^0\sigma^0 + \frac23\Psi_1^0 + \frac12\xbar\eth\sigma^0 (\Psi_2^0 + \xbar \Psi_2^0) \right]-P_s\int\;dv\; \frac{\eth(\Psi_2^0 + \xbar \Psi_2^0)}{2} \\
&\hspace{7.5cm}- P_s\xbar\eth\sigma^0 V_{1I}^u - 2\sigma^0 V_{2I}^{\bar{z}} + V_{3I}^z(z,\bar{z}),\nonumber
\end{split}
\end{equation}
where $V_{1I}^u,\;\;V_{2I}^u,\;\;V_{2I}^z,\;\;V_{3I}^z$ are integration constants that indicate the initial velocity of the observer. We will now set all of them to be zero as we require the observer is static initially.
At $r_0^{-2}$ order, $V$ has angular components due to the presence of gravitational waves characterized by $\sigma^0$. In other words, gravitational radiation force{}{s} free falling time-like particles {}{to rotate}. Since $V$ is time-like, the infinitesimal change of the proper time of the observer can be derived from the co-vector. It is just:
\begin{equation}\begin{split}
d\chi=&du + \frac{1}{2r_0}\left((\Psi^0_2+\xbar\Psi^0_2) du + \frac{\int \;\xbar\eth (\Psi^0_2+\xbar\Psi^0_2) \;dv}{P_s} dz + \frac{ \int \;\eth (\Psi^0_2+\xbar\Psi^0_2) \;dv}{P_s} d\bar{z} \right)+O(r_0^{-2})\\
=&d\left(u + \frac{1}{2r_0} \int\;(\Psi^0_2+\xbar\Psi^0_2) \;dv\right)+O(r_0^{-2}),
\end{split}\end{equation}
where $\chi$ is the proper time. Defining
\begin{equation}
\cM=\frac{1}{2}\int\;(\Psi^0_2+\xbar\Psi^0_2) \;dv,
\end{equation}
between two space-time points $(u_i,z_i,{\bar{z}}_i)$ and $(u_f,z_f,{\bar{z}}_f)$ on this geodesic, the change {}{in} proper time is
\begin{equation}
\Delta \chi=\Delta u + \frac{1}{r_0} \Delta \cM+O(r_0^{-2}).
\end{equation}
Clearly, the observer receives {}{a} time delay at order $\frac1r$. We want to emphasize that $\Delta \cM$ is {}{angle} dependent. Thus, both the Bondi mass aspect of {}{a} massive object and gravitational radiation contribute to the time delay. $\Delta \cM$ is constrained by the time evolution equation
\begin{equation}
\partial_u\Psi^0_1=\eth\Psi^0_2 - 2\sigma^0 \partial_u \eth{\xbar\sigma^0},
\end{equation}
and it is completely fixed by the change of the angular momentum aspect as\footnote{See also \cite{Pasterski:2015tva} for the relation of the angular momentum flux {}{to} the spin memory.}
\begin{equation}\label{memory}
\xbar\eth\eth \Delta\cM =\Delta(\frac12\xbar\eth\Psi_1^0 + \frac12\eth\xbar\Psi_1^0 + \xbar\eth\sigma^0 \eth\xbar\sigma^0) + \int\;(\sigma^0\xbar\eth\dot\sigma^0 + \xbar\sigma^0\eth\xbar\eth\dot\sigma^0)\;dv,
\end{equation}
up to a real constant. This is very similar to the displacement memory \cite{Christodoulou:1991cr,Frauendiener} in the sense that it includes the linear piece
\begin{equation}
\Delta(\frac12\xbar\eth\Psi_1^0 + \frac12\eth\xbar\Psi_1^0 + \xbar\eth\sigma^0 \eth\xbar\sigma^0),
\end{equation}
and the non-linear piece
\begin{equation}
\int\;(\sigma^0\xbar\eth\dot\sigma^0 + \xbar\sigma^0\eth\xbar\eth\dot\sigma^0)\;dv.
\end{equation}
The effect of the integration constant in \eqref{memory} can be eliminated by choosing a ring of freely falling observers who are initially static and synchronized\footnote{{}{It is meaningful to point out that displacement memory and spin memory happen at the same time when $\Delta \cM$ is {}{angle} dependent. There is only displacement memory no spin memory when $\Delta \cM$ has no angular dependence. In such case, the changes {}{in} proper time of {}{a} ring of freely falling observers are the same. In this sense, we say displacement memory does not contribute to a relative time delay for a ring of free falling observers (see also Appendix A of \cite{Strominger:2014pwa}).}}. The difference {}{in} the proper time of every observer on this ring at {}{a} later time can \textit{memorize} the waveform of the gravitational waves. This is another observational effect of the spin memory.
Since we have $V=n$ when $r\rightarrow\infty$, the leading piece of the shear of the time-like geodesic congruence is just $\frac{\lambda^0}{r}$. The ring of free falling observers will observe two memory effects: the displacement memory that will squash and stretch the ring and the time delay that will cause the difference of the proper time of nearby observers.
\section{Conjugate points}
To measure the time delay effect, one may expect an even simpler ideal experiment if there are conjugate points on the time-like geodesics, which is possible in curved space-time. Two free falling observers are launched with certain initial velocities at the same point. Then they will meet again at the conjugate point where they can compare their proper time. However, such {}{an} experiment is extremely hard to arrange. Because conjugate points on a time-like geodesic are very very far from each other in the asymptotic region in our set-up, though a time-like geodesic does have conjugate points \cite{Hawking:1973uf} in such case{}{s}. In order to show that, we will consider part of a time-like geodesic, namely the length of this part of the geodesic is much smaller than $r_0$. Then the $\frac1r$ expansion can {}{be applied}. In the end, we will prove that there {}{are} no conjugate points on this part of the geodesic.
A solution $T^c$ of the geodesic deviation equation
\begin{equation}
\xbar\nabla_V(\xbar\nabla_V T^c)=-{R_{abd}}^c\,T^bV^aV^d.
\end{equation}
is called {}{a} Jacobi field on the geodesic that $V$ is tangent to. Two points $p$ and $q$ are conjugate points along the geodesic if there exists a non-zero Jacobi field $T^c$ along the geodesic that vanishes at $p$ and $q$ \cite{Hawking:1973uf}. According to the induced metric \eqref{metric}, ${R_{abd}}^c\,V^aV^d=O(r^{-1})$, so the leading piece of the Geodesic deviation equation is
\begin{equation}
\partial_u^2 T^a_0(u,z,\bar{z})=0,
\end{equation}
where $T^a_0(u,z,\bar{z})$ is the leading term of $T^a$ in the $\frac1r$ expansion. Then $T^a_0$ is given by
\begin{equation}
T^a_0=uT^a_{01}(z,\bar{z})+T^a_{02}(z,\bar{z}).
\end{equation}
Naively, one can find {}{an infinite number of possible choices of} $T^a_0$ that allow two points $(u_i,z_i,\bar{z}_i)$ and $(u_f,z_f,{\bar{z}}_f)$ from the geodesic to be conjugate points, namely
\begin{equation}\begin{split}
&u_i\,T^a_{01}(z_i,\bar{z}_i)+T^a_{02}(z_i,\bar{z}_i)=0,\\
&u_f\,T^a_{01}(z_f,\bar{z}_f)+T^a_{02}(z_f,\bar{z}_f)=0.
\end{split}\end{equation}
However, the change of the angular coordinates on this geodesic is very tiny and proportional to $\frac1r$. Then the second equation $u_f\,T^a_{01}(z_f,\bar{z}_f)+T^a_{02}(z_f,\bar{z}_f)=0$ becomes
\begin{equation}
u_f\,T^a_{01}(z_i,\bar{z}_i)+T^a_{02}(z_i,\bar{z}_i)+O(r^{-1})=0.
\end{equation}
Hence the condition of having $(u_i,z_i,\bar{z}_i)$ and $(u_f,z_f,{\bar{z}}_f)$ be conjugate points is reduced to
\begin{equation}\begin{split}
u_i\,T^a_{01}(z_i,\bar{z}_i)+T^a_{02}(z_i,\bar{z}_i)=0,\\
u_f\,T^a_{01}(z_i,\bar{z}_i)+T^a_{02}(z_i,\bar{z}_i)=0,
\end{split}\end{equation}
but $u_i\neq u_f$. Obviously, there is no such solution.
Thus we can never find two points that are not very far from each other on one geodesic where $T^a=0$ at leading order. Conceptually, this is expected since geodesics in flat (Minkowski) space-times do not have conjugate points. Now we {}{have just shown} that conjugate points {}{are} not ``close'' to each other in the asymptotic region in asymptotically flat space-times though they do exist.
\section{Discussions}
In this work, we first derived the formula {}{for the} displacement memory effect for the case with an arbitrary 2 surface boundary topology. Via a Weyl transformation, it can be mapped into the unit 2 sphere. Then the standard formula {}{for the} displacement memory {}{applies}. This leads us to a direct derivation of the displacement memory formula in the original form of the solutions. Secondly, we proposed a new observational effect of the spin memory. It is a time delay of time-like free falling observer{}{s}.
The discovery of spin memory was originally inspired {}{by} the connection between {}{the} gravitational memory effect and Weinberg's soft graviton theorem. The displacement memory and spin memory correspond to the leading and sub-leading soft graviton theorem, respectively. The novel results in \cite{Cachazo:2014fwa} show that soft graviton theorems exist even at third order in the low-energy expansion. There are indeed some positive signs indicating a third gravitational memory effect. On the one hand, the two known memory formulas are completely controlled by the time evolution equations of the Weyl tensors:
\begin{equation}
\partial_u\Psi^0_2=- \partial_u\eth^2 \xbar\sigma^0 - \sigma^0 \partial_u^2\xbar\sigma^0,
\end{equation}
and
\begin{equation}
\partial_u\Psi^0_1=\eth\Psi^0_2 - 2\sigma^0 \partial_u \eth{\xbar\sigma^0},
\end{equation}
in unit 2 sphere case. There is indeed a third time evolution equation
\begin{equation}
\partial_u\Psi^0_0=\eth\Psi^0_1+3\sigma^0\Psi^0_2.
\end{equation}
The asymptotic shear $\sigma^0$ is constrained by the imaginary part of this equation through some tedious but not difficult calculations (it is more clear in {}{the} linearized gravity case \cite{Conde:2016rom}). On the other hand, displacement and spin memories are related to the energy flux and the angular momentum flux through null infinity. Newman and Penrose \cite{Newman:1965ik,Newman:1968uj} discovered more gravitationally-conserved quantities that may {}{account} for the possible third gravitational memory (see also recent relevant development{}{s} \cite{Compere:2017wrj,Godazgar:2018vmm,Godazgar:2018qpq,Godazgar:2018n}).
We have shown the power of Weyl transformation{}{s} with a precise example in this work. Actually, the action of the full BMS$_4$ group combined with Weyl transformation{}{s} on the Newman-Unti solution space was given in \cite{Barnich:2016lyg}. It would be very meaningful to compute the transformation law of the BMS$_4$ current, especially the action of {}{a} Weyl transformation elsewhere \cite{tocome}.
\section*{Acknowledgments}
The authors thank Glenn Barnich and Jun-Bao Wu for useful discussions and additionally Jun-Bao Wu again for critical comments on our computations. This work is supported in part by the China Postdoctoral Science Foundation (Grant No. 2017M620908), by the National Natural Science Foundation of China (Grant Nos. 11575286, 11731001, 11475179, and 11575202).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 290 |
Mehron at The Makeup Show NYC 2018
The fun brand Mehron was in the building at The Makeup Show again this year! It's always a pleasure to catch up and see what new products they will launch.Their color palettes are fabulous and rich in color and is perfect for those that are makeup artists looking to work in film and television. The pigments are always so bold and true to color and the quality is fierce.
This season it was all about their Sweet and Spicy Palettes, the super gorgeous Echo Glitter Pressed Palette, the Highlight Pro 3 Color Palette (which I now have) and the signature Liquid Vinyl. All these products are great to try for a glam look or special occasion!
Get the beauty scoop from their Creative Director Stephanie Koutikas in the video feature below.
Posted by CPD at Saturday, May 12, 2018
Labels: Beauty, Interviews, mehron, The Makeup Show
Chastity Palmer-Davis
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,145 |
{"url":"https:\/\/www.physicsforums.com\/threads\/are-these-two-functions-equivalent-when-y-0.544959\/","text":"# Are these two functions equivalent when y = 0?\n\n1. Oct 28, 2011\n\n### nobahar\n\nHello!\n\nAm I right to conclude that:\n$$y^2(A(\\frac{x}{y})^2 + B(\\frac{x}{y})+ C) = Ax^2 + Bxy + Cy^2$$\n\nOnly when y does not equal zero. I'm guessing I could evaluate the function lim y -> 0 but this is not the same as y explicitly equalling zero, is it? On the RHS, y can equal zero, on the LHS, y cannot equal zero. So I am guessing they are only equal when y does not equal zero.\nIs this true?\n\nMany thanks.\n\n2. Oct 28, 2011\n\n### micromass\n\nStaff Emeritus\nIndeed, you cannot evaluate the expression in 0. It is undefined in 0.\n\n3. Oct 29, 2011\n\n### nobahar\n\nThanks Micro!\n\nKnow someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook","date":"2017-12-13 14:03:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5382746458053589, \"perplexity\": 1705.3297418418358}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948523222.39\/warc\/CC-MAIN-20171213123757-20171213143757-00383.warc.gz\"}"} | null | null |
Where are the Toons Now?
Attaining success beyond their wildest dreams. But what happens after the cute theme songs end.
When the Saturday mornings are over and reality sets in.
This is a look at our favorite Latin Marsupial, Speedy Gonzalez. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,059 |
[MY GO 🇲🇾] Malaysian Army ✊🏻 BTS X MEDIHEAL Special Set (50%) 1. Hydrating 2. Brightening 3. Moisture barrier 4. Skin soothing Can choose which set you want Rm78 to WM, add rm5 to EM Limited slots while stock last! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,851 |
\section[]{Sample}
This Appendix presents the full list of damped absorbers identified in the EUADP, including estimates of their metallicity from various references.
XXX mettre a jour la table avec les nouvelles metallicite solaire
\onecolumn
\begin{longtable}{lcccccc}
\hline
\hline
QSO name & $z_{\rm em}$& $z_{\rm abs}$ & log NHI & [X/H] & X & Reference for metallicity measurements\\
\hline
\endfirsthead
\multicolumn{7}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline\hline
QSO name & $z_{\rm em}$ & $z_{\rm abs}$ & log NHI & [X/H] & X & Reference for metallicity measurements \\
\hline
\endhead
\hline \multicolumn{7}{r}{\textit{Continued on next page}} \\
\endfoot
\hline
\endlastfoot
QSO J0003-2323 & 2.1870 & $19.60\pm0.40$ & $-1.76\pm0.42$ & OI & \cite{Richter2005} \
Q0005+0524 & 0.8514 & $19.08\pm0.04$ & $-0.43\pm0.18$ & FeII & \cite{Meiring2009} \
QSO J0006-6208 & 2.9700 & $20.70\pm0.00$ & $0.00\pm0.00$ & no & \cite{Peroux2005} \
QSO J0006-6208 & 3.2020 & $20.80\pm0.10$ & $0.00\pm0.00$ & no & \cite{Peroux2005} \
QSO J0006-6208 & 3.7760 & $21.00\pm0.00$ & $0.00\pm0.00$ & no & \cite{Peroux2005} \
QSO J0006-6208 & 4.1450 & $19.37\pm0.15$ & $0.00\pm0.00$ & no & \cite{Peroux2005} \
QSO J0008-2900 & 2.2540 & $20.22\pm0.10$ & $-1.58\pm0.21$ & FeII & This work \
QSO J0008-2901 & 2.4910 & $19.94\pm0.11$ & $-1.38\pm0.21$ & SII & This work \
QSO J0008-0958 & 1.7680 & $20.85\pm0.15$ & $-0.31\pm0.17$ & ZnII & Herbert-Fort06 \
LBQS 0009-0138 & 1.3860 & $20.26\pm0.02$ & $-1.32\pm0.04$ & SiII & \cite{Meiring2009} \
LBQS 0010-0012 & 2.0250 & $20.95\pm0.10$ & $-1.26\pm0.12$ & ZnII & Ledoux03,Srianand05 \
Q0012-0122 & 1.3862 & $20.26\pm0.02$ & $-1.34\pm0.08$ & SiII & \cite{Meiring2009} \
LBQS 0013-0029 & 1.9730 & $20.83\pm0.05$ & $-0.65\pm0.06$ & ZnII & Petitjean02 \
LBQS 0018+0026 & 0.5200 & $19.54\pm0.03$ & $-1.51\pm0.19$ & FeII & Ellison08 \
LBQS 0018+0026 & 0.9400 & $19.38\pm0.15$ & $0.10\pm0.24$ & FeII & Ellison08 \
J001855-091351 & 0.5840 & $20.11\pm0.10$ & $-1.38\pm0.21$ & FeII & This work \
Q0021+0104 & 1.3259 & $20.04\pm0.11$ & $-0.49\pm0.21$ & FeII & \cite{Meiring2009} \
Q0021+0104 & 1.5756 & $20.48\pm0.15$ & $-1.11\pm0.15$ & SiII & \cite{Meiring2009} \
J0021+0043 & 0.9424 & $19.38\pm0.13$ & $0.54\pm0.26$ & FeII & Dessauges09 \
QSO B0027-1836 & 2.4020 & $21.75\pm0.10$ & $-1.52\pm0.10$ & ZnII & Noterdaeme07 \
QSO B0039-3354 & 2.2240 & $20.60\pm0.10$ & $-1.27\pm0.11$ & SiII & Noterdaeme08 \
J004054.7-091526 & 4.7400 & $20.55\pm0.15$ & $-1.93\pm0.15$ & SiII & Becker11,Rafelski12 \
J0040-0915 & 4.7394 & $20.30\pm0.15$ & $-1.39\pm0.24$ & FeII & \cite{Rafelski2012} \
QSO J0041-4936 & 2.2480 & $20.46\pm0.13$ & $-1.32\pm0.16$ & ZnII & This work \
LBQS 0042-2930 & 1.8090 & $20.40\pm0.10$ & $-1.21\pm0.12$ & SiII & \cite{Fox2009} \
LBQS 0042-2930 & 1.9360 & $20.50\pm0.10$ & $-1.23\pm0.11$ & SiII & \cite{Fox2009} \
QSO B0058-292 & 2.6710 & $21.10\pm0.10$ & $-1.43\pm0.11$ & ZnII & Ledoux03,Srianand05 \
QSO B0100+1300 & 2.3090 & $21.35\pm0.08$ & $-1.42\pm0.08$ & ZnII & Prochaska98,Dessauges03 \
QSO J0105-1846 & 2.3700 & $21.00\pm0.08$ & $-1.79\pm0.14$ & ZnII & Ledoux03,Srianand05 \
QSO J0105-1846 & 2.9260 & $20.00\pm0.10$ & $-1.56\pm0.13$ & OI & Noterdaeme08 \
QSO B0112-30 & 2.4180 & $20.50\pm0.08$ & $-2.24\pm0.11$ & OI & Petitjean08 \
QSO B0112-30 & 2.7020 & $20.30\pm0.10$ & $-0.44\pm0.13$ & SiII & Ledoux03,Srianand05 \
QSO B0122-005 & 1.7610 & $20.78\pm0.07$ & $-0.87\pm0.11$ & SiII & Ellison09 \
QSO B0122-005 & 2.0100 & $20.04\pm0.07$ & $-1.88\pm0.09$ & SiII & Ellison09 \
QSO J0123-0058 & 1.4090 & $20.08\pm0.09$ & $-0.41\pm0.13$ & ZnII & \cite{Peroux2008} \
QSO J0124+0044 & 2.9880 & $19.18\pm0.10$ & $-0.57\pm0.16$ & SiII & Peroux07 \
QSO J0124+0044 & 3.0780 & $20.21\pm0.10$ & $-0.59\pm0.40$ & SiII & Peroux07 \
QSO B0128-2150 & 1.8570 & $20.21\pm0.09$ & $-1.00\pm0.09$ & SII & This work \
J013209-082349 & 0.6470 & $20.60\pm0.12$ & $-0.78\pm0.23$ & FeII & This work \
QSO J0133+0400 & 3.1390 & $19.01\pm0.10$ & $0.00\pm0.00$ & no & Peroux07 \
QSO J0133+0400 & 3.6920 & $20.68\pm0.15$ & $-0.96\pm0.16$ & SiII & Prochaska03 \
QSO J0133+0400 & 3.7730 & $20.55\pm0.13$ & $-0.59\pm0.13$ & SiII & Prochaska03 \
QSO J0133+0400 & 3.9950 & $19.94\pm0.15$ & $-1.54\pm0.20$ & SiII & Peroux07 \
QSO J0133+0400 & 3.9990 & $19.16\pm0.15$ & $0.00\pm0.00$ & no & Peroux07 \
QSO J0133+0400 & 4.0210 & $19.09\pm0.15$ & $0.00\pm0.00$ & no & Peroux07 \
QSO J0134+0051 & 0.8420 & $19.93\pm0.13$ & $-0.60\pm0.22$ & FeII & Peroux06 \
QSO B0135-42 & 3.1010 & $19.81\pm0.10$ & $-1.21\pm0.27$ & SiII & Peroux07 \
QSO B0135-42 & 3.6650 & $19.11\pm0.10$ & $-2.42\pm0.16$ & OI & Peroux07 \
QSO J0138-0005 & 0.7820 & $19.81\pm0.09$ & $0.32\pm0.10$ & ZnII & \cite{Peroux2008} \
QSO J0153+0009 & 0.7710 & $19.70\pm0.09$ & $0.00\pm0.00$ & no & \cite{Peroux2008} \
QSO J0157-0048 & 1.4160 & $19.90\pm0.07$ & $-0.34\pm0.10$ & ZnII & Ellison08 \
QSO B0201+113 & 3.3850 & $21.26\pm0.08$ & $-1.17\pm0.14$ & SII & Ellison01 \
QSO J0209+0517 & 3.6660 & $20.47\pm0.10$ & $-1.97\pm0.21$ & FeII & Prochaska03 \
QSO J0209+0517 & 3.7070 & $19.24\pm0.10$ & $0.00\pm0.00$ & no & \cite{Peroux2005} \
QSO J0209+0517 & 3.8630 & $20.55\pm0.10$ & $-2.60\pm0.11$ & SiII & Prochaska03 \
QSO B0216+0803 & 1.7690 & $20.20\pm0.10$ & $-0.86\pm0.12$ & ZnII & Lu96 \
QSO B0216+0803 & 2.2930 & $20.45\pm0.16$ & $-0.54\pm0.17$ & ZnII & Lu96,Vladilo11 \
QSO J0217+0144 & 1.3450 & $19.89\pm0.09$ & $-1.11\pm0.10$ & MgII & Bergeron86,Blades82 \
QSO B0237-2322 & 1.3650 & $19.30\pm0.30$ & $0.08\pm0.30$ & SiII & Srianand07 \
QSO B0237-2322 & 1.6720 & $19.65\pm0.10$ & $-0.37\pm0.13$ & ZnII & Ellison12 \
QSO B0244-1249 & 1.8630 & $19.48\pm0.18$ & $-0.79\pm0.27$ & SiII & Ellison09 \
QSO B0253+0058 & 0.7250 & $20.70\pm0.17$ & $-0.07\pm0.17$ & ZnII & Peroux06 \
QSO B0254-404 & 2.0460 & $20.45\pm0.08$ & $-1.43\pm0.09$ & SII & Noterdaeme08 \
QSO B0307-195B & 1.7880 & $19.00\pm0.10$ & $0.49\pm0.10$ & SiII & This work \
QSO J0332-4455 & 2.6560 & $19.82\pm0.05$ & $-1.67\pm0.06$ & OI & Fox07 \
QSO B0335-122 & 3.1780 & $20.80\pm0.07$ & $-2.77\pm0.10$ & SiII & Akerman05,Noterdaeme08 \
QSO B0336-017 & 3.0620 & $21.20\pm0.09$ & $-1.33\pm0.09$ & SII & Prochaska01 \
0338-0005 & 2.9090 & $21.10\pm0.10$ & $-1.03\pm0.10$ & SII & Prochaska07 \
QSO B0347-383 & 3.0250 & $20.63\pm0.09$ & $-0.96\pm0.15$ & ZnII & Prochaska01,Ledoux03 \
QSO B0347-2111 & 1.9470 & $20.30\pm0.10$ & $0.00\pm0.00$ & no & Akerman05 \
QSO J0354-2724 & 1.4050 & $20.18\pm0.15$ & $-0.01\pm0.15$ & ZnII & \cite{Meiring2007} \
QSO J0407-4410 & 1.9130 & $20.80\pm0.10$ & $-0.92\pm0.11$ & ZnII & Ledoux06 \
QSO J0407-4410 & 2.5510 & $21.13\pm0.10$ & $-1.25\pm0.11$ & ZnII & Lopez03,Srianand05 \
QSO J0407-4410 & 2.5950 & $21.09\pm0.10$ & $-0.97\pm0.10$ & ZnII & Lopez03,Srianand05 \
QSO J0407-4410 & 2.6210 & $20.45\pm0.10$ & $-1.95\pm0.10$ & OI & Lopez03,Srianand05 \
QSO J0422-3844 & 3.0820 & $19.37\pm0.02$ & $-0.69\pm0.04$ & OI & Carswell96 \
QSO J0427-1302 & 1.5620 & $19.35\pm0.10$ & $-2.26\pm0.21$ & FeII & This work \
QSO J0427-1302 & 1.4080 & $19.04\pm0.04$ & $-0.99\pm0.06$ & SiII & \cite{Meiring2009} \
QSO B0432-440 & 2.2970 & $20.78\pm0.10$ & $0.00\pm0.00$ & no & Akerman05 \
QSO B0438-43 & 2.3470 & $20.78\pm0.12$ & $-0.62\pm0.12$ & ZnII & Akerman05 \
PKS 0439-433 & 0.1012 & $19.63\pm0.15$ & $0.28\pm0.15$ & SII & Kulkarni14 \
QSO B0449-1645 & 1.0070 & $20.98\pm0.07$ & $-0.92\pm0.10$ & ZnII & \cite{Peroux2008} \
QSO B0450-1310B & 2.0670 & $20.50\pm0.07$ & $-2.13\pm0.08$ & OI & Dessauges05,Zafar13 \
PKS 0454-220 & 0.4740 & $19.45\pm0.03$ & $0.47\pm0.04$ & SII & This work, Kulkarni14, Rao06, Som15 \
4C-02.19 & 2.0400 & $21.70\pm0.10$ & $-1.13\pm0.10$ & ZnII & Heinmuller06 \
QSO B0512-3329 & 0.9310 & $20.49\pm0.08$ & $-1.16\pm0.21$ & FeII & Lopez05 \
QSO B0515-4414 & 1.1510 & $19.88\pm0.05$ & $-0.22\pm0.06$ & ZnII & Quast08,Zych08 \
QSO B0528-2505 & 2.1410 & $20.70\pm0.08$ & $-0.26\pm0.09$ & ZnII & Lu96,Centurion03 \
QSO B0528-2505 & 2.8110 & $21.35\pm0.07$ & $-0.82\pm0.10$ & ZnII & Srianand98,Lu96 \
QSO B0551-36 & 1.9620 & $20.70\pm0.08$ & $-0.24\pm0.09$ & ZnII & Ledoux02 \
J060008.1-504036 & 2.1490 & $20.40\pm0.12$ & $-0.85\pm0.12$ & ZnII & This work \
QSO B0642-5038 & 2.6590 & $20.95\pm0.08$ & $-1.01\pm0.10$ & ZnII & Noterdaeme08,Zafar13 \
J0747+4434 & 4.0196 & $20.95\pm0.15$ & $-2.40\pm0.25$ & NiII & \cite{Rafelski2012} \
J0759+1800 & 4.6577 & $20.85\pm0.15$ & $-1.71\pm0.16$ & SII & \cite{Rafelski2012} \
J0817+1351 & 4.2584 & $21.30\pm0.15$ & $-1.12\pm0.15$ & SII & \cite{Rafelski2012} \
J0825+3544 & 3.2073 & $20.30\pm0.10$ & $-1.67\pm0.21$ & FeII & \cite{Rafelski2012} \
J0825+3544 & 3.6567 & $21.25\pm0.10$ & $-1.83\pm0.13$ & SiII & \cite{Rafelski2012} \
J0825+5127 & 3.3180 & $20.85\pm0.10$ & $-1.67\pm0.14$ & SiII & \cite{Rafelski2012} \
Q0826-2230 & 0.9110 & $19.04\pm0.04$ & $1.11\pm0.09$ & ZnII & \cite{Meiring2009} \
QSO B0827+2421 & 0.5180 & $20.30\pm0.04$ & $-0.85\pm0.19$ & FeII & \cite{Meiring2006} \
J0831+4046 & 4.3440 & $20.75\pm0.15$ & $-2.36\pm0.15$ & SiII & \cite{Rafelski2012} \
J0834+2140 & 3.7102 & $20.85\pm0.10$ & $-1.55\pm0.21$ & FeII & \cite{Rafelski2012} \
J0834+2140 & 4.3900 & $21.00\pm0.20$ & $-1.27\pm0.20$ & SII & \cite{Rafelski2012} \
J0834+2140 & 4.4610 & $20.30\pm0.15$ & $-1.86\pm0.16$ & SiII & \cite{Rafelski2012} \
J0839+3524 & 4.2800 & $20.30\pm0.15$ & $-1.14\pm0.24$ & FeII & \cite{Rafelski2012} \
QSO B0841+129 & 1.8640 & $21.00\pm0.10$ & $-1.30\pm0.11$ & SII & Ledoux06 \
QSO B0841+129 & 2.3750 & $21.05\pm0.10$ & $-1.50\pm0.11$ & ZnII & Prochaska98,Zafar13 \
QSO B0841+129 & 2.4760 & $20.80\pm0.10$ & $-1.67\pm0.11$ & ZnII & Prochaska98,Dessauges05 \
J0909+3303 & 3.6584 & $20.55\pm0.10$ & $-1.16\pm0.11$ & SII & \cite{Rafelski2012} \
QSO B0913+0715 & 2.6180 & $20.35\pm0.10$ & $-2.41\pm0.10$ & OI & Pettini08,Petitjean08 \
J0925+4004 & 0.2477 & $19.55\pm0.15$ & $-0.29\pm0.17$ & OI & \cite{Battisti2012} \
J0928+6025 & 0.1538 & $19.35\pm0.15$ & $0.41\pm0.25$ & FeII & \cite{Battisti2012} \
QSO B0933-333 & 2.6820 & $20.50\pm0.10$ & $-1.22\pm0.12$ & SiII & Akerman05,Noterdaeme08 \
QSO B0951-0450 & 3.2350 & $20.25\pm0.10$ & $-1.97\pm0.10$ & SiII & Prochaska03 \
QSO B0951-0450 & 3.8580 & $20.60\pm0.10$ & $-1.47\pm0.10$ & SiII & Prochaska98 \
QSO B0951-0450 & 4.2030 & $20.55\pm0.10$ & $-2.63\pm0.34$ & OI & Prochaska98 \
QSO B0952+179 & 0.2380 & $21.32\pm0.05$ & $-0.95\pm0.06$ & ZnII & Kulkarni05 \
QSO B0952-0115 & 4.0240 & $20.55\pm0.10$ & $-2.61\pm0.11$ & SiII & Prochaska01 \
J1001+5944 & 0.3035 & $19.32\pm0.10$ & $-0.37\pm0.10$ & OI & \cite{Battisti2012} \
QSO J1009-0026 & 0.8400 & $20.20\pm0.07$ & $-0.97\pm0.19$ & FeII & \cite{Meiring2007} \
QSO J1009-0026 & 0.8800 & $19.48\pm0.08$ & $0.34\pm0.09$ & ZnII & \cite{Meiring2007} \
J1009+0713 & 0.1140 & $20.68\pm0.10$ & $-0.55\pm0.16$ & SII & \cite{Battisti2012} \
J1013+4240 & 4.7979 & $20.60\pm0.15$ & $-2.14\pm0.15$ & SiII & \cite{Rafelski2012} \
J1017+6116 & 2.7684 & $20.60\pm0.10$ & $-2.71\pm0.10$ & OI & \cite{Rafelski2012} \
LBQS 1026-0045B & 0.6320 & $19.95\pm0.07$ & $-0.05\pm0.17$ & ZnII & Ellison08 \
LBQS 1026-0045B & 0.7090 & $20.04\pm0.06$ & $-0.08\pm0.19$ & FeII & Ellison08 \
J1028-0100 & 0.6321 & $19.95\pm0.07$ & $-0.01\pm0.21$ & FeII & Dessauges09 \
J1028-0100 & 0.7089 & $20.04\pm0.06$ & $-0.06\pm0.20$ & FeII & Dessauges09 \
QSO B1036-2257 & 2.5330 & $19.30\pm0.10$ & $-1.33\pm0.10$ & MgII & This work \
QSO B1036-2257 & 2.7770 & $20.93\pm0.05$ & $-1.26\pm0.05$ & SII & Prochaska03 \
Q1037+0028 & 1.4244 & $20.04\pm0.12$ & $-0.46\pm0.12$ & SiII & \cite{Meiring2009} \
QSO J1039-2719 & 2.1390 & $19.70\pm0.05$ & $-0.17\pm0.06$ & ZnII & Srianand01 \
J1042+3107 & 4.0865 & $20.75\pm0.10$ & $-1.95\pm0.10$ & SiII & \cite{Rafelski2012} \
QSO B1045+056 & 0.9510 & $19.28\pm0.02$ & $-0.93\pm0.20$ & FeII & \cite{Meiring2006} \
J1051+3107 & 4.1392 & $20.70\pm0.20$ & $-1.96\pm0.22$ & SII & \cite{Rafelski2012} \
J1051+3545 & 4.3498 & $20.45\pm0.10$ & $-1.88\pm0.10$ & SiII & \cite{Rafelski2012} \
J1051+3545 & 4.8206 & $20.35\pm0.10$ & $-2.28\pm0.10$ & SiII & \cite{Rafelski2012} \
Q1054-0020 & 0.8301 & $18.95\pm0.18$ & $0.24\pm0.25$ & FeII & \cite{Meiring2009} \
Q1054-0020 & 0.9514 & $19.28\pm0.02$ & $-0.76\pm0.18$ & FeII & \cite{Meiring2009} \
QSO B1055-301 & 1.9040 & $21.54\pm0.10$ & $-1.19\pm0.10$ & ZnII & Akerman05 \
J1100+1122 & 4.3947 & $21.74\pm0.10$ & $-1.67\pm0.22$ & FeII & \cite{Rafelski2012} \
QSO B1101-26 & 1.8380 & $19.50\pm0.05$ & $-1.64\pm0.10$ & OI & \cite{Dessauges-Zavadsky2003} \
J1101+0531 & 4.3446 & $21.30\pm0.10$ & $-1.07\pm0.12$ & SiII & \cite{Rafelski2012} \
QSO B1104-181 & 1.6610 & $20.85\pm0.01$ & $-0.93\pm0.01$ & ZnII & Lopez99 \
QSO J1107+0048 & 0.7400 & $21.00\pm0.04$ & $-0.50\pm0.16$ & ZnII & Peroux06,Zych08 \
Q1107+0003 & 0.9542 & $20.26\pm0.14$ & $0.00\pm0.00$ & no & \cite{Nestor2008} \
QSO B1108-07 & 3.4820 & $19.95\pm0.07$ & $-1.57\pm0.09$ & SiII & Ledoux06 \
QSO B1108-07 & 3.6080 & $20.37\pm0.07$ & $-1.69\pm0.08$ & OI & Prochaska01,Petitjean08 \
J1111+3509 & 4.0520 & $20.80\pm0.15$ & $-1.95\pm0.16$ & SiII & \cite{Rafelski2012} \
QSO J1113-1533 & 3.2650 & $21.30\pm0.05$ & $-1.61\pm0.07$ & ZnII & Ledoux06,Zafar13 \
QSO B1122-168 & 0.6820 & $20.45\pm0.05$ & $-3.23\pm0.06$ & MgII & Ledoux02b \
QSO B1151+068 & 1.7750 & $21.30\pm0.08$ & $-1.52\pm0.11$ & ZnII & \cite{Pettini1997b} \
J115538.6+053050 & 3.3270 & $21.00\pm0.10$ & $-0.81\pm0.10$ & SII & This work \
J1155+3510 & 2.7582 & $21.00\pm0.10$ & $-1.35\pm0.10$ & SII & \cite{Rafelski2012} \
J1200+4015 & 3.2200 & $20.85\pm0.10$ & $-0.55\pm0.11$ & ZnII & \cite{Rafelski2012} \
J1200+4618 & 4.4765 & $20.50\pm0.15$ & $-1.37\pm0.24$ & FeII & \cite{Rafelski2012} \
J1201+2117 & 3.7975 & $21.35\pm0.15$ & $-0.75\pm0.15$ & SiII & \cite{Rafelski2012} \
J1201+2117 & 4.1578 & $20.60\pm0.15$ & $-2.38\pm0.15$ & SiII & \cite{Rafelski2012} \
QSO B1202-074 & 4.3830 & $20.49\pm0.16$ & $-1.59\pm0.17$ & SiII & Lu96a \
J1202+3235 & 4.7955 & $21.10\pm0.15$ & $-2.34\pm0.24$ & FeII & \cite{Rafelski2012} \
J1202+3235 & 5.0647 & $20.30\pm0.15$ & $-2.66\pm0.16$ & SiII & \cite{Rafelski2012} \
J120550.2+020131 & 1.7470 & $20.40\pm0.10$ & $-0.88\pm0.13$ & ZnII & \cite{Fox2009} \
QSO B1209+0919 & 2.5840 & $21.40\pm0.10$ & $-0.98\pm0.11$ & ZnII & Prochaska07 \
LBQS 1210+1731 & 1.8920 & $20.70\pm0.08$ & $-0.89\pm0.09$ & ZnII & Prochaska01,Dessauges05 \
Q1215-0034 & 1.5543 & $19.56\pm0.02$ & $-0.31\pm0.18$ & FeII & \cite{Meiring2009} \
PG1216+069 & 0.0063 & $19.32\pm0.03$ & $-1.69\pm0.06$ & OI & Tripp05 \
QSO B1220-1800 & 2.1120 & $20.12\pm0.07$ & $-0.71\pm0.08$ & SII & Noterdaeme08 \
Q1220-0040 & 0.9746 & $20.20\pm0.07$ & $-1.00\pm0.19$ & FeII & \cite{Meiring2009} \
LBQS 1223+1753 & 2.4660 & $21.40\pm0.10$ & $-1.41\pm0.10$ & ZnII & Prochaska01,Srianand05 \
LBQS 1223+1753 & 2.5570 & $19.32\pm0.15$ & $-0.45\pm0.15$ & SiII & \cite{Dessauges-Zavadsky2003} \
PHL 1226 & 0.1602 & $19.48\pm0.10$ & $0.24\pm0.15$ & SII & Kulkarni14 \
QSO B1228-113 & 2.1930 & $20.60\pm0.10$ & $-0.15\pm0.11$ & ZnII & Akerman05 \
Q1228+1018 & 0.9376 & $19.41\pm0.02$ & $0.03\pm0.18$ & FeII & \cite{Meiring2009} \
QSO B1230-101 & 1.9310 & $20.48\pm0.10$ & $-0.10\pm0.11$ & ZnII & Akerman05 \
LBQS 1232+0815 & 1.7200 & $19.48\pm0.13$ & $-0.58\pm0.13$ & SiII & This work \
LBQS 1232+0815 & 2.3340 & $20.70\pm0.04$ & $-0.62\pm0.10$ & ZnII & Srianand00,Ge01,Zafar13 \
J1238+3437 & 2.4714 & $20.80\pm0.10$ & $-2.01\pm0.15$ & SII & \cite{Rafelski2012} \
J1241+4617 & 2.6674 & $20.70\pm0.10$ & $-2.18\pm0.10$ & SiII & \cite{Rafelski2012} \
LBQS 1246-0217 & 1.7810 & $21.45\pm0.00$ & $-1.00\pm0.05$ & ZnII & Herbert-Fort06 \
J1248+3110 & 3.6973 & $20.60\pm0.10$ & $-1.63\pm0.21$ & FeII & \cite{Rafelski2012} \
J1253+1046 & 4.6001 & $20.30\pm0.15$ & $-1.35\pm0.24$ & FeII & \cite{Rafelski2012} \
J1257-0111 & 4.0208 & $20.30\pm0.10$ & $-1.56\pm0.10$ & SiII & \cite{Rafelski2012} \
J1304+1202 & 2.9131 & $20.55\pm0.15$ & $-1.62\pm0.16$ & SII & \cite{Rafelski2012} \
J1304+1202 & 2.9289 & $20.30\pm0.15$ & $-1.51\pm0.16$ & SII & \cite{Rafelski2012} \
J1323-0021 & 0.7160 & $20.21\pm0.20$ & $0.66\pm0.21$ & ZnII & Peroux06a \
QSO J1330-2522 & 2.6540 & $19.56\pm0.13$ & $-1.73\pm0.21$ & AlII & This work \
Q1330-2056 & 0.8526 & $19.40\pm0.02$ & $-0.74\pm0.18$ & FeII & \cite{Meiring2009} \
QSO B1331+170 & 1.7760 & $21.15\pm0.07$ & $-1.10\pm0.07$ & ZnII & Prochaska98,Dessauges03 \
QSO J1342-1355 & 3.1180 & $20.05\pm0.08$ & $-1.22\pm0.08$ & OI & Pretitjean08 \
J1353+5328 & 2.8349 & $20.80\pm0.10$ & $-1.35\pm0.10$ & SII & \cite{Rafelski2012} \
QSO J1356-1101 & 2.3970 & $19.85\pm0.08$ & $-1.55\pm0.20$ & FeII & This work \
QSO J1356-1101 & 2.5010 & $20.44\pm0.05$ & $-1.29\pm0.10$ & SII & Akerman05,Noterdaeme08 \
QSO J1356-1101 & 2.9670 & $20.80\pm0.10$ & $-1.35\pm0.12$ & SiII & Akerman05,Noterdaeme08 \
QSO B1409+0930 & 2.0190 & $20.65\pm0.10$ & $-1.58\pm0.14$ & ZnII & Ledoux08 \
QSO B1409+0930 & 2.4560 & $20.53\pm0.08$ & $-2.07\pm0.10$ & OI & Pettini02 \
QSO B1409+0930 & 2.6680 & $19.80\pm0.08$ & $-1.18\pm0.14$ & OI & Pettini02,dessauges03 \
J1412+0624 & 4.1095 & $20.40\pm0.15$ & $-1.71\pm0.25$ & FeII & \cite{Rafelski2012} \
QSO J1421-0643 & 3.4480 & $20.40\pm0.10$ & $-1.29\pm0.13$ & SiII & Akerman05,Noterdaeme08 \
J1435+3604 & 0.2026 & $19.80\pm0.10$ & $-0.32\pm0.16$ & SII & \cite{Battisti2012} \
Q1436-0051 & 0.7377 & $20.08\pm0.11$ & $0.03\pm0.12$ & ZnII & \cite{Meiring2009} \
J1438+4314 & 4.3990 & $20.89\pm0.15$ & $-1.28\pm0.15$ & SII & \cite{Rafelski2012} \
QSO J1439+1117 & 2.4180 & $20.10\pm0.10$ & $0.27\pm0.11$ & ZnII & Noterdaeme08 \
QSO J1443+2724 & 4.2240 & $20.95\pm0.10$ & $-0.52\pm0.10$ & ZnII & Prochaska01,Noterdaeme08,Ledoux06 \
LBQS 1444+0126 & 2.0870 & $20.25\pm0.07$ & $-0.69\pm0.17$ & ZnII & Ledoux03,dessauges03 \
Q1455-0045 & 1.0929 & $20.08\pm0.06$ & $-0.95\pm0.12$ & SiII & \cite{Meiring2009} \
J1507+4406 & 3.0644 & $20.75\pm0.10$ & $-1.90\pm0.14$ & SII & \cite{Rafelski2012} \
J1541+3153 & 2.4435 & $20.95\pm0.10$ & $-1.48\pm0.15$ & ZnII & \cite{Rafelski2012} \
J1553+3548 & 0.0830 & $19.55\pm0.15$ & $-0.84\pm0.16$ & SiII & \cite{Battisti2012} \
PHL 1598 & 0.4297 & $19.18\pm0.03$ & $0.01\pm0.09$ & SII & Kulkarni14 \
J1607+1604 & 4.4741 & $20.30\pm0.15$ & $-1.71\pm0.15$ & SiII & \cite{Rafelski2012} \
J1616+4154 & 0.3211 & $20.60\pm0.20$ & $-0.35\pm0.23$ & SII & \cite{Battisti2012} \
J1619+3342 & 0.0963 & $20.55\pm0.10$ & $-0.59\pm0.13$ & SII & \cite{Battisti2012} \
QSO J1621-0042 & 3.1040 & $19.70\pm0.20$ & $-1.43\pm0.20$ & SiII & This work \
J1626+2751 & 4.3110 & $21.34\pm0.15$ & $-1.15\pm0.24$ & FeII & \cite{Rafelski2012} \
J1626+2751 & 4.4975 & $21.39\pm0.15$ & $-2.45\pm0.24$ & FeII & \cite{Rafelski2012} \
J1626+2751 & 5.1791 & $20.94\pm0.15$ & $-1.46\pm0.15$ & SII & \cite{Rafelski2012} \
4C 12.59 & 0.5310 & $20.70\pm0.09$ & $-1.58\pm0.22$ & FeII & This work \
4C 12.59 & 0.9000 & $19.70\pm0.04$ & $-0.67\pm0.19$ & FeII & \cite{Meiring2009} \
J1654+2227 & 4.0022 & $20.60\pm0.15$ & $-1.65\pm0.24$ & FeII & \cite{Rafelski2012} \
QSO J1723+2243 & 3.6970 & $20.35\pm0.10$ & $0.00\pm0.00$ & no & Prochaska03 \
QSO B2000-330 & 3.1720 & $19.75\pm0.15$ & $-2.29\pm0.15$ & OI & Prochter10 \
QSO B2000-330 & 3.1880 & $19.80\pm0.15$ & $-1.34\pm0.15$ & SiII & Prochter10 \
QSO B2000-330 & 3.1920 & $19.10\pm0.15$ & $-0.48\pm0.15$ & SiII & Prochter10 \
Q2051+1950 & 1.1157 & $20.00\pm0.15$ & $0.34\pm0.18$ & ZnII & \cite{Meiring2009} \
LBQS 2114-4347 & 1.9120 & $19.50\pm0.10$ & $-0.70\pm0.10$ & MgII & This work \
QSO J2119-3536 & 1.9960 & $20.10\pm0.07$ & $-0.36\pm0.11$ & ZnII & \cite{Dessauges-Zavadsky2003} \
QSO B2126-15 & 2.6380 & $19.25\pm0.15$ & $-0.09\pm0.15$ & SiII & This work \
QSO B2126-15 & 2.7690 & $19.20\pm0.15$ & $0.08\pm0.15$ & SiII & This work \
LBQS 2132-4321 & 1.9160 & $20.74\pm0.09$ & $-0.61\pm0.09$ & ZnII & This work \
LBQS 2138-4427 & 2.3830 & $20.60\pm0.05$ & $-1.11\pm0.09$ & ZnII & Ledoux06 \
LBQS 2138-4427 & 2.8520 & $20.98\pm0.05$ & $-1.55\pm0.07$ & ZnII & Ledoux03,Srianand05 \
Q2149+212 & 1.0023 & $19.30\pm0.05$ & $0.00\pm0.00$ & no & \cite{Nestor2008} \
LBQS 2206-1958A & 1.9210 & $20.67\pm0.05$ & $-0.32\pm0.05$ & ZnII & Prochaska01,Vladilo11 \
LBQS 2206-1958A & 2.0760 & $20.44\pm0.05$ & $-2.08\pm0.06$ & OI & Pettini08 \
QSO B2222-396 & 2.1540 & $20.85\pm0.10$ & $-1.72\pm0.12$ & SII & Noterdaeme08 \
LBQS 2230+0232 & 1.8640 & $20.90\pm0.10$ & $-0.66\pm0.10$ & ZnII & Prochaska98,Dessauges05 \
QSO B2237-0607 & 4.0790 & $20.55\pm0.10$ & $-1.79\pm0.10$ & SiII & Lu96 \
J223941.8-294955 & 1.8250 & $18.84\pm0.14$ & $1.36\pm0.15$ & ZnII & Zafar14_prep \
J2252+1425 & 4.7475 & $20.60\pm0.15$ & $-1.76\pm0.26$ & FeII & \cite{Rafelski2012} \
QSO B2311-373 & 2.1820 & $20.48\pm0.13$ & $-1.45\pm0.15$ & SiII & Akerman05,Noterdaeme08 \
QSO B2318-1107 & 1.6290 & $20.52\pm0.14$ & $-1.52\pm0.23$ & FeII & This work \
QSO B2318-1107 & 1.9890 & $20.68\pm0.05$ & $-0.74\pm0.06$ & ZnII & Noterdaeme07 \
QSO J2328+0022 & 0.6520 & $20.32\pm0.07$ & $-0.45\pm0.17$ & ZnII & Peroux06 \
QSO B2332-094 & 3.0570 & $20.50\pm0.07$ & $-1.24\pm0.07$ & OI & Ledoux03,Prochaska03,Petitjean08,Zafar13 \
J233544.2+150118 & 0.6800 & $19.70\pm0.30$ & $0.11\pm0.30$ & ZnII & \cite{Peroux2008} \
QSO B2343+125 & 2.4310 & $20.40\pm0.07$ & $-0.76\pm0.10$ & ZnII & Noterdaeme07 \
QSO J2346+1247 & 2.5690 & $20.98\pm0.04$ & $-0.66\pm0.07$ & ZnII & Rix07 \
QSO B2348-0180 & 2.4260 & $20.50\pm0.10$ & $-0.56\pm0.14$ & SII & Noterdaeme07a,Dessaugses05 \
QSO B2348-0180 & 2.6150 & $21.30\pm0.08$ & $-1.92\pm0.11$ & SiII & Prochaska01 \
QSO B2348-147 & 2.2790 & $20.56\pm0.08$ & $-1.95\pm0.14$ & SII & Prochaska98,Dessauges07 \
Q2352-0028 & 0.8730 & $19.18\pm0.09$ & $-0.84\pm0.20$ & FeII & \cite{Meiring2009} \
Q2352-0028 & 1.0318 & $19.81\pm0.13$ & $0.17\pm0.13$ & SiII & \cite{Meiring2009} \
Q2352-0028 & 1.2467 & $19.60\pm0.24$ & $-0.53\pm0.30$ & FeII & \cite{Meiring2009} \
LBQS 2359-0216 & 2.0950 & $20.65\pm0.10$ & $-0.61\pm0.10$ & ZnII & Prochaska98 \
LBQS 2359-0216 & 2.1540 & $20.30\pm0.10$ & $-1.49\pm0.10$ & SiII & Prochaska98 \
\end{longtable}
\twocolumn
\section[]{Individual objects}
\label{ann:individual}
This Appendix summarizes a description of the individual systems as well as figures and tables providing the Voigt profile parameters for the low, intermediate and high-ionization species when available.
For the following figures, we use $\rm z_{abs}$ from the literature as the zero velocity component.
\subsection{QSOJ0008-2900 z$_{\rm em}=2.645$, z$_{\rm abs}=2.254$, $\rm \log N(HI)=20.22\pm0.10$}
\input{appendix/QSOs/QSOJ0008-2900.tex}
\subsection{QSOJ0008-2901z$_{\rm em}=2.607$, z$_{\rm abs}=2.491$, $\rm \log N(HI)=19.94\pm0.11$}
\input{appendix/QSOs/QSOJ0008-2901.tex}
\subsection{QSO J0018-0913 z$_{\rm em}=0.75593$, z$_{\rm abs}=0.584$, $\rm \log N(HI)=20.11\pm0.10$}
\input{appendix/QSOs/QSOJ0018-0913.tex}
\subsection{QSOJ0041-4936 z$_{\rm em}=3.24$, z$_{\rm abs}=2.248$, $\rm \log N(HI)=20.46\pm0.13$}
\input{appendix/QSOs/QSOJ0041-4936.tex}
\subsection{QSO B0128-2150 z$_{\rm em}=1.9$, z$_{\rm abs}=1.857$, $\rm \log N(HI)=20.21\pm0.09$}
\input{appendix/QSOs/QSOB0128-2150.tex}
\subsection{QSO J0132-0823 z$_{\rm em}=1.121$, z$_{\rm abs}=0.6467$, $\rm \log N(HI)=20.60\pm0.12$}
\input{appendix/QSOs/QSOJ0132-0823.tex}
\subsection{QSO B0307-195B z$_{\rm em}=2.122$, z$_{\rm abs}=1.788$, $\rm \log N(HI)=19.0\pm0.10$}
\input{appendix/QSOs/QSOB0307-195B.tex}
\subsection{QSO J0427-1302 z$_{\rm em}=2.166$, z$_{\rm abs}=1.562$, $\rm \log N(HI)=19.35\pm0.10$}
\input{appendix/QSOs/QSOJ0427-1302.tex}
\subsection{QSO PKS0454-220 z$_{\rm em}=0.534$, z$_{\rm abs}=0.474$, $\rm \log N(HI)=19.45\pm0.03$}
\input{appendix/QSOs/QSOPKS0454-220.tex}
\subsection{QSOJ0600-5040 z$_{\rm em}=3.13$, z$_{\rm abs}=2.149$, $\rm \log N(HI)=20.4\pm0.12$}
\input{appendix/QSOs/QSOJ0600-5040.tex}
\subsection{QSO B1036-2257 z$_{\rm em}=3.13$, z$_{\rm abs}=2.533$, $\rm \log N(HI)=19.3\pm0.10$}
\input{appendix/QSOs/QSOB1036-2257.tex}
\subsection{QSO J115538.6+053050 z$_{\rm em}=3.475$, z$_{\rm abs}=3.327$, $\rm \log N(HI)=21.0\pm0.10$}
\input{appendix/QSOs/QSOJ115538.6+053050_3.32.tex}
\subsection{QSO LBQS 1232+0815 z$_{\rm em}=2.57$, z$_{\rm abs}=1.72$, $\rm \log N(HI)=19.48\pm0.13$}
\input{appendix/QSOs/QSOLBQS1232+0815.tex}
\subsection{QSO J1330-2522 z$_{\rm em}=3.91$, z$_{\rm abs}=2.654$, $\rm \log N(HI)=19.56\pm0.13$}
\input{appendix/QSOs/QSOJ1330-2522_2.654.tex}
\subsection{QSO J1356-1101 z$_{\rm em}=3.006$, z$_{\rm abs}=2.397$, $\rm \log N(HI)=19.85\pm0.08$}
\input{appendix/QSOs/QSOJ1356-1101.tex}
\subsection{QSO J1621-0042 z$_{\rm em}=3.7$, z$_{\rm abs}=3.104$, $\rm \log N(HI)=19.7\pm0.20$}
\input{appendix/QSOs/QSOJ1621-0042.tex}
\subsection{QSO 4C12.59 z$_{\rm em}=1.792$, z$_{\rm abs}=0.531$, $\rm \log N(HI)=20.7\pm0.09$}
\input{appendix/QSOs/QSO4C12.59.tex}
\subsection{QSO LBQS2114-4347 z$_{\rm em}=2.04$, z$_{\rm abs}=1.912$, $\rm \log N(HI)=19.5\pm0.10$}
\input{appendix/QSOs/QSOLBQS2114-4347.tex}
\subsection{QSO B2126-15 z$_{\rm em}=3.268$, z$_{\rm abs}=2.638$, $\rm \log N(HI)=19.25\pm0.15$}
\input{appendix/QSOs/QSOB2126-15_2.638.tex}
\subsection{QSO B2126-15 z$_{\rm em}=3.268$, z$_{\rm abs}=2.769$, $\rm \log N(HI)=19.2\pm0.15$}
\input{appendix/QSOs/QSOB2126-15_2.769.tex}
\subsection{QSO LBQS2132-4321 z$_{\rm em}=2.42$, z$_{\rm abs}=1.916$, $\rm \log N(HI)=20.74\pm0.09$}
\input{appendix/QSOs/QSOLBQS2132-4321.tex}
\subsection{QSO B2318-1107 z$_{\rm em}=2.96$, z$_{\rm abs}=1.629$, $\rm \log N(HI)=20.52\pm0.14$}
\input{appendix/QSOs/QSOB2318-1107.tex}
\section[]{Multi-Element Analysis}
\subsection{Derivation of $\rm F_{*}$}
\label{ann:jenkins}
\cite{Jenkins09} proposed to use the abundances of different elements (namely C, N, O, Mg, Si, P, Cl, Ti, Cr, Mn, Fe, Ni, Cu, Zn, Ge, Kr and S) to compare the dust depletion of dense neutral hydrogen systems to that of the Interstellar Medium (ISM) of our Galaxy.
Specifically, the depletion $\rm [X_{gas}/H]$ described in \cite{Jenkins09} is the difference between the observed abundance of element X normalized to the total hydrogen abundance in both neutral and molecular phases, $\rm N(H)=N(H\,I)+2N(H_{2})$, and its intrinsic abundance (assumed to be solar, initially):
\begin{equation}
\rm [X_{gas}/H] = \log (X/H)_{obs} - log (X/H)_{\odot}
\label{eq:def_depl}
\end{equation}
Using $243$ sight lines in our Galaxy, \cite{Jenkins09} linearly fits the following formula:
\begin{equation}
\rm [X_{gas}/H]_{fit} = [X_{gas}/H]_{0} + A_{X}F_{*}
\end{equation}
where $\rm F_{*}$ is defined as the line-of-sight depletion factor\footnote{We stress that this definition is based on calibration of $\rm F_{*}$ against the descriptive summary definitions reported in the review article by \cite{Savage1996}.}, $\rm A_{X}$ as the propensity of the element X to increase the absolute value of its particular depletion level as $\rm F_{*}$ becomes larger and $\rm[X_{gas}/H]_{0}$ as the depletion for element X when $\rm F_{*}=0$. This equation can linearly be rewritten as:
\begin{equation}
\rm [X_{gas}/H]_{fit} = B_{X} + A_{X}(F_{*}-z_{X})
\label{eq:depl_eq}
\end{equation}
\label{eq:jenkins}where $\rm F_{*}$ has its zero-point reference displaced to an intermediate value $\rm z_{X}$ (unique to element X), $\rm B_{X}$ being the depletion at $\rm F_{*}=z_{X}$.
The three parameters, $\rm A_{X}$ $\rm B_{X}$ and $\rm z_{X}$, are then solved for each element \citep[Table 4 of][]{Jenkins09}.
In the case of DLAs and sub-DLAs\footnote{We note that the measured molecular hydrogen fraction for DLAs is rather low and that this fraction is not correlated with the HI column density \citep{Ledoux2003,Noterdaeme2008a}. Therefore, the definition $\rm N(H)=N(H\,I)+2N(H_{2})$ still holds for the present study.}, releasing the hypothesis of solar metallicity, equation \ref{eq:def_depl} can be rewritten as follows:
\begin{equation}
\rm [X_{gas}/H] = \log (X/H)_{obs} - log (X/H)_{intrinsic}
\end{equation}
where $\rm log (X/H)_{intrinsic}$ is the abundance of element X in the absence of depletion.
Equation \ref{eq:depl_eq} then becomes:
\begin{equation}
\rm [X/H]_{obs} - B_{X}+A_{X}z_{X}=[X/H]_{intrinsic}+A_{X}F_{*}
\label{eq:eq_dep_fit}
\end{equation}
where $\rm [X/H]_{obs}$ is the metallicity compared to solar as we measure it (hence affected by depletion) and $\rm [X/H]_{intrinsic}$ is the intrinsic metallicity compared to solar of the system derived from element X (corrected for depletion).
We derive $\rm F_{*}$ by linearly fitting the left hand side of equation \ref{eq:eq_dep_fit} as a function of $\rm A_{X}$, thus providing an estimate of the intrinsic metallicity.
\subsection{Derivation of $\rm A_{V}$}
\label{Av}
Our study relates to the rest frame extinction $\rm A_{V}$ through equation 3 of \cite{Vladilo2006}, which scales $\rm A_{V}$ to the dust-phase column density of iron $\rm \widehat{N_{Fe}}$. Assuming that zinc is completely undepleted, and that the intrinsic ratio $\rm Zn/Fe$ ratio in DLAs and sub-DLAs is solar, we can express $\rm \widehat{N_{Fe}}$ as the following:\begin{equation}
\rm \widehat{N_{Fe}} = f_{Fe}N_{Zn}\left(\frac{Fe}{Zn}\right)_{\odot}
\end{equation}
where $\rm f_{Fe}=1-10^{\delta_{Fe}}$ is the fraction of iron in dust form. We can assume $\rm \delta_{Fe}=[Fe/Zn]_{obs}$. To recover $\rm N_{Zn}$, we use the assumption that $\rm [Zn/H]=[X/H]$, with $\rm [X/H]$ being the metallicity derived for each system using the ion X (when Zn is not detected). We obtain the following expression:\begin{equation}
\rm \log \widehat{N_{Fe}}=\log\left(1-10^{[Fe/Zn]_{obs}}\right)+\log(N_{X})-\log\left(\frac{N_{X}}{N_{Fe}}\right)_{\odot}
\end{equation}
where $\rm N_{X}$ is the column density of the ion used for the metallicity derivation for a given system.
We can use equation \ref{eq:eq_dep_fit} to derive an estimate of the quantity $\rm [Fe/Zn]_{J}$ from the Jenkins analysis:\begin{equation}
\rm [Fe/Zn]_{J} = ( -1.01\pm 0.07 ) + (-0.68\pm0.08)F_{*}
\end{equation}
This results in $\rm \mean{[Fe/Zn]}_{J,\, sub-DLA} = -0.78\pm0.15$ and $\rm \mean{[Fe/Zn]}_{J, \,DLA} = -0.53\pm0.10$.
Using a bootstrap method, we derive $\rm \mean{\log \widehat{N_{Fe}}}_{sub-DLA}=14.38\pm0.80$ and $\rm \mean{\log \widehat{N_{Fe}}}_{DLA}=14.73\pm0.85$. Based on Fig. 4 from \cite{Vladilo2006}, we can estimate the rest frame extinction $\rm \log A_{V}$ from $\rm \log \widehat{N_{Fe}}$. For both populations, the value of $\rm \mean{\log \widehat{N_{Fe}}}$ falls below the range of MW data used to derive the correlation, giving low values of $\rm \mean{\log A_{V}}\lesssim -2$ mag. This suggests that the dust reddening is not observed in the current quasar selection.
\subsection{QSO J0008-2900}
The EUADP spectrum for this absorber covers multiple low-ionization transitions including FeII $\lambda\lambda\lambda\lambda$ 2374, 2382, 2586, 2600, SiII $\lambda$ 1526, and the saturated MgII doublet $\lambda\lambda$ 2796, 2803. The velocity profiles are well fitted with 5 components. The fit to the Fe lines is found to be consistent with the non-detection of FeII $\lambda$ 2260. The resulting total column density of Fe is \abundance{FeII}{13.78}{0.01}. By comparing the detected SiII $\lambda$ 1526 line with the weak SiII $\lambda$ 1808 transition, we deduce that the former is slightly contaminated. An estimation on the column density based on SiII $\lambda$ 1808 gives \abundance{SiII}{14.40}{0.1}. This total column density is confirmed with the apparent optical depth method applied on SiII $\lambda$ 1808 from $\rm v=-72$ km/s and $\rm v=+10$ km/s.
Because it is saturated, the MgII doublet is fitted fixing the number of components and parameters to the low-ionization lines. The result provides a lower limit to the Mg column density of \lowerlimit{MgII}{15.1}. The non-detection of CrII $\lambda$ 2062, ZnII $\lambda$ 2026 and MnII $\lambda$ 2576 leads to the determination of the following upper limits: \upperlimit{CrII}{12.37}, \upperlimit{ZnII}{11.68} and \upperlimit{MnII}{12.02}.
In addition to these low-ionization ions, the quasar spectrum covers high-ionization species including the SiIV doublet $\lambda\lambda$ 1393 and 1402. However, the bluest SiIV line lies on the Ly$\alpha$ emission line of the quasar thus complicating the quasar continuum placement. Therefore the reddest component of the SiIV doublet is used to model the Voigt profile. In addition, the intermediate-ionization lines of AlIII $\lambda\lambda$ 1854 and 1862 present a similar velocity profile to the SiIV doublet. Therefore, the fit is performed using these transitions simultaneously. A satisfactory 3-component fit leads to the following column densities: \abundance{SiIV}{13.72}{0.03} and \abundance{AlIII}{12.39}{0.04}.
The parameter fits are summarised in Table \ref{ind:0008-2900} and Voigt profile fits are shown in Fig. \ref{img:0008-2900}.
\input{appendix/abundances/QSOJ0008-2900.tex}
\onecolumn
\begin{figure}
\begin{center}
\hspace*{-.8in}
\includegraphics[width=1.2\textwidth]{eps/fits/0008-2900-eps-converted-to.pdf}
\caption[QSOJ0008-2900]{Fit to the low-ionization transitions of the z$_{\rm abs} = 2.254$, log N(H\,I) = $20.22\pm0.10$ absorber towards QSOJ0008-2900 (see
Table \ref{ind:0008-2900}). In this and the following figures, the Voigt profile fits are overlaid in red above the observed quasar spectrum (black) and the green horizontal line indicates the normalised flux level to one. The zero velocity corresponds to the absorption redshift listed in Table \ref{table:subsample} and the vertical dotted lines correspond to the redshift of the fitted components. We warn the reader that the y-axis varies from one panel to another in order to optimise for each transitions so that weaker lines can be readily seen. }
\label{img:0008-2900}
\end{center}
\end{figure}
\twocolumn
\section[]{Sample}
\label{ann:tableau}
This Appendix presents the full list of damped and sub-damped absorbers identified in the EUADP+. We compile estimates of abundances from various references in the literature, as specified in the last column.
We list the column density of ZnII, FeII and SII, as well as estimates of their metallicity based on the ion specified. The online version of this table includes a more complete list of ions.
\onecolumn
\setlength\LTleft{0pt}
\setlength\LTright{0pt}
\begin{longtable}{cccccccc>{\centering\arraybackslash}p{1cm}}
\caption{Full list of damped and sub-damped absorbers identified in the EUADP+, with the column densities of ZnII, FeII and SII, as well as estimates of their metallicity based on the ion specified.}\\
\hline
\hline
QSO name &$z_{\rm abs}$ & log N(HI) & log N(Zn) & log N(Fe) & log N(S) & [X/H] & X & Ref\\
\hline
\endfirsthead
\multicolumn{9}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline\hline
QSO name &$z_{\rm abs}$ & log N(HI) & log N(Zn) & log N(Fe) & log N(S) & [X/H] & X & Reference\\
\hline
\endhead
\hline \multicolumn{9}{r}{\textit{Continued on next page}} \\
\endfoot
\hline
\endlastfoot
CTQ418 & 2.5100 & $20.50\pm0.07$ & - & $14.07\pm0.03$ & $13.99\pm0.06$ & $-1.63\pm0.09$ & SII & 43\\
CTQ418 & 2.4300 & $20.68\pm0.07$ & - & $13.91\pm0.05$ & $13.97\pm0.03$ & $-1.83\pm0.08$ & SII & 43\\
QXO0001 & 3.0000 & $20.70\pm0.05$ & - & $<15.09$ & - & $-1.62\pm0.05$ & OI & 92\\
Q0000-262 & 3.3900 & $21.41\pm0.08$ & $12.01\pm0.05$ & $14.87\pm0.03$ & - & $-1.96\pm0.09$ & ZnII & 62, 90\\
QSO J0003-2323 & 2.1870 & $19.60\pm0.40$ & $<11.04$ & $13.22\pm0.12$ & $<13.12$ & $-1.76\pm0.42$ & OI & 103\\
Q0005+0524 & 0.8514 & $19.08\pm0.04$ & $<11.24$ & $13.79\pm0.01$ & - & $-0.47\pm0.18$ & FeII & 60\\
PSS0007+2417 & 3.5000 & $21.10\pm0.10$ & $<12.39$ & $>14.63$ & - & $-1.53\pm0.11$ & SiII & 94\\
PSS0007+2417 & 3.8400 & $20.85\pm0.15$ & - & $13.91\pm0.03$ & - & $-2.12\pm0.24$ & FeII & 94\\
J0007+0041 & 4.7300 & $20.65\pm0.20$ & - & - & - & $-2.19\pm0.20$ & SiII & 99\\
QSO J0008-2900 & 2.2540 & $20.22\pm0.10$ & $<11.68$ & $13.78\pm0.01$ & - & $-1.33\pm0.14$ & SiII & 114\\
QSO J0008-2901 & 2.4910 & $19.94\pm0.11$ & $<12.12$ & $13.65\pm0.02$ & $13.68\pm0.18$ & $-1.32\pm0.26$ & OI & 114\\
J0008-0958 & 1.7700 & $20.85\pm0.15$ & $13.31\pm0.05$ & $15.62\pm0.05$ & $15.84\pm0.05$ & $-0.10\pm0.16$ & ZnII & 44, 3, 4, 5\\
LBQS 0009-0138 & 1.3860 & $20.26\pm0.02$ & $<11.55$ & $14.25\pm0.01$ & - & $-1.32\pm0.04$ & SiII & 60\\
LBQS 0010-0012 & 2.0250 & $20.95\pm0.10$ & $12.19\pm0.05$ & $15.18\pm0.03$ & $14.98\pm0.05$ & $-1.32\pm0.11$ & ZnII & 52, 110\\
Q0012-0122 & 1.3862 & $20.26\pm0.02$ & $<11.55$ & $14.24\pm0.01$ & - & $-1.34\pm0.08$ & SiII & 60\\
LBQS 0013-0029 & 1.9730 & $20.83\pm0.05$ & $12.74\pm0.05$ & $14.81\pm0.03$ & $15.28\pm0.03$ & $-0.65\pm0.07$ & ZnII & 76, 110, 44\\
LBQS 0018+0026 & 0.5200 & $19.54\pm0.03$ & - & $13.17\pm0.04$ & - & $-1.55\pm0.19$ & FeII & 32\\
LBQS 0018+0026 & 0.9400 & $19.38\pm0.15$ & $<11.64$ & $14.62\pm0.04$ & - & $0.06\pm0.24$ & FeII & 32\\
J001855-091351 & 0.5840 & $20.11\pm0.10$ & $<12.41$ & $13.87\pm0.03$ & - & $-1.42\pm0.21$ & FeII & 114\\
Q0019-15 & 3.4400 & $20.92\pm0.10$ & - & $>14.79$ & - & $-1.01\pm0.11$ & SiII & 89, 90\\
Q0021+0104 & 1.3259 & $20.04\pm0.11$ & $<11.48$ & $14.69\pm0.01$ & - & $-0.53\pm0.21$ & FeII & 60\\
Q0021+0104 & 1.5756 & $20.48\pm0.15$ & $<11.95$ & $14.61\pm0.02$ & - & $-1.11\pm0.15$ & SiII & 60\\
J0021+0043 & 0.9424 & $19.38\pm0.13$ & $<11.64$ & $15.06\pm0.14$ & - & $0.50\pm0.26$ & FeII & 26\\
QSO B0027-1836 & 2.4020 & $21.75\pm0.10$ & $12.79\pm0.02$ & $14.97\pm0.02$ & $15.23\pm0.02$ & $-1.52\pm0.10$ & ZnII & 64\\
J0035-0918 & 2.3400 & $20.55\pm0.10$ & - & $13.07\pm0.04$ & $<13.13$ & $-2.69\pm0.17$ & OI & 17\\
QSO B0039-3354 & 2.2240 & $20.60\pm0.10$ & - & $14.41\pm0.03$ & - & $-1.27\pm0.11$ & SiII & 66\\
J004054.7-091526 & 4.7400 & $20.55\pm0.15$ & - & $14.05\pm0.06$ & - & $-1.93\pm0.15$ & SiII & 98\\
J0040-0915 & 4.7394 & $20.30\pm0.15$ & - & $14.05\pm0.06$ & - & $-1.43\pm0.24$ & FeII & 98\\
QSO J0041-4936 & 2.2480 & $20.46\pm0.13$ & $11.70\pm0.10$ & $14.42\pm0.04$ & $<14.82$ & $-1.32\pm0.16$ & ZnII & 114\\
LBQS 0042-2930 & 1.8090 & $20.40\pm0.10$ & - & - & - & $-1.21\pm0.12$ & SiII & 37\\
LBQS 0042-2930 & 1.9360 & $20.50\pm0.10$ & - & - & - & $-1.23\pm0.11$ & SiII & 37\\
J0044+0018 & 1.7300 & $20.35\pm0.10$ & $<12.61$ & $>14.77$ & $15.27\pm0.05$ & $-0.20\pm0.11$ & SII & 3, 4, 5\\
Q0049-2820 & 2.0700 & $20.45\pm0.10$ & - & $14.50\pm0.02$ & - & $-1.26\pm0.11$ & SiII & 67\\
QSO B0058-292 & 2.6710 & $21.10\pm0.10$ & $12.23\pm0.05$ & $14.75\pm0.03$ & $14.92\pm0.03$ & $-1.43\pm0.11$ & ZnII & 51, 110\\
J0058+0115 & 2.0100 & $21.10\pm0.15$ & $12.95\pm0.05$ & $15.18\pm0.05$ & $15.40\pm0.05$ & $-0.71\pm0.16$ & ZnII & 3, 4, 5\\
QSO B0100+1300 & 2.3090 & $21.35\pm0.08$ & $12.49\pm0.02$ & $15.10\pm0.04$ & $15.09\pm0.06$ & $-1.42\pm0.08$ & ZnII & 89, 22\\
QSO J0105-1846 & 2.3700 & $21.00\pm0.08$ & $11.77\pm0.11$ & $14.47\pm0.10$ & $14.30\pm0.04$ & $-1.79\pm0.14$ & ZnII & 51, 110\\
QSO J0105-1846 & 2.9260 & $20.00\pm0.10$ & - & $13.80\pm0.03$ & $13.82\pm0.03$ & $-1.56\pm0.13$ & OI & 66\\
B0105-008 & 1.3700 & $21.70\pm0.15$ & $12.93\pm0.04$ & $15.59\pm0.03$ & - & $-1.33\pm0.16$ & ZnII & 35\\
QSO B0112-30 & 2.4180 & $20.50\pm0.08$ & - & $13.33\pm0.04$ & $14.44\pm0.03$ & $-2.24\pm0.11$ & OI & 77\\
QSO B0112-30 & 2.7020 & $20.30\pm0.10$ & - & $14.77\pm0.07$ & - & $-0.44\pm0.13$ & SiII & 51, 110\\
Q0112+030 & 2.4200 & $20.90\pm0.10$ & - & $14.85\pm0.01$ & $14.79\pm0.05$ & $-1.23\pm0.11$ & SII & 110, 52, 67\\
QSO B0122-005 & 1.7610 & $20.78\pm0.07$ & - & $15.10\pm0.10$ & - & $-0.87\pm0.11$ & SiII & 33\\
QSO B0122-005 & 2.0100 & $20.04\pm0.07$ & $<11.40$ & $13.69\pm0.07$ & - & $-1.88\pm0.09$ & SiII & 33\\
QSO J0123-0058 & 1.4090 & $20.08\pm0.09$ & $12.23\pm0.10$ & $14.98\pm0.02$ & - & $-0.41\pm0.13$ & ZnII & 74\\
QSO J0124+0044 & 2.9880 & $19.18\pm0.10$ & - & $<13.55$ & $<14.27$ & $-0.57\pm0.16$ & SiII & 73\\
QSO J0124+0044 & 3.0780 & $20.21\pm0.10$ & - & $<14.13$ & - & $-0.59\pm0.40$ & SiII & 73\\
QSO B0128-2150 & 1.8570 & $20.21\pm0.09$ & $<12.26$ & $14.44\pm0.01$ & $14.33\pm0.03$ & $-1.00\pm0.09$ & SII & 114\\
J013209-082349 & 0.6470 & $20.60\pm0.12$ & - & $14.96\pm0.07$ & - & $-0.82\pm0.23$ & FeII & 114\\
QSO J0133+0400 & 3.6920 & $20.68\pm0.15$ & - & $13.51\pm0.07$ & - & $-0.96\pm0.16$ & SiII & 93\\
QSO J0133+0400 & 3.7730 & $20.55\pm0.13$ & $<13.10$ & $>14.87$ & - & $-0.59\pm0.13$ & SiII & 93\\
QSO J0133+0400 & 3.9950 & $19.94\pm0.15$ & - & $<13.43$ & - & $-1.54\pm0.20$ & SiII & 73\\
PSS0133+0400 & 3.6900 & $20.70\pm0.10$ & - & $13.57\pm0.04$ & $<13.35$ & $-2.31\pm0.21$ & FeII & 93, 77\\
PSS0133+0400 & 3.7700 & $20.60\pm0.10$ & $<13.10$ & $>14.87$ & - & $-0.65\pm0.10$ & SiII & 93, 67\\
QSO J0134+0051 & 0.8420 & $19.93\pm0.13$ & $<12.17$ & $14.47\pm0.01$ & - & $-0.64\pm0.22$ & FeII & 72\\
PSS0134+3317 & 3.7600 & $20.85\pm0.08$ & - & - & - & $-2.69\pm0.09$ & AlII & 93\\
QSO B0135-42 & 3.1010 & $19.81\pm0.10$ & - & $13.67\pm0.11$ & - & $-1.21\pm0.27$ & SiII & 73\\
QSO B0135-42 & 3.6650 & $19.11\pm0.10$ & - & $<13.47$ & - & $-2.42\pm0.16$ & OI & 73\\
Q0135-273 & 2.8000 & $21.00\pm0.10$ & - & $14.77\pm0.03$ & $14.80\pm0.02$ & $-1.32\pm0.10$ & SII & 110, 52, 67\\
Q0135-273 & 2.1100 & $20.30\pm0.15$ & - & - & $14.38\pm0.06$ & $-1.04\pm0.16$ & SII & 52\\
QSO J0138-0005 & 0.7820 & $19.81\pm0.09$ & $12.69\pm0.05$ & $<15.17$ & - & $0.32\pm0.10$ & ZnII & 74\\
J0140-0839 & 3.7000 & $20.75\pm0.15$ & - & $<12.73$ & $<13.33$ & $-2.75\pm0.15$ & OI & 34\\
UM673A & 1.6300 & $20.70\pm0.10$ & $11.43\pm0.15$ & $14.59\pm0.03$ & $14.53\pm0.00$ & $-1.83\pm0.18$ & ZnII & 15\\
J0142+0023 & 3.3500 & $20.38\pm0.05$ & $<11.50$ & $13.70\pm0.10$ & $13.26\pm0.06$ & $-2.24\pm0.08$ & SII & 34\\
Q0149+33 & 2.1400 & $20.50\pm0.10$ & $11.50\pm0.10$ & $14.20\pm0.02$ & $<14.80$ & $-1.56\pm0.14$ & ZnII & 89, 11, 90\\
Q0151+0448 & 1.9300 & $20.36\pm0.10$ & $<11.81$ & $13.70\pm0.01$ & $<13.47$ & $-1.86\pm0.11$ & SiII & 34, 118\\
QSO J0157-0048 & 1.4160 & $19.90\pm0.07$ & $12.12\pm0.07$ & $14.57\pm0.03$ & - & $-0.34\pm0.10$ & ZnII & 32\\
QSO B0201+113 & 3.3850 & $21.26\pm0.08$ & - & $15.35\pm0.05$ & $15.21\pm0.11$ & $-1.17\pm0.14$ & SII & 29\\
Q0201+365 & 2.4600 & $20.38\pm0.05$ & $12.47\pm0.05$ & $15.01\pm0.01$ & $15.29\pm0.01$ & $-0.47\pm0.07$ & ZnII & 87, 79, 90, 92, 3, 4, 5\\
Q0201+1120 & 3.3900 & $21.26\pm0.10$ & - & $15.35\pm0.10$ & $15.21\pm0.10$ & $-1.17\pm0.14$ & SII & 29\\
QSO J0209+0517 & 3.6660 & $20.47\pm0.10$ & - & $13.63\pm0.05$ & - & $-2.01\pm0.21$ & FeII & 93\\
QSO J0209+0517 & 3.8630 & $20.55\pm0.10$ & - & $<13.34$ & - & $-2.60\pm0.11$ & SiII & 93\\
PSS0209+0517 & 3.6700 & $20.45\pm0.10$ & - & $13.64\pm0.05$ & - & $-1.99\pm0.21$ & FeII & 93\\
J0211+1241 & 2.6000 & $20.60\pm0.15$ & - & $15.06\pm0.05$ & - & $-0.58\pm0.17$ & SiII & 3, 4, 5\\
QSO B0216+0803 & 1.7690 & $20.20\pm0.10$ & $11.90\pm0.06$ & $14.53\pm0.09$ & - & $-0.86\pm0.12$ & ZnII & 57\\
QSO B0216+0803 & 2.2930 & $20.45\pm0.16$ & $12.47\pm0.05$ & $14.88\pm0.02$ & $15.04\pm0.02$ & $-0.54\pm0.17$ & ZnII & 57, 117\\
QSO J0217+0144 & 1.3450 & $19.89\pm0.09$ & - & $14.38\pm0.10$ & - & $-1.11\pm0.10$ & MgII & 6, 7\\
SDSS0225+0054 & 2.7100 & $21.00\pm0.15$ & $12.89\pm0.11$ & $15.30\pm0.08$ & - & $-0.67\pm0.19$ & ZnII & 44\\
J0233+0103 & 1.7900 & $20.60\pm0.15$ & - & $14.62\pm0.05$ & - & $-1.34\pm0.16$ & SiII & 3, 4, 5\\
J0234-0751 & 2.3200 & $20.90\pm0.10$ & - & $14.18\pm0.03$ & $14.18\pm0.03$ & $-1.84\pm0.10$ & SII & 28\\
AO0235+164 & 0.5200 & $21.70\pm0.10$ & - & $15.30\pm0.40$ & - & $-1.58\pm0.45$ & FeII & 13\\
QSO B0237-2322 & 1.3650 & $19.30\pm0.30$ & - & $14.13\pm0.01$ & - & $0.08\pm0.30$ & SiII & 111\\
QSO B0237-2322 & 1.6720 & $19.65\pm0.10$ & $11.84\pm0.09$ & $14.57\pm0.02$ & - & $-0.37\pm0.13$ & ZnII & 35\\
Q0242-2917 & 2.5600 & $20.90\pm0.10$ & - & $14.36\pm0.03$ & $14.11\pm0.02$ & $-1.91\pm0.10$ & SII & 67\\
QSO B0244-1249 & 1.8630 & $19.48\pm0.18$ & $<11.50$ & $<13.90$ & - & $-0.79\pm0.27$ & SiII & 33\\
QSO B0253+0058 & 0.7250 & $20.70\pm0.17$ & $13.19\pm0.04$ & $15.13\pm0.30$ & - & $-0.07\pm0.17$ & ZnII & 72\\
QSO B0254-404 & 2.0460 & $20.45\pm0.08$ & - & $14.17\pm0.03$ & $14.14\pm0.04$ & $-1.43\pm0.09$ & SII & 66\\
J0255+00 & 3.9200 & $21.30\pm0.05$ & - & $14.75\pm0.09$ & $14.72\pm0.01$ & $-1.70\pm0.05$ & SII & 90\\
J0255+00 & 3.2500 & $20.70\pm0.10$ & - & $14.76\pm0.01$ & - & $-0.89\pm0.11$ & SiII & 90\\
Q0300-3152 & 2.1800 & $20.80\pm0.10$ & - & $14.21\pm0.02$ & $14.20\pm0.03$ & $-1.72\pm0.10$ & SII & 67\\
Q0302-223 & 1.0100 & $20.36\pm0.11$ & $12.45\pm0.06$ & $14.67\pm0.05$ & - & $-0.47\pm0.13$ & ZnII & 82\\
QSO B0307-195B & 1.7880 & $19.00\pm0.10$ & $<12.18$ & $14.48\pm0.00$ & - & $0.49\pm0.10$ & SiII & 114\\
J0307-4945 & 4.4700 & $20.67\pm0.09$ & - & $14.21\pm0.17$ & $<15.46$ & $-1.45\pm0.19$ & OI & 20\\
TXS0311+430 & 2.2900 & $20.30\pm0.00$ & $<12.50$ & $14.85\pm0.20$ & - & $-0.63\pm0.27$ & FeII & 32, 32\\
J0311-1722 & 3.7300 & $20.30\pm0.06$ & - & $<13.76$ & - & $-2.29\pm0.10$ & OI & 16\\
QSO J0332-4455 & 2.6560 & $19.82\pm0.05$ & - & $13.51\pm0.05$ & - & $-1.67\pm0.06$ & OI & 36\\
QSO B0335-122 & 3.1780 & $20.65\pm0.07$ & $<12.25$ & $13.70\pm0.04$ & - & $-2.44\pm0.10$ & SiII & 1, 67\\
QSO B0336-017 & 3.0620 & $21.20\pm0.09$ & - & $14.90\pm0.03$ & $14.99\pm0.01$ & $-1.33\pm0.09$ & SII & 90\\
0338-0005 & 2.9090 & $21.10\pm0.10$ & $<12.47$ & $15.02\pm0.06$ & $15.19\pm0.01$ & $-1.03\pm0.10$ & SII & 95\\
QSO B0347-383 & 3.0250 & $20.63\pm0.09$ & $12.23\pm0.12$ & $14.47\pm0.01$ & $14.73\pm0.01$ & $-0.96\pm0.15$ & ZnII & 90, 51\\
QSO J0354-2724 & 1.4050 & $20.18\pm0.15$ & $12.73\pm0.03$ & $15.10\pm0.03$ & - & $-0.01\pm0.15$ & ZnII & 59\\
B0405-331 & 2.5700 & $20.60\pm0.10$ & $<12.74$ & $14.31\pm0.00$ & - & $-1.40\pm0.10$ & SiII & 1\\
Q0405-443 & 2.5500 & $21.13\pm0.10$ & $12.44\pm0.05$ & $14.95\pm0.06$ & $14.82\pm0.06$ & $-1.25\pm0.11$ & ZnII & 55, 110\\
Q0405-443 & 2.6200 & $20.47\pm0.10$ & - & $13.60\pm0.02$ & $<14.34$ & $-1.97\pm0.10$ & OI & 55, 110, 68, 112\\
QSO J0407-4410 & 1.9130 & $20.80\pm0.10$ & $12.44\pm0.05$ & - & - & $-0.92\pm0.11$ & ZnII & 52\\
QSO J0407-4410 & 2.5510 & $21.13\pm0.10$ & $12.44\pm0.05$ & $14.95\pm0.06$ & $14.82\pm0.06$ & $-1.25\pm0.11$ & ZnII & 55, 110\\
QSO J0407-4410 & 2.5950 & $21.09\pm0.10$ & $12.68\pm0.02$ & $15.15\pm0.02$ & $15.19\pm0.05$ & $-0.97\pm0.10$ & ZnII & 55, 110\\
QSO J0407-4410 & 2.6210 & $20.45\pm0.10$ & - & $13.60\pm0.02$ & $<14.34$ & $-1.95\pm0.10$ & OI & 55, 110\\
Q0421-2624 & 2.1600 & $20.65\pm0.10$ & - & $13.97\pm0.01$ & - & $-1.81\pm0.10$ & SiII & 67\\
QSO J0422-3844 & 3.0820 & $19.37\pm0.02$ & - & $13.96\pm0.10$ & - & $-0.69\pm0.04$ & OI & 10\\
Q0425-5214 & 2.2200 & $20.30\pm0.10$ & - & $13.96\pm0.03$ & $14.07\pm0.03$ & $-1.35\pm0.10$ & SII & 67\\
BRJ0426-2202 & 2.9800 & $21.50\pm0.15$ & $<12.17$ & $14.15\pm0.07$ & - & $-2.53\pm0.24$ & FeII & 93\\
QSO J0427-1302 & 1.5620 & $19.35\pm0.10$ & $<11.75$ & $12.23\pm0.04$ & - & $-2.30\pm0.21$ & FeII & 114\\
QSO J0427-1302 & 1.4080 & $19.04\pm0.04$ & $<11.09$ & $13.33\pm0.02$ & - & $-0.99\pm0.06$ & SiII & 60\\
Q0432-4401 & 2.3000 & $20.95\pm0.10$ & $<12.20$ & $14.87\pm0.10$ & - & $-1.18\pm0.12$ & SiII & 1, 67, 119\\
QSO B0438-43 & 2.3470 & $20.78\pm0.12$ & $12.72\pm0.03$ & - & - & $-0.62\pm0.12$ & ZnII & 1\\
PKS 0439-433 & 0.1012 & $19.63\pm0.15$ & - & $14.92\pm0.03$ & $15.03\pm0.03$ & $0.28\pm0.15$ & SII & 105\\
QSO B0449-1645 & 1.0070 & $20.98\pm0.07$ & $12.62\pm0.07$ & $15.09\pm0.01$ & - & $-0.92\pm0.10$ & ZnII & 74\\
QSO B0450-1310B & 2.0670 & $20.50\pm0.07$ & - & $14.29\pm0.03$ & $14.18\pm0.06$ & $-2.13\pm0.08$ & OI & 24, 119\\
PKS 0454-220 & 0.4740 & $19.45\pm0.03$ & - & $14.71\pm0.01$ & $15.06\pm0.04$ & $0.49\pm0.05$ & SII & 114\\
Q0454+039 & 0.8600 & $20.69\pm0.06$ & $12.33\pm0.08$ & - & - & $-0.92\pm0.10$ & ZnII & 82\\
4C-02.19 & 2.0400 & $21.70\pm0.10$ & $13.13\pm0.05$ & $15.38\pm0.05$ & - & $-1.13\pm0.11$ & ZnII & 42\\
QSO B0512-3329 & 0.9310 & $20.49\pm0.08$ & - & $14.47\pm0.06$ & - & $-1.20\pm0.21$ & FeII & 56\\
QSO B0515-4414 & 1.1510 & $19.88\pm0.05$ & $12.22\pm0.04$ & $14.31\pm0.03$ & - & $-0.22\pm0.06$ & ZnII & 97, 122\\
HE0512-3329A & 0.9300 & $20.49\pm0.08$ & - & $14.47\pm0.06$ & - & $-1.20\pm0.21$ & FeII & 56\\
HE0515-4414 & 1.1500 & $20.45\pm0.15$ & $12.11\pm0.04$ & $14.31\pm0.20$ & - & $-0.90\pm0.16$ & ZnII & 123\\
QSO B0528-2505 & 2.1410 & $20.70\pm0.08$ & $13.00\pm0.03$ & $14.94\pm0.26$ & $14.83\pm0.04$ & $-0.26\pm0.09$ & ZnII & 57, 12\\
QSO B0528-2505 & 2.8110 & $21.11\pm0.07$ & $13.27\pm0.03$ & $15.47\pm0.02$ & $15.56\pm0.02$ & $-0.40\pm0.08$ & ZnII & 107, 57, 12\\
QSO B0551-36 & 1.9620 & $20.50\pm0.08$ & $13.02\pm0.05$ & $15.05\pm0.05$ & $15.38\pm0.11$ & $-0.04\pm0.09$ & ZnII & 49\\
J060008.1-504036 & 2.1490 & $20.40\pm0.12$ & $12.11\pm0.03$ & $14.84\pm0.03$ & - & $-0.85\pm0.12$ & ZnII & 114\\
QSO B0642-5038 & 2.6590 & $20.95\pm0.08$ & $12.50\pm0.06$ & $15.10\pm0.04$ & - & $-1.01\pm0.10$ & ZnII & 66, 119\\
Q0738+313 & 0.0900 & $21.18\pm0.06$ & $<12.66$ & $15.02\pm0.15$ & - & $-1.34\pm0.24$ & FeII & 58, 47\\
HS0741+4741 & 3.0200 & $20.48\pm0.10$ & - & $14.05\pm0.01$ & $14.00\pm0.02$ & $-1.60\pm0.10$ & SII & 90, 92\\
J0747+4434 & 4.0196 & $20.95\pm0.15$ & - & $>14.32$ & - & $-2.50\pm0.20$ & NiII & 98\\
FJ0747+2739 & 3.9000 & $20.50\pm0.10$ & $<12.40$ & $<13.80$ & $<14.36$ & $-1.98\pm0.10$ & SiII & 93\\
J0759+1800 & 4.6577 & $20.85\pm0.15$ & - & $<15.16$ & $14.26\pm0.05$ & $-1.71\pm0.16$ & SII & 98\\
SDSS0759+3129 & 3.0300 & $20.60\pm0.10$ & - & $13.80\pm0.20$ & - & $-2.01\pm0.32$ & SiII & 70\\
PSSJ0808+52 & 3.1100 & $20.65\pm0.07$ & $<12.13$ & $14.17\pm0.04$ & - & $-1.56\pm0.14$ & SiII & 91, 93\\
FJ0812+32 & 2.0700 & $21.00\pm0.10$ & $12.21\pm0.02$ & $14.89\pm0.02$ & - & $-1.35\pm0.10$ & ZnII & 95, 43\\
FJ0812+32 & 2.6300 & $21.35\pm0.10$ & $13.15\pm0.05$ & $15.09\pm0.05$ & $15.63\pm0.07$ & $-0.76\pm0.11$ & ZnII & 93, 95, 3, 4, 5\\
J0815+1037 & 1.8500 & $20.30\pm0.15$ & - & $>14.87$ & - & $-0.43\pm0.47$ & SiII & 3, 4, 5\\
J0816+1446 & 3.2900 & $22.00\pm0.10$ & $13.53\pm0.00$ & $15.89\pm0.00$ & - & $-1.03\pm0.10$ & ZnII & 41\\
J0817+1351 & 4.2584 & $21.30\pm0.15$ & - & $15.45\pm0.06$ & $15.30\pm0.02$ & $-1.12\pm0.15$ & SII & 98\\
J0824+1302 & 4.4700 & $20.65\pm0.20$ & - & $13.60\pm0.08$ & - & $-2.32\pm0.21$ & SiII & 99\\
J0825+3544 & 3.2073 & $20.30\pm0.10$ & - & $13.77\pm0.03$ & - & $-1.71\pm0.21$ & FeII & 98\\
J0825+3544 & 3.6567 & $21.25\pm0.10$ & - & $>14.65$ & - & $-1.83\pm0.13$ & SiII & 98\\
J0825+5127 & 3.3180 & $20.85\pm0.10$ & - & $14.22\pm0.01$ & - & $-1.67\pm0.14$ & SiII & 98\\
Q0826-2230 & 0.9110 & $19.04\pm0.04$ & $12.71\pm0.08$ & $13.57\pm0.06$ & - & $1.11\pm0.09$ & ZnII & 60\\
Q0827+243 & 0.5200 & $20.30\pm0.05$ & $<12.80$ & $14.59\pm0.02$ & - & $-0.89\pm0.19$ & FeII & 58, 45\\
J0831+4046 & 4.3440 & $20.75\pm0.15$ & - & $13.79\pm0.07$ & - & $-2.36\pm0.15$ & SiII & 98\\
J0834+2140 & 3.7102 & $20.85\pm0.10$ & - & $14.44\pm0.02$ & - & $-1.59\pm0.21$ & FeII & 98\\
J0834+2140 & 4.3900 & $21.00\pm0.20$ & - & $14.76\pm0.02$ & $14.85\pm0.04$ & $-1.27\pm0.20$ & SII & 98\\
J0834+2140 & 4.4610 & $20.30\pm0.15$ & - & $13.71\pm0.07$ & $<14.13$ & $-1.86\pm0.16$ & SiII & 98\\
Q0836+11 & 2.4700 & $20.58\pm0.10$ & $<12.12$ & $14.68\pm0.01$ & $<14.66$ & $-1.10\pm0.11$ & SiII & 90, 92\\
J0839+3524 & 4.2800 & $20.30\pm0.15$ & - & $14.30\pm0.04$ & - & $-1.18\pm0.24$ & FeII & 98\\
QSO B0841+129 & 1.8640 & $21.00\pm0.10$ & - & - & $14.82\pm0.05$ & $-1.30\pm0.11$ & SII & 52\\
QSO B0841+129 & 2.3750 & $21.05\pm0.10$ & $12.12\pm0.05$ & $14.76\pm0.11$ & $14.69\pm0.15$ & $-1.50\pm0.11$ & ZnII & 89, 119\\
QSO B0841+129 & 2.4760 & $20.80\pm0.10$ & $11.69\pm0.05$ & $14.43\pm0.03$ & $14.48\pm0.12$ & $-1.67\pm0.11$ & ZnII & 89, 24\\
SDSS0844+5153 & 2.7700 & $21.45\pm0.15$ & - & $15.29\pm0.06$ & - & $-0.99\pm0.15$ & SiII & 44\\
J0900+42 & 3.2500 & $20.30\pm0.10$ & - & $14.54\pm0.01$ & $14.65\pm0.01$ & $-0.77\pm0.10$ & SII & 95, 43\\
J0909+3303 & 3.6584 & $20.55\pm0.10$ & - & $14.43\pm0.01$ & $14.51\pm0.04$ & $-1.16\pm0.11$ & SII & 98\\
QSO B0913+0715 & 2.6180 & $20.35\pm0.10$ & $<11.90$ & $12.99\pm0.01$ & $13.88\pm0.03$ & $-2.41\pm0.10$ & OI & 84, 77\\
B0913+003 & 2.7400 & $20.74\pm0.10$ & $<12.82$ & $14.60\pm0.00$ & - & $-1.47\pm0.10$ & SiII & 1\\
Q0918+1636 & 2.4100 & $21.26\pm0.06$ & $13.23\pm0.18$ & $15.51\pm0.23$ & - & $-0.59\pm0.19$ & ZnII & 39\\
Q0918+1636 & 2.5800 & $20.96\pm0.05$ & $13.40\pm0.01$ & $15.43\pm0.01$ & $15.82\pm0.01$ & $-0.12\pm0.05$ & ZnII & 38\\
J0925+4004 & 0.2477 & $19.55\pm0.15$ & - & $14.22\pm0.09$ & $<14.72$ & $-0.29\pm0.17$ & OI & 2\\
J0927+5823 & 1.6400 & $20.40\pm0.25$ & $13.29\pm0.05$ & $>15.27$ & $15.61\pm0.05$ & $0.33\pm0.25$ & ZnII & 3, 4, 5\\
J0928+6025 & 0.1538 & $19.35\pm0.15$ & - & $14.90\pm0.08$ & $<14.65$ & $0.37\pm0.25$ & FeII & 2\\
SDSS0928+0939 & 2.9100 & $20.75\pm0.15$ & - & $14.10\pm0.30$ & - & $-1.83\pm0.38$ & FeII & 70\\
Q0930+28 & 3.2400 & $20.35\pm0.10$ & - & $13.49\pm0.03$ & - & $-2.07\pm0.10$ & SiII & 92, 93\\
QSO B0933-333 & 2.6820 & $20.50\pm0.10$ & $<11.99$ & $14.46\pm0.08$ & - & $-1.22\pm0.12$ & SiII & 1, 66\\
Q0933+733 & 1.4800 & $21.62\pm0.10$ & $12.71\pm0.02$ & $15.19\pm0.01$ & - & $-1.47\pm0.10$ & ZnII & 101\\
Q0935+417 & 1.3700 & $20.52\pm0.10$ & $12.26\pm0.02$ & $14.82\pm0.10$ & - & $-0.82\pm0.10$ & ZnII & 61, 79, 100\\
Q0948+433 & 1.2300 & $21.62\pm0.06$ & $13.15\pm0.01$ & $15.56\pm0.01$ & - & $-1.03\pm0.06$ & ZnII & 101\\
QSO B0951-0450 & 3.2350 & $20.25\pm0.10$ & - & $13.49\pm0.03$ & - & $-1.97\pm0.10$ & SiII & 93\\
QSO B0951-0450 & 3.8580 & $20.60\pm0.10$ & - & $14.06\pm0.06$ & - & $-1.47\pm0.10$ & SiII & 89\\
QSO B0951-0450 & 4.2030 & $20.55\pm0.10$ & - & $13.07\pm0.19$ & $<13.89$ & $-2.55\pm0.10$ & OI & 89\\
BR0951-04 & 3.8600 & $20.60\pm0.10$ & - & $14.06\pm0.06$ & - & $-1.46\pm0.10$ & SiII & 89, 90\\
QSO B0952+179 & 0.2380 & $21.32\pm0.05$ & $12.93\pm0.04$ & - & - & $-0.95\pm0.06$ & ZnII & 47\\
QSO B0952-0115 & 4.0240 & $20.55\pm0.10$ & - & $14.19\pm0.08$ & - & $-2.61\pm0.11$ & SiII & 90\\
PC0953+4749 & 4.2400 & $20.90\pm0.15$ & - & $13.90\pm0.07$ & - & $-2.18\pm0.15$ & SiII & 106, 93\\
PC0953+4749 & 3.8900 & $21.20\pm0.10$ & - & $15.09\pm0.10$ & - & $-1.29\pm0.23$ & FeII & 106, 93\\
PSSJ0957+33 & 3.2800 & $20.45\pm0.08$ & $<12.13$ & $14.37\pm0.02$ & $<14.58$ & $-1.08\pm0.09$ & SiII & 90, 93\\
PSSJ0957+33 & 4.1800 & $20.70\pm0.10$ & - & $14.13\pm0.05$ & $14.39\pm0.06$ & $-1.43\pm0.12$ & SII & 90, 93\\
J0958+0145 & 1.9300 & $20.40\pm0.10$ & $<12.00$ & $14.23\pm0.05$ & $14.44\pm0.05$ & $-1.08\pm0.11$ & SII & 3, 4, 5\\
J1001+5944 & 0.3035 & $19.32\pm0.10$ & - & $14.30\pm0.04$ & $<14.53$ & $-0.37\pm0.10$ & OI & 2\\
SDSS1003+5520 & 2.5000 & $20.35\pm0.15$ & - & $12.90\pm0.30$ & - & $-2.06\pm0.34$ & SiII & 70\\
J1004+0018 & 2.6900 & $21.39\pm0.10$ & - & $14.71\pm0.04$ & $14.70\pm0.02$ & $-1.81\pm0.10$ & SII & 28\\
J1004+0018 & 2.5400 & $21.30\pm0.10$ & - & $15.13\pm0.02$ & $15.09\pm0.01$ & $-1.33\pm0.10$ & SII & 28\\
Q1007+0042 & 1.0400 & $21.15\pm0.20$ & $13.27\pm0.04$ & - & - & $-0.44\pm0.20$ & ZnII & 63\\
Q1008+36 & 2.8000 & $20.70\pm0.05$ & - & $<15.11$ & - & $-1.75\pm0.05$ & SiII & 43\\
QSO J1009-0026 & 0.8400 & $20.20\pm0.07$ & $<11.85$ & $14.37\pm0.03$ & - & $-1.01\pm0.19$ & FeII & 59\\
QSO J1009-0026 & 0.8800 & $19.48\pm0.08$ & $12.38\pm0.04$ & $15.33\pm0.06$ & - & $0.34\pm0.09$ & ZnII & 59\\
J1009+0713 & 0.1140 & $20.68\pm0.10$ & - & $15.29\pm0.17$ & $15.25\pm0.12$ & $-0.55\pm0.16$ & SII & 2\\
Q1010+0003 & 1.2700 & $21.52\pm0.07$ & $12.96\pm0.06$ & $15.26\pm0.05$ & - & $-1.12\pm0.09$ & ZnII & 58, 63, 3, 4, 5\\
J1013+4240 & 4.7979 & $20.60\pm0.15$ & - & - & - & $-2.14\pm0.15$ & SiII & 98\\
J1013+5615 & 2.2800 & $20.70\pm0.15$ & $13.56\pm0.05$ & $>15.45$ & - & $0.30\pm0.16$ & ZnII & 3, 4, 5\\
BRI1013+0035 & 3.1000 & $21.10\pm0.10$ & $13.33\pm0.02$ & $15.18\pm0.05$ & - & $-0.33\pm0.10$ & ZnII & 95\\
J1017+6116 & 2.7684 & $20.60\pm0.10$ & - & $13.76\pm0.05$ & - & $-2.71\pm0.10$ & OI & 98\\
Q1021+30 & 2.9500 & $20.70\pm0.10$ & $<12.23$ & $14.04\pm0.01$ & $13.87\pm0.07$ & $-1.95\pm0.12$ & SII & 93, 95\\
J1024+0600 & 1.9000 & $20.60\pm0.15$ & - & $15.27\pm0.08$ & $15.45\pm0.05$ & $-0.27\pm0.16$ & SII & 3, 4, 5\\
LBQS 1026-0045B & 0.6320 & $19.95\pm0.07$ & $12.46\pm0.16$ & $15.11\pm0.06$ & - & $-0.05\pm0.17$ & ZnII & 32\\
LBQS 1026-0045B & 0.7090 & $20.04\pm0.06$ & $<12.51$ & $15.10\pm0.03$ & - & $-0.12\pm0.19$ & FeII & 32\\
J1028-0100 & 0.6321 & $19.95\pm0.07$ & $<12.38$ & $15.08\pm0.08$ & - & $-0.05\pm0.21$ & FeII & 26\\
J1028-0100 & 0.7089 & $20.04\pm0.06$ & $<12.49$ & $15.12\pm0.07$ & - & $-0.10\pm0.20$ & FeII & 26\\
SDSS1031+4055 & 2.5700 & $20.55\pm0.10$ & - & $13.80\pm0.20$ & - & $-1.93\pm0.29$ & FeII & 70\\
QSO B1036-2257 & 2.5330 & $19.30\pm0.10$ & $<11.74$ & $12.93\pm0.01$ & - & $-1.33\pm0.10$ & MgII & 114\\
QSO B1036-2257 & 2.7770 & $20.93\pm0.05$ & $<12.36$ & $14.68\pm0.01$ & $14.79\pm0.02$ & $-1.26\pm0.05$ & SII & 120, 52, 67\\
Q1037+0028 & 1.4244 & $20.04\pm0.12$ & $<12.04$ & $14.84\pm0.02$ & - & $-0.46\pm0.12$ & SiII & 60\\
J1037+0139 & 2.7000 & $20.50\pm0.08$ & - & $13.53\pm0.02$ & - & $-2.13\pm0.09$ & OI & 16, 70\\
QSO J1039-2719 & 2.1390 & $19.70\pm0.05$ & $12.09\pm0.04$ & $14.56\pm0.02$ & $14.82\pm0.04$ & $-0.17\pm0.06$ & ZnII & 109\\
J1042+3107 & 4.0865 & $20.75\pm0.10$ & - & $14.22\pm0.03$ & - & $-1.95\pm0.10$ & SiII & 98\\
J1042+0628 & 1.9400 & $20.70\pm0.15$ & - & $15.00\pm0.15$ & $15.08\pm0.05$ & $-0.74\pm0.16$ & SII & 3, 4, 5\\
SDSS1042+0117 & 2.2700 & $20.75\pm0.15$ & $<12.74$ & $15.08\pm0.13$ & - & $-0.79\pm0.17$ & SiII & 44\\
SDSS1043+6151 & 2.7900 & $20.60\pm0.15$ & - & $14.00\pm0.20$ & - & $-2.01\pm0.34$ & SiII & 70\\
QSO B1045+056 & 0.9510 & $19.28\pm0.02$ & $<11.70$ & $13.49\pm0.08$ & - & $-0.97\pm0.20$ & FeII & 58\\
SDSS1048+3911 & 2.3000 & $20.70\pm0.10$ & - & $13.70\pm0.20$ & - & $-2.31\pm0.32$ & SiII & 70\\
J1049-0110 & 1.6600 & $20.35\pm0.15$ & $13.14\pm0.05$ & $15.17\pm0.05$ & $15.47\pm0.05$ & $0.23\pm0.16$ & ZnII & 3, 4, 5\\
J1051+3107 & 4.1392 & $20.70\pm0.20$ & - & $13.95\pm0.03$ & $13.86\pm0.08$ & $-1.96\pm0.22$ & SII & 98\\
J1051+3545 & 4.3498 & $20.45\pm0.10$ & - & $13.66\pm0.05$ & - & $-1.88\pm0.10$ & SiII & 98\\
J1051+3545 & 4.8206 & $20.35\pm0.10$ & - & - & - & $-2.28\pm0.10$ & SiII & 98\\
Q1054-0020 & 0.8301 & $18.95\pm0.18$ & $<11.76$ & $14.33\pm0.01$ & - & $0.20\pm0.25$ & FeII & 60\\
Q1054-0020 & 0.9514 & $19.28\pm0.02$ & $<11.70$ & $13.66\pm0.01$ & - & $-0.80\pm0.18$ & FeII & 60\\
J1054+1633 & 3.8400 & $20.65\pm0.20$ & - & $13.58\pm0.07$ & - & $-2.25\pm0.28$ & FeII & 99\\
J1054+1633 & 4.8200 & $20.65\pm0.20$ & - & - & - & $-2.17\pm0.20$ & SiII & 99\\
J1054+1633 & 4.1400 & $20.65\pm0.20$ & - & - & - & $-0.35\pm0.20$ & SiII & 99\\
QSO B1055-301 & 1.9040 & $21.54\pm0.10$ & $12.91\pm0.03$ & - & - & $-1.19\pm0.10$ & ZnII & 1\\
Q1055+46 & 3.3200 & $20.34\pm0.10$ & - & $13.94\pm0.06$ & - & $-1.60\pm0.15$ & SiII & 91, 43\\
J1056+1208 & 1.6100 & $21.45\pm0.15$ & $13.76\pm0.05$ & $15.81\pm0.05$ & $>16.15$ & $-0.25\pm0.16$ & ZnII & 3, 4, 5\\
J1100+1122 & 4.3947 & $21.74\pm0.10$ & - & $15.21\pm0.09$ & - & $-1.71\pm0.22$ & FeII & 98\\
QSO B1101-26 & 1.8380 & $19.50\pm0.05$ & $<11.27$ & $13.51\pm0.02$ & $13.66\pm0.11$ & $-1.64\pm0.10$ & OI & 22\\
J1101+0531 & 4.3446 & $21.30\pm0.10$ & - & $15.19\pm0.14$ & - & $-1.07\pm0.12$ & SiII & 98\\
QSO B1104-181 & 1.6610 & $20.85\pm0.01$ & $12.48\pm0.01$ & $14.77\pm0.02$ & - & $-0.93\pm0.01$ & ZnII & 53\\
J1106+1044 & 1.8200 & $20.50\pm0.15$ & - & $>15.15$ & $15.33\pm0.05$ & $-0.29\pm0.16$ & SII & 3, 4, 5\\
QSO J1107+0048 & 0.7400 & $21.00\pm0.04$ & $13.06\pm0.15$ & $15.53\pm0.02$ & - & $-0.50\pm0.16$ & ZnII & 72, 122\\
QSO B1108-07 & 3.4820 & $19.95\pm0.07$ & - & - & - & $-1.57\pm0.09$ & SiII & 52\\
QSO B1108-07 & 3.6080 & $20.37\pm0.07$ & - & $13.88\pm0.01$ & - & $-1.69\pm0.08$ & OI & 90, 77\\
J1111+3509 & 4.0520 & $20.80\pm0.15$ & - & $14.13\pm0.05$ & $<14.34$ & $-1.95\pm0.16$ & SiII & 98\\
Q1111-152 & 3.2700 & $21.30\pm0.05$ & $12.32\pm0.10$ & $14.81\pm0.01$ & $14.62\pm0.04$ & $-1.54\pm0.11$ & ZnII & 52, 67, 120, 119\\
SDSS1116+4118A & 2.6600 & $20.48\pm0.10$ & $12.40\pm0.20$ & $14.36\pm0.10$ & - & $-0.64\pm0.22$ & ZnII & 31\\
BR1117-1329 & 3.3500 & $20.84\pm0.12$ & $12.26\pm0.03$ & $14.83\pm0.03$ & - & $-1.14\pm0.12$ & ZnII & 71, 110\\
HE1122-1649 & 0.6800 & $20.45\pm0.05$ & $<11.76$ & $14.55\pm0.01$ & - & $-0.60\pm0.13$ & SiII & 123, 50\\
Q1127-145 & 0.3100 & $21.70\pm0.08$ & $13.53\pm0.13$ & $>15.16$ & - & $-0.73\pm0.15$ & ZnII & 45\\
J1131+6044 & 2.8800 & $20.50\pm0.15$ & - & $13.76\pm0.03$ & $<13.29$ & $-1.52\pm0.20$ & SiII & 34\\
HS1132+2243 & 2.7800 & $21.00\pm0.07$ & $<11.99$ & $14.02\pm0.01$ & $14.07\pm0.06$ & $-2.05\pm0.09$ & SII & 93\\
J1132+1209 & 4.3800 & $20.65\pm0.20$ & - & $13.78\pm0.07$ & - & $-2.05\pm0.28$ & FeII & 99\\
J1132+1209 & 5.0200 & $20.65\pm0.20$ & - & $<13.55$ & - & $-2.66\pm0.20$ & SiII & 99\\
J1135-0010 & 2.2100 & $22.05\pm0.10$ & $13.62\pm0.03$ & $15.76\pm0.03$ & $>16.19$ & $-0.99\pm0.10$ & ZnII & 48, 69\\
Q1137+3907 & 0.7200 & $21.10\pm0.10$ & $13.43\pm0.05$ & $15.45\pm0.05$ & - & $-0.23\pm0.11$ & ZnII & 58\\
J1142+0701 & 1.8400 & $21.50\pm0.15$ & $13.29\pm0.05$ & $15.47\pm0.05$ & - & $-0.77\pm0.16$ & ZnII & 3, 4, 5\\
QSO B1151+068 & 1.7750 & $21.30\pm0.08$ & $12.34\pm0.08$ & - & - & $-1.52\pm0.11$ & ZnII & 80\\
J115538.6+053050 & 3.3270 & $21.00\pm0.10$ & - & - & $15.31\pm0.00$ & $-0.81\pm0.10$ & SII & 114\\
J1155+3510 & 2.7582 & $21.00\pm0.10$ & - & $<14.73$ & $14.77\pm0.01$ & $-1.35\pm0.10$ & SII & 98\\
J1155+0530 & 2.6100 & $20.37\pm0.11$ & - & - & - & $-1.57\pm0.16$ & SiII & 119\\
J1155+0530 & 3.3300 & $21.05\pm0.10$ & $12.89\pm0.07$ & $15.37\pm0.05$ & $15.40\pm0.05$ & $-0.72\pm0.12$ & ZnII & 3, 4, 5\\
Q1157+014 & 1.9400 & $21.70\pm0.10$ & $13.11\pm0.06$ & $15.49\pm0.05$ & $>15.16$ & $-1.15\pm0.12$ & ZnII & 75, 24, 25, 3, 4, 5\\
J1200+4015 & 3.2200 & $20.85\pm0.10$ & $12.86\pm0.04$ & $15.31\pm0.04$ & $15.36\pm0.01$ & $-0.55\pm0.11$ & ZnII & 98\\
J1200+4618 & 4.4765 & $20.50\pm0.15$ & - & $14.27\pm0.02$ & - & $-1.41\pm0.24$ & FeII & 98\\
J1201+2117 & 3.7975 & $21.35\pm0.15$ & - & $15.56\pm0.04$ & - & $-0.75\pm0.15$ & SiII & 98\\
J1201+2117 & 4.1578 & $20.60\pm0.15$ & - & $13.76\pm0.03$ & - & $-2.38\pm0.15$ & SiII & 98\\
QSO B1202-074 & 4.3830 & $20.55\pm0.16$ & - & $13.88\pm0.11$ & - & $-1.49\pm0.17$ & OI & 57, 106, 27\\
J1202+3235 & 4.7955 & $21.10\pm0.15$ & - & $13.90\pm0.03$ & - & $-2.38\pm0.24$ & FeII & 98\\
J1202+3235 & 5.0647 & $20.30\pm0.15$ & - & - & - & $-2.66\pm0.16$ & SiII & 98\\
J1204-0021 & 3.6400 & $20.65\pm0.20$ & - & $13.85\pm0.04$ & - & $-1.98\pm0.27$ & FeII & 99\\
J120550.2+020131 & 1.7470 & $20.40\pm0.10$ & $12.08\pm0.08$ & - & - & $-0.88\pm0.13$ & ZnII & 37\\
J1208+0010 & 5.0800 & $20.65\pm0.20$ & - & $13.27\pm0.08$ & - & $-2.41\pm0.20$ & SiII & 99\\
QSO B1209+0919 & 2.5840 & $21.40\pm0.10$ & $12.98\pm0.05$ & $15.25\pm0.03$ & - & $-0.98\pm0.11$ & ZnII & 95\\
LBQS 1210+1731 & 1.8920 & $20.70\pm0.08$ & $12.37\pm0.03$ & $14.95\pm0.06$ & $14.96\pm0.03$ & $-0.89\pm0.09$ & ZnII & 90, 24\\
Q1215-0034 & 1.5543 & $19.56\pm0.02$ & $<11.63$ & $14.39\pm0.01$ & - & $-0.35\pm0.18$ & FeII & 60\\
Q1215+33 & 2.0000 & $20.95\pm0.07$ & $12.33\pm0.05$ & $14.75\pm0.05$ & $<15.36$ & $-1.18\pm0.09$ & ZnII & 89, 11, 90, 91\\
PG1216+069 & 0.0063 & $19.32\pm0.03$ & - & $13.23\pm0.14$ & - & $-1.69\pm0.06$ & OI & 115\\
J1219+1603 & 3.0000 & $20.35\pm0.10$ & - & $13.80\pm0.10$ & - & $-2.52\pm0.35$ & OI & 70\\
QSO B1220-1800 & 2.1120 & $20.12\pm0.07$ & - & $14.36\pm0.04$ & $14.53\pm0.04$ & $-0.71\pm0.08$ & SII & 66\\
Q1220-0040 & 0.9746 & $20.20\pm0.07$ & $<11.69$ & $14.34\pm0.02$ & - & $-1.04\pm0.19$ & FeII & 60\\
J1221+4445 & 4.8100 & $20.65\pm0.20$ & - & $14.35\pm0.06$ & - & $-2.21\pm0.20$ & SiII & 99\\
LBQS 1223+1753 & 2.4660 & $21.40\pm0.10$ & $12.55\pm0.03$ & $15.16\pm0.02$ & $15.14\pm0.04$ & $-1.41\pm0.10$ & ZnII & 90, 110\\
LBQS 1223+1753 & 2.5570 & $19.32\pm0.15$ & $<11.51$ & $13.98\pm0.03$ & - & $-0.45\pm0.15$ & SiII & 22\\
Q1224+0037 & 1.2300 & $20.88\pm0.05$ & $<11.89$ & $>15.11$ & - & $-1.29\pm0.09$ & SiII & 59\\
Q1225+0035 & 0.7700 & $21.38\pm0.11$ & $<13.01$ & $15.69\pm0.03$ & - & $-0.87\pm0.21$ & FeII & 58, 63\\
PHL 1226 & 0.1602 & $19.48\pm0.10$ & - & $14.76\pm0.18$ & $14.84\pm0.11$ & $0.24\pm0.15$ & SII & 105\\
QSO B1228-113 & 2.1930 & $20.60\pm0.10$ & $13.01\pm0.04$ & - & - & $-0.15\pm0.11$ & ZnII & 1\\
Q1228+1018 & 0.9376 & $19.41\pm0.02$ & $<11.67$ & $14.58\pm0.01$ & - & $-0.01\pm0.18$ & FeII & 60\\
PKS1229-021 & 0.4000 & $20.75\pm0.07$ & $12.92\pm0.10$ & $<14.95$ & - & $-0.39\pm0.12$ & ZnII & 8\\
QSO B1230-101 & 1.9310 & $20.48\pm0.10$ & $12.94\pm0.05$ & - & - & $-0.10\pm0.11$ & ZnII & 1\\
LBQS 1232+0815 & 1.7200 & $19.48\pm0.13$ & $<11.58$ & $13.50\pm0.01$ & $<14.19$ & $-0.58\pm0.13$ & SiII & 114\\
LBQS 1232+0815 & 2.3340 & $20.90\pm0.04$ & $12.64\pm0.09$ & $14.68\pm0.08$ & $14.83\pm0.10$ & $-0.82\pm0.10$ & ZnII & 108, 40, 12, 120\\
J1238+3437 & 2.4714 & $20.80\pm0.10$ & - & $14.06\pm0.03$ & $13.91\pm0.11$ & $-2.01\pm0.15$ & SII & 98\\
J1240+1455 & 3.1100 & $21.30\pm0.20$ & $12.90\pm0.07$ & $14.60\pm0.03$ & $15.56\pm0.02$ & $-0.96\pm0.21$ & ZnII & 34\\
J1241+4617 & 2.6674 & $20.70\pm0.10$ & - & $14.02\pm0.04$ & - & $-2.18\pm0.10$ & SiII & 98\\
LBQS1242+0006 & 1.8200 & $20.45\pm0.10$ & - & - & - & $-1.20\pm0.15$ & SiII & 119\\
J1245+3822 & 4.4500 & $20.65\pm0.20$ & - & $<13.93$ & - & $-2.14\pm0.20$ & SiII & 99\\
LBQS 1246-0217 & 1.7810 & $21.45\pm0.00$ & $13.01\pm0.05$ & $15.47\pm0.02$ & - & $-1.00\pm0.05$ & ZnII & 44\\
J1248+3110 & 3.6973 & $20.60\pm0.10$ & - & $14.11\pm0.03$ & - & $-1.67\pm0.21$ & FeII & 98\\
SDSS1249-0233 & 1.7800 & $21.45\pm0.15$ & $13.15\pm0.05$ & $15.47\pm0.02$ & $15.53\pm0.05$ & $-0.86\pm0.16$ & ZnII & 44, 3, 4, 5\\
SDSS1251+4120 & 2.7300 & $21.10\pm0.10$ & - & $14.20\pm0.30$ & - & $-2.71\pm0.32$ & SiII & 70\\
J1253+1046 & 4.6001 & $20.30\pm0.15$ & - & $14.09\pm0.03$ & - & $-1.39\pm0.24$ & FeII & 98\\
PSS1253-0228 & 2.7800 & $21.85\pm0.20$ & $12.77\pm0.07$ & $15.36\pm0.04$ & - & $-1.64\pm0.21$ & ZnII & 93\\
J1257-0111 & 4.0208 & $20.30\pm0.10$ & - & $13.65\pm0.07$ & $<13.90$ & $-1.56\pm0.10$ & SiII & 98\\
J1304+1202 & 2.9131 & $20.55\pm0.15$ & $<11.83$ & $13.72\pm0.04$ & $14.05\pm0.05$ & $-1.62\pm0.16$ & SII & 98\\
J1304+1202 & 2.9289 & $20.30\pm0.15$ & $<11.95$ & $13.85\pm0.03$ & $13.91\pm0.04$ & $-1.51\pm0.16$ & SII & 98\\
J1305+0924 & 2.0200 & $20.40\pm0.15$ & - & $15.21\pm0.14$ & $15.39\pm0.05$ & $-0.13\pm0.16$ & SII & 3, 4, 5\\
J1310+5424 & 1.8000 & $21.45\pm0.15$ & $13.57\pm0.05$ & $15.64\pm0.05$ & $>16.05$ & $-0.44\pm0.16$ & ZnII & 3, 4, 5\\
J1323-0021 & 0.7160 & $20.21\pm0.20$ & $13.43\pm0.05$ & $15.15\pm0.03$ & - & $0.66\pm0.21$ & ZnII & 72\\
Q1323-0021 & 0.7200 & $20.54\pm0.15$ & $13.29\pm0.21$ & - & - & $0.19\pm0.26$ & ZnII & 63\\
SDSS1325+1255 & 3.5500 & $20.50\pm0.15$ & - & $<13.69$ & - & $-2.51\pm0.25$ & SiII & 70\\
Q1328+307 & 0.6900 & $21.25\pm0.10$ & $12.72\pm0.10$ & $14.98\pm0.10$ & - & $-1.09\pm0.14$ & ZnII & 79, 8, 50\\
QSO J1330-2522 & 2.6540 & $19.56\pm0.13$ & - & - & - & $-1.83\pm0.13$ & AlII & 114\\
Q1330-2056 & 0.8526 & $19.40\pm0.02$ & $<11.96$ & $13.80\pm0.01$ & - & $-0.78\pm0.18$ & FeII & 60\\
QSO B1331+170 & 1.7760 & $21.15\pm0.07$ & $12.61\pm0.01$ & $14.60\pm0.00$ & $15.08\pm0.11$ & $-1.10\pm0.07$ & ZnII & 89, 22\\
J1335+0824 & 1.8600 & $20.65\pm0.15$ & - & $>15.17$ & $15.29\pm0.05$ & $-0.48\pm0.16$ & SII & 3, 4, 5\\
Q1337+113 & 2.8000 & $21.00\pm0.08$ & $<12.25$ & $14.33\pm0.01$ & $14.33\pm0.02$ & $-1.95\pm0.11$ & OI & 93, 110, 52, 95, 77, 67\\
J1340+1106 & 2.8000 & $21.00\pm0.06$ & - & $14.32\pm0.01$ & $14.30\pm0.02$ & $-1.65\pm0.07$ & OI & 16\\
J1340+3926 & 4.8300 & $20.65\pm0.20$ & - & $14.33\pm0.05$ & - & $-1.50\pm0.27$ & FeII & 99\\
QSO J1342-1355 & 3.1180 & $20.05\pm0.08$ & - & $13.93\pm0.03$ & $13.83\pm0.03$ & $-1.22\pm0.08$ & OI & 77\\
J1345+2329 & 5.0100 & $20.65\pm0.20$ & - & - & $14.66\pm0.05$ & $-1.11\pm0.21$ & SII & 99\\
BRI1346-03 & 3.7400 & $20.72\pm0.10$ & - & $<14.13$ & - & $-2.28\pm0.10$ & SiII & 89, 90\\
SDSS1350+5952 & 2.7600 & $20.65\pm0.10$ & - & $13.50\pm0.20$ & - & $-2.33\pm0.29$ & FeII & 70\\
J1353+5328 & 2.8349 & $20.80\pm0.10$ & - & $>14.46$ & $14.57\pm0.02$ & $-1.35\pm0.10$ & SII & 98\\
Q1354+258 & 1.4200 & $21.54\pm0.06$ & $12.59\pm0.08$ & $15.01\pm0.04$ & - & $-1.51\pm0.10$ & ZnII & 81\\
PKS1354-17 & 2.7800 & $20.30\pm0.15$ & - & $13.37\pm0.08$ & - & $-1.83\pm0.16$ & SiII & 93\\
QSO J1356-1101 & 2.3970 & $19.85\pm0.08$ & $<12.38$ & $13.44\pm0.01$ & - & $-1.59\pm0.20$ & FeII & 114\\
QSO J1356-1101 & 2.5010 & $20.44\pm0.05$ & $<11.70$ & $14.36\pm0.08$ & $14.27\pm0.09$ & $-1.29\pm0.10$ & SII & 1, 66\\
QSO J1356-1101 & 2.9670 & $20.80\pm0.10$ & $<11.93$ & $14.63\pm0.05$ & - & $-1.35\pm0.12$ & SiII & 1, 66\\
J1358+0349 & 2.8500 & $20.50\pm0.10$ & - & $13.01\pm0.05$ & - & $-2.81\pm0.27$ & OI & 70\\
J1358+6522 & 3.0700 & $20.35\pm0.15$ & - & $<12.80$ & - & $-3.01\pm0.17$ & OI & 70, 18\\
QSO B1409+0930 & 2.0190 & $20.65\pm0.10$ & $11.63\pm0.10$ & - & - & $-1.58\pm0.14$ & ZnII & 52\\
QSO B1409+0930 & 2.4560 & $20.53\pm0.08$ & - & $13.74\pm0.02$ & - & $-2.07\pm0.10$ & OI & 83\\
QSO B1409+0930 & 2.6680 & $19.80\pm0.08$ & $<11.22$ & $14.02\pm0.13$ & $13.54\pm0.06$ & $-1.18\pm0.14$ & OI & 83, 22\\
J1412+0624 & 4.1095 & $20.40\pm0.15$ & - & $13.83\pm0.08$ & - & $-1.75\pm0.25$ & FeII & 98\\
J1417+4132 & 1.9500 & $21.45\pm0.25$ & $13.55\pm0.05$ & $15.58\pm0.05$ & $>15.80$ & $-0.46\pm0.25$ & ZnII & 3, 4, 5\\
J1418+3142 & 3.9600 & $20.65\pm0.20$ & - & $<15.78$ & - & $-0.39\pm0.20$ & SiII & 99\\
J1419+0829 & 3.0500 & $20.40\pm0.03$ & - & $13.54\pm0.03$ & - & $-1.92\pm0.04$ & OI & 16, 86\\
QSO J1421-0643 & 3.4480 & $20.40\pm0.10$ & $<11.98$ & $14.18\pm0.08$ & - & $-1.29\pm0.13$ & SiII & 1, 66\\
Q1425+6039 & 2.8300 & $20.30\pm0.04$ & $12.18\pm0.04$ & $14.48\pm0.01$ & - & $-0.68\pm0.06$ & ZnII & 57, 91, 92, 95\\
J1431+3952 & 0.6000 & $21.20\pm0.10$ & $13.03\pm0.19$ & $15.15\pm0.11$ & - & $-0.73\pm0.21$ & ZnII & 35\\
PSS1432+39 & 3.2700 & $21.25\pm0.10$ & $<12.65$ & $>14.93$ & - & $-1.09\pm0.11$ & SiII & 93\\
J1435+3604 & 0.2026 & $19.80\pm0.10$ & - & $14.20\pm0.08$ & $14.60\pm0.12$ & $-0.32\pm0.16$ & SII & 2\\
SDSS1435+0420 & 1.6600 & $21.25\pm0.15$ & $<13.21$ & $15.70\pm0.07$ & - & $-0.84\pm0.17$ & SiII & 44\\
J1435+5359 & 2.3400 & $21.05\pm0.10$ & - & - & $14.78\pm0.05$ & $-1.39\pm0.11$ & SII & 43\\
Q1436-0051 & 0.7377 & $20.08\pm0.11$ & $12.67\pm0.05$ & $14.94\pm0.02$ & - & $0.03\pm0.12$ & ZnII & 60\\
J1437+2323 & 4.8000 & $20.65\pm0.20$ & - & - & - & $-2.34\pm0.20$ & SiII & 99\\
J1438+4314 & 4.3990 & $20.89\pm0.15$ & - & $14.42\pm0.01$ & $14.73\pm0.01$ & $-1.28\pm0.15$ & SII & 98\\
QSO J1439+1117 & 2.4180 & $20.10\pm0.10$ & $12.93\pm0.04$ & $14.28\pm0.05$ & $15.27\pm0.06$ & $0.27\pm0.11$ & ZnII & 66\\
SDSS1440+0637 & 2.5200 & $21.00\pm0.15$ & - & $14.50\pm0.30$ & - & $-2.31\pm0.34$ & SiII & 70\\
QSO J1443+2724 & 4.2240 & $20.95\pm0.10$ & $12.99\pm0.03$ & $15.33\pm0.03$ & $15.52\pm0.01$ & $-0.52\pm0.10$ & ZnII & 90, 66, 52\\
LBQS 1444+0126 & 2.0870 & $20.25\pm0.07$ & $12.12\pm0.15$ & $14.41\pm0.03$ & $14.62\pm0.08$ & $-0.69\pm0.17$ & ZnII & 51, 22\\
Q1451+123 & 2.4700 & $20.39\pm0.10$ & - & $13.36\pm0.07$ & $<13.55$ & $-1.90\pm0.16$ & SiII & 75, 110\\
Q1451+123 & 2.2600 & $20.30\pm0.15$ & $11.85\pm0.11$ & $14.33\pm0.07$ & - & $-1.01\pm0.19$ & ZnII & 22\\
J1454+0941 & 1.7900 & $20.50\pm0.15$ & $12.72\pm0.05$ & $15.02\pm0.12$ & $15.25\pm0.06$ & $-0.34\pm0.16$ & ZnII & 3, 4, 5\\
Q1455-0045 & 1.0929 & $20.08\pm0.06$ & $<11.91$ & $14.57\pm0.01$ & - & $-0.95\pm0.12$ & SiII & 60\\
J1456+0407 & 2.6700 & $20.35\pm0.10$ & - & $13.00\pm0.10$ & - & $-2.49\pm0.30$ & OI & 70\\
Q1501+0019 & 1.4800 & $20.85\pm0.13$ & $12.93\pm0.06$ & - & - & $-0.48\pm0.14$ & ZnII & 58\\
Q1502+4837 & 2.5700 & $20.30\pm0.15$ & - & $14.15\pm0.12$ & - & $-1.57\pm0.17$ & SiII & 93\\
PSS1506+5220 & 3.2200 & $20.67\pm0.07$ & $<12.11$ & $13.71\pm0.03$ & - & $-2.30\pm0.07$ & SiII & 93\\
J1507+4406 & 3.0644 & $20.75\pm0.10$ & - & $14.03\pm0.03$ & $13.97\pm0.10$ & $-1.90\pm0.14$ & SII & 98\\
J1509+1113 & 2.0300 & $21.30\pm0.15$ & - & $15.48\pm0.07$ & $15.69\pm0.05$ & $-0.73\pm0.16$ & SII & 3, 4, 5\\
PSS1535+2943 & 3.7600 & $20.40\pm0.15$ & - & - & - & $-1.97\pm0.16$ & SiII & 94\\
J1541+3153 & 2.4435 & $20.95\pm0.10$ & $12.03\pm0.11$ & $14.50\pm0.11$ & - & $-1.48\pm0.15$ & ZnII & 98\\
SBS1543+393 & 0.0100 & $20.42\pm0.04$ & - & - & $15.19\pm0.04$ & $-0.35\pm0.06$ & SII & 9\\
J1553+3548 & 0.0830 & $19.55\pm0.15$ & - & $14.01\pm0.07$ & $<14.24$ & $-0.84\pm0.16$ & SiII & 2\\
J1555+4800 & 2.3900 & $21.50\pm0.15$ & $<13.95$ & $15.84\pm0.05$ & $>15.88$ & $-0.46\pm0.16$ & SiII & 3, 4, 5\\
SDSS1557+2320 & 3.5400 & $20.65\pm0.10$ & - & $13.50\pm0.30$ & - & $-2.24\pm0.15$ & OI & 70\\
SDSSJ1558+4053 & 2.5500 & $20.30\pm0.04$ & - & $13.07\pm0.06$ & - & $-2.45\pm0.06$ & OI & 85\\
J1558-0031 & 2.7000 & $20.67\pm0.05$ & - & $14.11\pm0.03$ & $14.07\pm0.02$ & $-1.72\pm0.05$ & SII & 43\\
PHL 1598 & 0.4297 & $19.18\pm0.03$ & - & - & $14.36\pm0.05$ & $0.06\pm0.06$ & SII & 105\\
J1604+3951 & 3.1600 & $21.75\pm0.20$ & $13.12\pm0.05$ & $15.47\pm0.05$ & $15.71\pm0.05$ & $-1.19\pm0.21$ & ZnII & 34, 3, 4, 5\\
J1607+1604 & 4.4741 & $20.30\pm0.15$ & - & $14.03\pm0.06$ & - & $-1.71\pm0.15$ & SiII & 98\\
SDSS1610+4724 & 2.5100 & $21.15\pm0.15$ & $13.56\pm0.05$ & $15.62\pm0.05$ & $>16.01$ & $-0.15\pm0.16$ & ZnII & 44, 3, 4, 5\\
J1616+4154 & 0.3211 & $20.60\pm0.20$ & - & $15.02\pm0.05$ & $15.37\pm0.11$ & $-0.35\pm0.23$ & SII & 2\\
J1619+3342 & 0.0963 & $20.55\pm0.10$ & - & $14.38\pm0.15$ & $15.08\pm0.09$ & $-0.59\pm0.13$ & SII & 2\\
QSO J1621-0042 & 3.1040 & $19.70\pm0.20$ & - & $13.30\pm0.04$ & - & $-1.43\pm0.20$ & SiII & 114\\
3C336 & 0.6600 & $20.36\pm0.10$ & - & $14.59\pm0.11$ & - & $-0.95\pm0.23$ & FeII & 113, 14, 50\\
J1623+0718 & 1.3400 & $21.35\pm0.10$ & $12.91\pm0.09$ & $15.28\pm0.05$ & - & $-1.00\pm0.13$ & ZnII & 35\\
J1626+2751 & 4.3110 & $21.34\pm0.15$ & - & $15.33\pm0.06$ & - & $-1.19\pm0.24$ & FeII & 98\\
J1626+2751 & 4.4975 & $21.39\pm0.15$ & - & $14.08\pm0.02$ & - & $-2.49\pm0.24$ & FeII & 98\\
J1626+2751 & 5.1791 & $20.94\pm0.15$ & - & $>14.59$ & $14.60\pm0.02$ & $-1.46\pm0.15$ & SII & 98\\
J1626+2858 & 4.6100 & $20.65\pm0.20$ & - & - & - & $-2.73\pm0.22$ & SiII & 99\\
J1629+0913 & 1.9000 & $20.80\pm0.10$ & $12.68\pm0.08$ & $>14.93$ & $15.24\pm0.05$ & $-0.68\pm0.13$ & ZnII & 3, 4, 5\\
4C 12.59 & 0.5310 & $20.70\pm0.09$ & - & $14.26\pm0.08$ & - & $-1.62\pm0.22$ & FeII & 114\\
4C 12.59 & 0.9000 & $19.70\pm0.04$ & $<12.18$ & $14.17\pm0.03$ & - & $-0.71\pm0.19$ & FeII & 60\\
J1637+2901 & 3.5000 & $20.70\pm0.10$ & - & $13.84\pm0.10$ & - & $-3.10\pm0.22$ & OI & 70\\
J1654+2227 & 4.0022 & $20.60\pm0.15$ & - & $14.09\pm0.03$ & - & $-1.69\pm0.24$ & FeII & 98\\
SDSS1709+3417 & 2.5300 & $20.45\pm0.15$ & - & $14.30\pm0.20$ & - & $-1.46\pm0.25$ & SiII & 70\\
SDSS1709+3417 & 3.0100 & $20.40\pm0.10$ & - & $13.90\pm0.20$ & - & $-1.68\pm0.29$ & FeII & 70\\
J1712+5755 & 2.2500 & $20.60\pm0.10$ & - & $14.49\pm0.02$ & - & $-1.19\pm0.12$ & SiII & 43\\
Q1715+4606 & 0.6500 & $20.44\pm0.10$ & $<12.87$ & $14.94\pm0.03$ & - & $-0.68\pm0.21$ & FeII & 58\\
PSS1715+3809 & 3.3400 & $21.05\pm0.12$ & $<12.11$ & $13.74\pm0.04$ & - & $-2.49\pm0.22$ & FeII & 94\\
Q1727+5302 & 1.0300 & $21.41\pm0.15$ & $12.76\pm0.24$ & $14.81\pm0.01$ & - & $-1.21\pm0.28$ & ZnII & 116, 63\\
Q1727+5302 & 0.9400 & $21.16\pm0.10$ & $13.25\pm0.11$ & $15.29\pm0.01$ & - & $-0.47\pm0.15$ & ZnII & 116, 63\\
Q1733+5533 & 1.0000 & $20.70\pm0.10$ & $<12.11$ & - & - & $-0.73\pm0.12$ & SiII & 58, 63\\
SDSS1737+5828 & 4.7400 & $20.65\pm0.10$ & - & $13.30\pm0.10$ & - & $-2.53\pm0.23$ & FeII & 106\\
J1737+5828 & 4.7400 & $20.65\pm0.20$ & - & - & - & $-2.23\pm0.21$ & SiII & 99\\
Q1755+578 & 1.9700 & $21.40\pm0.15$ & $13.85\pm0.05$ & $15.79\pm0.05$ & $>16.12$ & $-0.11\pm0.16$ & ZnII & 3, 4, 5\\
Q1759+75 & 2.6300 & $20.76\pm0.05$ & $>11.65$ & $15.08\pm0.02$ & $15.24\pm0.01$ & $-0.64\pm0.05$ & SII & 89, 90, 43\\
PSS1802+5616 & 3.8100 & $20.35\pm0.20$ & - & $13.67\pm0.10$ & - & $-1.99\pm0.22$ & SiII & 94\\
PSS1802+5616 & 3.5500 & $20.50\pm0.10$ & $<12.63$ & $14.08\pm0.06$ & - & $-1.60\pm0.21$ & FeII & 94\\
PSS1802+5616 & 3.3900 & $20.30\pm0.10$ & $<12.41$ & $14.26\pm0.04$ & - & $-1.22\pm0.21$ & FeII & 94\\
QSO B2000-330 & 3.1720 & $19.75\pm0.15$ & - & $<12.86$ & - & $-2.29\pm0.15$ & OI & 96\\
QSO B2000-330 & 3.1880 & $19.80\pm0.15$ & - & $13.69\pm0.04$ & - & $-1.34\pm0.15$ & SiII & 96\\
QSO B2000-330 & 3.1920 & $19.10\pm0.15$ & - & $13.49\pm0.07$ & - & $-0.48\pm0.15$ & SiII & 96\\
J2036-0553 & 2.2800 & $21.20\pm0.15$ & - & $14.68\pm0.11$ & - & $-1.67\pm0.16$ & SiII & 43\\
Q2051+1950 & 1.1157 & $20.00\pm0.15$ & $12.90\pm0.10$ & $15.02\pm0.02$ & - & $0.34\pm0.18$ & ZnII & 60\\
SDSS2059-0529 & 2.2100 & $20.80\pm0.20$ & $12.94\pm0.11$ & $15.00\pm0.11$ & - & $-0.42\pm0.23$ & ZnII & 44\\
Q2059-360 & 3.0800 & $20.98\pm0.08$ & - & $14.52\pm0.07$ & $14.41\pm0.04$ & $-1.58\pm0.09$ & OI & 75, 110, 77\\
SDSS2100-0641 & 3.0900 & $21.05\pm0.15$ & $13.24\pm0.05$ & $15.37\pm0.05$ & $15.49\pm0.05$ & $-0.37\pm0.16$ & ZnII & 44, 3, 4, 5\\
LBQS 2114-4347 & 1.9120 & $19.50\pm0.10$ & $<12.17$ & $14.02\pm0.01$ & $<13.97$ & $-0.70\pm0.10$ & MgII & 114\\
QSO J2119-3536 & 1.9960 & $20.10\pm0.07$ & $12.30\pm0.09$ & $14.77\pm0.09$ & $<14.95$ & $-0.36\pm0.11$ & ZnII & 22\\
QSO B2126-15 & 2.6380 & $19.25\pm0.15$ & $<11.58$ & $14.05\pm0.01$ & - & $-0.09\pm0.15$ & SiII & 114\\
QSO B2126-15 & 2.7690 & $19.20\pm0.15$ & $<11.95$ & $14.17\pm0.00$ & - & $0.08\pm0.15$ & SiII & 114\\
LBQS 2132-4321 & 1.9160 & $20.74\pm0.09$ & $12.66\pm0.02$ & $15.03\pm0.02$ & $>14.90$ & $-0.64\pm0.09$ & ZnII & 114\\
LBQS 2138-4427 & 2.3830 & $20.60\pm0.05$ & $12.05\pm0.07$ & - & - & $-1.11\pm0.09$ & ZnII & 52\\
LBQS 2138-4427 & 2.8520 & $20.98\pm0.05$ & $11.99\pm0.05$ & $14.65\pm0.02$ & $14.50\pm0.02$ & $-1.55\pm0.07$ & ZnII & 51, 110\\
J2144-0632 & 4.1300 & $20.40\pm0.15$ & - & $<13.51$ & - & $-2.37\pm0.47$ & OI & 70\\
PSSJ2155+1358 & 3.3200 & $20.50\pm0.15$ & $12.05\pm0.32$ & $14.51\pm0.13$ & - & $-1.01\pm0.35$ & ZnII & 19, 93\\
LBQS 2206-1958A & 2.0760 & $20.44\pm0.05$ & $<11.20$ & $13.33\pm0.01$ & - & $-2.08\pm0.06$ & OI & 84\\
Q2206-199 & 1.9200 & $20.68\pm0.03$ & $12.91\pm0.01$ & $15.30\pm0.02$ & $15.42\pm0.02$ & $-0.33\pm0.03$ & ZnII & 88, 90, 91, 117\\
QSO B2222-396 & 2.1540 & $20.85\pm0.10$ & - & $14.42\pm0.03$ & $14.08\pm0.02$ & $-1.89\pm0.10$ & SII & 66\\
SDSS2222-0946 & 2.3500 & $20.50\pm0.15$ & $<12.78$ & $15.06\pm0.08$ & $15.37\pm0.05$ & $-0.25\pm0.16$ & SII & 44, 3, 4, 46, 5\\
Q2223+20 & 3.1200 & $20.30\pm0.10$ & - & $13.32\pm0.06$ & - & $-2.17\pm0.11$ & SiII & 93\\
Q2228-3954 & 2.1000 & $21.20\pm0.10$ & $12.51\pm0.06$ & $15.17\pm0.02$ & - & $-1.25\pm0.12$ & ZnII & 67\\
LBQS 2230+0232 & 1.8640 & $20.83\pm0.10$ & $12.80\pm0.03$ & $15.19\pm0.02$ & $15.29\pm0.10$ & $-0.59\pm0.10$ & ZnII & 89, 90, 91, 24, 25, 3, 4, 5\\
Q2231-00 & 2.0700 & $20.53\pm0.08$ & $12.30\pm0.05$ & $14.83\pm0.03$ & $15.10\pm0.15$ & $-0.79\pm0.09$ & ZnII & 89, 90, 21, 23\\
QSO B2237-0607 & 4.0790 & $20.55\pm0.10$ & - & $13.85\pm0.11$ & - & $-1.79\pm0.10$ & SiII & 57, 106, 43\\
J223941.8-294955 & 1.8250 & $19.84\pm0.14$ & $12.76\pm0.06$ & $14.33\pm0.04$ & - & $0.36\pm0.15$ & ZnII & 24\\
J2241+1225 & 2.4200 & $21.15\pm0.15$ & - & $15.02\pm0.08$ & $>15.01$ & $-1.31\pm0.25$ & FeII & 3, 4, 5\\
PSS2241+1352 & 4.2800 & $21.15\pm0.10$ & - & $>14.65$ & $14.58\pm0.03$ & $-1.69\pm0.10$ & SII & 93\\
HE2243-6031 & 2.3300 & $20.67\pm0.02$ & $12.22\pm0.03$ & $14.92\pm0.01$ & $14.88\pm0.01$ & $-1.01\pm0.04$ & ZnII & 54\\
J2252+1425 & 4.7475 & $20.60\pm0.15$ & - & $13.98\pm0.11$ & $<14.41$ & $-1.80\pm0.26$ & FeII & 98\\
QSO B2311-373 & 2.1820 & $20.48\pm0.13$ & $<11.82$ & $14.23\pm0.04$ & - & $-1.45\pm0.15$ & SiII & 1, 66\\
B2314-409 & 1.8600 & $20.90\pm0.10$ & $12.52\pm0.10$ & $15.08\pm0.10$ & $15.10\pm0.15$ & $-0.94\pm0.14$ & ZnII & 30\\
PSS2315+0921 & 3.4300 & $21.10\pm0.20$ & - & $>14.63$ & - & $-1.46\pm0.21$ & SiII & 94\\
QSO B2318-1107 & 1.6290 & $20.52\pm0.14$ & $<11.74$ & $14.14\pm0.02$ & $<14.54$ & $-1.56\pm0.23$ & FeII & 114\\
QSO B2318-1107 & 1.9890 & $20.68\pm0.05$ & $12.50\pm0.03$ & $14.91\pm0.01$ & $15.09\pm0.02$ & $-0.74\pm0.06$ & ZnII & 64\\
J2321+1421 & 2.5700 & $20.70\pm0.05$ & $<11.84$ & $14.18\pm0.03$ & $<13.60$ & $-1.76\pm0.06$ & SiII & 34\\
PSS2323+2758 & 3.6800 & $20.95\pm0.10$ & - & $13.32\pm0.13$ & - & $-2.54\pm0.10$ & SiII & 93\\
QSO J2328+0022 & 0.6520 & $20.32\pm0.07$ & $12.43\pm0.15$ & $14.84\pm0.01$ & - & $-0.45\pm0.17$ & ZnII & 72\\
QSO B2332-094 & 3.0570 & $20.50\pm0.07$ & $<12.17$ & $14.34\pm0.03$ & $14.34\pm0.18$ & $-1.24\pm0.07$ & OI & 51, 93, 77, 119\\
J233544.2+150118 & 0.6800 & $19.70\pm0.30$ & $12.37\pm0.04$ & $14.83\pm0.03$ & - & $0.11\pm0.30$ & ZnII & 74\\
J2340-00 & 2.0500 & $20.35\pm0.15$ & $12.63\pm0.07$ & $14.98\pm0.05$ & $14.95\pm0.05$ & $-0.28\pm0.17$ & ZnII & 95, 3, 4, 5\\
Q2342+34 & 2.9100 & $21.10\pm0.10$ & $<12.60$ & $14.91\pm0.07$ & $15.19\pm0.05$ & $-1.03\pm0.11$ & SII & 93, 95, 3, 4, 120, 5\\
QSO B2343+125 & 2.4310 & $20.40\pm0.07$ & $12.20\pm0.07$ & $14.52\pm0.02$ & $14.66\pm0.02$ & $-0.76\pm0.10$ & ZnII & 64\\
Q2344+12 & 2.5400 & $20.36\pm0.10$ & - & $14.03\pm0.03$ & $<14.20$ & $-1.69\pm0.10$ & SiII & 90, 92\\
PSSJ2344+0342 & 3.2200 & $21.25\pm0.08$ & $12.23\pm0.30$ & $15.06\pm0.15$ & - & $-1.58\pm0.31$ & ZnII & 19, 93\\
QSO J2346+1247 & 2.5690 & $20.98\pm0.04$ & $12.88\pm0.06$ & $15.24\pm0.04$ & $15.38\pm0.05$ & $-0.66\pm0.07$ & ZnII & 104\\
QSO B2348-0180 & 2.4260 & $20.50\pm0.10$ & $<11.20$ & $14.83\pm0.07$ & $15.06\pm0.10$ & $-0.56\pm0.14$ & SII & 65\\
QSO B2348-0180 & 2.6150 & $21.30\pm0.08$ & $<11.87$ & $14.57\pm0.09$ & - & $-1.92\pm0.11$ & SiII & 90\\
QSO B2348-147 & 2.2790 & $20.56\pm0.08$ & $<11.28$ & $13.79\pm0.02$ & $13.72\pm0.12$ & $-1.95\pm0.14$ & SII & 89, 24\\
Q2352-0028 & 0.8730 & $19.18\pm0.09$ & $<11.67$ & $13.48\pm0.02$ & - & $-0.88\pm0.20$ & FeII & 60\\
Q2352-0028 & 1.0318 & $19.81\pm0.13$ & $<11.93$ & $14.91\pm0.01$ & - & $0.17\pm0.13$ & SiII & 60\\
Q2352-0028 & 1.2467 & $19.60\pm0.24$ & $<11.53$ & $14.21\pm0.01$ & - & $-0.57\pm0.30$ & FeII & 60\\
Q2353-0028 & 0.6000 & $21.54\pm0.15$ & $13.25\pm0.29$ & - & - & $-0.85\pm0.33$ & ZnII & 63\\
B2355-106 & 1.1700 & $21.00\pm0.10$ & $12.76\pm0.17$ & $15.08\pm0.10$ & - & $-0.80\pm0.20$ & ZnII & 35\\
LBQS 2359-0216 & 2.0950 & $20.65\pm0.10$ & $12.60\pm0.03$ & $14.51\pm0.03$ & - & $-0.61\pm0.10$ & ZnII & 89\\
LBQS 2359-0216 & 2.1540 & $20.30\pm0.10$ & $<11.90$ & $13.89\pm0.03$ & - & $-1.49\pm0.10$ & SiII & 89\\
\end{longtable}
\textbf{References: }1: \citealt{Akerman2005}, 2: \citealt{Battisti2012}, 3: \citealt{Berg2013}, 4: \citealt{Berg2014}, 5: \citealt{Berg2015}, 6: \citealt{Bergeron1986}, 7: \citealt{Blades1982}, 8: \citealt{Boisse1998}, 9: \citealt{Bowen2005}, 10: \citealt{Carswell1996}, 11: \citealt{Centurion2000}, 12: \citealt{Centurion2003}, 13: \citealt{Chen2005}, 14: \citealt{Churchill2000}, 15: \citealt{Cooke2010a}, 16: \citealt{Cooke2011}, 17: \citealt{Cooke2011a}, 18: \citealt{Cooke2012}, 19: Dessauge-Zavadsky (unpublished), 20: \citealt{Dessauges-Zavadsky2001}, 21: \citealt{Dessauges-Zavadsky2002}, 22: \citealt{Dessauges-Zavadsky2003}, 23: \citealt{Dessauges-Zavadsky2004}, 24: \citealt{Dessauges-Zavadsky2006}, 25: \citealt{Dessauges-Zavadsky2007}, 26: \citealt{Dessauges-Zavadsky2009}, 27: \citealt{Dodorico2004}, 28: \citealt{Dutta2014}, 29: \citealt{Ellison2001}, 30: \citealt{Ellison2001b}, 31: \citealt{Ellison2007}, 32: \citealt{Ellison2008}, 33: \citealt{Ellison2009}, 34: \citealt{Ellison2010}, 35: \citealt{Ellison2012}, 36: \citealt{Fox2007}, 37: \citealt{Fox2009}, 38: \citealt{Fynbo2011}, 39: \citealt{Fynbo2013}, 40: \citealt{Ge2001}, 41: \citealt{Guimaraes2012}, 42: \citealt{Heinmueller2006}, 43: \citealt{Henry2007}, 44: \citealt{Herbert-Fort2006}, 45: \citealt{Kanekar2014}, 46: \citealt{Krogager2013a}, 47: \citealt{Kulkarni2005}, 48: \citealt{Kulkarni2012}, 49: \citealt{Ledoux2002}, 50: \citealt{Ledoux2002b}, 51: \citealt{Ledoux2003}, 52: \citealt{Ledoux2006}, 53: \citealt{Lopez1999}, 54: \citealt{Lopez2002}, 55: \citealt{Lopez2003}, 56: \citealt{Lopez2005}, 57: \citealt{Lu1996}, 58: \citealt{Meiring2006}, 59: \citealt{Meiring2007}, 60: \citealt{Meiring2009}, 61: \citealt{Meyer1995}, 62: \citealt{Molaro2000}, 63: \citealt{Nestor2008}, 64: \citealt{Noterdaeme2007}, 65: \citealt{Noterdaeme2007a}, 66: \citealt{Noterdaeme2008}, 67: \citealt{Noterdaeme2008a}, 68: \citealt{Noterdaeme2012a}, 69: \citealt{Noterdaeme2012b}, 70: \citealt{Penprase2010}, 71: \citealt{Peroux2002}, 72: \citealt{Peroux2006}, 73: \citealt{Peroux2007}, 74: \citealt{Peroux2008}, 75: \citealt{Petitjean2000}, 76: \citealt{Petitjean2002}, 77: \citealt{Petitjean2008}, 78: \citealt{Pettini1994b}, 79: \citealt{Pettini1997a}, 80: \citealt{Pettini1997b}, 81: \citealt{Pettini1999}, 82: \citealt{Pettini2000}, 83: \citealt{Pettini2002}, 84: \citealt{Pettini2008}, 85: \citealt{Pettini2008a}, 86: \citealt{Pettini2012}, 87: \citealt{Prochaska1996}, 88: \citealt{Prochaska1997a}, 89: \citealt{Prochaska1999}, 90: \citealt{Prochaska2001}, 91: \citealt{Prochaska2001b}, 92: \citealt{Prochaska2002a}, 93: \citealt{Prochaska2003}, 94: \citealt{Prochaska2003a}, 95: \citealt{Prochaska2007}, 96: \citealt{Prochter2010}, 97: \citealt{Quast2008}, 98: \citealt{Rafelski2012}, 99: \citealt{Rafelski2014}, 100: \citealt{Rao2000}, 101: \citealt{Rao2005}, 102: \citealt{Rao2006}, 103: \citealt{Richter2005}, 104: \citealt{Rix2007}, 105: \citealt{Som15}, 106: \citealt{Songaila2002}, 107: \citealt{Srianand1998}, 108: \citealt{Srianand2000}, 109: \citealt{Srianand2001}, 110: \citealt{Srianand2005}, 111: \citealt{Srianand2007}, 112: \citealt{Srianand2012}, 113: \citealt{Steidel1997}, 114: This work, 115: \citealt{Tripp2005}, 116: \citealt{Turnshek2004}, 117: \citealt{Vladilo2011}, 118: \citealt{Zafar2011}, 119: \citealt{Zafar2014}, 120: \citealt{Zafar2014a}, 121: Zafar et al. (in prep), 122: \citealt{Zych2009}, 123: \citealt{delaVarga2000}
\twocolumn
\section{Conclusion}
We present in this paper physical properties of 15 new sub-damped Lyman-$\rm \alpha$ absorbers seen in absorption in background quasar's high resolution UVES spectra. These systems cover a wide redshift range (\zabsbetween{0.584}{3.104}). The metallicity measurements were performed using Voigt profile fitting of the normalized high resolution UVES quasar spectra.
Our sub-DLA measurements add significantly to previous studies since high resolution spectroscopy is required to study these systems.
We apply a multi-element based method to assess the level of dust depletion in the line of sight to the quasar. This study appears to be promising as it uses the combined information from several ions, and is relative to measurements of our Galaxy's ISM. With a survival analysis, we derive the best fit of the depletion factor $\rm F_{*}$ for the DLAs and sub-DLAs and find negative values, statistically different, for both groups: $\rm F^{sub-DLA}_{*}=-0.34\pm0.19$ and $\rm F^{DLA}_{*}=-0.70\pm0.06$.
In comparison with values derived in our Galaxy, DLAs lie outside the halo and the sub-DLAs are associated with Halo like stars, in terms of depletion patterns. This is counter-intuitive as we expect DLAs to be more self-shielded to the UV background than sub-DLAs. We conclude that quasar absorbers differ from the Galactic depletion patterns or alternatively have a different nucleosynthetic history. Future analysis with the Small Magellanic Cloud will enable comparisons to a galaxy more in line with the morphology or the $\rm H_{2}$ fraction of DLAs. Moreover, we derive the averaged rest frame extinction $\rm A_{V}$ for both populations to be below 0.01, suggesting that dust reddening is not observed in the current quasar selection.
We then examine the relative abundances of Fe and $\rm \alpha$-elements.
We derived an offset in $\rm [\alpha/Fe]${} for the DLAs in our sample of $0.32\pm0.18$, excluding systems with $\rm [\alpha/H]>-1.5$ to be less sensitive to dust depletion. This value is similar to that derived by \cite{Rafelski2012}.
However, we cannot derive a similar parameter for the sub-DLA population as we can not disentangle dust depletion effects from the $\alpha$-enhancement. We therefore apply the DLA corrections to the sub-DLAs.
We study the evolution of the cosmic metallicity $\Omega_{m}$, also described by the mean H\,I-weighted metallicity $\rm \mean{Z}${}. We confirm the steeper evolution of sub-DLAs than DLAs eventually reaching a solar metallicity at low redshifts as expected from chemical evolution models. We note that a third of the newly derived sub-DLA abundances appear as outliers from the previous data.
We measure the velocity width of the absorption systems in our new sample, $\rm \Delta V_{90}${}, with a new method using the information from the Voigt profile fits. We confirm that there is a correlation between $\rm \Delta V_{90}${} and metallicity for sub-DLAs. Indeed, sub-DLAs are potentially probing a different mass range than DLAs. Sub-DLAs could have a more important feedback mechanism than DLAs, thus increasing the scatter and weakening the possible velocity width/metallicity correlation.
Finally, we look at the metallicity distribution of sub-DLAs. At low redshifts, \zabslt{1.25}, we see a hint of a bimodal distribution which peaks at $\sim -1.1$ and $\sim -0.3$. This indicates that low-redshift sub-DLAs are tracing different mechanisms at play within the CGM, such as cold-mode accretion and outflows.
Larger samples of sub-DLAs and LLS abundances at low redshifts are required to better identify their connection to gas inflow/outflow processes.
\section{Introduction}
In depth studies of galaxy evolution require an understanding of the complex processes occurring at the interface of the galaxy and its nearby environment, the Circum-Galactic Medium (CGM).
On the one hand, the star formation process is believed to be fed in galaxies via accretion mechanisms \citep{Rees1977, White1978, Prochaska2009, Bauermeister2010}. For galaxies with masses typically below $\rm \sim10^{11-12}M_{\odot}$, the accreting gas follows cold flows ($\rm T\sim 10^{4-5}K$) while for more massive galaxies, a second mode of accretion appears, the "hot mode", where the gas is shock heated near the virial temperature ($\rm T\sim 10^{6}K$) \citep{Rees1977, Silk1977a, White1978, Birnboim2003, Keres2005, Dekel2006, Ocvirk2008}. Simulations show that about 40\% of the accretion may be genuinely smooth \citep{Genel2010}. These modes also differ in metallicity \citep{Fumagalli2011, Shen2013}. Indeed, \cite{Ocvirk2008} showed that the "cold mode" accreting gas can reach metallicities up to tenth solar, while the hot mode accreting gas metallicities are usually lower and are highly dependent on the distance to the center of the galaxy and on how well the gas is mixed. These accreting streams may also provide the galaxy with additional angular momentum \citep{Fall1980}.
Observational evidences for accretion have been challenging to gather due to the low surface brightness and low filling factor of the infalling gas and its expected low metallicity. Nevertheless, cold accretion has been recently detected in a few objects \citep{Steidel2000, Martin2012, Rubin2012, Bouche2013a}.
Similarly, early evidences for cold accretion onto quasars have been recently reported by \cite{Cantalupo2014} and \cite{Martin2014a}.
On the other hand, galaxies release energy and material in their environment (up to $\rm \sim 125 kpc$) via supernovae (SNe), stellar winds or Active Galactic Nuclei (AGN) activity. These outflows tend to chemically enriched the IGM \citep{Songaila1996, Simcoe2004, Adelberger2005,Ryan-Weber2009, DOdorico2013,Shull2014, Shull2015}, and can regulate the star formation process of galaxies. Indeed, as the gas is released, it will starve the galaxy from fresh gas accreting along the galaxy major axis, quenching the star formation. It will also enhanced the star formation by cooling the gas via metal line emissions. Fountains can also be created if the gas does not leave the potential well of the galaxy. In this scenario, the fully metal enriched gas recycles and falls back onto the galactic disk and contribute directly to the star forming processes as it can cool efficiently. Simulations have shown that fountains dominate the global accretion mechanism for $\rm z\lesssim1$ galaxies \citep{Oppenheimer2010}. Even though outflows are ubiquitous at all redshifts around star forming galaxies \citep{Shapley2003, Martin2005, Rubin2014} and their existence is confirmed by signatures of OVI found within the CGM of low redshift star forming galaxies \citep{Tumlinson2011}, they remain poorly understood in the context of galaxy formation models.
In the context of emission line study, \cite{Bertone2010UV, Bertone2010X} argued that the ionization state of elements provides valuable insight on the physical state of the CGM (mainly its temperature but also its ionizing process) and can be used to study the different feedback processes taking place, including metal pollution of accreting gas via galactic fountains. \cite{Fumagalli2011} also argued that kinematic analysis of absorption lines can be used in addition to the metallicity analysis to distinguish metal-rich outflowing material from metal-poor ($\rm \lesssim0.01Z_{\odot}$) accreting gas.
Therefore, the study of metal lines (kinematics, line strengths, ionization states) might be the key diagnostic to observationally disentangle outflows from inflows and assess the level of metal enrichment of the CGM and thus galaxy evolution.
Absorbers observed in background quasar spectra are a tool to probe the low density gas and its metallicity. Indeed, simulations predict that cold accretion onto galaxies can be observed in absorption via dense H\,I absorbing systems with $\rm \log N(H\,I)>15.5$ \citep{Faucher-Giguere2011, VandeVoort2012b, Shen2013}. They predict that the cold streams could be traced with metal-poor H\,I absorption systems, mostly in the Lyman Limit System (LLS) range \NHIbetween{17.2}{19.0}. Recently, \cite{Lehner2013} showed observational evidence for low redshift LLS presenting a bimodal metallicity distribution, which they associated with infalls and outflows.
However, the metallicities of LLS depend sensitively on model-dependent ionization corrections, since the LLS gas is highly ionized. This makes it harder to reliably detect the difference between inflows and outflows using the LLS. A more robust way of detecting the metallicity distribution of the gas around galaxies is by using the damped Lyman-$\alpha$ (DLA; $\rm \log N(H\,I) \geq 20.3$) and sub-damped Lyman-$\alpha$ (sub-DLA; $\rm 19.0 \leq \log N(H\,I) < 20.3$) absorbers.
These systems are the primary neutral gas reservoir at $\rm 0 < z < 5$ \citep[][]{StorrieLombardi2000, Peroux2005, Prochaska2005, Rao2006,Zafar2013a} and offer the most precise element abundance measurements in distant galaxies. In particular, at $\rm z\leq2$, \cite{Fumagalli2011} anticipate that almost half of the cross-section in the sub-DLA H\,I column density range is due to streams, while at $\rm z\sim3$, \cite{VandeVoort2012b} anticipate that it is more than 80\%.
In an era of large quasar surveys, with samples of thousands of DLAs available \citep[e.g.,][]{Noterdaeme2012}, sub-DLAs remain little studied.
Indeed, at low H\,I column densities, one requires a high spectral resolution and high signal to noise ratio (SNR) to derive element abundances.
The large quasar samples observed with the high resolution spectrographs VLT/UVES \citep{Zafar2013} and Keck/HIRES \citep{Omeara2015} are therefore crucial tools for our understanding of sub-DLA properties. Here, we present a detailed study of the metallicity and kinematics of a large sample of DLAs and sub-DLAs observed at high resolution with UVES.
The paper is organised as follows. In \S 2 we present the data sample and in \S 3 we describe the abundance measurements. The results are discussed in \S 4 followed by conclusions in \S 5.
\section[]{Metallicities}
\begin{table*}
\begin{minipage}{170mm}
\caption{Abundances with respect to solar for the 22 systems studied in this work. In column [X/H], (a) refers to Ar, (b) refers to O, (c) refers to N and (d) refers to C. For PKS 0454-220, the metallicities with the asterisk have been derived by Som et al. 2015.}
\label{result:metallicities}
\begin{center}
\begin{tabular}{lcccccccc}
\hline\hline
\textbf{QSO} & \textbf{$z_{abs}$} & \textbf{log $\rm N(H\,I)$} & \textbf{[S/H]} & \textbf{[Al/H]} & \textbf{[Si/H]} & \textbf{[Cr/H]} \\ \hline
QSO J0008-2900 & 2.254 & $20.22\pm0.10$ & - & - & $<-1.33$ & $<-1.49$ \\
QSO J0008-2901 & 2.491 & $19.94\pm0.11$ & $-1.38\pm0.21$ & - & - & $<-0.68$ \\
\textbf{QSO J0018-0913} & 0.584 & $20.11\pm0.10$ & - & - & - & $<-0.78$ \\
QSO J0041-4936 & 2.248 & $20.46\pm0.13$ & $<-0.75$ & $>-0.85$ & $-1.19\pm0.16$ & $-0.98\pm0.58$ \\
QSO B0128-2150 & 1.857 & $20.21\pm0.09$ & $-1.00\pm0.09$ & - & $-0.90\pm0.09$ & - \\
\textbf{QSO J0132-0823} & 0.647 & $20.60\pm0.12$ & - & - & - & $<-1.07$ \\
QSO B0307-195B & 1.788 & $19.00\pm0.10$ & - & - & $0.49\pm0.10$ & $<0.13$ \\
QSO J0427-1302 & 1.562 & $19.35\pm0.10$ & - & $-2.02\pm0.14$ & - & $<-0.60$ \\
PKS 0454-220 & 0.474 & $19.45\pm0.03$ & $0.49\pm0.04^{*}$ & - & $>-0.78^{*}$ & - \\
J060008.1-504036 & 2.149 & $20.40\pm0.12$ & - & $>-0.52$ & $-0.83\pm0.12$ & $-0.94\pm0.12$ \\
QSO B1036-2257 & 2.533 & $19.30\pm0.10$ & - & $-1.24\pm0.10$ & $-1.17\pm0.10$ & $<-0.40$ \\
J115538.6+053050 & 3.327 & $21.00\pm0.10$ & $-0.81\pm0.10$ & - & $-0.58\pm0.10$ & - \\
LBQS 1232+0815 & 1.720 & $19.48\pm0.13$ & $<-0.41$ & - & $-0.58\pm0.13$ & $<-0.74$ \\
QSO J1330-2522 & 2.654 & $19.56\pm0.13$ & - & $-1.83\pm0.13$ & - & - \\
QSO J1356-1101 & 2.397 & $19.85\pm0.08$ & - & - & - & $<-0.85$ \\
QSO J1621-0042 & 3.104 & $19.70\pm0.20$ & - & - & $-1.43\pm0.20$ & - \\
4C 12.59 & 0.531 & $20.70\pm0.09$ & - & - & - & - \\
LBQS 2114-4347 & 1.912 & $19.50\pm0.10$ & $<-0.65$ & $-0.95\pm0.10$ & $-0.62\pm0.10$ & $<-0.37$ \\
QSO B2126-15 & 2.638 & $19.25\pm0.15$ & - & - & $-0.09\pm0.15$ & - \\
QSO B2126-15 & 2.769 & $19.20\pm0.15$ & - & $>0.39$ & $0.08\pm0.15$ & $<-0.44$ \\
LBQS 2132-4321 & 1.916 & $20.74\pm0.09$ & $>-0.96$ & - & $-0.70\pm0.10$ & $-1.06\pm0.11$ \\
QSO B2318-1107 & 1.629 & $20.52\pm0.14$ & $<-1.10$ & $<-0.04$ & - & $<-1.69$ \\
\hline
\end{tabular}
\begin{tabular}{lcccccc}
\hline\hline
\textbf{QSO} & \textbf{[Fe/H]} & \textbf{[Ni/H]} & \textbf{[Zn/H]} & \textbf{[Mg/H]} & \textbf{[Mn/H]} & \textbf{[X/H]} \\ \hline
QSO J0008-2900 & $-1.94\pm0.10$ & - & $<-1.10$ & $>-0.81$ & $<-1.63$ & $<-1.55^{(a)}$ \\
QSO J0008-2901 & $-1.79\pm0.13$ & $<-0.87$ & $<-0.38$ & - & - & $-1.32\pm0.35^{(b)}$ \\
\textbf{QSO J0018-0913} & $-1.74\pm0.10$ & - & $<-0.26$ & - & - & - \\
QSO J0041-4936 & $-1.54\pm0.14$ & $-1.61\pm0.20$ & $-1.32\pm0.16$ & - & - & $-2.36\pm0.13^{(c)}$ \\
QSO B0128-2150 & $-1.27\pm0.09$ & $-1.17\pm0.10$ & $<-0.51$ & - & - & - \\
J013209-082349 & $-1.14\pm0.14$ & - & - & - & - & - \\
QSO B0307-195B & $-0.02\pm0.10$ & $<0.00$ & $<0.62$ & - & $<-0.30$ & - \\
QSO J0427-1302 & $-2.62\pm0.11$ & $<-0.34$ & $<-0.16$ & - & $<-0.94$ & - \\
PKS 0454-220 & $-0.24\pm0.03$ & $0.02\pm0.09^{*}$ & - & - & $-0.30\pm0.03$ & $-1.34\pm0.09^{*(c)}$ \\
\textbf{QSO J0132-0823} & $-1.06\pm0.12$ & $-1.00\pm0.12$ & $-0.85\pm0.12$ & - & - & - \\
QSO B1036-2257 & $-1.87\pm0.10$ & $<-0.59$ & $<-0.12$ & $-1.33\pm0.10$ & - & - \\
J115538.6+053050 & - & $-1.48\pm0.10$ & - & - & - & - \\
LBQS 1232+0815 & $-1.48\pm0.13$ & $<-0.65$ & $<-0.46$ & - & - & - \\
QSO J1330-2522 & - & $<-0.56$ & - & - & - & - \\
QSO J1356-1101 & $-1.91\pm0.08$ & $<-1.31$ & $<-0.03$ & - & $<-1.21$ & - \\
QSO J1621-0042 & $-1.90\pm0.20$ & - & - & - & - & $<-1.72^{(d)}$ \\
4C 12.59 & $-1.94\pm0.12$ & - & - & - & - &$-5.77^{(d)}$ \\
LBQS 2114-4347 & $-0.98\pm0.10$ & $<-0.84$ & $<0.11$ & $-0.70\pm0.10$ & $<-0.69$ & - \\
QSO B2126-15 & $-0.70\pm0.15$ & $-0.32\pm0.15$ & $<-0.23$ & - & - & - \\
QSO B2126-15 & $-0.53\pm0.15$ & - & $<0.19$ & - & $<-0.35$ & - \\
LBQS 2132-4321 & $-1.21\pm0.11$ & $-1.19\pm0.11$ & $-0.64\pm0.11$ & - & - & - \\
QSO B2318-1107 & $-1.88\pm0.14$ & - & $<-1.34$ & - & $-2.17\pm0.15$ & - \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table*}
\subsection{The Ionized Fraction of sub-DLAs}
\label{sec:ionization}
Given that observationally we are sensitive to the neutral gas in quasar absorbers, it is important to quantify the fraction of gas ionized in these systems. In the DLA column density range, the ionization corrections are below the typical abundance measurement errors \citep{Vladilo2001, Dessauges-Zavadsky2003}.
The situation might differ in the sub-DLA H\,I column density range given that the lower N(H\,I) might prevent complete self-shielding from the surrounding UV background. To address this issue, \cite{Dessauges-Zavadsky2003, Meiring2007, Meiring2009, Som2013, Som15} among others studied the ionized fraction of sub-DLAs based on photo-ionization CLOUDY modeling of individual systems. These studies show that the ionized fraction of hydrogen varies greatly within the sub-DLA H\,I column density range (see e.g. Fig 4 of
\citealt{Meiring2009} and Fig. 10 of \citealt{Lehner2014}).
Nevertheless, while sub-DLAs might have an important fraction of their gas ionized in some cases, the ionization corrections to the measured abundances for sub-DLAs are often low. The large majority of elements require an ionization correction $\epsilon<0.3$ dex, while it is negligible for FeII but important for ZnII \citep{Dessauges-Zavadsky2003}. Based on these past results and in order to be in line with abundance measurements from the literature reported here, we choose not to apply ionization correction to the new abundances presented.
A more statistical approach is now required. To this end, Fumagalli et al. (submitted) have recently built CLOUDY model grids to establish posterior probability distribution functions for different states of the gas with a Bayesian formalism and Markov Chain Monte Carlo algorithm.
While such an analysis is beyond the scope of the current paper, we plan to address these issues in further publications.
\subsection{Assessing the Dust-Content of Quasar Absorbers: a Multi-Element Analysis}
\label{sec:obsbias}
Refractory elements are easily incorporated onto dust (e.g. Fe, Cr, Ni), while volatile elements are less prone to locking up into dust grains (e.g. Zn, S). To estimate the level of depletion of a given line of sight, it is possible to compare the abundance of a volatile element with that of a refractive element. The quantity $\rm [Zn/Fe]$ is therefore an excellent tool to probe the quantity of Fe atoms locked into dust \citep{Vladilo1998}.
Indeed, Zn is thought to behave like Fe in different stages of chemical evolution, excluding the effects of dust depletion. From studies of low metallicity stars in our Galaxy, \citep{Saito2009, Barbuy2015}, [Zn/Fe] stays steady at $\rm [Zn/Fe]\sim 0$ down to metallicities $\rm[Fe/H] = -3$ and then increases for lower values of [Fe/H].
Hence, $\rm [Zn/H]$ provides a robust metallicity indicator. Unfortunately, its low cosmic abundance and long rest-frame wavelengths make it challenging to measure in sub-DLAs, preventing from a robust dust-metallicity derivation.
Here, we propose a different approach for the study of dust depletion based on the multi-element analysis proposed by \cite{Jenkins09} to assess the level of dust in a given line of sight. \cite{Jenkins09} proposed to use the abundances of different elements (namely C, N, O, Mg, Si, P, Cl, Ti, Cr, Mn, Fe, Ni, Cu, Zn, Ge, Kr and S) to compare the dust depletion of dense neutral hydrogen systems to that of the Interstellar Medium (ISM) of our Galaxy. Using a sample of 243 sight lines in our Galaxy, he established a connection between the line of sight depletion factor $\rm F_{*}$ and the different elements' abundances of each sight line. We refer the reader to Appendix \ref{ann:jenkins} for a mathematical description of the method.
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{eps/DLAs_Jenkins_median-eps-converted-to.pdf}
\includegraphics[width=0.49\textwidth]{eps/subDLAs_Jenkins_median-eps-converted-to.pdf}
\caption[Jenkins med]{Fits of $\rm F_{*}$ from equation \ref{eq:eq_dep_fit} for the DLAs (362 systems, left panel), and sub-DLAs (92 systems, right panel) in the EUADP+ sample. The blue crosses stand for the detections, the red triangles for the upper limits, the green triangles for lower lower limits, and the cyan points are the median of the detections for each element X. The fits are performed on the medians of the detections with a bisector fit (dashed line), and on the detections and the limits using a survival analysis technique, the Buckley-James method (solid line). We note that the $\rm \alpha$-elements (Mg and Si) are below the trend lines for both DLAs and sub-DLAs. We refer the reader to Appendix \ref{ann:jenkins} for a mathematical description of the fit.}
\label{img:exampleJenkins_med}
\end{center}
\end{figure*}
Fig. \ref{img:exampleJenkins_med} shows the fit for the line of sight depletion factor $\rm F_{*}$ (slope) for both populations of quasar absorbers from the EUADP+ sample. For each element, we plot in cyan the median of the detections if there is at least 4 systems measured. The vertical error bars represent the error on the median using a bootstrap technique with a confidence level of 95\%.
We are confronted with a large number of non-detections, creating a bias in the sample towards metal-rich systems. A large fraction of these upper (resp. lower) limits falls below (resp. above) the associated median. To address this issue, a survival analysis is considered. A Buckley-James linear regression, from the stsdas.statistics package in IRAF, results in $\rm F_{*}=-0.34\pm0.19$ for sub-DLAs and $\rm F_{*}=-0.70\pm0.06$ for DLAs.
On the one hand, both populations show negative values for $\rm F_{*}$, suggesting that sub-DLAs and DLAs arise in galaxies with a lower dust content than the Milky Way. On the other hand, the derived $\rm F_{*}$ values for both populations are different at the 1.8 $\rm \sigma$ level. The sub-DLA population is consistent with the Halo like ISM from our Galaxy\footnote{$\rm F_{*}=-0.28$ for \textit{Halo} like ISM, $\rm F_{*}=-0.08$ for \textit{Disk+Halo} like ISM, $\rm F_{*}=0.12$ for \textit{Warm Disk} like ISM and $\rm F_{*}=0.90$ for \textit{Cool Disk} like ISM} while the DLAs are described by an $\rm F_{*}$ value well below the ones measured in the Milky Way. This is counter intuitive as we expect DLAs to be self-shielded from the UV background towards the center of the galaxy. Indeed, numerous cosmological simulations predict DLAs to be closer to the center of the galaxy than sub-DLAs \citep{Fumagalli2011, Faucher-Giguere2015}. They should therefore exhibit an $\rm F_{*}$ value corresponding to regions within the halo.
But DLAs and sub-DLAs might not be systematically associated with spiral galaxies. They might arise from a mixture of galaxy types, hence the non-physical values of $\rm F_{*}$.
In addition, the method described here is based on measurements in our Galaxy at log \NHIgt{19.5} to limit photo-ionization effects, while our quasar absorber sample goes down to $\rm \log N (H\,I) = 19.0$.
Furthermore, the ionization levels of the sub-DLAs and DLAs in our EUADP+ sample are higher than in the Milky Way ISM, as $\rm F_{*}$ is quite different between ionized and neutral gas \citep[$\rm F_{*}=-0.1$ for the warm ionized medium and $\rm F_{*}=0.1$ for the warm neutral medium, e.g.][]{Draine2011}.
We derive $\rm F_{*}$ for log \NHIgt{19.5} sub-DLAs, and find similar results, suggesting that ionization effects do not affect the results much.
Moreover, the quasar absorbers trace gas at high redshifts, which may differ from the Milky Way properties as a local galaxy.
Overall, these results suggest that quasar absorbers differ from the Galactic depletion patterns or alternatively have a different nucleosynthetic history.
Also, the current QSO sample may suffer from dust selection bias. Indeed, it is possible that quasars in the background of dusty absorbers are not being accounted for in current selection techniques \citep{Boisse1998}. Programs to observe reddened quasars might bring valuable insights to the dust content of quasar absorbers \citep{Maddox2012, Krogager2015, Krogager2016}. Using the analysis from \cite{Vladilo2006}\footnote{see Appendix \ref{Av} for details of the calculation}, we recover estimates for the average extinction in our quasar absorber samples to be below 0.01, in line with results from \cite{Frank2010a} or \cite{Khare2012}. This suggests that the dust reddening is not observed in the current quasar selection.
Given these limitations, we do not apply dust corrections to the measured abundances.
There is work underway \citep{Tchernyshyov2015} to derive the parameters $\rm A_{X}$, $\rm B_{X}$ and $\rm z_{X}$ for the Small Magellanic Cloud, which is more in line with the expected morphological type or $\rm H_{2}$ fraction of DLAs.
\subsection{$\alpha$-elements}
The production of $\alpha$-elements (O, N, Mg, Si, S, Ti, Ca...) and Fe-peak elements (V, Cr, Mn, Fe, Co, Ni...) has different origins in the history of star formation. $\alpha$-elements are mainly created during core-collapse Type II supernovae (SNe), whereas Fe-peak elements originate mainly from thermonuclear Type Ia SNe. These two processes have different time scales, as they originate from distinct stellar populations: the Type II SNe occur from short-lived massive stars while Type Ia SNe are thought to involve binary pairs containing a white dwarf exchanging material over longer periods of time.
Observations of different objects suggest an excess of $\alpha$-elements with respect to Fe-peak elements (from \cite{Wallerstein1962} for G-dwarf stars, to \cite{Timmes1995} for QSO absorption line systems and \cite{Rafelski2012} for DLAs).
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{eps/alpha_Fe_vs_alpha_H-eps-converted-to.pdf}
\caption[]{The $\alpha$-enhancement of DLAs (red) and sub-DLAs (blue) versus metallicity, $\rm [\alpha/Fe]$ versus $\rm[\alpha/H]$, with $\alpha=$ OI, SII or SiII. For clarity, only one in ten data point displays error bars and the limits have faint colors.
\label{img:alphaEnhancement_evolution}
\end{center}
\end{figure}
In Fig. \ref{img:alphaEnhancement_evolution}, we plot $\rm [\alpha/Fe]${} versus metallicity using $\alpha=$ OI, SII, MgII and SiII for the sub-DLA (blue) and DLA (red) populations.
We observe a correlation between $\rm [\alpha/Fe]${} and $\rm [\alpha/H]$ for sub-DLAs. A Spearman test gives $\rm \rho_{sub-DLA}=0.69$ with a probability of no correlation $\rm P(\rho_{sub-DLA})<10^{-7}$. This correlation spans from low- to high-metallicity systems.
The total number of DLA detections adds up to 227 systems. We do not see a flattening for DLAs with $\rm [\alpha/H]<-1$ as in \cite{Rafelski2012}, who attributed this flattening to the fact that the offsets in $\rm [\alpha/Fe]${} values for $\rm [\alpha/H]<-1$ are the effect of $\rm \alpha$-enhancement only. To avoid any dust extinction effects, we consider the 80 DLAs with $\rm [\alpha/H]<-1.5$ to derive a correction for $\rm \alpha$-enhancement for the EUADP+ DLAs \citep{DeCia2013}. We note that these systems might still be affected by dust depletion, as a trend is still visible between $\rm [\alpha/Fe]${} and $\rm [\alpha/H]$, even for $\rm [\alpha/H]$ down to $-3$ dex.
We derive a mean value of $\rm [\alpha/Fe]=0.32\pm0.18$ using 80 DLAs with $\rm [\alpha/H]<-1.5$, which is consistent with the value found by \cite{Rafelski2012}.
DLAs and sub-DLAs may have different nucleosynthetic histories, as they may originate from galaxies of different masses \citep{Khare2007,Kulkarni2010} and hence experience different star formation rates. Indeed, different [Mn/Fe] vs. [Zn/H] trends for DLAs and sub-DLAs suggest different nucleosynthetic histories for the two populations \citep{Meiring2007, Som15}. Therefore, one expects their $\rm \alpha$-enhancement to be statistically different and probably higher for sub-DLAs, which may experience higher star formation rates. In Fig. \ref{img:alphaEnhancement_evolution}, there is no apparent plateau for sub-DLAs, probably due to the small number of detections at low metallicities. More observations of sub-DLAs are needed to obtain more definitive conclusions in this H\,I column density regime. Nevertheless, to address the question of $\rm \alpha$-enhancement for sub-DLAs at least partly, we make use of the value derived for DLAs.
These corrections add $0.32\pm0.18$ dex to every metallicity derived using element Fe. This doesn't include a correction for dust extinction.
We emphasize that such a trend of $\rm [\alpha/Fe]${} versus $\rm [\alpha/H]$ in DLAs/sub-DLAs does not necessarily imply nucleosynthetic $\alpha$-enhancement. This is because of the increasing dust depletion of Fe with increasing metallicity, a trend that is seen to hold even at metallicities below -1 dex.
$\rm [\alpha/Zn]$ is indeed less prone to depletion than $\rm [\alpha/Fe]${}, but our current sub-DLA sample has only a limited number of Zn detections (19/92).
Additional Zn observations in the future will help address this question better.
\section{Results}
\subsection{Evolution of metals with redshift}
Together, DLA and sub-DLA populations contain the majority of the neutral gas mass in the Universe \citep{Zafar2013a}. Therefore, they present a valuable tool to estimate the cosmic metallicity throughout the ages.
Models of cosmic chemical evolution claim that the global interstellar metallicity would rise with decreasing redshift, to reach near solar metallicity values at present day \citep{Lanzetta1995,Pei1995,Malaney1996,Pei1999,Tissera2001}. Sub-DLAs in particular contribute substantially to the cosmic metal budget. Indeed, \cite{Kulkarni2007} show that the contribution of sub-DLAs to the metal budget increases with decreasing redshift considering a constant relative H\,I gas in DLAs and sub-DLAs at low and high redshifts. \cite{Bouche2007} anticipate that $\lesssim17$ per cent of the metals are in sub-DLAs at $\rm z\sim2.5$ but this estimate is highly dependent on the ionized fraction of the gas. It is therefore highly important to compare sub-DLA metallicities with those of DLAs.
Our study adds 15 new measurements of sub-DLA metallicity. We chose to use ZnII as our main metallicity indicator \citep{Pettini1994b} as it is nearly undepleted onto interstellar dust. Moreover, ZnII lines are usually unsaturated, and since ZnII is the dominant ionization state in neutral regions, it does not require strong ionization correction.
However, Zn has an overall low cosmic abundance and the stronger lines $\lambda\lambda$ 2026 and 2062 can be blended with MgI $\lambda$ 2026 and CrII $\lambda$ 2062. When these ZnII lines are undetected, we use the dominant ions of other elements in the following order: OI, SII, SiII, MgII, FeII (corrected for the $\rm \alpha$ analysis) and NiII.
Fig. \ref{img:M/HVSzcolor} shows the evolution of the metallicity [M/H] with redshift of the systems for the 92 sub-DLAs (bottom panel) and the 362 DLAs (top panel) from the EUADP+ sample, color-coded with respect to the element used to derive the metallicity. We note that Zn is only detected up to $\rm z = 3$ (but for one DLA measured at $\rm z\sim4$), and O is only derived for metal-poor systems ($\rm [M/H] <-1$) because OI $\lambda$ 1302, the only OI line usually accessible to ground-based telescopes, is saturated otherwise.
\cite{Lanzetta1995} estimated the cosmic metallicity from the gas mass density $\Omega_{g}$ and metal mass density $\Omega_{m}$ via the H\,I-weighted mean metallicity $\rm \mean{Z}${}:
\begin{equation}
\rm <Z(z)>=\Omega_{m}(z)/\Omega_{g}(z)=\frac{\sum_{i}Z_{i}N(H\,I)_{i}}{\sum_{i}N(H\,I)_{i}}
\label{eq:HI-weight_Z}
\end{equation}
Fig. \ref{img:M/HVSz} shows the metallicity derived in our sample, as well as the H\,I-weighted mean metallicity $\rm \mean{Z}${} for both populations (sub-DLAs in blue and DLAs in red)
The bins for $\rm \mean{Z}${} are chosen such that there is an almost constant number of systems in each bin, that is 16 for the sub-DLAs and 26 for the DLAs
The vertical error bars are derived from the consideration on sampling and measurements errors. The sampling errors are calculated from a bootstrap technique as described in \cite{Rafelski2012} and the measurement errors from the propagation formula. The total errors are the quadratic sums of these two quantities for each bin.
We note a large scatter for a third of the newly derived sub-DLA metallicities (blue dots). This points out to the need for a larger sample of sub-DLA measurements at all redshifts.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{eps/_M_H_vsZperelement_DLA-eps-converted-to.pdf}
\includegraphics[width=0.49\textwidth]{eps/_M_H_vsZperelement_subDLA-eps-converted-to.pdf}
\caption[Evolution of metallicity with redshift]{Evolution of [M/H] with redshift, color-coded with respect to the element used to derive the metallicity for DLAs (top panel) and sub-DLAs (bottom panel) from the EUADP+ sample. }
\label{img:M/HVSzcolor}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{eps/meanZvsredshift-eps-converted-to.pdf}
\caption[Evolution of metallicity with redshift]{Evolution of the N(HI)-weighted mean metallicity $\rm \mean{Z}$ with redshift. Clearly, both DLA and sub-DLA populations show an increase of $\rm \mean{Z}${} with decreasing redshift, but sub-DLAs have a steeper evolution of $\rm \mean{Z}${} with redshift than DLAs. We also note a floor at $\rm [X/H]=-3$ below which no metals are detected.}
\label{img:M/HVSz}
\end{center}
\end{figure}
We measure an anti-correlation between redshift and metallicity for both populations. The Spearman coefficient for sub-DLAs is $\rm \rho=-0.49$ and $\rm \rho=-0.55$ for DLAs, with probabilities of no correlation $\rm P(\rho)<10^{-6}$ for both populations. The Kendall's $\rm \tau$ is $-0.34$ for sub-DLAs and $-0.38$ for DLAs, with a probability of no-correlation of below $10^{-5}$ for both populations. The dotted lines in Fig. \ref{img:M/HVSz} show the best bisector fits for the metallicity evolution with redshift of both populations. We measure\begin{equation}
\rm <Z>_{DLAs}=(-0.15\pm0.03)~ z - (0.6\pm0.13)
\end{equation}
\begin{equation}
\rm <Z>_{sub-DLAs}=(-0.30\pm0.07) ~z + (0.15\pm0.31)
\end{equation}
The fit has been performed shifting the y-axis to $\rm z=3$ to minimize the error on the intercept and ignoring the last DLA bin which presents a rapid decline in metallicity \citep{Rafelski2014}.
The evolution with redshift is steeper for sub-DLAs than for DLAs.
Previous authors \citep{Khare2007, Kulkarni2010} argued that this effect might arise from the fact that sub-DLAs are more massive than DLAs.
Our results are in agreement with previous work \citep{Kulkarni2007,Kulkarni2010, Som15}, with a more significant result in the sub-DLA regime thanks to the larger sample presented here. The slope remains unchanged with respect to earlier studies. However, the significance of the result increases indicating a convergence towards a realistic value of the slope.
\subsection{Kinematics}
In addition to the different abundances derived from Voigt profile fitting, information on the kinematics of the absorbers can be derived from the UVES high resolution spectra.
\subsubsection{Voigt Profile Optical Depth Method}
\label{sec:deltaVmethod}
We use the definition of the velocity interval $\rm \Delta V_{90}${} as defined by \cite{Prochaska1997}, based on the integrated optical depth $\tau_{tot}=\int \tau(v)dv$ and considering the velocity interval from $5\%$ to $95\%$ of this quantity.
In this paper, we do not consider the apparent optical depth (AOD) $\tau_{app}=-\log (I/I_{c})$ to derive the velocity interval, as is usually done, but we use instead the optical depth derived from the Voigt profile fits (see appendix \ref{ann:individual} for a description of the fits for every system individually).
This method, which we refer to as Voigt profile optical depth (VPOD) method, makes use of the information gathered from the fits. The saturation and contamination issues are then considered when deriving $\rm \Delta V_{90}${}. This is the main difference with the AOD method, which might provide $\rm \Delta V_{90}${} measurements affected by blends. In the VPOD method, we use simultaneously the information on several transitions to derive the velocity interval for any ion. Indeed, the only quantity that differs between transitions of the same ion is the oscillator strength, which has no impact on the velocity axis. Fig. \ref{img:VPOD} shows an example of the derivation of the velocity interval for an FeII line.
Table \ref{result:deltav90} summarizes the $\rm \Delta V_{90}${} measurements for the 22 systems studied here. For 20 of them, we use the information from the FeII lines as it is the ion most detected in our sample.
One of the sub-DLA in our sample, towards PKS 0454-220, has already been studied by \cite{Som15}. They use SII $\lambda$ 1250 from an HST/COS spectrum and derive $\rm \Delta V_{90}${}=155 km/s based on the AOD method. However, we find with the VPOD method described above a value almost twice smaller. We use the AOD method on the UVES spectrum with FeII $\lambda$ 2374 and derive $\rm \Delta V_{90}${}$\sim$85.0 km/s, consistent with the result from the VPOD method. We note that the Line Spread Function (LSF) derived from the COS consortium is responsible for the reported large value.
To overcome this problem, we exclude COS measurements from our analysis.
In conclusion, the VPOD $\rm \Delta V_{90}${} values are not sensitive to blending and saturation effects, to the shape of the instrument's LSF, its resolution and to the SNR of the derived spectrum. We note that depletion of refractory elements contributes to the error in the $\rm \Delta V_{90}${} because the different components can be affected differently by dust depletion. In the present study, FeII has been used because it is uniformly detected among the 22 new systems presented here.
In the remaining of the sample there is no object in common between the EUADP and the already derived $\rm \Delta V_{90}${} found in the literature.
\subsubsection{$\rm \Delta V_{90}${} versus Metallicity Relation}
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{eps/0454_v90-eps-converted-to.pdf}
\caption[]{An example illustrating the computation of the velocity interval $\rm \Delta V_{90}${}. The black curve is the normalized spectrum of PKS 0454-220 centered on the FeII $\lambda$ 2374 line, the red curve is the Voigt profile fit of the absorption and the blue curve is the integrated optical depth derived from the Voigt profile. The vertical dotted lines indicates the 5\% and 95\% thresholds for the integrated optical depth, defining the velocity width $\rm \Delta V_{90}${}.}
\label{img:VPOD}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{eps/M_v90_DLA-eps-converted-to.pdf}
\includegraphics[width=.45\textwidth]{eps/M_v90_subDLA-eps-converted-to.pdf}
\caption[]{[X/H] versus $\rm \Delta V_{90}${} for the newly derived systems, the systems from \cite{Ledoux2006}, \cite{Neeleman2013}, \cite{Som15}, \cite{Meiring2009b}, and \cite{Moller2013} in blue, red, green, cyan, magenta and yellow, respectively. The upper panel shows the DLAs and the bottom panel the sub-DLAs. The data points measured with COS (stars) are not considered for the fit due to the discussion in section \ref{sec:deltaVmethod}. The open triangles represent the sub-DLAs and the dots the DLAs. The dashed blue line reproduces the bisector fit of the sub-DLAs, the dashed red line is the bisector fit for all the DLAs and the green dashed line in the lower panel represents the \cite{Som15} sub-DLA fit.}
\label{img:v90_M_HI}
\end{center}
\end{figure}
\begin{table}
\caption{Measures of $\rm \Delta V_{90}${} in our sample derived from Voigt profile fits to the FeII lines (except for two systems with no Fe coverage, for which we used SiII (a) and AlII (b)).}
\label{result:deltav90}
\begin{center}
\begin{tabular}{lcccccccc}
\hline\hline
\textbf{QSO} & \textbf{$z_{abs}$} & log $\rm N(H\,I)$ [cm$^{-2}$] & $\Delta v_{90}$ [km/s] \\ \hline
QSO J0008-2900 & 2.254 & 20.22 & $53.7$ \\
QSO J0008-2901 & 2.491 & 19.94 & $12.4$ \\
\textbf{QSO J0018-0913} & 0.584 & 20.11 & $192.6$ \\
QSO J0041-4936 & 2.248 & 20.46 & $21.2$ \\
QSO B0128-2150 & 1.857 & 20.21 & $53.7$ \\
\textbf{QSO J0132-0823} & 0.647 & 20.60 & $76.5$ \\
QSO B0307-195B & 1.788 & 19.00 & $204.8$ \\
QSO J0427-1302 & 1.562 & 19.35 & $7.8$ \\
PKS 0454-220 & 0.474 & 19.45 & $78.7$\\
J060008.1-504036 & 2.149 & 20.40 & $79.9$ \\
QSO B1036-2257 & 2.533 & 19.30 & $220.4$ \\
J115538.6+053050 & 3.327 & 21.00 & $186.8^{a}$ \\
LBQS 1232+0815 & 1.72 & 19.48 & $167.6$ \\
QSO J1330-2522 & 2.654 & 19.56 & $21.0^{b}$ \\
QSO J1356-1101 & 2.397 & 19.85 & $337.7$ \\
QSO J1621-0042 & 3.104 & 19.70 & $161.9$ \\
4C 12.59 & 0.531 & 20.70 & $62.4$ \\
LBQS 2114-4347 & 1.912 & 19.50 & $135.9$ \\
QSO B2126-15 & 2.638 & 19.25 & $62.3$ \\
QSO B2126-15 & 2.769 & 19.20 & $121.2$ \\
LBQS 2132-4321 & 1.916 & 20.74 & $233.4$ \\
QSO B2318-1107 & 1.629 & 20.52 & $13.8$ \\
\hline
\end{tabular}
\end{center}
\end{table}
Recently, \cite{Som15} compared for the first time the sub-DLA metallicity versus velocity width trend over a statistically significant sample of 31 sub-DLAs at \zabsbetween{0.1}{3.1}.
We propose here to extend their analysis to a wider sub-DLA sample using our new 15 sub-DLAs.
We consider a different sample than the one used in the remaining of the paper (EUADP+) as the velocity widths are not provided by all authors. We consider the data from \cite{Ledoux2006} (52 DLAs and 14 sub-DLAs at redshifts \zabsbetween{1.7}{4.3}, corrected for \cite{Asplund2009} photospheric solar abundances),
observations from \cite{Meiring2009b} (29 sub-DLAs at redshifts \zabslt{1.5}, corrected for \cite{Asplund2009} photospheric solar abundances),
observations from \cite{Neeleman2013} (98 DLAs at redshifts \zabsbetween{1.6613}{5.0647}),
observations from \cite{Moller2013} (4 DLAs at redshifts \zabsbetween{1.9}{3.1}),
as well as results from this study (see Table \ref{result:deltav90}). We only consider the systems with detected $\rm [\alpha/H]$ (11/15 sub-DLAs and 4/7 DLAs).
The resulting sample gathers 54 sub-DLAs and 162 DLAs.
Fig. \ref{img:v90_M_HI} shows the trend between the metallicity and the velocity width $\rm \Delta V_{90}${} for sub-DLAs (bottom panel) and DLAs (top panel) from this $\rm \Delta V_{90}${} sample.
We fit both populations with the best bisector fits:
\begin{equation}
\rm [X/H]_{DLA}=(1.52\pm0.08)~\log \Delta V_{90} -(4.20\pm0.16)
\end{equation}
\begin{equation}
\rm [X/H]_{sub-DLA}=(1.61\pm0.22)~\log \Delta V_{90} -(3.94\pm0.45)
\end{equation}
The fits are performed shifting the y-axis to $\rm \log \Delta V_{90}=2$ to minimize the error on the intercept.
\cite{Som15} find a higher intercept and a shallower slope for the sub-DLA population, using only Zn and S with ionization corrections. Although this result is free from dust depletion effect on the metallicity estimation, it might be biased towards higher metallicity sub-DLAs, where Zn or S can be measured. A larger Zn-based metallicity sub-DLAs samples is required to recover the metallicity-$\rm \Delta V_{90}${} relation free from the effects of dust bias.
A larger Zn-based metallicity sub-DLAs samples is required to recover the metallicity-Delta v relation free from the effects of dust bias.
As in previous studies, the DLA sample is well correlated. The Spearman coefficient is $\rm \rho=0.63$, with a probability of no correlation $<10^{-6}$. The Kendall's $\tau$ is $0.46$ with a probability of no correlation $<10^{-6}$.
The sub-DLA sample is less correlated than the DLA sample, in agreement with \cite{Som15}. The Spearman coefficient is $\rm \rho=0.39$, with a probability of no correlation of $\rm 0.004$.
The Kendall's $\tau$ is $0.25$ with a probability of no correlation of $0.007$.
Adding more sub-DLAs to the $\rm \Delta V_{90}${}-metallicity relation does not improve the correlation. \cite{Som15} showed that the ionization correction also does not improve this correlation.
This indicates a larger spread for the sub-DLAs, which may originate from more complex kinematic behaviors in sub-DLA clouds.
However, the determination of $\rm \Delta V_{90}${} appears to be sensitive to the resolution, the LSF and the SNR of the data. This might contribute to the observed scatter, although an intrinsic scatter is expected from CGM regions.
\subsubsection{Is $\rm \Delta V_{90}${} a Reliable Tracer of Mass?}
A mass-metallicity relation (hereafter MZR) has been reported at low redshifts \citep{Lequeux1979, Tremonti2004}, intermediate redshifts \citep{Savaglio2005} and high redshifts \citep{Erb2006}. It relates the stellar mass of galaxies to the metallicity of their ISM. This relation is crucial in our understanding of galaxy evolution as it supports the theory of metal ejection from galactic outflows in low-mass (and hence low potential well) galaxies and their enrichment with accreting metal-poor IGM gas, diluting the galactic metallicity.
For quasar absorbers, some simulations indicate that the origin of the velocity width, $\rm \Delta V_{90}${}, could be strongly related to the gravitational potential well of the absorption system's host galaxy \citep[e.g.][]{Prochaska1997, Haenelt1998, Pontzen2008a}. Similarly, assuming a scaling of the galaxies luminosity with dark matter haloes, \cite{Ledoux2006} and later \cite{Moller2013} proposed to interpret the $\rm \Delta V_{90}${} versus metallicity relation of quasar absorbers as a MZR. Such a picture does not take into account the complex gas processes at play now known to take place in CGM regions. In other words, the $\rm \Delta V_{90}${} may reflect bulk motions of the absorbing gas rather than motions governed by the gravitational potential well.
Observationally, a measurement of mass and $\rm \Delta V_{90}${} has been possible in few individual systems. Infra-red IFU SINFONI observations of the galaxy hosts of 3 DLAs and 2 sub-DLAs in \cite{Peroux2011} and \cite{Peroux2014} allow one to determine the mass of the systems from a detailed kinematic study.
In addition, \cite{Christensen2014} has used photometric information of the galaxy hosts and Spectral Energy Distribution (SED) fits to estimate the stellar mass of 13 DLAs. Combined together, these findings suggest that, individually, the absorption systems align well with the MZR reported at these redshifts.
In addition to these measurements in a few specific systems, several authors have put constraints on the mass estimates of quasar absorbers in a statistical manner. Interestingly, the local analogues to DLAs, the 21cm $\rm z=0$ emitting galaxies studied with HIPASS by \cite{Zwaan2008} show that the quantity $\rm \Delta V_{90}${} correlates little with mass. Similarly, \cite{Bouche2007} \citep[and later][]{Lundgren2009, Gauthier2014} have used the ratio of MgII systems auto-correlation with a correlation with Luminous Red Galaxies (LRG) at the same redshifts to derive an estimate of the overall mass of quasar absorbers. Their findings show an anti-correlation between equivalent width, a proxy for $\rm \Delta V_{90}${}, and metallicity. Admittedly, the populations of absorbers do not completely overlap, the \cite{Ledoux2006} sample contains mostly DLAs, while the MgII sample of \cite{Bouche2007} might have at most 25\% of DLAs (according to the criterion of \cite{Rao2006}: 50\% meet the FeII/MgII criteria and 35-50\% of these are DLAs). In fact, \cite{Bouche2007} and \cite{Schroetter2015} argue that MgII absorbers can be used to trace superwinds as they are not virialized in the gaseous halo of the host-galaxies.
\cite{Bouche2012} also show that the inclination of the galaxy has a direct impact on the absorption profile and therefore on the velocity width. Put together, these many lines of evidence question the interpretation of the velocity width as a proxy for the mass of the host galaxy and the interpretation of the $\rm \Delta V_{90}${}/metallicity correlation as a MZR for quasar absorbers.
\subsection{Tracing the Circum-Galactic Medium with sub-DLAs}
\begin{figure}
\begin{center}
\includegraphics[width=.4\textwidth]{eps/histoLehner_alpha_LLS-eps-converted-to.pdf}
\includegraphics[width=.4\textwidth]{eps/histoLehner_alpha_subDLA_all_z-eps-converted-to.pdf}
\includegraphics[width=.4\textwidth]{eps/histoLehner_alpha_DLA_all_z-eps-converted-to.pdf}
\caption[Metallicity distribution in LLSs, sub-DLAs and DLAs]{Metallicity [$\alpha$/H] distribution of LLSs (top panel), sub-DLAs (middle panel) and DLAs (bottom panel). The histogram for LLSs has been taken from \cite{Lehner2013} and indicates a bimodality in the metallicity distribution for LLS at $\rm z<1$. The black vertical dashed lines represent the mean values derived from the \zabslt{1} LLS sub-groups by \cite{Lehner2013}.}
\label{img:bimodality_alpha}
\end{center}
\end{figure}
Our understanding of galaxy formation and evolution is tightly linked with the study of two opposite processes that take place within the CGM. Indeed, to create stars, the galaxy requires a continuous input of cold gas, that is believed to accrete along the filamentary structures from the cosmic web. In addition, cosmological simulations fail to reproduce the observed SFR without invoking feedback processes from star formation itself or AGN activity.
These outflowing processes and their large scale impact have been confirmed observationnally \citep{Steidel2010,Bouche2012,Kacprzak2014}, but there is still little observational evidence for accretion of cool material \citep{Bouche2013a, Cantalupo2014, Martin2014a}.
Quasar absorbers with H\,I column densities in the range of LLS and sub-DLAs are believed to be good probes of this CGM \citep{Fumagalli2011, VandeVoort2012a}. \cite{Lehner2013, Lehner2014} report a bimodality in the metallicity distribution of 29 $z<1$ LLS, which they interpret as the signatures of outflows (metal-rich) and infalls (metal-poor).
\cite{Lehner2013} extended their analysis on 29 sub-DLAs and 26 DLAs, but do not report a bimodality distribution in the metallicity of these systems based on $\rm \alpha$-elements. Clearly, larger samples of quasar absorbers are required to perform such studies.
Here, we perform similar analysis on a larger sample of sub-DLAs and DLAs with a broad redshift range.
\begin{figure*}
\begin{center}
\includegraphics[width=.4\textwidth]{eps/histoLehner_alpha_subDLA_high_z-eps-converted-to.pdf}
\includegraphics[width=.4\textwidth]{eps/histoLehner_Fe_subDLA_high_z-eps-converted-to.pdf}\\
\includegraphics[width=.4\textwidth]{eps/histoLehner_alpha_subDLA_inter_z-eps-converted-to.pdf}
\includegraphics[width=.4\textwidth]{eps/histoLehner_Fe_subDLA_inter_z-eps-converted-to.pdf}\\
\includegraphics[width=.4\textwidth]{eps/histoLehner_alpha_subDLA_low_z-eps-converted-to.pdf}
\includegraphics[width=.4\textwidth]{eps/histoLehner_Fe_subDLA_low_z-eps-converted-to.pdf}
\caption[Metallicity distribution in sub-DLAs at different redshift bins]{Metallicity (left panels: $\rm [\alpha/H]$, right panels: [Fe/H]) distribution of sub-DLAs for different redshift bins: $\rm z>2.4$ (top panels), $\rm 1.25<z<2.4$ (middle panels) and $\rm z<1.25$ (bottom panels). The black vertical dashed lines represent the mean values derived from the \zabslt{1} LLS sub-groups by \cite{Lehner2013}. The black areas represent upper limits. The metallicity distribution is a strong function of redshift and only the lowest redshift range presents hints of a bimodal distribution for the [Fe/H] metallicity.}
\label{img:bimodality}
\end{center}
\end{figure*}
Fig. \ref{img:bimodality_alpha} shows the bimodal metallicity distribution in $z<1$ LLS by \cite{Lehner2013} and the $\alpha$-element metallicity distribution for DLAs (316 systems) and sub-DLAs (68 systems) derived from our EUADP+ sample at all redshifts.
The sub-DLA $\rm [\alpha/H]$ distribution in the middle panel of Fig. \ref{img:bimodality_alpha} also suggests bimodality.
In Fig. \ref{img:bimodality}, we plot the distribution of the $\rm \alpha$ abundances (left panels) and Fe abundances (right panels) for the EUADP+ sub-DLAs in 3 redshift bins (\zabsgt{2.4} for the upper panels, \zabsbetween{1.25}{2.4} for the middle panels and \zabslt{1.25} for the bottom panels).
We consider the metallicity traced by FeII as we have more detections with this ion and it is little affected by photo-ionization effect, even though Fe has an inclination to lock up onto dust grains. These histograms reveal the strong metallicity evolution with redshift for sub-DLAs.
The sub-DLAs in the high redshift bin, \zabsgt{2.4}, present an unimodal distribution centered around [M/H]$\sim-1.6$, similar to the metal-poor LLS population derived by \cite{Lehner2013}.
At intermediate redshifts, \zabsbetween{1.25}{2.4}, a transition from low to higher metallicities appears.
At low redshifts, \zabslt{1.25}, however, the distribution presents hints of a bimodal distribution. This trend is more pronounced for the [Fe/H] distribution. A DIP test rejects the unimodal distribution at a significance level of 83\%, taking the upper limit as a detection. The peaks of the distribution are located at $\rm [Fe/H]=$ -1.12 and $\rm [Fe/H]=$ -0.29, from a Gaussian Mixture Modeling. These values are compared with what is expected from simulations in terms of metallicity of accreting or outflowing gas.
The prediction for the cold-mode accretion metallicity is above a hundredth solar, which is in line with the metal poor population in our distribution \citep{Ocvirk2008, Shen2013}. Therefore, the metal rich population should trace either outflowing gas or gas directly associated with the galaxy's ISM.
However, this is not seen in the $\rm [\alpha/H]$ distribution, where the DIP test rejects the unimodal distribution at a significance level of 31\%, still taking upper limits as detections. But these limits are located at the high metallicity end of the distribution, and therefore do not contradict the possible bimodal distribution seen in $\rm [Fe/H]$.
These results indicate a similar behavior for low redshift sub-DLAs as for low redshift LLS. However, we expect the position of the peaks to be higher than those derived for the $\rm z<1$ LLS, as the $\rm [Fe/H]$ metallicity is underestimated due to depletion of Fe onto dust grains. Lehner et al. (in prep.) show that the bimodal distribution for LLS disappears at higher redshifts, similarly to what we find with sub-DLAs at higher redshifts. We note however that the bimodality in LLS could perhaps be incorrect, given that there are larger uncertainties in the metallicity determination of LLS (due to ionization corrections).
We note that the effect of redshift plays an essential role in such an analysis, as illustrated in Fig. \ref{img:bimodality}.
Altogether, larger samples of both LLS and sub-DLAs at low-redshifts are required to distinguish between metal-poor gas accreting onto the galaxy and metal-rich gas being expelled.
\section{The Data}
\label{sec:data}
\subsection{New absorbers}
In order to put together a significant sample of sub-DLAs observed at high spectral resolution, we make use of the ESO UVES Advanced Data Products (EUADP) sample from \cite{Zafar2013}. This sample consists of 250 high-resolution ($\rm R\sim42,000$) quasars spectra covering a total of 196 damped absorbers (with log \NHIgt{19.0}).
This dataset has motivated a number of studies including a report of new H\,I systems \citep{Zafar2013} and how they can be used to constrained the neutral gas mass density of sub-DLAs in particular \citep{Zafar2013a}, the nucleosynthetic history of Nitrogen \citep{Zafar2014} and the low Argon abundances observed in DLAs \citep{Zafar2014a}.
Most of the absorbers in the EUADP sample have their metallicity abundances published in the literature \citep[][and reference therein]{Peroux2006, Peroux2006a,Peroux2008,Zafar2013a}. We present here the analysis of 14 new EUADP sub-DLAs covering a redshift range \zabsbetween{0.584}{3.104}. We also include 6 new DLAs for completeness.
The measurements of HI column densities and redshifts of each system in the EUADP sample are reported in \cite{Zafar2013} and references therein.
In addition to these 14 new sub-DLAs from the EUADP sample, we present the UVES spectra of two other systems: one sub-DLA at \zabseq{0.584} and one DLA at \zabseq{0.647}. These two low-redshift absorbers have been observed with the HST ACS grism from which an estimate of their H\,I column densities has been derived \citep{Turnshek2015}. The quasars were subsequently observed with UVES on VLT under the programme 91.A-0300 (PI: C. P\'eroux) in Service Mode in August and September 2013. Each object was observed using a combined 346$+$564 nm setting with two different observations with exposure times lasting 4500\,+\,3600 sec (QSO J0018$-$0913) and 2 x 4500 sec (QSO J0132$-$0823). The data were reduced using the most recent version of the UVES pipeline in MIDAS (uves/5.4.3). Master bias and flat images were constructed using calibration frames taken closest in time to the science frames. The science frames were extracted with the ``optimal" option and corrected to the vacuum heliocentric reference.
To combine the resulting spectra, we choose to weight them by the signal-to-noise ratio, as for the remaining of the EUADP sample \citep{Zafar2013}, in line with standard practice at this spectral resolution \citep{Omeara2015}.
The absorption redshifts, which are based on the $\rm N(H\,I)${} or MgII features, are used to analyse the associated metal lines. Table \ref{table:subsample} summarises the properties of the quasars and absorbers in the sample studied here. The two additional objects which were not originally published by \cite{Zafar2013} are shown in bold.
\subsection{Literature sample}
In addition to these 15 new sub-DLA measurements (+7 DLAs), we gather metallicity estimates of sub-DLAs from the remaining part of the EUADP sample as well as other recently published samples \citep{Meiring2006, Meiring2009, Dessauges-Zavadsky2009, Battisti2012, Som15}.
In order to compare the properties of sub-DLAs to that of DLAs, we add to the sample a collection of DLA metallicity measurements from the EUADP sample as well as from the literature (see earlier references and \citealt{Berg2015}). Altogether, this literature sample is the largest and most up-to-date sub-DLA sample published today.
The table in Appendix \ref{ann:tableau} lists the metallicity estimates of the full sample of absorbers and associated references. Fig. \ref{img:hist_samplelandfig} illustrates the distribution in redshift of the absorber sample studied for both DLAs and sub-DLAs (top and middle panels respectively). The bottom panel presents the N(H\,I) distribution of the sample.
We stress that the additional systems are consistent with the parent sample as they are not selected on their metal content or redshift but solely on their H\,I column density (see also Fig. \ref{img:M/HVSz}).
In conclusion, the final sample, referred to as the EUADP+ sample, contains 92 sub-DLAs (with 15 new measurements) and 362 DLAs (7 new measurements).
Clearly, the data presented here contribute most in the sub-DLA H\,I column density range.
\input{body/subsampleTable.tex}
\begin{figure}
\begin{center}
\includegraphics[width=.4\textwidth]{eps/histoZ_bin=20_DLA-eps-converted-to.pdf}
\includegraphics[width=0.4\textwidth]{eps/histoZ_bin=20_subDLA-eps-converted-to.pdf}
\includegraphics[width=0.4\textwidth]{eps/histoHI_bin=20-eps-converted-to.pdf}
\caption[Sample descriptive histograms]{Absorption redshifts and N(H\,I) distributions of the DLAs and sub-DLAs in our sample compared with the remaining absorbers covered by the EUADP survey and the literature (referred to as EUADP+ sample). The vertical line in the bottom panel indicates the canonical DLA definition. Clearly, the data presented here contribute most in the sub-DLA H\,I column density range at low redshift where few systems have been studied so far.}
\label{img:hist_samplelandfig}
\end{center}
\end{figure}
\section{Analysis}
\subsection{Method}
The continua of the quasar spectra are fitted using a spline function connecting the regions of the spectrum free from absorption features as described in \citet{Zafar2013a}. The Voigt profile fits are performed with the FIT/LYMAN package within the MIDAS environment \citep{Fontana1995}. The routine calculates a $\chi^{2}$ Hessian minimization and enables fits of up to 50 free parameters including the central wavelength, the column density and the Doppler parameter of each component of the fit. This allows fitting several ions simultaneously as well as several transitions of the same species, thus making maximum use of the information available from the velocity profiles. The low-ionization species (OI, FeII, SiII, ...) are fitted as a separate group from the high-ionization species (CIV, SiIV, ...) \citep[e.g.,][]{Wolfe2005, Fox2007, Milutinovic2010, Crighton2013a}. The intermediate-ionization species AlIII are fitted either on its own, or with the low-ionization or high-ionization species, depending on the similarity in the absorption velocity profiles.
This process allows us to identify possible blends of interloping absorbers at the positions of the features under study. In case of blending, the profiles are fitted using information on central wavelengths and Doppler parameters from other un-blended profiles, thus leading to upper limits in the column density determination. In addition, saturated transitions or components are avoided because the column density information cannot be recovered in that case. The quasar continuum solution is iteratively refined when necessary during the Voigt profile fitting process. The fits are performed minimizing the number of components. In cases where a transition is not detected, we derive a 3-$\sigma$ upper limit from an estimate of the SNR of the spectra at the expected position of the line. The laboratory wavelengths and oscillator strengths used throughout the fits are taken from \cite{Morton2003}\footnote{Recently, a new set of oscillator strengths for SII and ZnII lines has been derived for studies of the Inter-Stellar Medium (ISM), DLAs and sub-DLAs \citep{Kisielius2014, Kisielius2015}. A change from \cite{Morton2003} oscillator strengths to this new study would lower [Zn/H] by about 0.1 dex.}.
We estimate the abundance for various elements of each absorbing system by summing the column densities of the different components found in the velocity profile described above. The metallicity $\rm [X/H]$ of an element X with respect to solar metallicity is derived from the following expression:
\begin{equation}
\rm [X/H] = \log \left(\frac{N(X)}{N(H)}\right) - \log (X/H)_{\odot}
\end{equation}
where $\rm (X/H)_{\odot}$ is the photospheric solar abundance from \cite{Asplund2009} and $\rm N(X)$ is the column density of element X. The column density of each element is taken to be that of the dominant ion, and ionization correction is ignored here (see section \ref{sec:ionization} for further discussion on this point). The error estimate on the total column density $\rm \log N$ is calculated from the error on individual column density $\rm \log N$ of each component through the error propagation formula:
\begin{equation}
\rm \sigma_{\log (N(X))}=\frac{\sqrt{\sum_{i}(N(X)_{i}\sigma_{\log(N(X))_{i}})^{2}}}{N(X)}
\end{equation}
The global uncertainty on the abundance determination is then calculated from a quadratic sum of $\rm \sigma_{\log (N(X))}$ and $\rm \sigma_{\log (N(H))}$ since the errors in the solar abundances would introduce systematic effects which can be neglected in studies of relative abundances.
The resulting Voigt profile parameters and corresponding velocity plots for the low-, intermediate- and high-ionization species as well as a detailed description of the 22 individual systems mentioned earlier are provided in Appendix \ref{ann:individual}. The column densities and abundances derived for these systems are gathered in tables \ref{result:columndensities} (for total column densities) and \ref{result:metallicities} (for abundances).
For the different H\,I and metals column densities presented in this paper, the associated error on the abundances are based on $\chi^{2}$ minimization. The continuum placement error is not taken into account to be consistent with other measurements from the literature.
\input{body/resultatsFinaux.tex}
\input{body/resultatsFinauxMetallicity.tex}
\section*{Acknowledgements}
We thank N. Lehner, C. Howk, M. Fumagalli, M. Rafelski, M. Pieri, R. Bordoloi, J. O'Meara, D. Som and P. M\o ller for useful discussions. This work has been funded within the BINGO! (`history of Baryons: INtergalactic medium/Galaxies cO-evolution') project by the Agence Nationale de la Recherche (ANR) under the allocation ANR-08-BLAN-0316-01 as well as within the REGAL ('what REgulates the growth of GALaxies? ') project by the Labex (Laboratoire d'Excellence) OCEVU ('Origines, Constituants et Evolution de l'Univers'). This work has been carried out thanks to the support of the OCEVU Labex (ANR-11-LABX-0060) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French government program managed by the ANR. We thanks Pierre Mege for contributing in the development of the VPOD method. SQ acknowledges CNRS and CNES support for the funding of his PhD. CP thanks the ESO science visitor program for support. VPK acknowledges partial support from the NSF grant AST/1108830, with additional support from NASA grant NNX14AG74G and NASA/STScI grant for program GO 12536. EJ thanks Aix-Marseille University and C\'ecile Gry for a visit where part of this work was undertaken.
\bibliographystyle{mn2e}
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The Disrobing of Christ or El Expolio () is a painting by El Greco begun in the summer of 1577 and completed in the spring of 1579 for the High Altar of the sacristy of the Cathedral of Toledo, where it still normally hangs. In late 2013 it was on temporary display at the Prado in Madrid (with the other El Grecos), following a period of cleaning and conservation work there; it was returned to Toledo in 2014. It is one of El Greco's most famous works. A document dated July 2, 1577 which refers to this painting is the earliest record of El Greco's presence in Spain. The commission for the painting was secured thanks to El Greco's friendship from Rome with Luis, the son of Diego de Castilla, the dean of the Cathedral of Toledo. De Castilla senior also arranged El Greco's other major commission, on which he worked simultaneously, the paintings for the Toledan church of Santo Domingo el Antiguo.
Description
The painting shows Christ looking up to Heaven with an expression of serenity; His idealized figure seems segregated from the other people and the violence surrounding him. A figure dressed in black in the background points at Christ accusingly, while two others argue over who will have His garments. A man in green to Christ's left holds Him firmly with a rope and is about to rip off His robe in preparation for his crucifixion. At the lower right, a man in yellow bends over the cross and drills a hole to facilitate the insertion of a nail to be driven through Christ's feet. The radiant face of the Savior is violently juxtaposed to the coarse figures of the executioners, who are amassed around Him creating an impression of disturbance with their movements, their gestures and lances.
Christ is clad in a bright red robe; it is on this red tunic that El Greco concentrated the full expressive force of his art. The purple garment (a metonymic symbol of the divine passion) is spread out in a light fold; only the chromatic couple of yellow and blue in the foreground raises a separate note which approaches, in power, the glorifying hymn of the red.
In the left foreground, the three Marys contemplate the scene with distress. Their presence was objected to by the Cathedral authorities, since they are not mentioned as present at this point in the Gospels. Greco probably took this detail, with some others like the rope around Christ's wrists, from the account in the Meditations on the Passion of Jesus Christ by Saint Bonaventure. The placement of the tormentors higher than the head of Christ also was cited by the commissioners of the Cathedral in the arbitration process over the price.
In designing the composition vertically and compactly in the foreground El Greco seems to have been motivated by the desire to show the oppression of Christ by his cruel tormentors. The figure of Christ, robust, tall and tranquil, dominates the center of the composition which is built vertically like a wall. El Greco chose a method of space elimination that is common to middle and late 16th-century Mannerists. According to Harold Wethey, El Greco "probably recalled late Byzantine paintings in which the superposition of heads row upon row is employed to suggest a crowd".
Critical analysis
Wethey regards the painting as a "masterpiece of extraordinary originality". The powerful effect of the painting especially depends upon his original and forceful use of colour. Something of the effect of the grand images of the Saviour in Byzantine art is recalled; by this date the disrobing was a rare subject in Western art. The motif of the crowding round Christ suggests an acquaintance with the works of Northern artists, like Bosch (the best collection of whose works belonged to Philip II); the figure preparing the Cross could be derived from the similar figure bending forward in Raphael's tapestry and cartoon of the Miraculous Draught of Fishes, which he would have known from Rome. This is, however, the last time that there are any hints of specific borrowings. The original altar of gilded wood that El Greco designed for the painting has been destroyed, but his small sculpted group of the Miracle of St. Ildefonso still survives on the lower center of the frame.
Arbitration
The Disrobing of Christ was a subject of a dispute between the painter and the representatives of the Cathedral regarding the price of the work; El Greco was forced to have recourse to legal arbitration and eventually received only 350 ducats, when his own appraiser had valued it at 950. He was also supposed to remove some of the figures objected to, which he never did.
Variants
Despite the complaints of the commissioners of the Cathedral the painting was hugely successful; currently, more than 17 versions of the painting are known. Two greatly reduced versions are generally accepted as from the hand of El Greco himself; possibly one may have been an oil sketch study, or, more likely, a studio record of the composition. Other replica versions may also be in whole or part by the master himself. A 1581-1586 autograph or studio copy has been in the Museum of Fine Arts of Lyon since 1886.
A c.1600 variant of the work is in the National Gallery in Oslo, Norway — this is attributed to the artist. Another in the Alte Pinakothek, Munich is thought to be an autograph preparatory sketch for the main work.
A version is also on display at Upton House in Warwickshire, England.
References
Sources
Brown, Jonathan (ed.) (1982). "El Greco and Toledo", El Greco of Toledo (catalogue). Little Brown.
Clark, Kenneth. Looking at Pictures. New York: Holt Rinehart and Winston, 1960
Paintings by El Greco
1570s paintings
Altarpieces
Toledo, Spain
Paintings depicting the Passion of Jesus
Paintings of the Virgin Mary
Paintings in Toledo, Spain
Paintings depicting Mary Magdalene
Paintings in the collection of the Museum of Fine Arts of Lyon
Torture in art | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,209 |
\section{Introduction}
TITUS (Tokai Intermediate Tank with Unoscillated Spectrum) is a proposal for an intermediate detector as part of the Hyper-Kamiokande~(HK) experiment~\cite{Abe:2011ts}. It will be located approximately 2~km from the J-PARC neutrino beam. TITUS is a cylindrical Cherenkov detector, filled with about 2~ktonne of gadolinium (Gd) doped water, aligned with \(2.5^{\circ}\) off-axis with respect to the neutrino beam. A magnetised iron muon detector is located at the downstream part of the tank to measure muons ranging out of the detector.
The Cherenkov effect allows detection and identification of electrons and muons produced in neutrino Charged Current (CC) interactions, and Gd allows the detection of possible outgoing neutrons. The primary goal of TITUS is to constrain the neutrino flux from the J-PARC beam that directly affects the sensitivity to CP violation at the far detector. The selection of the neutrino flux at the near detector is improved with respect to water Cherenkov-only tanks thanks to the Gd, that allows the capture of the final state neutrons from the neutrino-nucleon interaction and allows neutrinos and antineutrinos to be distinguished. A precise measurement of neutrino cross sections in water further helps the selection. TITUS can also be used for other physics purposes including detection of supernovae neutrinos, sterile neutrino studies and understanding the background for proton decay searches.
\section{Hyper-Kamiokande}
Hyper-Kamiokande is a proposed next generation neutrino oscillation experiment using a 1~megatonne water Cherenkov detector. It aims to study CP violation in the lepton sector by comparing the oscillation probabilities for neutrinos and antineutrinos~\cite{Abe:2015zbg}. To achieve this, the systematic uncertainties need to be greatly reduced from the current neutrino oscillation experiment at T2K. HK will also measure other neutrino oscillation parameters; it is expected to probe the proton life-time at an order of magnitude beyond the current limit~\cite{Abe:2011ts,Abe:2015zbg}, and it will be able to study astrophysical neutrinos.
\section{TITUS}
To achieve the required reduction of the systematics uncertainties, precise measurements are required of both the unoscillated neutrino flux and the neutrino cross section. This can be achieved using TITUS, an intermediate detector between the J-PARC beam source and HK. By using the same target (water) and flux as HK, it is possible to cancel many of the differences between the near and far detectors. A smart selection using Gd helps to reduce the background. Figure~\ref{fig:fluxratios} shows the ratio of the flux for different baselines for TITUS and the flux at HK. The flux at \(\simeq 2\)~km is very similar to that at HK.
\begin{figure}[!ht]
\begin{center}
\subfloat[\(\nu_\mu\) flux for beam in neutrino mode]{
\includegraphics[width=0.33\textwidth]{p320_nm_rebin2.pdf}
}
\hspace*{0.05\textwidth}
\subfloat[\(\bar{\nu}_\mu\) flux for beam in antineutrino mode]{
\includegraphics[width=0.33\textwidth]{m320_nmb_rebin2.pdf}
}
\caption{Unoscillated flux ratios (Nominal HK / Near Detector) at baselines of 1000m, 1828m, and 2036m, for \(\nu_\mu\) with neutrino enhanced beam (left) and \(\bar{\nu}_\mu\) with antineutrino enhanced beam (right).}
\label{fig:fluxratios}
\end{center}
\end{figure}
\subsection{Gadolinium doping}
Doping the water with 0.1\% of Gd allows the detection of neutrons produced in neutrino interactions; this is realised because the neutron capture on Gd has a very high cross-section and produces a cascade of photons with total energy of about 8~MeV producing 4-5~MeV of visible energy that can be detected~\cite{Dazeley:2008xk}. For an oscillation analysis this can provide a very pure sample of CCQE interaction events both when a neutrino is interacting (producing no neutrons) or an anti-neutrino (producing 1 neutron). The effect on the spectrum of the selection can be seen in Figure~\ref{fig:selections}. R\&D is ongoing to monitor the feasibility and response of the detector when the water is doped with Gd~\cite{Renshaw:2012np, Anghel:2015xxt}.
\begin{figure}[!ht]
\begin{center}
\subfloat[Before neutron tagging]{
\includegraphics[width=0.28\textwidth]{titus_1R_polrhc.pdf}
}
\subfloat[No tagged neutron]{
\includegraphics[width=0.28\textwidth]{titus_1R_polrhc_noneutron.pdf}
}
\subfloat[Tagged neutron]{
\includegraphics[width=0.28\textwidth]{titus_1R_polrhc_hasneutron.pdf}
}
\caption{The composition of the one muon-like ring sample in TITUS during anti-neutrino mode running. The effect of different neutron selections is shown. \label{fig:selections}}
\end{center}
\end{figure}
\subsection{Photosensors}
Different types of photosensors are currently under investigation. Along with Photomultiplier Tubes (PMT), TITUS may include LAPPDs (Large Area Picosecond Photo Detectors), the next generation photosensors with improved timing resolution of the order of few tens of picoseconds and can reconstruct the hit position on the detector surface to within a few centimetres~\cite{Anghel:2013zxa}. Adding LAPPDs greatly improves the event reconstruction for low energy events (neutron capture on Gd). These detectors are currently being developed.
\subsection{Magnetised Muon Range Detector}
Due to the size of TITUS, about 18\% of the muons coming from beam neutrino interactions escape the tank. These muons come from neutrinos in the higher end of the spectrum. It is therefore important to quantify their energy after they ranged out of the detector to help in understanding the high energy background.
A Magnetised Muon Range Detector (MMRD) with magnetic field of 1.5~T can provide energy and charge reconstruction.
Figure~\ref{fig:MMRD} shows the charge reconstruction efficiency dependent on neutrino energy.
Combined with the neutron tagging this could give very high purity samples as well as providing a method for validating and calibrating the neutron tagging.
\begin{figure}[!ht]
\centering
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=0.8\columnwidth]{MRDplot2.png}
\captionof{figure}{MMRD charge reconstruction efficiency for muons coming from the interaction of the neutrinos with the tank.}
\label{fig:MMRD}
\end{minipage}
\hspace*{0.05\textwidth}
\begin{minipage}{.45\textwidth}
\centering
\includegraphics[width=0.9\columnwidth]{CPplot1.png}
\captionof{figure}{Scaling of the error on \(\delta_{\text{CP}}\), where a POT scale of one is after 10~years of operation with a beam of 750~kW and equal splitting between neutrino and anti-neutrino mode beam. The study was realised assuming \(\sin^2(2\theta_{13})=0.095\) and \(\delta_{\text{CP}}=0\).}
\label{fig:CPsensitivity}
\end{minipage}
\end{figure}
\section{CP violation sensitivity}
Due to the very high discrimination of TITUS described in the previous section, the sensitivity of HK to CP violation is increased. The addition to the CP violation fit at HK of the intermediate detector sample, and in particular with the neutron-tagged sample, where detector systematics including neutrino cross-section and nucleon final state interaction uncertainties have been included, leads to a significantly decreased time to discovery by a reduction in the total systematic error at the far detector, as seen in Figure~\ref{fig:CPsensitivity}.
\section{Summary}
The addition of TITUS, a 2~ktonne Gd-doped water Cherenkov detector with magnetised muon range detector, located approximately at 2~km from J-PARC, to the HK project will allow precise measurements of the unoscillated spectrum of the J-PARC neutrino beam. The 0.1\% doping of Gd allows for a detectable signal from neutron capture, and thus the discrimination of neutrino and antineutrino interactions as well as inputs into neutrino cross-section measurements. The downstream MMRD provides a second method for neutrino/antineutrino discrimination through charge reconstruction of muons exiting the tank and also provides energy reconstruction for these muons. These features allow TITUS to significantly reduce systematic errors in CP violation measurements at HK, providing increased sensitivity and reduced time to discovery.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,575 |
from __future__ import (absolute_import, division,
print_function, unicode_literals)
import unittest
from cython.functions import (
p0x88_to_tuple,
tuple_to_0x88,
chess_notation_to_0x88,
p0x88_to_chess_notation,
chess_notation_to_tuple
)
from cython.importer import Board
from consts.colors import WHITE, BLACK
from consts.pieces import PAWN, KING
class TestBoardEvaluation(unittest.TestCase):
def test_initial_get_value(self):
board = Board(True)
self.assertEqual(board.get_value(), 0)
def test_get_value_when_white_is_winning(self):
board = Board(False)
board.load_fen(
"rnbqkbnr/ppppp1pp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1")
self.assertGreater(board.get_value(), 0)
def test_get_value_when_black_is_winning(self):
board = Board(False)
board.load_fen(
"rnbqkbnr/pppppppp/8/8/8/8/PPPP1PPP/RNBQKBNR w KQkq - 0 1")
self.assertLess(board.get_value(), 0)
class TestBoardPiecesCount(unittest.TestCase):
def test_white_pawns_count(self):
board = Board(True)
self.assertEqual(board.count(WHITE, PAWN), 8)
def test_black_pawns_count(self):
board = Board(True)
self.assertEqual(board.count(BLACK, PAWN), 8)
def test_white_king_count(self):
board = Board(True)
self.assertEqual(board.count(WHITE, KING), 1)
def test_black_king_count(self):
board = Board(True)
self.assertEqual(board.count(BLACK, KING), 1)
class TestAI(TestBoardEvaluation,
TestBoardPiecesCount):
pass | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,689 |
\section{Introduction}
Sunspots are dark areas on the solar surface and are associated with strong
magnetic fields. The magnetic field inhibits the convective flow of plasma in
the region and as this is the primary mechanism for heat transport at the
surface, the sunspot is cooler and darker. Study of sunspots started around the
early 1600s although there are records of observations in China going back for
2000 years \citep{Yau1988,Eddy1989}. Since the discovery of the magnetic field
in sunspots \citep{Hale1908} they have been a primary indicator of solar
activity and detailed records have been kept. By studying the evolution of
sunspot characteristics (area, field strength, etc), on timescales of days we
can gain insight into their formation and dispersal, while studies on longer
timescales (months and years) can reveal the longer-term behaviour of the Sun's
large-scale magnetic field, naturally of great importance for constraining
models of the solar dynamo. For example, the North-South asymmetry of sunspot
numbers and areas is well-established and has been studied for many decades
\citep[see e.g. ][ and references
therein]{1993A&A...274..497C,2006AdSpR..38..868Z,2007A&A...476..951C} and may
indicate a phase lag between the magnetic activity in the northern and southern
hemispheres, possibly hinting at non-linear behaviour, such as random
fluctuations of the dynamo terms and strong high order terms
\citep[e.g.][]{2003A&ARv..11..287O}.
The sunspot cycle variation of many solar parameters is of course well
established, however it was reported by \citet{Penn2006} that Zeeman
splitting observations of the strongest fields in sunspot umbrae show a secular
decrease between 1998 and 2005, apparently without a clear cyclic variation.
This goes hand-in-hand with an increase in the umbral brightness. Such a secular
change, if verified, would have striking implications for the coming sunspot
cycles - \citet{2010arXiv1009.0784P} suggest that if the trend continues there
would be virtually no sunspots at the time of cycle 25. It is one of the main
goals of the present study to automatically examine the MDI data for such
behaviour. In creating the dataset necessary to do this we also obtain and
report on the cycle-dependent behaviour of sunspot areas and locations. In
particular, the total projected area of sunspots present on the visible disk is
of interest in solar spectral irradiance studies
\citep{1982JGR....87.4319W,1985SoPh...97...21P,1997SoPh..173..427F} where it
enters as a parameter in spectral irradiance calculations.
We are fortunate now to have long and consistent series of solar observations
from which such parameters can be extracted, and the computational capacity to
do it automatically. Image processing and feature recognition/tracking in solar
data is now a very active field \citep{2010SoPh..262..235A}, and sunspot
detection is a well-defined image processing problem that has been studied by
several authors
\citep{2005SoPh..228..377Z,2008SoPh..248..277C,2008SoPh..250..411C,Watson2009}.
It is the purpose of this article to detail some physical properties of
sunspots detected in the continuum images from the SOHO/MDI instrument
\citep{Scherrer1995} and how they vary throughout solar cycle 23. We have used
an image processing algorithm based on mathematical morphology
\citep{Watson2009}.
The article proceeds with section 2 detailing the generation of the sunspot
catalogue and the results of looking at evolution in sunspot area and locations
over solar cycle 23. Then, section 3 details the evolution of magnetic fields in
sunspots, particularly in the umbra where the fields are strongest. Finally, in
section 4 we finish with our discussion and conclusions.
\begin{figure}[ht]
\centerline{\includegraphics[width=0.45\textwidth,clip=]{sunspot_number_plot_smoother.eps}}
\caption{The solid line shows the number of sunspots detected by the STARA code,
scaled to match the magnitude of the international sunspot number near the peaks
as calculated by the SIDC, shown by the dashed line.}
\label{fig:emergences}
\end{figure}
\section{Creating a catalogue of sunspots}
In order to analyse the sunspots over solar cycle 23, the STARA (Sunspot
Tracking and Recognition Algorithm) code developed by \cite{Watson2009} was
used, and readers are referred there for information on the method and its
testing. This is an automated system for detecting and tracking sunspots through
large datasets and also records physical parameters of the sunspots detected. It involves using techniques from the field of morphological image processing to detect the outer boundaries of sunspot penumbrae. This is achieved by means of the top-hat transform which allows us to remove any limb-darkening profile from the data and to perform the detections in one step. In
addition to the method given in \citet{Watson2009} the code had to be developed
further to separate the umbra and penumbra of spots as we would be looking at
the magnetic fields present in the umbra. When visually inspecting the
data there is a clear intensity difference between the umbra and
penumbra in sunspots. This difference
is due to the magnetic structure of susnpots. The umbra has a higher density of
magnetic flux which inhibits convection more than in the penumbral region. This
causes the umbra to be cooler and therefore appear darker. However, as sunspots
move towards the limb both the umbra and penumbra are limb-darkened. For this reason, we cannot use a single threshold value to define the
outer edge of the umbra. The algorithm we use removes all limb darkening effects at the same time as sunspot detection, greatly increasing speed as these two steps are carried out together. This problem has been approached by other authors
using different techniques, for example the inflection point
method of \cite{1997SoPh..171..303S}, the cumulative histogram method of
\cite{1997SoPh..175..197P}, the fuzzy logic approach of
\cite{2009SoPh..260...21F}, and the morphological approach of
\cite{2005SoPh..228..377Z}. Our method begins with the sunspots (which includes
umbrae and penumbrae) detected by STARA, and then produces a histogram of
sunspot pixel intensities for each spot. This clusters in two peaks, the local
minimum between which corresponds to the intensity value at the edge of the
umbra. A similar histogram-based approach was implemented by
\cite{2009SoPh..260...21F} who then used concepts from fuzzy logic to assign
membership to umbra or penumbra; they showed that particularly the pixel
membership of the penumbra can vary significantly (tens of percent) depending on
a parameter known as the membership function, but this is apparently less of a
problem for low-resolution data, in which brightness variations within the
penumbra are smeared out. We have not adopted such a method, but have instead
identified the local minimum for each sunspot's histogram, and created a mask
for umbral pixels. We normally find that the umbra region of sunspots has an MDI pixel value of less than 7000 - 8000. However, our algorithm does have the benefit of being applied
consistently across the entire data series, and being able to deal with the
varying intensity across the solar disk due to limb darkening which eases the
problems of sunspot detection and area estimation that occur if a
straightforward intensity threshold is used.
The data used in this study are taken from the MDI instrument
\citep{Scherrer1995} on the SOHO spacecraft. We use the level 1.8 continuum data
as well as the level 1.8 magnetograms to analyse magnetic fields present in the
spots. Our dataset uses 15 years of data and we analyse daily measurements taken
at 0000UT when co-temporal continuum images and magnetograms are recorded. The
STARA code takes around 24 hours to process the approximately 5000 days of data
available to generate the sunspot catalogue used in this article and holds
30\,084 separate sunspot detections. The same sunspot will be detected in many
different images and tracked from image to image allowing them to be associated
with one another. The physical parameters obtained from this analysis are the
sunspot total area and `centre of mass' location, number and area of umbrae;
mean, maximum and minimum magnetic fields in the umbrae and penumbra; total and
excess flux in the umbrae and penumbra and the information relating to the
observation itself such as time and instrument used.
\subsection{Number of sunspots}
The trend of sunspot number throughout a solar cycle is well documented and
generally rises rapidly at the start of a solar cycle before a slower decrease
towards the end of the cycle. The Solar Influences Data Center (SIDC,
\url{http://www.sidc.be/sunspot-data/}) keeps records on the sunspot index and
so we compare
the results of our detections with the findings of the SIDC as an initial test.
It must be noted that both indicators are not measuring the same thing as the
international sunspot number recorded by the SIDC weights the sunspots seen in
groups so that it becomes a stronger proxy for solar activity whereas STARA only
gives us the raw number of observed sunspots. However, it is beneficial to see
if the same trends are present. The data used here are the smoothed monthly
sunspot number \citep{SIDC} and so our daily measurements have been treated in
the same way to give a fair comparison.
In Fig.~\ref{fig:emergences} we can see that both curves share several features.
The STARA output has been scaled up to the same level as the International
Sunspot Number around 2001 - 2003 when sunspot count rates were higher and the
general trends are more important here than absolute values due to the
differences in counting methods (this scaling is permissible due to the somewhat
arbitrary factors present in the SIDC sunspot numbers - see
Equation~\ref{eq:1}.) We see that both datasets exhibit the same patterns of
increasing and decreasing at the same time and the agreement is very good in the
declining phase of the cycle. This also continues into cycle 24 shown at the
right hand side of the plot with both curves rising at the same time and we will
continue to track the agreement of these further into the next cycle.
The SIDC data \citep{Clette2007}, shown as a dashed line on the plot has a
smooth rise up to the first maximum sometime in the year 2000 and falls before
reaching a second maximum in 2002. This `double maximum' feature, separated by
the `Gnevyshev gap' \citep{1967SoPh....1..107G} is also seen in the STARA output
although the first maximum is weaker when compared to the second, in contrast
with the SIDC data in which the first maximum is larger than the second.
However, both sets of data scale well with one another after this second maximum
with very little deviation and this continues from 2002 up to the current day.
The differences in the first peak, and indeed in the rise before that are most
likely due to the method of counting sunspots as mentioned previously. In fact,
the SIDC sunspot number is calculated using the formula
\begin{equation}\label{eq:1}
T = k ( 10g + s )
\end{equation}
where $T$ is the total sunspot number for that measurement, $g$ is the number of
sunspot groups observed and $s$ is the number of individual sunspots observed.
It is based on the assumption that sunspot groups have an average of 10 sunspots
in them and so even in poor observing conditions, this would be a good
substitute. The coefficient $k$ is a number that represents the seeing
conditions from the observing site and is usually less than 1.
What Fig.~\ref{fig:emergences} suggests is that the SIDC observers are either
detecting more sunspots than STARA in the first half of the cycle, or that they
are detecting groups that have fewer than 10 sunspots in them, on average. This
second explanation is more likely. Inspecting the STARA data we find it is rare
to see a sunspot group with as many as ten spots in this stage of the cycle,
which would account for the SIDC number being an overestimate for the actual
sunspot number at this time. This in itself has interesting implications for the
solar cycle, suggesting that very complex magnetic groups - and the heightened
activity that accompanies them - are more likely to appear in the second part of
the overall solar maximum.
\subsection{Sunspot locations}
The locations of sunspots were also recorded by the STARA code and this allows
us to produce a butterfly diagram of sunspot locations. The `butterfly' shape is
produced by the pattern of sunspot emergences seen in each cycle. At the start
of a cycle sunspots tend to appear at high latitudes, between 20 and 40 degrees
above and below the solar equator. But as the cycle progresses, the spot
emergences are observed closer to the equator. The cycle then ends before the
sunspots are seen to emerge at the equator and as a result of this it is very
rare to see a sunspot forming within a few degrees of the solar equator.
\citet{Zharkov2007} have observed a `standard' butterfly pattern in sunspot
emergences in cycle 23 and our results are shown in Fig.~\ref{fig:butterfly}
\begin{figure}[ht]
\centerline{\includegraphics[width=0.45\textwidth,clip=]{butterfly_diag_final.eps}}
\caption{The latitude of all 30\,084 sunspot detections from solar cycle 23. The
end of solar cycle 22 can be seen as well as the onset of cycle 24. Note that
there is a much larger `gap' between cycle 23 and 24 than between cycles 22 and
23. This confirms the lack of solar activity from mid 2008 to early 2010.}
\label{fig:butterfly}
\end{figure}
The butterfly shape can be clearly seen as can some other features. There are
gaps in 1998 as the SOHO spacecraft was lost for some time and no data were
recorded. Also, the vertical line in early 1999 corresponds to the failure of
the final gyroscope onboard and a rescue using gyroless control software. This
caused the spacecraft to roll and so all data recorded at this time does not
have a consistent sun orientation. These artifacts have been left in the figure
(although corrected for in our subsequent analysis) to illustrate some of the
potential problems with using long term data sets.
To enable the continuation of the mission the spacecraft is rotated
approximately every three months to allow the high gain antenna to point at the
Earth as it can no longer be moved. This means that the data are rotated and
this introduces further small errors in position detection as the roll angle is
not known exactly but the algorithm assumes that the data is either 'north up'
or 'south up'.
We can see from Fig.~\ref{fig:butterfly} that the end of solar cycle 23
exhibited asymmetric behaviour with very few spots appearing on the north
hemisphere compared to the south. \citet{Hathaway2010} shows that a north-south
asymmetry in sunspot area during a cycle is very common but he also states that
any systematic trend in the asymmetry during a solar cycle is found to change in
the next cycle and so is not particularly useful for predictions of activity or
for solar dynamo modelling. This asymmetry was studied in more detail by
\citet{Carbonell1993} using a variety of statistical methods and they found that
a random component was dominant in determining the trend of hemispheric
asymmetry in sunspots.
\subsection{Sunspot areas}
As was the case with the number of sunspots detected, the area of the largest
visible sunspot also follows the activity of the solar cycle with a clear rising
phase and a slower declining phase. When calculating the area of a
sunspot or umbra the number of pixels within the spot or umbral boundary is
corrected to take into account the geometrical foreshortening effects that
change the observed area relative to its position on the solar disk. We show
this in Fig.~\ref{fig:areas}. The variation is larger as sunspot sizes have a
larger range than the number of spots that are present. Again, this has been
smoothed to give a fair comparison to the international sunspot number
calculated by the SIDC. An interesting feature of this plot is that at the start
of cycle 24 there is no significant increase in the areas of observed spots so
we can say that there are more spots beginning to appear but the spot magnetic
fields are still weak.
\begin{figure}[ht]
\centerline{\includegraphics[width=0.45\textwidth,clip=]{spot_area_evolution_w_SIDC_num.eps}}
\caption{The area of the largest sunspot observed is shown here, smoothed over 3
months to minimise the effect of very large sunspots and days where no spots
were visible. This roughly follows the international sunspot number as well as
the activity seen throughout solar cycle 23.}
\label{fig:areas}
\end{figure}
\begin{figure}[ht]
\centering
\begin{tabular}{ l
}\includegraphics[width=0.45\textwidth,clip=]{total_areas_w_error.eps} \\
\includegraphics[width=0.45\textwidth,clip=]{area_ratio_w_error.eps}
\end{tabular}
\caption{top panel : The upper line shows the total observed sunspot area and
the lower line shows the total umbra area smoothed over three month periods and
corrected for foreshortening effects. Only sunspots within 60$^\circ$ of the
centre of the disk were used to minimise errors from this correction. bottom
panel : the ratio of total umbral area to total sunspot area. This ratio is
fairly constant, with the umbral area consiting of 30 - 40\% of the total
sunspot area and does not vary rapidly throughout the cycle. The errors are
shown by the shaded area and are lower between 1999 and 2005 due to the
increased number of sunspots at that time.}
\label{fig:ratio_areas}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{total_deproj_area_w_error.eps}
\caption{We show the total sunspot (solid line) and umbra (crosses) area here as
a percentage of the area of the projected solar disk. The data are smoothed over
a three month period.}
\label{fig:deproj_areas}
\end{figure}
In addition to looking at the largest sunspot areas observed, we are also able
to examine the total area of the solar surface covered by sunspots at any one
time. This is shown in Fig.~\ref{fig:ratio_areas}. Both the total sunspot and
umbral areas are shown and, yet again, they both follow the overall trend of the
solar cycle with increases and decreases at the same times. More interesting
than this however, is the ratio of umbral area to sunspot area, shown in the
bottom panel. We observe that the umbral area is 20-40\% of the total observed
sunspot area and the ratio stays within this range throughout the cycle. Even
though a large variety of sunspot shapes and configurations are seen, the
fractional area of associated umbra does not show high amplitude fluctuations
unlike the maximum sunspot area observed - the dominant characteristic is a
relatively smooth variation. Note that this does not hold for individual
sunspots due to the variety of configurations seen, only to the large scale
distribution of sunspots over time. There are also interesting features present,
most of all the dip in the year 1999. At this time, the sunspot area is
increasing more quickly than the area of the associated umbrae. This soon
changes and the umbral areas start to occupy more of the sunspot again, rising
by a few percent by 2004 before starting to drop off again. During the first
peak in solar activity in 2000 we see that the umbra is occupying a lower
fraction of the sunspot and from Fig.~\ref{fig:emergences} this is when the
International Sunspot Number was higher than the STARA sunspot count. This could
indicate that there are sunspot groups with lower than ten sunspots present in
them. This suggests that there is more space in these groups for the sunspot
penumbrae to grow. In comparison to this, in the second peak of activity in 2002
we see that the fraction of sunspot area occupied by umbrae has grown and that
the STARA count rate is above the International Sunspot Number. This suggests
that we are seeing sunspot groups with more than ten spots in them. These would
be very complex groups and so it may be the case that the sunspots have multiple
umbrae present within them which would likely increase the fractional umbral
area.
In Fig.~\ref{fig:areas} and Fig.~\ref{fig:ratio_areas} we show the error in the
areas measured as a shaded band surrounding the line representing the data
points. Estimating the errors involved is done by examining the output of the
STARA algorithm. When detecting sunspots and sunspot umbrae, the centroid of the
region is determined with good accuracy. However, when defining the perimeter of
the region, we believe that there is an error of 1 pixel both towards and away
from the centre of the region. This means that large sunspots will have a
smaller fractional error than small spots, even though the absolute value of the
error will be greater for large spots.
We also show the percentage of the projected solar disk covered by sunspots from
the viewpoint of the SOHO spacecraft in Fig.~\ref{fig:deproj_areas}. The trend
is very similar to that of the absolute total area of sunspots looked at
previously. We see the fraction of the solar disk covered by sunspots rise to
about 0.35\% at the peak of activity in cycle 23 which is equivalent to 3500 MSH
(millionths of a solar hemisphere). This is comparable to some of the largest
sunspots ever detected. There are significant short-term fluctuations in this
series, in addition to the overall solar cycle variation.
\section{The evolution of sunspot magnetic fields}
As the detection algorithm is directly linked with the MDI magnetograms recorded
at the same time, we are also able to track the evolution of the magnetic field
present in sunspots throughout the cycle. We assume that the magnetic field
within sunspot umbrae is in the local vertical direction. As the MDI data only
gives the line of sight magnetic field we apply a cosine correction to account
for this. The amplification of magnetic field strength due to the cosine
correction becomes very large as sunspots approach the limb, so making an
incorrect assumption about the field being vertical can lead to vastly wrong B
values at the limb. To minimise these effects we only include sunspots with a
value of $\mu > 0.95$ where $\mu$ is the cosine of the angle between the local
solar vertical and the observers line of sight. In addition to this, the
observed line of sight field is corrected with the assumption that the true
field direction is
perpendicular to the local photosphere. As we are looking at the strongest
fields in sunspot umbrae this is a reasonable approximation.
Fig.~\ref{fig:rawspots} shows the maximum sunspot umbral fields measured daily
from 1996 - 2010. The first thing to notice is the spread of magnetic fields
measured. We also see that the majority of measurements fall between 1500 and
3500 Gauss. It is very difficult to see any kind of trend in the data due to the
spread of values but we can observe a lack of strong sunspots from 2008 - 2010
when the most recent solar minimum occured.
A similar study has been undertaken by \citet{Penn2006} using the McMath-Pierce
telescope on Kitt Peak which includes umbra measurements going further back, to
1991. The method is different as they use the Zeeman splitting of the Fe I line
(1564.8nm) to infer a magnetic field strength at the location of the
measurement. Measurements are made in the darkest part of the umbra, where this
is identified in the image using a brightness meter. The Zeeman splitting
identified at that location is used to determine the true magnetic field as the
splitting of the spectral line observed is not dependent on the angle between
the magnetic field and the observers line of sight. Very small spots were
excluded from their dataset, as the small size of the umbra increases the risk
of scattering of penumbral radiation into the umbral area, and consequent
distortion of the line profile. Pore fields correspond to the range 1600-2600~G,
with a mean of 2100~G. When this dataset of maximum measured umbral field is
binned and averaged by year, and plotted as a function of time, a decrease is
visible which can be fitted with a linear trend equivalent to around -52 Gauss
per year. We repeat the analysis carried out by \cite{Penn2006} on our dataset,
both including and excluding all spots with a vertical magnetic field component
below 1500~G to minimise the possible effects of pores being included in the
analysis, for a direct comparison with the \cite{Penn2006} result. The results
are shown in Fig.~\ref{fig:spottrend}.
\begin{figure}[ht]
\centerline{\includegraphics[width=0.45\textwidth]{raw_spot_data.eps}}
\caption{Maximum sunspot umbra field from 1996-2010. Measurements are taken
daily.}
\label{fig:rawspots}
\end{figure}
\begin{figure}[ht]
\centering
\begin{tabular}{ l
}\includegraphics[width=0.45\textwidth]{mean_stdev_all_spots.eps} \\
\includegraphics[width=0.45\textwidth]{mean_stdev_penn_spots.eps}
\end{tabular}
\caption{The data shown in Fig.~\ref{fig:rawspots} have been binned by year and
the mean of each bin is plotted here. Top panel: all data from
Fig.~\ref{fig:rawspots} are included. Bottom panel: only measurements with a
field above 1500 Gauss are included. The error bars correspond to the standard
error on the mean. The solid line shows the evolution of the international
sunspot number over the same period for reference. Assuming a linear trend gives
a gradient of -23.6 $\pm $ 3.9 Gauss per year and -22.3 $\pm$ 3.9 Gauss per year
respectively.}
\label{fig:spottrend}
\end{figure}
The top panel includes all sunspots detected whereas the bottom panel excludes
any spots with a maximum field strength of less than 1500 Gauss. The error bars
are calculated as the standard error on the mean of all measurements in the bin.
The data are in line with a picture in which the umbral fields are simply
following a cyclical variation pattern, as the increases and decreases follow
the international sunspot number. This cannot be confirmed with the current data
and we will need to wait until the next cycle is well under way to see if the
trends continues to be present. If we do a straight line fit as in
\citet{Penn2006}, then the gradient of the best fitting line gives a decrease in
umbral fields of 23.6 $\pm$ 3.9 Gauss per year which, although still decreasing,
is a far slower decline than seen by Penn and Livingston. Repeating the analysis
excluding sunspots with fields below 1500 Gauss gives a long term decrease in
field strength of 22.4 $\pm$ 3.9 Gauss per year. This is even further from the
result they observed, although as the sunspots with fields below 1500 Gauss make
up such a small fraction of the population we observe, we would not expect a
significant change in the result. Other studies have also cast doubt on the long
term decrease of umbral magnetic fields. The \citet{Penn2006} article suggests
that a decrease of 600 Gauss over a solar cycle would cause a change in mean
umbral radius as a relationship between these two quantities has been shown by
\citet{Kopp1992} and \citet{Schad2010} but follow up observations by
\citet{Penn2007} could not see this in their data. It has also been suggested by
\citet{Mathew2007} that a small sunspot sample may introduce a bias into results
if the size distribution of sunspots used is not calculated in advance.
However, the long term decline in sunspot magnetic fields does agree with the
lack of an increase in sunspot area as shown in Fig.~\ref{fig:areas}. If the
magnetic field is now weaker than at the same time in the last cycle we would
expect sunspots to be smaller and this is currently what is observed.
Interestingly, if the data from only the declining phase of the cycle (from 2000
to 2010) are used, then the maximum umbral field strengths are seen to decrease
by around 70 Gauss per year which is far greater than the \citet{Penn2006}
study.
This then leads to the question of how valid this comparison is. In
fact, instruments such as MDI and the new Helioseismic and Magnetic Imager on
SDO do not measure the true value of magnetic field strength in a pixel. The
value they return is an average magnetic field strength with a resolution
determined by pixel size. However, if the filling factor of spatially unresolved
magnetic elements within the pixel is close to unity, then the pixel value is a
good approximation for the true line of sight magnetic field strength. This is
thought to be the case deep in the umbrae of strong sunspots and so for these
measurements we can say that our observations are good approximations for the
true line of sight magnetic fields. In addition to this, we have only used
sunspots with $\mu > 0.95$ which corresponds to 18.2 degrees from solar disk
centre in an effort to minimise any corrections to the magnetic field
measurements but still assume that the field in the core of sunspot umbrae is
perpendicular to the local photosphere.
Also, MDI has problems with saturation in magnetic field measurements
with a peak value of between 3000 and 3500 Gauss depending on when the observation was made (the saturation value has lowered as the instrument degrades). This
has a greater effect on measurements made at solar maximum and so has the effect
of reducing the long term field strength decrease. However, this does not fully
account for the discrepancy between our value of the rate of long term field decrease and that of other studies.
We have not only compared the trends seen but also the data points used
in calculating these trends. The latest Livingston and Penn data is kept up to
date by Leif Svalgaard and can be viewed at his own website
(see \url{www.leif.org/research}). With the exception of a single data point in
1994, the Livingston and Penn yearly averages are similar to ours. Sadly, there are no
yearly averages in the Livingston and Penn data between 1994 and 2001 to better
compare the two studies.
\section{Discussion and Conclusions}
Using a catalogue of sunspot detections created by the STARA code provides a
reliable way to analyse the long term variation of certain physical parameters
relating to sunspots. We found that the number of sunspots detected compared
very well with the international sunspot number, even through the period of
2008-2010 when sunspot detections have been more sparse and difficult due to the
decreased magnetic field strengths that are causing them. When looking at the
locations of sunspots a traditional butterfly pattern is seen which also shows
the end of cycle 22 as well as the period of almost no sunspots from late 2008
to early 2010 before cycle 24 started. Fig.~\ref{fig:butterfly} also shows some
of the problems of a long term observing run, such as spikes in early 1999
caused by failure of the gyroscopes onboard SOHO. In addition to this, the high
gain antenna on SOHO malfunctioned in mid 2003.
The area of sunspots was then examined with the maximum spot area being first
observed. The rough pattern of an initial steep rise and gradual fall associated
with a solar cycle was seen but with many other features present. However, when
the total observable sunspot area was plotted, a much smoother evolution was
seen. The same smooth evolution was also present in the total observable umbral
area. We also found that throughout the whole of solar cycle 23, if smoothed
over a three month period, the area of umbra visible was between 20 and 40\% of
the visible sunspot area once corrections for geometric foreshortening had been
applied.
We then continued to show the evolution of magnetic fields in sunspot umbrae and
Fig.~\ref{fig:rawspots} shows the large spread of sunspot magnetic fields
observed. Once the spot magnetic field data had been binned by year, a long term
cyclical trend could be observed but it is yet unknown whether this is a
cyclical variation around a long term linear decrease as suggested by other
studies. Our data supports
stronger fields near solar maximum and weaker fields at solar minimum. When
compared with other similar studies, the rate of magnetic field decrease is very
different and is likely due to the wide range of sunspot fields. The next solar
cycle should bring a more definitive answer to the question of whether a secular
trend in sunspot fields exists over multiple solar cycles. We will continue to
track this for as long as SOHO still flies and also plan to incorporate data
from the new Helioseismic and Magnetic Imager on the Solar Dynamics Observatory
spacecraft which serves as the successor to SOHO.
\begin{acknowledgements}
F.T.W. acknowledges the support of an STFC Ph. D. studentship. This work was
supported by the European Commission through the SOLAIRE Network
(MRTN-CT-2006-035484) and by STFC rolling grant STFC/F002941/1. SOHO is a
project of international cooperation between ESA and NASA. We acknowledge
useful discussions with M. Hendry and would like to thank our anonymous referee
for their thought provoking comments. Thanks also to Leif Svalgaard for allowing
us
to use his plots for comparing with Livingston and Penn data.
\end{acknowledgements}
\bibliographystyle{aa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,379 |
\section{Introduction}
There has been a lot of interest in quantum field theories on curved
spacetime background\cite{BiDa}. In general, quantum field theories
may be sensitive to both the local and global structure of spacetime.
The structure can alter the physical parameters in quantum loop
corrections, which can not be absorbed by simple redefinition of the
parameter after removing the ultraviolet divergences. A well-known
example is the finite temperature field theory, where the
renormalized parameters can become temperature dependent. This
phenomenon could be of particular interest for the models in which the
physical parameters get non-vanishing values through quantum loop
corrections\cite{Coleman}.
In this paper we will study the three dimensional nonlinear $O(N)$
sigma model\cite{Aref'eva} on $S^2\times R^1$ Euclidean spacetime by
evaluating the effective potential in the large-$N$
limit\cite{Coleman,Song}. In the model on $R^3$, there exist a
critical coupling constant $g_c$ which separates the strongly coupled
case (strong-coupled regime) and weakly coupled case (weak-coupled
regime), while dynamical mass generation takes place only in the
strongly coupled case \cite{Aref'eva,Song}. Due to $S^2$, the
spacetime should have nonvanishing scalar curvature\footnote{ The
author thanks Professor P. Gilkey and Dr. J.H. Park for discussions on
this point.}. For the sake of explicit evaluations, we will restrict
our discussions on the case of the canonical metric for $S^2$ and the
flat metric on $R^1$ which gives the constant scalar curvature,
$R=2/\rho^2$, with the radius $\rho$ of $S^2$. In the presence of
scalar curvature, we may add a gravitational interaction term
$\xi R n^i n_i$ $(=\xi R n^2)$ to Lagrangian, where $\xi$ is a
gravitational coupling constant and $n^i$ $(i=1,2,\cdots,N)$ is a
boson field. Taking the above arguments into consideration, the
Lagrangian of the nonlinear $O(N)$ sigma model on $S^2\times R^1$ may
be written as:
\begin{eqnarray}
L=\int d\theta d\varphi dz\rho^2\sin\theta
[&&\frac{1}{\rho^2} \partial_\theta n^i\partial_\theta n^i
+\frac{1}{\rho^2\sin^2\theta}
\partial_\varphi n^i\partial_\varphi n_i \nonumber\\
&&+\partial_z n^i\partial_z n_i +\xi Rn^2
+\sigma(n^2-\frac{N}{g_0^2})],
\end{eqnarray}
where $\sigma$ is a Lagrange multiplier to coerce the constraint
$n^2=N/g_0^2$ and $g_0$ is the bare coupling constant of the model.
In the literatures\cite{BiDa} special emphases have been made on the
conformally coupled cases: $\xi=\frac{d-2}{4(d-1)}$ on $d$-dimensional
spacetime or $\xi=\frac{1}{8}$ on three dimension. On $S^d$ or
$S^{d-n}\times R^n$ there have been some analyses for other
models\cite{DruSh,RT&KM}, and the $\zeta$-function method has been
mostly used in isolating divergences of quantum loop
correction\cite{BiDa,DruSh,RT&KM,DS,O'Connor&Hu&Shen}, while we will
use the cut-off method.
In the next section we will evaluate the effective potential, $V$, on
$S^2\times R^1$ in the leading order of $N$. For $\xi\geq
\frac{1}{8}$, the renormalized effective potential will be explicitly
found. In Sec.III, we will study whether the dynamical mass generation
takes place by investigating the stationary point in $V$. It will be
shown that for $\xi=\frac{1}{8}$ dynamical mass generation will take
place for any value of $\rho$ if $g$ is larger than $g_c$, as in the
model on $R^3$. For $\xi>\frac{1}{8}$, there exists a critical
curvature $R_c$ $(=2/\rho_c^2)$; For $R\geq R_c$, dynamical mass
generation does not occur even in the strongly coupled case. For
$\xi>\frac{1}{8}$, we found the analytic expression of $R_c$, which
shows that $R_c$ decreases monotonically as $\xi$ increases. The final
section will be devoted on discussions.
\section{Effective potential}
In the leading order of the $1/N$ expansion, the effective potential
for constant $\sigma$ is given by the tree and one-loop diagrams with
external $\sigma$ lines\cite{Coleman&Weinberg,Coleman}. Or, in order
to make use of functional integral\cite{Jackiw}, one can write the
Lagrangain density with the quantum mechanical angular momentum
operator ${\bf L}$ as follows;
\begin{equation}
{\cal L}=n^i Dn_i-N\sigma/g_0^2,
\end{equation}
where
\begin{equation}
D=-\partial_z^2 +\frac{{\bf L}^2}{\rho^2} +\frac{2\xi}{\rho^2}
+\sigma.
\end{equation}
By performing the Gaussian functional integral, one can find the
effective potential per unit volume as
\begin{equation}
\frac{V_0}{N}=-\frac{\sigma}{g_0^2}
+\frac{1}{4\pi \rho^2}(\text{Tr}\ln D+C).
\end{equation}
$C$ is a constant which may arise from the functional integral and it
will be fixed by demanding that $V_0\mid_{\sigma=0}=0$. As is
well-known, the diagrams which have $\sigma$ propagators as internal
lines give contributions of next to the leading order in the $1/N$
expansion. Therefore, the effective potential of the leading order per
unit volume can be written as;
\begin{equation}
\frac{V}{N}=-\frac{\sigma}{g_0}+\frac{1}{4\pi\rho^2}\sum_{l=0}^{I}
\int_{|k|<\Lambda_1} \frac{dk}{2\pi}(2l+1)
\ln(1+\frac{\sigma}{k^2+\frac{l(l+1)}{\rho^2}+\frac{2\xi}{\rho^2}
+i\epsilon}).
\end{equation}
In Eq.(5), we introduce the cut-off $\Lambda_1$ and $I$. While the
mass dimension of $\Lambda_1$ is that of a momentum as usual, the $I$
is a pure number since it is a cut-off for the quantum number of the
operator ${\bf L}$. If sapcetime is a product of two manifolds with
different topologies, it is necessary to introduce different cut-offs.
Although this should happen in finite-temperature field theories, this
could be implicit if summation or integral along some directions are
finite.
In order to compare the effective potential in Eq.(5) with that on
$R^3$, one can use the following formula:
\begin{equation}
\sum_{l=0}^N f(l)=\frac{1}{2}f(0)+\int_0^{N+1}f(x)dx
+\sum_{l=0}^N\int_0^1 f'(x+l)(x-\frac{1}{2})dx
-\frac{1}{2}f(N+1),
\end{equation}
which comes from the Euler-Maclaurin formula
\begin{equation}
\int_0^1 f(x+l) dx =\frac{1}{2}(f(l+1)+f(l))-\int_0^1 f'(x+l)
(x-\frac{1}{2})dx
\end{equation}
for any differentiable function $f(x)$.
Making use of the equality in Eq.(6), the effective potential can be
written as
\begin{eqnarray}
\frac{V}{N}&=&-\frac{\sigma}{g_0^2}
+\frac{1}{8\pi^2\rho^2}\int_{|k|<\Lambda_1}
\int_{x=0}^{I} dx (2x+1)
\ln(1+\frac{\rho^2\sigma}{(x+\frac{1}{2})^2+2(\xi-\frac{1}{8})
+\rho^2k^2+i\epsilon})\nonumber\\
&&+\frac{1}{16\pi^2 \rho^2} \int_{-\infty}^{\infty} dk
\ln(1+\frac{\rho^2\sigma}{\rho^2 k^2 +2\xi +i\epsilon})\nonumber\\
&&+\frac{1}{8\pi^2}\int_{-\infty}^{\infty}dk\sum_{l=0}^\infty
\int_0^1 dx (x-\frac{1}{2})
\left[
\begin{array}{l}
\frac{2}{R^2}\ln (1+\frac{\rho^2\sigma}{(x+\frac{1}{2}+l)^2
+2(\xi-\frac{1}{8})+\rho^2k^2+i\epsilon})\\
-\frac{4\sigma(x+\frac{1}{2}+l)^2}
{[(x+\frac{1}{2}+l)^2+2(\xi-\frac{1}{8})+\rho^2k^2+i\epsilon]
[(x+\frac{1}{2}+l)^2+2(\xi-\frac{1}{8})+\rho^2k^2
+\rho^2\sigma+i\epsilon] }
\end{array}\right]\nonumber\\
&&+O(1/\Lambda_1)+O(1/I)
\end{eqnarray}
It is convenient to divide $V$ into two pieces so that
\begin{equation}
V=V_\xi+V_\Delta+O(1/\Lambda_1)+O(1/I),
\end{equation}
where
\begin{equation}
\frac{V_\xi}{N}=-\frac{\sigma}{g_0^2}
+\frac{1}{8\pi^3}\int_{|k|<\Lambda_1}dk
\int_{k_2=0}^{\Lambda_2} dk_2~2\pi k_2
\ln(1+\frac{\sigma}{k^2+k_2^2+\frac{2(\xi-\frac{1}{8})}{\rho^2}
+i\epsilon})
\end{equation}
and
\begin{eqnarray}
\frac{V_\Delta}{N}&=&
\frac{1}{16\pi^2 \rho^2} \int_{-\infty}^{\infty} dk
\ln(1+\frac{\sigma}{k^2 +\frac{2\xi}{\rho^2}+i\epsilon})
\nonumber\\
&&-\frac{1}{4\pi^2\rho^2}\int_0^\frac{1}{2} dt~t
\int_{-\infty}^\infty dk
\ln(1+\frac{\sigma}{k^2+(\frac{t}{\rho})^2
+\frac{2}{\rho^2}(\xi-\frac{1}{8})+i\epsilon})\nonumber\\
&&+\frac{1}{8\pi^2}\sum_{l=0}^\infty\int_{-\frac{1}{2}}^{\frac{1}{2}}
dt~t \int_{-\infty}^{\infty}dk
\left[
\begin{array}{l}
\frac{2}{R^2}\ln (1+\frac{\sigma}{k^2+\frac{(t+l)^2}{\rho^2}
+\frac{2(\xi-\frac{1}{8})}{\rho^2}+i\epsilon})\\
-\frac{4\sigma(t+l)^2/\rho^4}
{[k^2+\frac{(t+l)^2}{\rho^2}
+\frac{2(\xi-\frac{1}{8})}{\rho^2}+i\epsilon]
[k^2+\frac{(t+l)^2}{\rho^2}+\sigma
+\frac{2(\xi-\frac{1}{8})}{\rho^2}+i\epsilon] }
\end{array}\right].
\end{eqnarray}
In Eq.(10), the $\Lambda_2$ whose dimension is of momentum is
defined as
\[\Lambda_2=\frac{I+\frac{1}{2}}{\rho}.\]
{}From now on we will restrict our attention on the case
$\xi\geq\frac{1}{8}$, for which logarithm functions or their integrals
in Eqs.(8,10,11) are well defined even when $\epsilon=0$. For
discussions on the case of $\xi<\frac{1}{8}$, we should develop some
analytic continuations for logarithm functions, which is beyond the
scope of this paper.
Making use of the formulae\cite{Rtable}
\begin{equation}
\int_{-\infty}^\infty \ln\frac{\alpha^2+x^2}{\beta^2+x^2}dx
=2(|\alpha|-|\beta|)\pi,
\end{equation}
\begin{equation}
\int_{-\infty}^\infty \frac{dx}{(\gamma+x^2)(\delta+x^2)}
=\frac{\pi}{\sqrt{\gamma\delta}(\sqrt{\gamma}+\sqrt{\delta})}
=\frac{\pi}{\sqrt{\gamma\delta}}\frac{\sqrt{\delta}-\sqrt{\gamma}}
{\delta-\gamma},
\end{equation}
we can find a simpler form of $V_\Delta$;
\begin{eqnarray}
V_\Delta
&=&-\frac{1}{2\pi \rho^2}\int_0^{\frac{1}{2}}dt~t
[\sqrt{(\frac{t}{\rho})^2+\frac{2(\xi-\frac{1}{8})}{\rho^2}+\sigma}
-\sqrt{(\frac{t}{\rho})^2+\frac{2(\xi-\frac{1}{8})}{\rho^2}}]
\nonumber\\
&&+\frac{1}{8\pi \rho^2}[\sqrt{\frac{2\xi}{\rho^2}+\sigma}
-\frac{\sqrt{2\xi}}{\rho}]
\nonumber\\
&&+\frac{1}{2\pi}\sum_{l=1}^\infty\int_{-\frac{1}{2}}^{\frac{1}{2}}
dt~ t\left[
\begin{array}{l}
\frac{1}{\rho^2}[\sqrt{\frac{(t+l)^2}{\rho^2}
+\frac{2(\xi-\frac{1}{8})}{\rho^2}+\sigma}
-\sqrt{\frac{(t+l)^2}{\rho^2}
+\frac{2(\xi-\frac{1}{8})}{\rho^2}} ] \\
-\frac{\sigma(t+l)^2}
{\rho\sqrt{[(t+l)^2+2(\xi-\frac{1}{8})+\rho^2\sigma]
[(t+l)^2+2(\xi-\frac{1}{8})]}}\\
\times\frac{1}{\sqrt{[(t+l)^2+2(\xi-\frac{1}{8})+\rho^2\sigma}
+\sqrt{[(t+l)^2+2(\xi-\frac{1}{8})}}
\end{array}\right] \\
&=&-\frac{1}{2\pi \rho^3}\int_0^{\frac{1}{2}}dt~t
[\sqrt{t^2+2(\xi-\frac{1}{8})+\rho^2\sigma}
-\sqrt{t^2+2(\xi-\frac{1}{8})}]
\nonumber\\
&&+\frac{1}{8\pi \rho^3}[\sqrt{2\xi+\rho^2\sigma}-\sqrt{2\xi}]
\nonumber\\
&&+\frac{1}{2\pi \rho^3}\sum_{l=1}^\infty
\int_{-\frac{1}{2}}^{\frac{1}{2}}dt~ t\left[
\begin{array}{l}
[\sqrt{(t+l)^2+2(\xi-\frac{1}{8})+\rho^2\sigma}
-\sqrt{(t+l)^2+2(\xi-\frac{1}{8})}]\\
\times[1-
\frac{(t+l)^2}{\sqrt{(t+l)^2+2(\xi-\frac{1}{8})+\rho^2\sigma}
\sqrt{(t+l)^2+2(\xi-\frac{1}{8})}}]
\end{array}\right],
\end{eqnarray}
which clearly shows that $V_\Delta$ is finite for any $\rho$,
$\sigma$. Furthermore $V_\Delta$ is zero in the limit $\rho$
approaches to infinity (in the $R^3$ limit), which shows that it is an
effect of topology. In this limit, the integral of Eq.(10) for $V_\xi$
which diverges with infinite cut-offs, in fact, corresponds to that of
the model on $R^3$:
\[
\frac{1}{8\pi^3}
{\int dk \int dk_2}_{k_2>0,k^2+k_2^2 < \Lambda^2}~2\pi k_2
\ln(1+\frac{\sigma}{k^2+k_2^2+i\epsilon}).
\]
The difference in the $R^3$-limit is that we have two cut-offs,
because our spacetime $S^2\times R^1$ is the product of two spaces
with different topologies. As mentioned earlier this should happen,
for example, in finite-temperature field theory, if we use cut-off
regularization. In renormalizing the model on $S^2\times R^1$ we will
assume - as usual - that we could deform the momentum space of
integration.
That is, the $V_\xi$ which does not vanish in the $R^3$-limit could be
written as:
\begin{eqnarray}
\frac{V_\xi}{N}&=&-\frac{\sigma}{g_0^2}
+\frac{1}{8\pi^3}
{\int dk \int dk_2}_{k_2>0,k^2+k_2^2 < \Lambda^2}~2\pi k_2
\ln(1+\frac{\sigma}{k^2+k_2^2+\frac{2(\xi-\frac{1}{8})}{\rho^2}
+i\epsilon}) \\
&=&-\frac{\sigma}{g_0^2} +\frac{\sigma\Lambda}{2\pi^2}-\frac{1}{6\pi}
\sqrt{[\frac{2(\xi-\frac{1}{8})}{\rho^2}+\sigma]^3}
+\frac{1}{3\pi \rho^3}\sqrt{2(\xi-\frac{1}{8})^3}\nonumber\\
&&+O(1/\Lambda),
\end{eqnarray}
which shows that the divergence of the potential on $S^2\times R^1$
(or $\Lambda$ dependence) is the same as that on $R^3$ and we could
renormalize the model on $S^2\times R^1$ by using the relation on
$R^3$\cite{Song}:
\begin{eqnarray}
\frac{1}{g_0^2}&=&\frac{1}{g^2}+\int_{|p|<\Lambda}
\frac{d^3p}{(2\pi)^3}\frac{1}{p^2+M^2}\\
&=&\frac{1}{g^2}+ \frac{\Lambda}{2\pi}-\frac{M}{4\pi}+O(1/\Lambda),
\end{eqnarray}
where $M$ is the renormalization mass.
The fact that topological change of spacetime does not give rise to
new counterterms in the renormalization has been found explicitly for
some cases (for example, see Ref. \cite{DruSh}) and a more general
discussion has been given in Ref.\cite{Banach}.
Now we have the renormalized effective potential for $\xi\geq
\frac{1}{8}$
\begin{equation}
\frac{V}{N}=-\frac{\sigma}{g^2}+\frac{\sigma M}{4\pi}-\frac{1}{6\pi}[
\sqrt{[\frac{2(\xi-\frac{1}{8})}{\rho^2}+\sigma]^3}
+\frac{1}{3\pi \rho^3}\sqrt{2(\xi-\frac{1}{8})^3}]
+\frac{V_\Delta}{N}+O(1/\Lambda),
\end{equation}
which, in the $R^3$-limit, reduces to the potential on $R^3$, $V_0$
\cite{Song}:
\begin{equation}
\frac{V_0}{N}=-\frac{\sigma}{g^2}+\frac{\sigma M}{4\pi}
-\frac{\sigma^{3/2}}{6\pi} +O(1/\Lambda).
\end{equation}
As a preliminary of the next section, it may be good to recapitulate
the dynamical properties of the model on $R^3$. The first derivative
of $V_0$ with respect to $\sigma$
\[\frac{1}{N} \frac{\partial V_0}{\partial \sigma}=
\frac{M}{4\pi}-\frac{1}{g^2}-\frac{\sqrt{\sigma}}{4\pi}+O(1/\Lambda)
\]
shows that there exist a global stationary point only when $g^2 >
\frac{4\pi}{M}(=g_c^2)$. That is, dynamical mass generation takes
place only in the strongly coupled case ($g^2 > g_c^2$), and $g_c$
is the critical coupling constant. The dynamically generated mass is
$\sqrt{\sigma_0}$ $(=M-\frac{4\pi}{g^2})$ which reduces to zero as
$g$ goes to $g_c$ \cite{GRV}.
\section{Phase Structure}
Though the forms of $V_0$ and $V$ differ by $V_\Delta$ which is rather
complicated, they share the facts that $V_0\mid_{\sigma=0}=
V\mid_{\sigma=0}=0$ and
\[ V\simeq V_0\simeq -\frac{\sigma^{3/2}}{6\pi}
~~~\text{for large } \sigma;\]
That is, they start from 0 and decrease to $-\infty$ as $\sigma$
increases to $\infty$. To see the shape of $V$, we evaluate the first
derivative of $V$ with respect to $\sigma$;
\begin{equation}
\frac{\rho}{N}\frac{\partial V}{\partial \sigma}=
(\frac{M}{4\pi}-\frac{1}{g^2})\rho - f(\xi;y),
\end{equation}
where
\begin{eqnarray}
f(\xi;y)
&=&\frac{1}{4\pi}\sqrt{2(\xi-\frac{1}{8})+y}
-\frac{1}{16\pi}\frac{1}{\sqrt{2\xi+y}}
+\frac{1}{4\pi}\int_0^{\frac{1}{2}}dt~t
\frac{1}{\sqrt{t^2+2(\xi-\frac{1}{8})+y}}\nonumber\\
&&-\frac{1}{4\pi}\sum_{l=1}^\infty
\int_{-\frac{1}{2}}^{\frac{1}{2}}dt~t
\left[
\begin{array}{l}
\frac{1}{\sqrt{(t+l)^2+2(\xi-\frac{1}{8})+y}}
[1-\frac{(t+l)^2}{\sqrt{(t+l)^2+2(\xi-\frac{1}{8})+y}
\sqrt{(t+l)^2+2(\xi-\frac{1}{8})} }] \\
+\frac{(t+l)^2}{\sqrt{[(t+l)^2+2(\xi-\frac{1}{8})+y]^3
[(t+l)^2+2(\xi-\frac{1}{8})]} } \\
\times[\sqrt{(t+l)^2+2(\xi-\frac{1}{8})+y}-\sqrt{
(t+l)^2+2(\xi-\frac{1}{8})}]
\end{array} \right] \nonumber\\
&=&\frac{1}{4\pi}\sqrt{2(\xi-\frac{1}{8})+y}
-\frac{1}{16\pi}\frac{1}{\sqrt{2\xi+y}}
+\frac{1}{4\pi}\int_0^{\frac{1}{2}}dt~t
\frac{1}{\sqrt{t^2+2(\xi-\frac{1}{8})+y}}\nonumber\\
&&-\frac{2(\xi-\frac{1}{8})+y}{4\pi}\sum_{l=1}^\infty
\int_{-\frac{1}{2}}^{\frac{1}{2}}dt~t
\frac{t}{[(t+l)^2+2(\xi-\frac{1}{8})+y]^{3/2}}
\end{eqnarray}
with $y=\rho^2\sigma$.
Making use of the formulae in the appendix, a further simplification
of $f$ is possible;
\begin{eqnarray}
f(\xi;y)=\frac{1}{4\pi}{\cal F}_{\frac{1}{2}}
(\sqrt{2(\xi-\frac{1}{8})+y})
&=&\frac{1}{4\pi}[{\cal F}(2\sqrt{2(\xi-\frac{1}{8})+y})
-{\cal F}(\sqrt{2(\xi-\frac{1}{8})+y})],
\end{eqnarray}
where for $\alpha >0$ ${\cal F}_{\frac{1}{2}}(\alpha)$ and
${\cal F}(\alpha)$ are defined by
\begin{eqnarray}
{\cal F}_{\frac{1}{2}}(\alpha) &=&
\sum_{l=0}^\infty[1-\frac{l+\frac{1}{2}}
{\sqrt{\alpha^2+(l+\frac{1}{2})^2}}],\nonumber\\
{\cal F}(\alpha)&=&\sum_{l=1}^\infty[1
-\frac{1}{\sqrt{1+(\frac{\alpha}{l})^2}}].
\end{eqnarray}
It is easy to find that ${\cal F}(0)=0$ and
${\cal F}_{\frac{1}{2}}(\alpha)$ monotonically increases as $\alpha$
increases, which proves that if there is a stationary point of $V$
then it must be global. Since the dynamical mass generation is denoted
by the presence of stationary point of the effective potential, the
above facts reveal two properties of the model on $S^2\times R^1$
for $\xi\geq \frac{1}{8}$. The first is that if dynamical mass
generation takes place it occurs only in the strong-coupled regime
($g^2 >g_c^2$), since $f(\xi;y)$ is never less than 0. Secondly, even
in the strong-coupled regime dynamical mass generation takes place
only if $\rho$ $(R)$ of 2-sphere is larger (smaller) than $\rho_c$
($R_c$).
Since $f(\xi;y)$ is monotonically increasing function of $y$ for a
fixed $\xi$, the critical condition is given by
\begin{equation}
\frac{\partial V}{\partial \sigma}\mid_{\sigma=0}=0,
\end{equation}
which gives the $\rho_c$ as;
\begin{equation}
\rho_c \sqrt{\sigma_0}={\cal F}_{\frac{1}{2}}(\sqrt{2\xi-\frac{1}{4}})
={\cal F}(2\sqrt{2\xi-\frac{1}{4}})
- {\cal F}(\sqrt{2\xi-\frac{1}{4}}).
\end{equation}
The smallest $\rho_c$ (largest $R_c$) which is in the the conformal
coupling case $(\xi=\frac{1}{8})$, is $0$ ($\infty$) since
${\cal F}(0)$ is zero. In other words, the dynamical mass generation
takes place for any size of sphere in the strong-coupled regime of
the conformal coupling case.
\section{Conclusion}
We have studied the large-$N$ nonlinear $O(N)$ sigma model on
$S^2\times R^1$ by evaluating the effective potential with cut-off
method. By analysing the (renormalized) effective potential for
$\xi\geq \frac{1}{8}$, we find that in the strongly coupled case there
exists critical size or curvature of 2-sphere. Even in the strongly
coupled case the dynamical mass generation does not take place when
the radius (curvature) of two sphere is smaller (larger) than the
critical one.
As the change of temperature may cause phase transition in
finite-temperature field theory, it has been known that the change of
curvature of background spacetime could give rise to phase transition
\cite{Ford&Toms,O'Connor&Hu&Shen}, while our results provide another
explicit example. The smallest critical radius (largest critical
curvature) for the model we considered is $0$ ($\infty$) in conformal
(gravitational) coupling case. Thus on $S^2\times R^1$ dynamical mass
generation of the model on $R^3$ always takes place in the conformal
coupling case.
The value of $\sigma$ at which the potential is stationary is the
square of the dynamically generated mass. Though we cannot
analytically find the value of dynamically generated mass, the fact
that $f(\xi;y)$ is a monotonically increasing function of $y$ suggests
that the phase transition would be of second order, since the mass
continuously approach zero as $\rho$ goes to $\rho_c$ from the above
($\rho>\rho_c$). For the conformal coupling case $(\xi=1/8)$,
$f(\frac{1}{8};y)$ is also a monotonically increasing function of
$y$. Therefore, at $\xi=1/8$ the dynamically generated mass
approaches 0 as $g$ goes to $g_c$, which is support the
ansatz made in Ref. \cite{GRV}.
It would be interesting to analyse the case $\xi<\frac{1}{8}$ which is
beyond the scope of this paper. It would be also of great interest to
analyse the model on different topologies. A similar analysis of the
model on $S^2$ will be published elsewhere\cite{Song2}.
\acknowledgments
The author thanks Professor L.H. Ford for fruitful discussion on his
works and Professor G. 't Hooft for discussion on treatment of
cut-offs. He also thanks Dr. W. Bietenholz for kind comments on the
first version of this paper. This work was supported in part by the
KOSEF and also by the U.S. Department of Energy under Contract No.
DE-AC02-76-ER03130.
\section*{Appendix}
In Sec. III the following facts are used;
\begin{equation}
\sum_{l=1}^\infty\int_{-\frac{1}{2}}^{\frac{1}{2}} dt
\frac{t}{[(t+l)^2+\alpha^2]^{3/2}}= I_1+ I_2,
\end{equation}
where
\begin{equation}
I_1=\sum_{l=1}^\infty\int_{l-\frac{1}{2}}^{l+\frac{1}{2}}
\frac{y}{[y^2+\alpha^2]^{3/2}} dy
=\frac{1}{\sqrt{\frac{1}{4}+\alpha^2}}
\end{equation}
and
\begin{eqnarray}
\alpha^2 I_2
&=&\sum_{l=1}^\infty l[\frac{l-\frac{1}{2}}
{\sqrt{\alpha^2+(l-\frac{1}{2})^2}}
-\frac{l+\frac{1}{2}}{\sqrt{\alpha^2+(l+\frac{1}{2})^2}}]
\nonumber\\
&=&\frac{1}{2\sqrt{\alpha^2+\frac{1}{4}}} + \lim_{N\rightarrow\infty}
[\sum_{l=1}^N
\frac{l+\frac{1}{2}}{\sqrt{\alpha^2+(l+\frac{1}{2})^2 }}
-(N+1)\frac{N+\frac{3}{2}}{\sqrt{\alpha^2+(N+\frac{3}{2})^2}}]
\nonumber\\
&=&\sum_{l=0}^\infty
[\frac{l+\frac{1}{2}}{\sqrt{\alpha^2+(l+\frac{1}{2})^2}} -1]
=-{\cal F}_{\frac{1}{2}}(\alpha).
\end{eqnarray}
${\cal F}_{\frac{1}{2}}(\alpha)$ and ${\cal F}(\alpha)$ are related
as follows;
\begin{equation}
{\cal F}_{\frac{1}{2}}(\alpha) ={\cal F}(2\alpha)-{\cal F}(\alpha).
\end{equation}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,480 |
{"url":"https:\/\/talmidimway.org\/commentary\/revelation\/rev07\/","text":"# The 144,000 - Revelation 7\n\nWatch the video\n\nRevelation 7 Cross-reference Verses PDF\n\nRev 7:1-17\n\nAfter this, I saw four angels standing at the four corners of the earth, holding back the four winds of the earth, so that no wind would blow on the land, on the sea or on any tree. I saw another angel coming up from the east with a seal from the living God, and he shouted to the four angels who had been given power to harm the land and the sea, \u201cDo not harm the land or the sea or the trees until we have sealed the servants of our God on their foreheads!\u201d I heard how many were sealed\u2014144,000 from every tribe of the people of Isra\u2019el: From the tribe of Y\u2019hudah 12,000 were sealed, from the tribe of Re\u2019uven 12,000, from the tribe of Gad 12,000, from the tribe of Asher 12,000, from the tribe of Naftali 12,000, from the tribe of M\u2019nasheh 12,000, from the tribe of Shimon 12,000, from the tribe of Levi 12,000, from the tribe of Yissakhar 12,000, from the tribe of Z\u2019vulun 12,000, from the tribe of Yosef 12,000, from the tribe of Binyamin 12,000. After this, I looked; and there before me was a huge crowd, too large for anyone to count, from every nation, tribe, people and language. They were standing in front of the throne and in front of the Lamb, dressed in white robes and holding palm branches in their hands; and they shouted, \u201cVictory to our God, who sits on the throne, and to the Lamb!\u201d All the angels stood around the throne, the elders and the four living beings; they fell face down before the throne and worshipped God, saying, \u201cAmen! \u201cPraise and glory, wisdom and thanks, honor and power and strength belong to our God forever and ever! \u201cAmen!\u201d One of the elders asked me, \u201cThese people dressed in white robes\u2014who are they, and where are they from?\u201d \u201cSir,\u201d I answered, \u201cyou know.\u201d Then he told me, \u201cThese are the people who have come out of the Great Persecution. They have washed their robes and made them white with the blood of the Lamb. That is why they are before God\u2019s throne. \u201cDay and night they serve him in his Temple; and the One who sits on the throne will put his Sh\u2019khinah upon them. \u201cThey will never again be hungry, they will never again be thirsty, the sun will not beat down on them, nor will any burning heat. \u201cFor the Lamb at the center of the throne will shepherd them, will lead them to springs of living water, and God will wipe every tear from their eyes.\u201d\n\n### Rev 7:1\n\nAfter this, I saw four angels standing at the four corners of the earth, holding back the four winds of the earth, so that no wind would blow on the land, on the sea or on any tree.\n\n\u2022 \u201cAfter this,\u201d could mean that this vision came after the previous one; not necessarily that the events in chapter 7 necessarily follow those in Chapter 6.\n\u2022 Keener sees 6:12-17 as representing the end of the age, so chapter 7 with the sealing of the faithful must occur beforehand.1\n\u2022 The seven seals\/trumpets\/bowls could be different visions describing the same things; not necessarily 21 different events in sequence.\n\u2022 Four corners is clearly symbolic \u2013 no one in that day believed the earth was square. Here it represents the four directions on the compass.\n\u2022 The four winds\/horsemen\/angels are apparently connected:\n\u2022 Zec 6:1-8 Again I lifted my eyes and saw, and behold, four chariots came out from between two mountains. And the mountains were mountains of bronze. (2) The first chariot had red horses, the second black horses, (3) the third white horses, and the fourth chariot dappled horses\u2014all of them strong. (4) Then I answered and said to the angel who talked with me, \u201cWhat are these, my lord?\u201d (5**) And the angel answered and said to me, \u201cThese are going out to the four winds of heaven, after presenting themselves before the Lord of all the ear**th. (6) The chariot with the black horses goes toward the north country, the white ones go after them, and the dappled ones go toward the south country.\u201d (7) When the strong horses came out, they were impatient to go and patrol the earth. And he said, \u201cGo, patrol the earth.\u201d So they patrolled the earth.\n\n### Rev 7:2-3\n\nI saw another angel coming up from the east with a seal from the living God, and he shouted to the four angels who had been given power to harm the land and the sea, \u201cDo not harm the land or the sea or the trees until we have sealed the servants of our God on their foreheads!\"\n\n\u2022 This seal is God\u2019s signet ring; i.e., His stamp of authentication\n\u2022 From Rev 14:1, we know that the seal was the name of the Lamb and of his Father. Rev 14:1 Then I looked, and behold, on Mount Zion stood the Lamb, and with him 144,000 who had his name and his Father\u2019s name written on their foreheads.\n\u2022 This is a challnge to interpret since we believe there are more than 144,000 servants of God. See discussion at Rev 7:4-8.\n\u2022 Additionally, we know from the 5th seal that there are many martyrs so we cannot say that the 144,000 is a global seal of protection for all believers; it keeps the people sealed from God\u2019s wrath but not wrath in general.\n\n\u2022 This recalls the preparations for the Exodus and the application of the lamb\u2019s blood to the doorpost prior to the final plague of the death of the firstborn\n\u2022 Exo 12:12-13 For I will pass through the land of Egypt that night, and I will strike all the firstborn in the land of Egypt, both man and beast; and on all the gods of Egypt I will execute judgments: I am the LORD. The blood shall be a sign for you, on the houses where you are. And when I see the blood, I will pass over you, and no plague will befall you to destroy you, when I strike the land of Egypt.\n\u2022 Seal on the foreheads recalls Exodus, Deuteronomy, and Ezekiel:\n\u2022 Exo 13:9 And it shall be to you as a sign on your hand and as a memorial between your eyes, that the law of the LORD may be in your mouth. For with a strong hand the LORD has brought you out of Egypt.***\n\u2022 We are to be sealed by the Word of God: Deu 6:6-8 And these words that I command you today shall be on your heart. 7 You shall teach them diligently to your children, and shall talk of them when you sit in your house, and when you walk by the way, and when you lie down, and when you rise. 8 You shall bind them as a sign on your hand, and they shall be as frontlets between your eyes.\n\u2022 Eze 9:4-6 And the LORD said to him, \u201cPass through the city, through Jerusalem, and put a mark on the foreheads of the men who sigh and groan over all the abominations that are committed in it.\u201d 5 And to the others he said in my hearing, \u201cPass through the city after him, and strike. Your eye shall not spare, and you shall show no pity. Kill old men outright, young men and maidens, little children and women, but touch no one on whom is the mark. And begin at my sanctuary.\u201d So they began with the elders who were before the house.\n\u2022 This mark is for those who oppose the abominations done in Jerusalem.\n\u2022 In Ezekiel, the word for \u201cmark\u201d is the Hebrew letter Tav. \u05ea,\n\u2022 In Ezekiel\u2019s day it would have looked like an x.\n\u2022 It bears some similarity to a cross but it is unlikely a Cross, the way we might think of it, was in view.\n\u2022 Rev 9:3-4 Then from the smoke came locusts on the earth, and they were given power like the power of scorpions of the earth. 4 They were told not to harm the grass of the earth or any green plant or any tree, but only those people who do not have the seal of God on their foreheads.\n\u2022 This stands in obvious contrast with the mark of the beast\n\u2022 Rev 13:15-17 And it was allowed to give breath to the image of the beast, so that the image of the beast might even speak and might cause those who would not worship the image of the beast to be slain. 16 Also it causes all, both small and great, both rich and poor, both free and slave, to be marked on the right hand or the forehead, 17 so that no one can buy or sell unless he has the mark, that is, the name of the beast or the number of its name.\n\n### Rev 7:4-8\n\nI heard how many were sealed\u2014144,000 from every tribe of the people of Isra\u2019el: From the tribe of Y\u2019hudah 12,000 were sealed, from the tribe of Re\u2019uven 12,000, from the tribe of Gad 12,000, from the tribe of Asher 12,000, from the tribe of Naftali 12,000, from the tribe of M\u2019nasheh 12,000, from the tribe of Shimon 12,000, from the tribe of Levi 12,000, from the tribe of Yissakhar 12,000, from the tribe of Z\u2019vulun 12,000, from the tribe of Yosef 12,000, from the tribe of Binyamin 12,000.\n\n\u2022 We either must read the 144,000 as consistently literally or consistently symbolically.\n\u2022 Either it\u2019s literally 12,000 male Jewish virgins \u2013 or specifically, those who have not ritually defiled themselves in any fashion (per Rev 14:4) from each of the 12 listed tribes, or it\u2019s all the spiritual servants of God (Rev 7:3, 9), but not literally 144,000.\n\u2022 Saying there are literally 144,000 sealed, but not literally Jews who are sealed is an incoherent argument. If someone claims to be a member of the 144,000, a scriptural response is to ask which tribe they are from.\n\u2022 As Revelation fits squarely within the genre of Jewish apocalyptic literature, perhaps we shouldn\u2019t be so quick to allegorize this, despite the apparent relatively small number of those who are sealed.\n\u2022 If we are to allegorize, why is the list of 12 tribes enumerated?\n\u2022 Why does this exact number of 144,000 reappear in Rev 14?\n\u2022 Lancaster sees the 144,000 as is a portion of the whole amount of Israel, specifically an army as we see in 1st Samuel.\n\u2022 1 Samuel 21:3-5 Now then, what do you have on hand? Give me five loaves of bread, or whatever is here.\u201d 4 And the priest answered David, \u201cI have no common bread on hand, but there is holy bread\u2014if the young men have kept themselves from women.\u201d 5 And David answered the priest, \u201cTruly women have been kept from us as always when I go on an expedition. The vessels of the young men are holy even when it is an ordinary journey. How much more today will their vessels be holy?\u201d\n\u2022 \u201chave not defiled themselves with women\u201d is an idiom for complete Levitical ritual purity.\n\u2022 When the army of Israel went out to war, they had to be in a state of ritual purity.\n\u2022 This is because the Lord was in their midst.\n\u2022 Therefore the implication of the 144,000 is that this is an army of the Lord, the fighting men of the son of King David.\n\u2022 Usually, when there is a census in the Bible, it is to determine the number of fighting men.\n\u2022 Only the men are numbered.\n\u2022 Numbers 31:2-5 \u201cAvenge the people of Israel on the Midianites. Afterward you shall be gathered to your people.\u201d 3 So Moses spoke to the people, saying, \u201cArm men from among you for the war, that they may go against Midian to execute the LORD\u2019s vengeance on Midian. 4 You shall send a thousand from each of the tribes of Israel to the war.\u201d 5 So there were provided, out of the thousands of Israel, a thousand from each tribe, twelve thousand armed for war.\n\u2022 Num 31:48-49 Then the officers who were over the thousands of the army, the commanders of thousands and the commanders of hundreds, came near to Moses 49 and said to Moses, \u201cYour servants have counted the men of war who are under our command, and there is not a man missing from us.\"\n\u2022 So there\u2019s a miracle in that they all survived the battle, just as the 144,000 will be sealed.\n\u2022 The context of Numbers 31 is the preservation of Israel, which should be viewed as the theme of Revelation.\n\u2022 In Revelation, we see a mathematical amplification.\n\u2022 Peter asked, \u201cshould I forgive seven times,\u201d Jesus answered \u201c70 times seven\u201d\n\u2022 In the same way we can say of Revelation 7, \u201cnot 12,000 as against the Midianites, but 12 times 12,000.\u201d\n\u2022 Lancaster quoted an 18th-century Messianic Jew who believes the 144,000 faithful represent a first-fruits offering.\n\u2022 In fact, we are told in Revelation 14:4 It is these (144,000) who have not defiled themselves with women, for they are virgins. It is these who follow the Lamb wherever he goes. These have been redeemed from mankind as firstfruits for God and the Lamb,\n\u2022 A first fruits offering is 1\/50th of the total harvest\n\u2022 This small portion is said to sanctify the whole\n\u2022 Paul mentions this in Romans 11, which is the chapter on the ultimate sanctification of Israel:\n\u2022 Rom 11:16, 26: If the dough offered as firstfruits is holy, so is the whole lump, and if the root is holy, so are the branches\u2026 And in this way all Israel will be saved, as it is written, \u201cThe Deliverer will come from Zion, he will banish ungodliness from Jacob\u201d;\n\u2022 According to this approach, when the 144,000 are sealed and Messiah is revealed, this will lead to faith of all Israel.\n\u2022 The relatively modern Messianic Jewish movement may be FAR MORE significant than anyone realizes. These 144,000 are, by definition, Messianic Jews.\n\u2022 God who will be doing the selection knows who is from which tribe, even though we may not see evidence of the 12 tribes today.\n\u2022 On that note, remember the legend of the \u201c10 lost tribes\u201d is just that. There is no scriptural basis.\n\u2022 James addressed his letter to the \u201c12 tribes\u201d in the diaspora.\n\u2022 As the northern kingdom was furthering their journey into idolatry, we know the faithful migrated to Judah (2 Chr 15:9)\n\u2022 In acts 2 there are separate references to the men of Judah (Acts 2:14) and the men of Israel (Acts 2:22, 36).\n\u2022 In the days of the apostles, the 10 tribes were considered absent, not missing or lost.\n\u2022 By the way, if you know anyone named Cone, Cohen, Levin, Lewis, or perhaps Lewinsky and Epstein, they might be descended from the tribe of Levi, though certainly many people with these names do not act with the holiness required of a Levitical priest.\n\u2022 Missler and many other pre-trib commentators would see the 144,000 as the preservation of the Jewish people through the Tribulation, whereas the church is kept from the Tribulation.\n\u2022 Everything on earth pertains to Jews, while everything in Heaven pertains to the church.\n\u2022 As messianic commentators point out, this position gets difficult to defend from scripture once one starts peeling back the layers.\n\u2022 Ultimately it is an assumption that the \u201crapture\u201d occurs between chapters 3 and 4.\n\u2022 If we assume a pre-trib rapture, are we saying that Messianic Jews will NOT be raptured because they are Jewish? That\u2019s nonsensical.\n\u2022 I\u2019m not picking a side here \u2013 we encourage everyone to do their Acts 17:11 homework.\n\n#### The list of the 12 tribes\n\n\u2022 It\u2019s easy to miss that, overall, there are actually 13 tribes from which to compose the various lists of the 12 tribes that occur in the Bible. Why is this? Joseph received a double-portion in Ephraim and Manasseh.\n\u2022 When listing the 12 tribes and we want to include Levi, Joseph is just Joseph: Reuben, Simeon, Levi, Judah, Dan, Naphtali, Gad, Asher, Issachar, Zebulun, Joseph, and Benjamin. o- In certain lists, the priestly tribe of Levi is excluded. Yet the list still contains 12 tribes because Joseph is split into Ephraim and Manasseh: Reuben, Simeon, Judah, Dan, Naphtali, Gad, Asher, Issachar, Zebulun, Manasseh, Ephraim, and Benjamin.\n\u2022 Where is Dan?\n\u2022 It\u2019s important to note that we aren\u2019t told anywhere why Dan is omitted, but the usual explanation is that Dan is generally associated with idolatry and apostasy\n\u2022 Jewish Sages explained that the north was the literal dark side (since in the Northern Hemisphere, the Sun can be east, west, or south, but never north).\n\u2022 Dan was positioned on the north side of the camp of Israel because \u201cDan darkened the world by idolatry.\u201d\n\u2022 This may underlie a warning that even among God\u2019s \u201cchosen\u201d people (whether it\u2019s the Jewish people or we, who have been chosen by Jesus), there is a danger of these things creeping in.\n\u2022 Joh 6:70 Jesus answered them, \u201cDid I not choose you, the twelve? And yet one of you is a devil.\u201d\n\u2022 Iraneus claims to be quoting Jewish apocalyptic literature which says the anti-messiah will come from Dan.\n\u2022 There is no biblical evidence to support, but since Iraneus was a disciple of Polycarp, who was a disciple of John himself, perhaps this tradition also shouldn\u2019t be quickly dismissed.\n\u2022 Listing Manasseh and Joseph is odd. Manasseh was a part of Joseph. While the Bible is fundamentally inerrant in its original transmissions, we do know that there are a few scribal errors - John 5:3b-4 is a well-known example; it is omitted from most modern translations. - Some suggest that naming Manasseh instead of Dan was such an error.\n\u2022 Ephraim is indirectly referenced, likely because they were also associated with Jeroboam\u2019s Idolatry (Hos 4:17)\n\u2022 Jeroboam set up two pagan altars: one was in the land of Ephraim and the other was in Dan.\n\u2022 Archaeologists have excavated what they believe are the remains of the alter at Tel Dan.\n\n### Rev 7:9\n\nAfter this, I looked; and there before me was a huge crowd, too large for anyone to count, from every nation, tribe, people and language. They were standing in front of the throne and in front of the Lamb, dressed in white robes and holding palm branches in their hands;\n\n\u2022 It is possible that this huge crowd is another picture of those who were sealed now in heaven, or a different group.\n\u2022 An example of the former is found in Genesis 41, where Pharoah had two different visions of the same famine event.\n\u2022 We saw a great multitude in Rev 5:9\n\u2022 Again we see the white robes we saw the martyrs wearing in chapter 6 \u2013 may or may not be the same group\n\u2022 If it is the same group, it is now \u201ca little while longer\u201d (Rev 6:11)\n\u2022 Palm branches reminiscent of Palm Sunday and the Feast of Tabernacles\n\n### Rev 7:10-12\n\nand they shouted, \u201cVictory to our God, who sits on the throne, and to the Lamb!\u201d All the angels stood around the throne, the elders and the four living beings; they fell face down before the throne and worshipped God, saying, \u201cAmen! \u201cPraise and glory, wisdom and thanks, honor and power and strength belong to our God forever and ever! \u201cAmen!\u201d\n\n\u2022 In Greek, the word for \u201cvictory\u201d is \u201csoteeria\u201d, literally \u201csalvation\u201d; but \u201cSalvation to our God\u201d is awkward.\n\u2022 Salvation FROM our God makes more sense - Psa 98:2 The LORD has made known his salvation; he has revealed his righteousness in the sight of the nations.\n\u2022 In Hebrew, the word would be Yeshua, the Hebrew name for Jesus.\n\u2022 \u05d4\u05b4\u05e0\u05bc\u05b5\u05d4 \u05d0\u05b5\u05dc \u05d9\u05b0\u05e9\u05c1\u05d5\u05bc\u05e2\u05b8\u05ea\u05b4\u05d9, \u05d0\u05b6\u05d1\u05b0\u05d8\u05b7\u05d7 \u05d5\u05b0\u05dc\u05b9\u05d0 \u05d0\u05b6\u05e4\u05b0\u05d7\u05b8\u05d3, \u05db\u05bc\u05b4\u05d9 \u05e2\u05b8\u05d6\u05bc\u05b4\u05d9 \u05d5\u05b0\u05d6\u05b4\u05de\u05b0\u05e8\u05b8\u05ea \u05d9\u05b8\u05d4\u05bc \u05d9\u05b0\u05d9\u05b8, \u05d5\u05b7\u05d9\u05b0\u05d4\u05b4\u05d9 \u05dc\u05b4\u05d9 \u05dc\u05b4\u05d9\u05e9\u05c1\u05d5\u05bc\u05e2\u05b8\u05d4\n\u2022 \u201cBehold, God is my salvation, I will trust God and not be afraid, for my strong faith and song of praise for God will be my salvation.\u201d Jewish folk song\n\n### Rev 7:13\n\nOne of the elders asked me, \u201cThese people dressed in white robes\u2014who are they, and where are they from?\u201d \u201cSir,\u201d I answered, \u201cyou know.\u201d Then he told me, \u201cThese are the people who have come out of the Great Persecution. They have washed their robes and made them white with the blood of the Lamb.\n\n\u2022 Asking a question a student wouldn\u2019t know was a common rabbinical technique.\n\u2022 Eze 37:3 And he said to me, \u201cSon of man, can these bones live?\u201d And I answered, \u201cO Lord GOD, you know.\u201d\n\u2022 Great Persecution\/Tribulation refers to Daniel\n\u2022 Dan 12:1 \u201cAt that time shall arise Michael, the great prince who has charge of your people. And there shall be a time of trouble, such as never has been since there was a nation till that time. But at that time your people shall be delivered, everyone whose name shall be found written in the book.\n\u2022 This time, there IS a definite article in Greek. This is THE great tribulation, the mega pressure, literally.\n\u2022 As we mentioned at the beginning of the study, these words are intended to offer comfort to those 1st century believers undergoing persecution.\n\u2022 A blood stain is notoriously difficult to remove, and a figurative bloodstain moreso than a literal one (as Shakespeare referenced in Macbeth)\n\u2022 We are made pure by His blood and our being faithful to His commands\n\u2022 These are clearly martyrs, perhaps we can use the term \u201cTribulation Saints.\u201d\n\n### Rev 7:15-17\n\nThat is why they are before God\u2019s throne. \u201cDay and night they serve him in his Temple; and the One who sits on the throne will put his Sh\u2019khinah upon them. \u201cThey will never again be hungry, they will never again be thirsty, the sun will not beat down on them, nor will any burning heat. \u201cFor the Lamb at the center of the throne will shepherd them, will lead them to springs of living water, and God will wipe every tear from their eyes.\u201d\n\n\u2022 The heavenly throne is again described as a temple (Rev 4:6), and continues the theme of Jesus the great High Priest serving in the Heavenly temple (Hebrews 8)\n\u2022 Shekinah = protection and provision\n\u2022 He spreads his covering over us, which is a theme that runs through scripture as the Shekinah inhabited the Tent of Meeting.\n\u2022 The two words are linguistically related in Hebrew:\n\u2022 Shekinah = dwelling; God\u2019s glory dwelling with mankind\n\u2022 Mishkan \u2013 tent, tabernacle\n\u2022 Heb 1:3-4 He is the radiance of the glory of God and the exact imprint of his nature, and he upholds the universe by the word of his power. After making purification for sins, he sat down at the right hand of the Majesty on high, 4 having become as much superior to angels as the name he has inherited is more excellent than theirs.\n\u2022 Eze 37:27 My dwelling place shall be with them, and I will be their God, and they shall be my people.\n\u2022 Rth 3:9-11 He said, \u201cWho are you?\u201d And she answered, \u201cI am Ruth, your servant. Spread your wings over your servant, for you are a redeemer.\u201d 10 And he said, \u201cMay you be blessed by the LORD, my daughter. You have made this last kindness greater than the first in that you have not gone after young men, whether poor or rich. 11 And now, my daughter, do not fear. I will do for you all that you ask, for all my fellow townsmen know that you are a worthy woman.\n\u2022 Isa 4:5-6 Then the LORD will create over the whole site of Mount Zion and over her assemblies a cloud by day, and smoke and the shining of a flaming fire by night; for over all the glory there will be a canopy. There will be a booth for shade by day from the heat, and for a refuge and a shelter from the storm and rain.\n\u2022 Isa 25:8 He will swallow up death forever; and the Lord GOD will wipe away tears from all faces, and the reproach of his people he will take away from all the earth, for the LORD has spoken.\n\u2022 Isa 49:10 they shall not hunger or thirst, neither scorching wind nor sun shall strike them, for he who has pity on them will lead them, and by springs of water will guide them.\n\u2022 Joh 10:11 I am the good shepherd. The good shepherd lays down his life for the sheep.\n\u2022 The culmination of wiping every tear and springs of living water will be in Revelation 21:4,6.\n\n### Pause to Review\n\n\u2022 In this gap between the 6th and 7th seal, this is a good spot to recap\n\u2022 We began with the apocalypse, the revelation, the unveiling of Jesus Christ\n\u2022 John was imprisoned on Patmos during the Domitian persecution.\n\u2022 Domitian considered Jews tax evaders, though they were legally exempted, but he considered gentile believers in Jesus traitors for disavowing the Roman gods.\n\u2022 Revelation is entirely consistent with extra-biblical Jewish apocalyptic literature: Although terrible things are happening to God\u2019s people, God is not absent. For the wicked, a day of judgment; for the righteous a day of rewards is yet to come\n\u2022 There is an order of transmission: God to Jesus, to His Angel, to John, to the seven assemblies.\n\u2022 We are reading it indirectly.\n\u2022 The letter is written for us but not to us.\n\u2022 Nonetheless, there is a blessing for ANYONE who reads, hears, AND KEEPS the words (Rev 1:3)\n\u2022 Before John gets into the crazy stuff he saw, he sends specific messages to the seven assemblies.\n\u2022 \u201cI see your deeds\u201d; five of the churches were told to repent!\n\u2022 A promise to the overcomer\n\u2022 Chapter 4 \u2013 after this a door was open in heaven\n\u2022 Lancaster sees Rosh Hashana imagery\n\u2022 Isaiah and Ezekiel saw similar views as John with the Ancient of Days on His throne\n\u2022 John sees the Seraphim from Isaiah (John saw seven torches) representing what is above and we see the four living creatures from Ezekiel representing the earthly realm\n\u2022 Chapter 5 opened with the Ancient of Days with a scroll in His hand with seven seals\n\u2022 There is a moment of drama as no one, seemingly is found to open the scroll, until the One like a Son of Man, the same One Daniel saw in chapter 7 presents Himself.\n\u2022 He is both the Lion of the Tribe of Judah and He is the Lamb standing as though it had been slain.\n\u2022 Every creature in heaven, on earth, and under the earth and in the sea and all that is in them praised, \u201cto him who sits on the throne and to the lamb be blessing and honor and glory and might forever and ever. Amen!\u201d\n\u2022 Chapter 6 has Lamb opening the first six seals\n\u2022 The first seal is white horse of conquest, perhaps by deception and likely without armed conflict\n\u2022 The second seal is the red horse of armed conquest\n\u2022 The third seal is the black horse of famine and inflation\n\u2022 The fourth seal is the pale horse of death\n\u2022 The fifth seal is not a judgment poured on earth, but it is the blood of the martyrs poured on the alter. They cried for God\u2019s justice but were told to wait a little while longer. This little while for us, has been lasting for 1900 years since John first saw the vision.\n\u2022 The sixth seal brings great calamity, the wrath of the Lamb, the Day of the Lord.\n\u2022 Under the sixth seal we will have the remaining judgments in Revelation.\n\u2022 It\u2019s reasonable to ask, \u201cwho on Earth will be able to survive this? How is God going to fulfill His promises He made to Israel?\u201d\n\u2022 The answer comes in Chapter 7\n\u2022 Chapter 7 sees an angel putting a seal on 144,000 Jews which assures that they will survive to the end and will be able to usher in the Messianic Era","date":"2023-01-28 07:52:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35008910298347473, \"perplexity\": 4704.753368934165}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499524.28\/warc\/CC-MAIN-20230128054815-20230128084815-00442.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.physicsforums.com\/threads\/wedge-product-question.275983\/","text":"# Wedge product question\n\n1. Nov 30, 2008\n\n### Diffy\n\n1. The problem statement, all variables and given\/known data\nPlease, I just am trying to understand the question. I wish to prove it on my own, but the way the question is phrased makes no sense.\n\nSo here it is:\n\nLet us define the linear map\n$$\\phi : V^{*} \\otimes \\bigwedge^{i} V \\rightarrow \\bigwedge^{i-1} V$$\n\nby the formula\n\n$$\\ell \\otimes v_1 \\wedge ... \\wedge v_i \\mapsto \\sum_{s=1}^{i} (-1)^{s-1} \\ell (v_s) v_1 \\wedge ... \\wedge \\hat{v_s} \\wedge ... \\wedge v_s$$\n\nProve that the map $$\\phi$$ is well defined and does not depend on the choice of basis.\n\n2. Relevant equations\nWell all the usual definition of exterior algebras, and tensor products are needed.\n\n3. The attempt at a solution\n\nAs I stated, I haven't started solving yet, I am simply trying to understand the question. I don't see how it goes to wedge i-1. What exactly is v hat sub s? Does that make i wedges?\nI don't think this formula is going to i-1 wedges. Please help me to understand what is going on here.\n1. The problem statement, all variables and given\/known data\n\n2. Relevant equations\n\n3. The attempt at a solution\n1. The problem statement, all variables and given\/known data\n\n2. Relevant equations\n\n3. The attempt at a solution\n\n2. Nov 30, 2008\n\n### Dick\n\nWhat they are trying to notate by v hat sub s, is that the vector v_s is NOT included in the wedge product. All the other v_i's are. That's why it's an i-1 form.\n\n3. Nov 30, 2008\n\n### Diffy\n\nNow it makes sense!\n\nNow I just have to solve the problem. Any hints are appreciated.\n\n4. Nov 30, 2008\n\n### lurflurf\n\nWell there are two main ways to prove something is invariant\n1 show it does not change with form\nie\nlet x and x' be different forms of x\nif\nf(x)=f(x')\nf is invariant\n\n2 provide an invariant definition\nie\ndefine\nf(x)\nso that the form of x is not a factor\n\nSince you have been provided a noninvariant definition\nlet Av be a change of basis\nyou want to show\nl(v1)^v2^...^vn=l(Av1)^Av2^...^Avn=det(a)*l(v1)^v2^...^vn\nwith det(A)=1","date":"2017-01-23 02:50:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6886184215545654, \"perplexity\": 737.9438922356857}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560281746.82\/warc\/CC-MAIN-20170116095121-00088-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.ni.com\/documentation\/en\/labview-comms\/1.0\/mt-node-ref\/mt-map-bits-to-msk-symbols\/","text":"Maps an incoming bit stream to symbols comprised of MSK frequency deviations.\n\n## input bit stream\n\nThe incoming bit stream to be mapped to symbols.\n\n## symbol rate\n\nThe symbol rate, in hertz (Hz), of the input bit stream.\n\nNote\n\nFor MSK modulation, the frequency deviation is \u00bc the symbol rate. For nondifferential MSK, the symbol-to-bit mapping is performed by mapping bit 0 $\\to$ -0.25 \u00d7 symbol rate, and mapping bit 1 $\\to$ +0.25 \u00d7 symbol rate.\n\n## error in\n\nError conditions that occur before this node runs. The node responds to this input according to standard error behavior.\n\nDefault: no error\n\n## differential encoding enable\n\nA value that indicates whether to enable differential encoding of the bit stream prior to mapping to MSK symbols.\n\n disable Disables bit stream encoding. enable Enables bit stream encoding.\n\nDefault: enable\n\n## reset?\n\nA Boolean that determines whether stored state information is cleared on each call to this node.\n\nWhen the input bit stream is not comprised of an integer number of symbols, the carryover bits are buffered.\n\n TRUE Clears the buffered data, checks the input parameters on a first call, and reflects any change in the input parameter values during subsequent iterations. FALSE Adds the buffered data to the beginning of data from next iteration, in continuous operations.\n\nDefault: TRUE\n\n## symbols\n\nAn array of MSK frequency deviations with a one-to-one mapping to the input bit stream.\n\nNote\n\nWire this parameter to MT Apply Pulse Shaping Filter (MSK, CPM) to generate the oversampled baseband complex waveform.\n\n## error out\n\nError information. The node produces this output according to standard error behavior.","date":"2018-01-23 00:29:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.48681679368019104, \"perplexity\": 5156.322449505852}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084891546.92\/warc\/CC-MAIN-20180122232843-20180123012843-00102.warc.gz\"}"} | null | null |
#pragma once
#include <cstdint>
#include <memory>
#include <string>
class ConfigItem;
class FileBuffer;
class DataBuffer;
constexpr uint32_t ROM_INFO_HID = 0x1;
constexpr uint32_t ROM_INFO_HID_MX23 = 0x2;
constexpr uint32_t ROM_INFO_HID_MX50 = 0x4;
constexpr uint32_t ROM_INFO_HID_MX6 = 0x8;
constexpr uint32_t ROM_INFO_HID_SKIP_DCD = 0x10;
constexpr uint32_t ROM_INFO_HID_MX8_MULTI_IMAGE = 0x20;
constexpr uint32_t ROM_INFO_HID_MX8_STREAM = 0x40;
constexpr uint32_t ROM_INFO_HID_UID_STRING = 0x80;
// Omitted value: 0x100
// Omitted value: 0x200
constexpr uint32_t ROM_INFO_HID_NO_CMD = 0x400;
constexpr uint32_t ROM_INFO_SPL_JUMP = 0x800;
constexpr uint32_t ROM_INFO_HID_EP1 = 0x1000;
constexpr uint32_t ROM_INFO_HID_PACK_SIZE_1020 = 0x2000;
constexpr uint32_t ROM_INFO_HID_SDP_NO_MAX_PER_TRANS = 0x4000;
constexpr uint32_t ROM_INFO_AUTO_SCAN_UBOOT_POS = 0x8000;
constexpr uint32_t ROM_INFO_HID_ROMAPI = 0x10000;
struct ROM_INFO
{
const char * m_name;
uint32_t free_addr;
uint32_t flags;
};
const ROM_INFO * search_rom_info(const std::string &s);
const ROM_INFO * search_rom_info(const ConfigItem *item);
size_t GetContainerActualSize(std::shared_ptr<DataBuffer> p, size_t offset, bool bROMAPI=false);
size_t GetFlashHeaderSize(std::shared_ptr<DataBuffer> p, size_t offset = 0);
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Q: O que significa PSR? Olá pessoal recentemente encontrei um termo chamado PSR na area de PHP, mas o topico não foi muito esclarecedor em definir o real significado da palavra PSR, vi que ele estava relacionado com a area de Orientaçao a Objetos, fiz uma pesquisa no google e não encontrei muita coisa, acredito que pesquisei pelos termos errados.
Mas enfim, alguem poderia me explicar o significado do PSR e sua real aplicação?
A: Achei uma resposta:
O PHP Framework Interoperability Group é um grupo formado por membros com poder de voto e que representam frameworks PHP e membros não votantes que podem participar e que definiu, até o momento, três padrões: PSR-0, PSR-1 e PSR-2.
A norma PSR-0, dita a respeito de autoloader. A PSR-1, de normas básica de códificação e a PSR-2, vai além e do básico de normatização do PSR-1.
Como acessar o guia de estilos PSR
Via URL, através de
PSR-0: https://github.com/php-fig/fig-standards/blob/master/accepted/PSR-0.md
PSR-1: https://github.com/php-fig/fig-standards/blob/master/accepted/PSR-1-basic-coding-standard.md
PST-2: https://github.com/php-fig/fig-standards/blob/master/accepted/PSR-2-coding-style-guide.md
Via Git, recomendo
git clone git://github.com/php-fig/fig-standards.git
A: As PSR (do inglês PHP Standards Recommendation) são especificações de projetos propostos pelo PHP-FIG (PHP Framework Interop Group), um grupo composto por representantes de expressivos projetos em PHP.
Esses padrões tem como objetivo facilitar a reutilização de código entre os diversos projetos que implementem determinado padrão.
Um exemplo é a PSR-3, que sugere uma especificação para Interface de Logs de Aplicação. Qualquer projeto que suporte a PSR-3 pode simplesmente substituir o módulo de logs por outro compatível que também suporte a PSR-3 sem nenhum impacto no projeto original (Ai temos a ideia da interoperabilidade entre os projetos).
Além do padrão de Logs, existem PSRs para implementações de autoload (PSR-0 e PSR-4), sugestões de estilos de código, como posição de chaves, indentação (Usar tabulações ou espaços?) (PSR-1 e PSR-2).
Existem também propostas em draft para padronização dos docblock de documentação (PSR-5) e uma interface para requisições HTTP (PSR-7)
Mais informações leia o FAQ e visite o repositório no GitHub com os padrões já aceitos pelo grupo.
É importante lembrar que a adoção desses padrões no seu projeto é opcional.
Ninguém é obrigado a implementar funcionalidade "X" de certa maneira, porém é recomendado implementar a partir de um padrão já conhecido e adotado pela comunidade para não causar dores de cabeça futuras com seu código.
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{"url":"https:\/\/chat.stackexchange.com\/transcript\/9417?m=50381090","text":"12:15 AM\n@YuriSulyma you\u2019re welcome to do some house cleaning. it looks like \u201cother books\u201d is a decent target, but less so misc: sort the files by size to see what\u2019s actually going to improve the situation\n\n19 hours later\u2026\n7:36 PM\nDoes anyone know a reference that proved, in detail, that giving a monoidal category is equivalent to giving a Grothendieck opfunctor over \\Delta^op satisfying the Segal condition?\n\n8:05 PM\n@Dedalus so I think that this is (to some extent) proven in my paper with Liang Ze Wong, in the setting of simplicially enriched monoidal categories, but of course you can just take something with a discrete simplicial enrichment. arxiv.org\/pdf\/1808.08020.pdf\n(also that's in the strict case, but this isn't a big deal because any monoidal category can be rigidified to a strict monoidal category up to monoidal equivalence)\ncf. basically Prop 4.1.4 and the things that follow it\nit relies on an enriched version of the grothendieck construction, which, again, restricts to the unenriched version if you take a discrete enrichment\nAlthough I guess we don't really say anything explicitly about the Segal condition. You could restrict to subcategories on both sides though and I think prove your statement pretty easily, as the equivalence is described explicitly.\n\nThanks! I want a reference of the fact that if we have a Grothendieck opfibration over \\Delta^op, then the pentagon axiom holds.\n\n1 hour later\u2026\n9:37 PM\n@Dedalus the pentagon axiom follows immediately from a certain equality inside of the hom-set $hom_{\\Delta^{op}}([4],[1])$\n\n9:51 PM\n@AaronMazel-Gee I agree that is the way to do it, but when I do it I get something quite messy. One works out the different factorizations of the map [1] \\to [4] etc. It is clear what to do, but messy (at least when I write it down carefully). Is this what you meant by immediate?\n\n2 hours later\u2026\n11:36 PM\n@Dedalus well, that is the argument i was referring to (and i couldn't imagine a fundamentally different argument); but as for whether it's messy, i guess that depends what you're taking as given. one subtlety here is that an opfibration over C only determines a functor C --> Cat up to natural equivalence.\nif you are working homotopy-coherently, you can just say that the functor $\\bDelta^{op} \\to Cat$ takes this equality to a natural isomorphism (a path in a hom-space), but if you want to think about categories at the point-set level (i.e. having a set of objects, etc.) then you may have to be more careful, and indeed things may get messy.\nthis is starting to get out of my wheelhouse, but you might be interested in the notion of a \"cloven opfibration\", which is essentially an opfibration equipped with choices of cocartesian morphisms (which are a priori only well-defined up to a contractible groupoid of choices). every opfibration admits a cleaving, and if i recall correctly, this gives you a well-defined choice of straightening at the point-set level.","date":"2019-09-18 16:48:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9493058919906616, \"perplexity\": 498.4593629991424}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573309.22\/warc\/CC-MAIN-20190918151927-20190918173927-00306.warc.gz\"}"} | null | null |
{"url":"https:\/\/ckms.kms.or.kr\/journal\/view.html?doi=10.4134\/CKMS.2013.28.3.535","text":"- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations \u318dPaper Submission \u318dPaper Reviewing \u318dPublication and Distribution - Code of Ethics - For Authors \u318dOnline Submission \u318dMy Manuscript - For Reviewers - For Editors\n Jordan Derivations on Prime Rings and Their Applications in Banach Algebras,~I Commun. Korean Math. Soc. 2013 Vol. 28, No. 3, 535-558 https:\/\/doi.org\/10.4134\/CKMS.2013.28.3.535Printed July 1, 2013 Byung-Do Kim Gangneung-Wonju National University Abstract : The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let $A$ be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A\\to A$ such that $D(x)^3[D(x),x]\\in \\mbox{rad}(A)$ for all $x\\in A.$ In this case, we show that $D(A)\\subseteq \\mbox{rad}(A).$ Keywords : prime and semiprime ring, (Jacobson) radical, Jordan derivation Downloads: Full-text PDF\n\n Copyright \u00a9 Korean Mathematical Society. (Rm.1109) The first building, 22, Teheran-ro 7-gil, Gangnam-gu, Seoul 06130, Korea Tel: 82-2-565-0361\u00a0\u00a0| Fax: 82-2-565-0364\u00a0\u00a0| E-mail: paper@kms.or.kr\u00a0\u00a0 | Powered by INFOrang Co., Ltd","date":"2022-05-23 05:41:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19106413424015045, \"perplexity\": 3419.995464707192}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662555558.23\/warc\/CC-MAIN-20220523041156-20220523071156-00432.warc.gz\"}"} | null | null |
Der Martin Cirque ist ein 1,5 km breiter und markanter Bergkessel im ostantarktischen Viktorialand. In der Asgard Range liegt er 3 km nordwestlich des Mount Newall und nimmt die Südwand des Wright Valley zwischen dem Denton-Gletscher und dem Nichols Ridge ein. Der Bergkessel liegt in einer Höhe von rund und ist nahezu eisfrei.
Das Advisory Committee on Antarctic Names benannte ihn 1997 nach Craig J. Martin, der unter anderem von 1977 bis 1987 an Bau- und Ingenieursprojekten auf der Siple-Station, der Amundsen-Scott-Südpolstation, der McMurdo-Station und diversen Forschungscamps beteiligt war.
Weblinks
(englisch)
Martin Cirque auf geographic.org (englisch)
Asgard Range
Tal im Transantarktischen Gebirge | {
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Q: CouchDB on windows says No 'Access-Control-Allow-Origin' header is present on the requested resource I am trying to use CouchDB as a back-end for my application. I tried to install couchDB on my windows OS machine. It works fine.
When i tried to run this example after creating the db on "Projects", the console displays an error.
No 'Access-Control-Allow-Origin' header is present on the requested resource. Origin 'http://fiddle.jshell.net' is therefore not allowed access. The response had HTTP status code 405.
I tried to do fix it by this suggestion, how to add cors in couchDB, but i don't find [cors] section
I manually created [cors] section, that too did not work.
UPDATE:
I am using the latest version 1.6.1
A: you need to enable CORS on COUCHDB administration side and allow all sites
| {
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Romsley & Hunnington History Society -
You are here: HomeLocal HistoryHalesowen AbbeyChapter 6: Abbey Hospitality
Chapter 6: Abbey Hospitality
A glance at the ground plan of Halesowen Abbey will show that there was an extensive guest house occupying the whole of the west side of the cloister. There the successive Abbots of Hales carried out the injunction of the founder, Peter de Roches, in dispensing hospitality. This they did in common with other Premonstratensian houses, for one of the Order's rules was that a cella hospitum or guest chamber must be in existence before a new house was colonised.
Here, then, at Halesowen was accommodation for visiting abbots and priors of other houses of the Order, for the Bishop of Worcester claiming hospitality on the occasions of his visitations, for nobles and gentry having business with the convent, for royal messengers carrying letters under privy seal, and even for the monarch himself, should he find himself more than a days journey from the manor he proposed to visit. Halesowen was once thus honoured, for Edward III was entertained there on the occasion of one of his forays into Wales.
Admission of guests to the Abbey was the responsibility of the canon who held the office of porter. On a guest's arrival, his task was to open the great gate, ask the visitor his name, take him into the Abbey church for prayer and then hand him over to the Hospitaller. The latter would show the guest his accommodation, offer him refreshment and make all other necessary provisions for his comfort. This important member of the staff had, of course, such assistants and servants as were necessary for the efficient discharge of his duties. To return for a moment to the porter, another of his responsibilities was to dispense alms to poor and needy persons at the Abbey gates. He also had to collect scraps from the abbey kitchens after meals and distribute them to indigent callers.
While there is no doubt that in a somewhat primitive society, England's abbeys provided the rudiments of a welfare system, it is also abundantly clear that their hospitality was primarily extended to members of the upper classes. The heavy expenditure incurred in this way was frequently cited as evidence of the poverty of a house when exemption was sought from taxation, or when the appropriation of yet another parish church was sought.
Halesowen's records contain examples of somewhat lavish entertainment of the nobility. Early in 1366, the kitchen accounts record the expenditure of 3s. 7d. on luxuries (specialia) for the visit of Sir Richard and Lady Fitton. Later that year my Lord of Dudley and his lady stayed at the abbey. They seem to have been welcome guests, for what was a large sum in those days - twelve pence - was given to the boy who heralded the party's approach. During the week of the stay, the kitchen used the carcase of a cow (6s.), a calf (2s. ld), pork (4s.), a sheep (2s. 2d), three sucking pigs (4s. 6d.), ten geese (1s. l0 ½ d.), herrings (5d.) and the astonishing total of 750 eggs (3s. 4d.). Wine, too, was provided to the tune of 6s. 8d. The staple drink of the canons was, of course, beer, and ten shillings was spent on the provision of this beverage during the period 6th May - 30th September 1366.
It can be imagined that, at a time when 750 eggs could be purchased for 3s. 4d., ten shillings would procure quite a gallonage of ale!
We find that in 1343 the Abbot of Hales was complaining of the heavy expenses the house incurred in providing for the needs of strangers and wayfarers. Some evidence in support of this can be gleaned from the record of a much later visitation in 1489 when the following figures for food consumption are recorded (and it should be borne in mind that the Abbey's regular inmates never numbered more than about 17):
20 bushels of wheat and rye for bread used weekly.
1,110 quarterns of barley used anually
60 oxen used anually
40 sheep used anually
30 swine used anually
24 calves used anually
A bushel was a measure of capacity equalling eight gallons. A quartern equalled four pounds.
There would appear to be no lack of protein in the monastic diet of those times! Much, if not all, of this food would come from the Abbey's farms at Home Grange (now Goodrest Farm), Warley Grange, Hill Grange, Owley Grange, Farley Grange, Witley Grange, Uffmore Grange, Rudhall Grange and Blakeley Grange.
It is clear from monastic records in general that, among the country's nobility, a short stay at a monastery was a recognised form of holiday, and those who were liberal to the house were sometimes given an open invitation to come and stay whenever they chose. From this it was only a small step to the state of affairs where a religious house would grant permanent accommodation to a benefactor (or to a benefactor's nominee) where suitable payment had been made in cash or in land. Such grants of food and lodging for life were known as corrodies and during its history, Halesowen was burdened by a number of such "paying guests".
What is perhaps significant in summing up this aspect of the place of monastic houses in mediaeval society is that when dissolution was imminent, very few protesting voices were raised, and of these the majority did so on grounds of the value of the institutions as places of hospitality.
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Tiffany gets the listeners through their day with her quirkiness and bubbly personality. Tiffany is not only an on-air personality but she is also a mix DJ that performs under the name DJ Tiffalicious.
From Noon to 1PM, she is the host of the All Request Eat 2 The Beat. This is a long standing feature on Hot Z95 and listeners love what they hear each day. DJ Tiffany is our female voice, through the afternoon. | {
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1949 Johnny 2022
Johnny Winberry Butts
October 26, 1949 — September 22, 2022
Johnny Winberry Butts, 72, of Sanford, passed away on Thursday, September 22, 2022 at Central Carolina Hospital.
Mr. Butts was born in Kinston, North Carolina on October 26, 1949, the son of Jesse R. Butts, Sr. and Willie Winberry Butts.
Johnny was a stand-out athlete at J.F. Webb High School in Oxford, North Carolina, where he was named the North Carolina Coca-Cola Football Player of the Year and All-Conference in football and baseball. He was also heavily involved with the Future Farmers of America during high school serving as a chapter and state officer.
Johnny was a proud U.S. Marine Veteran serving his country from 1969 -1971. He worked as a licensed real estate appraiser. A former long-time Siler City resident, he was an active member of the Chatham County Democratic Party throughout the 1990's, serving as the group's Chairman for several years. Johnny was an avid reader and devoted NC State and sports fan. He volunteered many years as a youth sports coach for football, basketball, baseball and softball teams in Chatham and Lee Counties.
In addition to his parents, he is preceded in death by his brother, Jesse R. Butts, Jr., and niece, Jessica Butts.
He is survived by his daughters, Natalie Simpson and husband Jimmy of Chapel Hill, and Caroline Butts of Morehead City; son, Johnathan Butts of Myrtle Beach; brothers, Will Butts of Greensboro, and Tim Butts of Sanford; grandchildren, Jack, Drew, and Charlotte Simpson, and several nieces and nephews.
A private family service will be held at a later date.
Smith & Buckner Funeral Home is assisting the Butts family.
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Puffin Island (iriska: Oileán na gCánóg) är en ö i republiken Irland. Den ligger i grevskapet Ciarraí och provinsen Munster, i den sydvästra delen av landet, km sydväst om huvudstaden Dublin.
Klimatet i området är tempererat. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är juli, då medeltemperaturen är °C, och den kallaste är januari, med °C.
Källor
Externa länkar
Öar i Munster | {
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PLATO
_Julia Annas_
New York / London
**www.sterlingpublishing.com**
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Published by Sterling Publishing Co., Inc.
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© 2003 by Julia Annas
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Frontispiece: Bust of Plato
# CONTENTS
•
**ONE** Arguing with Plato
**TWO** Plato's Name, and Other Matters
**THREE** Drama, Fiction, and the Elusive Author
**FOUR** Love, Sex, Gender, and Philosophy
**FIVE** Virtue, in Me and in My Society
**SIX** My Soul and Myself
**SEVEN** The Nature of Things
_References_
_Further Reading_
_Picture Credits_
**In reading Plato,** it is important to pay attention to the role of argument. This undated mural painting by Pierre Puvis de Chavannes showing Plato conversing with a student at the Academy hangs in the Boston Public Library.
# ONE
# Arguing with Plato
## The Jury's Problem
Imagine that you are on a jury, listening to Smith describe how he was set upon and robbed. The details are striking, the account hangs together, and you are completely convinced; you believe that Smith was the victim of a violent crime. This is a true belief; Smith was, in fact, attacked.
Do you _know_ that Smith was attacked? This might at first seem like an odd thing to worry about. What better evidence could you have? But you might reflect that this is, after all, a courtroom, and that Smith is making a case which his alleged attacker will then try to counter. Can you be sure that you are convinced because Smith is telling the truth, or might it be the way the case is being presented that is persuading you? If it is the latter, then you might be worried; for then you might have been convinced even if Smith had not been telling the truth. Besides, even if he is telling the truth, is his evidence conclusive as to his being attacked? For all you know, he might have been part of a setup, and it's not as though you had been there and seen it for yourself. And so it can seem quite natural to conclude that you don't actually _know_ that Smith was attacked, though you have a belief about it which is true, and no actual reason to doubt its truth.
• • • • •
### THE _THEAETETUS_
The _Theaetetus_ is one of Plato's most appealing dialogues, but also one of his most puzzling. In it, Socrates says that he is a midwife like his mother: he draws ideas out of people, before testing them to see whether they hold up to reasoned examination. Refusing to put forward his own ideas about what knowledge is (though displaying sophisticated awareness of the work of other philosophers), he shows faults in all of the accounts of knowledge suggested by young Theaetetus. Pursuing the thought that if you _know_ something, you can't be wrong, Theaetetus suggests that knowing might be perceiving; then having a true belief; then, having a true belief and being able to defend or "give an account of" it. All these suggestions fail, and the dialogue leaves us better off only in awareness of our own inability to sustain an account of knowing. Socrates' insistence on arguing only against the positions of others, not for any position of his own, made the dialogue a key one for the Platonic tradition which took Plato's inheritance to be one of seeking truth by questioning those who claim to have it (as Socrates often does in the dialogues) rather than by making his own philosophical claims. Others, noting that in other dialogues we find positive, ambitious claims about the nature of knowledge, thought of the _Theaetetus_ as clearing away only accounts of knowledge that Plato took to be mistaken. Socrates here, the midwife of others' ideas who has no "children" of his own, seems very different from the Socrates of other dialogues such as the _Republic,_ who puts forward positive ideas quite confidently. Readers have to come to their own conclusions about this (some ancient and modern solutions are discussed in Chapter 3).
**In the _Theaetetus,_** rather than espousing and supporting with philosophical argument his own idea of what knowledge is, Socrates simply refutes the many attempts of the young Theaetetus to define knowledge. Socrates is shown here discussing a philosophical premise with two pupils in a manuscript from the first half of the thirteenth century.
**The Areopagus (Hill of Ares),** the original four-hundred-member supreme tribunal of Athens (later increased to 501 members), is believed to have met in the open air on this rocky hill in Athens.
• • • • •
In his dialogue _Theaetetus_ Plato raises this issue. What can knowledge be, young Theaetetus asks, other than true belief? After all, if you have a true belief you are not making any mistakes. But Theaetetus is talking to Socrates (of whom more in Chapter 2) and, as often, the older man finds a problem. For persuading people in public is something that can be skillfully done. He means the skill of what we would call lawyers, although he is talking about a system in which there are no professional lawyers. The victim had to present his own case, though many people hired professional speechwriters, especially since they had to convince a jury of not 12 but 501 members.
• • • • •
### HOW WE REFER TO PLATO'S WORKS
In 1578 the publisher Henri Etienne, the Latin form of whose surname is Stephanus, produced the first printed edition of Plato's works in Paris. The new technology enabled a much greater number of people than hitherto to read Plato. And for the first time it became possible to refer precisely to passages within dialogues, since readers were for the first time using the same pagination. We still refer to the page on which the passage appeared in Stephanus's edition (for example, 200), together with one of the letters _a_ to _e_ , which served to divide the page into five areas from top to bottom. "Stephanus numbers" are printed in the margins of most Plato texts and translations, and a reference such as "200e" enables readers to find a passage no matter what the pagination of the book they are using.
• • • • •
Socrates continues:
**_SOCRATES_ :** These men, at any rate, persuade by means of their expertise, and they don't teach people, but get them to have whatever beliefs they wish. Or do you think that there are any teachers so clever as to teach the truth about what happened adequately, in the short time allowed, to people who weren't there when others were robbed of their property or violently attacked?
**_THEAETETUS_ :** No, I don't think they could at all, but I think they could persuade them.
**_SOCRATES_ :** And by persuading them don't you mean getting them to have a belief?
**_THEAETETUS_ :** Of course.
**_SOCRATES_ :** Well, when a jury has been persuaded fairly about something about which you could only have knowledge if you were an eyewitness, not otherwise, while they judge from what they've heard and get a true belief, haven't they then judged without knowledge, though they were persuaded of what's correct, since they made a good judgment?
**_THEAETETUS_ :** Absolutely.
**_SOCRATES_ :** But look, if true belief and knowledge were the same thing, then an excellent juryman wouldn't have a correct belief without knowledge. As it is, the two appear to be distinct.
_(Theaetetus 201a—c)_
This sounds convincing, indeed perhaps blindingly obvious. But, like the jury, we can raise the question of whether we should be convinced. Why _don't_ the jury know that Smith was robbed?
**Socrates challenges Theaetetus** in his attempt to provide a definition of knowledge, using the example of a trial jury being persuaded to adopt a belief, and then judging based on that belief. In this Jéan-Léon Gerôme painting from 1861, the courtesan Phryne stands trial for profaning a ceremony dedicated to the goddesses Demeter and Persephone.
## What Is Required for Knowledge?
One reason put forward by Plato for the claim that the jury lack knowledge is that they have been persuaded, by someone whose main aim it is to get them to believe what he wants them to believe. In this case he has persuaded them of the truth, but we may think that he would have been able to persuade them even if his story hadn't been true. At first this worry may seem far-fetched: if you have acquired a true belief in a certain way, why worry that you _might have been_ persuaded of something false in that same way? How can what didn't happen cast doubt on what did? But, in fact, this worry about the power of persuasion is serious, because it casts doubt on the route by which the belief is acquired. If it is a route by which I can acquire false beliefs as readily as true ones, then it cannot guarantee me only true beliefs. And this does raise a doubt in most people's minds that a belief that I have acquired by that route could amount to knowledge.
Another reason put forward in the passage is that the sort of fact the jury have been persuaded of, namely that Smith was attacked, is not the sort of fact that you could have knowledge of anyway unless you had been there and seen it for yourself. However convinced we are that Smith is telling the truth, all we are getting is a version that is secondhand, and conveyed by an entirely different kind of route from Smith's own. He experienced and saw the robbery; we are only being told about it. However vivid the telling, it's still just a telling; only somebody who was there and saw it can have knowledge of it. Again, this may at first seem far-fetched. If we limit knowledge to what we can actually experience firsthand for ourselves, then there won't be much that we can know; nothing that we read or hear secondhand will count. Yet there is a powerful thought being appealed to here, one that can be expressed by saying that nobody else can know things for you or on your behalf. Knowledge requires that you acquire the relevant belief for yourself. What it is to acquire a belief for yourself will differ depending on the kind of belief it is, but with the belief that Smith was robbed the only way you can acquire it for yourself with no intermediary is, it seems, to be there yourself and actually see it.
## A Problem for Us
Plato has given us two kinds of reason for rejecting the idea that the jury's true belief could amount to knowledge. Both are strong, but how well do they go together? The problem with persuasion was that it turned out to be a route that could not guarantee that the beliefs we acquired from someone else would be true. But for this to be a problem with _persuasion_ there has to be the possibility of a route of this kind that did have such a guarantee. Socrates complains that the victim has to convince the jury in too short a time, and in circumstances that are too emotional and fraught, for their acquisition of beliefs to be the right kind for knowledge. This complaint is pointless unless there could be a way of acquiring beliefs that didn't have these disadvantages—say, one where there were no time constraints, and each member of the jury could examine witnesses and victim as much as they required to satisfy every last scruple. So it looks as though we are assuming that there is a way of conveying beliefs that could amount to knowledge, though it isn't persuasion.
The second point, however, suggested that _no_ way of conveying beliefs, however careful and scrupulous, could amount to knowledge, since any belief conveyed to you from another will be secondhand, and thus something that you cannot know, because you cannot know it for yourself. Relying on someone else's testimony, however sound, is never the same as experiencing the fact for yourself.
The problem now is that the second objection seems to conflict with the first. The second supposes that knowledge cannot be conveyed, but must be acquired by each person in their own case; but the first found fault with persuasion in a way suggesting that there _could be_ a way of acquiring a belief from someone else which would amount to knowledge, so that knowledge _is_ conveyable.
**The reader of the _Theaetetus_** must consider whether Plato himself knew that he has given Socrates mutually conflicting demands on knowledge and whether Plato intended that Socrates realize that as well. Plato, who was a student of Socrates', is shown here with his teacher in this English manuscript illumination from the thirteenth century.
## The Reader Comes In
At this point the reader is forced to think for herself about the passage, and about what Plato is doing. The simplest response would be to conclude that Plato has given Socrates mutually conflicting demands on knowledge because Plato is himself confused; he just hasn't noticed that he is requiring knowledge to be both conveyable and not conveyable. Unsympathetic readers may stop at this point.
We might probe a little further, however. For one thing, Socrates in this dialogue repeatedly stresses that he is not putting forward positions of his own, only arguing against those of others. He produces two objections to Theaetetus's suggestion that true belief might amount to knowledge. Each is powerful against that suggestion. Do we have to suppose that Plato, the author, was unaware that these objections run up against each other? Not necessarily (and if we do not have to suppose the author unaware of this, we also do not have to suppose that he intended to portray Socrates as unaware of this problem—though this is a further matter, on which readers may disagree). And given the sophisticated level of argument in the _Theaetetus,_ the reasonable course is to suppose that Plato was aware of how these two objections are related.
Why then does Plato not appear to think that it matters? Here we have to take seriously Socrates' stress in the dialogue that he is only arguing against the views of others. This does not mean that he has no ideas on the subject himself, but it does mean that the point of the dialogue is not to put these forward. The problem we find when we reflect on Socrates' two grounds for rejecting Theaetetus's suggestion doesn't undermine the conclusion that that suggestion won't do; they do show that when we, or Plato, are working on a positive account of knowledge we need to be aware of this problem.
In another dialogue, the _Meno,_ we find the claim that knowledge is teachable (87b—c), where this is a firmly accepted point. But it is also in the _Meno_ that we find one of Plato's most famous ideas, that knowledge is really a sort of "recollection." Socrates engages in a conversation with a boy who knows no geometry, taking him through a geometrical proof which, though simple to follow, contains a step that the boy will find counterintuitive. Having walked him through the proof, Socrates says (85c) that the boy is now in the state of having true beliefs on the subject, but "if someone asks him these same things many times and in many ways, you know that in the end he will have knowledge about them as accurate as anyone's." Socrates has taught the boy in the sense of presenting the proof to him in such a way that the boy can come to have knowledge of it for himself. The boy will not actually have knowledge until he has done something for himself—making the effort to understand the proof. The boy has to come to know the proof for himself, because only he can come to understand it for himself. Socrates can't do that. But Socrates can convey knowledge in that he can convey the proof to the boy in a way that will enable the boy to make the effort for himself. Hence we can see how knowledge can be teachable while it is still true that knowledge is something each person can achieve only for himself. In a further move, Plato calls this recollection; for when the boy comes to understand the proof, Plato holds that his soul has come to recollect knowledge it had prior to embodiment, and thus prior to the boy's actual experience. Clearly, though, the further step about recollection is not required by the argument itself; it is Plato's bold and exciting way of interpreting the results of the argument.
**In the _Meno,_** another of Plato's dialogues, Socrates takes a young boy through a geometric proof in such a way that the boy can understand and have knowledge of it himself. This detail of an illustration at the beginning of Euclid's _Elementa_ (1309—16), in the translation attributed to twelfth-century English scholar Adelard of Bath, shows a woman, apparently a geometry teacher, and possibly the personification of geometry, instructing a group of students with a set square, dividers, and a compass.
## Arguing with Plato
In many ways, the jury passage in the _Theaetetus_ provides a good introduction to Plato's way of writing. We find right away that it is important to pay attention to the way in which Plato writes, particularly to the role of argument in supporting one's own position or attacking those of others. We find also that the reader is drawn into the argument herself, needing to challenge Plato's arguments even where Socrates in the dialogue easily wins.
The brief mention I have made of the _Meno_ argument introduces us to another feature of Plato's writing. In the _Theaetetus,_ Plato uses the point that knowledge is conveyable, and also the point that knowledge requires firsthand experience of one's own. If we follow this through with an everyday example, like the jury's judgment about the crime, we find problems. In the _Meno_ we find both points in a context in which they are not in conflict. But the context there is a geometrical proof—an example of knowledge that is very different from the jury's judgment. A geometrical proof is something articulated, abstract, and far removed from everyday experience. There is something substantial to understand, and to convey. It is no accident that when Plato struggles with the concept of knowledge, he tends to conclude that what meets his standards for knowledge is far more restricted than the range of things we normally assume that we know. If we think about the differences between the jury example and a geometrical proof, we can see why he tends to do this. For example, the notion of understanding has less scope with an everyday example in which knowledge just comes down to seeing the crime.
Plato is perhaps best known for what is often called the "Theory of Forms," a set of striking claims about what is real and what we can know. Forms, of which we shall see more, do not figure in the _Meno_ or _Theaetetus,_ but we can detect in these works lines of thought that make Plato's claims about Forms, when we encounter them, more understandable.
Plato writes in a way which involves us in argument with him. He also puts forward philosophical claims that have seldom been matched for their boldness, and for the imaginative manner in which they are expressed. (The idea that knowledge is "recollection" is one of the most famous of these.) Interpretations of Plato tend to overemphasize one of these aspects at the expense of the other. At times, he has been read as interested solely in engagement with the reader, and distanced from any positive ideas. At other times, he has been read as a bold theorist striding dogmatically ahead, indifferent to argument. What is difficult and also rewarding to bear in mind about Plato is that he is intensely concerned both with argument and with bold ideas, in a way that is subtle and hard to capture without simplification. This introduction to Plato does not pretend either to cover all of Plato's ideas or to provide a recipe for interpreting him, but rather aims to introduce you to engagement with Plato in a way that will, I hope, lead you to persist.
**A geometrical proof,** in contrast to a jury's judgment, is articulated, abstract, and far removed from everyday experience. Nino Pisano's allegory of geometry with Euclid at his desk is shown in this marble panel from the east side of the lower basement of the bell tower at the Museo dell'Opera del Duomo in Florence, Italy.
**Biographies of Plato** did not turn up until several generations after the philosopher's death, so their information is unreliable at best. This late-fifteenth-century vellum from _Histoire des Philosophes_ depicts Plato at his desk.
# TWO
# Plato's Name, and Other Matters
## Name or Nickname?
Plato's name was probably Plato. The "probably" may surprise you; how can there be any doubt? Plato's writings have come down to us firmly under that name. But within the ancient biographical tradition there is a surprisingly substantial minor tradition according to which "Plato" was a nickname which stuck, while the philosopher's real name was Aristocles. This is credible; Plato's paternal grandfather was called Aristocles, and it was a common practice to call the eldest son after the father's father. We have, however, no independent evidence that Plato was the eldest son. And "Plato" does not appear to be a nickname; it turns up frequently in the period. Further, the explanations we find for it as a nickname are unconvincing. Because "Plato" suggests _platus_ , "broad," we find the suggestion that Plato had been a wrestler known for his broad shoulders, or a writer known for his broad range of styles! Clearly this is just guessing, and we would be wise not to conclude that Plato changed his name or had it changed by others. But then what do we make of the Aristocles stories? We don't know, and can't tell. And this is frustrating. A change of name is an important fact about a person, but this "fact" slips through our fingers.
Our ancient sources about Plato often put us in this position. There are plenty of stories in the ancient biographies of Plato, and frequently they would, if we could rely on them, give us interesting information about Plato as a person. But they nearly always dissolve at a touch.
## Facts and Factoids
Plato was born in Athens in 427 BCE and died in 347; we are fairly well informed about his family.
• • • • •
### PLATO'S FAMILY
Both Plato's father, Ariston, and his mother, Perictione, came from old Athenian families. Plato in the _Critias_ makes much of his family's descent from the sixth-century statesman Solon, who brought about reforms that put Athens on the road to eventual democracy. Plato had two full brothers, Glaucon and Adeimantus, to whom he allots parts in the _Republic._ After Ariston's death, Perictione married Pyrilampes, who was already the father of a son called Demos (referred to in the _Gorgias)._ By Pyrilampes Perictione had a son Antiphon, Plato's half brother, who took up philosophy but quickly lost interest; he is given the role of narrating the entire conversation of the _Parmenides._ Pyrilampes had strong democratic sympathies (Demos is Greek for "The People"). After Athens's utter defeat in the long-drawn-out Peloponnesian War in 404 BCE, antidemocratic sympathizers brought about a coup and set up a government of thirty (known as the Thirty Tyrants). Perictione's brother Critias and cousin Charmides (both of whom have parts in the _Charmides)_ were among them. Plato thus came from a family divided by the civil war. We do not know his own political views, though this has not stopped much speculation about them. It is plausible that he was alienated from the restored democracy by Socrates' execution under it.
**Plato was a descendent** through his mother of the Athenian statesman lawmaker Solon (ca. 630—ca. 560 BCE), who is shown here with three students in an Arabic manuscript from the first half of the thirteenth century.
• • • • •
**Although we do not** know much about Plato as a person, we do know that he came from a family divided by civil war. Following Athens's defeat by Sparta in the Peloponnesian War, a battle of which is depicted in this ca. 1900 drawing of a naval conflagration in the harbor of Syracuse, his mother's uncle and cousin served in the antidemocratic Thirty Tyrants government, which was established by coup.
He was regarded as an outstanding philosophical and literary figure from early on, someone around whom stories gathered. However, it was not until several generations had passed that we find what we would call biographies, claiming to give narratives about Plato the individual; in Plato's own time this kind of interest had not developed. By then very few facts about Plato would have been accurately recoverable, but people had begun to want to know about the person behind the dialogues (as many of us still do). So we find narratives of Plato's life in which facts about his life are appealed to, often in order to explain why a passage in one of the dialogues says what it does, particularly if there is no other obvious reason for its being there. Thus we find, for example, the claim that Plato went on a journey to Egypt seeking wisdom. There is nothing implausible about this. On the other hand, it is a claim made about many ancient philosophers, particularly in later antiquity with the growth of the idea that Greek wisdom originally came from older, Eastern countries. A passage in the dialogue _Laws_ may suggest that Plato had actually seen the stylized Egyptian art which he prefers to the innovations of Greek art, but it does not compel us to that conclusion. We simply do not know whether we have a fact that sheds light on the _Laws_ passage, or a factoid created later from that passage.
• • • • •
### PLATO ON GREEK AND EGYPTIAN ART
The Athenian in the _Laws,_ the dialogue's main speaker, claims that the Greeks have much to learn from the way the Egyptians codified artistic styles and stuck to them, as opposed to the restless craving for originality and new styles marking Greek art of his day.
**_ATHENIAN_ :** Long ago, it seems, [the Egyptians] recognized this principle of which we are now speaking, namely that the movements and songs that young people in cities practice habitually should be fine ones. They drew up a list of what these are and what they are like and displayed it in the temples. Painters and others who produce any kinds of forms were forbidden to innovate or invent anything nontraditional; and it still is forbidden both to them and in the arts in general. If you look, you will find that things painted or sculpted there ten thousand years ago—and I mean literally ten thousand—are not at all better or worse than what is produced now, but are produced according to the very same skill.
**_CLEINIAS_ :** It's amazing, what you say.
**_ATHENIAN_ :** Rather, an exceptional product of legislative and statesmanlike skill.
_(Laws 656d—657a)_
Some of this suggests that Plato had seen Egyptian art; some suggests that he had not. It does not matter for his point: fixed stylization in art is preferable to a developing tradition valuing originality.
**Whether Plato had traveled** to Egypt seeking wisdom and had seen Egyptian art while he was there is a matter of conjecture. We must garner what knowledge we can of the philosopher from his dialogues. In his dialogue _Laws,_ for instance, he indicates a preference for the fixed, stylized art of ancient Egypt, shown, for example, in the triad statue of the pharaoh Menkaura, accompanied by the goddess Hathor (on his right) and Parva, the personification of the nome (province) of Diospolis (on his left), over the developing tradition valuing originality of ancient Greece, shown, for example, in the statue of the goddess Eirene (the personification of peace) bearing Plutus (the personification of wealth), a Roman copy after a Greek votive statue by Kephisodotos (ca. 370 BCE).
• • • • •
This matters chiefly in that we do not have independent access to Plato's individual personality as we do for more recent philosophers. In the dialogues he never speaks in his own voice. Whatever we make of this, we cannot evade it by appealing to his life; our views of his life are irrevocably contaminated by the dialogues.
## Different Interpretations
Two very differing interpretations are nearly contemporary with Plato himself. His nephew Speusippus, who succeeded him as head of his philosophical school, held that Plato's real father was not Ariston but the god Apollo. A whole corresponding tradition grew up: Plato was born on Apollo's birthday; bees came and sat on his infant lips; his teacher Socrates dreamed of a swan, Apollo's bird, just before meeting Plato. Thinking of Plato as semidivine, alien to us, is not so startling in a world in which great families claimed descent from the gods. It makes the point that would be made in later times by saying that Plato was a genius, somebody altogether out of the ordinary, with talents that transcend the historical circumstances of his birth and upbringing. A similar tradition grew up at some point around Pythagoras. Plato is seen as a more than human figure because of the profundity of his thought and the grandeur of his philosophical conceptions. In this way of looking at him, what matter most are the large pronouncements, rather than arguments and the idea of seeking for the truth. In late antiquity, particularly, Plato was seen as this kind of towering figure, a superhuman Sage. It is not too hard to find passages in Plato's writings that can inspire this sort of interpretation (particularly in the _Timaeus)._
**Plato's nephew Speusippus,** who succeeded him as head of Plato's philosophical school, held that Plato's father was not Ariston, but the god Apollo, which led to a tradition of thinking of Plato as semidivine. Apollo, who is the Greek and Roman god of sunlight, prophecy, music, and poetry, is shown here with two muses in an oil on canvas from 1741 by Italian painter Pompeo Batoni.
Probably contemporary with the "son of Apollo" interpretation is the utterly different one found in the so-called Seventh Letter. Among the body of works by Plato that have come down to us are thirteen works purporting to be letters by him to various people. Most of them are of a much later date, but two, the seventh and eighth, contain no definitive anachronisms. The "Seventh Letter" contains what purports to be an autobiographical account by Plato of his early disillusionment with politics, and his attempts, during mysterious visits to the Sicilian city of Syracuse, to persuade the tyrant Dionysius II to submit to constitutional rule. Whether authentic or not, the letter was accepted by many in the ancient world as illuminating Plato's own very idealistic approach to political philosophy. In the last two centuries it has formed the basis for a stronger view, that Plato's impetus to philosophy in the first place was basically political, but this claim is clouded by the persistent authenticity problems. It is a mistake, in any case, to think of it as a psychologically revealing account of an individual experience; it is a rhetorical exercise in defending Plato and Dionysius's opponent Dion, part of a debate of which we have only one side.
We can easily see why the "political" interpretation has seemed more credible and appealing to modern scholars than the "son of Apollo" interpretation, and the former has been widespread for many years. It fits our ways of thinking better to see Plato's philosophy as politically motivated than it does to see it as the work of a transcendent genius (let alone a god!). We should hesitate, however, to claim that the "Seventh Letter" takes us behind the dialogues and gives us the "real" Plato in a way that suggests that his own nephew was wrong. Interpretations of Plato are contested. They were probably contested before he was dead.
## Socrates and the Academy
We do have two relatively firm points to grasp in approaching Plato. One is the great influence on him of the Athenian Socrates, and the other is his founding of the Academy, the first philosophical school.
**One perspective** from which to approach Plato is the great influence that the Athenian philosopher Socrates had on him. This marble bust of Socrates, a Roman copy after a Greek original, is from the fourth century BCE.
### SOCRATES
Socrates (about 468–399 BCE) was the son of a stonemason and a midwife. His wife, Xanthippe, has an aristocratic name, and at one point he had the money to serve as a heavy-armed soldier, but by the end of his life he was poor. Plato has Socrates in his _Apology (Defense)_ ascribe this to his devotion to philosophy, to the neglect of his own affairs. He had three sons; later tradition gives him a second wife, Myrto.
Socrates was tried and executed under the restored democracy in 399. It has often been suspected that he was unpopular because of his association with people who had overthrown the democracy, but the circumstances are unrecoverable. He was found guilty on vague charges of introducing new divinities and corrupting the youth. The first charge probably relates to Socrates' "divine sign" _(daimonion)_ , which at times held him back.
Socrates quickly became the symbolic figure of the Philosopher, the person devoting his life to philosophical inquiry and willing to die for it. He became a figurehead for many different schools of philosophy; each could find their own ideas or methods in Socrates, who left no writings. He was a controversial person, inspiring both dislike and devotion. The comic dramatist Aristophanes wrote an unpleasant play, _The Clouds,_ about him, and he was attacked after his death. Many of his associates produced "Socratic writings" to defend his memory. We have some fragments by his followers Aeschines and Antisthenes, who, along with another follower, Aristippus, went on to develop very different kinds of philosophy. Our main sources, however, are Xenophon and Plato. Disputes as to who gives the "truer" picture of Socrates are futile; Socrates was from the first a figure onto whom very different positions could be projected, and the differences between Xenophon's Socrates and Plato's are best seen as differences between Xenophon and Plato.
**After his death,** Socrates became the symbolic figure of the Philosopher, one willing to die rather than compromise his values, and for this, Plato held him in high esteem, using him as the main figure in most of his dialogues. Italian Baroque painter Salvator Rosa's late-seventeenth-century painting shows Socrates taking poison as ordered by the Athenian court.
In Plato's dialogues Plato himself is never a character, and Socrates is usually the chief figure, in dialogue which is sometimes direct and sometimes narrated, by others or by Socrates himself. Plato's Socrates varies enormously between dialogues. Sometimes he is a persistent questioner of others' positions; sometimes he puts forward his own views confidently and at length; sometimes he is merely a bystander. Plato was always inspired by Socrates as the ideal figure of the philosopher, but his views as to what the tasks and methods of philosophy should be are not constant, and so Socrates appears in a variety of roles. In the dialogues in which Socrates is marginal, Plato's conception of the philosopher goes beyond what he thinks Socrates can plausibly represent. And where Socrates is the main figure it is wiser to think of Plato as developing different aspects of what Socrates represents for him than to ask how close he is to (or far from) the "real" Socrates.
**Socrates was a** controversial figure, inspiring both dislike and devotion. Comic dramatist Aristophanes, shown here in an undated bust portrait, fell into the former camp; his play about Socrates, _The Clouds,_ was not flattering.
**Socrates left no** writings of his own. Our main sources for his ideas and methods are Plato and Greek historian Xenophon (ca. 431—ca. 352 BCE), who is shown here in an engraving by George Cooke published in 1810 in _The Historical Gallery of Portraits and Paintings._
• • • • •
Socrates thought of himself as seeking for the truth. He looked for it, however, in a radically new way. Refusing to produce grand theories of the world, or philosophical treatises—refusing, indeed, to write anything philosophical—he sought the truth by talking to individuals and pressing on them the importance of understanding what was being talked about. Plato was obviously impressed by Socrates' insistence that the grander tasks of philosophy will have to wait until we achieve understanding of what we take for granted—courage, justice, and other virtues, the idea of living a good life, our own claims to understanding. Socrates identified the philosophical life as one of continuing inquiry and investigation, into others' beliefs and one's own. Plato was profoundly impressed by Socrates' insistence on putting inquiry before doctrine, and the search for understanding before ambitious claims. Socrates also took the philosophical life as one to be lived seriously, and died rather than compromise his values in defending his life. The best measure of Plato's respect for Socrates is the fact that in most of the dialogues he wrote Socrates is the main figure, and there is only one dialogue (the Laws) in which he does not appear at all. Rather than write in his own person, Plato chose always to present Socrates as the figure of the philosopher searching for truth.
At some point in his life, which we cannot pinpoint accurately, Plato made two momentous decisions. He rejected his family and civic duty of marrying and producing heirs. (Modern readers are unsurprised that Plato never married, because his writings seem so obviously homosexual in temperament. But in ancient Athens marriage was a duty for the continuance of the family and the city, and had nothing to do with personal sexuality. In not marrying, Plato was giving up having posterity of his own, a great loss in his society.) And he founded the first school of philosophy, called the Academy after the gymnasium where it met.
**Plato made two critical decisions** at some point in his life. First, he decided to forgo marriage and a family, a dramatic choice in Athenian society at the time because he would be left with no heirs. Second, he founded the Academy, the first school of philosophy, and though this painting portrays Plato as a teacher in the Academy he founded, we know very little about the school's organization.
**Greek philosopher Aristotle** (384-322 BCE) was a student of Plato's, and although he disagreed fundamentally with some of Plato's ideas, he remained at the Academy for twenty years. This marble bust of Aristotle is a Roman copy after a Greek bronze original by Lysippos from 330 BCE; the alabaster mantle is a modern addition.
We know very little about the organization of the Academy, and academics of every generation have been tempted to see in it some of the structure of their own university system. Aristotle was there for twenty years, and when we hear of him teaching we are tempted to think of him as an advanced graduate student or junior professor; but we should remember that the Academy was always a public gymnasium, and that it is unlikely that Plato's school had many of the institutional features of a modern university. Plato did not charge fees, but only those wealthy enough to spend time on philosophy were able to attend for long. We know of one public lecture Plato gave, on "The Good," which was a fiasco because the audience came to hear about the good life, while Plato talked about mathematics. We have a parody of students in the Academy defining a vegetable. Otherwise, the picture we get of the Academy is of a center for discussions, with no indication that students went there to learn Platonic doctrines. Indeed, perhaps "students" is a misnomer; the first center of further education was in a world without degrees, grades, credentials, or tenure.
It is easy to see the founding of a philosophical school as being in tension with Plato's devotion to the memory of Socrates. Socrates, after all, rejected everything in philosophy that could be thought of as academic. Yet as Plato presents Socrates, seeking the truth through inquiry does not, as we shall see, preclude having positive opinions of your own. And the Academy was not a place where those who came had to learn to agree with Plato. Not only Plato's greatest pupil, Aristotle, but the next two heads of the Academy disagreed quite fundamentally with some of Plato's ideas. So the Academy can be seen as a school for learning to think philosophically, and so to continue in the tradition of Socrates.
In one respect, however, Plato can be said to have moved on quite decisively from Socrates, who lost interest in the theories of his time about the nature of the world and focused on questions of how to live. In the ancient world Plato was thought of as the first systematic philosopher, the first to see philosophy as a distinctive approach to what were later to be called logic, physics, and ethics. If we look at the dialogues as a whole, we can indeed see a large and systematic set of concerns—systematic in that they are a continuing set of concerns, though not a set of organized dogmas. Both in antiquity and later, some have further systematized Plato's ideas as a set of doctrines, generally referred to as "Platonism," but this is a step Plato himself never takes. He leaves us with the dialogues, and we have to do for ourselves the work of extracting and organizing his thoughts.
Plato is the first thinker to demarcate philosophy as a subject and method in its own right, distinct from other approaches such as rhetoric and poetry. He is sometimes said to have been the inventor of philosophy because of this insistence on its difference from other forms of thought, and he seems to have been the first to use the word _philosophia_ , "love of wisdom," to capture what he has in mind. He is certainly the inventor of philosophy as a subject, as a distinctive way of thinking about, and relating to, a wide range of issues and problems. Philosophy in this sense is still taught and learned in schools and universities today.
**In one sense,** the founding of a philosophical school could be seen as in being in tension with Plato's devotion to the memory of Socrates, since Socrates eschewed all things academic in philosophy. This image from 1874 shows Plato meditating at the grave of Socrates.
**Plato abuses cultural forms** that use persuasion to attract the audience to a conclusion rather than relying the intellectual force of the argument alone. Among the targets of his criticism was publicly performed dramatic and epic poetry, an example of which is shown in this hand-colored engraving of a performance at the Theater of Dionysus in ancient Athens from the 1891 German encyclopedia _Pierers Konversationslexikon,_ edited by Joseph Kürschner.
# THREE
# Drama, Fiction, and the Elusive Author
## Theory and Practice
Plato goes out of his way many times to insist that philosophy is the search for truth, using methods of argument. At different times he puts forward different candidates for the best philosophical method, which he often calls "dialectic," but he never compromises on the point that philosophy has a different (and higher) aim, and a more austere method, than what he sees as its main cultural competitors. There has always been hostility, he says at the end of the _Republic,_ between philosophy and poetry (he means publicly performed dramatic and epic poetry, not the private reading of short poems). And in the _Gorgias_ and _Phaedrus_ he establishes, in different ways, strong opposition between philosophy and the practice of rhetoric. Philosophy aims only at the truth, not at mere persuasion regardless of truth, which is a dubious enterprise in both its intentions and its methods. (Recall the jury's problem in Chapter 1.) Perhaps Plato is not so much building on already recognized distinctions between philosophy and other kinds of intellectual activity, as actually establishing them, by his pioneering of the idea that philosophy has its own aims and methods, that it forms a distinct, and distinctive, subject which we should demarcate from other ways of thinking. In any case, few philosophers have stressed as much as Plato the need to distinguish philosophy's procedures sharply from procedures that produce agreement by persuasive, nonrigorous means.
And yet Plato is the most "literary" philosopher, the philosopher most accessible to nonspecialists because of the readability and charm of (at least some of) his writings. Some of his works are as famous for their literary as for their philosophical aspects. Even the more subdued contain metaphors, comic passages, and other attention grabbers.
One of the most striking things about his works, moreover, is that they are all cast in a dramatic form—either a dialogue between two or more people or a monologue, sometimes reporting others' dialogue. Many of these writings characterize various speakers, guide the discussion, and keep the reader involved with great skill. Nothing could seem further from the specialized, often technical, and academic form in which most philosophers have written. Moreover, such "literary" devices seem obviously open to the objections Plato brings against the purveyors of mere persuasion: they attract the reader to the conclusions, rather than relying on the bare intellectual force of argument. How can so literary a writer be against what literature does? Is he not undermining what he himself is doing?
• • • • •
### SOCRATIC "IRONY"
Socrates is talking to Hippias of Elis, a traveling "sophist" who sets up as a professional "wise man," taking money for lessons in private and public rhetoric, and managing public business himself. How, Socrates asks, does Hippias explain the fact that wise men in the old days were not rich public figures?
**_HIPPIAS_ :** What do you think it could be, Socrates, other than that they were incompetent and not capable of using their wisdom to achieve in both areas, public and private?
**_SOCRATES_ :** Well, other skills have certainly improved, and by comparison with modern craftsmen the older ones are worthless. Are we to say that your skill—sophistry—has improved in the same way, and that the ancients who practiced wisdom were worthless compared to you?
**_HIPPIAS:_** Yes—you're completely right!...
**_SOCRATES_ :...** None of those early thinkers thought it right to demand money as payment, or to make displays of their own wisdom before all sorts of people. That's how simpleminded they were; they didn't notice how valuable money is. But each of the modern people you mention [Gorgias and Prodicus] has made more money from his wisdom than any other craftsman from any skill. And Protagoras did it even before they did.
**_HIPPIAS:_** Socrates, you have no idea just how fine this is. If you knew how much money _I've_ made, you'd be amazed!... I'm pretty sure that I've made more money than any two sophists you like put together!
**_SOCRATES_ :** What a fine thing to say, Hippias! It's very indicative of your own wisdom, and of what a difference there is between people nowadays and the ancients.
_(Hippias Major 281d-283b)_
Hippias thinks Socrates is complimenting him. The reader, however, sees clearly that Socrates despises the use of intellect to make money, rather than to search for the truth, and hence has complete contempt for Hippias. Socrates is often "ironical" in this way, operating at the level of his interlocutor in such a way that the reader can see that he does not share it. This is not always an attractive trait, but it makes for many vivid and comic passages in Plato's writing.
### PLATO'S WORKS
Unusually for an ancient philosopher, we can be fairly confident that we have all Plato's "published" works, including one unfinished fragment _(Critias)_ and some short works which were attributed to Plato after his death but contain later style and vocabulary (these are marked by *). Works about which there is less consensus, which may be by Plato, are marked by †.
We have no external indications of the order in which Plato wrote his dialogues (except that the _Laws_ seems to have been unfinished at his death). In the ancient world there was no one privileged order either for teaching the dialogues or for regarding them as a presentation of "Plato's philosophy"; much depended on the reader's interests, aptitude, and level of philosophical sophistication.
**Although we can be fairly** certain that we have all of Plato's "published" works, there is no record of the order in which they were written, and there appears to have been no established order in which the works were discussed or taught in the ancient world. Many modern-day collections follow an organization created by the Platonist philosopher Thrasyllus for the works of the ancient Greek philosopher, who is shown here in an illustration from the _Nuremberg Chronicle._
The following order of the dialogues was established by Thrasyllus, a Platonist philosopher who was also the Emperor Tiberius's private astrologer. Thrasyllus put the dialogues in groups of four for reasons which are not always clear. His order has been used by many editions of Plato's text, as well as by the Hackett translation of the complete works of Plato.
_Euthyphro, Apology (Socrates' Defense), Crito, Phaedo, Cratylus, Theaetetus, Sophist, Statesman, Parmenides, Philebus, Symposium, Phaedrus, Alcibiades, Second Alcibiades *, Hipparchus, Lovers†, Theages†, Charmides, Laches, Lysis, Euthydemus, Protagoras, Gorgias, Meno, Greater Hippias, Lesser Hippias, Ion, Menexenus, Clitophon, Republic, Timaeus, Critias, Minos*, Laws, Epinomis*, Letters†, Definitions*, On Justice*, On Virtue*, Demodocus*, Sisyphus*, Halycon*, Eryxias*, Axiochus*, Epigrams†_
• • • • •
## Detachment and Authority
We can answer this by the thought that Plato is, indeed, undermining his own philosophical activity, systematically denouncing the form he uses. We can take him to be doing this either naively, simply not noticing that he uses persuasive techniques to abuse persuasion, or else with a sophisticated theory in mind of upsetting the reader's expectations. But there is a simpler, less extreme explanation which fits much better with the content of Plato's views on knowledge.
In presenting his works in the form of dialogue (direct or reported), Plato is detaching himself, as the possessor of philosophical views, from the views of the characters. The author is obviously present in all the characters in the dialogue, since Plato is writing all the parts. The reader is presented with the development of a debate between two or more people, and so with an argument, but then it is up to her to make what she can of it; the author does not present her with conclusions to be accepted on grounds that have the author's authority.
This point has sometimes been ignored, by interpreters who abstract Plato's ideas from the dialogue form and treat them as though they were written out in treatise form. And it has sometimes been exaggerated, by interpreters who refuse to move from the dialogue to ascribing any positive ideas to Plato at all. So it is worth examining first what does _not_ follow from recognizing that Plato detaches himself from the characters' views in all his works by writing in dramatic form.
It doesn't follow that Plato is detached in the way that the author of an actual play is; he is not constructing a dramatic world in which the figures interact for our entertainment. Plato's works raise serious issues for the reader to engage with; they are meant to get the reader involved in doing philosophy, not just enjoying the drama. Hence, Plato doesn't present all the characters as equally deserving of our time and attention. Some are obnoxious or ridiculous, and others are colorless. The main character in many dialogues is Socrates, and it is obvious that he is often idealized, and put forward as the embodiment of philosophical activity in contrast to other kinds of life (what this is differs between different dialogues).
Plato's use of the dialogue form is perfectly consistent with his having a position on the issues discussed, and with his sometimes ascribing that position to Socrates. In some dialogues Socrates argues with another person, showing him that he lacks understanding of some matter on which he thought he was an expert, but Socrates himself puts forward no positive views on the issue, and may even declare that he also lacks understanding. It does not at all follow that Plato has no position on the matter. Plato uses the character Socrates in many ways, not simply to put forward his own views.
Why does Plato distance himself in this way? If he does have positions, and if it is clear enough to the reader that if anybody in the dialogues presents these views it will be Socrates, then what is the point of writing in a dramatic form? Why doesn't Plato just come out and tell us what his position is?
Plato very much wants not to present his own position for the reader to accept on Plato's authority. He was aware of philosophers who wrote authoritative treatises, telling their readers what to think about a number of large and important matters. Plato has very substantial and strongly held views on a number of issues; that is why he is so prominent in Western philosophy. But he also sees himself as a follower of Socrates, who wrote nothing, but examined the views of others, trying to get them to understand for themselves. Plato wants the reader to come to understand what is said for himself or herself. As we shall see in more detail when we consider his views on knowledge and understanding (and as we have already had a glimpse in the jury passage—recall Chapter 1), the reader is made to do his or her own work to come to understand what Plato is saying. Plato is sure that he is right on a number of issues, but he doesn't want the reader to pick up these views just because Plato says so.
It is easy to miss this point, because in some of Plato's most famous dialogues Socrates is made to expound positive positions confidently and at some length, while the people he is talking to (the "interlocutors") are given only comments like, "Quite true, Socrates." We may think that in these passages there is no real distancing; what Socrates says is just what Plato thinks. But Plato cannot know anything for you; you have to do your own work to achieve understanding of what is going on. Sometimes, indeed, the reader is aided in this by finding that Socrates' claims are contested, or that he is on the defensive, or that the overall intention of a passage, or a dialogue, is obscure. Further, formal detachment of Plato from what is being said by Socrates (or, in some works, by a Visitor from Elea) is always important, even where it is not dramatically very lively. For, even if you have worked out what Plato thinks, there is still work to do; it isn't _your_ thought, as opposed to Plato's, until you have thought it through for yourself, rather than just passively taking it in as being what Plato says. Only then can it become something you understand.
In one famous passage, Plato shows us Socrates comparing himself to a midwife, who delivers other people's ideas and tests them, rather than having "children" of his own. The metaphor doesn't imply that Socrates has no ideas of his own; it implies that he keeps two things separate: having his own ideas, and testing the ideas of others. Plato writes philosophy as he does because he is concerned to keep two things apart also: presenting his own positions, and getting the reader to come to understand them for herself. Few philosophers have presented their ideas as passionately as Plato. But he never confuses this with foisting his ideas on the reader; formally, the reader never faces Plato's own ideas, only ideas he presents in a detached way through other people.
• • • • •
### SOCRATES THE MIDWIFE
Socrates, the son of a midwife, Phaenarete, claims to practice a kind of midwifery himself.
This at least is true of me as well as of midwives: I am barren of wisdom, and it's a true reproach that many people have made about me, that I ask other people questions but never put forward my own position about anything, because I don't have anything wise to say. This is the reason for it: the god compels me to be a midwife, but has forbidden me to give birth. So I myself am hardly a wise person, and I have no such discovery either that has been born as the offspring of my soul. Take people who associate with me, however. At first some of them seem quite stupid, but as the association goes on all those to whom the god grants it turn out to make amazing progress, as others think as well as themselves. But this is clear: they have never learned anything from me; rather they have discovered within themselves many fine things, and brought them to birth. And for the delivery the god and I myself are responsible.
_(Theaetetus 150c–d)_
Some think, on the basis of passages like these, that Plato is an Academic, having no beliefs.
_(Anonymous ancient commentator on the Theaetetus)_
Why _did_ god tell Socrates, in the _Theaetetus,_ to be a midwife to others, but not to give birth himself?... Suppose that nothing can be apprehended and known by humans: then it was reasonable for god to prevent Socrates giving birth to bogus beliefs, false and baseless, and to force him to test others who had opinions of that kind. Argument that rids you of the greatest evil—deception and pretentiousness—is no small help, rather a major one.... This was Socrates, healing, not of the body but of the festering and corrupted soul. But suppose there _is_ knowledge of the truth, and that there is one truth—then this is had not just by the person who discovers it but no less by the person who learns from the discoverer. But you are more likely to get it if you are not already convinced that you have it, and then you get the best of all, just as you can adopt an excellent child without having given birth yourself.
**Greek biographer** and moralist Plutarch (ca. 46 CE—after 119) is shown in this French line engraving from 1541.
_(Plutarch, Platonic Question 1)_
## Two Traditions
In the ancient world there were two traditions of reading Plato, and of identifying yourself as one of his philosophical followers. The less familiar to us came first. After a period following Plato's death when his successors in the Academy developed their own ideas about metaphysics and morality, the Academy was (around 268 BCE) recalled by a new head, Arcesilaus, to the method of argument exemplified in the dialogues in which Socrates is shown arguing with someone but not positively stating or arguing for his own position. Arcesilaus identified this feature of Socratic argument—arguing with the other person _on his own terms,_ showing him that _he_ has a problem regardless of what you think—as the most important aspect of doing philosophy in Plato's way. He probably appealed to Plato's use of dialogue to detach himself from the positions put forward in order to hold that the positive claims we find in Plato, however confidently stated, always have the status merely of positions put forward for discussion, even where it is relatively clear that Plato thinks them correct. At any rate, he put Plato's school on a course which is, in ancient terms, "skeptical"—that is, inquiring and questioning the credentials of others' views, rather than committed to particular philosophical beliefs of one's own. This "New" or Skeptical Academy continued as Plato's school, teaching people to argue against current dogmas, until the institution came to an end in the first century BCE.
Not until Plato's own school had ended do we find a tradition starting, called "Platonist" as opposed to the inquiring "Academy," in which interpreters think of Plato's works as putting forward a system of ideas, taken to be "Platonism." For this tradition, it is Plato's positive claims that are interesting, not just his insistence on argument to demolish the claims of others and to enable one's own understanding of others' positions. From the first century BCE to the end of antiquity we find philosophers producing commentaries on Plato's dialogues, designed to help readers with the language, the details, and the arguments. They also wrote introductions to Plato, in which Plato's thought is set out as a philosophical system, often in the later ancient format of three parts: logic (and epistemology), physics (and metaphysics), and ethics (and politics). When Plato's thought is treated in this way, the dialogues are thought of as sources for his position on various issues.
**Plato's Academy** continued to operate after his death, and around 268 BCE, its head at the time, Arcesilaus, had the school return to its former method of argument, in which the philosopher espouses no point of his own but argues with another person on his or her own terms, thus beginning the so-called Skeptical School. The remains of the Academy, shown here in a recent photograph, are believed to have been found in the Akadimia Platonos subdivision of Athens, Greece.
This second tradition has been divided by modern interpreters into the "Middle Platonists," who produced on the whole dutiful and academic but unexciting work, and "Neo-Platonists," who, beginning from Plotinus's brilliant rethinking of Plato in the third century CE, developed Plato's thought in original and innovative ways. But this is a modern distinction; in the ancient world the only real distinction was seen as that between two traditions. On the one hand, there was the "skeptical," inquiring Academy tradition of taking from Plato the practice of arguing on the opponent's terms and detaching yourself from commitment to your conclusions as authoritative pronouncements. On the other, there was the Platonist tradition, "doctrinal" or "dogmatic," for which what mattered were Plato's actual ideas about the soul, the cosmos, virtue, and happiness. For thinkers in this second tradition, philosophical activity took the form of lovingly studying Plato's works, developing his ideas further in contemporary terms, or both.
**Egyptian-born Roman philosopher Plotinus** (205—270 CE), shown here in an illustration from the _Nuremberg Chronicle,_ is credited with the founding of Neoplatonism, a school of thought that developed Plato's philosophy in original and innovative ways.
It is the "dogmatic" Platonist tradition which is most familiar to us. We find it natural for there to be editions and translations of Plato's texts, commentaries on them, and both scholarly and popular books about his ideas (such as this one, of course), even if we are less likely to expect modern philosophers to develop Platonic themes. The alternative tradition, that it is Plato's method of doing philosophy that he wants us to engage with rather than his own ideas, has been present only fitfully in the twentieth century, and has usually taken eccentric forms that have prevented its being taken seriously. It has become better known in the last few years, as students of ancient philosophy have taken more interest in ancient methods of arguing.
Do these traditions have to be mutually hostile? At times they have been; but it is possible for them to coexist and even learn from each other. Even if you think that what is interesting about Plato is his ideas about the soul, Forms, or the good life, you can learn a lot from the way Plato distances himself from commitment and stresses the importance of arguing on the opponent's terms. And even if you think that what is compelling in Plato is his picture of Socrates, always inquiring and never claiming knowledge, it is interesting to work out the positive views within which Plato has Socrates function in this way.
### PLATO THE SKEPTIC?
Is Plato a skeptic—that is, in ancient terms, does he identify philosophical activity with questioning the claims of others, rather than putting forward conclusions as justified?
Cicero puts the case for saying yes:
The skeptical Academy is called the New Academy, but it seems to me we can also call it the Old Academy, if we ascribe Plato to the New as well as the Old Academy. In his works nothing is stated firmly, and there are many arguments on both sides of a question. Everything is subject to inquiry, and nothing is stated as certain.
Sextus Empiricus, a different kind of skeptic, says no:
**This bust of Roman statesman,** orator, and author Marcus Tullius Cicero (106—43 BCE) was created by Danish/Icelandic sculptor Bertel Thorvaldsen in 1799 or 1800.
As for Plato, some have said that he is dogmatic, others aporetic, others partly aporetic and partly dogmatic (for in the gymnastic works, where Socrates is introduced either as playing with people or as contesting with sophists, they say that his distinctive character is gymnastic and aporetic; but that he is dogmatic where he makes assertions seriously through Socrates or Timaeus or someone similar....) Here... we say... that when Plato makes assertions about Forms or about the existence of Providence or about a virtuous life being preferable to a life of vice, then if he assents to these things as being really so, he is holding beliefs; and if he commits himself to them as being more plausible, he has abandoned the distinctive character of Skepticism....
**Greek physician and philosopher** Sextus Empiricus (ca. 160—ca. 210 CE) is shown here in a 1692 copper engraving after an ancient coin.
• • • • •
## Many Voices?
"Plato has many voices, not, as some think, many doctrines." So says Arius Didymus, an ancient scholarly philosopher, aware that when we read the dialogues, we become progressively more puzzled as to how they are supposed to add up. Even if we assume that the positions defended in some dialogues by Socrates, or the Visitor from Elea, are all at least provisionally accepted by Plato, we find differences of emphasis and perspective which make it difficult to judge how important a given theme is, as well as radically different treatments of similar ideas and sometimes what look like outright conflicts between the positions in different dialogues.
Over the centuries there have been many reactions to this. One is to hold that Plato wrote his dialogues to be read separately, and that it is mistaken to try to build up a system of ideas from them jointly. It is hard to refute this position, but it is also revealingly hard to carry it through, to read _Apology, Crito,_ and _Gorgias,_ for example, as though the claims about goodness and happiness in them were quite unconnected. And when we read what is said about pleasure in the _Protagoras_ and then go on to find an apparently conflicting position in the _Gorgias,_ it is unsatisfactory just to reflect that these are different dialogues. There are strands of thought which run through many of Plato's dialogues, and encourage us to try to put the ideas together.
What kind of unity do we find in these ideas, however? Some interpreters find a very high degree of unity, but at the cost of dismissing, or downgrading, what look like different approaches in different dialogues. The ancient Platonists tend to do this. The extreme version of this view sees "Platonism" as a monolithic set of ideas in Plato's mind independent of his presentation of them in the dialogues, and also independent of his development of arguments for them. Proponents of this view have given Plato a bad name among philosophers, as being more interested in dogma than in argument. In the twentieth century more attention has been paid to the details of Plato's arguments, and interpreters have been more open to the thought that he may have returned to the same idea more than once, not always in the same way. Until recently it was a standard assumption of Plato scholarship that Plato's works display a "development" of his thought, from early dialogues which represent Socrates as arguing without coming to conclusions, to the "middle" and "late" dialogues in which Plato puts forward his own ideas. The developmental view rests on questionable assumptions about Plato's life, about the possibility of dating texts, and about reading Socrates as simply a mouthpiece for Plato, and is nowadays much queried. It does have answers for some problems created by apparently conflicting passages, but there are other ways of meeting these problems.
Plato's ideas can be seen as hanging together tightly or loosely. They can also be seen as more or less dogmatically put forward. Many doctrinal Platonists have been insensitive to Plato's refusal to commit himself in person; they too have given Plato a bad name among philosophers, as though he were simply using Socrates, or the Eleatic Visitor, as a mouthpiece to pontificate. But we can respect Plato's refusal to dogmatize while remaining interested in his ideas. Many people find that as they read through the dialogues they get an increasingly cumulative impression of a distinctive set of ideas; they can also recognize that Plato's statements of these ideas is never more than provisional.
## Fiction, Myth, and Philosophy
The philosopher aims at truth—and so should have no use for the kind of enterprise we call fiction, where we entertain ourselves by stories we know are not true. Plato goes further, and is notoriously hostile to the fictions popular in his culture, mainly taking the form of publicly performed drama and recitation. He is aware of the power that such narratives have to shape our conceptions of ourselves and of the social world we live in. He is strongly against such power when used thoughtlessly to propagate traditional ideas, which can be harmful. In the _Republic_ especially, Plato makes the case that the traditional cultural education of his time leaves people with false beliefs about the gods and false ideals to live up to. The stories found in Homer and the ancient dramatists (which played the role taken in our society by popular entertainment) glamorize the values of a warrior society, and are bound to unfit people for living in civic society, where they must act in cooperation with others.
**This Hellenistic** terra cotta figurine (second century BCE), shows an actor wearing the mask of a rustic.
**Especially in the _Republic,_** a fragment of which is shown here on a papyrus from Oxyrhynchus, Egypt, Plato condemned the traditional cultural education of his time as leaving people with false beliefs and false ideals to live up to.
Plato is intensely hostile to the way that what we would call creativity and imagination are thoughtlessly put to trivial or damaging ends. But he is, as already noted, a creative and imaginative writer himself, and hardly unaware of this. His commitment to the philosophical search for truth alters his attitude to his own gifts in two ways.
Firstly, he thinks of their role as limited. Some of the dialogues are written in ways that will draw in the unphilosophical, but this is a level at which we are not encouraged to stay. Even in the easier, attractive dialogues there is always a clear message that philosophy goes on to argue, to examine, and to test claims in a way that leaves behind their appeal to the imagination.
**According to Plato,** the work of Greek epic poet Homer (ninth—ca. eighth century BCE) and other ancient dramatists, which glamorized the warrior society, were bound to make people unfit for living in a civilized society. Homer is shown here in a 1663 oil-on-canvas portrait by Rembrandt.
And further, Plato rejects the idea that imagination and creativity have value of their own; he uses them only in the service of furthering what he takes to be true positions. One of his most notorious views, one that has recommended him to puritans in every age, is his rejection of the idea of harmless entertainment. For him the appeal of a good story is valuable if it encourages us to think of, and think further about, good values; otherwise it is harmful, since it encourages us to feel satisfied with the unquestioned values of our culture.
**Despite its intended purpose** —to get the reader to examine the ideas of government and power—Plato's unfinished fictional story of Atlantis, which appears in _Timaeus_ and the fragment _Critias,_ has instead appealed to people who are determined to undercover a previously hidden version of history. As a result, the "real" Atlantis has been found in locations throughout the world. This idealistic 1928 illustration by J. Augustus Knapp, from _The Secret Teachings of All Ages: An Encyclopedic Outline of Masonic, Hermetic, Qabbalistic and Rosicrucian Symbolical Philosophy,_ by Manly P. Hall, depicts the Atlantean Mystery Temple.
Hence Plato is quite ready, in his own writings, to use traditional forms such as narrative, descriptive images, and myth, stories involving the superhuman. Their content, though, is thoroughly transformed, particularly with respect to myth, where Plato rejects his culture's acceptance of a plurality of mutually indifferent or hostile gods interfering in human life, replacing it by a form of monotheism in which god is responsible only for what is good. Plato's elaborate myths, in the _Gorgias, Phaedo, Republic, Phaedrus,_ and _Statesman,_ underline the points made through argument in the dialogue by using them as materials for an imaginative narrative.
One irony here is that in terms of sheer numbers of people affected, probably the most influential thing Plato ever wrote was his unfinished story of Atlantis, in the introduction to _Timaeus_ and the fragment _Critias._ He begins a narrative about ancient Athens, which embodied an ideal form of government, and a threatened invasion by Atlantis, a rich, sophisticated civilization to the west of the known Greek world. Atlantis itself was originally Utopian also, but it is flawed, in ways that lead it to seek imperialist conquest. Even the beginnings of this story have inspired a genre of Utopian writing, as well as romances, action stories, and movies about exotic outsiders threatening "our" civilization. (Most of these are cruder than Plato's, which offers its readers no easy identification with "the good guys," and no straightforwardly optimistic ending.)
Most interesting, however, is that Plato has his narrator begin the story with a long preamble about getting it from Egyptian priests, who have, he says, far older records than the Greeks, whose civilization has frequently been destroyed and risen again, so that they are ignorant of their own history. This idea has a deep appeal for many people determined to uncover a hitherto hidden version of "our history." The "real" Atlantis has been "discovered" in the Mediterranean, on the island of Thera and at the site of Troy, and west of the Mediterranean, in prehistoric Britain, Ireland, Denmark, South America, the Yucatán, the Bahamas, North America, and as a lost continent now sunk in the Atlantic.
The continuing industry of discovering Atlantis illustrates the dangers of reading Plato. For he is clearly using what has become a standard device of fiction—stressing the historicity of an event (and the discovery of hitherto unknown authorities) as an indication that what follows is fiction. The idea is that we should use the story to examine our ideas of government and power. We have missed the point if instead of thinking about these issues we go off exploring the seabed. The continuing misunderstanding of Plato as historian here enables us to see why his distrust of imaginative writing is sometimes justified.
**This detail from the tondo** of a red-figure Attic cup, ca. 480 BCE, by the so-called Briseis Painter, depicts one aspect of the socially acceptable sexual and erotic relations between men in ancient Greek society, an adult man, the "lover," and an adolescent boy, his "beloved," sharing a kiss.
# FOUR
# Love, Sex, Gender, and Philosophy
## Not Seeing Plato Whole
Plato is, according to Saint Augustine, the pagan philosopher who comes nearest to Christianity. In their eagerness to co-opt Plato's authority in the intellectual development of the church, however, Augustine and other church fathers looked away from something in Plato which was anathema to Judaism and Christianity, and thus began an unfortunate tradition of selective and sometimes dishonest attention to Plato's works.
Plato wrote in a society in which sexual and erotic relations between men were taken for granted, and were often socially acceptable, particularly between an adolescent boy and an adult man, where the older "lover" served as the younger "beloved's" mentor and guide to the adult world. Such relationships were romanticized, and not regarded as competitors to more prosaic relationships like marriage.
Plato's treatment of love as background to and possibly part of philosophy is mostly to be found in the dialogues _Symposium_ and _Phaedrus,_ although it forms part of the setting of some other dialogues. In what follows (and for the rest of the book), I shall talk of Plato's views, assuming that the reader will not need constant repetition of the points we have noted about the distancing produced by the dialogue form.
Plato goes beyond accepting homoerotic relationships as part of his social world. He takes the romantic view of them, and takes it further, in two ways. He stresses the mentoring aspect of the lover-beloved relation, elevating it to an idealized relation between teacher and pupil which is above physical attraction and consists in concern for the other's soul—that is, their psychological and mental well-being. This is what is often labeled "Platonic love"—love with the form of a romantic relation, but transformed by concern with the soul rather than the body. Socrates is often depicted as concerned with the well-being of young boys with whom he hangs out at the gymnasia. Indeed, sometimes he claims to be an expert on love ( _ta erotika,_ love of the sexual and romantic sort).
This is, of course, liable to misunderstanding. Older men who hang round gymnasia are usually, after all, interested in young men's bodies, not their souls. In the _Symposium_ there is a passage (215a—222b) designed to show what Socrates' love really is. Alcibiades, a beautiful, brilliant, and rich young Athenian, is used to being pursued by older men, and becomes fascinated by the way Socrates refuses to be drawn by his glamour. He discovers that only Socrates is capable of getting him to feel ashamed of his superficial way of life and to aspire to be a better person. Wanting Socrates as his mentor, he resolves to seduce him into a sexual relationship. But, humiliatingly, he fails, even when he moves from flirtation to spending the night with Socrates under the same cloak. Socrates merely comments that, if he could indeed make Alcibiades a better person, this would be a prize worth a great deal more than mere sex.
**In Plato's _Symposium_ ,** one character is the beautiful, brilliant, and rich young Athenian Alcibiades, shown here in a marble bust, a Roman copy after a Greek original of the fourth century BCE. The young Athenian, used to the attentions of older men but fascinated by Socrates refusal to be drawn to his glamour, finds that only Socrates can make him feel ashamed of his superficial life and inspire him to try to become a better person.
Despite the eloquence of this passage, misunderstanding was not always averted. The later satirist Lucian has a Platonist philosopher reassure a father nervous about having him as a tutor for his teenage son: it is the soul that interests him, he says, not the body, and even when his pupils spend the night under the same cloak—they never complain!
## Love and Sex
Indeed, some passages, particularly in the _Phaedrus,_ suggest that sex is not totally excluded from a continuing philosophical relationship (not, however, the highest sort), once it has progressed beyond the mentor-pupil relationship to one of a more equal philosophical companionship. For Plato sex as such is not the problem here; the issue is the extent to which lives can be dedicated to the study of philosophy without becoming indifferent to the demands of everyday life.
There is a second way in which Plato uses the language of homoerotic romantic love. Most notably in the _Symposium,_ he represents the urge to philosophical inquiry and understanding as itself being a transformation of sexual desire. In a passage on the "ascent of love," Socrates describes how erotic urging can become sublimated and transfigured, leading the person to move beyond particular gratifications, finding satisfaction only in the transformation from individual possession to contemplation and understanding universal truths. Plato's ideas here have been compared to Freud's, though they are arguably less reductive: the human urge to understand is traced to a basic drive we all share, but one which can, while retaining its energy and urgency, be transformed into something with intellectual structure and complexity.
**In his _Symposium,_** Plato suggests that the urge to philosophical inquiry is a transformation of sexual desire. The 1873 oil on canvas _Plato's Symposium_ is Anselm Feuerbach's interpretation of Plato's philosophical gathering, in which the guests give speeches in honor of love.
Why does Plato do anything as unlikely as trace the drive for philosophical understanding to the energy of love? Perhaps because he is attracted, as often, by an explanation which has the promise of harmonizing two very different demands on what is to be explained. The drive to do philosophy has to come from within you, and be genuine. Plato is struck by its likeness to the lover's desire: it comes from within you in a way that cannot be deliberately produced, and, like love, it drives you to focus all your efforts to achieve an aim which you feel you cannot live without, however impossible attainment may seem. But philosophy is also a joint activity; and few have stressed as much as Plato the importance of mutual discussion and argument; philosophical achievement is produced from the conversations of two or more, not just the intense thoughts of one. Plato stresses at times the way that love can produce a couple with joint concerns which transcend what each gets separately out of the relationship; philosophy similarly requires the stimulus and cooperation of joint discussion and argument. Philosophy and love thus share puzzling features. How far love illuminates philosophy is another matter; certainly Plato's discussion locates the place of both in human life in a way that is original and inspiring.
## Gender Trouble
Inspiring to men, perhaps. But isn't there a problem for women reading these works, in which romantic and erotic love is discussed entirely in homoerotic terms, and women are not considered, or brought in only as an inferior or rejected option? Plato talks of love between men producing intellectual "offspring" which are far to be preferred to the mere physical offspring that men and women produce together. Here he is probably just picking up contemporary contempt for the feminine sphere in taking love between men to be superior, intellectually and otherwise, to heterosexual love; though he probably exaggerates this contempt, as well as the significance of homoerotic love in his society. (Love between women does not interest him much; probably he knew little about it.) However, Plato's attitude to women is complex. He is obviously not concerned about women's sensitivities in his writings. But in the _Symposium_ the account of the "ascent" of love is actually put into the mouth of a woman, a priestess called Diotima. And alongside the misogyny, Plato perceives that there is a problem about women's lives and their expectations, a problem philosophers have until recently rarely appreciated.
## Women's Potential, and the Family
Plato's _Republic,_ and to a lesser extent _Laws,_ are famous for the idea that in an ideally governed society the nuclear family would be either abolished or severely limited. Plato is struck by the way that families often serve as schools of selfishness and a competitive and hostile attitude to outsiders, and that this often closes off the spread of attachment to wider groups. Cities will have citizens with real attachment to their city and its ideals, he thinks, only if the kind of influences provided within the nuclear family are reined in. Among the benefits of this idea he sees a release of the potential in women, who will exchange a narrow life of caring for husband and children at home for one in which their physical and mental capacities can be developed in wider contexts, just as those of men are.
**A Greek woman** is depicted grinding wheat in this terra cotta figurine (ca. 450 BCE) from Kameiros, Rhodes, which is housed in the British Museum in London.
In the _Republic_ this idea is developed in a very idealized context in which it is assumed that women can become both warriors and philosophers in the way that men do. In the _Laws_ the context is nearer to that of Plato's world, and women are allowed some expansion of role beyond traditional ones, though the nuclear family is retained. These ideas, even in a narrower version, were revolutionary in Plato's day, calling forth ridicule and misunderstanding.
In a period when the issues have been thoroughly debated in an organized way, we can clearly see many defects in Plato's approach. It is entirely unempirical, resting on a priori claims about human nature, and hence has no clear application to actual societies. As a heroic but unrealistic ideal, it has made little actual impact through the centuries. Further, despite being theoretically committed to equality between the sexes, Plato persists in thinking that women will on the whole perform at a lower standard than men, both physically and mentally. And there is a reason for this: he thinks of improving the lot of women by enabling them to do what men do, and to play the roles that men play. He sees nothing in women and their activities as they are in his society that is worthy of respect, or of retention as something that both men and women should do. This is a major reason why he continues to refer to women in misogynistic terms.
So we can see why some have thought of Plato as the first feminist, because he sees no reason why women should be barred from activities that men do, while others have seen in him a deeply antifeminist strain, holding that women are worth thinking about only to the extent that they can be socially reconstructed as men. Considering the difficulty of the issue, and the way that feminism tends to divide on the subject of whether traditionally feminine activities and traits should be rejected or valued, we can appreciate why Plato sends mixed messages here. It is open to us to attack him for his lack of appreciation for what women actually are and do. Or we can be impressed by the fact that Plato does in fact see that the position of women in society is a problem, and that ideally something would be done about it. It is one of the marks of his originality that almost no other philosophers have thought this. Aristotle, for example, with greater respect for existing views, finds no problem at all in the fact that women run domestic homes, lack political rights, and are not educated as men are; and until recently he has been typical.
There is a story that there were two women pupils in the Academy, Lastheneia, and Axiothea, who came to the school disguised as a man after reading the _Republic._ The story may be an invention in the light of the _Republic,_ but, whether historically true or not, it illustrates the way in which Plato was seen as holding that gender is irrelevant to intellectual development.
## Sex and Gender
Until the twentieth century, while Plato has often been prominent in the Western philosophical tradition, his views on sex, love, and gender have been, for different reasons, regarded as off-limits to philosophical discussion, and this has resulted in a curious willed blindness to what is in the texts. Though not invented then, the hypocrisy involved was particularly apparent in the nineteenth century, when Plato's works became prominent in university education.
• • • • •
### VICTORIAN EVASION OF THE HOMOEROTIC ELEMENT IN PLATO
Tom Stoppard's play _The Invention of Love_ captures the ambivalence of Victorian Oxford's attitude to Plato. Here we meet Walter Pater, a repressed homosexual whose book _Plato and Platonism_ brought some aspects of Plato's love of male beauty almost to the surface, and Benjamin Jowett, the Master of Balliol College, who translated Plato into English and pioneered the study of Plato, particularly the _Republic,_ at Oxford. In Stoppard's play Jowett charges Pater with writing inappropriately fervid letters to a Balliol student.
**_PATER:_**... I am astonished that you should take exception to an obviously Platonic enthusiasm.
**John Wood and Ben Parker** starred in Tom Stoppard's _The Invention of Love_ (1997) at the Theatre Royal Haymarket (November 1998).
**_JOWETT:_** A Platonic enthusiasm as far as Plato was concerned meant an enthusiasm of the kind that would empty the public schools and fill the prisons where it is not nipped in the bud. In my translation of the Phaedrus it required all my ingenuity to rephrase his description of paederastia into the affectionate regard as exists between an Englishman and his wife. Plato would have made the transposition himself if he had had the good fortune to be a Balliol man.
**_PATER:_** And yet, Master, no amount of ingenuity can dispose of boy-love as the distinguishing feature of a society which we venerate as one of the most brilliant in the history of human culture, raised far above its neighbours in moral and mental distinction.
**_JOWETT_ :** You are very kind but one undergraduate is hardly a distinguishing feature, and I have written to his father to remove him.... The canker that brought low the glory that was Greece shall not prevail over Balliol!
• • • • •
Homosexuality was literally unspeakable, and Plato was made available in bowdlerized and misleading translations. At the same time, there was a general anxious half awareness that Platonic love was not socially approved heterosexual love.
The idea that men's social roles should be available to women, while not literally unspeakable, was regarded as a joke, until women's movements in the nineteenth century turned it into a serious subject of political discourse. For 150 years the _Republic_ in particular has been discussed with this issue in mind. By this point, studying Plato has little to contribute to modern feminist discussion: his starting points and many of his assumptions are too remote from ours for him to be a profitable partner in debate for very long.
**Plato's idea** that men's social roles should be available to women was regarded as revolutionary in his time, and considered a joke until more than two thousand years later. The caption to this late-nineteenth-century cartoon says, "Lady Customer—'Mr. Smith—ah, ah—have you any Her-books?' Bookseller—(Slightly surprised) —'HER-books, ma'am? I really—' Lady Customer—'Ah, well, you naughty men call them Hymn-books. But, as we of the angelic sex are resolved on freeing ourselves from the chains imposed on us by tyrant man, we want Her-books in future.' [Bookseller faints.]"
**By the time** women's rights began to be taken seriously, Plato's work had little to add to the discussion, resting as it did on a priori claims about human nature. Among those at the forefront of the women's rights movement in the nineteenth century was Lucy Stone, who with social reformer Henry Blackwell founded the weekly newspaper the _Woman's Journal_ in furtherance of the cause; the front page of the March 8, 1913, issue is shown here.
Yet it is in his attitude to women that Plato is most radical and pioneering. Even to have the idea that there is nothing natural about women's social roles, that they can do what men do, is a surprising breakthrough. However, original though his ideas about love and philosophy are, his focus on homoerotic love, when we look at it dispassionately, required much less originality. It has been the troubled attitude of so many later readers to this topic that has inflated it to the status of a major issue.
**According to Plato,** to lead a happy life, you must lead a virtuous life. Andrea Andreani's allegorical chiaroscuro woodcut _Virtue Chained by Love, Error, Ignorance, and Opinion_ is from 1585.
# FIVE
# Virtue, in Me and in My Society
## How to Be Happy
In many dialogues Plato grapples with the question of how we are to live a good life. He begins from an assumption which he shares with the rest of his society, namely that we all seek happiness ( _eudaimonia)._ What we think of as ethics emerges as the concern not just to _live_ one's life, but to do it _well,_ to make a good job of it. We all seek to be happy, in the sense of living a good life (something to be sharply distinguished from modern notions of happiness, which identify it with feeling good; happiness in all ancient thinkers is the achievement of someone who lives an admirable, enviable life). Plato never doubts that this is where ethical concern starts. He gives, however, a radically different answer than most people, and most other philosophers, to the question of what it is to live an admirable, enviable life, and so to achieve happiness.
Many people, in the modern as much as in the ancient world, find it natural to say that a happy life is one in which you are successful; the happy person will be, typically, the rich, secure person who has achieved something in life. It sounds odd, indeed perverse, to say that someone could be happy, could be living a life you admire and try to emulate, if he or she turned out to be rejected and unsuccessful. But Plato was influenced by the example of Socrates, who gave up worldly success for philosophy, and who ended up condemned as a criminal and executed—yet who clearly seemed to Plato to have lived an admirable life. And so, most people must be wrong about how to achieve a happy life.
Where do most people go wrong? They think that their life will go well, and that they will be happy, if they have the things that most people think are good—health, wealth, good looks, and so on. But are these things good? Do they do you any good—do they benefit you? Surely, thinks Plato, you are here like a craftsperson with tools and material—they do not do you any good until you put them to _use,_ that is, _do_ something with them. Moreover, you have to do the _right_ thing with them, put them to use which is expert and intelligent, or they will not benefit you—indeed may do you harm. Someone who wins the lottery, for example, may well not be made any happier by just having the money. Unless she puts it to intelligent use, the money may do nothing for her, or even ruin her life. Happiness cannot just be the stuff you have; you have to put it to good use, deal with it in the way that a craftsperson deals with her materials, before it will benefit you, and so make your life better.
Hence we find that the virtues, which enable us to deal well with the material advantages of our life, are called (in the _Laws_ ) "divine goods," in contrast to the "human goods" constituted by those material advantages. Without the divine goods, we will lose the benefit of the human ones. So the value for us of health, wealth, and the like depends on our possession of virtues like courage and justice. And the virtues depend in turn for their value in a human life on the practical reasoning which forms them and guides their application. Hence in the _Euthydemus_ the virtues which make something out of the stuff of our lives are identified with wisdom, the practical intelligence which guides virtuous living.
**According to Plato,** happiness cannot be simply having things that people think of as good—winnings from the lottery, for example; you must do something—the right thing—with those winnings for you to benefit from them. In this photographic print from an 1853 wood engraving published in the _Illustrated News,_ hopeful lottery ticketholders await the results.
We obviously have a bold thought here, but just how bold? Is Plato saying that things like health and wealth do not just by their presence make my life better, but do make it better if practical wisdom puts them to good use? If so, he thinks that they are good only conditionally—only in the context of a well-lived life. Or does he think, more austerely, that things like health and wealth are not good at all, and that it is only the intelligent use I make of health, wealth, and other goods of fortune that makes my life better, while their presence does not?
**The value of our material advantages,** such as health and wealth, depends on our possession of the divine goods, such as courage and justice. This detail of a 1508 ceiling fresco by Italian master Raphael in the Room of the Segnatura in the Vatican shows an allegory of Justice holding a sword in her right hand and scales in her left.
Plato seems not to have thought through the difference between these positions, since we find language supporting both. Later ethical theories distinguished them, and the second, more austere position, that of the Stoics, was generally thought to have won in claiming Plato as its ancestor. One reason for this is that the more austere view implies that being virtuous is in itself sufficient for a happy life, and this is a position that finds support elsewhere in Plato.
**The position that being virtuous** itself is sufficient for a happy life is supported by more than one of Plato's works and was espoused by the later philosophical school Stoicism, whose founder, Zeno of Citium (ca. 335—ca. 263 BCE), is pictured here in a bust cast at the Pushkin museum from an original in Naples.
## What Matters
In _Apology_ (Socrates' defense speech), _Crito,_ and _Gorgias_ we find explicit statements of a very uncompromising kind. Socrates claims that all that is relevant to the issue of whether someone is happy or not is whether they are virtuous. If we know that a course of action is wrong, then we should not do it, and no amount of anything we could gain or lose by doing the action makes any impact on this point. Even if your life is at risk, you should not try to save it by compromising your values.
• • • • •
### UNCOMPROMISING VIRTUE
In the _Crito_ (48c—d) Socrates, waiting for execution, examines why he should or should not try to escape from prison.
**_SOCRATES_ :** We should now examine this—whether it is just for me to try to escape [from prison], or not. If it turns out to be just, let us try, and if not, let's drop it. But these considerations you mention, about spending money, and reputation, and bringing up my children, I suspect, Crito, that these are in truth considerations that appeal to... most people. But for us, since the argument demands it, there is nothing else to examine except what we just said, namely, whether we shall be acting justly [if we arrange my escape] or whether we shall in truth be acting unjustly if we do all this. And if this will clearly be an unjust action for us to do, then there is no need at all for us to take into account whether I will have to die if I stay and do nothing, or have to suffer anything else whatever rather than do wrong.
• • • • •
Why is Socrates so sure that the claims of virtue cannot be compromised—cannot indeed be weighed up against considerations like those of money, security, and so on? We have seen that virtue is not just one good thing for me to have, something that might be measured against other good things, such as wealth or security. Rather, virtue is a "divine" good—it is either the only unconditional good, or the only thing which is good at all. And it holds this position because it is virtue which enables us to put other conventionally good things to good use—hence, it is what makes the difference between having things like health and wealth benefit us or do us no good, or even ruin our lives. Hence virtue is often thought of as a kind of skill or expertise—a kind of practical knowledge which is applied in making materials into a unified and finished product.
**According to Socrates,** all that is relevant to the issue of whether someone is happy is whether he or she is virtuous. This detail of a medieval tapestry is from the _book Zahm und wild, Basler und Straßburger Bildteppiche des 15. Jahrhunderts_ (Tame and wild, Basel and Strasbourg tapestries of the fifteenth century); it depicts a virtuous woman using her expertise not only for her own good but for that of others: in taming the mythological figure of the woodwose (wild man) she protects herself from his advances, helps the woodwose himself (by civilizing him, not a controversial goal at the time), and saves other women from being accosted by him.
The idea here is a powerful one. By the time I start thinking about how to live my life well, I already, as we say, have a life—I have a set of commitments and relationships, such as my family and my job, and a set of goals, my ambitions and dreams. I also, typically, want to be a good person, to be courageous rather than cowardly, fair rather than unjust, and the like. Plato tells us, uncompromisingly, that virtue has a special role, and a special kind of value. To be virtuous is not just to have some goods like wealth, health, and so on, and also virtue. Rather, virtue is the _controlling_ and _defining_ element in your life; everything else is just materials for it to work on, and it produces a result which is either a well-organized whole or, if it fails, a mess. If we look at things this way, we can appreciate why Plato sees the role of virtue as so crucial in a life. He does not, however, articulate the kind of precise theory that later philosophers did produce as a result of thinking about, and refining, this idea of virtue as the controlling element in a life.
## Becoming Like God
This may already strike modern readers as a demanding view. Most of us probably have more sympathy with Aristotle's commonsensical position, which allows that virtue is important as the basic organizing factor in your life, but insists that conventional goods like health and wealth are also good and make your life better if you have them (and, if you lose them, disrupt your life sufficiently that you are no longer happy).
Plato's is without doubt a very demanding position, and was recognized as such in the ancient world (as already indicated, it was generally identified with the austere Stoic position). If he is right, my life should be lived very differently from the way I now live it; instead of pursuing goals like wealth or power I should do all I can to have my life organized and controlled by virtue—and for most people this will make a tremendous difference.
Sometimes, however, we find Plato putting forward the idea that it is not enough to transform your life by getting virtue to direct your priorities. Rather, you should recognize that all our everyday concerns and worries are really petty and unimportant. You should try to take the perspective from which the things that people get worked up about are seen as merely trivial. Virtue requires, in other words, _detachment_ from everyday concerns, and hence from the mixture of good and bad that is inevitable in ordinary life. For in life as it is, there is no such thing as really being virtuous, being _perfect_ —"that is why we should try to flee as fast as we can from the world here to the world there. This flight is coming to be like god as far as is possible, and this coming to be like god is coming to be just and pious, with understanding" _(Theaetetus_ 176a—b.)
The idea of becoming like god would strike Plato's audience as shocking. Gods are a different kind of being from humans, just as the other animals are. Traditionally, for a human to seek to become a god was a transgression (one that the traditional gods were quick to punish). What Plato has in mind is naturally not this, but a philosophically refined view of what god is. God is purely good, wholly without evil (unlike the traditional Greek gods), and to become like god is to aspire to get as near to perfection as a human can.
The ideal of virtue as becoming like god runs against the main current of ancient ethical thought, which takes virtue to be an ideal fulfillment of human nature and its potential, not an attempt to transcend it and to become another kind of being altogether in a quest for perfection that can be attained only in a withdrawal from everyday life. Sidelined for many hundreds of years, the otherworldly ideal had a new lease on life in late antiquity, in the Neoplatonist interpretations of Plato and the impact these had on the intellectual development of Christianity.
## Educating Good People
Attracted as he at times is to this idea, however, Plato for the most part thinks of virtue as _a practical kind_ of knowledge, exercised in and on the agent's life. Moreover, as we have seen, he thinks that becoming virtuous is crucial for someone hoping to achieve what everyone hopes to achieve, namely, happiness. How, though, is a person to become virtuous? Aristotle, Plato's pupil, later thinks that we start by taking as role models the virtuous people in our community, and proceed to emulate and to criticize the content of their deliberations. If we develop well, we achieve virtue that is richer, more reflective and unified than what we start with; but we will not go far wrong in beginning from our community's standards. Plato wholly disagrees; some of his most vivid passages present the person who aspires to virtue as being quite at odds with their community, finding little sympathy or support for their own ideas. The more talented and sensitive a person is, he suggests in one passage, the more they will be molded by the various kinds of pressure that society brings to bear.
**The performance** of dramas and epic poetry, such as works by the Greek poet Homer, were to ancient Athenian society what films, television, and books are to us today. Plato took such influential media seriously, refusing to regard them as harmless. This mural painting by a Roman master from the first century BCE depicts a scene from one of Homer's two major works, the _Odyssey._
Plato recognizes that these pressures are not all of an overtly moral or political kind. What we call a society's culture affects people in lots of ways. In particular, Plato is the first to emphasize the importance of what we call the arts in forming the values of the members of a society. The role played in our society by films, television, and books was played in Plato's Athens by the performance of dramas in the theater, by festivals, and by the learning and performance of various kinds of poetry—epic (notably Homer's _Iliad_ and _Odyssey)_ and lyric. Plato takes these very seriously, refusing to regard them as mere harmless entertainment.
**In the improved city of Plato's _Laws,_** in which Plato insists on a complete reform of his society's culture, there is none of the drama that made up such a large part of Greek popular culture (what we now call "Greek tragedy"). Melpomene, the Greek muse of tragedy, is shown holding a tragic mask in this statue from the second century CE.
In two of his longest works, the _Republic_ and the _Laws,_ the latter a work in which he sketches a legal code for a new city, Plato insists on radical reform of his community's culture, in the interests of the moral growth of its members. The content of traditional culture, notably poetry, is to be thoroughly reformed, and purged of passages which encourage selfish and uncooperative behavior. And Plato is suspicious of the very idea of dramatic representation. He thinks, as have puritans in a number of traditions, that acting parts makes the actor's own self weak and pliable. Moreover, he distrusts the effect of drama on the audience; it encourages them to feel serious emotions lightly, weakening their control over their own emotions. In the improved city of the _Laws_ there is none of the drama which made up so large a part of Greek popular culture (and which has come down to us as "Greek tragedy"). Plato is unrepentant about the impoverishment of people's creative and imaginative side; for him what matters is moral development, and the energies on which the arts elsewhere draw are in Plato's ideal community strictly focused on that.
• • • • •
### THE LEVELING EFFECT OF POPULAR OPINION
Plato's distrust of the effects of popular culture in stifling individual thought comes out vividly in this passage from the _Republic_ (492a—c).
**_SOCRATES_ :** The nature of the person who loves wisdom, as we laid it down, will necessarily arrive as it grows at every virtue, if, that is, it gets appropriate teaching. But if it is sown, and nurtured as it grows, in one that is inappropriate, then, unless some god happens to rescue it, it must turn out quite the opposite. Or do you too think what most people do, namely that some young people are corrupted by sophists, and that it's some sophists, private people, who do the corrupting to any great extent? Don't you think that it's the very people who say this who are the greatest sophists of all, and who do the most complete educating, producing people to be the way they want them, young and old, men and women?
When? he said.
When many of them are sitting together in an assembly, the law courts, the theater, the camp or some other general meeting of a lot of people; they make a huge uproar as they criticize some things said or done and praise others—excessively in both cases—by yelling and banging, and as well as them, the rocks and the surrounding place echo the uproar of praise and blame and return it doubled. When things are like this, what heart will a young man have, as the saying goes? What kind of individual education of his will hold out and not be swept away by criticism and praise of this sort, being carried off by the flood wherever it goes, so that he agrees with them about fine and base things, practices what they do, and becomes just like them?
• • • • •
## The Individual and the State
So far I have talked of community rather than state, but for Plato there is no sharp boundary between the cultural and the political. His ideas on how states should be organized reject the idea that politics provides a framework within which individuals can develop as they see best in pursuing their own goals. Indeed, Plato's political ideals are throughout driven by the thought that it is competitive individualism which is the main political problem. People want to "drag" things into their own houses and enjoy whatever they achieve privately, instead of wanting to cooperate in the production of shared goods, which all can enjoy publicly. In an avowedly fantastic sketch of an "ideal state" in the _Republic,_ and in a more detailed account in the _Laws_ of how a new Greek city could be organized on idealized lines, Plato reforms both political and educational institutions to produce a person whose self-conception will be primarily that of a citizen, someone whose life goals are shared with those of his fellow citizens—and her fellow citizens, for even in the _Laws_ Plato thinks that women should think of themselves as citizens, sharing in public space rather than trapped in individual domestic drudgery. In the _Republic_ fantasy these ideas go to the lengths of abolishing the nuclear family altogether; in the _Laws_ Plato moves rather to strengthening it as a basis for educating a communally minded citizenry.
**In Plato's _Republic,_** the philosopher bans the nuclear family, but in _Laws,_ he strengthens it as a basis for educating a communally minded citizenry. This Greek family scene on a funerary stele is from the third century BCE.
What does Plato think is the justification for such radical ideas, which would alter institutions relentlessly in the interests of producing more socially minded people? This is, he thinks, the only rational way of organizing society so as to function as a whole rather than consisting in a bunch of conflicting individuals. These ideas are always presented as an expert's solution, and constantly compared with the authoritative pronouncements of the expert navigator or doctor. In contrast, democracy, the accepted position in Plato's Athens, is presented as a chaotic scramble of competing voices, each shouting for a selfish individual claim with no expert grasp of the needs of the whole.
• • • • •
### DEMOCRACY AND BUREAUCRACY
Plato sees democracy as imposing stifling bureaucracy on gifted individuals. Here _(Statesman_ 298c—299b) he satirically describes what navigation and medicine would be like if run by Athenian democracy. He later admits that democratic control is useful as a safeguard against abuse of power in our actual world.
**_VISITOR FROM ELEA_ :** So suppose we were to make it our policy... no longer to allow [either navigation or medicine] to have full control over anyone, slave or free, but to call ourselves together as an assembly.... We permit both laymen and other craftsmen to contribute their opinion about sailing and diseases, as to how we should use drugs and the doctor's instruments on the sick, and even as to ships themselves... and when this is written on notice boards and stone blocks... this is how for all future time ships are to be sailed and the sick taken care of.
**In Plato's** _Statesman,_ he gives voice to his belief that democracy stifles gifted individuals with bureaucracy. Pictured here is a fragment of a _kleroterion,_ an instrument of Athenian democracy by which public offices and juries were filled. In jury selection, for example, citizens' _pinakia_ (tokens) were inserted into predetermined slots in the _kleroterion,_ and a group of white and black balls was dropped through a tube mounted on the side of the device. A crank device ejected one ball at a time, and its color determined, row by row, who would be chosen for jury duty that day.
**_YOUNG SOCRATES_ :** What you've described is very peculiar.
**_VISITOR FROM ELEA_ :** And we'd also set up yearly officials from the people... selected by lottery; and these on taking office should fulfill it by steering the ships and curing the sick according to the written rules.
**_YOUNG SOCRATES_ :** This is even harder to accept.
**These _pinakia,_** or tokens, housed at the Ancient Agora Museum in Athens, were inserted by citizens into predetermined slots in the _kleroterion,_ part of the democratic jury-selection process of ancient Athens.
**_VISITOR FROM ELEA_ :** Consider also what follows after this. When each official's year ends, courts will have to be set up... and ex-officials have to be tried and investigated. Anyone who wants to can accuse one of not steering the ships that year according to the written rules... and the same goes for those curing the sick. The court has to assess how those condemned should be punished or pay restitution.
**_YOUNG SOCRATES_ :** Well, anyone willing voluntarily to hold office in these conditions would fully deserve any punishment and restitution!
• • • • •
As Plato sees it, democracy is a menace because it rejects the idea that society should be directed by expertise, and thus blocks changes that would encourage people to think less individualistically. It drags gifted people down to the lowest level of shared understanding. On the other hand, in the world as it is, the bureaucracy and splitting up of power that democracy encourages do prevent abuse of power by uncontrolled, misguided individuals who merely think that they are experts. In the _Republic_ fantasy, absolute power is given to perfect people. But in other works where Plato is thinking more about actual conditions the expert ruler remains an ideal, but democracy is accepted, unenthusiastically, as the best working option. In the _Laws_ the institutions of Athenian democracy are taken over as a basis to be modified in a community-minded direction; no other kind of institution is envisaged as a place to start. For Plato, democracy is the worst form of government except for all the others. Only in an ideal world could we do better, and live not merely alongside one another but together, with shared lives and ideals. Plato is, as we have seen, utterly uncompromising about the individual's commitment to virtue, whatever the state of the actual world. But he also thinks, more or less hopefully, that the actual world could be improved in the interests of virtue.
**The Pnyx (ruins shown here in a recent photograph)** was the seat of democracy in ancient Athens. It was the site of the ecclesia, the popular assembly, where Athenian citizens who had the right to vote (Athenian-born men over the age of eighteen) met periodically to conduct public business.
**One of the questions posed** in Greek thought is whether the soul is indissolubly united to the body, so that when the body dies the soul dies with it. This painting by Dutch artist Hieronymus Bosch (ca. 1450—1516), which appeared in _The Smithsonian Institution: An Establishment for the Increase and Diffusion of Knowledge Among Men,_ by Walter Karp, shows a miser on his deathbed between an angel and Death, who entreat the miser for his soul.
# SIX
# My Soul and Myself
## Problems About the Soul
In Greek thought, the soul ( _psyche)_ is what causes living things ( _empsucha)_ to be alive. This leaves a large range of questions about the soul open. Our bodies are animated; is what animates them itself some kind of physical body, or is it something of an entirely different kind? If the latter, how is its nature to be understood? Is the soul indissolubly united to the body it animates, so that at death it perishes when the body ceases to be animated—or could it carry on in some other form? Am I the animated body, or am I really to be identified with the soul rather than with the animated body? If so, is there some sense in which _I_ could survive death, the cessation of the body's animation?
**Plato was a model of dualism** when it came to thought about the body and the soul; he believed they were two completely separate entities. This late-nineteenth-century lithograph, _Dr. Alford's Biblical Chart of Man,_ contains an "explanatory key showing the relation of the soul to the body, the senses to the attributes, mortality to immortality."
By Plato's time there had been a variety of answers to these questions, and his works appear at first to offer a spread of answers themselves, not always consistently. On two points he always appears firm. He always takes as a starting point the thought that the soul is a different kind of thing from the body. Indeed, he is often regarded as a paradigm of dualism, the position that soul and body (in modern versions mind and body) are radically different kinds of entity. Further, Plato never doubts that when I ask what I, myself, really am, the answer will be that I am my soul, rather than my animated body. Hence Socrates, on his deathbed, jokingly reminds his friends that they will not be burying _him,_ only his body.
• • • • •
### SOCRATES ON HIS DEATHBED
In the _Phaedo_ (115c–116a) Socrates prepares to drink the hemlock:
"How shall we bury you?" asked Crito.
"However you like," Socrates said, "—if you catch me and I don't get away from you." He laughed quietly and said, looking at us, "See, I can't convince Crito that what _I_ am is Socrates here, the person talking to you now and drawing up the arguments. He thinks that I am what he will shortly see as a corpse, and asks how he shall bury me. I seem to have wasted my words on him, though consoling both you and myself, in the argument I have long been making, that when I drink the poison I shall no longer remain here with you, but will go away to some kind of happiness of the blessed.... You must cheer up, and say that you are burying my body, and bury it however you like and in what you think is the most customary way."
• • • • •
But Plato appears to offer different and sometimes conflicting answers to further questions about the soul. Sometimes he insists that the soul is a simple nature, while in other passages we find that it is divided, indeed has parts which are metaphorically represented as individual humans and animals. Sometimes what is essential to the soul appears to be its power of thinking and reasoning; sometimes it is the power of self-motion. And, while Plato in general defends the idea that the soul is immortal, so that its relation to the body is merely temporary, we find conflicting suggestions about the nature of this relation. Sometimes the soul appears as the body's ruler and director; sometimes as its unhappily trapped prisoner.
There is no one consistent account, however general, uniting everything that Plato says about the soul. Some scholars have pointed to this as evidence for development in Plato's thought, but it is difficult to find a single line of development here. It is more natural to find in Plato several lines of inquiry which have common themes but do not always turn out to lead in the same direction.
## Simple or Complex?
One of the most famous passages in Plato is his division of the soul into three "parts" or aspects in the _Republic._ As an animated body, I function as a unity, but I contain distinct sources of motivation, something which becomes apparent when they conflict. Plato imagines a thirsty person who desires to drink but refrains, because drinking would be bad for him. (The reason is not specified; there are many ways in which it might be bad for him.) This is not simply the kind of conflict that arises from wanting to do two things in time adequate for only one of them. Rather, the conflict here is between two different _kinds_ of motivation; desire just goes for what I want _now,_ without regard for what will happen later, whereas the motivation to refrain comes from a realization of what is good for me over time. This is _reason,_ which enables me to grasp and understand the idea of my life as a whole, and which motivates me to pursue this, notably by opposing desires whose gratification would interfere with it.
Reason is not just an intellectual faculty that can work out what is best for you overall, as a person with a continuing life, a past and a future. It also motivates you without the help of desire. Desire moves you to get its object _here and now;_ reason is what gets you to resist this gratification when it is not in your best interests overall.
The contrast between short-term desires and long-term reasoned motivation is clear enough, but Plato does not find it adequate as an explanation of all of our behavior. There is also _thumos,_ which is variously translated as "spirit," "the passionate part," and the like. It is distinguished from reason by the fact that it can be inarticulate, as in children and animals, and it can also come into conflict with desires. Plato has picked on the interesting point that we can sometimes overrule particular desires without having an articulate rationale for so doing. Sometimes we are motivated by a sense of self which is unified and responsive to ideals and aspirations that conflict with particular desires, without being able to reason out the basis for this. (Soldiers responding to their country's need form one of Plato's examples.) This is the part of the soul where we find emotions, more complex and cognitively responsive than desires but falling short of the reflective abilities of reason.
**In the _Republic,_** Plato divides the soul into three parts, or aspects. The first is desire, _epithumia,_ the pursuit of what the individual wants now. This is sometimes represented by the ancient Greeks as Pothos, god of longing, yearning, and desire. It is countered by reason, represented by the ancient Greeks as Athena, goddess of wisdom and reason, which understands the long-term ramifications of our choices and persuades us to resist immediate gratification. This marble statue of Pothos is a Roman copy after a Greek original from around 300 BCE; Athena is shown in a marble statue, a Roman copy after a Greek original of the late fifth century BCE.
In the _Republic_ the main function of the theory of the soul's parts is to show that the good life is one in which reason rules the whole soul, allowing each part to flourish as it should. Reason's rule is justified by its grasp of the good of the whole person, while the other parts grasp only their own good, and therefore lead to dysfunction if they are in charge of the whole.
We find the same model in the _Phaedrus,_ where the person is depicted as a two-horse chariot whose driver, reason, tries to control the force of two horses, one (spirit) cooperative and one (desire) which tries to rebel and drag the whole chariot in the wrong direction.
Although spirit and desire are here battling animal forces, we also find that they communicate in language. Plato represents them as talking horses (one of which is deaf!). He is thinking of the soul's parts both as conflicting forces with varying strengths, and also as aspects of a person which are all responsive to reason in varying degrees. Spirit and desire are rational enough to communicate, but not rational enough to be depicted in human form. In the _Timaeus_ the soul's parts are located in different parts of the body, in ways which encourage reason (in the head) to dominate spirit (in the upper body) and desire (in the lower body).
**The description in the _Phaedrus_** of the soul as a winged two-horse chariot, strange as it is, has proved attractive to artists throughout the centuries. Here we find it on a medallion worn by the subject of a portrait bust by Donatello (1386-1466). It identifies the subject as interested in the revived Platonism (greatly influenced by the later ancient school of Neoplatonism) which was influential in Renaissance Italy.
We also, however, find in the _Phaedo_ (78b-84b) and toward the end of the _Republic_ arguments which actually depend on the position that the soul is a _simple_ unity. Both arguments claim that the soul is immortal, and that this would be impossible if its true nature were composite. The underlying thought is that anything composed of distinguishable parts is liable to dissolution into those parts; and if what is so liable will be dissolved at some point it cannot be immortal. (This thought can of course be challenged.) How does this idea relate to the soul's division into "parts"? Since both occur in one dialogue, the _Republic,_ it is to be hoped that they can be reconciled, and this seems to be the purpose of the phrase "its true nature." What makes the soul even apparently divided is its association with the body. It is the soul's embodiment (a problematic relation, we shall find) which explains how our motivations can be conflicted; the soul itself is not affected by divisions which arise from the nature of our existence as animated bodies.
**In Plato's** _Timaeus,_ the soul's three parts are located in different parts of the body: the reason in the head, the spirit in the upper body, and desire in the lower body. Shown here is a page from the _Timaeus_ from the 1578 Stephanus edition of Plato's works.
**If the soul's true nature** is to be unaffected by the body, then what animated Socrates' living body will not survive Socrates' death; only the aspects that are unaffected by the body will survive. This depiction of the dead Socrates is by nineteenth-century Russian sculptor Mark Antokolski.
If the soul's true nature is to be unaffected by the body, however, then what is it that survives Socrates' death? It will not be what animated Socrates' living body, but only the aspects of that which are unaffected by the body. Should Socrates be so sure that this will be _his_ survival?
## Mind or Mover?
Plato tends to contrast the soul with the body; in describing our psychological life and quest for knowledge he often sees these as competing forces, always to the disadvantage of the body. This is one reason why his ideas appealed to the ascetic church fathers, who interpreted the scriptural contrast of spirit and flesh as the Platonic contrast of sharply opposed soul and body, thus having a drastic effect on Western Christianity's attitude to the body.
We have seen, though, that the soul is not simply opposed to the body; when it animates the body, parts of it are in some way affected by and involved with the body. So, while sometimes Plato refers simply to the ordinary contrast between the body and what animates it, in other passages what he has in mind is the contrast between the _animated,_ ensouled body and the aspect of the soul which is unaffected by the body. In some passages about knowledge this contrast is developed as a contrast between the senses and the soul; the senses give us information, but the soul is stimulated not just to receive and process this information but to reflect on it and go beyond it. In the _Republic_ (523a—525b) the soul finds that the senses give mutually conflicting reports and is stimulated to reflect on what an adequate grasp of the world would require. In the _Theaetetus_ (184c—186e) Socrates gets young Theaetetus to discover for himself that the senses on their own cannot account for the way that we not only take in sensory information but interpret it and go beyond it.
• • • • •
### PERCEPTION, BODY, AND MIND
_SOCRATES:_ Take hot, hard, light and sweet—do you think that each of the things by which you perceive these belongs to the body?... And are you also ready to agree that when you perceive something through one power, it's impossible to perceive it through another? For example, you can't perceive by sight what you perceive through hearing, and vice versa?
**_THEAETETUS_ :** How could I not be ready to agree to that?
**_SOCRATES_ :** So, if you think something about both of them, you wouldn't be having a perception about both of them through either one of these instruments?
**_THEAETETUS_ :** No.
**_SOCRATES_ :** Take a sound and a color. First, don't you think _this_ about them, that both of them _are?_
**_THEAETETUS_ :** Yes.
**_SOCRATES_ :** And that each of them is different from the other one, and the same as itself?
**_THEAETETUS_ :** Of course.
**_SOCRATES:_** And that both together are two, and each is one?
**_THEAETETUS_ :** That too.
**_SOCRATES_ :** And you are capable of investigating whether they are unlike each other, or like each other?
**_THEAETETUS:_ I** suppose so.
**_SOCRATES:_** Well, through what is it that you think all these things about them? It isn't through hearing or through sight either that you can grasp what's common to them.... Through what does the power work which makes clear to you what is common to everything, including these things, to which we apply the words "is" and "is not" and the others we just used in the questions? What instruments are you going to assign to all these through which the perceiving aspect of us perceives all of them?
**_THEAETETUS_ :** You mean being and not being, and likeness and unlikeness, and same and different, and one and any other number they have. And clearly you're also asking about odd and even and everything that follows them, and asking through what bodily instrument we perceive these with the soul.... Well, really, Socrates, I couldn't say, except that it seems to me that they just don't have their own instrument the way the others do; the soul seems to me to consider the things that are common to everything itself, through itself.
**_SOCRATES_ :** Theaetetus... you've saved me a lengthy argument, since it seems to you that the soul considers some things itself, through itself, and others through the body's powers. That was what I thought myself, but I wanted you to believe it too.
_(Theaetetus 184e-185e)_
• • • • •
There is difficulty in sorting out a consistent overall account of just what in our sensory judgments Plato ascribes to the body and what to the soul working through the body, but one thing is quite clear from such passages: the soul here is what we would now call the mind or understanding. Our psychological resources include not just the ability to take in sensory information about the world, but the distinct cognitive ability to unify and make sense of it. Moreover, the understanding is not limited to interpreting the senses; its reflections lead it to go beyond what the senses provide and to discover objects that it can grasp without the senses. Such independent working of the mind is often opposed in the sharpest terms to our sensory experience. They are seen as competing for psychological space and energy, and reliance on the senses is disparaged as passive dozing along, while for the person to wake up is for her to start using her mind independently of what sense experience provides. Some of Plato's most vivid passages disparage the body and reliance on it for knowledge: this is called dreaming, as opposed to waking up.
**It is not always clear** which of our sensory experiences Plato attributes to the body and which to the soul working through the body. This depiction of the sense of taste is one of a five-part series of engravings by Dutch engraver Jan Saenredam (1565—1607).
One important point frequently stressed about the objects of this kind of pure thinking is that they are stable and unchanging. They are the objects of mathematics and what Plato calls Forms, to which we shall return in the next chapter. In the _Phaedo_ there is even a passage (78b—84b) in which we find Socrates emphasizing that the soul is akin to the unchanging Forms, the objects of pure thought which are unaffected by any of the sources of change in the world of our sensory experience. The soul's immortality is inferred from its likeness to its unchanging, stable, and simple objects—objects of pure thought and understanding.
However, in the _Phaedrus_ (245c—246a) we find that the soul is said to be immortal because it is always in motion (or change), and that its motion never fails because it moves itself, while everything else is moved by it. The argument is about "all soul," and this introduces a difficulty: it is not clear whether this means every individual soul, or soul as a kind of stuff—"soul" being used as a mass term like "snow" or "gold" which picks out not individuals but quantities or amounts of something. Certainly when we find related ideas in the _Timaeus_ and _Laws_ (893b-899d) we also find that the world as a whole has a soul, of which our souls are individual portions; so Plato has at least moved his main focus away from the individual ensouled person.
The idea that what defines the soul is self-motion is a deep and interesting one, which Aristotle was to develop further. It is, after all, an obvious fact about living, as opposed to nonliving, things that their sources of motion and change are internal to them. Further, with the thought that all other kinds of motion require a self-mover to account for them, Plato makes the first start in an argument which leads eventually to Aristotle's idea of an _unmoved_ mover. It is clear, however, that in arguing to the soul's immortality from its self-motion Plato is thinking of a different aspect of the soul from that where he argued to its immortality from its likeness to unchanging objects. Clearly it is our intellect which Plato is taking to be akin to its unmoving objects, and this is not the soul which is always in motion. Further, these are not just different aspects of my individual embodied soul. Rather, Plato develops two very different ideas of what characterizes the soul as opposed to the body. It is my soul which enables me to aspire to knowledge that is beyond what sensory experience can provide. But my soul is also a portion of a cosmic force which is itself actively in motion. Later Platonists found more or less academic ways of reconciling these strands of thought; Plato never does this in the dialogues.
**Plato's idea** that what defines the soul is self-motion was picked up and developed further by his student Aristotle, leading him eventually to his idea of the unmoved mover. This image of Aristotle is from the _Nuremberg Chronicle_.
## Ruler or Prisoner?
The soul, we often find, stands to the body as ruler to ruled; it is the body's superior and its organizing principle. Rulers need subjects and (Plato thinks) vice versa; this looks like a stable, if unequal, relationship. Yet we also find, notably in the _Phaedo,_ that we should try to "purify" ourselves from the body, and that philosophy is to be properly understood as practice for dying, the soul's final escape from the prison of the body. The body is an evil which drags the soul down, pestering it with its needs; death is a welcome release for the soul from its infection by the body.
The conflicts we find here come from emphasis and rhetoric rather than substance. Plato always thinks that soul and body are fundamentally unlike entities, and he has different, vivid ways of bringing this out. One is to represent the body as a hindrance to the soul; another is to emphasize the soul's activities as guiding the body. These are different ways of laying stress on what has for good reason come to be called "Platonic dualism": the idea that soul and body are such different kinds of entity that their relation is problematic and difficult to understand. But Plato makes things harder for us, and for himself, than he needs, by failing to focus on just where the line distinguishing soul from body should be drawn. As we have seen, sometimes the contrast is between the body to be animated and what animates it, sometimes between the animated body and either intellectual functions or a power of self-motion, belonging to the soul over and above its embodiment. It is because this line shifts, as well as because his conception of the soul's nature is not always constant, that we find such diverging pictures of the soul-body relation.
**Plato follows two lines of thought** in his discussion of the relationship between the body and the soul, one assigning the soul the role of ruler over the body, the other representing the body as a hindrance to the soul. This romantic lithograph published by Currier & Ives in 1876 espouses the second line of thought. The verse below the image reads, "I will sing you a song of that beautiful land. / The far away home of the soul. / Where no storms ever beat on the glittering strand. / While the years of eternity roll. / Oh, that home of the soul in my visions and dreams. / Its bright jasper walls I can see; Till I fancy but thinly the veil intervenes / Between this fair city and me."
## Reincarnation, Myth, and Argument
One of the most strongly marked themes in the dialogues is that the soul survives the person's death; but we have seen that it is not clear what this soul is. Especially where the emphasis is on leaving the body behind, it is hard to see how what survives could be the individual soul—Socrates' soul, say—for everything pertaining to the history of Socrates as an embodied individual will have been shed. How can Socrates' soul retain its individuality while retaining none of this history?
Plato struggles with this issue rather than resolving it. In some dialogues we find stories of postmortem judgment, with rewards for virtuous lives and punishments for the wicked. These rewards and punishments, moreover, are often said to have an effect on the soul with respect to further lives it will live in an embodied state. Sometimes we get a full-blown story of reincarnation, present lives being the fruit of past lives and having within themselves consequences for future lives. All this presupposes that an individual soul can remain one and the same soul through many lives, capable of improvement or degeneration as a result of them (though not, of course, conscious of them when in the embodied state).
**In his dialogues,** Plato makes clear that an individual's soul survives his or her death. What is not clear is what the soul in fact is. Plato struggles with rather than resolves this issue, bringing the ideas of both reincarnation and judgment into his work. Plato, of course, was not the only philosopher to develop a line of thinking on reincarnation. This Buddhist reincarnation wheel, which outlines the cycle of reincarnation, and is held in the clutches of Mara, Lord of Death and Desire, was built sometime between 1177 and 1249.
The status of these judgment and reincarnation stories in Plato is very disputed. Some have hailed these narrative "myths" as poetic invocations of insights that go beyond argument; others have seen them as ways of introducing ideas which evade argument. It is likely that they do not all have the same tone, or function. Some seem ironic (particularly when describing humans as reincarnated as animals), others quite serious.
We should remember that Plato avoids presenting his ideas as dogma, in treatises; he employs various strategies of indirectness. Clearly the idea of a judgment after death on the way a life has been lived was important to him, as was the idea of one life as a fitting outcome of the way another life has been lived. The stories illustrating these ideas can be interpreted as vivid ways of stressing the ethical importance of the way we live now, or as indicating, though not arguing for, a particular metaphysical view of the soul and the self.
Or, of course, as both. Plato's way of writing leaves us to extract ideas from different dialogues, put them together, and work out his position on a given issue. This can leave us frustrated, nowhere more so than with his views on the soul. We will be less frustrated if we think of him as coming up with different kinds of answers as he keeps returning to the nature of the soul. He never doubts that the soul is so different a kind of thing from the body that their relationship is problematic. Nor does he doubt that the soul is immortal—that in some way what I really am is not given by the boundaries of my embodied human life. His explorations of the soul's nature do not all go in the same direction because Plato, while sticking firmly to some points, follows more than one argument about the soul where it leads, and seeks truth about difficult issues rather than attempting to arrive at a tidy finished position.
**For Plato, the best way to understand** the natural world is as a product made by a craftsman. This illustration from William Blake's _Europe: A Prophecy_ (1794) depicts God as an architect creating the world.
# SEVEN
# The Nature of Things
## Chaos and Order
The natural world, despite disruptions, displays a striking degree of order and regularity. For Plato the best model for understanding it is to think of it as a product made by a craftsman, who does the best job he can in imposing order on otherwise unruly materials.
In the _Timaeus_ Plato describes the creation of the world as work done by a divine Craftsman, who does the job by reference to a model—a system of rational principles which are to be embodied in materials to produce a unified result. To the extent that the world can be seen to display rational structure, we can understand it as being the work of Reason; to the extent that it is embodied in materials which constrain reason and make failures possible, we have to take into account the effects of what Plato calls Necessity, the way things just have to be, whether there is a good reason for it or not.
**According to Plato,** God not only created order from chaos—as depicted in this 1731 illustration by engraver Bernard Picart, showing the heavens in chaos at the origin of the world, with constellations (including the twelve signs of the zodiac), fire, clouds, and wind—but created only _good_ from that chaos.
Plato's account, fanciful in detail and often obscure, raises a number of issues about what we would call his metaphysics. The divine Craftsman creates a _good_ world; why? Mathematics plays an important role in the _Timaeus's_ account of the world's structure—what role does this play in Plato's view both of the world and of the kind of knowledge that we might achieve of it? And finally, the _Timaeus_ makes prominent one of Plato's most famous ideas, that the real world is not, as we uncritically take it to be, the world around us that our senses report to us; the real world is rather what we grasp in thought when exercising our minds in abstract philosophical argument, in particular arguments which lead to what Plato calls Forms.
The _Timaeus_ was seen as central to Plato's metaphysical thinking until the nineteenth century, when obsession with Plato's political thinking replaced it by the _Republic,_ still Plato's most frequently read work. As often with Plato, both works are important, and point up different aspects of his thinking in ways that encourage both unification and contrast.
## God and Goodness
The Craftsman God made the best world possible because he is good _(Timaeus_ 29d—30c) and so wanted what he made to be as good as it could be. And, being free of jealousy because he is good, he wanted the world, in being as good as it could be, to be as much like him as it could be.
Coming to this idea under the influence of two thousand years of monotheism (Judaism, Christianity, Islam), we may be unsurprised by the idea that God is good and that his creation is good because he is. Here we should remember two points. One is that Plato is going out on a limb in his own culture. The other is that even so, Plato's position is in an important way still weaker than the monotheistic views we are accustomed to.
**In a dramatic departure** from the beliefs of his own culture, Plato tells us in _Timaeus_ that because God is free of jealousy, he made the world as good as it could be. German painter Albrecht Dürer's oil on wood, created around 1504, shows Christ blessing the world.
Ancient popular religion—various forms of polytheism—did not claim that God, or the gods, were good. This would have seemed naive and unrealistic; the divine, superhuman forces in the world, and in humans, appear to present a mixture of good and bad. The Greek gods of popular religion are capable of petty and destructive behavior. They are, moreover, extremely jealous where humans are concerned.
Plato's idea that God is good, and produces only good, is one that alienates him decisively from popular religion. He never rejects the forms and practices of the religion he knew, but he develops a theology which is radically at odds with most people's understanding of that religion. In the _Republic_ he insists that the gods are responsible only for good, and accepts that in a well-organized society this will require a radical censorship of most of the popular stories that people tell about the gods. (As we have seen, Plato does not care about suppressing people's creative and imaginative thinking, in this case about the divine.)
In the _Laws,_ he goes further. Although public religion remains that of an ordinary Greek city-state, repressive measures are introduced that have no parallel in the ancient pagan world. Citizens are to have no private shrines or worship of their own; the standard public rites are to be the only ones they take part in. And it matters not just what they do but what they believe; all citizens are to believe that there really are gods, that these gods care for humans, and that they cannot be bribed to overlook wrongdoing. Citizens who deny these beliefs are to be re-educated, or, if unpersuadable, executed. Plato is unique among ancient philosophers in holding it important for everyone to have the right beliefs about God (or the gods) and for these beliefs prominently to include the belief that God is responsible only for good, not for evil.
**The Greek gods** of Plato's culture were capable of petty and destructive behavior. In one myth, the mortal hero Actaeon saw Artemis, the goddess of the hunt, at her bath. The goddess was so angry that the mortal had seen her naked that she turned him into a stag, upon which he was killed by his own hounds. This sculpture of Actaeon, half man, half stag, is in the gardens of the magnificent Royal Palace at Caserta in Italy.
No other ancient philosopher rejects popular religion to this extent, and it is no surprise that ancient Christian thinkers found Plato by far the most congenial of the pagan philosophers. His concern with ordinary people's beliefs about God, or the gods, was as important to them as his insistence that God, or the gods, are good, and in no way evil.
Yet there is a barrier separating Plato from later Jews and Christians who took over much of his thought, and in particular spent huge amounts of energy in trying to assimilate the _Timaeus_ to the creation story in _Genesis._
Plato's God is a workman who does the best he can with the materials he has to work with; he creates order from chaos, but he does not create the original materials from nothing. (An already long tradition in Greek philosophy held that creation from nothing was an incoherent idea.) As a result, Plato does not face the "problem of evil" troubling the Judeo-Christian tradition; if God creates the world from nothing, then why does he create evil as part of it? Plato's God is a creator in the way a craftsman is; he makes the product, which is an excellent one, but he is not responsible for the effects of "Necessity," the unavoidable defects of the materials.
**In Plato's _Laws,_** we learn that the philosopher thinks that citizens should participate only in public rites and have no private shrines of their own, and that they must believe both that there are gods and that the gods care for humans. This view of the temple of Hephaestus and Athena Ergane, dedicated to the patron-god of metalworking and the patron-goddess of pottery and crafts, respectively, and inaugurated in 416—15 BCE, is from an 1851 engraving by Henry Winkles.
## Mathematics and Knowledge
In the _Timaeus_ great emphasis is placed on the mathematically calculable nature of the heavenly bodies' motions, even the apparently irregular ones. Plato also sustains the by then familiar view that there are four basic elements, but adds that their mutual transformations are due to the different geometrical figures in their underlying structure. For Plato it is basic to our world's being an ordered one that mathematics is the key to it.
In many dialogues mathematics is an important model for Plato for understanding knowledge. Sometimes, especially in the shorter dialogues where Socrates is depicted examining various types of virtue, the model for having knowledge is that of having a skill or expertise, and what is at issue is _practical_ knowledge. Nonetheless, some conditions emerge which for Plato always hold of knowledge (as discussed in Chapter 1). Knowledge can be communicated, and the person with knowledge can "give an account," explain and justify what she knows. And knowledge requires using your mind to think for yourself about things, rather than taking over opinions secondhand without examining them. In contrast, beliefs, even if true, are inferior in at least two ways. They can be produced by "persuasion," techniques for producing conviction which bypass explanation and justification, and result in a person's holding a view without understanding. The person with knowledge, however, understands what he knows and can "give an account" of it. In some works Plato thinks of this giving of an account as being comparable to the articulation an expert could give of her practical expertise.
Mathematics, however, takes hold as a model where Plato puts more stress on two features of knowledge. One is the idea of knowledge as _structured,_ not just a mass of information but an organized system of basic truths and others derived from them. To Plato, this ideal of systematization, allowing deployment of what there was to be understood, could be seen in geometry, the best-developed branch of mathematics that he knew. In geometry we can discern the starting points, the derived results, and a transparent account of the way they were derived. This ideal of knowledge appears in the _Meno_ and _Phaedo,_ but is seen at its most ambitious in the central books of the _Republic._ And in works like the _Timaeus_ and _Philebus_ we find Plato insisting that it is mathematics which provides us with whatever is organized and reliable in our knowledge.
**In the _Timaeus,_** Plato trusts in the mathematical calculability of the heavenly bodies. This seventeenth-century engraving depicts Polish astronomer Johannes Hevelius and his assistant using a six-foot sextant to measure the angular distances between stars.
The second impressive point about mathematics is quite simply its objects. When we learn Pythagoras's theorem, we are grasping something in our thinking, which is not made true (or false) by the particular diagrams we draw to illustrate it; any irregularities in these are irrelevant to the mathematical truth. Although it is not to be encountered in the world of experience, it is certain; having proved it, we know it to be true. It is clear that Plato was deeply impressed by this feature of mathematics: not only can we be certain of the results we prove, we realize that it is only by exercising a certain kind of abstract thinking that we can understand them. We learn that the evidence of our senses may be irrelevant to the results we can prove in thought, which may even conflict with them. For Plato this is the beginning of philosophical wisdom, the right way to think for ourselves about things. Although his views about knowledge vary, and he sometimes thinks that we can know items of experience (compare Chapter 1), Plato is sympathetic to the idea that progress toward knowledge properly begins when we come to think of the world of our experience as irrelevant, and appreciate that it is abstract thinking that produces understanding. Mathematics is a powerful influence on him as an excellent example of this progress.
However, both in its objects and in its way of thinking, mathematics is itself inferior to, and thus merely a good preparation for, the thinking done by the people Plato calls philosophers.
**Greek philosopher and mathematician** Pythagoras (ca. 580—ca. 500 BCE) is pictured here with Greek mathematician and inventor Archimedes (ca. 287—212 BCE) on the title page of a geometry and surveying manual. Pythagoras holds a right triangle, illustrating his theorem that the square of the length of the hypotenuse of a triangle is equal to the sum of the squares of the other two sides, a piece of knowledge that, once we have possessed it, we know to be true, even though we will not encounter it in the world of experience. _Geometriae practicae novae et auctae_ (New and enlarged practical geometry), by mathematician Daniel Schwenter (1585—1636), was published posthumously in 1641.
## The Forms
Philosophers, according to Plato, employ a kind of thinking which he calls _dialectic._ His account of what this is differs strikingly in different works, but one aspect remains: it develops in _dialegesthai,_ discussion. Philosophy always involves argument and discussion, ideally with others, and requires you to be able to defend your position against the arguments of others. It is not obvious what the best methods are for philosophers to use, and this is where we find the most variation, but again Plato is always sure that philosophical thinking is superior to all other kinds. Even mathematicians do not genuinely understand their own results; it is philosophers who make use of, and examine, others' results to make sense of them and establish the kind of justification they require. This conception of philosophy, which sounds astonishingly arrogant to others, has been one many philosophers aspire to, in spite of periods when philosophy has been bound in advance to answer to the discoveries of science, or theology.
The most famous aspect of Plato's view of philosophy has generally been his claim that philosophical thinking grasps what he calls "Forms" (though he has no technical term, often using a Greek idiom, "the F itself," which conveys little in English). Sometimes his philosophy is presented as though Forms were the high point and centerpiece, which is a tribute to the power of the idea, since Plato, in keeping with the way he writes in the dialogue form, has no sustained presentation of any "theory" of Forms. Forms appear at various points in the dialogues as an idea already familiar to Socrates and others, but there is no positive introduction of this idea, supposed to be so familiar. In the first part of the _Parmenides,_ however, six serious objections are brought against Forms, with the conclusion that the idea is a good one but needs further work to be viable.
**According to Plato,** philosophy always involves argument and discussion and requires you to be able to defend your position against the arguments of others. This image from the fourteenth-century _Liber de herbis_ (Book of herbs), by Monfredo de Monte Imperiali, shows an imaginary debate between twelfth-century Spanish-Arab philosopher and physician Averroës and third-century Phoenician Neoplatonic philosopher Porphyry.
The oblique and scanty appearances of Forms have not stopped readers from building a "Theory of Forms" out of these few passages, and from confronting this theory (successfully or not) with Plato's own criticisms. This is probably what Plato wanted us to do, but we should be cautious about making definite or final claims about an idea which is deliberately presented in such an elusive way.
In the _Timaeus_ Forms are presented in a very general way, as implied by our recognition of the differences between knowledge and true belief. (Plato, we should notice, does not consider the option that our conception of knowledge might not answer to anything; he assumes that the knowledge we aspire to have is, at least in principle, attainable.) This, however, leaves wide open what kind of thing Forms are, and Plato's treatments are not easy to unify.
In the _Timaeus_ itself Forms function as patterns for the Craftsman as he makes our world. Things in our world—species and kinds of thing, and the four primary elements—are embodied in matter and spatially situated (Plato is very obscure on this point, and was criticized for this by Aristotle) and, crucially, they "come to be," whereas Forms "are, without coming to be." This is the important metaphysical difference between Forms on the one hand and, on the other, the items around us which are said to "participate in" Forms, or to be "likenesses" or "images" of them. This difference is stressed forcibly also in the _Phaedo, Republic,_ and _Symposium,_ in some of Plato's most memorable passages. But we do not always get the same answer to the question of what it is for items in our world to "come to be," and, correspondingly, to the question of what items are "participants" in Forms.
**In Plato's _Parmenides,_** six serious objections are brought against Forms. The _Parmenides_ is a discussion between Socrates and the philosophers Parmenides of Elea, shown here in a ca. fifth-century BCE bust, and Zeno of Elea.
• • • • •
### THE FORMS
**_TIMAEUS_ :** Now it's with argument that we should make these distinctions and inquire about them. So: is there such a thing as Fire itself by itself, and so on for all the things of which we always say that each is "itself by itself"? Or are the things we see, and whatever we perceive through the body, the only things that have this kind of reality, and is there nothing else at all in any way over and beyond them, so that our claim in each case that there is a thinkable form for each of them is lost labor, nothing after all but words?
**The four primary elements,** as described by Greek pre-Socratic philosopher Empedocles, are earth, air, fire, and water, as pictured here in a colored woodcut from a 1472 edition of Lucretius's _De rerum natura_ (On the nature of things).
Well, it is not appropriate for us either to dismiss the present question without judgment or verdict, simply insisting that things _are_ like this, or to throw into an already long discourse a digression itself lengthy. But if a large distinction drawn briefly could be presented, that would be most suitable of all.
So this is how I myself cast my vote. If understanding and true belief are two different kinds, then absolutely there are these things "by themselves," forms that are not perceivable by us, only thinkable. But if, as some think, true belief is not at all different from understanding, then we have to take everything we perceive through the body as being the most stable items. But we _do_ have to say that they are two different things, because they come into being separately, and are unlike each other. We come to have understanding through teaching, while true belief is brought about in us by persuasion. Understanding always involves a true account, while true belief has no account to give. Understanding is not movable by persuasion, while true belief can be changed by it. And we have to say that everybody has a share of true belief, while of understanding only the gods do, and the human race to a small extent.
_(Timaeus 51b—e)
_
• • • • •
One surprisingly common answer is definitely wrong, namely that there is a Form for every word that we apply to a number of individuals, and so a Form for every general term (making Forms into what were later called universals). This view is based on a mistranslation of a passage in the _Republic_ (596a), which actually says that wherever there is a Form there is only one. The principle of a Form for every general term would be completely trivial, and make it baffling why Forms are objects of understanding, items we have to use our minds, with effort, to grasp. Moreover, it runs against Plato's firm view that our use of language embodies convention and prejudice and on its own is no good guide to philosophical truth _(Cratylus, Statesman_ 262—63).
"Participants" "come to be," while Forms "are." One way in which things come to be is that they change; at one time a thing has one property, and later it comes to have another, and may even come to have a property excluding or opposed to the original one. Certainly it is not hard to find passages where Plato stresses the mutability of the world we experience around us, contrasting this to the changelessness of the Forms. And this connects with understanding; we have a better grasp of what a thing is if we are not forced to characterize it in ways that have to be changed as it changes. (And one feature of mathematics is that its truths do not change over time.) But the mere fact that things around us change is a remarkably weak reason for insisting on their metaphysical inferiority to items that do not change. Fortunately it is not Plato's only reason.
More interesting is the "argument from opposites," which is the most prominent way in which Forms are discussed in the _Phaedo, Republic,_ and _Hippias Major._ This focuses on the point that, while we can make a true claim that something in the world of our experience is F, for some property F, we can also find some perspective from which we can also claim truly that it is the opposite of F. Sticks which are equal, say, in length are also unequal in, say, width; a girl who is beautiful among other girls is unappealing compared to goddesses; an action which is right in being the fulfilling of a promise is also wrong in being irresponsibly dangerous; and so on. Sometimes the perspective from which we find the opposite property to F is far-fetched in the extreme, but the point is that it can always be found. Hence, none of the items in the world of our experience can be really or truly F—F in a way that excludes ever, in any way, being the opposite. But we do have a grasp of what it is for something to be really and truly F, for this is what we grasp when we understand what it is to be F. So we find that the objects of our understanding are not the items in the world of our experience, which can always turn out to be the opposite of F as well as F, but rather "the F itself," the Form which we grasp in thought when we understand what it is to be F.
**While we can truly claim** that something in our world of experience is F, for some property F, we can also find a perspective from which we can truly claim that it is the opposite of F. For example, a girl who is beautiful among other girls is unappealing compared to goddesses. This fragment of a fresco from the Villa Lemmi near Florence, Italy, showing the woman Giovanna degli Albizi with Venus and the Graces, was created between 1486 and 1490.
This argument shows why Plato connects the difference between being and coming to be so closely to the difference between knowledge and belief. It also gives a role to his emphasis on change, since a thing's changing is clearly one way in which it can turn out to be F from one perspective and the opposite of F from another. What has caused most difficulty is that the argument from opposites will produce Forms, obviously, only for terms with opposites, but that, while Plato sometimes appears to realize (and indeed build on) this, at other times he expands the "range" of Forms without argument.
This problem is one of the many that we are left with, along with Plato's own six objections, when we try to bring together all his views on Forms. Plato does not pretend to have a final version. He makes a respected older philosopher say to Socrates, in the _Parmenides,_ that the further work the theory needs is to be found in the practice of argument, and this is doubtless Plato's advice to us.
• • • • •
### A FAMOUS IMAGE OF PLATO
One of the most famous and often-reproduced images of Plato comes from Raphael's fresco _The School of Athens,_ painted for the library of Pope Julius II. This picture of ancient philosophy is heavily influenced by the revival of Platonism in the Renaissance, and dominated by the figures of Plato, who holds the _Timaeus_ and points upward, while Aristotle, holding his _Ethics,_ looks at Plato's upraised hand but also gestures outward. The contrasting gestures indicate that Aristotle is more concerned to understand the world around us in terms of philosophical principles, while Plato is more austerely focused on the abstract and theoretical principles themselves. In the fresco there is great stress on the _Timaeus'_ mathematization of the world's underlying structure. Plato is shown between Pythagoras and Euclid, and his features are not those of the ancient portrait busts, but those of a contemporary mathematician, Leonardo da Vinci. In the Renaissance, Plato was also important as the philosopher most influential on Christianity. On the wall opposite, Raphael's depiction of the Trinity is greatly influenced by contemporary Neoplatonic writers. Saint Justin, a Platonist philosopher of the second century CE, who converted to Christianity and was martyred, repeats Plato's upward gesture as he points toward the Incarnation. In Pope Julius's scheme, the highest achievement of pagan philosophy recurs on a reduced scale in the representation of the central ideas of Christianity.
• • • • •
**Pictured is the center detail of Plato** and Aristotle from Raphael's fresco _The School of Athens_ (1509—10), which is in the Room of the Segnatura at the Vatican.
**This detail of St. Justin Martyr** is from Raphael's fresco _Disputation over the Sacrament_ (1509—10), which appears on the wall opposite _The School of Athens_ in the Room of the Segnatura at the Vatican.
## Conclusion: Philosophy
The Japanese Plato scholar Noburu Notomi has pointed out that when Western philosophy was introduced to Japan in the nineteenth century, a new word ("tetsu-gaku") was coined for it, for, although the various branches of what we call philosophy (cosmology, logic, moral and political thought, for example) had been extensively developed in Eastern intellectual traditions, these studies had not been unified under the heading of "philosophy." They have not always been unified in Western intellectual traditions either, and Notomi is in good company in finding Plato to be the first thinker for whom philosophy is a unified endeavor, to be defined and defended against competitors as being the way for us to seek understanding and wisdom. Plato was the first to institutionalize philosophy (giving us the word "Academy") and to think of it as requiring both a systematic pursuit of truth and a radical dependence on argument, with others and with oneself. It is not surprising that he left a divided legacy of dogmatists and skeptical inquirers, or that his dialogues have lent themselves, over two thousand years, to the most divergent interpretations. For in the end, his deepest message is not that we should believe in Forms, or the importance of virtue, but that we should engage with him, and with our own contemporaries, in aspiring to understand these matters.
**Although the various branches** of what we call philosophy had been extensively developed in Eastern intellectual thought, they were not gathered under the heading of "philosophy" until Western philosophy was introduced to Japan in the nineteenth century. However, the poetry of Kakinomoto no Hitomaro (ca. 658–ca. 708), who is shown in this color ukiyo-e print, exemplifies a Shinto esthetic in which what we now recognize as the various branches of philosophy are brought together.
# REFERENCES
•
### CHAPTER 1
The issue discussed in connection with the jury passage in the _Theaetetus_ was first clearly raised, and its importance stressed, by Myles Burnyeat in "Socrates and the Jury: Paradoxes in Plato's Distinction Between Knowledge and True Belief," _Aristotelian Society_ Supplementary Volume LIV (1980), 173-91.
### CHAPTER 2
Alice Riginos, in _Platonica,_ the anecdotes concerning the life and writings of Plato (Brill, Leiden, 1976) shows the fragility of the ancient traditions about Plato. (See pp. 64–69 for Egypt stories; pp. 9–32 for Plato's "Apollonian nature"; pp. 35–40 for Plato's name; and pp. 70–85 for Plato's political involvements.) For details about Plato's family, see J. K. Davies, _Athenian Propertied Families_ (Oxford University Press, 1971). For Socrates, see C. C. W. Taylor, _Socrates,_ in the Oxford University Press Very Short Introduction series, and also the articles in Paul Vander Waerdt (ed.), _The Socratic Tradition_ (Cornell University Press, 1994).
### CHAPTER 3
See Andrea Nightingale, _Genres in Dialogue: Plato and the Construct of Philosophy_ (Cambridge University Press, 1995), for Plato's demarcation of philosophy from other literary genres.
The Anonymous Commentator on the _Theaetetus_ (quoted at column 54, 38—43) is a dogmatic Platonist who here records the position of the skeptical Academics. His date is uncertain, and may be from the first century BCE to the second CE. Plutarch of Chaeronea is a second-century CE dogmatic Platonist writer, best known for his historical biographies, who has sympathy for the skeptical tradition. The quotation from Cicero is from _Academica_ II 46; that from Sextus is from _Outlines of Scepticism_ I 221-23.
For an introduction to "Atlantis studies," see Richard Ellis, _Imagining Atlantis_ (New York, Random House, 1998).
### CHAPTER 4
For Augustine, see _City of God,_ Book VIII, especially Chapter 5. Serious recent study of ancient homosexuality begins with K. J. Dover's _Greek Homosexuality_ (London, Duckworth, 1978). For an up-to-date discussion, see James Davidson, _Courtesans and Fishcakes: The Consuming Passions of Ancient Athens_ (London, Fontana, 1998).
Tom Stoppard's _The Invention of Love_ is published by Grove Press, New York (1997).
### CHAPTER 5
Plato's assumptions about happiness are clear in the _Euthydemus_ and _Philebus,_ though he does not lay them out explicitly as his pupil Aristotle was to do in his _Nicomachean Ethics._ The _Euthydemus_ is the major passage in which Plato develops the idea that it is the use of things that matters, and that they don't have value in themselves; a modified version of this can be found in the first two books of the _Laws. Apology, Crito,_ and _Gorgias_ are the major sources for Socrates' uncompromising commitment to the position that virtue is sufficient for happiness. Plato's views about education and the relation of the individual to community and to political society are to be found in the _Statesman_ and _Laws,_ as well as the more familiar _Republic,_ whose "ideal state" has been read in a literal-minded way, and overemphasized, by many interpreters.
### CHAPTER 6
Plato's arguments about the soul can best be encountered in the _Phaedo, Republic, Phaedrus,_ and _Laws._ A collection of recent articles which forms a good introduction to the major issues is _Essays on Plato's Psychology,_ edited by Ellen Wagner (Lexington Books, 2001).
### CHAPTER 7
Plato's difficult dialogue _Timaeus_ is translated with a long introduction by Donald Zeyl (Hackett, Indianapolis, 2000). A short introduction to important issues in Plato's approach to cosmology is Gregory Vlastos, _Plato's Universe_ (Oxford University Press, 1975). Arguments about Forms, and about knowledge, are treated in papers reprinted in Gail Fine (ed.), _Plato_ I (Oxford University Press, 2000). Plato's metaphysics and epistemology are the subject of much of the introductory literature mentioned under "Further Reading."
The comment by Noburu Notomi is from the introduction to his book _The Unity of Plato's Sophist_ (Cambridge University Press, 1999). In my _Ancient Philosophy: A Very Short Introduction,_ Chapter 6, I say a little more about philosophy in the ancient world and about Plato's role as establishing philosophy as a subject.
# FURTHER READING
•
There are now available many recent translations of all of Plato's dialogues. The complete works are available in _Plato, Complete Dialogues,_ edited by John Cooper (Hackett, 1997). Several dialogues are also available in individual Hackett translations, and also in recent translations published by Penguin and in the Oxford World's Classics series. Individual dialogues are all available in inexpensive paperback editions. If you become interested in Plato, you are well advised to read a dialogue in several translations, to get some idea of difficulties in the text.
The Clarendon Plato series contains new translations accompanied by philosophical commentary; these are for someone with a more advanced interest.
Several recent collections highlight problems of method in reading Plato, including Charles Kahn, _Plato and the Socratic Dialogue_ (Cambridge University Press, 1996); C. Gill and M. M. McCabe (eds.), _Form and Argument in Later Plato_ (Oxford University Press, 1996); J. C. Klagge and N. D. Smith (eds.), _Methods of Interpreting Plato and His Dialogues, Oxford Studies in Ancient Philosophy_ Supplementary Volume, 1992; J. Annas and C. J. Rowe (eds.), _New Perspectives on Plato, Modern and Ancient_ (Harvard University Press, 2002).
There are many short introductions to Plato in standard reference works. (The articles on Plato in the new _Oxford Classical Dictionary_ and in _Greek Thought: A Guide to Classical Knowledge_ by Harvard University Press were written by me.)
Richard Kraut (ed.), _Cambridge Companion to Plato_ is a useful introduction to various aspects of Plato and has good bibliographies, both on individual dialogues and on Platonic topics. Christopher Rowe's _Plato_ is a good medium-length survey. Christopher Gill, _Greek Thought,_ is excellent background to Plato's ethical and social thought. Gail Fine, _On Ideas,_ is a thorough examination of the arguments for Forms and Aristotle's criticisms of them.
# PICTURE CREDITS
•
**ART RESOURCE: 108**
**THE BRIDGEMAN ART LIBRARY INTERNATIONAL: vi–vii; 16**
**CORBIS:** ii: © Gianni Dagli Orti/CORBIS; viii; 36; 50: © Bettmann/CORBIS; 29: © Christie's Images/CORBIS; 62: © Blue Lantern Studio/CORBIS; 76: © Robbie Jack/CORBIS; 139: © Mimmo Jodice/CORBIS; 147: © Ted Spiegel/CORBIS; 172appendix73: ©Jon Hicks/CORBIS
**GETTY IMAGES:** 20–21: Time & Life Pictures/Getty Images
**THE GRANGER COLLECTION: 10; 56; 141**
**MARY EVANS PICTURE LIBRARY: 33**
**COURTESY OF THE NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION/DEPARTMENT OF COMMERCE:** 136: Title page of Daniel Schwenter, _Geometriae practicae novae et auctae tractatus I[-IV]_
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**COURTESY OF RARE BOOK AND SPECIAL COLLECTIONS DIVISION, LIBRARY OF CONGRESS:** 124: Illustration from William Blake, _Europe: A Prophecy_ (1794)
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**COURTESY OF WIKIMEDIA COMMONS:** 3: Meister des al-Mubashshir-Manuskripts, _Socrates and Two Students_ /Yorck Project: 10.000 Meisterwerke der Malerei; 7: Jean-Léon Gérôme, _Phryne Before the Areopagus_ (1861); 13: Illustration at the beginning of Euclid's _Elementa_ (1309–16), in the translation attributed to Adelard of Bath; 15: _Euclid, or the Architecture_ (1334—36) by Nino Pisano/Upload by Jastrow; 19: Meister des al-Mubashshir-Manuskripts, _Solon and Students_ /Yorck Project: 10.000 Meisterwerke der Malerei; 22: Triad statue of pharaoh Menkaura, Hathor, and the personification of the nome of Diospolis Parva/Upload by Chipdawes; 23: _Eirene Bearing Plutus/Upload_ by Bibi Saint-Pol; 25: Pompeo Batoni, _Apollo and Two Muses_ (1741); 27: Bust of Socrates/Upload by Jastrow; 34: Bust of Aristotle from the Ludovisi Collection at the Palazzo Altemps (Museo Nazionale Romano), Rome, Italy/Upload by Jastrow; 43: Plato in the _Nuremberg Chronicle;_ 47: Reinhold Begas, _Kriegswissenschaft_ (1887)/Upload by Mutter Erde; 52: Plato's Academy archaeological site in Akadimia Platonos subdivision of Athens, Greece/Upload by Tomisti; 53: Plotinus in the _Nuremberg Chronicle/Upload by_ Zp at cs.wikipedia, transferred by Sevela.p; 55: Marcus Tullius Cicero, by Bertel Thorvaldsen, copy from Roman original/Upload by Gunnar Bach Pedersen; 59: Actor wearing the mask of a rustic from Canino, Italy/Upload by Jastrow; 60: Oxyrhynchus Papyrus, containing excerpts from _The Republic;_ 61: Rembrandt, _Homer/_ Yorck Project: 10.000 Meisterwerke der Malerei; 65: Parthenon/Upload by Thermos; 66: _Erastes_ (lover) and _eromenos_ (beloved) kissing/Upload by Jastrow; 69: Alcibiades, Palazzo dei Conservatori, Hall of the Triumphs / Upload by Jastrow; 70: Anselm Feuerbach, _Plato's Symposium_ /Yorck Project: 10.000 Meisterwerke der Malerei; 73: Woman grinding wheat/Upload by Jastrow; 84: Raphael, detail, tondo of Justice in the Room of the Segnatura in the Vatican/Yorck Project: 10.000 Meisterwerke der Malerei; 85: Zeno of Citium/Upload by Shakko; 87: _Virtuous Woman Tames Woodwose_ from _Zahm und wild, Basler und Straßburger Bildteppiche des 15. Jahrhunderts;_ 91: Landscape scene from the _Odyssey_ by a Roman master, ca. 60—40 BCE/Yorck Project: 10.000 Meisterwerke der Malerei; 92: Statue of Melpomene, muse of tragedy (second century CE) from Monte Calvo/Upload by Wolfgang Sauber; 95: Family scene with inscription "[daughter of] Mousiaos," Musée du Louvre/Upload by Jastrow; 97: Kleroterion/Upload by Xocolatl; 98: Pinakia/Upload by Marsyas; 99: Pnyx/Upload by Qwqchris; 106l: Statue of Pothos/Upload by Jastrow; 106r: Statue of Athena/Upload by Jastrow; 109: Image of _Timaeus_ pages from 1578 Stephanus edition of Plato's works; 110: Mark Antokolski, _Death of Socrates_ (1875)/Upload by Alex Bakharev; 115: Jan Saenredam, _Taste;_ 117: Aristotle in the _Nuremberg Chronicle;_ 121: Wheel of reincarnation/Upload by Calton; 128: Albrecht Dürer, _Salvator Mundi_ (1504)/Yorck Project: 10.000 Meisterwerke der Malerei; 130–31: Sculpture of Actaeon, Royal Palace at Caserta/Upload by Japiot; 138: Monfredo de Monte Imperiali, _Liber de herbis, Imaginary Debate Between Averroes and Porphyry;_ 144—45: Sandro Botticelli, Frescoes from the Villa Lemmi near Florence, scene: Giovanna degli Albizi with Venus and the Graces, fragment/Yorck Project: 10.000 Meisterwerke der Malerei; 148: Raphael, _Disputation over the Sacrament,_ detail/Yorck Project: 10.000 Meisterwerke der Malerei
# BRIEF INSIGHTS
•
A series of concise, engrossing, and enlightening books that explore
every subject under the sun with unique insight.
_Available now_ :
**THE AMERICAN PRESIDENCY**
Charles O. Jones
**ATHEISM**
Julian Baggini
**BUDDHISM**
Damien Keown
**THE CRUSADES**
Christopher Tyerman
**EXISTENTIALISM**
Thomas Flynn
**HISTORY**
John H. Arnold
**ECONOMICS**
Partha Dasgupta
**GALILEO**
Stillman Drake
**GANDHI**
Bhikhu Parekh
**GLOBALIZATION**
Manfred Steger
**INTERNATIONAL RELATIONS**
Paul Wilkinson
**LOGIC**
Graham Priest | **JUDAISM**
Norman Solomon
**LITERARY THEORY**
Jonathan Culler
**MODERN CHINA**
Rana Mitter
**PAUL**
E. P. Sanders
**PHILOSOPHY**
Edward Craig
**PLATO**
Julia Annas
**MARX**
Peter Singer
**MATHEMATICS**
Timothy Gowers
**NELSON MANDELA**
Elleke Boehmer
**POSTMODERNISM**
Christopher Butler
**SOCIAL AND CULTURAL ANTHROPOLOGY**
John Monaghan and Peter Just
**STATISTICS**
David J. Hand
---|---
• • • • •
| {
"redpajama_set_name": "RedPajamaBook"
} | 4,031 |
Babies!
It's like a YouTube video with high production values.
By Dana Stevens
May 07, 20104:37 PM
I wish there was a whole Babies TV channel (analogous to the short-lived cable Puppy Channel of the late '90s). It would be so relaxing to flip away from the cable news chatter and reality competitions and watch unnarrated footage of small preverbal people chewing their own feet. Babies (Focus Features) is barely even a movie—it's more of a 79-minute YouTube video with high production values and the vaguest of ethnographic pretensions. But it scratches an itch film audiences may not have known they had—of course! the big screen needed more babies!—and sends you home with your brain pleasantly awash in a bath of oxytocin. I predict it will be, by documentary standards, a runaway hit.
Directed by Thomas Balmès from an original idea by producer Alain Chabat, Babies couldn't be simpler in concept or execution: The film follows four babies from birth till first steps, each growing up in a radically different cultural context. There's Mari, a baby girl in Tokyo; Ponijao, a girl in a rural village in Namibia; Bayarjargal, a boy in the steppes of Mongolia; and Hattie, a girl in San Francisco. Balmès' cameras filmed for a total of more than 400 hours, observing the infants as they nurse, sleep, play, torment cats, chew on rolls of toilet paper, and grow. Any adult conversation that's captured in the process is purely random background noise and appears unsubtitled on the screen. In interviews, Balmès has said that his original idea was to make a wildlife film about human babies, and Babies is at times reminiscent of a nature documentary, right down to the manipulative music soundtrack that works too hard to reinforce the already-sufficient cuteness on-screen.
The chubby bald subjects of Babies seem, by moments, to be enacting small parables about childhood and human behavior. As Bayarjargal is tormented by his older brother, who repeatedly whacks him in the face with a scarf, the younger boy simultaneously wails at the injustice and waits eagerly for the next whack, proving the truism that negative attention is better than none at all. Ponijao, who has nothing to play with but animal bones, dirt, rocks, and the bodies of her mother and siblings, gives Western parents a lesson in how to turn lemons into lemonade; without a single toy, book, or mommy-and-me enrichment class, she manages to develop on schedule (for whatever that concept is worth) and have a grand-old time doing it. The segments concentrating on Hattie, a relatively privileged only child, are the closest the movie comes to social commentary, but the satiric brush Balmès wields is both gentle and broad. As her father plays with her in bed, Hattie's mother reads a book called Becoming the Parent You Want To Be, a title that I doubtflies off the shelves in Mongolia. Later her mother takes her to a children's music class where the teacher drones, "The earth is our mo-other," as Hattie, understandably, makes a break for the door.
Of the four babies, the Mongolian Bayarjargal is perhaps the most ready for his close-up, with a wide face and droopy jowls that give him the look of Bert Lahr as the Cowardly Lion. His deadpan stare as his yurt is invaded by a rooster, or a goat wanders over to drink bathwater from his tub, is among the movie's best comic effects. And while Babies does elicit the occasional coo (for example, when Balmès' camera captures the elusive phenomenon of the newborn "sleep smile"), by far the more common response to the shorties' shenanigans is laughter. As every parent of a young child knows, babies are hilariously funny to watch, most of all when they have no idea they're being funny. This movie's high point may be a multi-part tantrum thrown by Mari, the Japanese baby, as she tries and fails to master a stacking toy. After each frustrated attempt to fit the pieces together, she literally rolls on the floor in despair, then sits up, tries again, fails, takes another wailing dive. Who knew small-motor-skills development could provide both high tragedy and comic gold?
Babies has inspired an interesting schism in the Slate offices: Our editor, David Plotz, has been obsessed with the film since catching an advance screening. When anyone at Slate professes a lack of interest in seeing it, David (a father of three) chalks their indifference up to their childless (and presumably therefore heartless) state. I wish I could find a way to access my pre-parental brain, to test how I would have experienced the movie without four years of daily exposure to cuteness. I don't think you need to have a baby to appreciate Babies, but you need to be open to the part of your animal self that lights up at all things bright and beautiful, all creatures great and small. If you can watch all 17 seconds of the "surprised kitten" video on YouTube without even a twinge of longing to crush said kitten with love, skip Babies. If you find yourself clicking "replay" to watch the kitten again, pre-order your ticket now.
Slate V: The Critics on Iron Man 2, Babies, and Casino Jack
Like Slate on Facebook. Follow us on Twitter. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,406 |
import { Injectable } from '@angular/core';
import { Events } from 'ionic-angular';
import { Storage } from '@ionic/storage';
@Injectable()
export class UserData {
_favorites: string[] = [];
HAS_LOGGED_IN = 'hasLoggedIn';
HAS_SEEN_TUTORIAL = 'hasSeenTutorial';
constructor(
public events: Events,
public storage: Storage
) {}
hasFavorite(sessionName: string): boolean {
return (this._favorites.indexOf(sessionName) > -1);
};
addFavorite(sessionName: string): void {
this._favorites.push(sessionName);
};
removeFavorite(sessionName: string): void {
let index = this._favorites.indexOf(sessionName);
if (index > -1) {
this._favorites.splice(index, 1);
}
};
login(username: string): void {
this.storage.set(this.HAS_LOGGED_IN, true);
this.setUsername(username);
this.events.publish('user:login');
};
signup(username: string): void {
this.storage.set(this.HAS_LOGGED_IN, true);
this.setUsername(username);
this.events.publish('user:signup');
};
logout(): void {
this.storage.remove(this.HAS_LOGGED_IN);
this.storage.remove('username');
this.events.publish('user:logout');
};
setUsername(username: string): void {
this.storage.set('username', username);
};
getUsername(): Promise<string> {
return this.storage.get('username').then((value) => {
return value;
});
};
hasLoggedIn(): Promise<boolean> {
return this.storage.get(this.HAS_LOGGED_IN).then((value) => {
return value === true;
});
};
checkHasSeenTutorial(): Promise<string> {
return this.storage.get(this.HAS_SEEN_TUTORIAL).then((value) => {
return value;
});
};
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,129 |
Q: Can I safely copy an existing laravel project and rename it locally for another project Sometimes my internet is slow/off and I don't have the ability to download via composer,so can I safely create a laravel project via the command
composer create-project --prefer-dist laravel/laravel
and keep that raw project locally to use for another time, or should I use the composer each time I want to start a new project.
A: short answer is YES.
actually I do that and I keep a starting project on my local environment with all the vendors ready, npm modules installed and the vueJs ready to use. and every time I start a project all what I need to do is just a copy.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,111 |
also available
contents
cover
also available
title
introduction
indulgent
basics
copyright
introduction
I have had a very long love affair with pizza. It started at a young age when my mum would make pizzas on the weekends. My mates would come over to watch some movies and eat pizza after a day of surfing and kicking the football down at the beach. Mum would first make the dough, then top the bases with the best toppings she could find. The pizzas were quite simple and therein lies the beauty—you just need a few quality ingredients and the rest, as they say, is 'as easy as (pizza) pie'.
When I started my apprenticeship, at the age of seventeen, I was taught how to make 'authentic' pizza by an Italian fellow named Arturo (or Arthur as he was often called). Arturo was a real pizzaiolo (pizza maker). To watch him work was like watching an opera, a ballet and a rock concert all at the same time. Every part of the pizza-making process has a rhythm and feel to it, from the making of the dough, to the rolling out, tossing and flattening of the dough.
It wasn't until I was thirty years old that I took on the biggest challenge of my culinary career and opened up a pizza restaurant. At the time, I had opened three award-winning restaurants that specialised in contemporary global cuisine, but I wanted to open a very cool pizzeria. I spent a year researching and working in pizzerias to uncover the secrets to the very best pizza. I can remember locking myself away for more than three months in the kitchen, experimenting with my pizzas. I would spend a whole day on the margherita, making 50 different versions of it. I would end the day completely and utterly exhausted, but having perfected the topping. I did the same with all the pizzas that made it onto the menu (and even with those that didn't). I wanted them to be faultless and, most importantly, they needed to be able to be replicated and recreated without me present in the restaurant kitchen. I believe I have achieved this and the restaurant has gone on to win numerous awards, including the Best Pizza in the World award in New York City in 2005 and also a few Best in Australia titles.
But awards aside, what I am most thrilled about is that I have been able to write this book with the knowledge and trust that YOU can create my pizzas at home for family and friends and, most importantly, for yourself, with the same results that I achieve in my restaurants.
As with all recipes, these are just ideas that work for me and my tastebuds, so I hope that they inspire you, and I encourage you to experiment with different flavours. I look forward to hearing about your culinary triumphs. (I am always thrilled when one of my readers sends me their success stories!) Send your top pizza toppings to info@peterevanschef.com. If you take just one of these recipes and incorporate it into your cooking repertoire, then I know that all the work that has gone into making this book has been worth it.
Cheers, Pete
ingredients for the perfect pizza
the oven
The domestic countertop electric pizza ovens are brilliant as they can generate enough heat to make a commercial-quality pizza at home. This is because they have heat elements at the top and bottom, as well as a ceramic pizza stone, which helps to create the perfect crispy crust.
A conventional oven, with the aid of a pizza stone (a flat piece of unglazed stoneware) heated to its hottest setting will also give good results. All domestic ovens differ, but as a general rule, the higher you place the pizza stone in the oven, the better the result. If you have a hooded grill at home, then crank it up as hot as it goes and follow the steps that I have outlined for a conventional oven. And, of course, if you're lucky enough to have a wood-fired oven at home, this is the original and best way to cook pizza.
the dough
Making pizza dough is so easy; take the time to give it a go. You can store proved dough balls in the freezer, then defrost when ready to use. It's fun for the kids and very cost effective. You need to use strong flour to make a proper pizza dough (I like to use '00' flour). In my restaurants, and in this book, we use dry yeast which we activate with lukewarm water before mixing into the dough. Next step is proving. This is when the dough rests, giving it time to rise and for the yeast to ferment before baking. The ideal temperature is between 27°C (80°F) and 43°C (109°F). Any hotter and the yeast will be destroyed, resulting in a heavy and stodgy dough; any cooler and the yeast will not activate, resulting in an inferior dough. I roll out the pizza bases with semolina, polenta or flour as it gives the dough a beautiful crisp finish. I love a thin crust, so that I don't bloat my guests' bellies, but if you like a thicker dough, then by all means go for it. Finally, remember to dock the pizza (prick the base with a fork), to stop the dough from bubbling up when it's cooked.
the sauce
There are generally two types of pizza bases, red (rosso) and white (bianco). Red is obviously a tomato base. My tried and true recipe for a tomato pizza sauce is the simplest thing in the world—I just take the best-quality tinned, whole peeled tomatoes I can find, then blend them with some sea salt, freshly ground black pepper and dried oregano—that's it! I don't cook it, I don't add garlic, I don't add fresh herbs, I don't use fresh tomatoes. This allows the toppings on this very basic, sweet, unadulterated sauce to shine. A white pizza consists of a simple olive oil and seasoning base, then topped with cheese so it doesn't burn, followed by the toppings of your choice. Don't drown the base in sauce or it will never crisp up.
the cheese
In pizza making there are two types of mozzarella, the regular cow's milk stringy one, which is used on the base and acts like a glue to stick the toppings on. Less is more when it comes to this type of cheese. The second type is buffalo milk mozzarella, the food lover's choice. If you have the budget always use this cheese in place of the first one described.
the toppings
Splurge on the best-quality toppings you can afford. Go to a gourmet deli and invest in some paper-thin slices of San Daniele prosciutto or Parma ham—this will slightly melt as it is sprinkleed across the top of a hot pizza. Also, pick up some semi-dried tomatoes, roasted capsicum and good-quality olives. And again, don't smother your pizza with toppings. Remember that what you put on after the pizza comes out of the oven can also turn a good pizza into a great one, so think about using fresh cheeses, toasted nuts, fresh herbs and herb sauces, dressings, and edible leaves of any description.
Buon appetito!
pizza stone
A pizza stone is a piece of unglazed ceramic or earthenware specifically used to cook pizzas on. The nature of the pizza stone and preheating it from a cold oven helps to distribute the heat evenly across the pizza base to create a crisp crust. Always put a pizza stone in a cold oven and then turn the oven on, letting it heat up along with the oven; adding a cold pizza stone to a hot oven can cause it to crack. Remember that once a pizza stone has been heated, the stone is too hot to handle, even with an oven glove. If you don't have a pizza stone, it is possible to preheat a large oven tray for 30 minutes and use it in the same way.
wide pizza spatula
Sometimes called a pizza paddle or pizza peel, or even pizza shovel, the wide spatula is used to slide pizzas out of the oven after cooking. It has a flat carrying surface and a long handle. The spatula can be made of wood or metal. They are available from good kitchenware stores.
pizza docker
A pizza docker is used to uniformly prick holes in the dough to aerate the crust. If you don't prick the dough, steam can build up and create bubbles. It can easily be replaced with a fork.
pizza cutter
A rotary pizza cutter (roller) is designed to easily cut through the pizza.
PIZZA IS OFTEN CONSIDERED A VERY SIMPLE DISH, BUT IF YOU WANT TO IMPRESS SOMEONE SPECIAL, THROW ON SOME LUXURIOUS INGREDIENTS—SUCH AS SCALLOPS, PORCINI OR GOAT'S CHEESE—FOR AN INDULGENT TREAT.
berries with maple syrup pancake pizza
braised ham hock, fontina, cavolo nero and egg
carbonara pizza
scallop, foie gras and caramelised witlof
pork belly and porcini calzone
braised lamb shank and white bean calzone
bresaola with baby beetroot, watercress and horseradish
pulled pork and chipotle coleslaw
vitello tonnato
potato, egg and truffle with lardo
mum's bolognese calzone
peking duck
chilli crab
chocolate hazelnut and banana with vanilla gelato
black forest with chocolate fondant, cherries and Chantilly cream
berries with maple syrup
makes six 15 cm (6 inch) round pizzas
serves 6
Recent research has named blueberries a superfood due to their nutrient-dense make-up. This is great news—delicious as well as nutritious, what a winning combination. This pizza could also be made in a pie shape by folding up the edges, or made into a great calzone.
semolina or plain (all-purpose) flour, for dusting
6 x 85 g (3 oz) pizza dough balls
300 g (10½ oz) whipping cream
1 tablespoon icing (confectioners') sugar, sifted
½ vanilla bean, seeds scraped
125 ml (4 fl oz/½ cup) pure maple syrup
mint sprigs, to serve
berry compote
155 g (5½ oz/1 cup) blueberries
170 g (6 oz) caster (superfine) sugar
finely grated zest of 2 lemons
120 g (4¼ oz) raspberries
120 g (4¼ oz) strawberries, hulled and quartered
pancake mix
2 tablespoons plain (all-purpose) flour
½ teaspoon baking powder
1 teaspoon caster (superfine) sugar
2 eggs, separated
60 ml (2 fl oz/¼ cup) milk
80 g (2¾ oz) fresh ricotta cheese
To make the berry compote, place the blueberries, sugar and lemon zest in a saucepan over medium–high heat. Bring to the boil. Take off the heat and add the remaining berries. Gently stir through. Set aside to cool.
Place two pizza stones in the oven and preheat the oven to 250°C (500°F/Gas 9) or to its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stones to heat up.
To make the pancake mix, sift the flour and baking powder into a bowl, then add the sugar. In a separate bowl, mix the egg yolks and milk together, then slowly whisk into the dry ingredients. Break up the ricotta cheese and add to the mix. Whisk the egg whites in a separate bowl until stiff peaks form, then fold them into the ricotta mixture. Set aside.
Lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 15 cm (6 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled pizzas to the pizza stones. Prick the pizza bases all over with a fork or docker.
Evenly spread 1/3 cup of the pancake mix onto each pizza base, leaving a 2 cm (½ inch) border.
Transfer two pizza bases onto each pre-heated pizza stone. Cook the pizzas in the oven for 5–10 minutes or until golden and crisp.
Meanwhile, in a small bowl, whisk the cream, icing sugar and vanilla seeds to soft peaks. Set aside.
Using a pizza paddle or wide spatula, carefully remove the pizzas from the oven and transfer to a plate. Repeat for the other pizzas.
Drizzle each pizza with 1 tablespoon of maple syrup, add a dollop of the cream mixture, then divide the berry compote evenly between the pizzas. Serve with a sprig of mint.
braised ham hock, fontina, cavolo nero and egg
makes two 30 cm (12 inch) round pizzas
serves 2–4
Before I tried it, the concept of an egg on a pizza seemed strange to me. But then, when you think about it, a pizza is just a flatbread and of course once cooked you could actually call it a toasted flatbread and what goes perfectly with eggs? Toast! This recipe also includes another magical ingredient: ham hock. If you don't have the time to make the ham hock, simply use smoked leg ham or even pancetta or bacon and you'll get the same great results. And if you can't get the cavolo nero, try it with spinach instead— it will still be a magnificent pizza.
semolina or plain (all-purpose) flour, for dusting
2 x 170 g (6 oz) pizza dough balls
80 ml (2½ fl oz/1/3 cup) olive oil, plus 1 teaspoon extra
95 g (3¼ oz/2/3 cup) shredded mozzarella cheese
200 g (7 oz) fontina cheese, sliced
sea salt and freshly ground black pepper
2 free-range eggs
200 g (7 oz) cavolo nero
small handful of baby parsley or celery cress
1 tablespoon extra virgin olive oil
ham hock
2 teaspoons olive oil
1 brown onion, chopped
1 carrot, chopped
1 celery stalk, chopped
2 garlic cloves
800 g (1 lb 12 oz) smoked ham hock
freshly ground black pepper
To make the ham hock, heat the olive oil in a large saucepan over medium heat. Add the onion, carrot, celery and garlic, and cook, stirring often, for 7–8 minutes or until softened slightly. Add the ham hock and 2 litres (70 fl oz/8 cups) cold water. Bring to the boil, skimming off any scum that comes to the surface. Reduce the heat and simmer, partially covered, for 1½ hours, or until the ham meat starts to fall off the bone.
Carefully remove the ham hock from the saucepan and set aside to cool slightly. Remove all the meat from the bones and sinew. Discard the bone and sinew and shred the meat, keeping it warm until required.
Place two pizza stones in the oven and preheat the oven to 250°C (500°F/Gas 9) or its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stones to heat up.
Meanwhile, lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 30 cm (12 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled pizzas to the heated pizza stones. Prick the pizza bases all over with a fork or docker.
Brush the pizza bases with the olive oil. Spread the mozzarella evenly over the pizza bases, then add the fontina. Season with salt and black pepper.
Transfer the pizzas to the heated pizza stones. Cook the pizzas in the oven for 3 minutes. Crack the eggs into separate cups, add to the pizza and cook for a further 5 minutes or until the pizzas are golden and crisp.
Meanwhile, sauté the cavolo nero with the extra olive oil in a large frying pan over medium–high heat and season with sea salt and freshly ground black pepper.
Using a pizza paddle or wide spatula, carefully transfer the pizzas to a chopping board or plate. Add the cavolo nero, ham hock and baby parsley or celery cress, and drizzle with the extra virgin olive oil before serving.
carbonara
makes one 30 x 20 cm (12 x 8 inch) oval pizza
serves 1–2
One of the first pasta dishes I learned to cook was spaghetti carbonara. I was amazed at how simple it was to make, but more importantly how much flavour it packed from using only a few ingredients well. The heroes are, of course, the egg, cheese and pancetta, or bacon. For the pizza, I have just added some buffalo mozzarella to the base so that the egg, pancetta and parmesan can shine. Serve with lots of freshly ground black pepper.
semolina or plain (all-purpose) flour, for dusting
170 g (6 oz) pizza dough ball
1 tablespoon olive oil
50 g (1¾ oz) shredded mozzarella cheese
2 tablespoons onion confit
1 tablespoon chopped flat-leaf (Italian) parsley, plus 1 tablespoon extra, to serve
50 g (1¾ oz) buffalo mozzarella cheese, torn into pieces
8 very thin pancetta slices
freshly ground black pepper
3 free-range eggs
2 tablespoons finely grated parmesan cheese
shaved Parmigiano Reggiano, to serve
Place a pizza stone in the oven and preheat the oven to 250°C (500°F/Gas 9) or to its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Lightly dust a clean work surface with semolina or flour, then roll out the dough ball into a rough 30 x 20 cm (12 x 8 inch) oval that is about 3 mm (1/8 inch) thick. Transfer the pizza base onto a piece of baking paper; this is necessary for transferring the assembled pizza to the heated pizza stone. Prick the pizza base all over with a fork or docker.
Brush the pizza base with the olive oil, then sprinkle over the shredded mozzarella, onion confit, parsley and buffalo mozzarella pieces. Lay the pancetta over the top.
Crack the eggs into separate cups. Transfer the pizza onto the pizza stone, add the eggs to the pizza and top with the grated parmesan.
Cook the pizza in the oven for 5–10 minutes or until golden and crisp.
Using a pizza paddle or wide spatula, carefully transfer the pizza to a wire rack or chopping board. Serve sprinkled with the extra parsley, the Parmigiano Reggiano and plenty of freshly ground black pepper.
scallop, foie gras and caramelised witlof
makes two 15 cm (6 inch) round pizzas
serves 2
This is the pizza you make when you really want to show off. It could be the ultimate first-date meal for a guy to make for a girl he is trying to impress. The key ingredients are sexy and luxurious—scallops and foie gras—but with a simple dressed watercress salad added so it's not too heavy for the girl of your dreams. But the best part of it is that it is a pizza, immediately showing that you have a sense of humour and don't take yourself too seriously. And it is easy to cook, giving you more time to woo and charm your date—what more could you ask for? Let me know how you go fellas!
1 head of witlof (chicory/Belgian endive)
20 g (¾ oz) butter
3 tablespoons light brown sugar
60 ml (2 fl oz/¼ cup) orange juice
sea salt and freshly ground black pepper
14 large scallops, roe removed
semolina or plain (all-purpose) flour, for dusting
2 x 120 g (4¼ oz) pizza dough balls
1 tablespoon olive oil (for brushing pizza)
80 g (2¾ oz) buffalo mozzarella cheese, sliced
2 tablespoons olive oil (for tossing scallops)
60 ml (2 fl oz/¼ cup) foie grass vinaigrette
3 tablespoons salmon roe
watercress, picked and washed, to serve
Chop the witlof into 2.5 cm (1 inch) thick pieces. Heat a large frying pan over medium heat, melt the butter and cook the witlof until it softens. Add the sugar and cook until caramelised, then add the orange juice and cook until the liquid has evaporated. Strain any excess liquid. Season with sea salt and freshly ground black pepper, remove from the heat and leave to cool to room temperature.
Remove the scallops from the fridge and bring to room temperature before cooking. (They shouldn't be cold as they cook very quickly and you want to leave them just a little rare.)
Place a pizza stone in the oven and preheat the oven to 250°C (500°F/Gas 9) or its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Meanwhile, lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 15 cm (6 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled pizzas to the heated pizza stone. Prick the pizza bases all over with a fork or docker.
Brush the pizza bases with half the olive oil, then sprinkle over the buffalo mozzarella and cooked witlof, and season with sea salt and freshly ground black pepper.
Transfer the pizzas onto the heated pizza stone. Cook the pizzas in the oven for 5–10 minutes or until golden and crisp.
Meanwhile, heat a frying pan over high heat until very hot. Toss the scallops in a little oil and season lightly with sea salt. Cook them in the frying pan over high heat for about 30 seconds each side. They should still be a little rare.
Using a pizza paddle or wide spatula, carefully transfer the pizzas to a chopping board or plate. Add the seared scallops, foie gras vinaigrette, salmon roe and watercress to the pizza and serve.
pork belly and porcini calzone
makes four calzoni
serves 4
Of all the calzoni featured in this book this would have to be my favourite by far. If you have never cooked with pork belly then please, please, please give this a try. It is also great eaten on its own, or with some potato purée, or polenta, or tossed through some pasta or risotto, hell, you could pop some in a bread roll and it would still be incredible! Give it a go—I defy you not to love it.
175 g (6 oz) thick Italian pork sausages (100% pork)
125 g (4½ oz) pork belly, skin on, halved
1 fresh bay leaf
125 ml (4 fl oz/½ cup) white vinegar
1 tablespoon extra virgin olive oil, plus extra, for brushing
¼ brown onion, finely diced
¼ carrot, finely diced
¼ celery stalk, finely diced
1 small sprig rosemary, picked plus extra to serve
40 ml (1¼ fl oz) dry white wine
½ garlic clove, finely crushed (optional)
2 teaspoons tomato paste (concentrated purée)
250 ml (9 fl oz/1 cup) Italian tomato passata (puréed tomatoes) or ready-made tomato pasta sauce
70 g (2½ oz) fresh or frozen porcini mushrooms, roughly chopped
½ teaspoon sugar
125 ml (4 fl oz/½ cup) hot water
semolina or plain (all-purpose) flour, for dusting
4 x 90 g (3¼ oz) pizza dough balls
freshly grated parmesan cheese, to taste
Place the sausages and pork belly in a small deep saucepan over high heat. Cover with cold water and add the bay leaf and vinegar. Boil for about 10 minutes.
Remove the meats from the pan and allow to cool slightly. Cut the sausages into rough 1.5 cm (5/8 inch) pieces and the pork into 1 cm (½ inch) pieces.
Heat the olive oil in a saucepan over medium–low heat. Add the onion, carrot and celery and gently sauté for 5 minutes. Add the meats and gently sauté for 10 minutes. Pour in the wine to deglaze the pan, then stir, using a wooden spoon. Once the wine has evaporated add the garlic, if using, and tomato paste and gently cook for a couple more minutes. Add the passata, porcini, sugar and some sea salt and freshly ground black pepper. When the sauce comes to the boil, lower the heat, add the hot water and gently simmer for 1–1 ½ hours, slightly covered. Set aside to cool to room temperature.
Place a pizza stone in the oven and preheat the oven to 250°C (500°F/Gas 9) or its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up. Meanwhile, lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 15 cm (6 inch) round that is about 2 mm (1/16 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled calzoni to the heated pizza stone. Prick the pizza bases all over with a fork or docker.
Spoon a quarter of pork mixture on each pizza base leaving a 2 cm (¾ inch) border. Fold the other side over the filling and pinch the edges together, folding the edge over to prevent the mixture from escaping. Brush with the extra olive oil and sprinkle with the parmesan and salt.
Transfer the calzoni onto the heated pizza stone. Cook for 5–10 minutes or until golden and crisp. Using a pizza paddle or wide spatula, carefully transfer the calzoni to a chopping board or plate. Cut each calzoni in half and serve.
braised lamb shank and white bean calzone
makes five calzoni
serves 5
When creating a calzone, you have to use a filling that is wetter than what you would normally pop on a pizza. That's what this calzone is all about: a lovely rich lamb shank ragu (stew) beefed up with white beans and cooked for hours so the meat is falling off the bone and all that luscious sauce that you have cooked it in thickens up with the shredded meat to become the best pie filling in the world. You can do this type of recipe with any secondary cut of an animal whether it be duck, beef, pork or chicken. The key here is a lovely long slow braise generally on the bone to extract more flavour.
1 tablespoon olive oil, plus extra, for brushing
1 lamb shank, trimmed of excess fat
½ onion, diced
1 garlic clove, crushed
½ celery stalk, finely diced
½ small carrot, finely diced
½ red capsicum (pepper), seeds removed and diced
1 teaspoon thyme leaves, chopped
1 tablespoon tomato paste (concentrated purée)
60 ml (2 fl oz/¼ cup) red wine
400 g (14 oz) tin chopped tomatoes
40 g (1½ oz/¼ cup) green peas
1.5 litres (52 fl oz/6 cups) chicken or vegetable stock
85 g (3 oz) rinsed and drained tinned cannellini beans
sea salt and freshly ground black pepper
semolina or plain (all-purpose flour), for dusting
5 x 85 g (3 oz) pizza dough balls
Place two pizza stones in the oven. Preheat the oven to 160°C (315°F/Gas 2–3). Put a flameproof medium-sized casserole dish (with a fitted lid) over medium heat. Heat half the olive oil, add the shank and brown all over. Remove from the dish and set aside. Wipe the dish clean with kitchen paper. Return the dish to medium–low heat. Add the remaining oil, then the onion and garlic and sweat for 3 minutes. Add the celery and carrot and sweat for 3 minutes, or until softened. Stir in the capsicum and thyme, cook for 1 minute, add the tomato paste, and cook for 1 minute. Return the lamb to the pan with the wine, tomatoes, peas and stock. Bring to the boil.
Remove the dish from the heat, put the lid on, and place in the oven. Cook for 1¾ hours, or until the lamb is falling off the bone. Lift out the lamb, cool slightly, and remove the meat from the bone. Discard the bone. Return the pan to medium–high heat and cook for 15–20 minutes, until the sauce thickens. Stir in the beans, return the lamb to the sauce and season with sea salt and freshly ground black pepper to taste. Set aside to cool.
Increase the oven temperature to 250°C (500°F/Gas 9) or to its highest setting. Once it has reached the temperature, it will take about 15 minutes for the pizza stones to heat up. Lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 15 cm (6 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled calzoni to the pizza stones. Prick the pizza bases all over with a fork or docker.
Spoon ½ cup of the lamb ragu on one half of each pizza base, leaving a 2 cm (¾ inch) border. Fold the other side over the filling and pinch the edges together, pleating to prevent the mixture from escaping. Brush with olive oil and sprinkle with sea salt. Transfer the calzoni onto the heated pizza stones. Cook for 8–10 minutes or until golden and crisp. Using a pizza paddle or wide spatula carefully remove the calzoni from the oven and transfer to a chopping board. Serve.
bresaola with baby beetroot, watercress and horseradish
makes one 30 cm (12 inch) round pizza
serves 1–2
My good friends and right-handers in the kitchen, Monica Cannataci and her twin sister Jacinta, have been working with me for the past eight years. Monica recently won first place in a national pizza competition with this pizza. Bresaola is Italian air-dried beef, served in paper-thin slices. You can buy bresaola from delicatessens, but if you can't get your hands on any, you could also use thinly sliced roast beef or smoked salmon. Thank you girls ... I look forward to many more wonderful times cooking together.
3 baby beetroots (beetroot)
1 tablespoon olive oil
semolina or plain (all-purpose) flour, for dusting
170 g (6 oz) pizza dough ball
1½ tablespoons goat's curd
1½ tablespoons sour cream
1 tablespoon chopped flat-leaf (Italian) parsley
1 tablespoon crushed garlic confit
1 tablespoon onion confit
2 tablespoons tomato chutney, or use good-quality store bought
50 g (1¾ oz) shredded mozzarella cheese
sea salt and freshly ground black pepper
15 very thin slices bresaola
2.5 cm (1 inch) piece fresh horseradish, peeled
small handful of picked watercress sprigs
1 teaspoon lemon-infused oil
1 teaspoon apple balsamic or balsamic reduction
shaved truffled pecorino cheese, to serve
Place a pizza stone in the oven. Preheat the oven to 180°C (350°F/Gas 4). Place the beetroots on a piece of foil, drizzle with the olive oil and fold the foil to seal. Place on a baking tray and roast for 45 minutes or until the beetroots are tender. Cool slightly, then peel and cut into wedges.
Increase the oven temperature to 250°C (500°F/Gas 9) or to its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Lightly dust a clean work surface with semolina or flour, then roll out the dough ball into a 30 cm (12 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza base onto a piece of baking paper; this is necessary for transferring the assembled pizza to the heated pizza stone. Prick the pizza base all over with a fork or docker.
Combine the goat's curd and sour cream together in a bowl, then evenly spread the mixture over the pizza base. Sprinkle the parsley over the base, then add the garlic and onion confits, tomato chutney and mozzarella. Season with sea salt and freshly ground black pepper.
Transfer the pizza onto the heated pizza stone. Cook the pizza in the oven for 5–10 minutes or until golden and crisp.
Using a pizza paddle or wide spatula, carefully transfer the pizza to a chopping board or plate.
Fan the bresaola neatly on the hot pizza, then sprinkle with the beetroots and a small amount of finely grated horseradish. Serve topped with the watercress, lemon-infused oil, apple balsamic and shaved pecorino. Sprinkle with sea salt and serve.
pulled pork with chipotle coleslaw pizza
makes six 15 cm (6 inch) folded pizzas
serves 6
After a morning of skiing in Deer Valley, nothing beats a pulled pork roll or a smoky brisket roll with barbecue sauce when you're starving for lunch. On one such trip, I stumbled upon the best barbecue set-up I have ever seen. This fella had used a big old gas canister as the body of the barbecue, with an axe as his handle and a few other eccentricities to finish it off. And the smell that was coming off there—heaven to a hungry skier. He was cooking pork butt (that's pork shoulder) and some brisket on the side. Naturally I had to have both and bloody hell they were the best things in the world. So here's my take on it, to take me back to that mountain and those heavenly aromas.
semolina or plain (all-purpose) flour, for dusting
6 x 130 g (4½ oz) pizza dough balls
250 ml (9 fl oz/1 cup) Deer Valley Resort Barbecue Sauce, plus 125 ml (4½ fl oz/½ cup) extra
15 g (½ oz/½ cup) roughly chopped flat-leaf (Italian) parsley
480 g (1 lb 1 oz) pulled pork , mixed with 170 ml (5½ fl oz/2/3 cup) reserved liquid
1 serve chipotle coleslaw
3 tablespoons sliced dill pickles
Place two pizza stones in the oven and preheat the oven to 250°C (500°F/Gas 9) or its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stones to heat up.
Meanwhile, lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 15 cm (6 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled pizza to the heated pizza stones. Prick the pizza bases all over with a fork or docker.
Spread the barbecue sauce on the pizza bases and sprinkle with the parsley. Arrange the pork on one half of each base and fold the bases over to form a half moon shape.
Transfer the pizzas onto the heated pizza stones. Cook the pizzas in the oven for 5–10 minutes or until golden and crisp.
Using a pizza paddle or wide spatula, carefully transfer the pizzas to a plate. To finish, drizzle with the extra barbecue sauce, arrange the coleslaw over the pork then garnish with the pickles and serve.
vitello tonnato
makes one 30 x 25 cm (12 x 10 inch) oval pizza
serves 1–2
When creating this book, I took inspiration from the great classic Italian dishes that have stood the test of time, and, more importantly the ones I love to cook and eat and share with my family and friends. Vitello tonnato (veal and tuna) is one such recipe. Thank you Italia!
60 ml (2 fl oz/¼ cup) olive oil, plus extra, to serve
200 g (7 oz) veal fillet
70 g (2½ oz) drained tinned tuna
2 tablespoons aïoli
1 anchovy, finely chopped
¼ teaspoon grated lemon zest
1 teaspoon red wine vinegar
sea salt and freshly ground black pepper
semolina or plain (all-purpose) flour, for dusting
170 g (6 oz) pizza dough ball
1 tablespoon chopped flat-leaf (Italian) parsley
1 tablespoon crushed garlic confit
50 g (1¾ oz) buffalo mozzarella cheese
2 tablespoons baby capers, rinsed and drained
¼ cup baby herbs or picked watercress
2 tablespoons yellow celery leaves
crispy garlic
3 garlic cloves, sliced
200 ml (7 fl oz) vegetable oil
Place a pizza stone in the oven and preheat the oven to 140°C (275°F/Gas 1). Heat a frying pan with 1 tablespoon of olive oil and sear the veal fillet on all sides until golden. Transfer to a baking tray and cook in the oven for 10 minutes or until rare/medium-rare. Remove from the oven and rest for 20 minutes before slicing into 5 mm (¼ inch) pieces. Set aside at room temperature.
To make the crispy garlic, put the garlic and oil in a small saucepan and heat over medium heat until the garlic starts to turn golden. Remove the garlic with a slotted spoon and drain on kitchen paper.
Mix the tuna, aïoli, anchovy, lemon zest and vinegar together and season with sea salt and freshly ground black pepper.
Meanwhile, increase the oven temperature to 250°C (500°F/Gas 9) or its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Lightly dust a clean work surface with semolina or flour, then roll out the dough ball into a 30 cm (12 inch) long oval that is about 3 mm (1/8 inch) thick. Transfer the pizza base onto a piece of baking paper; this is necessary for transferring the assembled pizza to the heated pizza stone. Prick the pizza base all over with a fork or docker.
Spread the remaining olive oil on the pizza base then add the parsley, crushed garlic confit and mozzarella, and season with sea salt and freshly ground black pepper.
Transfer the pizza onto the heated pizza stone. Cook the pizza in the oven for 5–10 minutes or until golden and crisp.
Using a pizza paddle or wide spatula, carefully transfer the pizza to a chopping board or plate.
To finish, top with the veal, capers, crispy garlic, baby herbs, celery leaves and sea salt. Drizzle over the tuna aïoli and extra olive oil and serve.
potato, egg and truffle with lardo
makes one 30 cm (12 inch) round pizza
serves 1–2
Martha Stewart was recently in Australia for a visit and I was lucky enough to have her over to my mum's for a home-cooked dinner. Knowing Martha was a foodie, I couldn't resist serving her my ultimate pizza creation. Did she like it? You bet. Hope you do too.
½ large potato
semolina or plain (all-purpose) flour, for dusting
170 g (6 oz) pizza dough ball
1 tablespoon olive oil
1 tablespoon chopped flat-leaf (Italian) parsley
2 anchovies, chopped
45 g (1¾ oz/1/3 cup) shredded mozzarella cheese
50 g (1¾ oz) buffalo mozzarella cheese, torn into small pieces
8 thin slices lardo
1 free-range egg
1 teaspoon grated parmesan cheese
pinch of thyme leaves
12 thin black truffle shavings
To prepare the potatoes, leave the skin on and slice into pieces about 2 mm (1/16 inch) thick. Plunge the potato slices into salted boiling water for 30 seconds and refresh immediately in cold water. Strain, then lay out on a clean tea towel (dish towel) to dry.
Place a pizza stone in the oven and preheat the oven to 250°C (500°F/Gas 9) or to its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Lightly dust a clean work surface with semolina or flour, then roll out the dough ball into a 30 cm (12 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza base onto a piece of baking paper; this is necessary for transferring the assembled pizza to the heated pizza stone. Prick the pizza base all over with a fork or docker.
Brush the pizza base with the olive oil and sprinkle over the parsley, anchovies and shredded mozzarella. Starting from the outside of the pizza base, arrange the potato slices on the pizza, slightly overlapping the slices. Place the buffalo mozzarella pieces around the pizza, add the lardo and season with sea salt and freshly ground black pepper.
Transfer the pizza onto the heated pizza stone. Crack the egg on the centre of the pizza and sprinkle with the grated parmesan. Cook the pizza in the oven for 5–10 minutes or until golden and crisp.
Using a pizza paddle or wide spatula, carefully transfer the pizza to a chopping board or plate.
Cut the pizza into wedges, sprinkle with the thyme and shaved black truffle and serve.
mum's bolognese calzone
makes five 15 cm (6 inch) calzoni
serves 5
My mum is an adventurous cook, which is where my love of food comes from. She cooked Asian food before it was cool, and is quite handy at grilling a steak, but Mum's real secret weapon in the kitchen while I was growing up was her spaghetti bolognese. The day after having it for dinner, if there were any leftovers, we would put it inside a couple of pieces of bread and stick it in the toasted sandwich maker to make the best toasted sandwich. And now, a calzone.
1 tablespoon olive oil
¼ onion, finely chopped
1 garlic clove, finely chopped
250 g (9 oz) minced (ground) beef
¼ teaspoon dried oregano
60 ml (2 fl oz/¼ cup) red wine (shiraz is good)
1 teaspoon tomato paste (concentrated purée)
1 teaspoon tomato sauce (ketchup)
1 teaspoon sweet chilli sauce or pinch of dried chilli flakes
100 ml (3½ fl oz) tinned tomato soup
60 ml (2 fl oz/¼ cup) chicken stock
sea salt and freshly ground black pepper
1 tablespoon chopped flat-leaf (Italian) parsley
100 g (3½ oz/2/3 cup) macaroni, cooked
semolina or plain (all-purpose) flour, for dusting
5 x 85 g (3 oz) pizza dough balls
80 g (2¾ oz) finely grated parmesan cheese, plus 5 tablespoons extra
1 free-range egg, beaten
Heat the olive oil in a medium heavy-based saucepan over medium–high heat, add the onion and garlic and cook until soft but not coloured. Add the minced beef and cook for 3–4 minutes or until browned. Add the oregano and wine and cook until reduced and the wine almost evaporated. Stir in the tomato paste and sauces and cook for a further minute. Add the tomato soup, chicken stock and a pinch of sea salt and freshly ground black pepper to taste. Simmer over medium–low heat for 15 minutes (adding more stock if needed, but you don't want it too wet as it's going to be the calzone filling). Stir in the parsley and cooked macaroni and set aside to cool to room temperature.
Place two pizza stones in the oven and preheat the oven to 250°C (500°F/Gas 9) or to its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stones to heat up.
Lightly dust a clean work surface with semolina or flour, and then roll out each dough balls into a 15 cm (6 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled calzoni to the heated pizza stone. Prick the pizza bases all over with a fork or docker.
Sprinkle the parmesan evenly over one half of the pizza bases, leaving a 2 cm (¾ inch) border. Spoon ½ cup of the bolognese mixture over the cheese in the calzoni. Fold the other side over the filling and pinch the edges together, pleating to prevent the mixture from escaping. Brush with the beaten egg and sprinkle each with 1 tablespoon of the extra parmesan.
Transfer the calzoni onto the pizza stones. Cook the calzoni in the oven for 8–10 minutes or until golden and crisp.
Using a pizza paddle or wide spatula, carefully transfer the calzoni to a chopping board. Serve.
peking duck
makes eight 5 cm (2 inch) round pizzas
serves 8 as a canape
I've been taking my kids to Sydney's Chinatown since they were two years old. The first dish they loved there was the fried rice. Then it was the chicken and corn soup, then the prawn gow gee (dumplings), and then the whole steamed fish with ginger and soy (they actually only started to eat this when they could pick the live fish out of the tank) and their current favourites are pipis with xo sauce and barbecued pork. But from the tender age of three, both kids have loved Peking duck. They love it rolled up in pancakes, and guess what? They love it on a pizza too.
semolina or plain (all-purpose) flour, for dusting
8 x 30 g (1 oz) pizza dough balls
140 g (5 oz) Peking duck, breast part with skin on (see note)
1 tablespoon canola oil
80 ml (2½ fl oz/1/3 cup) hoisin sauce
2 tablespoons sliced shallots (scallions), white part only
½ small cucumber, peeled, seeds removed and shaved into 5 mm (¼ inch) strips
½ red chilli, seeds removed and julienned
picked small coriander (cilantro) leaves, to serve
½ teaspoon white sesame seeds, toasted
Place a pizza stone in the oven and preheat the oven to 250°C (500°F/Gas 9) or its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Meanwhile, lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 5 cm (2 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled pizzas to the heated pizza stone. Prick the pizza bases all over with a fork or docker. Brush with the canola oil.
Transfer the pizzas onto the heated pizza stone. Cook the pizzas in the oven for 5 minutes or until light golden.
Meanwhile, place the duck on a baking tray, skin side up, and heat the duck in the oven for 2–3 minutes before the pizza is ready to be served—it will only take a few moments with the oven at such high heat.
Using a pizza paddle or wide spatula, carefully transfer the pizzas to a plate. Spread the pizza bases with the hoisin sauce and then top with the shallots and cucumber strips, and season with sea salt and freshly ground black pepper. Slice the duck into pieces and arrange on the pizzas, then sprinkle over the chilli, coriander and toasted sesame seeds. Serve.
Note: Peking duck is available from Chinatown or from the frozen food section in some supermarkets.
chilli crab
makes two 15 cm (6 inch) round pizzas
serves 2
If anyone asks what my signature dish is, I'd have to say it's my chilli crab. You can use mud crab, blue swimmer or spanner crab and Lingham's chilli sauce works a treat. I was introduced to it some time ago in Singapore but have taken the liberty of changing a few of the elements. This is one of the things I love about cooking: there are very few rules that can't be broken. Chilli Crab Pizza—it breaks nearly every rule but it tastes so damn good!
semolina or plain (all-purpose) flour, for dusting
2 x 120 g (4¼ oz) pizza dough balls
4 tablespoons finely sliced shallots (scallions), sliced diagonally
105 g (3¾ oz/¾ cup) grated mozzarella cheese
12 cherry tomatoes, cut in half
sea salt and freshly ground black pepper
260 g (9¼ oz) cooked, picked crabmeat
handful of mixed mint, Vietnamese mint, Thai basil and coriander (cilantro) leaves
1 red chilli, julienned
lime wedges, to serve
chilli sauce
2 teaspoons oil
4 garlic cloves, chopped
1 teaspoon chopped chilli
2 teaspoons julienned fresh ginger
½ teaspoon finely chopped coriander (cilantro) root
60 ml (2 fl oz/¼ cup) tablespoons tomato sauce (ketchup)
3 teaspoons sweet chilli sauce
120 ml (3¾ fl oz) chicken stock
3 teaspoons hoisin sauce
1 teaspoon fish sauce
To make the chilli sauce, heat the oil in a small saucepan over medium heat. Add the garlic, chilli, ginger and coriander root and cook until aromatic. Add the tomato sauce, sweet chilli sauce, stock, hoisin sauce, fish sauce, sugar and some sea salt and stir well. Bring to the boil then reduce the heat and simmer for 10 minutes. Set aside to cool to room temperature.
Place a pizza stone in the oven and preheat the oven to 250°C (500°F/Gas 9) or its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Meanwhile, lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 15 cm (6 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper; this is necessary for transferring the assembled pizzas to the heated pizza stone. Prick the pizza bases all over with a fork or docker.
Spread two tablespoons of chilli sauce on each base, then add the shallots, mozzarella and tomatoes. Season with sea salt and freshly ground black pepper.
Transfer the pizzas onto the heated pizza stone. Cook the pizzas in the oven for 5–10 minutes or until golden and crisp.
Using a pizza paddle or wide spatula, carefully transfer the pizzas to a chopping board or plate. Top with the crabmeat and mixed herbs, add a dash of the chilli and serve with a squeeze of lime.
Note: You can keep any remaining chilli sauce in an airtight jar or container in the fridge for up to 2 months.
chocolate, hazelnut and banana with vanilla gelato
makes one 15 x 10 cm (6 x 4 inch) oval pizza
serves 2–4
This is the first dessert pizza I ever toyed around with. I wanted something on the menu that customers couldn't resist. It had to have chocolate, hazelnuts, bananas, honey and ice cream or gelato. I think this is one of the best pizzas I've ever created, and it's fantastically easy to make yourself. Buy the best-quality ice cream or gelato you can to top this special dessert pizza.
semolina or plain (all-purpose) flour, for dusting
120 g (4¼ oz) pizza dough ball
3 tablespoons chocolate-hazelnut paste
1 banana, sliced and coated in lemon juice
1 tablespoon fresh ricotta cheese, broken into pieces
1 scoop vanilla ice cream or gelato, to serve
2 tablespoons chocolate
2 tablespoons hazelnuts, toasted, peeled and chopped
2 teaspoons honey
icing (confectioners') sugar, for dusting (optional)
Place a pizza stone in the oven and preheat the oven to 250°C (500°F/Gas 9) or to its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Lightly dust a clean work surface with semolina or flour, and then roll out the dough ball into a 15 x 10 cm (6 x 4 inch) oval that is about 3 mm (1/8 inch) thick. Transfer the pizza base onto a piece of baking paper; this is necessary for transferring the assembled pizza to the pizza stone. Prick the pizza base all over with a fork or docker.
Spread the chocolate-hazelnut paste evenly over the pizza base. Place the sliced banana neatly on top, then the ricotta cheese.
Transfer the pizza onto the heated pizza stone. Cook the pizza in the oven for 5–10 minutes or until golden and crisp.
Shave the chocolate, at room temperature, with a knife. Regrigerate to prevent the shavings from melting when serving.
Using a pizza paddle or wide spatula, carefully transfer the pizza to a chopping board or plate. Dollop on a scoop of ice cream and sprinkle over the chocolate shavings and hazelnuts. Drizzle over the honey and dust with the icing sugar, if desired, before serving.
black forest with chocolate fondant, cherries and chantilly cream
makes two 15 cm (6 inch) round pizzas
serves 2
A special occasion pizza, this luscious treat will never fail to impress your friends and family. With the contrast between the tart cherries and rich dark chocolate, this is a satisfying and sumptuous dessert.
125 ml (4 fl oz/½ cup) kirsch or cherry liqueur
180 g (61/3 oz) fresh cherries, halved and pitted
100 ml (3½ fl oz) whipping cream
¼ teaspoon natural vanilla extract or ½ vanilla bean, seeds scraped
1 teaspoon caster (superfine) sugar
25 g (1 oz) dark chocolate
semolina or plain (all-purpose) flour, for dusting
2 x 120 g (4¼ oz) pizza dough balls
chocolate fondant
35 g (1¼ oz) dark chocolate (70% cocoa)
20 g (¾ oz) unsalted butter, chopped
1 free-range egg
2 tablespoons caster (superfine) sugar
1½ tablespoons plain (all-purpose) flour
Place the kirsch in a small saucepan and bring to the boil. Add the cherries, reduce the heat and simmer for 2 minutes. Remove the cherries with a slotted spoon and simmer the syrup for a further 1–2 minutes or until slightly reduced and syrupy. Set aside and allow to cool to room temperature.
To make the chocolate fondant, chop the chocolate into small pieces. Put it in a small heatproof bowl with the butter, and set it over a saucepan of gently simmering water, making sure the base of the bowl doesn't touch the water. When melted, remove from the heat.
Beat the egg and sugar with an electric hand beater until thick and tripled in volume, and then gently fold through the flour. Gradually fold in the melted chocolate and butter. Set aside at room temperature.
Place a pizza stone in the oven and preheat the oven to 250°C (500°F/Gas 9) or its highest temperature. Once it has reached the temperature, it will take about 15 minutes for the pizza stone to heat up.
Meanwhile, beat the cream with the vanilla and sugar until soft peaks forms then set aside. Shave the chocolate, at room temperature, with a knife. Refrigerate to prevent the shavings from melting when serving.
Lightly dust a clean work surface with semolina or flour, then roll out each dough ball into a 15 cm (6 inch) round that is about 3 mm (1/8 inch) thick. Transfer the pizza bases onto pieces of baking paper. Prick the pizza bases all over with a fork or docker. Spoon the chocolate fondant mixture onto both pizza bases.
Transfer pizzas, still on the baking paper, onto the heated pizza stone. Cook for 5–6 minutes or until the chocolate mixture is just cooked but still a little soft in the middle, and the base is light golden in colour. Using a pizza paddle or wide spatula, carefully transfer the pizzas to a chopping board or plate.
Cool for 10 minutes, then sprinkle with some cherries, spoon some cream in the centre and add more cherries. To finish, sprinkle with the chocolate shavings.
IF YOU LOVE MAKING PIZZA, IT'S GREAT TO HAVE A VARIETY OF TOPPINGS IN THE PANTRY OR THE FRIDGE THAT YOU CAN PULL OUT AND USE AT ANY TIME. BEFORE STARTING YOUR DOUGH, CHECK THE RECIPE QUANTITIES—ANY LEFTOVER DOUGH CAN BE KEPT IN THE FREEZER UNTIL REQUIRED.
dough
toppings
classic pizza dough
wholemeal pizza dough
gluten–free pizza dough
fried calzone dough
deep-dish pizza dough
classic pizza dough
makes 750 g (1lb 10 OZ) / roughly enough for four 30 cm (12 inch) pizzas
The following three dough recipes each make one large quantity of dough, which can then be split into separate portions. Weights vary for the different pizzas you'll find in this book. Most recipes call for 170 g (6 oz) balls of dough, but some call for other quantities, so consult the pizza recipe before you split the dough into separate portions. You can substitute wholemeal (whole-wheat) or gluten-free dough for the classic dough in most recipes. (Please see note at the end of the gluten-free dough recipe for any restrictions.)
250 ml (9 fl oz/1 cup) lukewarm water
2 teaspoons dry yeast
1¼ teaspoons sugar
1½ tablespoons olive oil, plus extra, for greasing
475 g (1 lb 1 oz) strong flour (see note), plus extra, for dusting
1¼ teaspoons salt
In a small bowl, mix the lukewarm water, yeast and sugar together until combined, then leave in a warm place for 5 minutes or until frothy. Stir in the olive oil.
Sift the strong flour and salt together into a large bowl. Pour the yeast mixture over the dry ingredients and use your hands to bring the mixture together to form a dough. Turn the dough out onto a work surface and use the heel of your hands to work the dough for 5 minutes until it is smooth and elastic.
Lightly grease the inside of a clean dry bowl with oil and place the dough inside. Place a tea towel (dish towel) over the dough and leave in a warm place to prove for 45–60 minutes or until doubled in size.
Dust a clean work surface lightly with the extra flour and tip out the dough. Use your fists to knock the dough back with one good punch to let any air out.
Before you portion the dough into separate balls, refer to the pizza recipe you want to make for correct measures.
Once you've separated your dough into portions, and working with one portion at a time, use the palm of your hands to cup the dough and roll it on the work surface in a circular motion to form a perfect ball. Repeat with the remaining dough portions.
Place the dough balls on a lightly greased baking tray, cover and leave in a warm place to prove for 15 minutes.
Note: Strong flour is very finely ground flour with a high gluten content. It is available from good delicatessens, gourmet food stores and some supermarkets. It is sometimes called '00' flour.
wholemeal pizza dough
makes 700 g (1 lb 9 oz) / roughly enough for four 30 cm (12 inch) pizzas
250 ml (9 fl oz/1 cup) lukewarm water
2 teaspoons dry yeast
1¼ teaspoons sugar
1½ tablespoons olive oil, plus extra, for greasing
425 g (15 oz) wholemeal (whole-wheat) flour, plus extra, for dusting
1¼ teaspoons table salt
In a small bowl, mix the lukewarm water, yeast and sugar together until combined, then leave in a warm place for 5 minutes or until frothy. Stir in the olive oil.
Sift the flour and salt together into a large bowl. Pour the yeast mixture over the dry ingredients and use your hands to bring the mixture together to form a dough. Turn the dough out onto a work surface and use the heel of your hands to work the dough for 5 minutes until it is smooth and elastic.
Lightly grease the inside of a clean dry bowl with oil and place the dough inside. Place a tea towel (dish towel) over the dough and leave in a warm place to prove for 45–60 minutes or until doubled in size.
Dust a clean work surface lightly with the extra flour and tip out the dough. Use your fists to knock the dough back with one good punch to let any air out.
Before you portion the dough into separate balls, refer to the pizza recipe you want to make for correct measures.
Once you've separated your dough into portions, and working with one portion at a time, use the palm of your hands to cup the dough and roll it on the work surface in a circular motion to form a perfect ball. Repeat with the remaining dough portions.
Place the dough balls on a lightly greased baking tray, cover and leave in a warm place for 15 minutes.
gluten-free pizza dough
makes 785 g (1 lb 11½ oz) / roughly enough for four 30 cm (12 inch) pizzas
450 g (1 lb/3 cups) gluten-free flour, plus extra, for dusting
¼ teaspoon bicarbonate of soda (baking soda)
1 teaspoon salt
2 teaspoons sugar
2 eggs, lightly beaten
80 ml (2½ fl oz/1/3 cup) extra virgin olive oil, plus extra, for greasing
140 ml (4¾ fl oz) lukewarm water
extra gluten-free flour, for dusting
Lightly grease a 30 cm (12 inch) pizza tray. Sift together the gluten-free flour, bicarbonate of soda, salt and sugar into a large bowl.
In a separate bowl, mix the eggs, olive oil and water. Add to the dry ingredients and use a fork to incorporate, then use your hands to bring the mixture together to form a dough.
Before you portion the dough into separate balls, refer to the pizza recipe you want to make for correct measures (see Note, below). Lay down a clean dry tea towel (dish towel) and dust with the extra gluten-free flour.
Working with one portion of dough at a time, use your hands to gently press and flatten the dough as much as you can. Try to keep the dough as round as possible and then using a rolling pin, roll out to fit the prepared pizza tray. You will find that as this dough has no gluten, the dough will not have the elasticity of regular pizza dough and will therefore not be as easy to handle.
Lay your pizza tray upside down on the rolled-out dough and quickly flip it over, using the tea towel (dish towel) to help, so that you end up with a dough-lined pizza tray. It is now ready for the topping of your choice.
Note: As there is less elasticity in this dough compared to a regular dough, when a recipe calls for 170 g (6 oz), increase the dough portions by about 10 per cent to 190 g (6¾ oz). This will help prevent tearing when you need to roll out the dough to fit a standard 30 cm (12 inch) pizza tray. The gluten-free dough cannot be used for any of the calzone recipes, the croque monsieur, reuben sandwich or pulled pork recipes as the dough is too delicate.
fried calzone dough
makes 390 g (13¾ oz) / roughly enough for thirteen small fried calzoni
125 ml (4 fl oz/½ cup) lukewarm water
1 teaspoon dried yeast
1 teaspoon brown sugar
2 teaspoons olive oil
250 g (9 oz/12/3 cup) strong flour
1 teaspoon table salt
In a small bowl, mix the lukewarm water, yeast and sugar together until combined, then leave in a warm place for 5 minutes or until frothy. Stir in the olive oil.
Sift the flour and salt together into a large bowl.
Pour the yeast mixture over the dry ingredients and use your hands to bring the mixture together to form a dough. Turn the dough out onto a work surface and use the heel of your hands to knead the dough for 5 minutes until it is smooth and elastic.
Lightly grease the inside of a clean dry bowl with oil and place the dough inside. Place a clean tea towel (dishtowel) over the dough and leave in a warm place to prove for 45–60 minutes or until doubled in size.
Dust a clean work surface lightly with flour and tip out the dough. Use your fists to knock the dough back with one good punch to let any air out.
Before you portion the dough into separate balls, refer to the fried calzone recipe you want to make for correct measures.
Once you've separated the dough into portions, and working with one portion at a time, use the palm of your hands to cup the dough and roll it on the work surface in a circular motion to form a perfect ball. Repeat with the remaining dough portions.
Place the dough balls on a lightly greased baking tray, cover, and leave in a warm place for 15 minutes.
Variation: To make the same quantity of sweet fried calzone dough, follow the instructions above but substitute 125 ml (4 fl oz/½ cup) lukewarm water with 70 ml (2¼ fl oz) lukewarm water and 60 ml (2 fl oz/¼ cup) Marsala, and omit the salt.
deep-dish pizza dough
makes 500 g (1 lb 2 oz) / roughly enough for one 20 cm deep dish pizza
1 teaspoon active dried yeast
185 ml (6 fl oz/¾ cup) lukewarm water
¾ teaspoon sugar
225 g (8 oz/1½ cups) strong flour
50 g (1¾ oz/¼ cup) polenta
½ teaspoon salt
1½ tablespoons olive oil, plus extra, for greasing
plain (all-purpose) flour, for dusting
Dissolve the yeast in 1½ tablespoons of lukewarm water. Add the sugar and the 1½ tablespoons of the flour and stir until combined. Cover with plastic wrap and let it prove in a warm area for 15 minutes.
Add the remaining water and flour, polenta, salt and olive oil to the mixture, and combine well. Use the heel of your hands to work the dough for 5 minutes or until it is smooth and elastic.
Lightly grease the inside of a bowl with oil and place the dough inside. Place a clean tea towel (dish towel) over the dough and leave in a warm place to prove for 45–60 minutes or until doubled in size. Dust a clean work surface with flour and tip out the dough. Use your fists to knock the dough back with one good punch to let any air out.
Use the palm of your hands to cup the dough and roll it on the work surface in a circular motion to form a perfect ball.
Place the dough ball on a lightly greased baking tray, cover, and leave in a warm place to prove for 15 minutes.
aioli
balsamic reduction
pesto
pizza sauce
salsa verde
balsamic onions
chilli Confit
garlic confit
onion confit
roasted capsicum
tomato chutney
caramelised onions
chermoula
fennel confit
romesco sauce
shallot vinaigrette
olive tapenade
foie gras vinaigrette
deer valley resort barbecue sauce
pulled pork
chipotle coleslaw
aioli
makes about 500 ml (17 fl oz/2 cups)
If you love making pizza, it's great to have a variety of toppings in the pantry or the fridge that you can pull out and use at any time.
2 egg yolks
1 tablespoon lemon juice
1 tablespoon white wine vinegar
3 garlic confit cloves, finely chopped
2 teaspoons dijon mustard
pinch of sea salt
250 ml (9 fl oz/1 cup) olive oil
250 ml (9 fl oz/1 cup) vegetable oil
sea salt and freshly ground black pepper
Using a blender or hand mixer, blend the yolks, lemon juice, white wine vinegar, garlic confit, dijon mustard and sea salt until combined. With the blender still running, slowly pour in the oils until the aïoli is creamy. Season with sea salt and freshly ground black pepper.
Note: Aïoli will keep in an airtight container in the fridge for up to 5 days.
Variation: To make saffron aïoli, add 10 threads of saffron to finished aïoli and allow to stand for 20 minutes.
balsamic reduction
makes 80 ml (2½ fl oz/1/3 cup)
250 ml (9 fl oz/1 cup) balsamic vinegar
Pour the balsamic vinegar into a small saucepan and place over high heat. Whisk briskly while the vinegar comes to the boil so that it does not stick on the bottom of the pan. Continue to boil, whisking occasionally, until reduced to 80 ml (2½ fl oz/1/3 cup). The vinegar will naturally sweeten as it reduces and will become syrupy. Allow to cool.
Note: Store in an airtight container in the fridge for up to 3 weeks.
pesto
makes 250 g (9 oz/1 cup)
50 g (1¾ oz/1 cup) basil leaves
2 garlic cloves
80 g (2¾ oz/½ cup) pine nuts
170 ml (5½ fl oz/2/3 cup) olive oil
35 g (1¼ oz/¼ cup) finely grated parmesan cheese
sea salt and freshly ground black pepper
lemon juice to taste (optional)
Roughly chop the basil and garlic and place in the bowl of a food processor with the pine nuts. Process until finely chopped. With the motor running, slowly add the olive oil and continue to process for a few seconds until it forms a smooth sauce.
Turn off the processor and stir in the cheese. Season with sea salt and freshly ground black pepper and add lemon juice to taste, if desired.
Note: To store the pesto, pour into an airtight container, cover the pesto with olive oil, then seal with a lid. It will keep in the fridge for 2–3 days.
pizza sauce
makes 420 ml (14½ fl oz/12/3 cups)
400 g (14 oz) tin whole peeled tomatoes
¼ teaspoon salt
1 teaspoon dried oregano
2 pinches of freshly ground black pepper
Place all the ingredients in a food processor and blend until smooth.
Note: This sauce can be stored in an airtight container in the fridge for up to a week or in the freezer for up to 3 months.
salsa verde
makes 330 ml (11¼ fl oz/11/3 cups)
½ slice of stale bread
125 ml (4 fl oz/½ cup) olive oil
1 cup basil leaves, firmly packed
40 g (1½ oz/2 cups) flat-leaf (Italian) parsley leaves
10 g (¼ oz/½ cup) mint leaves
2 anchovies
1½ tablespoons capers, rinsed and drained
2 teaspoons finely chopped cornichons (small pickles) (optional)
2 teaspoons lemon juice
2½ tablespoons pine nuts, toasted
sea salt and freshly ground black pepper
Soak the bread in half the olive oil in a small bowl for about 5 minutes. Place the soaked bread in the bowl of a food processor with all the remaining ingredients and season with sea salt and freshly ground black pepper (being mindful that the capers and cornichons are quite salty). Process until finely chopped.
Note: To store, place in a sterilised screw-top jar and cover with a thin layer of oil, then seal with the lid. Salsa verde can be stored in the fridge for up to 5 days.
balsamic onions
makes about 280 g (10 oz/1 cup)
80 ml (2½ fl oz/1/3 cup) olive oil
2 large onions, thinly sliced
75 g (2¾ oz/1/3 cup) caster (superfine) sugar
170 ml (5½ fl oz/2/3 cup balsamic vinegar
Heat the olive oil in a medium heavy-bottomed frying pan over medium–high heat. Add the onions and stir for 5 minutes, or until just starting to caramelise. Add the sugar and stir until dissolved. Add the vinegar and cook over low heat for 25–30 minutes, stirring occasionally, or until the mixture is a jam-like consistency.
Note: Balsamic onions can be stored in an airtight container in the fridge for up to a week.
chilli confit
makes about 125 g (4 ½ oz/½ cup)
150 g (5½ oz) fresh long red chillies, halved, seeded and thinly sliced
125 ml (4 fl oz/½ cup) olive oil
Place the chillies and olive oil in a small saucepan over the lowest heat possible on your stovetop (use a simmer pad if necessary) and cook for 1 hour or until the chilli is soft. You do not want the oil to boil. Remove from the heat and allow to cool.
Note: Chilli confit (with the oil) will keep in a sealed sterilised jar in the fridge for up to 3 months.
garlic confit
makes about 340 g (12 oz/1½ cups), including oil
1 cup garlic cloves, peeled
250 ml (9 fl oz/1 cup) olive oil
Place the garlic cloves and olive oil in a saucepan over the lowest setting possible on your stovetop (use a simmer pad if necessary) and cook for 1 hour or until the garlic is soft. You do not want the oil boiling at any time, you want it just past warm as this ensures the garlic becomes beautiful and soft—plus, you shouldn't get bad garlic breath if you cook it this way. Remove from the heat and allow to cool.
Note: The garlic confit (with the oil) will keep in a sealed sterilised jar in the fridge for up to 3 months. If you are short of time you could roast garlic instead: Preheat the oven to 180°C ( 350°F/Gas 4). Place whole garlic on a large piece of foil, drizzle with a little olive oil, and seal. Roast on a baking tray for 30–40 minutes or until tender. Cool slightly then squeeze the garlic from the skin.
onion confit
makes about 325 g (11½ oz/1½ cups), including oil
155 g (5½ oz/1 cup) chopped onion
250 ml (9 fl oz/1 cup) olive oil
Place the onion and olive oil in a saucepan over the lowest setting possible on your stovetop (use a simmer pad if necessary) and cook for 1 hour or until the onion is soft. You do not want the oil to boil. Remove from the heat and allow to cool.
Note: The onion confit (with the oil) will keep in a sealed sterilised jar in the fridge for up to 3 months.
roasted capsicum
makes 2 roasted capsicums
1 red capsicum (pepper)
1 yellow capsicum (pepper)
Preheat the oven to 230°C–240°C (450°F–475°F/Gas 8). Place the capsicum on a large baking tray. Roast for 30–40 minutes or until the skins blister and turn black. Remove from the oven and place in a bowl. Cover with plastic wrap or place the roasted capsicums in a plastic bag until cool. Peel away the skins. Discard the core, seeds and membrane. Use the roasted flesh as desired.
Note: Roasted capsicum will keep in an airtight container in the fridge for up to a week.
tomato chutney
makes 580 g (1 lb 4½ oz/2¼ cups)
1 tablespoon extra virgin olive oil
1 tablespoon yellow mustard seeds
1 onion, chopped
3 garlic cloves, finely chopped
1 fresh long red chilli, finely chopped
1 tablespoon ground ginger
1 tablespoon ground turmeric
6 ripe tomatoes, diced
2½ tablespoons red wine vinegar
3 tablespoons sugar
sea salt and freshly ground black pepper
Heat the olive oil in a small saucepan over medium–low heat, add the mustard seeds and onion, and cook until lightly brown. Add the garlic, chilli, ginger, and turmeric and cook for 1 minute or until fragrant. Add the diced tomatoes and cook, stirring occasionally, for 20 minutes. Add the vinegar and sugar. Cook, stirring occasionally, for about 15 minutes or until the liquid has reduced by half. Season with sea salt and freshly ground black pepper, then set aside to cool.
Note: Tomato chutney can be stored in an airtight container in the fridge for 1–2 weeks.
caramelised onions
makes 250 g (9 oz)
2 tablespoons olive oil
2 large brown onions, very thinly sliced
Heat a heavy-based frying pan over low heat. Add the olive oil and sliced onions.
Stir occasionally and slowly caramelise for about 15 minutes. Do not be tempted to turn the heat up! They will become a golden caramel brown.
Spoon the caramelised onions out onto absorbent paper. Leave to cool and store in an airtight container in the fridge for a week.
chermoula
makes about 325 g (11½ oz)
2 teaspoons cumin seeds
2 teaspoons coriander seeds
¼ bunch flat-leaf (Italian) parsley, leaves picked and washed
½ bunch mint, leaves picked
½ bunch coriander (cilantro), leaves picked and washed
1 garlic clove, crushed
2 teaspoons ground turmeric
1–2 bird's eye chillies, seeds removed and roughly chopped
100 ml (3½ fl oz) olive oil
zest and juice of ½ lemon
sea salt and freshly ground black pepper
Firstly prepare the spices. Heat a frying pan over medium heat. Toast the cumin and coriander seeds for a couple of minutes, stirring occasionally. Remove from the heat and grind in a mortar and pestle.
Put all the ingredients, apart from the olive oil and the lemon juice, in a blender. Add 125 ml (4 fl oz/½ cup) of the oil and blend. When all is combined, slowly add the rest of the oil. Season to taste and add some of the lemon juice if needed.
Place in an airtight container and add a thin layer of oil before sealing with the lid. Store in the fridge for up to 2 weeks.
fennel confit
makes 2 cups (375 g)
2 fennel bulbs
½ teaspoon roasted fennel seeds
250 ml (9 fl oz/1 cup) olive oil
Trim the fennel bulbs and slice. Combine the fennel, fennel seeds and olive oil in a small saucepan and place on the stove over the lowest heat possible. Cook for 45 minutes.
Note: Store leftovers in an airtight container in the fridge for up to 7 days.
romesco sauce
makes 2 cups (550 g)
2 red capsicums (peppers), halved, seeds and membrane removed
2 ripe tomatoes, halved
60 ml (2 fl oz/¼ cup) olive oil
sea salt
2 slices of crusty white bread, cubed
12 blanched almonds, lightly roasted
12 hazelnuts, roasted
1 teaspoon sweet paprika
½ bird's eye chilli, seeds removed
3 garlic cloves, finely chopped
2 tablespoons white wine vinegar
100 ml (3½ fl oz) extra virgin olive oil
Preheat the oven to 200°C (400°F/Gas 6). Place the capsicum and tomatoes on a baking tray, drizzle with 1 tablespoon of olive oil and sprinkle with sea salt. Cook for about 15–20 minutes, until the capsicums are coloured and the tomatoes soft. Cool a little, then peel the capsicums.
To make the croutons, toss the bread cubes in the remaining olive oil. Place on a baking tray and cook in the oven for 4 minutes or until light golden. Put the capsicums, tomatoes, almonds and hazelnuts in a blender and mix until smooth. Add the croutons and season with the paprika, chilli, garlic and vinegar and blend again and slowly add the extra virgin olive oil until evenly blended.
Note: Store in an airtight container in the fridge for up to 48 hours. This sauce is fantastic with seafood or chicken or as a dip.
shallot vinaigrette
makes 185 ml (6 fl oz/¾ cup)
1 spring onion (scallion), finely diced
60 ml (2 fl oz/¼ cup) red wine vinegar
125 ml (4 fl oz/½ cup) extra virgin olive oil
sea salt and freshly ground black pepper
Combine all the ingredients and let them steep for at least 2 hours. Whisk in a stainless steel bowl until emulsified.
Note: Best served on the day it's made.
olive tapenade
Makes about 175 g (6 oz/1 cup)
175 g (6 oz/1 cup) pitted kalamata or Sicilian green olives
2 anchovies
1 tablespoon capers, rinsed and drained
1 garlic clove, finely chopped
60 ml (2 fl oz/¼ cup) extra virgin olive oil, plus extra to store
zest of 1 lemon (optional)
1 tablespoon chopped flat-leaf (Italian) parsley
Place the olives, anchovy, parsley, capers and garlic in the bowl of a food processor and process until finely chopped. (Alternatively, place the ingredients in a mortar and pound with a pestle until well combined.)
With the motor running, gradually add the olive oil in a thin, steady stream until well combined and a smooth paste forms.
Transfer to an airtight container and pour the over extra oil to cover the tapenade surface (to prevent oxidisation).
foie gras vinaigrette
makes about 400 ml (14 fl oz)
2 teaspoons dijon mustard
50 ml (1¾ fl oz) red wine vinegar
100 ml (3½ fl oz) light olive oil
100 g (3½ oz) foie gras, chopped
50 ml (1¾ fl oz) hot chicken stock
sea salt and freshly ground black pepper
Put the mustard, vinegar and olive oil into a jug and mix with a stick blender. Gradually add the foie gras. Pour the chicken stock slowly in and blend until smooth. Season with sea salt and freshly ground black pepper to taste.
Keeps for up to 5 days in the fridge.
deer valley resort barbecue sauce
makes 500 ml (17 fl oz/2 cups)
1½ tablespoons finely diced brown onion
40 g (1½ oz/¼ cup) finely diced red capsicum (pepper)
1½ tablespoons finely diced green capsicum (pepper)
2 teaspoons chopped garlic clove
1 tablespoon vegetable oil
500 ml (17 fl oz/2 cups) tomato sauce (ketchup)
2 tablespoons molasses
3 teaspoons dijon mustard
2 teaspoons red wine vinegar
2 teaspoons honey
sea salt and freshly ground black pepper, to taste
pinch of cayenne pepper
zest and juice of ¼ lemon, to taste
Sauté the onions, capsicum and garlic in the oil in a small deep saucepan over low heat until the onions are translucent.
Add the remaining ingredients and simmer, stirring occasionally, for 30 minutes.
Note: The barbecue sauce can be stored in an airtight container in the fridge for up to 2 weeks or frozen for 3 months.
pulled pork
makes 550–750 g (1 lb 4 oz–1 lb 10 oz/3¼–41/3 cups) cooked pork
1 kg (2 lb 4 oz) boneless pork shoulder, known in the United States as pork butt
2 tablespoons olive oil
sea salt and freshly ground black pepper
45 g (1¾ oz/¼ cup) lightly packed brown sugar
60 g (2¼ oz/¼ cup) table salt
50 g (1¾ oz/¼ cup) garlic powder
2 tablespoons onion powder
½ teaspoon mustard powder
½ teaspoon ground white pepper
black pepper, to taste
125 ml (4 fl oz/½ cup) barbecue sauce
185 ml (6 fl oz/¾ cup) Worcestershire sauce
1.5 litres (52 fl oz/6 cups) water
Preheat the oven to 150°C (300°F/Gas 2). Pat the meat dry with a paper towel and cut in half. Rub roast with 1 tablespoon olive oil. Sprinkle with sea salt and freshly ground black pepper.
Heat remaining tablespoon oil in a casserole dish or large deep frying pan over medium heat. Cook the pork for 2 minutes on each side or until lightly browned. Place pork in a lightly greased deep roasting tray, fat side up, or into a greased casserole dish. Combine the remaining ingredients and pour over the pork.
Cover and cook in the oven for 1 hour. Reduce the heat to 100°C (200°F/Gas ½) and cook for a further 5–6 hours or until the meat is very tender and slices easily.
Remove the pork from the oven, reserving the liquid. Discard the fat and shred the meat. Add 125 ml (4 fl oz/½ cup) of the reserved liquid to pork to moisten. Reserve the remaining liquid.
Note: The pulled pork can be frozen for up to 3 months.
chipotle coleslaw
makes 2 cups
250 g (9 oz) red or green cabbage (or a mix of both)
85 g (3 oz/1/3 cup) mayonnaise
1 tablespoon sour cream
½ teaspoon chipotle chilli in adobo sauce, or to taste
sea salt and freshly ground black pepper, to taste
¾ teaspoon ground cumin, toasted
¾ teaspoon rice wine vinegar
2 teaspoons orange juice
Shred the cabbage. In a separate bowl, mix the remaining ingredients. Add the desired amount to the cabbage. Mix well.
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"redpajama_set_name": "RedPajamaBook"
} | 3,902 |
\section{Introduction}
A decision maker can formulate an action plan to achieve a specific goal.
The action plan is executed in sequential stages.
Planning under uncertainty considers already at the beginning random events which might happen while the plan is executed, taking the possibility of corrective (recourse) decisions into account. This ensures that the desired goal can be met, even in an uncertain environment where random events impact and influence the situation at each stage.
An important historical contribution in the development of decision making under uncertainty is \citet{Markowitz1952}, who considers an optimal asset allocation problem and developed what is nowadays called \emph{modern portfolio theory}. This example is often used to illustrate and outline the difficulty: an investment decision has to be made today, but the random return is observed after the investment horizon, in the future. In its simplest version, the problem has only one decision stage, but if a recourse decision is allowed after the random outcome is revealed, the problem is said to be \emph{two-stage}. However, if financial securities such as options with different maturities are included in the portfolio then more decision stages must be considered: intermediate decisions have to be made whenever options expire before reaching the investment horizon; the problem is then said to be \emph{multistage}.
Further problems dealing with decision making under uncertainty appear in logistics and inventory control. Goods must be shipped between various warehouses in view of random demands and unmatched demand triggers recourse actions by rapid new orders (cf.\ \citet{Dantzig1955} for a very early problem description).
This problem can be formulated as a decision problem in multiple stages. Optimal inventory and transportation policies constitute hedging strategies against unknown and varying demands.
As another problem with economic relevance we mention an optimal schedule to manage reservoirs of a hydropower plant. Its inflows are rain and perhaps melting snow, which are both random events. The price of electricity produced is highly volatile in addition, so that the optimal operation mode is a sequence of decisions, made at consecutive times and influenced by various random events (cf.\ \citet{Seguin2015} for a more concrete discussion).
\medskip
All problems mentioned above have uncertain scenarios and decisions at intermediate stages in common. The scenarios appear as the trajectories of a stochastic process in discrete time, $\xi=(\xi_1, \dots, \xi_T)$, with values $\xi_t\in\mathbb{R}^{m_t}$, say. For algorithmic treatment, such general processes must be approximated by simpler ones, in the same manner as functions are represented as vectors on finite grids on digital computers. In particular, a discrete time stochastic process is approximated by a \emph{scenario tree} or, in case of a Markovian process, by a \emph{scenario lattice}. Notice, however, that quite often the decisions at earlier stages influence the feasibility of profitability of later stages and in such a case the full history of the decision process has to be recorded. The history process is always a tree.
\medskip
An elementary component for generating scenario trees is the approximation of a random outcome, i.e., the approximation of random variables. An example is the random realization~$\xi_1$ at the first stage, which has to be approximated sufficiently well for an algorithmic treatment.
\citet{GrafLuschgy} provide a comprehensive study and theoretical background on this topic (called also {\it uncertainty vector quantification}), \citet{PagesFunctionalQuantization} and \citet{PagesBallyPrintems} present methods and algorithms, many of them are based on optimal clustering, cf.\ \citet{Hartigan1975}. It is an important observation that the approximation of a random outcome as $\xi_1$ suffers from the curse of dimensionality, a term coined by R.\ Bellman (cf.\ \citet{Bellman1961,Dudley1969}).
The generation of scenario trees is even more involved. Indeed, not only one random realization as $\xi_1$ has to be approximated, but the entire underlying stochastic process $(X_1,\dots,X_T)$, often with complicated interdependencies. Consequently, the approximation of the stochastic process is significantly more complex and computationally more expensive.
\medskip \citet{Hoyland2001} describe an early method to generate scenario trees, they employ selected \emph{statistical parameters} such as moments and correlations to characterize transitions and interdependences within a tree. Their method
requires solving a system of nonlinear equations to find representative discretizations of the scenario tree. However, the corresponding transition probabilities ignore all further indispensable characteristics of the governing distribution and for that the method is not appropriate in general. \citet{Klaassen2002} discusses the moment matching method in relation to arbitrage opportunities. In a broader context, the method is also outlined in \citet{CONSIGLI2000} including an algorithm employing sequential updates and a concrete managerial example. \citet{Kuhn2015} address an asset allocation problem in an ambiguous context.
Some decision problems exhibit specific problem characteristics. These properties are occasionally addressed directly by adjusting and designing the scenarios adequately, as \citet{HenrionRomisch2017} propose. The recent work \citet{Wallace2015}, e.g., particularly puts the tails of the distributions in focus, while \citet{KautWallace2011} involve copulas to properly capture the multivariate shape of the distributions.
\cite{Beyer2014} develop an evolution strategy to generate scenario trees. They employ swarm intelligence techniques to tackle related multistage optimization problems for optimal portfolio decisions.
\citet{Keutchayan2017} present a method to generate large scale scenario trees. They observe that the scenario tree has to be reduced for computational efficiency. The paper presents a heuristic which reduces the loss of information which comes along with reducing the scenario tree. \citet{HeitschRomisch2009} propose advanced heuristics as well to reduce large scale scenario to a computationally tractable size.
The quality of a scenario trees can be assessed if there is a distance available, which measures the distance to the genuine stochastic process to be approximated. \cite{Pflug2009} presents the first concept to measure the distance of different stochastic processes. \citet{KovacevicPichler} employ the method directly to generate scenario trees.
\medskip
This paper endorses nonparametric techniques for generating scenario trees. In contrast to most methods addressed above, they involve the full law (distribution) of the stochastic process rather than individual, selected statistical parameters. In this way the marginals, moments, covariance and other characteristics are correctly mirrored and asymptotic consistency can be shown. The methods do not require specific parametric assumptions on the underlying process and covers multivariate observations as well. To construct an approximating tree it is further enough to have samples from the process available, these samples are just trajectories or sample paths. We build the trees by employing stochastic approximation (see \citet{Kushner2008} for a survey). It is not necessary to specify and solve a system of corresponding equations.
\paragraph{Implementation.}
Our Julia implementation is available freely on GitHub in the package \texttt{ScenTrees.jl},\footnote{\texttt{ScenTrees.jl}: \href{https://github.com/kirui93/ScenTrees.jl}{https://github.com/kirui93/ScenTrees.jl}} which is released under the open-source MIT license. Description of various functions of the package and also different examples to illustrate the different methods can be found in the package's documentation.\footnote{\texttt{Documentation}: \href{https://kirui93.github.io/ScenTrees.jl/stable/}{https://kirui93.github.io/ScenTrees.jl/stable/}, cf.\ \citet{KiruiPichler}}
\paragraph{Structure of the paper.}
In~\cref{sec:Notation}, we introduce some prerequisites that are necessary for the discussion in this paper. This setting includes a discussion on the important concept of the quality of approximation in \cref{sec:Nested}.
\cref{sec:Trajectories} outlines the methods of scenario tree generation by nested clustering.
Successive improvements are obtained by adding further information or trajectories to a tree and \cref{sec:StochApproximation} presents the stochastic approximation procedure.
\cref{sec:Trees} addresses the situation of limited data available, as is the case in many situations of practical interest.
\cref{sec:Lattice} is dedicated specifically to stochastic processes with special properties as Markovian processes or processes with fixed states.
A brief introduction on trees without predefined structure is addressed in \cref{sec:Treeswithout}. Lastly, \cref{sec:application} presents a special application of scenario lattice generation methods on a limited electricity load data. We draw our conclusions on \cref{sec:Summary}.
\section{Mathematical setting}\label{sec:Notation}
In what follows we consider a general stochastic process $X$ in discrete time over $T$ stages, $X = (X_1,\ldots,X_T)$, where $X_1$ is a nonrandom starting value.
A scenario tree is a discrete time and discrete state process approximating the process~$X$. Throughout this paper, we employ the following terminology and conventions for scenario trees, scenario lattices and observed data.
\subsection{Scenario tree}
A finite scenario tree approximating~$X$ is denoted by $\tilde X=\left(\tilde X_1, \tilde X_2,\dots,\tilde X_T\right)$ and modelled as a scenario tree process.
\paragraph{Characterization of a scenario tree process.}
A directed, rooted and layered graph describes the topology of the scenario tree.
We enumerate all $n$ nodes of a scenario tree by $1,\dots,n$, the root node of the scenario tree is $1$ (the nodes $\mathcal N:=\{1,\dots,n\}$ are the vertices of the graph, cf.\ \cref{fig:Tree}).
A scenario tree process is then fully characterized by the following three lists:
\begin{enumerate}[label=(\roman*)]
\item The list of predecessors,
\[\operatorname{pred}(1),\dots,\operatorname{pred}(n),\]
determines the \emph{topology} of the tree. $\operatorname{pred}(i)$ is the predecessor of node number~$i$. For the root node we set $\operatorname{pred}(1):=0$. (For the exemplary tree in \cref{fig:Tree}, for instance, $\operatorname{pred}(2)=1$ and $\operatorname{pred}(8)=3$.) The pairs $(\operatorname{pred}(i),i)$, $i=2,\dots,n$, comprise all arcs of the graph.
\item The list of \emph{probabilities} is
\[\tilde p_1,\dots, \tilde p_n,\]
where $\tilde p_i$ denotes the conditional probability to reach the node $i$ from its predecessor along the edge of the tree. We set $\tilde p_1=1$ for the root node.
\item The list of \emph{states} (i.e., the values of the tree process at the nodes) is \[\tilde x_1,\dots,\tilde x_n.\]
The values $\tilde x_i$ constitute the outcomes of the process $\tilde X$ at stage $t$, they may be vectors of any dimension. We denote the dimension of the process at stage $t$ by $m_t$, i.e., $\tilde x_i\in\mathbb R^{m_t}=:\Xi_t$, whenever the node $i$ is at stage~$t$.\footnote{To allow a compact presentation below we shall assume, without loss of generality, that $\Xi_t=:\Xi$ for all $t\le T$.}
\end{enumerate}
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=0.7]
\node (1) at (0,0) [circle,shade,draw] {1};
\node (2) at (2,2)[circle,shade,draw] {2};
\node (3) at (2,0) [circle,shade,draw] {3};
\node (4) at (2,-1.5)[circle,shade,draw] {4};
\node (5) at (4,3.1) [circle,shade,draw] {5};
\node (6) at (4,2.1) [circle,shade,draw] {6};
\node (7) at (4,1.1) [circle,shade,draw] {7};
\node (8) at (4,0) [circle,shade,draw] {8};
\node (9) at (4,-1)[circle,shade,draw] {9};
\node (10) at (4,-2)[circle,shade,draw] {10};
\node (13) at (8,-2.5)[circle,shade,draw] {$\mathbf n$};
\node (12) at (8,-1.5)[circle,shade,draw] {...};
\node (14) at (8,3.5)[circle,shade,draw] {...};
\node (15) at (8,2.5)[circle,shade,draw] {...};
\node (16) at (8,0.5)[circle,shade,draw] {...};
\node (17) at (8,-.5)[circle,shade,draw] {...};
\draw[->, very thick] (1) to (2);
\draw[->, very thick] (2) to (5);
\draw[->, very thick] (2) to (6);
\draw[->, very thick] (2) to (7);
\draw[->, very thick] (1) to (3);
\draw[->, very thick] (3) to (8);
\draw[->, very thick] (1) to (4);
\draw[->, very thick] (4) to (9);
\draw[->, very thick] (4) to (10);
\draw[->, dotted, very thick] (5,-2) to (5.7,-2);
\draw[->, dotted, very thick] (6,-2) to (13);
\draw[->, dotted, very thick] (6,-2) to (12);
\draw[->, dotted, very thick] (5,0) to (5.7,0);
\draw[->, dotted, very thick] (6,0) to (16);
\draw[->, dotted, very thick] (6,0) to (17);
\draw[->, dotted, very thick] (5,3) to (5.7,3);
\draw[->, dotted, very thick] (6,3) to (14);
\draw[->, dotted, very thick] (6,3) to (15);
\node (t1) at (0,-3.5) {$t=1$};
\node (t2) at (2,-3.5) {$t=2$};
\node (t3) at (4,-3.5) {$t=3$};
\node (t4) at (8,-3.5) {$t=T$};
\end{tikzpicture}
\caption{Scenario tree process in $T$ stages\label{fig:Tree}}
\end{figure}
A sample scenario is a path $(j_1,\dots,j_T)$ with
$j_{t-1}=\operatorname{pred}(j_t)$.
The sample scenario connects the root $1$ and a leaf $j_T$. A scenario tree has as many scenarios as there are leaves. The probability of a scenario is $P\big((j_1,\dots,j_T)\big)=\tilde{p}_{j_1}\cdot\ldots\cdot \tilde{p}_{j_T}$.
The predecessor of the node $j\in\mathcal N$ at stage $t$ is $j_t$, that is, there is a sequence $(j_1,\dots,j_t,\dots, j_\tau)$ for some $\tau\ge t$ with $j_\tau=j$ and $\operatorname{pred}(j_{t^\prime})=j_{t^\prime-1}$ for all $t^\prime\le\tau$. (For the exemplary tree in \cref{fig:Tree}, for example, the predecessor of the node $n$ at stage $2$ is $4$, i.e., $n_2=4$.)
Filtrations model the flow of information associated with the tree process. The filtration is the sequence of $\sigma$-algebras $\mathcal F_t$ modeling the information available at stage $t$. The $\sigma$-algebra $\mathcal F_t$ is generated by
\[\mathcal F_t=\sigma\big( \left\{s=(j_1,\dots,j_T)\colon s\text{ is a scenario with } j_t=j\right\}, j\in\mathcal N\big).\]
We mention that the $\sigma$-algebras $\mathcal F_t$, $t=1,\dots,T$, uniquely characterize the topology, i.e., the branching structure of the process. Notice that $\tilde{X}$ is a tree process, if the $\sigma$-algebra generated by $\tilde{X}_t$ is the same as the one generated by $(\tilde{X}_1, \dots, \tilde{X}_t)$ for all $t$.
\medskip
\cref{fig:1} presents three scenario tree processes with different topology and filtrations. Nonetheless, the trajectories of their states are identical, their scenarios even have the same probabilities.
\begin{figure}[H]
\centering
\unitlength0.5cm
\hfill
\begin{subfigure}[t]{0.33\textwidth}
\begin{tikzpicture}[scale=0.5]
\node (1) at (0,0) [circle] {0};
\node (2) at (3.0,0) [circle] {10};
\node (3) at (7,1.1)[circle] {22};
\node (4) at (7,-0.7) [circle] {21};
\node (5) at (7,3.3) [circle] {28};
\node (6) at (7,-2.7) [circle] {20};
\foreach \x in {2} {
\draw[->,thin] (1) to (\x);
\foreach \y in {3,4,5,6} \draw[->, thin] (\x) to (\y);}
\draw (1.5,0.3) node{$1.0$};
\draw (5.0,2.6) node{$0.25$};
\draw (5.0,0.9) node{$0.25$};
\draw (5.0,0.0) node{$0.25$};
\draw (5.0,-2.0) node{$0.25$};
\end{tikzpicture}
\subcaption{Branching structure 1--4}\label{fig:t1}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.33\textwidth}
\begin{tikzpicture}[scale=0.5]
\node (1) at (0,0) [circle] {0};
\node (2) at (4,1.6) [circle] {10};
\node (3) at (4,-1.0)[circle] {10};
\node (4) at (8,3.8) [circle] {28};
\node (5) at (8,-0.2) [circle] {21};
\node (6) at (8,1.8) [circle] {22};
\node (7) at (8,-2.0) [circle] {20};
\foreach \x in {2,3}{\draw[->,thin] (1) to (\x);
\foreach \y in {5,7} \draw[->, thin] (3) to (\y);
\foreach \y in {4,6} \draw[->, thin] (2) to (\y);}
\draw (2,1.2) node{$0.5$};
\draw (2,-1.0) node{$0.5$};
\draw (6,3.0) node{$0.5$};
\draw (6,2.0) node{$0.5$};
\draw (6,-0.3) node{$0.5$};
\draw (6,-2.0) node{$0.5$};
\end{tikzpicture}
\subcaption{Branching 2--2}\label{fig:t2}
\end{subfigure}
\begin{subfigure}[t]{0.33\textwidth}
\begin{tikzpicture}[scale=0.5]
\node (1) at (0,0) [circle] {0};
\node (2) at (4,1.0) [circle] {10};
\node (3) at (4,-1.0)[circle] {10};
\node (8) at (4,3.0)[circle] {10};
\node (9) at (4,-3.0)[circle] {10};
\node (4) at (8,1.0) [circle] {22};
\node (5) at (8,-1.0) [circle] {21};
\node (6) at (8,3.0) [circle] {28};
\node (7) at (8,-3.0) [circle] {20};
\foreach \x in {2,3,8,9}{\draw[->,thin] (1) to (\x);
\foreach \y in {5} \draw[->, thin] (3) to (\y);
\foreach \y in {7} \draw[->, thin] (9) to (\y);
\foreach \y in {4} \draw[->, thin] (2) to (\y);
\foreach \y in {6} \draw[->, thin] (8) to (\y);}
\draw (2.3,1.0) node{$0.25$};
\draw (2.3,2.4) node{$0.25$};
\draw (2.3,-0.2) node{$0.25$};
\draw (2.3,-2.4) node{$0.25$};
\draw (6,3.5) node{$1.0$};
\draw (6,1.5) node{$1.0$};
\draw (6,-0.5) node{$1.0$};
\draw (6,-2.5) node{$1.0$};
\end{tikzpicture}
\subcaption{Branching 4--1, scenario fan}\label{fig:t3}
\end{subfigure}\hfill
\caption{Three scenario trees with identical paths but still different topologies}
\label{fig:1}
\end{figure}
\subsection{Scenario lattices}
Scenario lattices are a natural discretization of Markovian processes. While nodes of a scenario tree have one predecessor, nodes of a scenario lattice can have multiple predecessors as shown in \cref{fig:Lattice}. Many stochastic processes which occur in real world applications represent a Markovian process, i.e., the further evolution of the process depends on the state at the actual stage, but not on the history (the process is colloquially said to be \emph{memoryless}). This additional knowledge can and should be explored in implementations, the consequences are manifold.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale= 0.7]
\node (1) at (0,0) [circle,shade,draw] {1};
\node (2) at (2,2)[circle,shade,draw] {2};
\node (3) at (2,0.2) [circle,shade,draw] {3};
\node (4) at (2,-1.5)[circle,shade,draw] {4};
\node (5) at (4,3) [circle,shade,draw] {5};
\node (6) at (4,1.5) [circle,shade,draw] {6};
\node (7) at (4,0.2) [circle,shade,draw] {7};
\node (8) at (4,-1.0) [circle,shade,draw] {8};
\node (9) at (4,-2.5)[circle,shade,draw] {9};
\node (13) at (8,-2.5)[circle,shade,draw] {$n$};
\node (12) at (8,-0.5)[circle,shade,draw] {...};
\node (14) at (8,3)[circle,shade,draw] {...};
\node (15) at (8,1.5)[circle,shade,draw] {...};
\node (A) at (6,2) {...};
\node (B) at (6,-1) {...};
\foreach \x in {2, 3, 4} {
\draw[->,thin] (1) to (\x);
\foreach \y in {5,6,7,8,9} \draw[->, thin] (\x) to (\y);}
\foreach \x in {12,...,15} {
\draw[->,thin] (A) to (\x); \draw[->, thin] (B) to (\x);}
\node (t1) at (0,-3.5) {$t=1$};
\node (t2) at (2,-3.5) {$t=2$};
\node (t3) at (4,-3.5) {$t=3$};
\node (t4) at (8,-3.5) {$t=T$};
\end{tikzpicture}
\caption{A model of a lattice at $T$ stages and $n$ nodes\label{fig:Lattice}}
\end{figure}
In the Markovian case it is often \emph{not} necessary to build a complete tree for optimization purposes. Instead, a \emph{lattice} (i.e., a recombining tree) may be sufficient to model the whole process, cf.\ \cref{fig:Lattice}. The lattice is the structure of a finitely valued Markovian process, which is a rooted, layered, directed graph of height $T$ (all paths form the root to the leaves have same length $T$), such that edges can only exist between nodes of subsequent layers.
\begin{remark*}
The lattice model is not sufficient, e.g., if the decisions are path-dependent, which may happen even if the scenario model is Markovian. To give an example, suppose that the decisions contain to exercise American or Asian options or to deal with knock-out certificates. In these cases, the whole history of the price process and/\,or the decision process is relevant at each state of the process and a scenario tree has to be considered anyway.
\end{remark*}
\subsection{The decision problem}
Once a scenario tree or lattice is available which models the underlying process properly, then the tree can be used for decision making to meet a predefined managerial goal. The goal to be met, e.g., is to maximize the discounted, expected profit from today's perspective, at the beginning of the planning horizon.
To achieve this goal, decisions have to be made at each node within the tree. This tree has the same structure as the scenario tree, it is called the \emph{decision tree}. As \citet{RockafellarWets1991} elaborate, the decisions made on the decision tree reflect a hedging strategy against the unknowns. The purpose of the hedging strategy is to meet the initial, predefined goal.
\subsection{Quality of approximation}\label{sec:Nested}
The genuine process $X$ is often not eligible for decision making, this is the case if the process is available only from samples or because its structure is too complicated. An adequate approximation~$\tilde X$ of~$X$ should lead to reliable results, if decisions are based on $\tilde X$ instead of the initial process $X$.
The following subsections address the quality of approximation in two ways. First, we observe that the approximations $\tilde X$ obtained in the initial sections result by mapping the genuine process~$X$ to its tree approximation $\tilde X$, while the following section addresses the nested distance. Both perceptions make a distance available, which allows assessing the approximation quality in a quantitative way. More general results ensure that the policies, which are derived from the approximating process $\tilde X$, apply for the initial process $X$ as well and the model error can be stated explicitly in terms of the distances.
\subsubsection{Transportation maps}
Important procedures outlined below have in common that they fit individual trajectories into an existing tree structure.
This operation is a mapping, where trajectories are mapped on a tree by respecting the evolution of information over time (processes respecting the time evolution are said to be \emph{adapted}, or \emph{nonanticipative}).
To this end consider a map $\mathbf T$, mapping the state $\xi=(\xi_1,\dots,\xi_T)$ of the stochastic process $X=(X_1,\dots X_T)$ to the new scenario $\left(\tilde x_1,\dots\tilde x_T\right):=\mathbf T(\xi)$.
By the Doob--Dynkin lemma (cf.\ \citet{Kallenberg2002Foundations}), the \emph{transport map} $\mathbf T$ is adapted (i.e., it preserves the information) if its components are given by
\begin{equation*}\label{eq:T}
\mathbf T\big(\xi_1,\dots,\xi_T\big)=
\begin{pmatrix}\begin{array}{l}
\tilde x_1=\mathbf T_1(\xi_1)\\
\tilde x_2=\mathbf T_2(\xi_1,\xi_2)\\
\quad\vdots\\
\tilde x_T=\mathbf T_T(\xi_1,\dots,\xi_T)
\end{array}\end{pmatrix}.
\end{equation*}
The trajectory~$\xi$ and the path $\tilde {\mathbf x}=\mathbf T(\xi)$ within the tree have the distance
\begin{equation}\label{eq:dist}
d(\xi, \tilde{\mathbf x}):=\sum_{t=1}^T d_t\big(\xi_t,\tilde{\mathbf x}_t\big),
\end{equation}
where $d_t$ is the distance function on $\Xi_t$ for the stage $t$. This allows defining and assessing the quality of the approximation by
\begin{equation}\label{eq:distT}
\E_P\,d\big(\xi, \mathbf T(\xi)\big)= \int_{\Xi_1\times\dots\times\Xi_T} d\big(\xi,\,\mathbf T(\xi)\big)\,P(\mathrm d \xi),
\end{equation}
where the expectation (integration) is with respect to the law of the stochastic process~$X$.
The quantity~\eqref{eq:distT} is the \emph{average aberration} of the initial stochastic process $X$ in comparison with the tree approximation specified by the mapping $\mathbf T$. Eq.~\eqref{eq:distT} is a reliable quantity describing the quality of the approximation provided by $\mathbf T$. The objective of usual stochastic optimization problems can be shown to depend continuously on the quantity~\eqref{eq:distT}.
\medskip
To raise a conceptual issue associated with transport maps, consider an \emph{invertible} transport map $\mathbf T$ so that
\[ \E_P d\big(\xi,\mathbf T(\xi)\big)= \E_{P^{\mathbf T}} d\Big(\mathbf T^{-1}(\eta), \eta \Big)=\E_{P^{\mathbf T}} d\Big(\eta, \mathbf T^{-1}(\eta) \Big)\]
by the change of variable formula, where $P^{\mathbf T}:=P\circ \mathbf T^{-1}$ is the image measure.
A comparison with~\eqref{eq:distT} demonstrates that the inverse $\mathbf T^{-1}$ has to be adapted too to allow switching from the initial space to the image space.
This is, however, not the case in real-world application where $\mathbf T$ maps a stochastic process with continuous states to a tree process, which has only finitely many states.
The nested distance, presented in what follows, resolves this problem conceptually.
\subsubsection{Relation to the nested distance}
The \emph{nested distance} (or \emph{process distance}) is employed to measure the distance of two distinct stochastic processes directly. It is a symmetric version of~\eqref{eq:distT} and based on transportation distances derived from transportation theory, cf.\ \citet{Villani2003}.
The expression~\eqref{eq:distT} gives rise to rewrite the expectation as
\begin{equation}\label{eq:17}
\E_P d\big(\xi,\mathbf T(\xi)\big)=\E_\pi d(\xi,\eta)=\iint d(\xi,\eta)\,\pi(\mathrm d \xi,\mathrm d\eta),
\end{equation} employing the transport measure $\pi(A\times B):=P\big(A\cap\mathbf T^{-1}(B)\big)$.
As a generalization we consider general transportation measures $\pi$ in~\eqref{eq:17}, which are not necessarily of the form $P(A\cap\mathbf T^{-1}(B))$.
A general transport measure~$\pi$ is often termed \emph{transport plan} to distinguish it from the transportation map~$\mathbf T$.
This distance of probability measures obtained in this way is nowadays known as \emph{Kantorovich} or \emph{Wasserstein distance}, as \emph{earth mover}, or more generally as \emph{transportation distance}. We remark here that the notion of transportation distance extends obviously to non-discrete probability measures $P$ and $Q$ on a metric space $(\Xi,d)$.
The nested distance (see Appendix~\ref{sec:NestedA}) generalizes the transportation distance by additionally taking the increasing information at successive stages into account. In this way the nested distance generalizes transportation distances to a distance of stochastic processes, or to a distance of trees.
\section{Generation of scenario trees}\label{sec:Trajectories}
This section introduces various techniques of generating scenario trees. We also give examples of two stochastic processes that can be approximated via a scenario tree.
\subsection{Data and observations}
The basic data for tree generation is typically given by a set of independent trajectories or sample paths, these trajectories constitute ideally observed data. Time series of stocks or energy prices, demands or reservoir inflow data, to provide examples, can be collected throughout comparable time intervals (weeks, say) and these data represent a reliable collection of trajectories. Alternatively, an econometric model could also be estimated and representative sample paths can be generated by simulation to serve as basic data.
Observations of the process $X$ are trajectories, denoted by
\begin{equation}\label{eq:2}\xi_j, \qquad j=1,\dots,N,\end{equation} with $N\in\mathbb N$ (or $N=\infty$, if the samples are generated by an econometric model). Each observation is a complete outcome of the process over time, i.e., $\xi_j=(\xi_{j,1},\dots,\xi_{j,T})$.
\subsection{Description of the methods}
The sample paths are employed to construct the scenario tree. In order to learn as much as possible from the data, the number of trajectories observed ($N$, cf.~\eqref{eq:2}) should be huge and representative in terms of spanning the underlying stochastic phenomena.
A new independent trajectory may be used at every new step of our algorithm, if enough data are available or may be generated by simulation.
The corresponding algorithms are presented below first. In some real life situations, however, the sample size may not be large and that is why we discuss later also methods involving kernel estimates, which work with limited data and small sample sizes.
The algorithms we propose start with a tree, which is not more than a qualified guess (based on an expert opinion, e.g.). Drawing one sample after another we improve the initial scenario tree successively using a technique which derives from stochastic approximation.
Each iteration modifies the values of the tree without changing its structure. In this way the approximating quality of the tree improves gradually. The tree fluctuates at the beginning, but with more and more iterations the tree converges in probability. The resulting tree finally is optimal with the properties desired and is ready to be used for the decision making procedure.
We also discuss a construction algorithm which starts with a small tree and gradually adds new branches when necessary. Using this technique one does not even need to have an initial guess of the tree structure.
In this paper we consider several practical situations which require algorithms for scenario tree generation:
\begin{itemize}[noitemsep]
\item scenario tree generation from given stochastic models;
\item scenario tree generation from stochastic simulations;
\item scenario tree generation from observed trajectories.
\end{itemize}
\subsection{Nested clustering}
The \emph{nested clustering} algorithm, which we outline in what follows, generates a tree from a \emph{finite} sample of $N$ trajectories, $\xi_1,\dots,\xi_N$. A finite sample of trajectories is occasionally called a scenario \emph{fan}, reminding to the fact that the slats of a fan do not bundle in their nodes (see, e.g., \cref{fig:t3}). The nested clustering algorithm determines the nodes of the approximation tree process~$\tilde X$ sequentially by iterating over stages, starting at stage $t=1$ up to the terminal stage $t=T$.
The initial state $\tilde X_1$ at stage $t=1$ of the process is assumed to be known (i.e., deterministic), $\tilde X_1= \tilde x_1$. The observations at the following stages $t=2,\dots, T$ are random.
The subsequent stage $t=2$ can be handled well. Note that the random component~$X_2$ at stage $t=2$ follows some distribution, which is known by assumption or from which data are available. By employing clustering techniques it is possible to approximate the distribution~$X_2$ by representative points, or cluster means. The $k$-means clustering algorithm, e.g., finds cluster means $\tilde x_i$ ($i\in\mathcal S_1$) on the stage $t=2$ which minimize the average minimal distance,\footnote{One can avoid double counting in~\eqref{eq:kMeans} of $\xi_{j,2}$, equidistant to various means $\tilde x_i$, by counting the distance only once, for example for the smallest node index $i$. We shall refer to this rule as the \emph{tie-break rule}.\label{fn:tie-break}}
\begin{equation}\label{eq:kMeans}
\frac 1 N\sum_{j=1}^N \min_{i\in \mathcal S_1} d_2\left(\xi_{j,2},\, \tilde x_i\right),
\end{equation} where $d_2$ is the distance of outcomes at stage $t=2$ and \[\mathcal S_i:= \left\{\ell\in\mathcal N\colon\operatorname{pred}(\ell)=i\right\}\]
collects all successor-nodes of the node~$i$ (in \cref{fig:Tree}, $\mathcal S_1=\{2,3,4\}$ and $\mathcal S_2=\{5,6,7\}$, e.g.).
In this way,~$X_2$ is approximated by a discrete distribution $\tilde X_2$ located at $\tilde x_i$ ($i\in \mathcal S_1$), where each $\tilde x_i$ is reached with probability $P(\tilde X_2=\tilde x_i)=\tilde p_i$.
The probabilities $\tilde p_i$ are found by allocating each sample $\xi_{j,2}$, $j=1,\dots N$, to the closest of all $\tilde x_i$ at the first stage, $i\in \mathcal S_1$. The probability $\tilde p_i$ thus represents the relative count of how often $\tilde x_i$ is the mean closest to $\xi_{j,2}$ (see the tie-breaking rule in \cref{fn:tie-break}).
Note that the origin of the approximating tree is already found: the first two components of the approximating tree are $(\tilde X_1,\tilde X_2)$ with evaluations $(\tilde x_1, \tilde x_i)$ ($i\in \mathcal S_1$) at the first two stages.
The following stage displays a difficulty, which is not present at the previous stages $t=1$ and $t=2$. Indeed, to continue the tree to stage $t=3$ it is necessary to find locations $\tilde x_i$ at the second stage and corresponding transition probabilities $\tilde p_i$. These locations and probabilities represent a \emph{conditional} distribution, conditional on the process having previously passed a particular tree state at the first stage ($\tilde x_2$, say; cf.\ \cref{fig:ScenarioTree}).
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale= 1]
\node (1) at (0,0.2) {$\tilde{\boldsymbol x}_1$};
\node (2) at (2,2) {$\tilde{\boldsymbol x}_2$};
\node (3) at (2,-0.0) {$\tilde{\boldsymbol x}_3$};
\node (4) at (2,-1.5){$\tilde{\boldsymbol x}_4$};
\node (5) at (4,3) {$\tilde{\boldsymbol x}_5$};
\node (6) at (4,1.9) {$\tilde{\boldsymbol x}_6$};
\node (7) at (4,.4) {$\tilde{\boldsymbol x}_7$};
\node (8) at (4,-0.5) {};
\node (9) at (4,0) {};
\node (10) at (4,-1.5) {};
\draw[->, ultra thick] (1) to (2);
\draw[->, ultra thick] (2) to (5);
\draw[->, ultra thick] (2) to (6);
\draw[->, ultra thick] (2) to (7);
\draw[->, dashed, very thin] (1) -- (2,-.3) --(4,1)--(4.3,0.9);
\draw[->, dashed, very thin] (1) -- (2, .3)-- (4,0.1)--(4.3,0.3);
\draw[->, dashed, very thin] (1) to (4) -- (4,-1.6)--(4.3,-1.5);
\draw[->, dashed, very thin] (1) -- (2,-1.)-- (4,-.7)--(4.3,-1.0);
\draw (2,0.3) --(2,1.7);
\draw (2,2.3) --(2,3.5);
\draw (2,-.2) --(2,-1.1);
\draw (2,-1.8) --(2,-2.1);
\draw[ultra thick] (1.9,-.7) --(2.1,-.7);
\draw[ultra thick] (1.9,.8) --(2.1,.8);
\draw (4,.2) --(4,-2.1);
\draw (4,.8) --(4,1.6);
\draw (4,2.2) --(4,2.8);
\draw (4,3.3) --(4,3.6);
\draw[ultra thick] (3.9,1.2) --(4.1,1.2);
\draw[ultra thick] (3.9,2.5) --(4.1,2.5);
\node (A) at (4.7,3.1) [gray] {$\xi_1$};
\draw[gray, ->, thin](0.2,0.2) to (2,2.2) to (4,3.1) to (A);
\node (A) at (4.7,3.7) [gray] {$\xi_2$};
\draw[gray, ->, thin](0.2,0.2) to (2,1.4) to (4,3.4) to (A);
\node (A) at (4.7,2.0) [gray] {$\xi_3$};
\draw[gray, ->, thin](0.2,0.2) to (2,1.2) to (4,2.1) to (A);
\node (A) at (4.7,1.6) [gray] {$\xi_{12}$};
\draw[gray, ->, thin](0.2,0.2) to (2,1.9) to (4,1.4) to (A);
\node (A) at (4.7,2.5) [gray] {$\xi_5$};
\draw[gray, ->, thin](0.2,0.2) to (2,2.6) to (4,2.3) to (A);
\node (A) at (4.7,0.2) [gray] {$\xi_{15}$};
\draw[gray, ->, thin](0.2,0.2) to (2,1.6) to (4,-0.3) to (A);
\node (A) at (4.7,-.8) [gray] {$\xi_7$};
\draw[gray, ->, thin](0.2,0.2) to (2,1.3) to (4,-0.0) to (A);
\end{tikzpicture}
\caption{Observations associated with the cluster point $\tilde x_2$ (solid lines) and others (dashed line). Indicated as well is the Dirichlet tessellation (Voronoi regions).\label{fig:ScenarioTree}}
\end{figure}
The problem encountered here is that typically \emph{none} of the observed trajectories $\xi_j$ coincides with $\tilde x_2$ exactly at the second stage, so \emph{no} data is eligible to describe the conditional distribution at the third stage. This essential difficulty is present at all subsequent stages as well (and we elaborate on this intrinsic difficulty further in \cref{sec:Trees} below).
The problem can be resolved by considering all those samples, which were associated with $\tilde x_2$ at the previous stage (cf.\ \cref{fig:ScenarioTree}). Only this subset of samples is considered further and used to build new locations and probabilities at the next stage by using the same clustering techniques as outlined for the second stage.
As a result, the approximating process has the outcomes $(\tilde x_{i_1}, \tilde x_{i_2},\tilde x_{i_3})$, where $i_2$ is a successor of $i_1$ (i.e., $i_2\in \mathcal S_{i_1}$).
This procedure can be repeated for all other locations $\tilde x_i$, $i\in \mathcal S_{i_2}$ for ${i_2}\in\mathcal S_{i_1}$ at the second stage, resulting in a tree of height $2$ and outcomes $(\tilde x_{i_1}, \tilde x_{i_2}, \tilde x_{i_3})$, where $i_1=1$ and $i_2\in \mathcal S_{i_1}$ in a cascading, nested way.
\cref{fig:ScenarioTree} displays these locations within the approximating tree including those observed scenario paths, which are associated with $\tilde x_2$. The figure displays the corresponding cluster regions as well. The clustering is called Dirichlet tessellation, or Voronoi tessellation.
\medskip
The locations at the third and later stages are found by repeatedly applying the procedure outlined above, except that the sample paths considered further for the next $k$-means clustering are only those with common history. The paths in the approximating tree are $(\tilde x_{i_1},\dots, \tilde x_{i_T})$, where $i_{\ell+1}\in \mathcal S_{i_\ell}$ and the probabilities $\tilde p_{i_{\ell+1}}=\mathds P(\tilde X=\tilde x_{i_{\ell+1}}|\, \tilde X=\tilde x_{i_\ell})$ are conditional transition probabilities of reaching $\tilde x_{i_{\ell+1}}$ from $\tilde x_{i_\ell}$. The $k$-means problem to be solved for the node $i_t$ at stage $t$ is
\begin{equation}\label{eq:kMeans2}
\frac 1{|\mathcal N(\tilde x_{i_1},\dots,\tilde x_{i_t})|}\sum_{j \in \mathcal N(\tilde x_{i_1},\dots,\tilde x_{i_t})} \min_{\ell\in \mathcal S_{i_t}} d_{t+1}\left(\xi_{j,{t+1}},\, \tilde x_\ell\right),\qquad (\tilde x_\ell,\,\ell\in \mathcal S_{i_t}),
\end{equation}
where $d_{t+1}$ is the distance function at next stage $t+1$ and
\begin{equation*}
\mathcal N(\tilde x_{i_1},\dots,\tilde x_{i_t}):=
\left\{i=(i_1,\dots,i_t)\colon\
\begin{aligned}
&d_\tau(\xi_{j,\tau}, \tilde x_{i_\tau}) \le d_\tau(\xi_{j,\tau}, \tilde x_k)\\
&\text{ for all } k\text{ with } \operatorname{pred}(k)=\operatorname{pred}(i_\tau),\ \tau\le t
\end{aligned}
\right\}
\end{equation*}
collects the paths associated with nearest means $(\tilde x_{i_1},\dots,\tilde x_{i_t})$ up to stage $t$.
The total number of paths with common history is $|\mathcal N(\tilde x_{i_1},\dots,\tilde x_{i_t})|$ (see again the tie-break rule, Footnote~\ref{fn:tie-break}).
\medskip
The \emph{nested clustering} algorithm described above derives its name from this stage-wise, iterative procedure to identify an approximating process $\tilde X$ with a given branching structure.
We emphasize that clustering can be applied to data in higher dimension equally well (i.e., multivariate processes), the initial process $X$ and the approximating process $\tilde X$ do not have to be univariate (i.e., with outcomes on the real line).
Notice further that the clustering problems~\eqref{eq:kMeans} and~\eqref{eq:kMeans2} do not have to be solved to full precision in practice. Typically, a sufficient approximation is already obtained after a few subsequent iterations of the standard $k$-means clustering algorithm.
It is evident that the number $N$ of samples available should be significantly larger than $n$, the number of nodes in the approximating tree $\tilde X$. For asymptotic convergence we want to point out that the number of scenarios needed is huge, it has to hold that $N\gg n^{m_1+m_2+\dots +m_T}$, where $m_t$ is the dimension of the problem at stage $t$. This limits the use of general approximating trees in practice, especially with a higher number of stages. We refer to \citet{ShapiroNemirovski} for a more detailed analysis on the quality of approximations.
\begin{algorithm}[H]
\KwIn{Finite samples of $N$ trajectories, $\xi_1,\ldots,\xi_N$}
\KwOut{Scenario tree process $\tilde X$ with a fixed branching structure}
Stage $t=1$ is deterministic, set $\tilde X_1 = \tilde x_1$\\
\For{$t=1,\ldots,T-1$}{
Consider all paths $(\tilde x_{i_1},\ldots,\tilde x_{i_{t}})$ in the approximating tree
and find new cluster means $\tilde x_\ell$
at stage $t-1$ by solving the $k$-means problems
\[
\frac1{\left| \mathcal N(\tilde x_{i_1},\dots,\tilde x_{i_{t}})\right|}\sum_{j\in\mathcal N(\tilde x_{i_1},\dots,\tilde x_{i_{t}})} \min_{\ell \in \mathcal S_{i_{t}}} d_{t+1}\left(\xi_{j,t+1},\, \tilde x_\ell\right),\] where $d_{t+1}$ is the distance function of stage $t+1$.\\
Define the conditional probability at stage $t$ as
\[\tilde p_{i_{t+1}}= \mathds P(\tilde X_{t+1}=\tilde x_{i_{t+1}}\mid \tilde x_{i_1},\dots,\tilde x_{i_{t}})=
\frac{\left|\mathcal N(\tilde x_{i_1},\dots,\tilde x_{i_{t+1}})\right|}
{\left| \mathcal N(\tilde x_{i_1},\dots,\tilde x_{i_{t}})\right|}.\]
}
\KwResult{States $\tilde x_i$ of the approximating tree $(\tilde X_1,\tilde X_2,\ldots,\tilde X_T)$ for all nodes $i\in\mathcal N$.}
\caption{Nested clustering algorithm to determine the nodes of an approximating tree process $\tilde X$ sequentially over $T$ stages using, e.g., the $k$-means clustering algorithm.}
\label{alg0}
\end{algorithm}
\medskip
\subsection{Stochastic approximation}\label{sec:StochApproximation}
The preceding \cref{sec:Trajectories} and the nested clustering algorithm addressed therein is based on a finite number of $N$ samples $\xi_j$, $j=1,\dots N$. In practice, a (parametric) time series model is occasionally available which allows generating arbitrarily many new sample trajectories. Every additional sample represents new information, which can and should be used to improve the approximation quality of the scenario tree. This section discusses improvements, which can be obtained by adding new samples without starting the nested clustering algorithm from scratch. The method presented here is based on \emph{stochastic approximation} (cf.\ \citet{Pflug2001} for an early application of stochastic approximation to generate scenario trees).
Suppose that an initial, approximating tree $\tilde X$ is already given. The initial tree may represent a qualified expert opinion, or may result from nested clustering. The idea outlined in this section is to modify the present tree for every new sample path available. We use stochastic approximation to modify the actual tree given an additional, new sample.
Stochastic approximation requires us to choose a sequence of numbers $\alpha_1, \alpha_2,\dots$ beforehand, which satisfies $\alpha_k>0$, $\sum_{k=1}^\infty \alpha_k= \infty$ and $\sum_{k=1}^\infty \alpha_k^2< \infty$. A proposal, which has been proven useful in practice, is the sequence $\alpha_k=\frac 1{30+k}$, but different sequences may be more appropriate for a particular problem at hand.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale= 1.0]
\node (1) at (0,0.2) {$\mathbf{\tilde X_1}$};
\node (2) at (2,2) {$\mathbf{\tilde X_2}$};
\node (5) at (4,3) {$\tilde X_3$};
\node (6) at (4,1.9) {$\mathbf{\tilde X_4}$};
\node (7) at (4,.4) {$\tilde X_5$};
\node (xi) at (5.2,1.0) {$\xi_{K}$};
\draw[->, very thick] (1) to (2);
\draw[->, very thin] (2) to (5);
\draw[->, very thick] (2) to (6);
\draw[->, very thin] (2) to (7);
\draw (2,0.3) --(2,1.7);
\draw (2,2.3) --(2,3.5);
\draw[ultra thick] (1.9,.8) --(2.1,.8);
\draw[->,gray, ultra thick, dashed](1) --(2,3.2)--(4,1.3)--(xi);
\draw[->,ultra thick, red](2.1,2.2) --(2.1,2.6);
\draw[->,ultra thick, red](4.1,1.7) --(4.1,1.4);
\draw (4,.2) --(4,-0.1);
\draw (4,.8) --(4,1.6);
\draw (4,2.2) --(4,2.8);
\draw (4,3.3) --(4,3.6);
\draw[ultra thick] (3.9,1.2) --(4.1,1.2);
\draw[ultra thick] (3.9,2.5) --(4.1,2.5);
\end{tikzpicture}
\caption{Every stochastic approximation step modifies one path within the tree (thick) slightly towards the new observation $\xi_{K}$ (dashed), cf.~\eqref{eq:convex}.\label{fig:StochasticApproximation}}
\end{figure}
Suppose an additional trajectory is available, which we denote by $\xi_{k}=(\xi_{k,1},\dots,\xi_{k,T})$ for convenience. We modify the given tree $\tilde X^{(k-1)}$ based on the new information $\xi_{k}$ to get a new stochastic tree $\tilde{X}^{(k)}$. This is achieved by identifying the branch in the tree $\tilde X^{(k)}(i_1)$ which is closest to $\xi_{k,1}$ at the 1\textsuperscript{st} stage, then the node $i_2$ in the subtree which is closest to $\xi_{k,2}$ at the 2\textsuperscript{nd} stage, etc. Suppose the path $\left(\tilde X^{(k-1)}(i_1),\dots, \tilde X^{(k-1)}(i_T)\right)$ is sequentially closest to the observation $\xi_k$. The stochastic approximation step then updates the tree $\tilde X^{(k-1)}$ by shifting this path slightly. Each individual component of the trajectory is updated to
\begin{equation}\label{eq:convex}
\tilde X^{(k)}(i) \leftarrow (1-\alpha_{k})\cdot\tilde X^{(k-1)}(i)+ \alpha_{k}\cdot\xi_{k,i}
\end{equation}
for every $i\in \{i_1,\dots, i_T\}$.
Note that~\eqref{eq:convex} is a convex combination, shifting the path $\tilde X^{(k)}(i_t)$ towards the new observation $\xi_k$ with a force specified by $\alpha_{k}$.
\cref{fig:StochasticApproximation} illustrates a modifying step of the stochastic approximation procedure, the path is shifted along the horizontal arrows in direction of the new sample $\xi_{k}$.
The stochastic approximation procedure repeats the modifying step with new scenarios successively, until satisfactory convergence is obtained. We refer to \citet{Kushner2003} for details on stopping criteria.
\medskip
\citet{PflugPiSzenarioGen} present an extensive explanation for an algorithm for generating scenario trees. They also employ the concept of stochastic approximation procedure discussed in \cref{sec:StochApproximation}. The following example demonstrates generation of scenario trees using the concept presented therein.
\begin{example}[Running maximum]
Consider the running maximum process in \cref{fig:rmp} in 4 stages. This process is given by
\begin{equation}\label{eq:rmp}
M_t = \max \Big\{ \sum_{i=1}^{t^\prime} \xi_i : t^\prime \leq t\Big\},\ t=1,\dots,4\ \text{ with} \ \xi_i \sim \mathcal N(0,1) \ \text{i.i.d.}
\end{equation}
The process $M_t$ is non-Markovian since it depends on the history. Therefore, we can use this process to illustrate generation of scenario trees with a fixed branching structure and a certain number of iterations for the stochastic approximation algorithm.
\begin{figure}[H]
\centering
\includegraphics[height=6cm,width=10cm]{rmdata1.pdf}
\caption{Scenarios of a running maximum process in four stages}
\label{fig:rmp}
\end{figure}
\cref{fig:rmp2} below shows two scenario trees with branching structure\footnote{We refer to the number of nodes at each stage, $(|\mathcal N_1|,|\mathcal N_2|,\dots, |\mathcal N_T|)$, as the \emph{branching structure} of the process.} $(1,2,2,2)$ and $(1,3,3,3)$ approximating this process.
They are generated with \texttt{tree\_approximation()} function contained in the package
\href{https://kirui93.github.io/ScenTrees.jl/stable/}{\texttt{ScenTrees.jl}}.\textsuperscript{\ref{fn:ScenTrees}}
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width=\textwidth, trim=0 0 0 20, clip]{rmtree2.pdf}
\caption{Binary tree approximating process in \cref{fig:rmp} with a transportation distance of $0.313$}
\label{fig:rmtree2}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width=\textwidth, trim=00 00 00 20, clip]{rmtree3.pdf}
\caption{Tree with 3 branches at each node approximating process in \cref{fig:rmp} with a transportation distance of $0.107$}
\label{fig:rmtree3}
\end{subfigure}
\caption{Example of two scenario trees with different branching structures approximating process in \cref{fig:rmp}.}
\label{fig:rmp2}
\end{figure}
\cref{fig:rmtree3} has a better approximating quality than \cref{fig:rmtree2}. This is because \cref{fig:rmtree3} has more branches than \cref{fig:rmtree2} and therefore represents much more information. This can also be seen through the difference in the multistage distance. The multistage distance between the origin maximum running process $M_t$ (with realization in \cref{fig:rmp}) and \cref{fig:rmtree3} is $0.107$ while that of \cref{fig:rmtree2} is $0.313$.
\end{example}
\section{Scenario tree generation with limited data}\label{sec:Trees}
In real world applications it is not unusual that data available is limited, i.e., that the number of available samples $N$ is fixed and not large; \citet{Woodruff2015} report further on this difficulty. To generate a stochastic tree based on limited data it is necessary to modify the approach in comparison to the \cref{sec:Trajectories} above. In these previous sections we combine and compress scenarios with similar outcomes at each step to locate new, continuative nodes of the tree. These methods need many, or unlimited data to specify a good approximating tree, and thus cannot be applied in case of limited data.
In case of limited data it is necessary to learn as much as possible from the observations available. The methods presented below differ in the problem already mentioned in \cref{sec:Trajectories}, that is, how to deal with a specific history, which is not exactly met by any data observed.
To overcome this intrinsic difficulty we generate new and additional, but different samples based on the fixed number of observed samples $\xi_j$, $j=1,\dots N$, available. This is accomplished by estimating the density describing the transition.
\medskip
These additional samples can then be used to apply the procedures outlined above, but in what follows we describe further algorithms which make direct use of the estimated transition distributions as well.
\subsection{Scenario generation from limited data}
In contrast to nested clustering (cf.\ \cref{sec:Trajectories})
we estimate the distribution of the transition given the \emph{precise} history. This distribution can be estimated by non-parametric kernel density estimation. That is, the density at stage $t+1$ conditional on the history $x_{[t]}:=(x_1,x_{2},\dots,x_{t})$ is estimated by
\begin{equation}\label{eq:Conditioal}
\hat f_{t+1}(x_{t+1}\mid x_{[t]})=\sum_{j=1}^N w_j(x_{[t]}) \cdot k_{h_N}\Big(x_{t+1}-\xi_{j,t+1}\Big),
\end{equation}
where $k(\cdot)$ is a kernel function, $h_N$ is the bandwidth and $k_{h_N}(\cdot):=\frac{1}{h_N^{m_{t+1}}} k\left(\frac{\cdot}{h_N}\right)$. The weights
\begin{equation}\label{eq:weight}
w_j(x_{[t]})=\frac{k\left(\frac{x_{i_1}-\xi_{j,1}}{h_N}\right)\cdot\ldots\cdot k\left(\frac{x_{i_t}-\xi_{j,t}}{h_N}\right)}{\sum_{\ell=1}^N k\left(\frac{x_{i_1}-\xi_{\ell,1}}{h_N}\right)\cdot\ldots\cdot k\left(\frac{x_{i_t}-\xi_{\ell,t}}{h_N}\right)}
\end{equation}
depend on the history $x_{[t]}$ (for consistency, we set $w_j(x_{[1]}):=\frac 1 N$ for $t=1$).
It can be shown that the approximation is asymptotically optimal for \\ $h_N\sim N^{-1/(m_1+\dots+m_t+4)}$ (cf.\ Silverman's rule of thumb, see also \citet{PflugPichler2016}). Note as well that the weights sum to $1$ in~\eqref{eq:weight} ,i.e., for every $x_{[t]}$, $\sum_{j=1}^N w_j(x_{[t]})=1$. Further note that the estimator~\eqref{eq:Conditioal} for the conditional density involves \emph{all} samples $\xi_j$, $j=1,\dots, N$.
The weight $w_j(x_{[t]})$ is further large, if the first $t$ stages $(\xi_{j,1},\dots, \xi_{j,t})$ of the observation $\xi_j$ are similar to $x_{[t]}$, and negligible otherwise. In this way the weight $w_j(x_{[t]})$ determines the importance of the corresponding observation $\xi_j$ for $x_{[t]}$, and the estimator~\eqref{eq:Conditioal} takes this information adequately into account.
Univariate Kernel functions, which have proven useful are the Epanechnikov kernel $k(x)= \max\left\{\frac 3 4(1-x^2),\,0\right\}$ and the logistic kernel $k(x)= \frac 2{(e^x+e^{-x})^2}$, cf.\ \citet{Tsybakov}. For multivariate data (i.e.\ higher dimension, $m_t>1$) univariate kernels can be multiplied to obtain a multivariate kernel function, $k(x_1,\dots, x_{m_t})= k(x_1)\cdot\ldots\cdot k(x_{m_t})$.
\paragraph{Samples from the conditional distribution~\eqref{eq:Conditioal}~--- the composition method.}
The composition method allows to quickly find samples from the conditional distribution $f_{t+1}(\,\cdot\,|\,x_{[t]})$ in~\eqref{eq:Conditioal}. Indeed, pick a random number $U\in (0,1)$, where $U$ is uniformly distributed. Then there is a summation index $j^*\in \{1,2,\dots N\}$ so that
\begin{equation}\label{compeq}
\sum_{j=1}^{j^*-1}w_j(x_{[t]})< U\le \sum_{j=1}^{j^*}w_j(x_{[t]}).
\end{equation}
Note, that $j^*$ is distributed on $\{1,2,\dots N\}$ with probability mass function $P(j^*= j)= w_j(x_{[t]})$.
For this index $j^*$ then randomly pick an instant from the kernel density $\frac{1}{h_N^{m_{t+1}}} k\left(\frac{\,\cdot\ \, - \xi_{j^*,t+1}}{h_N}\right)$ shifted to $\xi_{j^*,\,t+1}$, cf.~\eqref{eq:Conditioal}. This sample is finally distributed with the desired density $\hat f_{t+1}(\,\cdot\,|\, x_{[t]})$. The composition method thus is a simple, quick and cheap method to make samples from the additively composed density~\eqref{eq:Conditioal} accessible.
\subsection{A direct tree generation approach with limited data}
Note that~\eqref{eq:Conditioal} describes a continuous distribution. To specify the tree process we may approximate the distribution again by finding a discrete distribution with locations and corresponding weights, which approximate~\eqref{eq:Conditioal} adequately.
A scenario tree can be generated directly by exploiting the conditional density estimates introduced above. To this end approximate the distribution in~\eqref{eq:Conditioal} with density $\hat f_1(\cdot)$ by the discrete distribution $\sum_{\ell\in \mathcal S_1} \tilde p_\ell \cdot \delta_{\tilde x_\ell}$ located at $\tilde x_\ell$ ($\ell\in \mathcal S_1$), where each $\tilde x_\ell$ is reached with probability $\tilde p_\ell$. These locations constitute the states at stage $t=1$, building the origin of the approximating tree.
For the next step define $\mathbf{\tilde{x}}_{i_1}:=(\tilde x_{i_1},\tilde x_{i_2})$ and consider the distribution with density $f_1(\cdot|\,\mathbf{\tilde x}_{i_1})$. By applying the same approximation procedure again one finds a discrete, approximating distribution $\sum_{\ell\in \mathcal S_{i_1}} p_\ell\cdot \delta_{\tilde x_\ell}$. The tree then is continued with $(\tilde x_{i_1}, \tilde x_{i_2},\tilde x_{i_3})$, where $i_2\in \mathcal S_{i_1}$ is a successor of $i_1$.
A tree is finally found by repeatedly applying the method outlined above at every node in the subtree already available.
\medskip
\begin{algorithm}[H]
\SetKwData{Init}{\textbf{Initialization:}}
\SetKwData{Iter}{\textbf{Iteration:}}
\KwIn{Samples $\xi_j$ with $j=1,\ldots,N$ and a kernel function $k(\cdot)$}
\KwOut{A random vector $x=(x_1,\ldots,x_T)$ following the distribution in~\eqref{eq:Conditioal}}
\Init $w=(1.0,\ldots,1.0)$ is an $N$-dimensional vector of sample weights, cf.~\eqref{eq:weight}.\\
\For{$t=1,\ldots,T$}{
Normalize the weights, set $w_j = \frac{w_j}{\sum_{\ell=1}^{N} w_\ell}$.\\
Set the bandwidth $h_t = \sigma_t \cdot N_t ^{-\frac{1}{m_t+4}}$, where $m_t$ is the dimension, $N_t = \frac{\big(\sum_{j=1}^{N} w_j\big)^2}{\sum_{j=1}^{N} w_{j}^{2}}$ is the effective sample size and $\sigma_{t}^{2} = \var\big(\xi_{j,t}\colon j=1,\ldots,N\big)$ is the variance at stage $t$.
\\
Draw a uniform random number $U \in [0,1]$.\\
Find an index $j^\ast$ such that $\sum_{j=1}^{j^\ast-1} w_j < U \leq \sum_{j=1}^{j^\ast} w_j$ (cf.~\eqref{compeq}).\\
Pick a random instance $K_t$ from the distribution with density $k(\cdot)$ and set $x_t = \xi_{j^\ast,t} + h_t \cdot K_t$.\\
Re-calculate the sample weights for the next stage:\\
\eIf{Markovian}{
\For{$j=1,\ldots,N$}{$w_j = k_{h_t}(x_t - \xi_{j,t})$}}{
\For{$j=1,\ldots,N$ }{$w_j = w_j \cdot k_{h_t}(x_t - \xi_{j,t})$}}}
\KwResult{A random trajectory $(x_1,\dots,x_T)$ following the distribution~\eqref{eq:Conditioal}}
\caption{Generate a new trajectory using conditional density estimation.
\href{https://kirui93.github.io/ScenTrees.jl/stable/}{\texttt{ScenTrees.jl}\textsuperscript{\ref{fn:ScenTrees}}} provides this function as \texttt{kernel\_scenarios().}
}
\label{alg3}
\end{algorithm}
\subsection{Stochastic approximation with limited data}
The stochastic approximation algorithm outlined in \cref{sec:StochApproximation} bases on additional sample paths, and each new sample path is used to improve the scenario tree. To obtain convergence of the method it is necessary to have an unlimited number of sample paths available, but this is certainly not the case for a limited number of data. However, the conditional density estimation methods described in this section allow generating new sample paths (cf.\ also \citet{Haerdle1997}).
\paragraph{New sample paths from observations.}
Every new sample path starts with $x_1$ at the first stage. Using~\eqref{eq:Conditioal} and the composition method one may find a new sample $x_2$ from $\hat f_2(\cdot\mid x_[1])$ at the second stage. Then generate another sample $x_3$ from $\hat f_3(\cdot\mid x_{[2]})$ at the third stage, a new sample $x_t$ from $\hat f_t(\cdot\mid x_{[t-1]})$ at the stage $t$, etc. Iterating the procedure until the final stage $T$ reveals a new sample path $x=(x_1,x_2,\dots,x_T)$, generated from the initial data $\xi_1,\dots,\xi_N$ directly~(cf.~\cref{alg3}).
\medskip
Generating new scenario paths can be repeated arbitrarily often to get new, distinct scenario paths. Each new sample path $x$ can be handed to the stochastic approximation algorithm (\cref{sec:StochApproximation}) to improve iteratively a tentative, approximating tree.
With this modification the stochastic approximation algorithm in \cref{sec:StochApproximation} is eligible even in case of limited data.
\begin{example}[Running Maximum using~\cref{alg3}]
Consider a case where we have just $N=100$ scenarios of running maximum data. We use this limited data and learn as much as possible so that we can generate new and additional samples based on this data. These generated samples are fed directly into stochastic approximation procedure to generate a scenario tree.
\cref{fig:rmdg} shows 100 trajectories generated from the data using \cref{alg3} and the resulting scenario tree with a branching structure of $(1,3,3,3)$.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width=\textwidth, trim=0 0 0 20, clip]{rmdatag.pdf}
\caption{100 trajectories generated from the process in \cref{fig:rmp} using \cref{alg3}}
\label{fig:rmdatag}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width=\textwidth, trim=00 00 00 20, clip]{rmtreeg.pdf}
\caption{Tree with 3 branches at each node approximating process in \cref{fig:rmdatag} with distance of $0.114$}
\label{fig:rmtreeg}
\end{subfigure}
\caption{100 trajectories generated from a limited running maximum data and a resulting scenario tree, generated with
\href{https://kirui93.github.io/ScenTrees.jl/stable/}{\texttt{ScenTrees.jl}\textsuperscript{\ref{fn:ScenTrees}}}}
\label{fig:rmdg}
\end{figure}
It is clear that the data generated (cf.\ \cref{fig:rmdg}) is similar to the original data in \cref{fig:rmp}. Therefore, this method proves efficient to generate new data for generation of scenario trees using stochastic approximation procedure (cf.\ \cref{fig:rmtreeg}).
\end{example}
\section{Generation of scenario lattices} \label{sec:Lattice}
In many situations of practical relevance the process to be approximated by a tree is known to have special properties. Of course, this information should be exploited when constructing the approximating tree in order to reduce the computational effort and time, and to obtain trees with better approximation quality.
\medskip
In what follows we discuss the situations of Markovian processes and processes with fixed states separately. A main characteristic of trees in finance is arbitrage. For this we discuss arbitrage opportunities as well, and we provide a new result for trees with stochastic dominance in addition.
\subsection{Markovian processes and lattices}\label{sec:Markov}
Estimating a lattice follows the same principle as estimating a tree. However, the lattice generation procedure is of significantly lower complexity. If $D_t =\{\tilde{x}_{t,1}, \dots, \tilde{x}_{t,i_t} \}$ is the set of states at stage $t$, then the induced distribution between the conditional lattice given $\tilde{x}_{t,i}$ and the Markovian process $\xi_t$ must be calculated only for all nodes from $D_t$ and not for all paths leading to them.
As in the tree case, the calculation of the transport distance requires a backward algorithm but can be based on the approximation of the one-period transition probabilities $P(\xi_{t+1}\in A \mid \xi_t = x)$.
The average aberration distance\footnote{This aberration distance incorporates the additional parameter $r\ge 1$.}
\begin{equation}\label{aad}
\left(\E\left(\sum_{t=1}^T \,d_t\big(\xi_t, \mathbf T_t(\xi_t)\big)\right)^r\right)^{\nicefrac1r}
\end{equation}
(cf.~\eqref{eq:distT}) is an upper bound for the nested distance, but this upper bound may be conservative.
One has to consider the full conditional future process for the estimation of a scenario lattice, however, one does not have to consider the past. This fact allows the following
simplifications for the algorithms:
\begin{enumerate}[label=(\roman*)]
\item When applying the nested clustering or stochastic approximation (\cref{sec:Trajectories,sec:StochApproximation} above), for example, all samples can be reused at intermediate stages, as they do not have to have the same history.
\item When applying the density estimator~\eqref{eq:Conditioal}, the weights in~\eqref{eq:weight} simplify to
\begin{equation*}
w_j(x_{[t]})=\frac{k\left(\frac{x_{i_t}-\xi_{j,t}}{h_N}\right)}{\sum_{\ell=1}^N k\left(\frac{x_{i_t}-\xi_{\ell,t}}{h_N}\right)}.
\end{equation*}
\end{enumerate}
Applications of lattice scenario models in an hydro storage management system can be found, e.g., in \citet{Wozabal2013}.
\medskip
\begin{example}[Gaussian random walk]
Consider the Gaussian random walk process $X_n = \sum_{k=1}^{n} Y_k$ on 5 stages, where the random variables $\big(Y_k\big)_{k\geq1}$ are i.i.d.\@ and the distribution of each $Y_k$ is normal.
Simple examples of this process would be a path traced by a molecule as it travels through a liquid or a gas, the price of a fluctuating stock and the financial status of a gambler.
Random walks are also simple representations of the Markov processes.
\cref{fig:gaussianL} employs \cref{alg2} to approximate this process using a scenario lattice.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width=\textwidth, trim=0 0 0 20, clip]{grwalk.pdf}
\caption{100 samples from Gaussian random walk process}
\label{fig:gaussian}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width=\textwidth, trim=00 00 00 20, clip]{grwlattice1.pdf}
\caption{Scenario lattice with branching structure $(1,3,4,5,6)$ approximating process in \cref{fig:gaussian} with a distance of $0.554$}
\label{poissonLattice}
\end{subfigure}
\caption{100 samples from Gaussian random walk process and the resultant scenario lattice, generated with \href{https://kirui93.github.io/ScenTrees.jl/stable/}{\texttt{ScenTrees.jl}}}
\label{fig:gaussianL}
\end{figure}
\end{example}
\begin{algorithm}[H]
\SetKwData{Init}{\textbf{Initialization:}}
\SetKwData{Iter}{\textbf{Iteration:}}
\KwIn{Let $K$ be the number of iterations.}
\KwOut{Scenario lattice with values $\tilde X^{(K)}$ and transition probabilities.
\Init Set the transportation distance $c_E=0$, set the counters $c(n) = 0$ for all nodes and let $\tilde X^{(0)}$ be an initial lattice chosen by expert opinion, e.g. Also, choose a non-negative and non-increasing sequence $\alpha_{k}$ such that $\sum_{k=1}^{\infty} \alpha_{k} = \infty$ and $\sum_{k=1}^{\infty} \alpha_{k}^{2} < \infty$.\\
\For{$k=1,\ldots,K$}{
Use a new and independent trajectory $\xi^{(k)} = (\xi_{1}^{(k)},\ldots,\xi_{T}^{(k)})$\\
Find a scenario $s=(i_1,\ldots,i_T)$ of nodes with the closest lattice entry such that
\begin{equation}\label{eq:si}
i_t\in \argmin_{n^\prime \in \mathcal{N}(t)} d_t\big(\tilde X^{(k)}(n^\prime),\xi_{t}^{(k)}\big),
\end{equation} where $\mathcal{N}(t)$ are all nodes in stage $t$.\\
\For{$t =1,\ldots,T$}{
Increase all counters $c(i_t) = c(i_t) + 1$ for all nodes in $s$ (cf.~\eqref{eq:si}).\\
Modify the values of the nodes in these entries to be \[\tilde X^{(k)} (i_t) = \tilde X^{(k-1)}(i_t) - \alpha_{k} \cdot r\cdot d_t\big(\tilde X^{(k-1)}(i_t),\xi_{t}^{(k)}\big)^{r-1}\cdot \big(\tilde X^{(k-1)}(i_t) - \xi_{t}^{(k)}\big).\]
The values on other node entries remain unchanged.
}
Set $c_E = c_E + \big(\sum_{t=1}^{T}d_t(\tilde X^{(k-1)}(i_t),\xi_{t}^{(k)})\big)^r.$}
\KwResult{Set the conditional probabilities $p(n) = \frac{c(n)}{K}$. The quantity approximating $\big(\E d(\xi,T(\xi))^r\big)^{\nicefrac1r}$ (cf.~\eqref{aad}) is given by $\big(c_E/ K\big)^{\nicefrac1r}$.}
\caption{Generating a scenario lattice with a fixed branching structure using stochastic approximation.
\href{https://kirui93.github.io/ScenTrees.jl/stable/}{\texttt{ScenTrees.jl}\textsuperscript{\ref{fn:ScenTrees}}} provides this algorithm as \texttt{lattice\_approximation()}}
\label{alg2}
\end{algorithm}
\subsection{Processes with fixed states}\label{sec:FixedStates}
In selected situations of practical relevance the distribution at stage $t$ is known approximately, or the support of this distribution or, for example, its quantiles can be described beforehand. In these situations it might make sense to fix the states beforehand and to use the algorithms introduced above to estimate the transition probabilities only, without modifying the states. Examples of such processes include mean reverting processes.
\begin{figure}[H
\centering
\begin{tikzpicture}[scale= 0.85]
\node (1) at (0,0) {$\tilde x_1$};
\node (2) at (2,1.7) {$\tilde x_2$};
\node (3) at (2,1) {$\tilde x_3$};
\node (4) at (2,0) {$\tilde x_4$};
\node (5) at (2,-1){$\tilde x_5$};
\node (6) at (2,-1.7){$\tilde x_6$};
\node (11) at (4,1.7) {$\tilde x_7$};
\node (10) at (4,1) {$\tilde x_8$};
\node (9) at (4,0) {$\tilde x_9$};
\node (8) at (4,-1){$\tilde x_{10}$};
\node (7) at (4,-1.7){$\tilde x_{11}$};
\foreach \x in {1.6,1,0,-1,-1.6}\node at (5,\x) {\dots};
\node (16) at (6,1.7) {$\tilde x_.$};
\node (15) at (6,1) {$\tilde x_.$};
\node (14) at (6,0) {$\tilde x_.$};
\node (13) at (6,-1){$\tilde x_.$};
\node (12) at (6,-1.7){$\tilde x_.$};
\node (21) at (8,1.7) {$\tilde x_.$};
\node (20) at (8,1) {$\tilde x_.$};
\node (19) at (8,0) {$\tilde x_.$};
\node (18) at (8,-1){$\tilde x_.$};
\node (17) at (8,-1.7){$\tilde x_.$};
\foreach \x in {2,...,6} \draw[->, ultra thin] (1) to (\x);
\foreach \x in {2,...,6} \foreach \y in {7,...,11} \draw[->, ultra thin] (\x) to (\y);
\foreach \x in {12,...,16}\foreach \y in {17,...,21}\draw[->, ultra thin] (\x) to (\y);
\node (t0) at (0.3,-2.5) {$t=1$};
\node (t1) at (2,-2.5) {$t=2$};
\node (t2) at (4,-2.5) {$t=3$};
\node (t3) at (7.7,-2.5) {$t=T$};
\end{tikzpicture}
\caption{A scenario lattice process with fixed states\label{fig:Fixed}}
\end{figure}
\Cref{fig:Fixed} displays a scenario lattice with 5, fixed states, which do not change over time. The points $\tilde x_2$ to $\tilde x_6$ can be determined beforehand, and they are kept fixed over time. For a univariate distribution they can be chosen to be the quantiles, e.g. That is,
$x_2$ is the $\nicefrac 1{10}$-quantile,
$x_3$ the $\nicefrac 3{10}$-quantile, \dots\ and
$x_6$ the $\nicefrac 9{10}$-quantile, resp.
\section{Trees lacking a predefined structure}\label{sec:Treeswithout}
Up to now we have assumed that the structure of the tree (i.e., the number of nodes and the predecessor vector $\operatorname{pred}$) has been chosen and is fixed. This a quite restrictive, since hardly anybody can decide beforehand what tree topology bes suits a given decision problem. Here is how to generate a tree with a given approximation quality without choosing the tree topology beforehand.
\begin{enumerate}[label=(\roman*)]
\item Choose (for each stage) a maximally acceptable distance between the estimated conditional densities and the discrete approximations.
\item Start with a simple tree (e.g., a binary one).
\item If during the tree construction the required distance cannot be achieved with the given number of successors, increase the branching factor by 1 until the distance condition can be satisfied.
\end{enumerate}
Details of this procedure can be found in \citet{PflugPichlerBuch}.
\section{Application of scenario lattice generation methods to an electricity pricing data}\label{sec:application}
To summarize the methods discussed in the preceding sections, we consider a limited electricity load data recorded for each hour of the day for the year 2017. The goal in what follows is to approximate the load of electricity for each hour of the day from this data using a scenario lattice. The scenario lattice has the same number of stages as the number of hours in the data. It also has a certain fixed branching structure (i.e., the number of nodes connected to each succeeding nodes in the lattice).
This computation has been done using the
\href{https://kirui93.github.io/ScenTrees.jl/stable/}{\texttt{ScenTrees.jl}}\textsuperscript{\ref{fn:ScenTrees}}
package which has been tested for Julia~$\geq 1.0$.
\paragraph{Brief summary of the data.}
The hourly actual load of electricity data consists of $52$ trajectories representing the $52$ weeks in a year (cf.\ \cref{fig:data}). Each trajectory consists of the 7 days of the week. The data is recorded from $12\colon00$\,a.m.\ to $11\colon59$\,p.m.\ of each day. This means that we are employing a $52\times168$ dimensional data where 168 stages represent the $24\times7$ hours of each week.
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{ActualData.pdf}
\caption{52 trajectories of the data}
\label{fig:data}
\end{figure}
The data shows a similar pattern in all the weeks. The actual total load of electricity tends to be higher during the weekdays than on the weekends. It is also higher during the daytime than in the nighttime. The maximum hourly mean of the data is \SI{70290.4}{\MW} and the minimum hourly mean is \SI{38775.3}{\MW}.
A special characteristic of the data is the presence of outliers which specifically represent holidays in Germany. For example, four of the holidays in the year 2017 fell on Monday (e.g., Easter Monday (April 17\textsuperscript{th}), Labor day (May 1\textsuperscript{st}), Whit day (June 5\textsuperscript{th}), Christmas (Dec 25\textsuperscript{th})). These Monday-outliers are visible in \cref{fig:data}.
\subsection{Approximation by a scenario lattice}
The data in \cref{fig:data} is limited. There are only $52$ trajectories in the data. We use the stochastic approximation procedure to discretize this data (\cref{alg2}). Ideally, to obtain convergence of the stochastic approximation procedure, it is necessary to have an unlimited number of sample paths available. The step size $\alpha_k = \frac{1}{3000+k}$,
where $k$ is the stochastic approximation iteration, turned out to be useful for this data
Based on this data and using \cref{alg3}, \cref{fig:generated} shows a sample of new and additionally generated scenarios.
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{generatedData.pdf}
\caption{\SI{1000} new trajectories generated from the sample in \cref{fig:data} by employing \cref{alg3}}
\label{fig:generated}
\end{figure}
These generated trajectories follow the same pattern as in the original data (cf.\ \cref{fig:data}). All the important and necessary characteristics of the original sample is also captured in this data. Therefore, these trajectories totally represent the original sample without any loss of information.
To use
\href{https://kirui93.github.io/ScenTrees.jl/stable/}{\texttt{ScenTrees.jl}}\textsuperscript{\ref{fn:ScenTrees}}
we fix the branching structure of the approximating scenario lattice and the number of iterations. In this case, we generate a scenario lattice with 5 nodes at each stage and $2.0\times10^6$ iterations.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[trim=00 18 00 36, clip, width=1.0\textwidth]{weekLattice.pdf}
\caption{Energy demand during the entire week}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[trim=15 18 55 55, clip, width=1.0\textwidth]{sat.pdf}
\caption{Detailed, hourly view of Saturday}
\end{subfigure}
\caption{Scenario lattice generated from \cref{fig:data} using \cref{alg3} and \cref{alg2}}
\label{fig:resLat}
\end{figure}
The scenario lattice in \cref{fig:resLat} has $5$ branches connected to each node in each stage. There are a total of $836$ nodes and approximately $5.4\times10^{116}$ scenarios possible in this lattice.
It is important to note that the generated scenario lattice is able to represent almost all the characteristics in the original data in \cref{fig:data}. This lattice recognizes the patterns in the original data and is able to discretize the data and recover this pattern. To determine the quality of approximation of the scenario lattice, we employ the transportation distance (cf.~\eqref{aad}). Ideally, one would want to find a scenario lattice which minimizes this distance. The generated scenario lattice in \cref{fig:resLat} has a transportation distance of \SI{1,203}{\MW} per stage.
\section{Conclusions}\label{sec:Summary}
In this paper, we present new and fast implementations to generate scenario trees and scenario lattices for decision making under uncertainty.
The package
\begin{center}
\href{https://kirui93.github.io/ScenTrees.jl/stable/}{\texttt{ScenTrees.jl}}\textsuperscript{\ref{fn:ScenTrees}}
\end{center}
collects efficient and freely available implementations of all algorithms.
We also provided explicit concepts to measure the quality of the approximation by employing distances, particularly the transportation distance. This distance constitutes an essential tool in stochastic optimization. We introduce several techniques to constructively obtain scenario tress from samples, which are observed trajectories.
We particularly elaborate on methods to find scenario trees and lattices with limited data only. We specifically apply scenario lattice generation methods on a limited electricity load data.
Our results prove that the new algorithms for generating scenario trees and scenario lattices are important as they are able to discretize the data such that all the information in the data is captured. These results also show that our implementation is highly competitive in terms of computational performance.
\nocite{Pflug2001}
\bibliographystyle{abbrvnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,357 |
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Geneva - Finland has breached the political rights of its Sami population, the UN Human Rights Committee ruled Monday, charging that Helsinki had violated the indigenous people's right to "internal self-determination".
Oslo - "Norway needs more children! I don't think I need to tell anyone how this is done," Norway's prime minister said cheekily, but she was raising a real concern.Too few babies are being born in the Nordic region.
Helsinki - Scientists in Finland have developed what they believe is the world's first vaccine to protect bees against disease, raising hopes for tackling the drastic decline in insect numbers which could cause a global food crisis.
Apo - Social media in Finland was ablaze with bemused comments on Monday after US President Donald Trump claimed the forest-covered nation prevents wildfires by raking its forest floors.
The leader of Finland denied on Sunday that he'd ever told President Donald Trump that the small Nordic nation relies upon "raking" its forests to prevent wildfires.
Apo - Finland has summoned Russia's ambassador to answer allegations that Moscow was behind the jamming of GPS signals in Lapland during recent NATO exercises, the foreign ministry in Helsinki confirmed Saturday.
Oslo - Oslo on Tuesday pointed a finger squarely at Russia, accusing it of jamming GPS signals in Norway's Far North when it hosted NATO's massive exercises in October and early November.
Apo - Russia denied Monday being behind the recent disruption to GPS signals across Lapland which put civil aviation at risk, after Finland's prime minister said the interference was "almost certainly deliberate".
Apo - Divided over how to handle ties to Hungary's populist prime minister and facing a future without their figurehead Angela Merkel, Europe's centre-right parties gather this week to seek a leader.
Strasbourg - Former Finnish prime minister Alexander Stubb launched a bid to become the next president of the European Commission on Tuesday.
Apo - Representatives of Finland's indigenous Sami said Tuesday they would urge holidaymakers to stop visiting igloos and riding husky sleds in Lapland, claiming non-Sami traditions are being passed off as authentic thereby exploiting their culture.
Apo - Finnish Foreign Minister Timo Soini clung on to his job Friday after winning a no-confidence vote in parliament over his attendance at an anti-abortion vigil during an official visit to Canada.
Apo - Finnish Foreign Minister Timo Soini faces a parliamentary confidence vote for taking part in a pro-life demonstration during an official visit to Canada earlier this year.
Apo - The United Nations may have judged Finland the happiest country in the world, but the Helsinki summit failed to make Donald Trump and Vladimir Putin smile.
Apo - The choice of Helsinki for Monday's first summit between Donald Trump and Vladimir Putin is a reminder of the Finnish capital's Cold War history when it hosted a number of key tete-a-tetes.
Apo - Finland may have largely shut down for the summer holidays but officials and police have been drafted back into work ahead of a historic summit in Helsinki between Donald Trump and Vladimir Putin.
Geologists and workers checking the bedrock in the tunnel.
The road in and out of the Onkalo Nuclear Waste Repository in Finland.
Help! Grasshoppers are destroying our grapes!
Jumping spider (Salticus cingulatus) filmed in the Burgwald, Hesse, Germany. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,966 |
A geleira Suvorov () é uma geleira de 5 milhas náuticas (9 km) de largura, fluindo para leste das colinas Wilson e desembocando no mar ao sul do cabo Northrup e do cabo Belousov. Foi mapeada pela Expedição Antártica Soviética, em 1958, e recebeu o nome de V. S. Suvorov, mecânico soviético que pereceu no Ártico.
Ver também
Inlandsis
Campo de gelo
Calota de gelo
Corrente de gelo
Referências
Terra de Oates
Costa Pennell
Suvorov | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,906 |
Jan Šimůnek (Praga, República Checa, 20 de febrero de 1987), es un futbolista checo. Juega de defensa y su actual equipo es el Vasas SC de la NBI.
Biografía
Šimůnek empezó jugando en diferentes equipos juveniles en Suiza y luego más tarde, en su país, en las categorías inferiores del Sparta Praga, aunque nunca llegó a debutar con el primer equipo. Su carrera profesional comenzó cuando el SK Kladno se fijó en él y consiguió que el Sparta lo cediera durante una temporada.
En 2007 se marcha a jugar al fútbol alemán con el VfL Wolfsburgo, equipo que pagó por él 4,5 millones de euros.
Selección nacional
Ha sido internacional con la Selección de fútbol de la República Checa en 4 ocasiones.
Participaciones en Copas del Mundo
Clubes
Palmarés
Campeonatos nacionales
Referencias
Enlaces externos
Futbolistas de la República Checa
Futbolistas de la selección de fútbol de la República Checa
Futbolistas del Sportovní Klub Kladno
Futbolistas del VfL Wolfsburgo II
Futbolistas del VfL Wolfsburgo en los años 2000
Futbolistas del VfL Wolfsburgo en los años 2010
Futbolistas del 1. FC Kaiserslautern en los años 2010
Futbolistas del 1. FC Kaiserslautern II
Nacidos en Praga | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,807 |
Meghan Addy (born 22 May 1978) is an American athlete who mainly competes in the 400 metres.
At the World Indoor Championships in Athletics 2004, Addy was part of the US team which finished third in the 4 x 400 metres relay.
Personal bests
Outdoor
400 metres: 53.70 seconds (2002)
400 metres hurdles: 55.70 seconds (2004)
Indoor
400 metres: 53.88 seconds (2003)
References
1978 births
Living people
American female sprinters
Place of birth missing (living people)
World Athletics Indoor Championships medalists | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 592 |
Q: invariance of angles under lattice automorphisms Let $L$ be a lattice, $x,y\in L$ and $g\in Aut(L)$.
Does this imply
$$<x,y>=<g(x),g(y)>,$$
in other words, are angles or inner products between any two lattice vectors invariant under lattice automorphisms?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,357 |
{"url":"http:\/\/jhealthscope.com\/en\/articles\/88454.html","text":"# Removal of Malathion by Sodium Alginate\/Biosilicate\/Magnetite Nanocomposite as a Novel Adsorbent: Kinetics, Isotherms, and Thermodynamic Study\n\nAUTHORS\n\nMehdi Hosseini 1 , * , Hossein Kamani 2 , Ali Esrafili 1 , Mojtaba Yegane Badi 1 , Mitra Gholami 1 , **\n\n1 Department of Environmental Health Engineering, Research Center for Environmental Health Technology, School of Public Health, Iran University of Medical Sciences, Tehran, Iran\n\n2 Department of Environmental Health, Health Promotion Research Center, Zahedan University of Medical Sciences, Zahedan, Iran\n\nCorresponding Authors:\n\nHow to Cite: Hosseini M , Kamani H, Esrafili A, Yegane Badi M, Gholami M. Removal of Malathion by Sodium Alginate\/Biosilicate\/Magnetite Nanocomposite as a Novel Adsorbent: Kinetics, Isotherms, and Thermodynamic Study, Health Scope. Online ahead of Print ; 8(4):e88454. doi: 10.5812\/jhealthscope.88454.\n\nARTICLE INFORMATION\n\nHealth Scope: 8 (4); e88454\nPublished Online: October 29, 2019\nArticle Type: Research Article\nRevised: August 1, 2019\nAccepted: February 18, 2019\nCrossmark\nCHEKING\n##### Abstract\n\nBackground: Organophosphorus pesticides are one of the widely consumed poisons in agriculture. The consumption of drinking water, which contains an excessive amount of poison, therefore, contributes to adverse health and hygiene outcomes in humans.\n\nMethods: In this study, a new sodium alginate\/biosilicate\/magnetite (SABM) nanocomposite made by the precipitation method was used to remove Malathion from aqueous solutions. The properties of MBSA were analyzed using XRD, SEM, EDX, and FTIR techniques. The possible impact of several parameters such as contact time, pH, initial Malathion concentration, temperature, and MBSA dosage on the adsorption process were investigated. The equilibrium isotherm and kinetic models were employed to evaluate the fitness of the experimental data.\n\nResults: The highest removal (94.82%) of MBSA was obtained at an optimum pH of 7, the contact time of 120 minutes, the adsorbent dosage of 4 g\/L, Malathion concentration of 10 mg\/L, and temperature of 318\u00b0K. The adsorption process followed the Freundlich isotherm model (R2 = 0.999), which implied that the adsorption process of Malathion molecules onto MBSA might be mainly a multi-molecular layer.\n\nConclusions: The results of this study showed that MBSA had a good removal efficiency, lower cost of processing, and as well as not producing substances harmful to the environment, which make it a promising adsorbent to remove Malathion from aqueous environments.\n\n## Keywords\n\nMalathion Removal Sodium Alginate Biosilicate Nanocomposite\n\nCopyright \u00a9 2019, Author(s). This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/) which permits copy and redistribute the material just in noncommercial usages, provided the original work is properly cited.\n\n### 1. Background\n\nIn the last few decades, pesticide contamination of water resources has emerged as a worldwide ecological concern. These compounds also have been detected in surface water and in bed sediments that its documents are available. Their concentration in aqueous sources is very variable and much higher concentrations of them have been reported in the effluents of farmlands.\n\nOrganophosphorus pesticides are the most common pesticides in the world. Unfortunately, their uncontrolled consumption in many parts of the world has contributed to their overabundance in the environment (1, 2). The pesticides used in agriculture can find a way into the surface water bodies through irrigation and precipitation which results in pollution of these waters (3). One of these pesticides is Malathion [(2dimethoxyphosphorothioyl) sulfanyl], which is a frequently used pesticide. The pesticides are widely used to increase the productivity of agricultural products as well as to control the diseases transmitted by arthropods (4). Organophosphorus pesticides such as Malathion are considered a serious threat to human health due to their effects on the cholinesterase activity and central nervous system disorder (5). Malathion may persist in water with a half-life of months or even years. The World Health Organization (WHO) has set the pesticide in drinking water at 0.1 \u00b5g L-1 (6).\n\nMalathion has a high solubility in water and its removal by conventional treatment processes such as sand filtration and coagulation is a really difficult process (7). Results from the previous studies indicate that in order to eliminate these pesticides, various methods such as photocatalytic degradation (8), biological oxidation (9), advanced oxidation processes (AOPs) (10), and adsorption (11, 12) have been employed. Among these techniques, adsorption process is a desirable method to remove pesticides. The adsorption technique is known for its simplicity, reliability, safety, and low costs that is an economical and effective method, and also a friendly environmental process (13-15).\n\n### 2. Objectives\n\nThe purpose of this study was to determine the efficiency of SABM nanocomposite in the removal of Malathion from aqueous environments. The possible impact of several parameters such as contact time, pH, initial Malathion concentration, temperature, and adsorbent dosage on the sorption process were investigated. The Langmuir and Freundlich adsorption isotherm was used to evaluate the adsorption capacity of SABM.\n\n### 3. Methods\n\n#### 3.1. Chemicals\n\nIn this study, sodium hydroxide (NaOH), hydrochloric acid (HCl), acetic acid (CH3CO2H), ferric chloride (FeCl3.6H2O), ferrous sulfate (FeSO4.7H2O), sodium triphosphate (Na5P3O10), ammonia solution, and sodium alginate were from Merck and Diatomite was from Sigma Aldrich. Also, Malathion, 95.0% of active ingredient was purchased from Sigma Aldrich. To make solutions needed by the experiments, deionized water was used.\n\nThere are two main steps in preparation of the adsorbent used in this study. First, making magnetite nanoparticles. Second, preparing the nanocomposite of SABM. These two steps are explained in the following.\n\n#### 3.3. Preparation of Magnetite Nanoparticles\n\nA chemical coprecipitation method was used to prepare the magnetite nanoparticles. In this method, FeSO4.7H2O and FeCl3.6H2 were first dissolved in a 1:1 ratio with a concentration of 3.2 M in the deionized water. The mixture was stirred in the presence of nitrogen gas at 80\u00b0C for 30 minutes. Then the ammonia solution with a purity of 25.0% was added to the mixture to reach the pH 10, and again it was washed under nitrogen gas for 60 minutes. The resulted magnetite nanoparticles were separated from the solution using a magnet, and then it was washed several times with ethanol and deionized water. The washed nanoparticles were dried at 70\u00b0C for 24 hours (25).\n\n#### 3.4. Preparation of Sodium Alginate\/Biosilicate\/Magnetite Nanocomposite Adsorbent\n\nTo prepare the nanocomposite, first 1 g of sodium alginate was added to 100 mL of acetic acid solution (1 M) and mixed for 2 hours. Then 1g of biosilicate and 1g of magnetite were added to the solution, and that was stirred by a stirrer at a fixed speed. To remove all bubbles in the solution and to obtain a no-bubble mixture, the resulting mixture was placed under a stable situation for 10 hours. In the next, a 100 mL mixture of NaOH (15.0%) and ethanol (95.0%) with ratio of 4:1 was prepared, and then the mixed solution of sodium alginate\/biosilicate\/magnetite was added to the mixture using a droplet, and then the solution was stored for 24 hours to form granular particles. Then the granular particles were washed with distilled water and dried at the ambient temperature to reach a constant weight. Finally, the dried mixture was chopped and then it was passed through a sieve to obtain the nanocomposite in a proper size (26). In the following, sodium alginate\/biosilicate\/magnetite nanocomposite adsorbent was signi\ufb01ed by abbreviation of SABM adsorbent.\n\n#### 3.5. Characterization of SABM Adsorbent\n\nA scanning electron microscope (SEM, Jeol ModelJsm-T330) with equipped an X-ray energy spectroscopy (EDX) under vacuum stable was used to determine the surface morphology and composition of the prepared SABM adsorbent. The crystal structure and purity of the SABM adsorbent particles in this work were characterized by X-ray diffraction (XRD) patterns, which were obtained on a Bruker D8 Advance X-ray diffractometer with Cu K\u03b1 radiation. The diffraction images were recorded at 40 mA and 40 kV in the 2\u03b8 range of 10\u00b0 - 80\u00b0. Also, to recognize the SABM adsorbent functional groups involved in the adsorption process, it was used a Fourier transform infrared (FTIR) adsorption spectrophotometer (JASCO, FT\/IR-6300Japan) using KBr disc method. The FTIR adsorption spectra were recorded in the range of 400 to 4000 cm-1.\n\n#### 3.6. The pH at Point Zero Charge (pHzpc)\n\nTo determine pH at zero pH point (pHZPC) of the SABM adsorbent, the following steps were conducted. At first of all, a suf\ufb01cient amount of 0.1 M NaNO3 solution was spilled into 250 mL flacks and their pH was adjusted between 2 and 11 by either 1 M HCl or NaOH. The whole volume of the solution in each flask was reached to 100 mL by adding NaNO3 solution of the same known concentration; meanwhile, the initial pH values of the solutions were accurately recorded. In the next step, 0.15 g of SABM adsorbent was added to each of the flasks and placed on a shaker at 200 rpm for 24 hours. Finally, the SABM adsorbent was separated from the suspensions, and then the pH values of the solution (final pH) were recorded. The pHZPC was obtained by plotting the initial pH values versus the final pH values (27).\n\nThis study was conducted in a batch system as a factor at the time. The effect of parameters, including contact time, initial pH values of the solution, SABM adsorbent dose, initial concentration of Malathion, and temperature of the solution was investigated on the adsorption of Malathion onto the SABM adsorbent. Also, adsorption kinetics and isotherms were studied.\n\nThe experiments were carried out in the following steps. In the first stage of experiments, a 100 mL suspension, including Malathion (5.0 mg L-1) and SABM adsorbent (0.5, 1, 1, 2 and 2.5 g L-1) were spilled into the 250 mL conical flasks and initial pH values were adjusted at 3, 5, 7, 9 and 11 using NaOH and HCl solution, and then it was placed on the thermoshaker at 200 rpm and 25\u00b0C, to shake for 120 minutes. The pH was measured with a pH meter (Aqua lytic (AL15)). To investigate Malathion concentration effect, the experiments were carried out at various concentrations (5, 25, 50 and 100 mgL-1) at pH 7 and SABM adsorbent dose 2 gL-1 at 25\u00b0C. Finely, effect of temperature on the adsorption process was carried out at various temperatures of solution (25, 35 and 40\u00b0C) and pH 7, adsorbent dose 2 gL-1 and Malathion concentration 5 mgL-1. The temperature was adjusted by incubator shaker.\n\n#### 3.8. Analysis\n\nAt the end of each experiment, a magnet was used to remove the SABM adsorbent from the suspension, (1.3 T), and then the residual of Malathion in the solution was measured with a UV-Vis spectrophotometer (DR 6000) at \u03bbmax of 236 nm. The removal efficiency of the adsorption process was calculated using Equation 1.\n\nWhere C0 and Ct are initial and final concentrations of Malathion, respectively.\n\nThe adsorption capacity was also calculated using Equation 2.\n\n$qt:(C0-Ct)Vm$\n\nWhere C0, Ct, V, and m, are initial concentration and the final concentration of Malathion, the volume of solution (L) and the mass of adsorbent particles (g), respectively.\n\n### 4. Results and Discussions\n\nThe scanning electron micrographs of SABM, sodium alginate, biosilica, and magnetic are shown in Figure 1. As shown in Figure 1D, the porosity of the SABM adsorbents is much more than other adsorbents. Such porosity level enhances the capacity and efficiency of Malathion adsorption onto SABM adsorbents. The elemental analysis of the adsorbent composition of SABM is shown in Figure 2. As can be seen, sodium, oxygen, iron, silica and aluminum presence in adsorbent structure. Moreover, the results revealed that silica can prevent the oxidation of iron nanoparticles by acid, which has been used in the process of adsorbent synthesis. These findings point out the suitable composition of the materials applied to the synthesis of SABM.\n\nFigure 1. SEM of adsorbent particles, A, sodium alginate, B, biosilica, C, magnetic, and D, SABM is shown\nFigure 2. EDX analytical results of SABM are shown\n\nThe XRD pattern of SABM is shown in Figure 3. In the magnetite pattern, the deflected peaks at the 2\u03b8 of 30.6, 36.04, 43.6, 54.2, 57.6, and 63.25 which are related to the crystalline plates (220, 311, 400, 422, 511, and 440) and they agree with the Fe3O4 cubic phase JCPDS (card No. 19-0629) (22). Also, there are some peaks in the SABM pattern indicating the presence of Fe3O4 in the SABM compound. As can be seen in Figure 3, the peaks obtained for the biosilica are in accordance with the pure silica phase (JCPDS ICDD File Card # 00-001-0647), and are quite obvious in the SABM pattern. Moreover, as shown in Figure 3 the intensity of the peaks in the composite SABM is reduced to the Fe3O4 and biosilica which can be related to the combination of these two substances with alginate because alginate have amorphous nature and it affects the pattern of SABM (26). FTIR spectroscopy is a powerful, well-developed method to determine the structure and identification of chemical species. It is mainly used to identify organic compounds because of the complexity of their spectra (5). The FTIR spectroscopy is a powerful, well-developed method to determine the structure and identification of chemical species. It is mainly used to identify organic compounds because of the complexity of their spectra (28). The FTIR spectrum of sodium alginate, magnetite, biosilica and SABM (before and after the adsorption of Malathion onto the SABM) is depicted in Figure 4. It\u2019s shown that some obvious changes take place in the spectrum of SABM in comparison with the pristine sodium alginate spectrum and bare magnetite. Also, considering Figure 4, the bands 1626 and 1453 are carboxylic anions (COO-). Owing to the polysaccharide property of the alginate, the band 1093 (C-O-C asymmetric traction) is visible. The strong and broadband 3442 is related to the stretching vibration of O-H groups (18). As known (Figure 4) in the magnetite spectrum, four major peaks are considerable. The 3450 band relates to the stretching vibration of O-H groups and the other three bands (635, 582 and 474) relate to the Fe-O vibrational bands (22). Comparison between the two spectra of the SABM adsorbent (before and after the adsorption of Malathion showed that the intensity of peaks at 3422, 2924, 2366, 627, 1453, 1093, 793, 627, and 454 was reduced after the adsorption of Malathion on the SABM, which indicated the impact of these functional groups on the adsorption process and confirmed that the magnetite nanoparticles were successfully coated with sodium alginate.\n\nFigure 4. FTIR spectra of sodium alginate, magnetite, biosilica, and SABM are shown\n\n#### 4.2. Effect of Contact Time\n\nTo investigate the adsorption behavior depended on time, the adsorption process was carried out at a determined statue for 4 hours. Figure 5 shows the result of the contact time effect on the adsorption process. As shown in Figure 5, the Malathion removal efficiency was increased immediately within 10 minutes (20.0%), then it was observed a stable pattern in the removal efficiency until 120 minutes, where the equilibrium was established and the removal efficiency was 92.1%.\n\n#### 4.3. Effect of pH\n\nThe effect of pH values on the removal of Malathion by the SABM adsorbent is shown in Figure 6A, where pH values were (3, 5, 7, 9, and 11), adsorbent dose was 1 g L-1, concentration of Malathion was 5 mg L-1, and temperature was 25\u00b0C at contact times 120 minutes. As can be seen, the highest removal efficiency of Malathion occurred at pH 7 and the lowest removal was at pH 11 (Figure 6). With an increase of pH from 3 to 7, the removal efficiency of Malathion has increased, but the removal efficiency has decreased with increasing pH from 7 to 9 and 11, respectively. In a previous study, a similar result has been reported by Kumar et al. (29) and by Zhang et al. (30) on the removal of Malathion by using both agricultural and commercial adsorbents. Concerning the effect of pH on the adsorption process, it is believed that determining pHZPC is important in the justification of the obtained results. Based on Figure 6B, pHZPCof SABM adsorbent was in 9.6. When the pH value is higher the pHZPC, the charge of adsorbent is negative and when it is lower the pHZPC, the charge of adsorbent is positive (15). Owing to the presence of electronegative centers (S and P) on the Malathion structure, and the SABM adsorbent pHZPC (9.6) (30), the Malathion can be adsorbed onto the SABM adsorbent at the acidic and natural pH best of alkaline pH values.\n\nFigure 6. Effects of pH on the removal of Malathion from the SABM (A) and pHZPC (B) are shown\n\n#### 4.4. Effect of the Adsorbent Dose\n\nEffect of various doses of adsorbent (0.5, 1, 1.5, 2, and 2.5 g L-1) in the Malathion adsorption onto the SABM adsorbent was shown in Figure 7A, where pH and concentration of Malathion were 7 and 5 mg L-1 and temperature was 25\u00b0C, respectively. As can be seen in Figure 7A, an increase in the SABM dose enhanced removal efficiency where the lowest and highest removal efficiency are in the SABM dose 0.5 and 2.5 g L-1, respectively. In a past study, a similar result has been observed by (7, 29). It is obvious that with increasing mass of adsorbent, the active site to adsorb pollutant increased that led to an increase in the removal efficiency of Malathion onto the SABM adsorbent (27).\n\nFigure 7. Effects of absorbent dose (A) and initial concentration of Malathion (B) on the removal of Malathion from the SABM are shown\n\n#### 4.5. Effect of the Initial Concentration of Malathion\n\nOne of the most important and influential factors in the adsorption process is the initial concentration of pollutants. Therefore, the effect of the initial different concentrations (5, 25, 50, and 100 mgL-1) of Malathion on the removal efficiency of the adsorption process was investigated, as pH and SABM dose were 7 and 2 g L-1 and temperature was 25\u00b0C, respectively, and the results were presented as Figure 7B. As can be seen in Figure 7B, when the Malathion initial concentration increased from 5 to 100 mg L-1, removal efficiency of Malathion decreased from 92.1% to 45.5%. This result agreed with the previous study by Kumar et al. (29) by which the Malathion was adsorbed onto the both agricultural and synthetic adsorbents. This phenomenon occurred because of a constant dose of SABM in contrast to the increased concentration of Malathion that reduced removal efficiency of the adsorption process. With an increase in the Malathion concentration, the active sites and surface area of the SABM become inadequate (29).\n\n#### 4.6. The Effect of Temperature\n\nThe results of the temperature effects on the removal efficiency of Malathion by SABM adsorbent are shown in Figure 8A. As can be seen, an increase in the temperature led to an increase in the adsorption of Malathion Figure 8A. The highest removal efficiency of Malathion is at 45\u00b0C (85.0%) and the lowest of it is at 25\u00b0C (92.0%). Hence, it can be explained by this fact that the adsorption process was endothermic in nature. This phenomenon can occur due to an increase in the displacement from the solubility phase of the molecules and their penetration within the pores of the SABM adsorbent (31, 32).\n\nFigure 8. Effects of temperature (A), isotherm of Longmuir (B), and pseudo-second kinetic (C) adsorption of Malathion onto the SABM are represented\n\nTo investigate the distribution of adsorbated molecules onto the adsorbent in equilibrium, the adsorption isotherm was employed. In this study, the relationship between the concentration of Malathion in solution and its adsorbed amount were determined by the Freundlich and Langmuir isotherms (15, 31).\n\nThe Langmuir isotherm model declares that the distribution of solute molecules onto the adsorbent surface has a monolayer pattern. As a solute molecule attaches to the active site placed on the adsorbent, no further adsorption can occur at that site (32, 33). The linear form of Langmuir isotherm model is expressed via Equation 3:\n\n$Ceqe=1Kaqm+Ceqm$\n\nSeparation factor (RL), which is a dimensionless parameter, is defined via Equation 4:\n\n$RL=11+KaC0$\n\nWhere Langmuir constants (Ce, qe, qm, and Ka) are attributed to the equilibrium concentration of Malathionin solution (mg L-1), amount of adsorbed Malathion (mg g-1), maximum monolayer adsorption capacity (mg L-1) and energy of adsorption (L mg-1), respectively, are calculated from plat of Ce\/qe versus Ce (15, 34). In Table 1, the results of Langmuir constants in modeling SABM adsorbent were presented calculated. With plotting Ce\/qe versus Ce for Langmuir isotherm model, not provided here, it appears that this isotherm model has a poor coefficient of determination (R2 = 0.852) in fit of the adsorption process. As can be seen in Table 1, Langmuir constants qm, RL, and Ka are 36.76 (mg g-1), 0.011 and 17.24 (L mg-1), respectively. This qm for SABM adsorbent was higher than what was reported in the previous study by Darvishi Cheshmeh Soltani et al. (26) conducted in desorption of a textile dye using bio-silica\/chitosan nanocomposite. Also, a study of Malathion removal by agricultural and commercial adsorbents that was carried out by Kumar et al. (29), showed a qm = 25 (mg g-1) which is lower than what reported in this study. The comparison of qm of SABM for Malathion removal with the other similar nanomaterial sorbents under similar experimental conditions is shown in Table 2. In this study, RL (dimensionless parameter) that indicates relative volatility in vapor-liquid equilibrium with a range between 0 and 1 for a favorable equilibrium (15), is at the favorable range. Therefore, it can be concluded that adsorption of Malathion onto the SABM adsorbent had a good favorable equilibrium.\n\nTable 1. Freundlich, Langmuir and Temkin Isotherm Parameters for the Adsorption of Malathion onto SABM Adsorbent\nLangmuir IsothermFreundlich IsothermTemkin Isotherm\nR2Ka (L mg-1)RLqm (mg g-1)R2nKfR2kt (L mg-1)b1\n0.85217.240.01136.760.99951.644.590.763661811.895\nTable 2. Various Parameters of Kinetic Models for the Malathion Adsorption onto the SABM\nqe, experimental (mg\/g)Pseudo-First OrderPseudo-Second OrderIntra-Particle Diffusion\nR2k1 (1\/min)qe, calculated (mg\/g)R2k2 (g\/mg. min)qe, calculated (mg\/g)R2C (mg\/g)Kid (mg\/g.min1\/2)\n24.80.7880.00530.98130.023250.859.650.22\n\nFreundlich isotherm model was used to determine the multilayer adsorption of adsorbate on the adsorbent surface. It also assumes that adsorption occurs on heterogeneous surfaces and can be expressed via Equation 5. (27).\n\nWhere Freundlich isotherm constants (Kf and n) are the extent of adsorption (mg g-1) and adsorption intensity of system calculated from plot of log qe versus log Ce. Figure 7B shows Freundlich isotherm model for the adsorption of Malathion onto the SABM adsorbent, where coefficient of determination (R2 = 0.9959) states that the adsorption process has good fit by Freundlich isotherm. Constants of Freundlich isotherm (Kf and n) prepared in Table 1 were 44.59 and 1.6, respectively. High amount of Kf constant represents very large extent of adsorption of Malathion onto the SABM adsorbent. Also, the value of n is larger than 1, indicating a favorable adsorption system and a multilayer physical process in the adsorption of Malathion by SABM adsorbent (35).\n\nThe Temkin model is employed to investigate the heat of the adsorption (adsorption energy) and adsorbent-adsorbate interactions. This isotherm assumes that the decrease of the adsorption energy of all the molecules in a layer linearly with the monolayer sorption on the active sites as a result of adsorbent-adsorbate interactions. The linear form of the Temkin model is given as follows (28, 36):\n\nWhere, B1, B1 = RT\/b1, denotes the Temkin constant (J\/mol). R is the universal gas constant and equal to 8.314 J\/mol.K. T is the absolute temperature (\u00b0K). kt and b1 represent the equilibrium binding constant (L\/g) and adsorption heat (kJ\/mol), respectively. Based on the data obtained, the magnitude of b1 value showed the fast removal of Malathion at the initial stage and the smallness of kt value implied the weak bonding of Malathion molecules onto the composite.\n\nTo study the mechanism of Malathion adsorption onto the SABM adsorbent, the transient behavior of the Malathion adsorption process was investigated using the pseudo-\ufb01rst-order and pseudo-second-order kinetics which are explained as follows.\n\nThe pseudo-\ufb01rst-order kinetic. Linear equation of pseudo-\ufb01rst order kinetic is shown in Equation 6 (37).\n\nWhere qe, qt, and k1 refers to the amount of adsorbed Malathion at equilibrium (mg g-1), the amount of adsorbed Malathion at time (t), and the equilibrium rate constant (min-1) of pseudo-\ufb01rst-order kinetic, respectively. The k1 is taken out from plotting Log (qe - qt) versus (t), where pseudo-\ufb01rst-order kinetic fitting for SABM adsorbent had a very poor coefficient of determination (R2 = 0.7884) (not shown). Calculated pseudo-\ufb01rst-order kinetic constants were provided in Table 3. As shown in Table 3, the equilibrium adsorption capacity qe (Cal) value was lower than the experimental qe (Exp) value, which indicated the inapplicability of this model.\n\nThe pseudo-second-order equation. Linear form of pseudo-second-order kinetic is given in Equation 7 (27).\n\n$tqt=1k2qe2+1qet$\n\nBy which, rate constant (k2) and adsorption capacity in equilibrium (qe) were calculated by plotting (t\/qt) versus (t) (Figure 7C). The initial adsorption rate (h) was calculated at zero time, by Equation 8.\n\n$h=k2qe2$\n\nAdditionally, the intraparticle diffusion model is conveniently employed to recognize the diffusion mechanism. The model can be epitomized as follows (38, 39):\n\nWhere, kid (mg\/g) is related to the intraparticle diffusion rate constant. C is the intercept and represents the thickness of the boundary layer (mg\/g) in which the effect of this layer depends on the value of the intercept.\n\nAs shown in Figure 7C, the pseudo-second-order kinetic fitting for adsorption of Malathion onto the SABM adsorbent have a good coefficient of determination (R2 = 0.9994). A similar behavior has been observed by Naushad et al. (31) on the removal of Malathion using amberlyst-15 resin. All the parameters of this kinetic model were prepared in Table 3. According to Table 3, the equilibrium adsorption capacity qe (Cal) value (25 mg g-1) was close to the experimental qe (Exp) value (24.8 mg g-1), which indicated the applicability of this kinetic model for the adsorption process behavior. Also, RL is 14.37 (min-1 mg g-1), which indicates the high initial adsorption rate. Based on the intra-particle diffusion model, the high values of C parameter (9.65) indicated that the boundary layer effect was also responsible for the adsorption. The multi-linearity of q versus t0.5 plot, and\/or deviation of the plots from the origin further confirms that the adsorption process is complex and some other mechanisms along with intraparticle diffusion control the process steps, as reported previously by Jerold et al. (40).\n\nTable 3. Adsorption Capacities of Various Adsorbents for the Uptake of Malathion\nMontmorillonite7.95(35)\nAmberlyst-15 cation exchange resin12.12(31)\npowdered activated carbon21.74(29)\nDe-Acidite FF-IP resin16.39(32)\nSABM nanocomposite36.86This work\n\nAbbreviation: SABM, sodium alginate\/biosilicate\/magnetite.\n\n#### 4.9. Thermodynamic Studies\n\nThe thermodynamic study was performed to reach a better understanding of the adsorptive behavior of Malathion toward nanocomposite. The free energy change (\u0394G0) (kJ mol-1), enthalpy change (\u0394H0) (kJ mol-1), and entropy change (\u0394S0) (kJ mol-1 K-1) for the adsorption of Malathion were calculated by Equations 9 (26).\n\n$\u0394G=-RTlnKD$\n$ln\u2061KD=(\u2206SR)-(\u2206HRT)$\n\nThe thermodynamic parameters of Malathion adsorption on MBAS are listed in Table 4. As represented in Table 4, the values of \u0394H and \u0394S are positive, and the standard free energy (\u0394G) is negative. The positive \u0394H value indicates that the sorption process was endothermic. In other words, the positivity of this parameter states that the increase in temperature has a positive effect on the adsorption of Malathion and, the adsorption of this pollutant at higher temperatures is more favorable. Furthermore, the negative values obtained for Gibbs free energy indicate that the adsorption of Malathion by the synthesized adsorbent is a spontaneous process (31).\n\nTable 4. Thermodynamic Parameters for the Adsorption of Malathion on MBAS\nTemperature (K)Ln KD\u2206G0 (kJ\/mol)\u2206H0 (kJ\/mol)\u2206S0 (kJ\/mol.K)\n2981.81-2.80602.600.1\n3082.05-4.05\n3183.19-5.78\n\n#### 4.10. The Mechanisms of the Adsorption\n\nThe mechanism of the adsorption of organic pollutants onto inorganic materials usually are a combination of electrostatic interaction, ion exchange, \u03c0-\u03c0 electron donor-acceptor (EDA) interaction, and hydrophobic surface interaction and so on.\n\nThe hydrophobic interaction is an important mechanism involved in the sorption of Malathion onto MBSA. Malathion is partially insoluble organics in water. As the pH increases, the Malathion molecules gain less water solubility and higher hydrophobicity; these results lead to a higher adsorption efficiency at pH 7. Therefore, the hydrophobic surface interactions should be a dominant mechanism in the adsorption process. In addition, electrostatic interaction can be a major mechanism governing adsorption of Malathion onto the MBSA (between the oppositely charged groups of adsorbate and adsorbent). Also, the acid-base interactions may be another significant factor involved in controlling Malathion adsorption.\n\n### References\n\n\u2022 1.\n\nHela DG, Lambropoulou DA, Konstantinou IK, Albanis TA. Environmental monitoring and ecological risk assessment for pesticide contamination and effects in Lake Pamvotis, northwestern Greece. Environ Toxicol Chem. 2005;24(6):1548-56. doi: 10.1897\/04-455r.1. [PubMed: 16117136].\n\n\u2022 2.\n\nWarren N, Allan IJ, Carter JE, House WA, Parker A. Pesticides and other micro-organic contaminants in freshwater sedimentary environments\u2014a review. Appl Geochem. 2003;18(2):159-94. doi: 10.1016\/s0883-2927(02)00159-2.\n\n\u2022 3.\n\nDehghani R, Moosavi SG, Esalmi H, Mohammadi M, Jalali Z, Zamini N. Surveying of pesticides commonly on the markets of Iran in 2009. J Environ Protect. 2011;2(8):1113-7. doi: 10.4236\/jep.2011.28129.\n\n\u2022 4.\n\nWard MH, Nuckols JR, Weigel SJ, Maxwell SK, Cantor KP, Miller RS. Identifying populations potentially exposed to agricultural pesticides using remote sensing and a Geographic Information System. Environ Health Perspect. 2000;108(1):5-12. doi: 10.1289\/ehp.001085. [PubMed: 10622770]. [PubMed Central: PMC1637858].\n\n\u2022 5.\n\nAl-Qurainy F, Abdel-Megeed A. Phytoremediation and detoxification of two organophosphorous pesticides residues in Riyadh area. World Appl Sci J. 2009;6(7):987-98.\n\n\u2022 6.\n\nWorld Health Organization. Guidelines for drinking-water quality: Recommendations. WHO; 2004.\n\n\u2022 7.\n\nOhno K, Minami T, Matsui Y, Magara Y. Effects of chlorine on organophosphorus pesticides adsorbed on activated carbon: Desorption and oxon formation. Water Res. 2008;42(6-7):1753-9. doi: 10.1016\/j.watres.2007.10.040. [PubMed: 18048077].\n\n\u2022 8.\n\nRamos-Delgado NA, Hinojosa-Reyes L, Guzman-Mar IL, Gracia-Pinilla MA, Hernandez-Ramirez A. Synthesis by sol-gel of WO3\/TiO2 for solar photocatalytic degradation of malathion pesticide. Catalysis Today. 2013;209:35-40. doi: 10.1016\/j.cattod.2012.11.011.\n\n\u2022 9.\n\nGetzin LW, Rosefield I. Partial purification and properties of a soil enzyme that degrades the insecticide malathion. Biochim Biophys Acta. 1971;235(3):442-53. doi: 10.1016\/0005-2744(71)90285-3. [PubMed: 5317645].\n\n\u2022 10.\n\nDoong RA, Chang WH. Photoassisted titanium dioxide mediated degradation of organophosphorus pesticides by hydrogen peroxide. J Photochem Photobiol Chem. 1997;107(1-3):239-44. doi: 10.1016\/s1010-6030(96)04579-0.\n\n\u2022 11.\n\nRani M, Shanker U. Effective adsorption and enhanced degradation of various pesticides from aqueous solution by Prussian blue nanorods. J Environ Chem Eng. 2018;6(1):1512-21. doi: 10.1016\/j.jece.2018.01.060.\n\n\u2022 12.\n\nYounis SA, Ghobashy MM, Samy M. Development of aminated poly(glycidyl methacrylate) nanosorbent by green gamma radiation for phenol and malathion contaminated wastewater treatment. J Environ Chem Eng. 2017;5(3):2325-36. doi: 10.1016\/j.jece.2017.04.024.\n\n\u2022 13.\n\nLiu Y, Chen M, Yongmei H. Study on the adsorption of Cu(II) by EDTA functionalized Fe3O4 magnetic nano-particles. Chem Eng J. 2013;218:46-54. doi: 10.1016\/j.cej.2012.12.027.\n\n\u2022 14.\n\nSrivastava M, Singh J, Yashpal M, Gupta DK, Mishra RK, Tripathi S, et al. Synthesis of superparamagnetic bare Fe(3)O(4) nanostructures and core\/shell (Fe(3)O(4)\/alginate) nanocomposites. Carbohydr Polym. 2012;89(3):821-9. doi: 10.1016\/j.carbpol.2012.04.016. [PubMed: 24750867].\n\n\u2022 15.\n\nNaghipour D, Taghavi K, Moslemzadeh M. Removal of methylene blue from aqueous solution by Artist's Bracket fungi: Kinetic and equilibrium studies. Water Sci Technol. 2016;73(11):2832-40. doi: 10.2166\/wst.2016.147. [PubMed: 27232421].\n\n\u2022 16.\n\nHossaini H, Moussavi G, Farrokhi M. The investigation of the LED-activated FeFNS-TiO2 nanocatalyst for photocatalytic degradation and mineralization of organophosphate pesticides in water. Water Res. 2014;59:130-44. doi: 10.1016\/j.watres.2014.04.009. [PubMed: 24793111].\n\n\u2022 17.\n\nDarvishi Cheshmeh Soltani R, Safari M, Rezaee A, Godini H. Application of a compound containing silica for removing ammonium in aqueous media. Environ Progr Sustain Energ. 2015;34(1):105-11. doi: 10.1002\/ep.11969.\n\n\u2022 18.\n\nOmidi Khaniabadi Y, Heydari R, Nourmoradi H, Basiri H, Basiri H. Low-cost sorbent for the removal of aniline and methyl orange from liquid-phase: Aloe Vera leaves wastes. J Taiwan Inst Chem Eng. 2016;68:90-8. doi: 10.1016\/j.jtice.2016.09.025.\n\n\u2022 19.\n\nDarvishi Cheshmeh Soltani R, Safari M, Maleki A, Godini H, Mahmoudian MH, Pordel MA. Application of nanocrystalline Iranian diatomite in immobilized form for removal of a textile dye. J Dispers Sci Tech. 2015;37(5):723-32. doi: 10.1080\/01932691.2015.1058715.\n\n\u2022 20.\n\nMauter MS, Elimelech M. Environmental applications of carbon-based nanomaterials. Environ Sci Technol. 2008;42(16):5843-59. doi: 10.1021\/es8006904. [PubMed: 18767635].\n\n\u2022 21.\n\nZhao Y, Xue Z, Wang X, Wang L, Wang A. Adsorption of congo red onto lignocellulose\/montmorillonite nanocomposite. J Wuhan UnivTech Mater Sci Ed. 2012;27(5):931-8. doi: 10.1007\/s11595-012-0576-2.\n\n\u2022 22.\n\nYi X, He J, Guo Y, Han Z, Yang M, Jin J, et al. Encapsulating Fe3O4 into calcium alginate coated chitosan hydrochloride hydrogel beads for removal of Cu (II) and U (VI) from aqueous solutions. Ecotoxicol Environ Saf. 2018;147:699-707. doi: 10.1016\/j.ecoenv.2017.09.036. [PubMed: 28938140].\n\n\u2022 23.\n\nDarvishi Cheshmeh Soltani R, Safari M, Maleki A, Rezaee R, Shahmoradi B, Shahmohammadi S, et al. Decontamination of arsenic(V)-contained liquid phase utilizing Fe3O4\/bone char nanocomposite encapsulated in chitosan biopolymer. Environ Sci Pollut Res Int. 2017;24(17):15157-66. doi: 10.1007\/s11356-017-9128-9. [PubMed: 28500548].\n\n\u2022 24.\n\nDarvishi Cheshmeh Soltani R, Khataee AR, Godini H, Safari M, Ghanadzadeh MJ, Rajaei MS. Response surface methodological evaluation of the adsorption of textile dye onto biosilica\/alginate nanobiocomposite: thermodynamic, kinetic, and isotherm studies. Desalin Water Treat. 2014;56(5):1389-402. doi: 10.1080\/19443994.2014.950344.\n\n\u2022 25.\n\nLou Z, Zhou Z, Zhang W, Zhang X, Hu X, Liu P, et al. Magnetized bentonite by Fe3O4 nanoparticles treated as adsorbent for methylene blue removal from aqueous solution: Synthesis, characterization, mechanism, kinetics and regeneration. J Taiwan Inst Chem Eng. 2015;49:199-205. doi: 10.1016\/j.jtice.2014.11.007.\n\n\u2022 26.\n\nDarvishi Cheshmeh Soltani R, Khataee AR, Safari M, Joo SW. Preparation of bio-silica\/chitosan nanocomposite for adsorption of a textile dye in aqueous solutions. Int Biodeterior Biodegrad. 2013;85:383-91. doi: 10.1016\/j.ibiod.2013.09.004.\n\n\u2022 27.\n\nPourkarim S, Ostovar F, Mahdavianpour M, Moslemzadeh M. Adsorption of chromium(VI) from aqueous solution by Artist\u2019s Bracket fungi. Separ Sci Tech. 2017;52(10):1733-41. doi: 10.1080\/01496395.2017.1299179.\n\n\u2022 28.\n\nMassoudinejad M, Rasoulzadeh H, Ghaderpoori M. Magnetic chitosan nanocomposite: Fabrication, properties, and optimization for adsorptive removal of crystal violet from aqueous solutions. Carbohydr Polym. 2019;206:844-53. doi: 10.1016\/j.carbpol.2018.11.048. [PubMed: 30553392].\n\n\u2022 29.\n\nKumar P, Singh H, Kapur M, Mondal MK. Comparative study of malathion removal from aqueous solution by agricultural and commercial adsorbents. J Water Proc Eng. 2014;3:67-73. doi: 10.1016\/j.jwpe.2014.05.010.\n\n\u2022 30.\n\nZhang Q, Jing Y, Shiue A, Chang CT, Ouyang T, Lin CF, et al. Photocatalytic degradation of malathion by TiO(2) and Pt-TiO(2) nanotube photocatalyst and kinetic study. J Environ Sci Health B. 2013;48(8):686-92. doi: 10.1080\/03601234.2013.778623. [PubMed: 23638896].\n\n\u2022 31.\n\nNaushad M, Alothman ZA, Khan MR, Alqahtani NJ, Alsohaimi IH. Equilibrium, kinetics and thermodynamic studies for the removal of organophosphorus pesticide using Amberlyst-15 resin: Quantitative analysis by liquid chromatography\u2013mass spectrometry. J Ind Eng Chem. 2014;20(6):4393-400. doi: 10.1016\/j.jiec.2014.02.006.\n\n\u2022 32.\n\nNaushad M, Alothman ZA, Khan MR. Removal of malathion from aqueous solution using De-Acidite FF-IP resin and determination by UPLC-MS\/MS: equilibrium, kinetics and thermodynamics studies. Talanta. 2013;115:15-23. doi: 10.1016\/j.talanta.2013.04.015. [PubMed: 24054556].\n\n\u2022 33.\n\nBalarak D, Mahdavi Y, Bazrafshan E, Mahvi AH, Esfandyari Y. Adsorption of fluoride from aqueous solutions by carbon nanotubes: Determination of equilibrium, kinetic, and thermodynamic parameters. Adsorption. 2016;49(1):71-83.\n\n\u2022 34.\n\nBazrafshan E, Zarei AA, Nadi H, Zazouli MA. Adsorptive removal of Methyl Orange and Reactive Red 198 dyes by Moringa peregrina ash. Indian J Chem Tech (IJCT). 2014;21(2):105-13.\n\n\u2022 35.\n\nPal OR, Vanjara AK. Removal of malathion and butachlor from aqueous solution by clays and organoclays. Separ Purif Tech. 2001;24(1-2):167-72. doi: 10.1016\/s1383-5866(00)00226-4.\n\n\u2022 36.\n\nBazrafshan E, Kord Mostafapour F, Rahdar S, Mahvi AH. Equilibrium and thermodynamics studies for decolorization of Reactive Black 5 (RB5) by adsorption onto MWCNTs. Desalin Water Treat. 2014;54(8):2241-51. doi: 10.1080\/19443994.2014.895778.\n\n\u2022 37.\n\nZhao J, Liu J, Li N, Wang W, Nan J, Zhao Z, et al. Highly efficient removal of bivalent heavy metals from aqueous systems by magnetic porous Fe3O4 -MnO2: Adsorption behavior and process study. Chem Eng J. 2016;304:737-46. doi: 10.1016\/j.cej.2016.07.003.\n\n\u2022 38.\n\nAlimohammadi M, Saeedi Z, Akbarpour B, Rasoulzadeh H, Yetilmezsoy K, Al-Ghouti MA, et al. Adsorptive removal of arsenic and mercury from aqueous solutions by Eucalyptus leaves. Water Air Soil Pollut. 2017;228(11). doi: 10.1007\/s11270-017-3607-y.\n\n\u2022 39.\n\nBalarak D, Kord Mostafapour F, Bazrafshan E, Mahvi AH. The equilibrium, kinetic, and thermodynamic parameters of the adsorption of the fluoride ion on to synthetic nano sodalite zeolite. Fluoride. 2017;50(2):223-34.\n\n\u2022 40.\n\nJerold M, Vasantharaj K, Joseph D, Sivasubramanian V. Fabrication of hybrid biosorbent nanoscale zero-valent iron-Sargassum swartzii biocomposite for the removal of crystal violet from aqueous solution. Int J Phytoremediation. 2017;19(3):214-24. doi: 10.1080\/15226514.2016.1207607. 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The VB31AT is a unique UTP power and video balun which provides an economical means of sending video and camera power over a standard category cable. Video is sent over one pair and camera power is sent over two of the remaining pairs. A mini-coax pigtail with male BNC is used on the VB31AT. A pair of wires are provided for power connection. Connections to the Category cable are made via an RJ45 connector.
The VB31AT passive UTP power and video balun combiner provides the same high immunity to noise and interference as all of the Nitek baluns for CCTV transmission. This simplified wiring scheme provides a convenient method of powering the camera and allowing for quicker and easier installations. The RJ45 modular jack uses standard 568B wiring so spare network cables can be used. The VB31AT is a UTP transceiver that provides another Security Transmission Solution from Nitek. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,689 |
Lahnda és el nom donat per iniciativa de Griersin a la seva obra Linguistic Survey of India, a una sèrie de llengües del grup de llengües indoàries parlades a la part sud-central-occidental d'Àsia que estarien prou properes entre elles per definir un macrollenguatge. Segons l'Ethnologue inclou:
Jakati (parlat a Ucraïna)
Nord Hindko (parlat a Pakistan)
Sud Hindko (parlat a Pakistan)
Khetrani (parlat a Pakistan)
Saraiki (parlat a Pakistan)
Panjabi occidental (parlat a Pakistan)
Mirpur Punjabi (parlat a Pakistan)
Potwari (parlat a Pakistan i a Azad Caixmir
Derawali (parlat a Pakistan)
No inclou, en canvi, el panjabi oriental parlat a l'Índia
Referències
Llengües indoàries | {
"redpajama_set_name": "RedPajamaWikipedia"
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package org.avaje.ebean.dbmigration.model;
import org.avaje.ebean.dbmigration.migration.Column;
/**
* Created by rob on 28/07/15.
*/
public class MColumn {
private final String name;
private final String checkConstraint;
private final String defaultValue;
private final String references;
private final String type;
public MColumn(Column column) {
this.name = column.getName();
this.type = column.getType();
this.checkConstraint = column.getCheckConstraint();
this.defaultValue = column.getDefaultValue();
this.references = column.getReferences();
}
public String getName() {
return name;
}
public String getType() {
return type;
}
public String getCheckConstraint() {
return checkConstraint;
}
public String getDefaultValue() {
return defaultValue;
}
public String getReferences() {
return references;
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}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,564 |
Ceratocyba umbilicaris, unique représentant du genre Ceratocyba, est une espèce d'araignées aranéomorphes de la famille des Linyphiidae.
Distribution
Cette espèce est endémique du Kenya.
Publication originale
Holm, 1962 : The spider fauna of the East African mountains. Part I: Fam. Erigonidae. Zoologiska Bidrag Från Uppsala, , .
Liens externes
genre Ceratocyba :
espèce Ceratocyba umbilicaris :
Notes et références
Linyphiidae
Espèce d'araignées (nom scientifique)
Faune endémique du Kenya | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,392 |
A Casa do Cemitério (Quella villa accanto al cimitero) é um filme de horror italiano sobrenatural de 1981, dirigido por Lucio Fulci. Sua trama gira em torno de uma série de assassinatos que ocorrem em uma casa na Nova Inglaterra, uma casa que passa a ter um segredo escondido dentro das paredes do porão. Temas e motivos dos filmes de terror como The Shining, The Amityville Horror de Frankenstein.
Sinopse
Um historiador recebe a notícia de que um amigo de profissão, a trabalho em Boston, cometera suicídio após ter assassinado a amante. Designado para o lugar do falecido, o homem muda-se para aquela cidade com a família e vai morar numa estranha mansão. Lá, coisas estranhas acontecem e logo se descobre que um assassino sedento por sangue vive escondido no porão.
Elenco
Catriona MacColl como Lucy Boyle
Paolo Malco como Dr. Norman Boyle
Giovanni Frezza como Bob Boyle
Ania Pieroni como Ann
Filmes da Itália de 1981
Filmes em língua italiana
Filmes de terror da Itália
Filmes de terror da década de 1980
Filmes de zumbis
Filmes dirigidos por Lucio Fulci | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,270 |
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\title{Coalgebraic Semantics for Nominal Automata}
\titlerunning{Coalgebraic Language Semantics for Nominal Automata}
\author{Florian Frank \and Stefan Milius\thanks{Supported by Deutsche Forschungsgemeinschaft (DFG) under project MI~717/7-1} \and Henning Urbat\thanks{Supported by Deutsche Forschungsgemeinschaft (DFG) under proj.~SCHR~1118/15-1}}
\authorrunning{F.~Frank, S.~Milius and H.~Urbat}
\institute{Friedrich-Alexander-Universit\"at Erlangen-N\"urnberg}
\begin{document}
\maketitle
\begin{abstract}
This paper provides a coalgebraic approach to the language semantics
of regular nominal non-deterministic automata (RNNAs), which were
introduced in previous work. These automata feature ordinary as well
as name binding transitions. Correspondingly, words accepted by RNNAs
are strings formed by ordinary letters and name binding
letters. Bar languages are sets of such words modulo
$\alpha$-equivalence, and to every state of an RNNA one associates
its accepted bar language. We show that this semantics arises both
as an instance of the Kleisli-style coalgebraic trace semantics as
well as an instance of the coalgebraic language semantics obtained
via generalized determinization. On the way we revisit coalgebraic
trace semantics in general and give a new compact proof for the main
result in that theory stating that an initial algebra for a functor
yields the terminal coalgebra for the Kleisli extension of the
functor. Our proof requires fewer assumptions on the functor than
all previous ones.
\end{abstract}
\section{Introduction}\label{S:intro}
Classical automata and their language semantics have long been
understood in the theory of coalgebras. For example, it is a well-known
exercise~\cite{Rutten98} that standard deterministic automata over a
fixed alphabet can be modelled as coalgebras, that the terminal
coalgebra is formed by all formal languages over that alphabet, and
the unique homomorphism into the terminal coalgebra assigns to each
state of an automaton the language it accepts. Non-deterministic
automata are also coalgebras for a functor extending the one for
deterministic automata in order to accomodate non-deterministic
branching. Their language semantics can be obtained coalgebraically in
two different ways. First, in the \emph{coalgebraic trace semantics}
by Hasuo et al.~\cite{HasuoEA07} one considers coalgebras for composed
functors $TF$ where $F$ is a set functor modelling the type of
transitions and $T$ is a set monad modelling the type of branching;
for example, for non-deterministic branching one takes the power-set
monad. Under certain conditions on $F$ and $T$, including that $F$ has
an extension $\mybar{0.6}{2.5pt}{F}$ to the Kleisli category of $T$, an initial
$F$-algebra is seen to lift to the terminal coalgebra for $\mybar{0.6}{2.5pt}{F}$. Its
universal property then yields the coalgebraic trace semantics. Among
the instances of this is the standard language semantics of
non-deterministic automata.
Second the \emph{coalgebraic language semantics}~\cite{bms13} is based on
generalized determinization by Silva et al.~\cite{SilvaEA13}. Here one
considers coalgebras for composed functors~$GT$ where $G$ models
transition types and $T$ again models the branching type. Assuming
that $G$ has a lifting to the Eilenberg-Moore category for $T$, generalized
determinization turns such a coalgebra into a $G$-coalgebra by taking
the unique extension of the coalgebra structure to the free
Eilenberg-Moore algebra on the set of states. Moreover, taking the
unique homomorphism from that coalgebra into the terminal
$G$-coalgebra yields the coalgebraic language semantics. In the
leading instance of non-deterministic automata, generalized
determinization is the well-known power-set construction and
coalgebraic language semantics the standard automata-theoretic
language semantics once again.
These two approaches were brought together by Jacobs et
al.~\cite{JacobsEA15} who study those species of systems which can be
modelled as coalgebras in both of the above ways. They show that
whenever there exists an \emph{extension} natural transformation
$TF \to GT$ satisfying two natural equational laws, then the two above
semantics are canonically related, and they agree in the
instances studied in op.~cit.
It is our aim in this paper to draw a similar picture for
non-deterministic automata for languages over infinite alphabets. Such
alphabets allow to model \emph{data}, such as nonces~\cite{KurtzEA07},
object identities~\cite{GrigoreEA13}, or abstract
resources~\cite{CianciaSammartino14}, and the ensuing languages are
therefore called \emph{data languages}. There are several species of
automata for data languages in the literature. We focus on two types
which are known to have a presentation as coalgebras:
non-deterministic orbit-finite automata (NOFA)~\cite{BojanczykEA14}
and regular non-deterministic nominal automata
(RNNA)~\cite{SchroderEA17}. For both of these types of automata one works
with the category of nominal sets and takes the set of \emph{names} as
the alphabet. While NOFAs are a straightforward nominal version of
standard non-deterministic automata, RNNAs feature \emph{binding
transitions}, which can be thought as storing an input name in a
`register' for comparison with future input names. Correspondingly,
they accept words including name binding letters and which are taken modulo
$\alpha$-equivalence; such words form \emph{bar languages} (the name
stems from the bar in front of name binding letters $\newletter
a$). However, while these automata are understood as coalgebras, their
semantics has not been studied from a coalgebraic perspective sofar.
We fill this gap here and prove that the data language accepted by a NOFA and the bar
language accepted by an RNNA arise as instances of both coalgebraic trace semantics
\renewcommand{\theoremautorefname}{Theorems}%
(\autoref{T:NOFA-Kl} and~\ref{T:RNNA-Kl})
\renewcommand{\theoremautorefname}{Theorem}%
and coalgebraic language semantics
\newcommand{\corollaryautorefname}{Corollaries}%
(\autoref{C:NOFA-EM} and~\ref{C:RNNA-EM}).
\renewcommand{\corollaryautorefname}{Corollary}%
The latter result is obtained by using canonical extension natural
transformations obtained from the result by Jacobs et al.~\cite{JacobsEA15}.
While these results will perhaps hardly surprise the cognoscenti, and
the treatment of NOFAs indeed appears as an(other) exercise in coalgebra, we
should like to point out that there are a number of technical
subtleties arising in the treatment of RNNAs. Essentially, what causes
some trouble is the presence of the binding functor in their type. We
solve all these difficulties by working with the uniformly finitely
supported power-set monad $\pow_{\ufs}$ on nominal sets in lieu of the
more common finitely supported power-set monad $\pow_{\fs}$ (which provides
the power objects of the topos of nominal sets). Note also that for a
nominal set $X$, neither $\pow_{\fs} X$ nor $\pow_{\ufs} X$ form cpos (so, in
particular, they do not form complete lattices). Hence, it may come as
a bit of a surprise that the Kleisli categories of both monads are
nevertheless enriched over complete lattices (\autoref{P:dcpo}), one of
the key requirements for coalgebraic trace semantics.
We present our
results in a modular way so that they may be resusable for the study
of coalgebraic semantics for other types of nominal systems, such as nominal tree automata. For
example, we show that all \emph{binding polynominal functors},
i.e.~those functors arising from a nominal algebraic (a.k.a.~binding)
signature in the sense of Pitts~\cite{Pitts13} have a canonical
extension to the Kleisli category of $\pow_{\ufs}$
(\autoref{cor:extension-pufs}). Analogously, we show a lifting result
for terminal coalgebras to the Eilenberg-Moore category for a subclass
of these functors (\autoref{C:lift-nu}).
Last but not least, on the way to the coalgebraic semantics of NOFAs
and RNNAs we take a fresh look at coalgebraic trace semantics in
general. We provide a new compact proof for the main theorem of that
theory. It states that for a functor $F$ and a monad $T$ satisfying
certain conditions, including that $F$ has an extension $\mybar{0.6}{2.5pt}{F}$ to the
Kleisli category of $T$, the initial $F$-algebra extends to a terminal
coalgebra for $\mybar{0.6}{2.5pt}{F}$ (\autoref{T:Kl}). We obtain this essentially as
a combination of Hermida and Jacobs' adjoint lifting
theorem~\cite[Thm.~2.14]{HermidaJ98} and an argument originally given
by Freyd~\cite{Freyd92} that for locally continuous endofunctors on
categories enriched in cpos an initial algebra yields a terminal
coalgebra. Here we adjust this argument to work for locally monotone
endofunctors on categories enriched in directed-complete partial
orders. As a consequence, our proof does not require the existence of
a zero object in the Kleisli category of $T$ and, notably, we only
need the mere existence of the initial algebra for $F$ and not that it is
obtained after $\omega$ steps of the initial-algebra chain given by
$F^n 0$ ($n <\omega$).
\section{Preliminaries}\label{S:prelim}
\subsection{Nominal Sets}
\hunote[inline]{Copy \& paste from our CONCUR paper.}
Nominal sets form a convenient formalism for dealing with names and
freshness; for our present purposes, names play the role of data. We
briefly recall basic notions and facts and refer to Pitts'
book~\cite{Pitts13} for a comprehensive introduction. Fix a countably
infinite set $\mathbb{A}$ of \emph{names}, and let $\mathrm{Perm}(\mathbb{A})$ denote
the group of finite permutations on $\mathbb{A}$, which is generated by
the \emph{transpositions} $(a\, b)$ for $a\neq b\in\mathbb{A}$ (recall
that $(a\, b)$ just swaps~$a$ and~$b$). A \emph{nominal set} is a
set~$X$ equipped with a (left) group action
$\mathrm{Perm}(\mathbb{A})\times X\to X$, denoted $(\pi,x)\mapsto \pi\cdot x$,
such that every element $x\in X$ has a finite
\emph{support}~$S\subseteq\mathbb{A}$, i.e.~$\pi\cdot x=x$ for every
$\pi\in \mathrm{Perm}(\mathbb{A})$ such that $\pi(a)=a$ for all $a\in S$. Every
element~$x$ of a nominal set $X$ has a least finite support, denoted
$\supp(x)$. Intuitively, one should think of $X$ as a set of syntactic
objects (e.g.~strings, $\lambda$-terms, programs), and of $\supp(x)$
as the set of names needed to describe an element $x\in X$. A name
$a\in\mathbb{A}$ is \emph{fresh} for~$x$, denoted $a\mathbin{\#} x$, if
$a\notin\supp(x)$. The \emph{orbit} of an element $x\in X$ is given by
$\{ \pi\cdot x: \pi\in\mathrm{Perm}(\mathbb{A})\}$. The orbits form a partition of
$X$. The nominal set $X$ is \emph{orbit-finite} if it has only
finitely many orbits.
A map $f\colon X\to Y$ between nominal sets is \emph{equivariant} if
$f(\pi\cdot x)=\pi\cdot f(x)$ for all $x\in X$ and
$\pi\in \mathrm{Perm}(\mathbb{A})$. Equivariance implies
$\supp(f(x))\subseteq \supp(x)$ for all $x\in X$. We denote by $\mathsf{Nom}$ the category of nominal sets and equivariant
maps.
Putting $\pi\cdot a = \pi(a)$ makes $\mathbb{A}$ into a nominal
set. Moreover,~$\mathrm{Perm}(\mathbb{A})$ acts on subsets $A\subseteq X$ of a
nominal set~$X$ by $\pi\cdot A = \{\pi\cdot x : x \in A\}$. A
subset $A\subseteq X$ is \emph{equivariant} if $\pi\cdot A=A$ for all
$\pi\in \mathrm{Perm}(\mathbb{A})$. More generally, it is \emph{finitely
supported} if it has finite support w.r.t.\ this action, i.e.~there
exists a finite set $S\subseteq \mathbb{A}$ such that $\pi\cdot A = A$ for all
$\pi\in \mathrm{Perm}(\mathbb{A})$ such that $\pi(a)=a$ for all $a\in S$. The set
$A$ is \emph{uniformly finitely supported} if
$\bigcup_{x\in A} \supp(x)$ is a finite set. This implies that $A$ is
finitely supported, with least support
$\supp(A)=\bigcup_{x\in A}
\supp(x)$~\cite[Theorem~2.29]{gabbay2011}. (The converse does not
hold, e.g.~the set $\mathbb{A}$ is finitely supported but not uniformly
finitely supported.) Uniformly finitely supported orbit-finite sets
are always finite (since an orbit-finite set contains only finitely
many elements with a given finite support). We denote by $\mathcal{P}_{\mathsf{ufs}}\colon \mathsf{Nom}\to \mathsf{Nom}$ and
$\pow_{\fs}\colon \mathsf{Nom}\to \mathsf{Nom}$ the endofunctors sending a nominal set $X$ the its set of (uniformly) finitely supported subsets and an equivariant map
$f\colon X\to Y$ to the map $A\mapsto f[A]$.
The coproduct $X+Y$ of nominal sets $X$ and $Y$ is given by their
disjoint union with the group action inherited from the two
summands. Similarly, the product $X \times Y$ is given by the
cartesian product with the componentwise group action; we have
$\supp(x,y) = \supp(x) \cup \supp(y)$. Given a nominal set $X$
equipped with an equivariant equivalence relation, i.e.~an equivalence
relation $\sim$ that is equivariant as a subset
$\mathord{\sim} \subseteq X \times X$, the quotient $X/\mathord{\sim}$
is a nominal set under the expected group action defined by
$\pi \cdot [x]_\sim = [\pi \cdot x]_\sim$.
A key role in the theory of nominal sets is played by
\emph{abstraction sets}, which provide a semantics for binding
mechanisms~\cite{GabbayPitts99}. Given a nominal set $X$, an equivariant equivalence relation $\sim$ on
$\mathbb{A} \times X$ is defined by $(a,x)\sim (b,y)$ iff
$(a\, c)\cdot x=(b\, c)\cdot y$ for some (equivalently, all)
fresh~$c$. The \emph{abstraction set} $[\mathbb{A}]X$ is the quotient
set $(\mathbb{A}\times X)/\mathord{\sim}$. The $\sim$-equivalence class
of $(a,x)\in\mathbb{A}\times X$ is denoted by
$\braket{a} x\in [\mathbb{A}]X$. We may think of~$\sim$ as an abstract notion of $\alpha$-equivalence,
and of~$\braket{a}$ as binding the name~$a$. Indeed we have
$\supp(\braket{a} x)= \supp(x)\setminus\{a\}$ (while
$\supp(a,x)=\{a\}\cup\supp(x)$), as expected in binding constructs.
The object map $X\mapsto [\mathbb{A}]X$ extends to an endofunctor
$[\mathbb{A}]\colon \mathsf{Nom} \to \mathsf{Nom}$
sending an equivariant map $f\colon X\to Y$ to the equivariant map $[\mathbb{A}]f\colon [\mathbb{A}]X\to [\mathbb{A}]Y$ given by $\braket{a}x\mapsto \braket{a}f(x)$ for $a\in \mathbb{A}$ and $x\in X$.
\subsection{Nominal Automata}\label{sec:nom-aut}
In this section, we recall two notions of nominal automata earlier
introduced in the literature: non-deterministic orbit-finite automata
(NOFAs)~\cite{BojanczykEA14} and regular non-deterministic nominal
automata (RNNAs)~\cite{SchroderEA17}. The former accept \emph{data
languages} (consisting of finite words over an infinite alphabet)
while the latter accept \emph{bar languages} (consisting of finite
words with , taken modulo
$\alpha$-equivalence).
\begin{notheorembrackets}
\begin{defn}[\cite{BojanczykEA14}]
(1)~A \emph{NOFA} $A=(Q,R,F)$ is given by an orbit-finite nominal set $Q$
of \emph{states}, an equivariant relation
$R\subseteq Q\times \mathbb{A} \times Q$ specifying \emph{transitions}, and
an equivariant set $F\subseteq Q$ of
\emph{final states}. We write $q \xra{a} q'$ in lieu of $(q,a,q')
\in R$.
\begin{enumerate}\stepcounter{enumi}
\item Given a string
$w=a_1a_2\cdots a_n\in \mathbb{A}^*$ and a state $q\in Q$, a \emph{run} for
$w$ from $q$ is a sequence of transitions
\(
q\xto{a_1}q_1\xto{a_2}\cdots \xto{a_n}q_n.
\)
The run is \emph{accepting} if $q_n$ is final. The state $q$
\emph{accepts} $w$ if there exists an accepting run for $w$ from
$q$. The data language \emph{accepted} by $q$ is given by $\{ w\in \mathbb{A}^*: \text{$q$ accepts $w$}\}$.
\end{enumerate}
\end{defn}
\end{notheorembrackets}
Note that in contrast to \cite{BojanczykEA14} we do not require NOFAs to have an explicit initial state $q_0\in Q$; this is more natural from a coalgebraic point of view. NOFAs are known to be expressively equivalent to \emph{finite memory
automata}~\cite{KaminskiFrancez94}.
\begin{rem}\label{rem:nofa-as-coalgebras}
\begin{enumerate}
\item%
\smnote{In general, we should not start sentences with `recall' if
we don't say \emph{from where} one should this recall; this is the
arrogant mathematicians way of assuming s.th.~about the knowledge
of the reader.} Given an endofunctor $F$ on a category $\mathscr{C}$, an
\emph{$F$-coalgebra} is a pair $(C,c)$ of an object $C$ and a
morphism $c\colon C\to FC$ on $\mathscr{C}$. A \emph{homomorphism} of
$F$-coalgebras from
$(C,c)$ to $(D,d)$ is a morphism $h\colon C\to D$ in
$\mathscr{C}$ such that $d\o h = Fh\o c$.
\item A NOFA corresponds precisely to an
orbit-finite coalgebra
\(
\langle f, \delta\rangle\colon Q\longrightarrow 2\times \pow_{\fs}(\mathbb{A}\times Q)
\)
for the functor on $\mathsf{Nom}$ given by
\[
Q\mapsto \pow_{\fs}(1+\mathbb{A}\times Q)\cong 2\times \pow_{\fs}(\mathbb{A}\times
Q).
\]
In fact, $f\colon Q\to 2$ defines the equivariant set $F\subseteq Q$ of
final states and $\delta\colon Q\to \pow_{\fs}(\mathbb{A}\times Q)$ defines
the transitions via
$q\xto{a}q'$ iff $(a,q')\in \delta(q)$.
\end{enumerate}
\end{rem}
In order to incorporate explicit name binding into the
automata-theoretic setting, we work with \emph{bar strings},
i.e. finite words over the infinite alphabet
\[
\barA := \mathbb{A} \cup \{\newletter a: a\in \mathbb{A}\}.
\]
We denote the nominal set of all bar strings by $\barA^*$, and we
equip it with the group action defined pointwise. The letter
$\newletter a$ is interpreted as binding the name~$a$ to the
right. Accordingly, a name $a\in \mathbb{A}$ is said to be \emph{free} in
a bar string $w\in \barA^*$ if (1)~the letter $a$ occurs in $w$, and
(2)~the first occurrence of $a$ is not preceded by any occurrence of
$\newletter a$. For instance, the name $a$ is free in
$a\newletter aba$ but not free in $\newletter aaba$, while the name
$b$ is free in both bar strings. This yields a natural notion of
$\alpha$-equivalence:
\begin{defn}[$\alpha$-equivalence]\label{def:alpha-fin}
Let $=_\alpha$ be the least equivalence relation on~$\barA^*$ such
that
$x\newletter av =_\alpha x\newletter bw$
for all $a,b\in \mathbb{A}$ and $x,v,w\in \barA^*$ such that
$\braket{a} v = \braket{b} w$.
We denote by $\barA^*/\mathord{=_\alpha}$ the sets of $\alpha$-equivalence classes of
bar strings, and we write $[w]_\alpha$ for the
$\alpha$-equivalence class of $w\in \barA^*$.
\end{defn}
\begin{rem}\label{rem:alpheq-technical}
\begin{enumerate}
\item By Pitts~\cite[Lem.~4.3]{Pitts13}, for every pair
$v,w\in \barA^*$ the condition $\braket{a} v = \braket{b} w$ holds
if and only if
\[
\text{$a=b$ and $v=w$,}
\qquad \text{or}\qquad
\text{$b\mathbin{\#} v$ and $(a\, b)\cdot v = w$.}
\]
\item The equivalence relation $=_\alpha$ is equivariant. Therefore,
$\barA^*/{=_\alpha}$ forms a nominal set with the group
action $\pi\cdot [w]_\alpha = [\pi\cdot w]_\alpha$ for
$\pi\in \mathrm{Perm}(\mathbb{A})$ and $w\in \barA^*$. The least support of
$[w]_\alpha$ is the set of free names of $w$.
\end{enumerate}
\end{rem}
\makeatletter
\newcommand\gobblepars{%
\@ifnextchar\par%
{\expandafter\gobblepars\@gobble}%
{}}
\makeatother
\begin{notheorembrackets}
\begin{defn}[\cite{SchroderEA17}]%
(1)~An \emph{RNNA} $A=(Q,R,F)$ is given by an orbit-finite
nominal set $Q$ of \emph{states}, an equivariant relation
$R\subseteq Q\times \barA \times Q$ specifying \emph{transitions}, and an equivariant set $F\subseteq Q$
of \emph{final states}. We write $q\xto{\sigma}q'$ if
$(q,\sigma,q')\in R$. The transitions are subject to two
conditions:
\begin{enumerate}[label=(\alph*)]
\item \emph{$\alpha$-invariance}: if $q\xto{\scriptnew a}q'$ and
$\braket{a} q'=\braket{b} q''$, then $q\xto{\scriptnew b}q''$.
\item \emph{Finite branching up to $\alpha$-invariance:} For every
$q\in Q$ the sets
\[
\set{(a,q') : q\xto{a}q'}
\qquad\text{and}\qquad
\set{\braket{a} q' : q\xto{\scriptnew a} q'}
\]
are finite (equivalently, uniformly finitely supported).
\end{enumerate}
\begin{enumerate}\stepcounter{enumi}
\item Given a bar string
$w=\sigma_1\sigma_2\cdots \sigma_n\in \barA^*$ and a state
$q\in Q$, a \emph{run} for $w$ from $q$ is a sequence of
transitions
\(
q\xto{\sigma_1}q_1\xto{\sigma_2}\cdots \xto{\sigma_n}q_n.
\)
The run is \emph{accepting} if $q_n$ is final. The state $q$
\emph{accepts} $w$ if there exists an accepting run for $w$ from
$q$. The bar language \emph{accepted} by $q$ is given
by $\{ [w]_\alpha : w\in \barA^*, \, \text{$A$ accepts $w$} \}$.
\end{enumerate}
\end{defn}
\end{notheorembrackets}
\begin{rem}\label{rem:rnna-as-coalgebras}
\begin{enumerate}
As for NOFAs, we do not equip RNNAs with explicit initial states. Similar to \autoref{rem:nofa-as-coalgebras}, RNNAs are seen to correspond to
coalgebras
\(
\langle f, \delta, \tau\rangle\colon Q\longrightarrow 2\times \mathcal{P}_{\mathsf{ufs}}(\mathbb{A}\times Q)\times
\mathcal{P}_{\mathsf{ufs}}([\mathbb{A}]Q)
\)
for the functor on $\mathsf{Nom}$ given by
\[
Q\mapsto \pow_{\ufs}(1+\mathbb{A}\times Q+[\mathbb{A}]Q)
\cong
2\times \pow_{\ufs} (\mathbb{A}\times Q) \times \pow_{\ufs}([\mathbb{A}] Q).
\]
Here $f$ and $\delta$ correspond to final states and free
transitions, and the equivariant map
$\tau\colon Q\to \pow_{\ufs}([\mathbb{A}]Q)$ defines the alpha-invariant
bound transitions via
$
q\xto{\scriptnew{a}} q'$ iff $\braket{a}q'\in
\tau(q)$.
The use of $\pow_{\ufs}$ (in lieu of $\pow_{\fs}$) ensures that if $Q$ is
orbit-finite, then the finiteness conditions in the definition of
an RNNA are met.
\end{enumerate}
\end{rem}
Our goal is to interpret the above ad-hoc definition of the data
languages of a NOFA and the bar languages of an RNNA within the
coalgebraic framework.
\subsection{Initial algebras in $\mathsf{DCPO}_\bot$-enriched categories}
\label{S:DCPOb}
For the Kleisli-style coalgebraic trace semantics we shall make use of
a result which shows that in categories where the hom-sets are
enriched over directed-complete partial orders, the initial algebra and
terminal coalgebra coincide.
Recall that a subset $D \subseteq P$ of a poset $P$ is \emph{directed}
if every finite subset of $D$ has an upper bound in $S$; equivalently,
$D$ is nonempty and for every $x,y \in D$, there exists a $z \in D$
with $x,y \leq z$. The poset $P$ is a \emph{dcpo with bottom} if it
has a least element and \emph{directed joins}, that is, every directed
subset has a join in $P$. We write $\mathsf{DCPO}_\bot$ for the category of dcpos
with bottom and continuous maps between them; a map is
\emph{continuous} if it is monotone and preserves directed joins.
\begin{defn}
\begin{enumerate}
\item A category $\mathscr{C}$ is \emph{left strictly $\mathsf{DCPO}_\bot$-enriched} provided that each
hom-set is equipped with the structure of a dcpo with bottom, and
composition preserves bottom on the left and is \emph{continuous}: for every morphism
$f$ and appropriate directed sets of morphisms $g_i$ ($i \in D$)
we have
\[
\textstyle
\bot \cdot f = \bot, \qquad
f \cdot \bigvee_{i\in D} g_i = \bigvee_{i\in D} f \cdot g_i,\qquad
\big(\bigvee_{i \in D} g_i\big) \cdot f = \bigvee_{i\in D} g_i
\cdot f.
\]
\item A functor on $\mathscr{C}$ is \emph{locally monotone} if its
restrictions $\mathscr{C}(A,B) \to \mathscr{C}(FA,FB)$ to the hom-sets are monotone.
\end{enumerate}
\end{defn}
\begin{notheorembrackets}
\begin{theorem}[{\cite[Prop.~5.6]{amm21}}]\label{T:DCPOb}\label{T:dcpo}
%
Let $F$ be a locally monotone functor on a left strictly $\mathsf{DCPO}_\bot$-enriched category.
If an initial algebra $(\mu F, \iota)$ exists, then $(\mu F, \iota^{-1})$ is a
terminal coalgebra.
\end{theorem}
\end{notheorembrackets}
\noindent
This result is an adaptation of an earlier related result proved by
Freyd~\cite{Freyd92} for locally continuous functors on
$\omega$-cpo-enriched categories. Note that preservation of bottom on
the right ($f \cdot \bot = \bot$) is not needed for this result.
\section{Coalgebraic Trace Semantics}
\label{S:trace}
In this section we shall see that the (bar) language semantics of NOFAs
and RNNAs is an instance of coalgebraic trace semantics. To this end we
first adapt and generalize the coalgebraic trace semantics for set
functors by Hasuo et al.~\cite{HasuoEA07} to arbitrary
categories. Here one considers coalgebras for composed functors~$TF$,
where $T$ is a monad modelling a branching type like non-determinism
or probabilistic branching, and $F$ models the type of transitions of
systems. We then instantiate this to coalgebras in $\mathsf{Nom}$ for functors
$TF$, where $T$ is $\pow_{\fs}$ and $F$ a polynominal functor or
$T=\pow_{\ufs}$ and $F$ a binding polynomial functor. Specifically, we
obtain the two desired types of nominal automata as instances.
\subsection{General Coalgebraic Trace Semantics Revisited}
We begin by recalling a few facts about extensions of functors to
Kleisli categories.
\begin{rem}\label{R:Kleisli}
Let $F$ be a functor and $(T, \eta, \mu)$ a monad both on
the category~$\mathscr{C}$.
\begin{enumerate}
\item\label{R:Kleisli:1} The \emph{Kleisli category} $\Kl T$ has the
same objects as $\mathscr{C}$ and a morphisms $f$ from $X$ to $Y$ is a
morphism $f\colon X \to TY$ of $\mathscr{C}$. The composition of $f$ with
$g\colon Y \to TZ$ is defined by $\mu_Z \cdot Tg \cdot f$ and the
identity on $X$ is $\eta_X \colon X \to TX$. We have the
identity-on-objects functor $J\colon \mathscr{C} \to \Kl T$ defined by
$J(f\colon X \to Y) = \eta_Y \cdot f$.
\item An endofunctor $\mybar{0.6}{2.5pt}{F}\colon \Kl T \to \Kl T$ \emph{extends}
the functor $F$ if $\bar F J = J F$. It is well-known and easy to
prove (see Mulry~\cite{Mulry94}) that extensions of $F$ to $\Kl T$ are in
bijective correspondence with \emph{distributive laws} of $F$ over
$T$; these are natural transformations $\lambda\colon FT \to
TF$ compatible with the monad structure of~$T$:
\[
\begin{tikzcd}
F
\arrow{r}{F\eta}
\arrow{rd}[swap]{\eta F}
&
FT
\arrow{d}{\lambda}
\\
&
TF
\end{tikzcd}
\qquad\qquad
\begin{tikzcd}
FTT
\arrow{r}{\lambda T}
\arrow{d}[swap]{F\mu}
&
TFT
\arrow{r}{T\lambda}
&
TTF
\arrow{d}{\mu F}
\\
FT
\arrow{rr}{\lambda}
&&
TF
\end{tikzcd}
\]
\item\label{R:Kleisli:3} Let $G$ be a quotient functor of $F$, which means that we have
a natural transformation with epimorphic components
$q\colon F \twoheadrightarrow G$. Suppose that $F$ extends to $\Kl T$
via a distributive law $\lambda\colon FT \to TF$. Then an
object-indexed family of morphisms $\rho_X\colon GTX \to TGX$ is a
distributive law of $G$ over $T$ provided that the following
squares commute
\[
\begin{tikzcd}
FTX
\ar{r}{\lambda_X}
\ar[->>]{d}[swap]{q_{TX}}
&
TFX
\ar[->>]{d}{Tq_X}
\\
GTX
\ar{r}{\rho_X}
&
TGX
\end{tikzcd}
\qquad
\text{for every object $X$ of $\mathscr{C}$.}
\]
\end{enumerate}
\end{rem}
\begin{expl}\label{E:ext}
\begin{enumerate}
\item\label{E:ext:1} Constant functors and the identity functor on $\mathscr{C}$ obviously
extend to $\Kl T$.
\item\label{E:ext:2} Suppose that $\mathscr{C}$ has coproducts. Then for a coproduct $F+G$
of functors $F$ and $G$ one uses that $J\colon \mathscr{C} \to \Kl T$
preserves coproducts. Given extensions $\mybar{0.6}{2.5pt}{F}$ and $\mybar{0.6}{2pt}{G}$, it is
then clear that $\mybar{0.6}{2.5pt}{F} + \mybar{0.6}{2pt}{G}$ extends $F+G$:%
\smnote{We will need the definition on morphisms in the proof of
\autoref{T:RNNA-Kl}; so we need to provide more details.}
for every morphism $f\colon X \to TY$ in $\Kl T$ one has
\[
\mybar{0.9}{2pt}{F + G} (f) = \big(FX + FY \xra{\bar Ff + \bar Gf} TFY+TGY
\xra{[T\mathsf{inl}, T\mathsf{inr}]} T(FY+GY)\big),
\]
where $FY \xra{\mathsf{inl}} FY + GY \xla{\mathsf{inr}} GY$ are the coproduct
injections. This works similarly for arbitrary coproducts.
\item\label{E:ext:3} Suppose that $\mathscr{C}$ has finite products. Then
finite products of functors with an extension can be extended when
the monad $T$ is \emph{commutative}; this notion was introduced by
Kock~\cite[Def.~3.1]{Kock70}. It is based on the notion of a
\emph{strong} monad, that is a monad $T$ equipped with a natural
transformation $s_{X,Y}\colon X \times TY \to T(X\times Y)$
(called \emph{strength}) satisfying four natural equational laws
(two w.r.t.~to~$1$ and~$\times$ on $\mathscr{C}$ and two w.r.t.~the monad
structure). We do not recall these laws explicitly since they are
not needed for our exposition. A strength gives rise to a
\emph{costrength} $t_{X,Y}\colon TX \times Y \to T(X\times Y)$
defined by
\[
t_{X,Y} = \big( TX\times Y\cong Y \times TX \xra{s_{Y,X}} T(Y
\times X) \xra{T(\cong)} T(X\times Y) \big).
\]
The monad $T$ is \emph{commutative} if the following diagram
commutes:
\[
\begin{tikzcd}[row sep = 0, column sep = 30]
&
T(TX \times Y)
\ar{r}{Tt_{X,Y}}
&
TT(X\times Y)
\ar{rd}[near start]{\mu_{X\times Y}}
\\
TX \times TY
\ar{ru}[near end]{s_{TX,Y}}
\ar{rd}[near end, swap]{t_{X,TY}}
\ar[dashed]{rrr}{d_{X,Y}}
&&&
T(X\times Y)
\\
&
T(X \times TY)
\ar{r}{Ts_{X,Y}}
&
TT(X \times Y
\ar{ru}[near start,swap]{\mu_{X\times Y}}
\end{tikzcd}
\]
The ensuing natural transformation $d$ in the middle is used to
extend the product $F \times G$ of endofunctors on $\mathscr{C}$
having extensions $\mybar{0.6}{2.5pt}{F}$ and $\mybar{0.6}{2pt}{G}$ on $\Kl T$: for every
morphism $f\colon X \to TY$ in $\Kl T$ one puts%
\smnote{Note that the barF and barG macros don't work on the label
arrows; TODO: perhaps fix this later.}
\[
\mybar{0.9}{2pt}{F \times G} (f) = \big(FX \times GX \xra{\bar F f \times \bar G f} TFY
\times TGY \xra{d_{FY,GY}} T(FY \times GY)\big).
\]
\end{enumerate}
\end{expl}
\begin{rem}
\begin{enumerate}
\item Every set monad is strong via a canonical strength; this
follows, for example, from Moggi's
result~\cite[Thm.~3.4]{Moggi91}. Commutative set monads are those
in whose algebraic theory every operation is a homomorphism. For
example, the power-set functor $\mathcal{P}\colon \mathbf{Set} \to \mathbf{Set}$ is
commutative via its canonical strength having the components
\begin{equation}\label{eq:pstr}
s_{X,Y}\colon X \times \mathcal{P} Y \to \mathcal{P}(X \times Y)
\quad
\text{defined by}
\quad
(x,S) \mapsto \set{(x,s) : s \in S}.
\end{equation}
\item As a consequence of what we saw in \autoref{E:ext} every
polynomial set functor has a canonical extension to the Kleisli
category of any commutative set monad
(cf.~\cite[Lem.~2.4]{HasuoEA07}).
\item More generally, this results holds for analytic set
functors~\cite[Thm.~2.9]{mps09}. That notion was introduced by
Joyal~\cite{Joyal81,Joyal86}, and he proved that analytic set
functors are precisely those set functors which weakly preserve wide
pullbacks.
\end{enumerate}
\end{rem}
With the help of Hermida and Jacobs' result~\cite[Thm.~2.14]{HermidaJ98} on
extending adjunctions to categories of algebras one easily obtains the
following extension result for initial algebras:
\begin{proposition}\label{P:ini}
Let $T$ be a monad on the category $\mathscr{C}$ and let $F\colon \mathscr{C} \to \mathscr{C}$
have an extension $\bar F$ on $\Kl T$. If $(\mu F, \iota)$ is an
initial $F$-algebra, then $\mu F$ is an initial $\mybar{0.6}{2.5pt}{F}$-algebra with
the structure $J\iota = \eta_{\mu F} \cdot \iota\colon \mu F
\to TF(\mu F)$.
\end{proposition}
Coalgebraic trace semantics can be defined when the extended initial
algebra above is also a terminal coalgebra for $\mybar{0.6}{2.5pt}{F}$.
\begin{theorem}\label{T:Kl}
Let $F$ be a functor and $T$ a monad on the category
$\mathscr{C}$. Assume that $\Kl T$ is left strictly $\mathsf{DCPO}_\bot$-enriched and that $F$ has a locally monotone extension
$\mybar{0.6}{2.5pt}{F}$ on $\Kl T$ and an
initial algebra $(\mu F, \iota)$. Then $(\mu F, J\iota^{-1})$ is a terminal
coalgebra for $\mybar{0.6}{2.5pt}{F}$.
\end{theorem}
\begin{proof}
Immediate from \autoref{P:ini} and \autoref{T:dcpo}.
\qed
\end{proof}
\noindent
Compared to the previous result for $\mathbf{Set}$~\cite[Thm.~3.3]{HasuoEA07}
our assumption on the enrichment of the Kleisli category is slightly
stronger; in op.~cit.~only enrichment in $\omega$-cpos is required. A
related result~\cite[Thm.~5.3.4]{Jacobs16} for general
base categories uses enrichment in directed-complete partial
orders. However, in contrast to both of these results, we do not
require that $\Kl T$ has a zero object and, most notably, we only need
the mere existence of $\mu F$ and not that the initial algebra for $F$
is obtained by the first $\omega$ steps of the initial-algebra chain,
that is, as the colimit of the $\omega$-chain given by $F^n 0$
($n < \omega$). The technical reason for this is that the proof of
\autoref{T:dcpo} does not make use of the classical limit-colimit
coincidence technique used e.g.~by Smyth and Plotkin in their seminal
work~\cite{SmythPlotkin82}. Consequently, our proof is easier and
shorter than the previous ones.
\begin{definition}[Coalgebraic Trace Semantics]\label{D:tr}
Given $F$ and $T$ on $\mathscr{C}$ satisfying the assumptions in
\autoref{T:Kl} and a coalgebra $c\colon X \to TFX$. The
\emph{coalgebraic trace map} is the unique coalgebra homomorphism
$\mathsf{tr}_c$ from $(X, c)$ to $(\mu F, J\iota^{-1})$; that is, the
following diagram commutes in $\Kl T$:
\begin{equation}\label{eq:trc}
\begin{tikzcd}[column sep = 30, row sep = 15]
X \ar{d}[swap]{c} \ar{r}{\mathsf{tr}_c}
&
\mu F
\ar{d}{J\iota^{-1}}
\\
\mybar{0.6}{2.5pt}{F} X
\ar{r}{\bar F \mathsf{tr}_c}
&
\mybar{0.6}{2.5pt}{F}(\mu F)
\end{tikzcd}
\end{equation}
\end{definition}
Among the instances of coalgebraic trace semantics are the trace
semantics of labelled transition systems with explicit
termination~\cite{HasuoEA07}, which are the coalgebras for the set
functor $\mathcal{P}(1 +\Sigma \times X)$ and that of labelled probabilistic
labelled transitions systems~\cite[Ch.~4]{Hasuo08}, which are the
coalgebras for the set functor $\mathcal D_\leq(1 + \Sigma \times X)$,
where $\mathcal D_\leq$ denotes the subdistribution monad.
\subsection{Coalgebraic Trace Semantics of Non-deterministic Nominal
Systems}
We will now work towards showing that the semantics of nominal
automata is an instance of the coalgebraic trace semantics. To this
end we will instantiate \autoref{T:Kl} to $\mathscr{C}=\mathsf{Nom}$,
$FX=1+\mathbb{A}\times X$ and $T=\pow_{\fs}$ (for NOFAs), or to
$FX = 1 + \mathbb{A} \times X + [\mathbb{A}] X$ and $T = \pow_{\ufs}$ (for RNNAs),
\newcommand{\remautorefname}{Remarks}%
cf.~\autoref{rem:nofa-as-coalgebras} and~\ref{rem:rnna-as-coalgebras}.
\renewcommand{\remautorefname}{Remark}%
More generally, we show that every endofunctor arising from a nominal
algebraic signature in the sense of Pitts~\cite[Def.~8.2]{Pitts13} has
a locally monotone extension to $\Kl\pow_{\ufs}$. For $T = \pow_{\fs}$ most
of the development works out, as we shall see. However, the
distributive law for the abstraction functor in the proof of
\autoref{P:abs-dist} is not well-defined for $\pow_{\fs}$.
But the first obstacle is that the nominal sets $\pow_{\fs} X$ and
$\pow_{\ufs} X$ are in general no complete lattices (and not even
$\omega$-cpos) since the union of a chain of (uniformly) finitely
supported sets may fail to be (uniformly) finitely supported.%
\smnote{TODO: We need to mention a concrete counterexampel (in the
appendix)!} In this light, the following result is slightly
surprising.
\begin{proposition}\label{P:dcpo}
For every pair $X,Y$ of nominal sets, the sets $\Kl\pow_{\fs}(X,Y)$ and $\Kl\pow_{\ufs}(X,Y)$
form complete lattices (whence dcpos with bottom).
\end{proposition}
\begin{corollary}\label{C:ini-ter}
If a locally monotone endofunctor $H$ on $\Kl\pow_{\fs}$ or $\Kl\pow_{\ufs}$ has an initial
algebra $(\mu H, \iota)$, then $(\mu H, \iota^{-1})$ is its terminal
coalgebra.
\end{corollary}
\noindent
This is a consequence of~\autoref{T:DCPOb} since the composition in
$\Kl\pow_{\fs}$ and $\Kl\pow_{\ufs}$ is easily seen to preserve the bottom
(empty set)
on the left and all joins (unions).
\mysubsec{Extending functors to $\Kl\pow_{\ufs}$.} We now show that
endofunctors arising from a nominal algebraic signature (with one
name and one data sort) have a canonical locally monotone extension
to $\Kl\pow_{\ufs}$. As an instance, we then obtain that the functor $F$
used for RNNAs has a locally monotone extension $\mybar{0.6}{2.5pt}{F}$ on
$\Kl\pow_{\ufs}$.
\begin{definition}
The class of \emph{binding polynomial functors} on $\mathsf{Nom}$ is the smallest
class of functors containing the constant, identity and abstraction
functors and being closed under finite products and coproducts.
\end{definition}
\noindent In other words, binding polynomial functors are formed
according to the following grammar:
\begin{equation}\label{eq:grammar}
\textstyle
F ::= C \mid \mathsf{Id} \mid [\mathbb{A}](-) \mid F \times F \mid \coprod_{i\in
I} F_i,
\end{equation}
where $C$ ranges over all constant functors on $\mathsf{Nom}$ and $I$ is an
arbitrary index set. These functors are precisely the functors
associated to a nominal algebraic signature with one name sort and one data
sort (see Pitts~\cite[Def.~8.12]{Pitts13}).
\begin{proposition}\label{P:comm}\smnote{Aren't there any results known in the literature that the
power object monad on a topos (satisfying s.th.) is commutative?}%
The monads $\pow_{\fs}$ and $\pow_{\ufs}$ are commutative.
\end{proposition}
\begin{proposition}\label{P:abs-dist}
The abstraction functor $[\mathbb{A}](-)$ has a locally monotone
extension on $\Kl \pow_{\ufs}$.
\end{proposition}
\begin{proof}[Sketch]
One uses \autoref{R:Kleisli}\ref{R:Kleisli:3}: the abstraction
functor is a quotient of the functor $FX = \mathbb{A} \times X$ which is
equipped with the canonical distributive law
$\lambda_X\colon \mathbb{A} \times \pow_{\ufs} X \to \pow_{\ufs}(\mathbb{A} \times
X)$ defined as in~\eqref{eq:pstr}. The maps $\rho_X\colon
[\mathbb{A}](\pow_{\ufs} X) \to \pow_{\ufs}([\mathbb{A}] X)$ are defined by
$\rho_X (\braket a S) = \set{\braket a s : s \in S}$.
\qed
\end{proof}
\begin{rem}
\smnote{Das ist Ex.~4.5 aus Üsames Projektarbeit.}%
For the monad $\pow_{\fs}$ our proof does not work. The problem is
that~$\rho_X$ above is not well-defined in general if $S$ is
not uniformly finitely supported. For example, for
$\mathbb{A} \in \pow_{\fs} \mathbb{A}$ we have
$\braket a \mathbb{A} = \braket b \mathbb{A}$ for every pair $a, b$ of
names. However, if $a \neq b$, then the sets
$\set{\braket a c : c \in \mathbb{A}}$ and
$\set{\braket b c : c \in \mathbb{A}}$ differ: $\braket a b$ is
contained in the former but not in the latter set. In fact, since $a \neq b$,
$\braket a b = \braket b c$ can hold only if $a\mathbin{\#}\set{b,c}$ and
$b = (a\, b)\cdot c$ (see Pitts~\cite[Lem.~4.3]{Pitts13}). The
latter means that $c = a$ contradicting freshness of $a$ for it.
\end{rem}
\begin{corollary}\label{cor:extension-pufs}
Every binding polynomial functor has a canonical locally monotone extension to
$\Kl\pow_{\ufs}$.
\end{corollary}
Unsurprisingly, an analogous result holds for polynomial functors and
$\pow_{\fs}$ by the same reasoning applied to a grammar as in~\eqref{eq:grammar}
that does not include the abstraction functor:
\begin{corollary}\label{cor:extension-pfs}
Every polynomial functor has a canonical locally monotone extension to
$\Kl\pow_{\fs}$.
\end{corollary}
\mysubsec{Nominal Coalgebraic Trace Semantics.} From
Pitts~\cite[Thm.~8.15]{Pitts13} we know that every binding polynomial
functor has an initial algebra carried by the nominal set $T_F$ of
terms modulo $\alpha$-equivalence (defined in Def.~8.6 of op.~cit.) of
the nominal algebraic signature associated to $F$. If $F$ is
polynomial, then $\alpha$-equivalence is trivial and $T_F$ the
usual set of terms. By \autoref{C:ini-ter} we have
\begin{corollary}
\begin{enumerate}
\item For every polynomial functor $F$ the terminal coalgebra of its canonical extension $\mybar{0.6}{2.5pt}{F}$ on $\Kl\pow_{\fs}$ is carried by the nominal set $T_F$.
\item For every binding polynomial functor $F$ the terminal coalgebra of
its canonical extension $\mybar{0.6}{2.5pt}{F}$ on $\Kl\pow_{\ufs}$ is carried by the
nominal set $T_F$.
\end{enumerate}
\end{corollary}
\noindent
According to \autoref{D:tr} we can thus define a coalgebraic trace
semantics for every coalgebra $X \to \pow_{\fs} FX$ with $F$ a polynomial functor, as well as for every coalgebra $X\to \pow_{\ufs} FX$ with $F$ a binding
polynomial functor. We now instantiate this to the two types of nominal automata introduced in
\autoref{sec:nom-aut}.
\mysubsec{Coalgebraic Trace Semantics of NOFAs.} Recall from
\autoref{rem:nofa-as-coalgebras} that NOFAs are coalgebras
$X\to \pow_{\fs} F X$ where $FX=1+\mathbb{A}\times X$ on $\mathsf{Nom}$.
\begin{proposition}\label{prop:ini-nofa}
The initial algebra for $F$ is the nominal set $\mathbb{A}^*$ with
structure $\iota\colon 1+\mathbb{A}\times \mathbb{A}^* \to \mathbb{A}^*$ defined
by $\iota(\ast)=\epsilon$ and $\iota(a,w)=aw$.
\end{proposition}
Indeed, the functor $F$ arises from from the algebraic signature with
a constant $\epsilon$ and unary operations $a(-)$ for every
$a \in \mathbb{A}$, and clearly the corresponding term algebra is
isomorphic to the algebra $\mathbb{A}^*$.
\begin{corollary}\label{C:ini}
The terminal coalgebra for the extension
$\mybar{0.6}{2.5pt}{F}\colon \Kl\pow_{\fs} \to \Kl\pow_{\fs}$ is $(\mathbb{A}^*, J\iota^{-1})$
for $\iota$ from~\autoref{prop:ini-nofa}.
\end{corollary}
\begin{theorem}\label{T:NOFA-Kl}
For every NOFA $c\colon X \to \pow_{\fs} FX$ its coalgebraic trace map
$\mathsf{tr}_c\colon X \to \pow_{\fs}(\mathbb{A}^*)$ assigns to every state of $X$
its accepted data language.
\end{theorem}
\mysubsec{Coalgebraic Trace Semantics of RNNAs.} Recall from
\autoref{rem:rnna-as-coalgebras} that RNNAs are coalgebras
$X \to \pow_{\ufs} FX$ where $FX = 1 + \mathbb{A} \times X + [\mathbb{A}] X$ on
$\mathsf{Nom}$.
\begin{proposition}\label{prop:ini}
The initial algebra for $F\colon \mathsf{Nom} \to \mathsf{Nom}$ is the nominal set
$\barA^*/\mathord{=_\alpha}$ of all bar strings modulo $\alpha$-equivalence with the
algebra structure $\iota\colon 1 +\mathbb{A} \times (\barA^*/\mathord{=_\alpha}) +
[\mathbb{A}](\barA^*/\mathord{=_\alpha}) \to \barA^*/\mathord{=_\alpha}$ defined by
\begin{equation}\label{eq:iniF}
\iota(*)= [\varepsilon]_\alpha,
\qquad
\iota(a,[w]_\alpha) = [aw]_\alpha,
\qquad
\iota(\braket a [w]_\alpha) = [|aw]_\alpha.
\end{equation}
\end{proposition}
\noindent
Indeed, the functor $F$ arises from a nominal algebraic signature with
a constant~$\varepsilon$, unary operations $a(-)$ for every $a \in \mathbb{A}$
and one unary name binding operation $\newletter{}$. Terms over this
signature are obviously the same as bar strings. Moreover, it is not
difficult to show that Pitts' notion of $\alpha$-equivalence for
terms~\cite[Def.~8.6]{Pitts13} is equivalent to $\alpha$-equivalence
for bar strings in~\autoref{def:alpha-fin}. Finally, the algebra
structure in~\eqref{eq:iniF} above corresponds to the one given by term formation by
Pitts~\cite[Thm.~8.15]{Pitts13}. Using \autoref{T:Kl} we thus
obtain the following result.
\begin{corollary}
The terminal coalgebra for the extension
$\mybar{0.6}{2.5pt}{F}\colon \Kl\pow_{\ufs} \to \Kl\pow_{\ufs}$ is $(\barA^*/\mathord{=_\alpha}, J\iota^{-1})$
for $\iota$ from~\eqref{eq:iniF}.\smnote{The next overfull goes away
if we switch the document from `draft' to `final'!}%
\end{corollary}
\begin{theorem}\label{T:RNNA-Kl}
For every RNNA $c\colon X \to \pow_{\ufs} FX$ its coalgebraic trace map
$\mathsf{tr}_c\colon X \to \pow_{\ufs}(\barA^*/\mathord{=_\alpha})$ assigns to every state of $X$
its accepted bar language.
\end{theorem}
\section{Coalgebraic Language Semantics}
\label{S:lang}
In this section we shall see that the language semantics of NOFAs
and RNNAs is an instance of coalgebraic language
semantics~\cite{bms13}. The latter is based on the generalized
determinization construction by Silva et al.~\cite{SilvaEA13}. Here
one considers coalgebras for a functor $GT$, where $T$ models a
branching type and $G$ models the type of transition of a system
(similarly as before in the coalgebraic trace semantics, but this time
the order of composition is reversed). Again, we will apply this to
coalgebras in $\mathsf{Nom}$ for functors $GT$, where $T
= \pow_{\fs}$ and $G$ is functor composed of products and exponentials, or to $T= \pow_{\ufs}$ and $G$ composed of products, exponentials and binding functors. Specifically, we obtain the two desired types
of nominal automata as instances.
\subsection{A Recap of General Coalgebraic Language Semantics}
\label{S:lan}
We begin by recalling a few fact about liftings of functors to
Eilenberg-Moore categories.
\begin{rem}\label{R:lift}
Let $G$ be a functor and $(T,\eta,\mu)$ be a monad
on the category $\mathscr{C}$.
\begin{enumerate}
\item The \emph{Eilenberg-Moore} category $\EM T$ consists of
algebras $(A,a)$ for $T$, that is, pairs formed by an object $A$ and a
morphism $a\colon TA \to A$ such that $a\cdot \eta_A=\mathit{id}_A$ and $a \cdot \mu_A = a
\cdot Ta$. A morphism in $\EM T$ from $(A,a)$ to $(B,b)$ is a
morphism $h\colon A \to B$ of $\mathscr{C}$ such that $h \cdot a = b \cdot
Th$. We write $U\colon \EM T \to \mathscr{C}$ for the forgetful functor
mapping an algebra $(A,a)$ to its underlying object $A$.
\item A \emph{lifting} of $G$ is an endofunctor $\widehat{G} \colon \EM T \to \EM
T$ such that $GU = U\widehat{G}$. As shown by Applegate~\cite{Applegate65} (see
also Johnstone~\cite{Johnstone75}), liftings of $G$ to $\EM T$ are in
bijective correspondence with distributive laws of $T$ over
$F$. The latter are natural transformations $\lambda\colon TG \to
GT$ compatible with the monad structure:
\[
G\eta = \lambda \cdot \eta G,
\qquad
\lambda \cdot \mu G = G\mu \cdot \lambda T \cdot T\lambda.
\]
\item\label{R:lift:3} Suppose that $G$ has the lifting $\widehat{G}$ on $\EM T$ via the
distributive law $\lambda$. It
follows from the work of Turi and Plotkin~\cite{TuriP97} (see also
Bartels~\cite[Thm.~3.2.3]{Bartels04}) that the terminal coalgebra for $G$ lifts to
a terminal coalgebra for $\widehat{G}$. In fact, one obtains a
canonical structure of a $T$-algebra on $\nu G$ by
taking the unique coalgebra homomorphism
$\alpha$ in the diagram below:
\[
\begin{tikzcd}
T(\nu G)
\ar{r}{T\tau}
\ar{d}[swap]{\alpha}
&
TG(\nu G)
\ar{r}{\lambda_{\nu G}}
&
GT(\nu G)
\ar{d}{G\alpha}
\\
\nu G
\ar{rr}{\tau}
&&
G(\nu G)
\end{tikzcd}
\]
It is then easy to prove that $\alpha$ is indeed the structure of an
algebra for $T$ and that $\tau\colon \nu G \to G(\nu G)$ is a
homomorphism of Eilenberg-Moore algebras (in fact, this is
expressed by the commutativity of the above diagram). Moreover,
$(\nu G, \tau)$ is the terminal $\widehat{G}$-coalgebra.
\end{enumerate}
\end{rem}
\begin{proposition}\label{P:adj-trans}
Let $T\colon \mathscr{C} \to \mathscr{C}$ be a monad and $L \dashv R\colon \mathscr{C} \to \mathscr{C}$
an adjunction with the counit $\varepsilon\colon LR \to \mathsf{Id}$. Given a distributive law
$\lambda\colon LT \to TL$, we obtain a
distributive law $\rho\colon TR \to RT$ as the adjoint
transpose of
\mbox{\(
LTR \xra{\lambda R} TLR \xra{T\varepsilon} T.
\)}
\end{proposition}
\begin{expl}\label{E:lift}
\begin{enumerate}
\item The identity functor on $\mathscr{C}$ obviously
lifts to $\EM T$, and so does a constant functor on the carrier object
of an Eilenberg-Moore algebra for~$T$.
\item Suppose that $\mathscr{C}$ has products. Then for a product $F \times
G$ of functors one uses that $U$ preserves products. Given
liftings $\widehat{F}$ and $\widehat{G}$, it is clear that $\widehat{F} \times
\widehat{G}$ is a lifting of $F\times G$. This works similarly for
arbitrary products.
\item Suppose that $\mathscr{C}$ is cartesian closed and that the monad $T$
is strong (cf.~\autoref{E:ext}\ref{E:ext:3}). Then the
exponentiation functor $(-)^A$ lifts to $\EM T$ for every object
$A$ of $\mathscr{C}$. In fact, we apply \autoref{P:adj-trans} to the
adjunction $A \times (-) \dashv (-)^A$ and use that two of the
axioms of the strength $A \times TX \to T(A \times X)$ state that
it is a distributive law of $A\times (-)$ over $T$.
\takeout
it is an easy exercise
(see~\cite[Exercise~5.2.16]{Jacobs16}) to prove that one obtains a
distributive law $\lambda$ with the component at an object $X$
given by currying the following morphisms
\[
Y \times T(X^Y) \xra{s_{Y,X^Y}} T(Y \times X^Y) \xra{T\mathsf{ev}} TX,
\]
where $s$ is the strength of $T$ and $\mathsf{ev}\colon Y \times X^\mathscr{Y} \to
X$ is the evaluation morphism (a component of the counit of the
adjunction $Y \times (-) \dashv (-)^Y$.
\end{enumerate}
\end{expl}
\begin{rem}\label{R:hom-ext}
Recall that, for every monad $(T,\eta,\mu)$ on $\mathscr{C}$, the pair
$(TX,\mu_X)$ is the free algebra for $T$ on $X$ with the
universal morphism $\eta_X\colon X \to TX$. Given an Eilenberg-Moore
algebra $(A,a)$ for $T$ and a morphism $f\colon X \to A$ in
$\mathscr{C}$, we have a unique morphism $f^\sharp\colon (TX,\mu_X) \to
(A,a)$ in $\EM T$ such that $f^\sharp \cdot \eta_X
= f$. We call $f^\sharp$ the \emph{homomorphic extension} of $f$
\end{rem}
\begin{construction}[Generalized Determinization~\cite{SilvaEA13}]\label{C:gen-det}
Let $T$ be a monad on the category $\mathscr{C}$ and $G$ an endofunctor on
$\mathscr{C}$ having a lifting $\widehat{G}$ on $\EM T$. Given a coalgebra
$c\colon X \to GTX$ its \emph{(generalized) determinization} is the
$G$-coalgebra obtained by taking the homomorphic extension
$c^\sharp\colon TX \to GTX$ using that $\mybar{0.6}{2pt}{G}(TX, \mu_X)$ is an
algebra for $T$ carried by $GTX$.
\end{construction}
\takeout
\begin{rem}\label{R:gen-det}
It is straightforward to establish that for every homomorphism
$h\colon (X,c) \to (Y,d)$ we obtain that $Th\colon (TX,c^\sharp) \to
(TY,d^\sharp)$ is a homomorphism of coalgebras for $\widehat{G}$. It
follows that generalized determinization
is the object assigment of a functor $D\colon \Coalg TF \to \Coalg
\widehat{G}$.
\end{rem}
\noindent
Among the instances of this construction are the well-known power-set construction of
deterministic automata~\cite{SilvaEA13} as well as the non-determinization of alternating
automata and that of Simple Segala systems~\cite{JacobsEA15}.
\begin{defn}[Coalgebraic Language Semantics~\cite{bms13}]\label{D:coalg-lan-sem}
Given $T$, $G$ and a coalgebra $c\colon X \to GTX$ as in \autoref{C:gen-det}, the
\emph{coalgebraic language morphism} $\ddagger c\colon X \to \nu G$ is the
composite of the unique coalgebra homomorphism $h$ from the
determinization of $(X,c)$ to $\nu G$ with the unit $\eta_X$ of the
monad $T$, which is summarized in the diagram below:
\[
\begin{tikzcd}
X
\ar{d}[swap]{c}
\ar{r}{\eta_X}
\ar[shiftarr = {yshift=15}]{rr}{\ddagger c}
&
TX
\ar{r}{h}
\ar{dl}[swap]{c^\sharp}
&
\nu G
\ar{d}{\tau}
\\
GTX
\ar{rr}{Gh}
&
&
G(\nu G)
\end{tikzcd}
\]
\end{defn}
\noindent
Among the instances of coalgebraic language semantics are, of course,
the language semantics of non-deterministic~\cite{SilvaEA13,JacobsEA15}, weighted and
probabiblistic automata, but also the languages generated by
context-free grammars~\cite{WinterEA13,mpw20}, constructively
$\mathds{S}$-algebraic formal power-series for a semiring $\mathds{S}$ (the
`context-free' weighted languages)~\cite{WinterEA15,mpw20}. Less
direct instances are the languages accepted by machines with extra
memory such as (deterministic) push-down automata and Turing
machines~\cite{gms20}.%
\smnote{Note that weighted and probabilistic automata do not seem to
appear in~\cite{SilvaEA13,JacobsEA15}; so the only source would be
our book.}
\mysubsec{Relation of Coalgebraic Trace and Language Semantics.}
Jacobs et al.~\cite{JacobsEA15} show how the coalgebraic trace
semantics and coalgebraic language semantics are connected in cases
where both are applicable. We give a terse review of this including a
proof (see appendix) of the result of op.~cit.~that we use here.
\begin{assumption}
We assume that $T$ is a monad and $F, G$ are endofunctors, all on
the category $\mathscr{C}$, such that $F$ has the extension $\mybar{0.6}{2.5pt}{F}$ on
$\Kl T$ via the distributive law $\lambda\colon FT \to TF$ and $G$
has the lifting $\widehat{G}$ on $\EM T$ via the distributive law
$\rho\colon TG \to GT$. Moreover, we assume that we have an
\emph{extension} natural transformation $\varepsilon\colon TF \to GT$
compatible with the two distributive laws:
\begin{equation}\label{diag:eps}
\begin{tikzcd}
TFT
\ar{r}{T\lambda}
\ar{d}[swap]{\varepsilon T}
&
TTF
\ar{r}{\mu F}
&
TF
\ar{d}{\varepsilon}
\\
GTT
\ar{rr}{G\mu}
&&
GT
\end{tikzcd}
\qquad\quad
\begin{tikzcd}
TTF
\ar{rr}{\mu F}
\ar{d}[swap]{T\varepsilon}
&&
TF
\ar{d}{\varepsilon}
\\
TGT
\ar{r}{\rho T}
&
GTT
\ar{r}{G \mu}
&
GT
\end{tikzcd}
\end{equation}
\end{assumption}
\begin{rem}\label{R:eps}
\begin{enumerate}
\item\label{R:eps:1} For every object $X$ of $\mathscr{C}$ the morphism $\varepsilon_X$ is
a homomorphism of Eilenberg-Moore algebras for $T$ from
$(TFX, \mu_{FX})$ to $\widehat{G}(TX, \mu_X)$. Indeed, this is precisely
what the commutativity of the diagram on the right
in~\eqref{diag:eps} expresses.
\item\label{R:eps:2} For every coalgebra $c\colon X \to TFX$ the
extension natural transformation yields a coalgebra
$\varepsilon_X \cdot c\colon X \to GTX$, and we take its determinization
$(TX, (\varepsilon_X \cdot c)^\sharp)$. This is the object assignment of
the functor $E\colon \Coalg \mybar{0.6}{2.5pt}{F} \to \Coalg \widehat{G}$ which maps an
$\mybar{0.6}{2.5pt}{F}$-coalgebra homomorphism $h\colon (X,c) \to (Y,d)$ to
$Eh = h^\sharp\colon TX \to TY$, the homomorphic extension of
$h\colon X \to TY$ (in $\mathscr{C}$). One readily proves that $h^\sharp$
is a $\widehat{G}$-coalgebra homomorphism using the naturality of $\varepsilon$
as well as the laws in~\eqref{diag:eps}. Functoriality follows
since $E$ is clearly a lifting of the canonical comparison functor
$\Kl T \to \EM T$; see Jacobs et al.~\cite[Thm.~2]{JacobsEA15} for
the proof, and we include a proof in the appendix for the
convenience of the reader.
\item\label{R:eps:3} We obtain a canonical morphism
$e\colon T(\mu F) \to \nu G$ by applying the functor $E$ to the
coalgebra $J\iota^{-1}\colon \mu F \to TF(\mu F)$
(cf.~\autoref{P:ini}) and taking the unique coalgebra homomorphism
from it to the terminal $\widehat{G}$-coalgebra
(\autoref{R:lift}\ref{R:lift:3}).
\end{enumerate}
\end{rem}
Now recall the coalgebraic trace semantics from \autoref{D:tr}. The
following result follows from Jacobs et al.'s
result~\cite[Prop.~5]{JacobsEA15}.
\begin{proposition}\label{P:relation}
For every coalgebra $c\colon X \to TFX$ we have
\[
\ddagger(\varepsilon_X\cdot c) = \big(X \xra{\mathsf{tr}_c} T(\mu F) \xra{e} \nu G \big).
\]
\end{proposition}
\subsection{Coalgebraic Language Semantics of Nominal Systems}
We will now work towards that the language semantics of nominal
automata is an instance of coalgebraic language semantics. To this end
we will instantiate the results of \autoref{S:lan} to $\mathscr{C}= \mathsf{Nom}$,
$GX = 2 \times X^\mathbb{A}$ and $T = \pow_{\fs}$ (for NOFAs), or to
$GX = 2 \times X^\mathbb{A} \times [\mathbb{A}]X$ and $T= \pow_{\ufs}$ (for
RNNAs). More generally, in the former case we show that certain
polynomial functors $G$ with exponentiation lift to $\EM\pow_{\fs}$, and in the latter
case, certain binding polynomial functors with exponentation lift to
$\EM\pow_{\ufs}$. For our specific instances of interest we show that
the terminal coalgebra $\nu G$ is given by (data or bar)
languages. The desired end result than follows by an application of
\autoref{P:relation}.
The class of functors $G$ we consider are formed according
to the grammar
\begin{equation}\label{eq:grammar-2}
G ::= A \mid \mathsf{Id} \mid [\mathbb{A}](-) \mid \prod_{i \in I} G_i \mid G^N,
\end{equation}
where $A$ ranges over all nominal sets equipped with the structure
$a\colon \pow_{\ufs} A \to A$ of an algebra for the monad $\pow_{\ufs}$, $I$
is an arbitrary index set, and $N$ ranges over all nominal sets.
Every such functor $G$ has a canonical lifting to $\EM \pow_{\ufs}$.
This can be proved by induction over the grammar using
\autoref{E:lift} and
\begin{proposition}\label{P:lift-abst}
The abstraction functor has a canonical lifting to $\EM \pow_{\ufs}$.
\end{proposition}
\begin{proof}
The abstraction functor $[\mathbb{A}](-)$ has a left-adjoint $\mathbb{A} *
(-)$, where $*$ denotes the \emph{fresh product} defined for two
nominal sets $X$ and $Y$ by
\[
X * Y = \set{(x,y) : x \in X,\, y\in Y,\, \supp(x) \cap\supp(y) = \emptyset},
\]
see \cite[Thm 4.12]{Pitts13}. The strength of $\pow_{\ufs}$ restricts to the fresh product; we
have
\[
s_{X,Y}\colon X * \pow_{\ufs} Y \to \pow_{\ufs}(X * Y)
\qquad
(x,S) \mapsto \set{(x,s) : s \in S}.
\]
Indeed, if $\supp x \cap \supp S = \emptyset$, then $\supp x \cap
\supp s = \emptyset$ for every $s\in S$ because $S$ is uniformly finitely supported and thus $\supp s \subseteq
\supp S$. It follows that
$s_{\mathbb{A}, X}\colon \mathbb{A} * \pow_{\ufs} X \to \pow_{\ufs} (\mathbb{A} * X)$
yields a distributive law of $\mathbb{A} *(-)$ over $\pow_{\ufs}$. By
\autoref{P:adj-trans} we thus obtain a distributive law of $\pow_{\ufs}$
over $[\mathbb{A}] (-)$.
\qed
\end{proof}
\begin{corollary}\label{C:lift-nu}
For every functor $G$ according to the grammar in
\eqref{eq:grammar-2} the terminal coalgebra $\nu G$ lifts to a
terminal coalgebra of $\widehat{G}$ on $\EM\pow_{\ufs}$.
\end{corollary}
\noindent
The terminal coalgebra $\nu G$ exists since every such $G$ is an
accessible functor on $\mathsf{Nom}$. This can be shown by induction on the
structure of $G$; for exponention in the induction step one argues
similarly as Wißmann~\cite[Cor.~3.7.4]{Wissmann20} has done for
orbit-finite sets: an exponentiation functor $(-)^N$ is
$\lambda$-accessible iff the set of orbits of $N$ has cardinality less than $\lambda$.
\smnote{Or is there a better reference for this?}
Now use \autoref{R:lift}\ref{R:lift:3}.%
Consequently, one can define a coalgebraic language semantics for
every functor $G$ according to the grammar~\eqref{eq:grammar-2}.
\begin{rem}
\begin{enumerate}
\item For $T = \pow_{\fs}$ one has the same results for functors $G$ on
$\mathsf{Nom}$ according to the reduced grammar obtained from the one
in~\eqref{eq:grammar-2} by dropping the abstraction functor
$[\mathbb{A}](-)$. In fact, a functor according to the reduced grammar
has a canonical lifting to $\EM T$ whenever $T$ is a strong monad
on a cartesian closed category (by \autoref{E:lift}).
\item We have dropped the abstraction functor in the previous item
because our proof of \autoref{P:lift-abst} does not work for
$\pow_{\fs}$. The problem is that the strength in~\eqref{eq:pstr} does
not restrict to the fresh product for all finitely supported
subsets. Indeed, even if $\supp(x)$ and $\supp(S)$ are disjoint,
the support of $x$ may not be disjoint from that of every element
$s\in S$, whence $(x,s)$ does not lie in $X * Y$. For example,
take $X = Y = \mathbb{A}$ and $S = \mathbb{A} \setminus \set {a}$ for some
$a \in \mathbb{A}$. Clearly, $\supp(S) = \set{a}$. Thus, for every
$b \neq a$, we see that $(b,S)$ lies in
$\mathbb{A} * \pow_{\fs}\mathbb{A}$. However, while $b \in S$ we do not
have that
$(b,b) \in \mathbb{A} * \mathbb{A} = \set{(a,a') : a,a' \in \mathbb{A},
a\neq a'}$, which means that $s_{\mathbb{A},\mathbb{A}}(b, S)$ does not
lie in $\pow_{\fs}(\mathbb{A} * \mathbb{A})$.
%
\takeout{
%
For the monad $\pow_{\fs}$ the above proof does not work. The problem
is that the isomorphism $\psi_X$ is not a distributive law because
the diagram
\[
\begin{tikzcd}
\pow_{\fs}\powfs({[\mathbb{A}]}X) \ar{rr}{\mu_{[\mathbb{A}]X}}
\ar{d}[swap]{\pow_{\fs}\psi_X}
& &
\pow_{\fs}({[\mathbb{A}]}X) \ar{d}{\psi_X}
\\
\pow_{\fs}({[\mathbb{A}]}(\pow_{\fs} X)) \ar{r}{\psi_{\pow_{\fs} X}}
&
{[\mathbb{A}]}(\pow_{\fs}\powfs X) \ar{r}{[\mathbb{A}]\mu_X}
&
{[\mathbb{A}]}\pow_{\fs} X
\end{tikzcd}
\]
does not commute. Indeed, fix $a \in \mathbb{A}$, then $\mathscr{S} = \set{\set{\braket{a}b}:
b \in \mathbb{A}} \in \pow_{\fs}\powfs({[\mathbb{A}]}\mathbb{A})$ with support $\set{a}$,
while every element of $\mathscr{S}$ has support $\set{b} \setminus \set{a}$.
If the diagram above commuted, the equality
\[
\braket{c}\set{x: \braket{c}x \in S \text{ for some } S \in \mathscr{S}}
=
\braket{d}\set{x: \braket{e}x \in S \text{ for some } S \in \mathscr{S},\ e \mathbin{\#} S}
\]
would need to hold, for any $c$ that is fresh for $\set{\braket{a}b: b \in \mathbb{A}}$
and $d$ that is fresh for $\pow_{\fs}\psi_X\mathscr{S}$. Because of Pitts'
Choose-a-Fresh-Name-Principle~\cite[p.~49]{Pitts13}, we can assume $d = c$,
and $d \neq a$, because $a$ supports $\set{\braket{a}b: b \in \mathbb{A}}$.
But then the equality cannot hold, because the set on left side does not
contain $a \in \mathbb{A}$: Since $d \neq a$, $\braket{d}a = \braket{a}b$ can
hold only if $d \mathbin{\#} \set{a,b}$ and $a = (d\,a)\cdot b$ (see
Pitts~\cite[Lem.~4.3]{Pitts13}). The latter means that $b = d$ contradicting
freshness of $d$ for it. However, the set of the right side does contain
$a \in \mathbb{A}$, since $a \mathbin{\#} \set{\braket{a}a}$.
\end{enumerate}
\end{rem}
\mysubsec{Coalgebraic Language Semantics of NOFAs} We now apply the
previous results to $T = \pow_{\fs}$ and $GX = 2 \times X^\mathbb{A}$.
\begin{rem}\label{R:exp}
We have a canonical isomorphism $\pow_{\fs}(\mathbb{A}
\times X) \cong (\pow_{\fs} X)^\mathbb{A}$ given by $S \mapsto (a \mapsto
\set{x: (a,x) \in S})$. This follows from the fact that
$\pow_{\fs}$ is the power object functor on the topos $\mathsf{Nom}$ and so we
have $\pow_{\fs} X \cong 2^X$.
\end{rem}
Consequently, a NOFA may be regarded as a coalgebra for $G\pow_{\fs}$:
\[
X\to \pow_{\fs}(1+\mathbb{A}\times X)\cong 2\times (\pow_{\fs} X)^A =
G\pow_{\fs} X.
\]
\begin{proposition}\label{prop:ter-nofa}
The terminal coalgebra for $G$ is the nominal set $\pow_{\fs}(\names^*)$ of all
data languages with the structure
\[
\pow_{\fs}(\names^*) \xra{\tau} 2 \times \pow_{\fs}(\names^*)^\mathbb{A},\quad
L \mapsto (b, a \mapsto a^{-1}L),
\]
where $b = 1$ if
$\epsilon\in L$ and $0$ else, and $a^{-1}L=\{w\in \mathbb{A}^* : aw\in L \}$.
\end{proposition}
The proof is analogous to the one that for every alphabet $A$ the set
functor $X\to 2\times X^A$ has the terminal coalgebra $\mathcal{P} (A^*)$,
see e.g.~Rutten~\cite{rutten00}.
We may thus define the coalgebraic language semantics for NOFAs as in
\autoref{D:coalg-lan-sem}.
\begin{rem}
We take $FX = 1 + \mathbb{A} \times X $ as in
\autoref{T:NOFA-Kl} and obtain $\mu F = \mathbb{A}^*$
(\autoref{prop:ini-nofa}) and $\nu G = \pow_{\fs}(\names^*)$
(\autoref{prop:ter-nofa}).
%
Moreover, analogous to ordinary non-deterministic
automata~\cite[Sec.~7.1]{JacobsEA15}, we have an extension natural
transformation
$\varepsilon_X\colon \pow_{\fs}(1 + \mathbb{A} \times X) \to 2\times (\pow_{\fs}
X)^\mathbb{A}$ given by
\[
\varepsilon_X(S) = (b, a \mapsto S_a),
\]
where $b = 1$ iff the element $*$ of $1$ lies $S$ and
$S_a = \set{x : (a,x) \in S}$. The ensuing canonical morphism
$e\colon \pow_{\fs}(\mu F) \to \nu G$ from \autoref{R:eps}\ref{R:eps:3}
is then easily seen to be just the identity map on $\pow_{\fs}(\names^*)$.
\end{rem}
\begin{corollary}\label{C:NOFA-EM}
The coalgebraic language semantics assigns to each state of a NOFA
the data language it accepts.
\end{corollary}
\noindent
Indeed, this follows from \autoref{T:NOFA-Kl} and \autoref{P:relation}
using that in the latter result
$e$ is the identity map on $\pow_{\fs}(\names^*)$.
\mysubsec{Coalgebraic Language Semantics of RNNAs} We now apply the
previous results to $T = \pow_{\ufs}$ and $GX = 2 \times X^\mathbb{A} \times
[\mathbb{A}]X$.
\begin{rem}
\begin{enumerate}
\item The canonical isomorphism from \autoref{R:exp}
restricts to an injection
$i\colon \pow_{\ufs}(\mathbb{A} \times X) \rightarrowtail (\pow_{\ufs}
X)^\mathbb{A}$. Indeed, take a uniformly finitely supported subset
$S \subseteq \mathbb{A} \times X$. Then for every $a \in \mathbb{A}$, every
element $x$ of the set $i(S)(a) = \set{x : (a,x) \in S}$ satisfies
$\supp x \subseteq \set{a} \cup \supp x = \supp (a,x) \subseteq \supp S$
and therefore that set lies in $\pow_{\ufs} X$. However, note that the
inverse of the isomorphism from \autoref{R:exp} does not restrict to
uniformly finitely supported subsets.
\item The components
$\rho_X\colon [\mathbb{A}]\pow_{\ufs} X \to \pow_{\ufs}([\mathbb{A}] X)$ of the distributive law from the
proof of \autoref{P:abs-dist} are in fact isomorphisms with
inverses $\psi_X\colon \pow_{\ufs}([\mathbb{A}] X) \to [\mathbb{A}]\pow_{\ufs} X$
defined by $\psi_X(S) = \braket a \set{x : \braket a x \in S}$,
where $a$ is fresh for $S$. These inverses can also be gleaned
from Pitts' result~\cite[Prop.~4.14]{Pitts13} which shows that the
abstraction functor preserves exponentials specializing to
$\pow_{\fs}([\mathbb{A}] X) \cong[\mathbb{A}]\pow_{\fs} X$. However, note that
$\rho_X$ has a more involved description in the case
of $\pow_{\fs}$.
\end{enumerate}
\end{rem}
It follows that for every nominal set $X$ we have an injection
\begin{equation}\label{eq:inj}
m_X\colon 2 \times \pow_{\ufs}(\mathbb{A} \times X) \times
\pow_{\ufs}([\mathbb{A}] X)
\rightarrowtail
2 \times (\pow_{\ufs} X)^\mathbb{A} \times [\mathbb{A}] (\pow_{\ufs} X).
\end{equation}
Thus every RNNA (\autoref{rem:rnna-as-coalgebras}) may be regarded as
a coalgebra for $G\pow_{\ufs}$.
A description of the terminal coalgebra for $G$ has previously been
given by Kozen et al.~\cite[Thm.~4.10]{KozenEA15}. We provide a
different (or course, isomorphic) description as a final ingredient
for our desired result.
\begin{proposition}\label{prop:ter}
The terminal coalgebra for $G$ is the nominal set $\pow_{\fs}(\barstr)$ of all
bar languages with the structure
\[
\pow_{\fs}(\barstr) \xra{\tau} 2 \times (\pow_{\fs}(\barstr))^\mathbb{A} \times
[\mathbb{A}]\pow_{\fs}(\barstr),\quad
S \mapsto (b, a \mapsto S_a, S_{\scriptnew a}),
\]
where $b = 1$ if
$[\varepsilon]_\alpha \in S$ and $0$ else,
$S_a = \set{[w]_\alpha : [aw]_\alpha \in S}$ and
$S_{\scriptnew a} = \braket a \set{[w]_\alpha : [\newletter a
w]_\alpha \in S}$ for any $a$ which is fresh for $S$.%
\smnote{Is the latter set really correct like this?}
\end{proposition}
We may thus define the coalgebraic language semantics for RNNAs as in
\autoref{D:coalg-lan-sem}.
\begin{rem}\label{R:epsX}
We take $FX = 1 + \mathbb{A} \times X + [\mathbb{A}] X$ as in
\autoref{T:RNNA-Kl} and obtain $\mu F = \barA^*/\mathord{=_\alpha}$
(\autoref{prop:ini}) and $\nu G = \pow_{\fs}(\barstr)$
(\autoref{prop:ter}). We also define a natural
transformation $\varepsilon\colon \pow_{\ufs} F \to G \pow_{\ufs}$ by composing the
canonical isomorphism $\pow_{\ufs}(1 + \mathbb{A} \times X + [\mathbb{A}] X)
\cong 2 \times \pow_{\ufs}(\mathbb{A} \times X) \times \pow_{\ufs}([\mathbb{A}] X)$
with the injection $m_X$ from~\eqref{eq:inj}.
For every uniformly finitely supported subset $S \subseteq 1 +
\mathbb{A} \times X + [\mathbb{A}] X$ we have
\(
\varepsilon_X(S) = (b, a \mapsto S_a, S_{\scriptnew a}),
\)
where $b = 1$ iff the element $*$ of $1$ lies in $S$, $S_a = \set{s :
(a,s) \in S}$ and $S_{\scriptnew a} = \braket a \set{s : \braket a
s \in S}$, where $a$ is fresh for (all elements $\braket b s$ in)
$S$.
\end{rem}
\begin{lemma}\label{L:eps-ext}
The natural transformation $\varepsilon\colon \pow_{\ufs} F \to G \pow_{\ufs}$ is
an extension.
\end{lemma}
\begin{lemma}\label{L:incl}
The canonical morphism $e\colon \pow_{\ufs}(\mu F) \to \nu G$ from
\autoref{R:eps}\ref{R:eps:3} is the inclusion map
$\pow_{\ufs}(\barA^*/\mathord{=_\alpha}) \hookto \pow_{\fs}(\barstr)$.
\end{lemma}
\begin{corollary}\label{C:RNNA-EM}
The coalgebraic language semantics assigns to each state of an RNNA
the bar language it accepts.
\end{corollary}
\noindent
Indeed, this follows from \autoref{T:RNNA-Kl} and \autoref{P:relation}
using that in the latter result
$e\colon\pow_{\ufs}(\barA^*/\mathord{=_\alpha}) \hookto \pow_{\fs}(\barstr)$ is the inclusion map by \autoref{L:incl}.
\section{Conclusions and Future Work}
We have worked out coalgebraic semantics for two species of
non-deterministic automata for data languages:
NOFAs~\cite{BojanczykEA14} and RNNAs~\cite{SchroderEA17}. We have seen
that their semantics arises both as an instance of the Kleisli style
coalgebraic trace semantics and from the Eilenberg-Moore style
coalgebraic language semantics, which is based on generalized
determinization. To see that both semantics coincide we have employed
the results by Jacobs et al.~\cite{JacobsEA15}.
We have also revisited coalgebraic trace semantics in general and
given a new compact proof of the main extension result
for initial algebras in that theory. Our proof avoids assumptions on
the convergence of the initial algebra chain; mere existence of an
initial algebra suffices.
Having provided coalgebraic semantics for non-deterministic nominal
systems makes the powerful toolbox of coalgebraic methods fully
available to those systems. For example, generic constructions like
coalgebraic $\varepsilon$-elimination~\cite{SilvaW13,bmsz15} can be
instantiated to them. Or coalgebraic up-to techniques
starting with the work by Rot et al.~\cite{RotEA13} might lead to new
proof principles and algorithms, cf.~\cite{BonchiPous13}.
Our general extension and lifting results for nominal systems may be
applied to related kinds of systems, e.g.~nominal transition
systems and the coalgebraic study of equivalences for them. Going a
step beyond the standard coalgebraic trace and language
semantics, graded semantics~\cite{DorschEA19} should lead to a nominal spectrum
of equivalences generalizing van Glabbeek's famous linear time -- branching time
spectrum~\cite{Glabbeek01}.
\smnote[inline]{Is there other future work we can think of? Didn't we
discuss something last week?}
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,398 |
Home PublicationsCommentary The GDPR Was Supposed to Boost Consumer Trust. It Has Failed.
The GDPR Was Supposed to Boost Consumer Trust. It Has Failed.
by Daniel Castro June 6, 2019
by Daniel Castro and Eline Chivot June 6, 2019
According to recently released survey data that was collected in November 2018, European trust in the Internet is at its lowest in a decade. These results show that the General Data Protection Regulation (GDPR)—which the EU has touted as the gold standard for data protection rules—has had no impact on consumer trust in the digital economy since it came into force last May. Moreover, these findings suggest that the conventional wisdom among EU policymakers—that more regulation is necessary to spur consumer trust and innovation in the digital economy—is fundamentally flawed and should be abandoned.
Read the full article on European Views.
Daniel Castro is the director of the Center for Data Innovation and vice president of the Information Technology and Innovation Foundation. Mr. Castro writes and speaks on a variety of issues related to information technology and internet policy, including data, privacy, security, intellectual property, internet governance, e-government, and accessibility for people with disabilities. His work has been quoted and cited in numerous media outlets, including The Washington Post, The Wall Street Journal, NPR, USA Today, Bloomberg News, and Businessweek. In 2013, Mr. Castro was named to FedScoop's list of "Top 25 most influential people under 40 in government and tech." In 2015, U.S. Secretary of Commerce Penny Pritzker appointed Mr. Castro to the Commerce Data Advisory Council. Mr. Castro previously worked as an IT analyst at the Government Accountability Office (GAO) where he audited IT security and management controls at various government agencies. He contributed to GAO reports on the state of information security at a variety of federal agencies, including the Securities and Exchange Commission (SEC) and the Federal Deposit Insurance Corporation (FDIC). In addition, Mr. Castro was a Visiting Scientist at the Software Engineering Institute (SEI) in Pittsburgh, Pennsylvania where he developed virtual training simulations to provide clients with hands-on training of the latest information security tools. He has a B.S. in Foreign Service from Georgetown University and an M.S. in Information Security Technology and Management from Carnegie Mellon University.
Eline Chivot
Eline Chivot is a former senior policy analyst at the Center for Data Innovation. Based in Brussels, Eline focuses on European technology policy issues and on how policymakers can promote digital innovation in the EU. Prior to joining the Center for Data Innovation, Eline Chivot worked for several years in the Netherlands as policy analyst in a leading think tank, where her work included research projects on defense, security and economic policy issues. More recently, Eline worked at one of Brussels' largest trade associations and managed its relations with representatives of the digital tech industry in Europe and beyond. Eline earned master's degrees in political science and economics from Sciences Po, and in strategic management and business administration from the University of Lille.
Visualizing the Homicide Crisis in Mexico
Tracking the Height of Glaciers | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 637 |
package org.panteleyev.money.app.settings;
import javafx.scene.paint.Color;
import org.w3c.dom.Element;
import java.io.InputStream;
import java.io.OutputStream;
import java.util.Map;
import java.util.concurrent.ConcurrentHashMap;
import static java.util.Objects.requireNonNull;
import static org.panteleyev.money.xml.XMLUtils.appendElement;
import static org.panteleyev.money.xml.XMLUtils.createDocument;
import static org.panteleyev.money.xml.XMLUtils.readDocument;
import static org.panteleyev.money.xml.XMLUtils.writeDocument;
final class ColorSettings {
private static final String ROOT_ELEMENT = "colors";
private static final String COLOR_ELEMENT = "color";
private static final String COLOR_ATTR_NAME = "name";
private static final String COLOR_ATTR_VALUE = "value";
private final Map<ColorName, Color> colorMap = new ConcurrentHashMap<>();
Color getColor(ColorName option) {
return colorMap.computeIfAbsent(option, key -> option.getDefaultColor());
}
String getWebString(ColorName option) {
var color = getColor(option);
return "#"
+ colorToHex(color.getRed())
+ colorToHex(color.getGreen())
+ colorToHex(color.getBlue());
}
void setColor(ColorName option, Color color) {
colorMap.put(
requireNonNull(option),
requireNonNull(color)
);
}
void save(OutputStream out) {
var root = createDocument(ROOT_ELEMENT);
for (var opt : ColorName.values()) {
var e = appendElement(root, COLOR_ELEMENT);
e.setAttribute(COLOR_ATTR_NAME, opt.name());
e.setAttribute(COLOR_ATTR_VALUE, getWebString(opt));
}
writeDocument(root.getOwnerDocument(), out);
}
void load(InputStream in) {
colorMap.clear();
var root = readDocument(in);
var colorNodes = root.getElementsByTagName(COLOR_ELEMENT);
for (int i = 0; i < colorNodes.getLength(); i++) {
var colorElement = (Element) colorNodes.item(i);
ColorName.of(colorElement.getAttribute(COLOR_ATTR_NAME).toUpperCase())
.ifPresent(option -> colorMap.put(option,
Color.valueOf(colorElement.getAttribute(COLOR_ATTR_VALUE))));
}
}
private static String colorToHex(double c) {
var intValue = (int) (c * 255);
var s = Integer.toString(intValue, 16);
if (intValue < 16) {
return "0" + s;
} else {
return s;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,922 |
{"url":"http:\/\/math.stackexchange.com\/questions\/251658\/unusual-compact-embeddings","text":"# Unusual Compact Embeddings\n\nCan anybody give a reference to the following two facts?\n\nThe embeddings $$H_0^{1,2}(\\mathbb R^n)\\to L^2(\\partial B_1(0))$$ and $$H^{1,2}(\\partial B_1(0))\\to H^{1\/2,2}(\\partial B_1(0))$$ are compact?\n\nHere\n\n\u2022 $B_1(0)$ denotes the unit ball centered at $0$ in $\\mathbb R^n$,\n\n\u2022 for a domain $\\Omega\\subset \\mathbb R^n$, $H_0^{1,2}(\\mathbb \\Omega)$ denotes the closure of $C^\\infty(\\Omega)$ functions with respect to the norm $\\int_\\Omega |u|^2\\mathrm d x+\\int_\\Omega|\\nabla u|^2\\mathrm d x$\n\n\u2022 $H^{1\/2,2}$ is the fractional Sobolev space.\n\nI've never met anything concerning this, so I am looking for good places to begin with the study. Thank you very much in advance\n\n-Guido-\n\n-","date":"2015-08-28 02:43:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9853317737579346, \"perplexity\": 277.9683577976359}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440644060173.6\/warc\/CC-MAIN-20150827025420-00113-ip-10-171-96-226.ec2.internal.warc.gz\"}"} | null | null |
Q: Algorithms for Search tree vs building tree say I want to have an algorithm which gets a solution as quick as possible, which consists of starting from a state in a tree, and going through all possible states in a tree-like strucutre, why would it be necessary to first build a tree, and then traverse it, instead of building a tree, and if during building a solution node has been found, to stop building and immediately backtrack to the root, noting down the path to this leaf?
Basically, is there an BF algorithm to 'generate' a tree Breadth-First, rather than creating a tree first, and then search through it in a breadth-first manner?
Kind of like the animated results here:
Thank you for reading
A: There is no advantage in a search algorithm to build a tree and then search it, exactly due to the reasons you have mentioned. Furthermore, there is no search-related advantage that is gained by building a tree breadth-first or depth-first.
Usually, there are trees already present and we traverse them using the Breadth-First approach or the Depth-First approach. The reason why we choose one of these approaches is due to a property of the search-element or of the tree.
If you have seen cases where we construct a tree and then search it, it's for pedagogical purposes where you have to write a standalone program that constructs the tree and then traverses it. This is similar to creating a linked-list of numbers and performing a linear search, or constructing a Binary-Search-Tree and then searching for a value. Typically this approach tends to teach about creation and traversal of a data-structure packaged into one program.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,912 |
\section{Introduction}
Gallium nitride (GaN) is a wide bandgap semiconductor with unique optical,
electrical and chemical properties \cite{morkocc2009handbook}.
It can by synthesized in the form of nanowires (NWs),
which are intensively investigated as active structures for various
optoelectronic applications, including light-emitting diodes
\cite{guo2010catalyst, hersee2009gan},
solar cells
\cite{tian2009single, tang2008vertically,
mozharov2015numerical, neplokh2016electron, shugurov2019study},
photodetectors \cite{rigutti2010ultraviolet, gonzalez2012room}
and piezogenerators \cite{jamond2016piezo, gogneau2016single, lu2018probing}.
Generally, GaN NWs are grown on bulk crystalline substrates, such as silicon,
\cite{guo2010catalyst, calarco2007nucleation, largeau2012n,
bolshakov2019effects, fedorov2018droplet, bolshakov2018dopant, gridchin2020selective}
sapphire \cite{hersee2006controlled, wang2006highly, avit2014ultralong}
or diamond \cite{hetzl2016reprint}.
In this case, the substrate imposes to NWs an epitaxial relationship,
which determines the NW growth direction and sidewall orientation.
Also, this type of nanostructures can be grown on alternative substrates such
as metal \cite{calabrese2016molecular}
or even amorphous materials like glass \cite{kumaresan2016self}.
While vertical NW orientation may be preserved in these cases, the in-plane orientation of
NWs is arbitrary.
In the last decade, a new class of virtual substrates made of two- dimensional
(2D) materials has attracted a strong attention \cite{hong2015van}.
It was observed that the transfer of 2D flakes or sheets to almost any substrate
can promote the epitaxial growth of both vertically and in-plane oriented NW arrays.
This approach, termed "van der Waals epitaxy", either reduces or completely
eliminates the influence of the host substrate on the crystal properties
of synthesized nanostructures.
Recently, graphene flakes have been explored as substrates to grow
selectively III-V NWs \cite{MNL, munshi2012vertically, gridchin2020selective}.
Graphene can be considered not only as a virtual substrate for the NW growth,
but in some cases also as a transparent electrode with simultaneously excellent mechanical and
electrical contact to NWs' bases \cite{KoreaPaper}.
Moreover, due to weak van der Waals interaction with the host substrate,
graphene with NWs encapsulated into polymer matrix can be exfoliated
from the rigid substrate providing very promising platform
for flexible optoelectronic applications.
Presumably, the contact of NWs to graphene can be preserved after
matrix exfoliation \cite{KoreaPaper}.
Today, a number of publications is devoted to van der Waals epitaxy of
both planar structures \cite{hong2015van, utama2013recent} and NWs' arrays,
including ZnO \cite{KoreaPaper}, Zn$_3$P$_2$ \cite{paul2020van},
GaAs \cite{munshi2012vertically,Berd2019,alaskar2014towards} , InAs \cite{Hong2011} and
GaN \cite{MNL, CGD, sundaram2019large}.
Note, that graphene seeding layers attract a lot of attention for
III-Nitrides NWs growth due to the hexagonal arrangement of the carbon
atoms with sp$^2$ hybridization, matched to the (0001) c-plane of wurtzite GaN
\cite{al2015impact}.
Several groups reported the synthesis of GaN NWs and column structures
on pristine or defective graphene flakes \cite{MorassiNWs,fernandez2017molecular}.
However, the detailed theoretical analysis the GaN NW nucleation on
virtual graphene substrates has not been reported so far.
Due to inversion asymmetry along the c-axis, wurtzite GaN nanostructures may
form N- or Ga-terminated facets corresponding to different polarity.
The specific facet polarity of nitride NWs impacts their electronic
\cite{stutzmann2001playing} and nonlinear optical properties \cite{hite2012development},
and therefore, affects the design of piezoelectric \cite{gogneau2016single}
and light-emitting \cite{carnevale2013mixed} applications.
N-polar surfaces were reported to have the advantages of low-resistivity ohmic
contacts and improved capability for large-scale processing for
high electron mobility transistors \cite{wong2013n}.
GaN NWs grown by molecular beam epitaxy on conventional
substrates commonly exhibit N-polarity \cite{hestroffer2011polarity},
however the question of the polarity of GaN NWs grown on graphene has not been addressed so far.
In this work, we employ the density functional theory (DFT) based analysis to study
and explain the experimentally observed polarity discrimination in GaN nanoislands
and NWs epitaxially grown on graphene.
The Kohn-Sham formalism \cite{kohn1965self, hohenberg1964inhomogeneous}
of DFT has proved itself as one of the most indispensable methods for
electronic structure calculations.
Nowadays, DFT is routinely used to predict equilibrium properties of the molecules,
bulk materials and nanostructures \cite{sholl2011density, dobson2013electronic}.
However, our work, for the first time to our knowledge,
addresses the mechanism of polarity discrimination within the study
of the preferred structure and polarity of GaN nanoclusters.
We tentatively explain this preference of N-polarity as being caused by dipole
interaction between the GaN nanocrystal and $\pi$-orbitals of the graphene.
Theoretical results are
fully agreed with the experimental observations on
self-induced vertically-oriented GaN NWs grown on CVD-graphene
by plasma assisted molecular beam epitaxy (PA-MBE).
\section{Experimental results}
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{figure1.pdf}
\caption{(a) 45° tilted SEM image of as-grown NWs on a graphene patch,
(b) SEM image of the same sample after KOH attack.
Yellow lines highlight \textit{exemplary} pencil-like top shapes.
Schematic illustration of (c) a N-polar GaN wurtzite crystal.
Brown spheres show the position of gallium atoms and
blue ones show the position of nitrogen atoms.
Lines are corresponding chemical bonds.}
\label{fig:sem_pol}
\end{figure*}
Many groups have demonstrated synthesis of GaN NWs on the graphene by different growth techniques
\cite{heilmann2016vertically, kumaresan2016epitaxy, fernandez2017molecular, lee2011flexible}.
In particular, in our previous experimental study we reported the growth of GaN NWs
on graphene transferred to an amorphous host substrate \cite{MNL, CGD}.
We used commercially available CVD polycrystalline graphene grown on a Cu foil
transferred to a Si/SiO$_2$ template.
It was demonstrated that vertical NWs can be selectively grown on graphene patches
and that the in-plane NW orientation is imposed by the graphene layer
in a way that the $\langle$2-1-10$\rangle$ directions of the wurtzite
GaN lattice are parallel to the directions of the zigzag chains of
the honeycomb carbon lattice \cite{kumaresan2016epitaxy}.
For the present investigation, we adopted the growth conditions yielding high
growth selectivity on the graphene with respect to the SiO$_2$ surface \cite{CGD}.
Graphene patches of 1 cm$^2$ were wet-transferred to Si(100)
substrates topped with a 300 nm layer of thermal oxide.
The substrate was outgassed in the growth chamber at 830$^{\circ}$C during 5 min,
then the temperature was stabilized at 815$^{\circ}$C.
Ga and N fluxes were provided simultaneously.
The V/III ratio was equal to 1.1 and the Ga flux was equivalent to a 2D GaN growth rate
of 0.7 monolayers per second. The NW nucleation was monitored by reflection
high-energy electron diffraction.
The first diffraction spots corresponding to the formation of GaN nuclei were observed
after 90 min time of exposure to the fluxes and then the growth was
continued for additional 40 min yielding the total growth time of 130 min.
Figure \ref{fig:sem_pol} (c) shows a scanning electron microscopy
(SEM) image of the resulting NW morphology.
The NW density is about 1.8$\times$10$^{10}$ cm$^{-2}$, the average
diameter and height are 25 nm and 160 nm, respectively.
It is a common result for different types of substrates,
that MBE-grown GaN NWs exhibit N-polarity \cite{kumaresan2016self,de2012polarity}.
Yet the polarity was not investigated for the specific case of NWs
on graphene/SiO$_2$ templates.
To determine the polarity in thin NWs, potassium hydroxide (KOH) etching
is a simple and effective method
\cite{hestroffer2011polarity, largeau2012n}.
Indeed, KOH selectively etches N-polar surface by forming hexagonal pyramids,
while Ga polar surface remains unaltered.
This selectivity is associated with the surface bonds configuration
for N- and Ga-polar surfaces \cite{li2001selective},
which is schematically illustrated in Figures \ref{fig:sem_pol} (c)
respectively: on Ga polar surface, OH$^-$ ions from KOH are strongly repelled
by the electronegativity of the three N dangling bonds, while for N polar surface,
OH- ions are adsorbed on the surface to form Ga$_2$O$_3$ by reacting with
Ga which later dissolves in the KOH solution.
To evaluate the polarity of GaN NWs grown on graphene, we used a KOH solution
of concentration of 0.5 mol/l to etch the NWs for 2 minutes at 40 $^{\circ}$C and,
then the sample was observed in SEM as shown in Figure \ref{fig:sem_pol} (b).
By comparing Figures \ref{fig:sem_pol} (a) and (b) displaying the images before and after
the KOH attack it can be seen, that all the as-grown NWs exhibit a flat top,
while after the KOH attack the top shape becomes pencil-like.
This transformation is a typical signature of N-polar NWs
\cite{hestroffer2011polarity, largeau2012n},
which confirms that GaN NWs grown on graphene present the same polarity
as NWs grown on other substrates.
\section{Computational analysis}
In order to find energetically favorable polarity of the NW,
we computationally analyse the equilibrium configuration for
the small GaN nanocrystals (NCs) on a graphene sheet,
which are the precursors for further GaN NWs growth.
Let us introduce the system under consideration.
Firstly, in our calculations only the graphene sheet was considered as the substrate.
Graphene is commonly considered to interact with silicon
oxide layer through weak van der Waals forces \cite{Fan_2012}
Indeed, as shown in experimental work \cite{CGD} a graphene flake transferred
to a relatively thick oxide layer substrate does not interact
strongly with the substrate underneath.
This is evidenced by the thermal expansion of the graphene
flake independent from the thermal expansion of the Si substrate supporting the flake.
In this case the graphene thermal expansion coefficient was found to be several
times lower compared to Si \cite{CGD}.
Thus, in our calculations we can neglect the GaN interaction with SiO$_2$ .
Next, we define the initial relative position of Ga and N atoms
in the NC with respect to the graphene lattice.
As proposed in \cite{MNL, CGD}, (3$\times$3) GaN and
(4$\times$4) graphene supercells adopt the in-plane alignment of GaN and graphene lattices.
To verify this assumption, we investigated the energy preferable positions of
individual Ga and N atoms on graphene sheet.
Similar to previous works \cite{munshi2012vertically,Nakada2011},
we analyzed the adsorption of Ga and N atoms at three types of sites
("top", "bridge" and "hollow") shown in Figure \ref{fig:top_view}.
For nitrogen atoms the "bridge" and "top" sites were found
to be energetically favorable.
While for gallium atoms all the three sites have shown very similar binding energies.
The detailed discussion is given in Supplementary materials, section 1.
Therefore the reported \cite{MNL} matching between (3$\times$3) GaN and
(4$\times$4) graphene supercells corresponds to the minimal binding
energy of Ga and N atoms.
In this case the $\langle$2-1-10$\rangle$
directions of the wurtzite GaN lattice are parallel to the directions
of the zigzag chains of the honeycomb carbon lattice.
To reflect this lattice symmetry the NC of at least ten Ga and ten N atoms
(Ga$_{10}$N$_{10}$) should be considered.
Figure \ref{fig:top_view} gives an example of the top view of Ga$_{10}$N$_{10}$
NC in the initial configuration.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{figure2.pdf}
\caption{\textit{Exemplary} top view of the system considered.
The $\langle 2-1-10 \rangle$ direction of the GaN cluster
was chosen to be aligned to the zigzag direction of graphene sheet.
Labels H6, T and B indicate the "hollow", "top" and "bridge"
positions on the graphene cell respectively.}
\label{fig:top_view}
\end{figure}
It is crucial that the chosen size of the cluster is sufficient to reconstruct
the symmetry of the wurtzite GaN on graphene.
So one can translate such NC to reconstruct the arbitrary large NW.
Meantime our approach also requires the considered GaN NC to be
larger than the size of the critical nucleus for a GaN nanoisland.
Previous reports on self-induced nucleation of GaN nanostructures
typically estimated the critical nucleus to a few III-V pairs
\cite{sobanska2016analysis, consonni2011physical, sobanska2019comprehensive}.
Such a small critical nucleus cannot be observed experimentally and
gives the ground to the assumption of the irreversible
growth of catalyst-free NWs on silicon substrates with
Si$_{\rm x}$N$_{\rm y}$ and Al$_{\rm x}$O$_{\rm y}$
amorphous layers \cite{sobanska2016analysis, fernandez2015monitoring}.
In Supplementary material we show that those typical assumptions of small
critical nucleus and irreversible growth are also reasonable
for the considered in this work catalyst-free GaN NCs on graphene.
Therefore, the size of the GaN NC considered in our model is not limited
by the size of the critical nucleus, though it should satisfy
the domain matching between graphene and GaN lattices.
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{figure3.pdf}
\caption{Examples of all possible crystalline combinations
in case of Ga$_{10}$N$_{10}$ nanocrystals.
For both polarities it is possible to construct nanocrystals
which start from layers (the closest to graphene layers)
of 7 N, 7 Ga, 3 N or 3 Ga atoms respectively.
First row (a) - (d) shows the Ga polar nanocrystals,
second row (c) - (h) shows the N polar nanocrystals.}
\label{fig:initial_configurations}
\end{figure*}
There are 8 possible configurations of the Ga$_10$N$_10$
NCs which are all shown in Figure \ref{fig:initial_configurations}:
four configurations are Ga-polar (upper row) and the other four are N-polar (lower row).
To reconstruct the wurtzite structure each hexagonal Ga$_{10}$N$_{10}$
NC should have four layers: two layers of seven
and two of three gallium or nitrogen atoms, labeled as 7Ga, 7N, 3Ga and 3N, respectively.
The layer closest to graphene determines the initial configuration
while the polarity of a crystal determines the order of remained layers.
For example in Figure \ref{fig:initial_configurations} (a)
the NC consisting of 7 N - 3 Ga - 3 N - 7 Ga layers is depicted.
Further, the full self-consistent optimization of the geometry of each systems
was performed by \textit{ab-initio} approach.
The \textit{ab-initio} calculations were carried out
using plane-wave basis set for projector augmented-wave method
implemented in GPAW package \cite{gpaw}.
The starting spatial structure of the systems under consideration
was made by ASE library \cite{ase}.
The simulation cell had periodic boundary conditions
and sizes 14.70 $\times$ 12.25 $\times$ 14.00 \AA$^3$.
The total number of atoms in the calculation was 80,
where 60 of them were carbon atoms forming graphene sheet.
To prevent interaction between NCs though the periodic
boundary the lateral distances between NCs was $\approx$ 6 \AA\
and the vacuum gap along $z$ direction was $\approx$ 8 \AA.
The full optimization of the spatial structure within DFT for the relaxation
of a stressed system and determination of the equilibrium configuration
was done by using the modified
Broyden – Fletcher – Goldfarb – Shanno algorithm \cite{nocedal2006numerical}
implemented in GPAW \cite{gpaw}.
The optimization process ends when mean force acting on the atoms become
less than a cutoff value (0.05 eV/\AA).
The graphene atoms were fixed during optimization, which allowed to
reduce the number of optimization steps without significant losses
in prediction accuracy.
For each step of geometry optimization the self-consistent calculation of the electronic
density was made with the use of the Perdew–Burke–Ernzerhof (PBE) generalized
gradient approximation (GGA) exchange-correlation functional.
The Monkhrost-Pack grid 4$\times$4$\times$4 was used in calculations
and the plane-wave cutoff was set to $450$ eV.
Calculations were performed with convergence threshold
for self consistency of the charge density $10^{-4}$.
We found that only two initial geometries,
(Ga-polar 7 N - 3 Ga - 3 N - 7 Ga Figure \ref{fig:initial_configurations} (a)
and N-polar 3 Ga - 7 N - 7 Ga - 3 N
Figure \ref{fig:initial_configurations} (h) ) remained stable after the optimisation procedure.
In the other systems it was more energetically favorable for nitrogen to form the
N$_2$ molecule and "evaporate" from the NC.
This nitrogen "evaporation" effect is common for calculations involving nitrogen
and special attention should be paid to account for it \cite{de1998structural}.
The snapshots of the system geometries during optimization are shown
in Supplementary material section 2.
with finite size of the systems considered.
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{figure4.pdf}
\caption{The snapshots of geometries of
NCs during optimization procedures.
Grey spheres and lines correspond to carbon atoms and bonds in graphene.
Brown ones to gallium and blue ones to nitrogen atoms.
(a) and (d) the initial configurations,
(b) and (e) the intermediate configurations and (c) and (f) final,
optimized structures of Ga and N-polar NCs, respectively.
Central panel shows the optimization trajectories
of Ga and N polar NC plotted in the dipole moment vs total system energy
($d_{\rm z}$ vs $\varepsilon$) coordinates.
Latin letters correspond to the system configurations shown on the figures
above and below. }
\label{fig:dipole_moment}
\end{figure*}
Let us now analyse the stable configurations of the NCs.
Figure \ref{fig:dipole_moment} shows the snapshots of the system geometry during the optimization
for Ga-polar (panels (a) - (c)) and N-polar (panels (d) - (f)) NCs.
It is seen, that the initial configurations of these systems have different orientation
of the Ga-N dipole.
For Ga-polar NC the initial configuration (Figure \ref{fig:dipole_moment} (a))
has the following order of atomic layers 7N - 3Ga - 3N - 7Ga.
The intermediate state is shown in Figure \ref{fig:dipole_moment} (b),
in that state 7N - 3Ga and 3N - 7Ga planes are almost at the same level.
At the final optimization step shown in Figure \ref{fig:dipole_moment} (c)
the position order of atomic planes along the growth direction
has changed to 3Ga - 7N - 7Ga - 3N.
It is worth to mention, that during the optimization procedure NCs maintain
the initial crystalline orientation with respect to the graphene sheet,
only a rearrangements along the $z$ axis was observed.
So, during the optimization procedure Ga-polar NC
changes the orientation of its Ga-N dipole, i.e. the polarity.
On the other hand, the N-polar NC only adjusts its shape,
without any significant rearrangement of the atomic planes
and changes in the Ga-N dipole orientation.
As a result, GaN NCs which had different polarities before optimization,
after optimization procedure have the same polarities
and resembling geometrical structures.
To further analyze the evolution of the GaN NCs
during the optimization one needs to define a physical quantity that characterizes
the polarity of the system.
The difference between Ga- and N-polarity structures
can be described by projection of the electric dipole moment
of the system on the $z$ axis.
The dipole moment $d$ of the system can be represented as
a sum of atomic dipole moments $d^{\rm at}$ and dipole moments
due to net charges which occur because of a non-uniform
distribution of the electron density of the system.
So the z-projection of the dipole moment $d_z$ can be written as:
\begin{equation}
d_z = \sum_{i=1}^{N_{\rm at}} \left( d^{\rm at}_{i} + q_{i} r_{i}\right ),
\label{eq:dipole_moment}
\end{equation}
where $r_{i}$ - the atomic positions, $q_i$ - effective electric net charges
corresponding to each atom with taking into account of
the valence electron charge density distribution and
$N_{at}$ - number of atoms of GaN system.
The values of the net charges are determined by the
charge transfer between different atoms into the system and
can be obtained by Bader analysis \cite{bader1, bader2, bader3}.
The atomic dipole moments were calculated through atomic
polarizability and effective electric field corresponding
to the current charge distribution.
The central panel of Figure \ref{fig:dipole_moment} shows the dependence
of the dipole moment of the system
along the $z$ axis on the total energy of the system for the two stable configurations.
One can notice that the initial configurations of Ga and N-polar systems
(Figure \ref{fig:dipole_moment} (a) and (d), respectively)
have very different positions on the plot.
N-polar system has a significantly lower total energy and dipole moment.
Despite different initial configurations both systems
in optimized state have the same dipole moment and similar total system energies.
The shape of the optimization curve depends strongly on the optimization
algorithm used, but the final geometries of the structures are the same.
Both optimization curves cross the zero level of dipole moment
(Figure \ref{fig:dipole_moment} (b) and (e)) and
final structures correspond to negative $d_z$ values
(Figure \ref{fig:dipole_moment} (c) and (f)).
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{figure5.pdf}
\caption{Sketch of the GaN nanocrystal placed on top of graphene sheet.
The bold arrow shows the direction of the nanocrystal dipole moment.
Yellow areas illustrate the position and shape of graphene $\pi$-bonds (not in scale).
z-axis is placed along the NW growth direction.
}
\label{fig:pi-bounds}
\end{figure}
Thus, experimental and numerical studies revealed the polarity discrimination
which can be explained by GaN interaction with graphene charge density.
The electron density of graphene extended along the $z$ -axis forms negative
charge density localized in the space between the GaN NC and the
graphene as illustrated in Figure \ref{fig:pi-bounds}.
Therefore, the electric dipole moment of the GaN NC tends to turn
towards the negative charge, which results in a negative dipole moment projection $d_z$.
Therefore, the lowest energy configuration has the electric dipole moment of
the GaN NC turned towards the negative charges. This makes the N-polar
configuration energetically stable while Ga-polar
configurations tend to switch the polarity.
\section{Conclusions}
In this work we have demonstrated experimentally the polarity discrimination
for GaN NWs MBE grown on graphene and explained the observations theoretically.
We have presented the DFT studies of a GaN cluster with a supercell matching
to graphene and a size that exceeds the critical nucleus for a GaN nanoisland.
We have considered all possible configurations of Ga$_{10}$N$_{10}$
NCs on graphene and have shown that only two of them
(Ga-polar starting from layer of 7 nitrogen atoms and
N-polar starting from layer of 3 gallium atoms
Figure~\ref{fig:initial_configurations} (a) and (h) respectively) are stable,
while others tend to dissociate.
The DFT energy optimization shows that the considered Ga-polar NC changes
the order of atomic layers turning into N-polar configuration.
We introduced the projection of the electric dipole moment of the system
as the physical quantity characterizing the NC evolution during the optimization.
Indeed, for both NCs the dipole moment projection is negative at minimum
energy configurations which corresponds to N-polarity.
Our results show that the observed phenomenon can be explained
by considering the interaction of the electric dipole of the NCs
with net charges formed by $\pi$-orbitals of the graphene sheet.
The stable GaN NC with the electric dipole moment turned towards
the graphene negative charges is preferred due to the lowest system energy.
Thus, the GaN nanostructures tend to show N-polarity.
These results can be applied for engineering of GaN NW-based optoelectronic
devices and crucial for design of piezogenerators where the opposite polarity of
different NWs can reduce the efficiency of pressure-to-voltage conversion.
\begin{acknowledgments}
The work was financially supported by the
Ministry of Science and Higher Education
of the Russian Federation (FSRM-2020-0005).
%
This work was done with the financial support of
the Russian Federation President Council
for grants (grants MK-2428.2020.2 and SP-2169.2021.1).
%
Y.B. acknowledges the support of growth modeling by
the Russian Science Foundation under the Grant 19-72-30004
%
C.B. and M.T. acknowledge the financial support from
the EU ERC project NanoHarvest (grant no. 639052)
and Labex "GaNeX" (ANR-11- LABX-2014).
%
We acknowledge the Supercomputing Center of
Peter the Great Saint-Petersburg Polytechnic University
(SPbPU) for providing the opportunities to
carry out large-scale simulations.
%
A.P. is grateful to Alexander Ustinov (SPbPU) for fruitful discussion.
%
The molecular structures in this article
were rendered using Ovito software package \cite{ovito}.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,030 |
\section{Introduction}
Quintessence is the name of one model put forward in order to explain
that the rate of expansion of the Universe increases.
The model modifies the equations of General Relativity by adding a
Lagrangian density for a massless scalar (quintessence)
field rolling down a potential minimally coupled
to the usual Einstein Hilbert Lagrangian density. \\
The equations of motion for the Friedmann--Robertson--Walker--Quintessence (FRWQ) system,
obtained from Einstein's equations modified by the addition of the quintessence field, simplified by symmetry considerations (which transforms them from partial to ordinary differential equations)
consist of a set of two ordinary dynamical second order differential equations which govern the evolution of the dynamical variables (the radius of
the universe and the scalar quintessence field) and one ordinary first order differential equation
which constraints the initial values and velocities of the dynamical variables. It is worth to mention that a similar Lagrangian to the one defined in quintessence models was put forward in a proposal to consider a modified form of torsion field (generated as the gradient of a scalar field) in Einstein--Cartan theory \cite{hrrs}. The torsion generating scalar field was called tlaplon.\\
Some of the articles which deal with the subject, write down a Lagrangian formulation for the FRWQ system.
Nevertheless, to the best of our knowledge, all of the work published up to now is based on a classical Lagrangian
which gives rise to the two dynamical equations but does not yield the contraint equation. \\
We show that the FRWQ system may be completely described
in terms of a Lagrangian similar to that of a relativistic particle
moving on a two dimensional gravitational field. The conformally flat two dimensional metric conformal factor is
a function of the radius of the universe and of the scalar quintessence (tlaplon) field which naturally play
the role of coordinates in this two dimensional (mini) superspace.\\
\section{Lagrangian for FRWQ system}
In general, the Lagrangian density $\mathcal{L}$ for the evolution of the
spacetime metric $g_{\mu \nu}(x^\alpha)$ in interaction with a
massless scalar field $\phi(x^\beta)$ (which could be quintessence) is written as
\begin{equation}
\mathcal{L}=\sqrt{-g}\left(\frac{R}{2K}+{\mathcal L}_\phi\right)\, ,
\label{lfrwq}
\end{equation}
where $g$ stands for the determinant of the metric, $K=8\pi G/c^4$ (with $G$ as the gravitational constant and $c$ the speed of light), and $\mathcal{L}_\phi$ is the Lagrangian density for the massless scalar field
\begin{equation}
\mathcal{L}_\phi=\epsilon\left(\frac{1}{2}g^{\mu \nu} \phi,_\mu
\phi,_\nu - V(\phi)\right)\, ,\label{lfrwq2}
\end{equation}
where $V(\phi)$ is (up to now) an unspecified potential for the
scalar field $\phi$ whereas $\epsilon$ is a parameter that classifies the nature of the scalar field, such that $\epsilon=1$ is for usual scalar fields and $\epsilon=-1$ defines the quintessence field.
As it is well known, the Lagrangian density $\mathcal{L}$ defined by \eqref{lfrwq} is
singular. The action $S$
\begin{equation}
S= \int \mathcal{L}\ d^4 x \label{action}\, ,
\end{equation}
gives rise (upon variation with respect to the metric tensor $g_{\mu
\nu}$) to gauge invariant (generally covariant) and constrained
(Einstein) field equations coupled to matter, i.e.,
\begin{equation}
G^{\mu \nu} =K T^{\mu \nu}\, , \label{ee}
\end{equation}
where $G^{\mu \nu}$ is the Einstein tensor and $T^{\mu \nu}$ is the
energy--momentum tensor of matter
\begin{equation}\label{}
T_{\mu\nu}=g_{\mu\nu} \mathcal {L}_\phi-2\frac{\delta \mathcal {L}_\phi}{\delta g^{\mu\nu}}\, .
\end{equation}
Variation of the action $S$ with respect to $\phi$ yields the Klein--Gordon equation for the massless scalar field
\begin{equation}
\Box \phi + \frac{d V(\phi)}{d \phi}= 0\, . \label{kge}
\end{equation}
Now, let us take the line element for an isotropic and
homogeneous Friedmann--Robertson--Walker (FRW) spacetime, with the metric defined by
\begin{equation}
ds^2=-dt^2+a(t)^2\left[\frac{dr^2}{1-kr^2} +r^2(d\theta^2+
\sin^2\theta d\phi^2)\right]\, , \label{frwm}
\end{equation}
where $a(t)$ is the (time dependent) radius of the Universe and
$k=-1, 0, +1$ measure its negative, zero or positive curvature.
When Einstein equations coupled to matter are written in terms of
the line element \eqref{frwm} and the scalar field $\phi$ is assumed
to depend on time only, they reduce to two (second order) dynamical
and one (first order) constraint. The dynamical equations are (choosing $K=1$, for convenience)
\begin{equation}
2\frac{\ddot a}{a}+\left(\frac{\dot a}{a}\right)^2 +\
\frac{k}{a^2} -\epsilon\left(\frac{1}{2}\ \dot \phi^2-V(\phi)
\right)= 0, \label{frwqe1}
\end{equation}
and
\begin{equation}
\ddot\phi + 3\frac{\dot a}{a} \dot\phi + \frac{d
V(\phi)}{d\phi}=0\, , \label{frwqe2}
\end{equation}
while the constraint equation is
\begin{equation}
3\left(\frac{\dot a}{a}\right)^2 +3\frac{k}{a^2}+\epsilon\left(\frac{1}{2}\dot \phi^2+ V(\phi) \right) =0\, . \label{frwqc}
\end{equation}
Also, another useful equation can be obtained manipulating \eqref{frwqc} and \eqref{frwqe1}
\begin{equation}\label{usefulEC}
\frac{\ddot a}{a}=\frac{\epsilon}{3}\left(\dot\phi^2-V\right)\, ,
\end{equation}
Notice how the set \eqref{frwqe1}-\eqref{usefulEC} becomes the FRWQ system for $\epsilon=-1$.
This system has been already studied and solved for quintessence by Capozziello and Roshan \cite{capozz} for different scenarios and configurations of matter.
Here, our aim is to explore the analogy of this system with a relativistic particle which implies, as we will show, a geometric unification. With this purpose in mind we focus our attention to the fact that the Lagrangian $L$ which gives
rise to the dynamical equations \eqref{frwqe1} and \eqref{frwqe2} is
\begin{equation}
L= 3 a {\dot a}^2 - 3 k a +\epsilon a^3\left(\frac{1}{2}\dot\phi^2-V(\phi) \right)\, . \label{lag1}
\end{equation}
However, it is important to emphasize that the Lagrangian \eqref{lag1} does
not produce the constraint equation \eqref{frwqc}, which is equivalent to
imposing that the Hamiltonian $H$ associated to $L$ vanishes, i.e.,
\begin{equation}
H \equiv \frac{\partial L}{\partial \dot a} \dot a + \frac{\partial
L}{\partial \dot \phi} \dot \phi- L=0\, . \label{H0}
\end{equation}
Despite of that, notice now that it is a remarkable fact that the change of variables
\begin{equation}
r={\frac{2\sqrt 6}{3}}\ a^{3/2}\, ,\qquad
\theta=\frac{3}{2\sqrt 6}\ \phi\, ,\label{theta}
\end{equation}
recasts the Lagrangian $L$ in a ``kinetic energy minus potential energy ($T-V$)'' form (disregarding the fact that both $T$ and $V$ have the ``wrong signs''
in the $\theta$ associated terms, in the case of quintessence)
\begin{equation}
{\bar L}=\frac{1}{2}(\dot r^2+\epsilon r^2\dot \theta^2)-\bar V(r,\theta)\, ,\label{lag2}
\end{equation}
where $\bar V(r, \theta)$ is a general potential
\begin{equation}
\bar V(r,\theta)=3 \left(\frac{3}{8}\right)^{\frac{1}{3}}k\, r^{2/3}
+\epsilon\frac{3}{8}r^2 V(\theta)\, .\label{pot1}
\end{equation}
Note that in the Lagrangian ${\bar L}$ defined in \eqref{lag2}, which
describes the evolution of a FRWQ Universe in the presence of geometry (represented by $r$ or $a$) and dark energy (represented by
$\epsilon=-1$, $\theta$ or $\phi$), shows that the Universe evolves as a
relativistic particle moving on a two dimensional surface under the influence
of the potential \eqref{pot1}, thus geometrically unifying gravity and dark energy.
Nevertheless, the Lagrangian \eqref{lag2} does not produce the
constraint equation \eqref{frwqc}. In order to construct a
Lagrangian which in fact gives rise to all three equations
\eqref{frwqe1}, \eqref{frwqe2} and \eqref{frwqc} it is enough to
recall that \cite{h2} Maupertuis and Fermat principles give rise to
identical equations of motion in classical mechanics and geometrical
(ray) optics except for the fact that Fermat principle also produces
a constraint equation. It is worth mentioning that exactly the same
results are reached for the case of a relativistic particle moving on a two
dimensional conformally flat spacetime.
From now on, we just focus in the case of $\epsilon=-1$ to be the most relevant case for the dark energy (quintessence) scenario.
The description of the FRWQ system in terms of a Fermat--type Lagrangian is established by defining the relation between
the potential $\bar V(r, \theta)$ and the conformal factor $n(r, \theta)$
\begin{equation}
\bar V(r, \theta) \equiv -\frac{1}{2} \left[n(r, \theta)\right]^2\, ,
\label{Vn}
\end{equation}
and the constraint
\begin{equation}
\bar H \equiv \frac{\partial \bar L}{\partial \dot r} \dot r +
\frac{\partial \bar L}{\partial \dot \theta} \dot \theta- \bar L=0\, .
\label{E0}
\end{equation}
The Fermat-like Lagrangian $L_F$ which gives rise to all three equations
\eqref{frwqe1}, \eqref{frwqe2} and \eqref{frwqc} is
\begin{equation}
L_F= n(r,\theta) \sqrt{\left(\frac{d r}{d\lambda}\right)^2-r^2 \left(\frac{d
\theta}{d\lambda}\right)^2}\, ,\label{LF}
\end{equation}
where $n(r,\theta) = \sqrt{-2 \bar V(r, \theta)}$,
with $\bar V(r, \theta)$ defined in \eqref{pot1} and $\lambda$ is,
in principle, an arbitrary parameter. Thus, the Lagrangian \eqref{LF} may be appropriately rewritten as
\begin{equation}
L_F= \sqrt{-2 \bar V(r, \theta)\left[ \left(\frac{d r}{d\lambda}\right)^2-r^2 \left(\frac{d
\theta}{d\lambda}\right)^2\right]}\, .\label{LF1}
\end{equation}
To reproduce the relativistic equations of motion, $\lambda$ is defined by Luneburg's parameter
choice \cite{lune,h2}
\begin{equation}
\sqrt{\left(\frac{d r}{d\lambda}\right)^2-r^2 \left(\frac{d
\theta}{d\lambda}\right)^2} = n(r,\theta)\, .\label{lune}
\end{equation}
\section{Quantization}
Having the description of a relativistic particle for the Universe, we can go further in our scheme and attempt to quantize this theory.
First, let us rewrite the Lagrangian \eqref{LF1} as
\begin{equation}
L_F= \sqrt{\bar V(\xi, \theta)\, e^{2\xi}\left( \dot\theta^2- \dot\xi^2\right)}\, ,\label{LF2}
\end{equation}
where we have introduced the new variable $\xi=\ln r$, and then $\bar V\equiv\bar V(\xi, \theta)=\bar V(r, \theta)$ and $\dot\theta={d\theta}/{d\lambda}$, $\dot\xi={d\xi}/{d\lambda}$. We can notice that the quinteessence field acts as a Super-time in this new description where the particle is moving in a two-dimensional conformally flat spacetime.
The conformally flat metric becomes
\begin{equation}\label{}
g_{00}=\Omega^2\, ,\, \, g^{00}=\frac{1}{\Omega^2}\, ,\, \,g_{11}=-\Omega^2\, ,\, \, g^{11}=-\frac{1}{\Omega^2}\, ,
\end{equation}
where we introduce the notation $\Omega\equiv\sqrt{\bar V}e^\xi$,
which will appear frequently in this work.
In order to avoid with the procedure of canonical quantization of the FRWQ system, we restrict ourselves to the cases where $\bar V >0$, and to consider only static manifolds \cite{saa}, where there exist a family of spacelike surfaces which are always orthogonal to a timelike Killing vector. This implies that ${\partial_\theta g_{\mu\nu}}=0$, or
\begin{equation}\label{}
\frac{\partial\bar V}{\partial\theta}=0\, ,
\end{equation}
implying that the original potential $V(\theta)$ is a constant. This means that for the current quantization process, $V(\theta)$ can only be the cosmological constant.
Classically, the Hamiltonian for the system described in the previous section is
\begin{equation}\label{}
H=m\sqrt{g_{00}}\sqrt{1-g^{11}\pi^2}=\sqrt{g_{00}}\sqrt{1+\frac{\pi^2}{\Omega^{2}}}\, ,
\end{equation}
where we have included $m$ as a new parameter (representing the analogue of a mass) the $\pi$ is the canonical momentum.
We use this Hamiltonian to construct the quantum theory for the FRWQ system. The quantum equation will be in the form
\begin{equation}\label{quantumeq}
i\hbar\frac{\partial\Psi}{\partial\theta}=\mathbf 1 \mathcal H\Psi\, ,
\end{equation}
where $\mathbf 1$ is the unit matrix and the Hamiltonian operator is
\begin{equation}\label{momentumoperator1}
\mathcal H=\Omega^{1/2}\sqrt{1+\hat p^2}\, \Omega^{1/2}
\end{equation}
where $\hat p$ is the momentum operator. This Hamiltonian $\mathcal H$ is constructed to avoid problems with the ordering.
In the following we proceed to quantize the FRWQ theory for two different and specific cases. First, we quantize the system as it were a spin-0 particle. This will produce a Klein-Gordon equation for the wavefunction of the Universe. The second case correspond to the quantization of the FRWQ system as a spin-1/2 particle. Despite of the possible arguments against this path, we perform this kind of quantization in an exploratory spirit, and because currently we do not know if the Universe has spin or not.
\subsection{Quantization of the FRWQ system as a spin-0 particle}
One way to canonically quantize the relativistic spinless particle can be obtained following Gavrilov and Gitman's method \cite{gavrilov}. This procedure is consistent as construct the quantum theory along the Dirac's theory for gauges and constraints \cite{diracT, Teitel}.
We will not reproduce here the calculations for the quantization of relativistic spin-0 particle, but just show that the quantization for the FRWQ system (considered as a spinless relativistic particle) produces the quantum equation \cite{gavrilov}
\begin{equation}\label{KG1G}
i\partial_\theta\Psi=\hat h \Psi\, ,
\end{equation}
where $\Psi$ is the spinor
\begin{equation}\label{}
\Psi=\left(
\begin{array}{c}
\chi \\
\psi \\
\end{array}
\right)\, ,
\end{equation}
and $\hat h$ is the matrix Hamiltonian
\begin{equation}
\hat h=\left(
\begin{array}{cc}
0 & -\partial_\xi^2+m^2 \Omega^{2} \\
1 & 0 \\
\end{array}
\right)\, ,
\end{equation}
where $m$ is anew a quantity analogue to a "mass" for the FRWQ system. From now on we will take $\hbar=1$ for convenience.
Eq.~\eqref{KG1G} gives rise to the equation
\begin{equation}\label{KGfinalG}
0=\frac{\partial^2\psi}{\partial \theta^2}-\frac{\partial^2\psi}{\partial \xi^2}+m^2 \Omega^2\psi\, ,
\end{equation}
where we have used that $i\partial_\theta\psi=\chi$. This is a Klein-Gordon equation for the wavefunction $\psi$ with a mass-corrected term due to the metric. This general equation represents the quantization of the FRWQ system as a spinless particle.
It can be simplified assuming the dependence $\psi=\phi\, e^{E\theta}$, with $\partial_\theta\phi=0$. Thus, we can find the equation
\begin{equation}\label{}
\frac{\partial^2\phi}{\partial \xi^2}=(m^2 \Omega^2+E^2)\phi\, .
\end{equation}
For the simplest case of $k=0$ and $\Omega^2=\lambda e^{4\xi}$ (where we have choosen $V(\theta)=-8\lambda/3$, with a constant positive $\lambda$), the general solution is
\begin{eqnarray}\label{}
\psi(\xi)&=&C_1 (-1)^{-E/4}I_{-E/2}\left(\frac{m\sqrt{\lambda}}{2}e^{2\xi}\right)\Gamma\left(1-\frac{E}{2}\right)\nonumber\\
&+&C_2 (-1)^{E/4}I_{E/2}\left(\frac{m\sqrt{\lambda}}{2}e^{2\xi}\right)\Gamma\left(1+\frac{E}{2}\right)\, ,
\end{eqnarray}
where $C_1$ and $C_2$ are constants, $\Gamma$ is the Euler gamma function, and $I_n$ is the modified Bessel function of the first kind of order $n$.
Notice that $\lambda$ is related to the cosmological constant.
A solution for $k\neq 0$ is not simple to obtain.
As a final point, we would like to remark an interesting consequence of this quantization scheme. We can write the Eq.~\eqref{KGfinalG} in a Super-Hamiltonian formalism, being less restrictive with the assumption that the potential $V(\theta)$ is constant.
For a closed universe $k=1$ (and the case quinteessence $\epsilon=-1$) it is possible to rewrite Eq.~\eqref{KGfinalG} as
\begin{equation}\label{}
\frac{\partial^2\psi}{\partial \alpha^2}-\frac{\partial^2\psi}{\partial \varphi^2}+\left(\hat m^2\varphi^2 e^{6\alpha}-e^{4\alpha}\right)\psi\equiv {\cal H}\psi =0
\end{equation}
where $\alpha=\ln a$, and $\varphi=2\theta/3$. To obtain this equation we chose $m^2=1/18$ and $V(\theta)=3\hat m^2\varphi^2$, where $\hat m$ is the mass of the field. Here, ${\cal H}$ is usually called the Wheeler-DeWitt Super-Hamiltonian \cite{hawk1,hawk2}. Thus, the quantization of the FRWQ system as a spinless relativistic particle proposed here can reproduce known results of quantization using the Super--Hamiltonian to construct the equivalent of Schr\"{o}dinger equation.
\subsection{Quantization of the FRWQ system as a spin-1/2 particle}
We can now propose a different way to quantize the FRWQ system. The problem is to solve the square root in the Hamiltonian \eqref{momentumoperator1}. This can be done as in the previous section, or with Dirac matrices (as it is done for the Dirac equation). This means that the FRWQ system is considered in an analogue way to a spin-1/2 particle. There is no restriction to this ansatz.
The momentum operator in \eqref{momentumoperator1}, $\hat p=\sqrt{-g^{11}}\hat\pi={\hat\pi}/{\Omega}$, must be defined such that
\begin{equation}\label{operDD}
\hat\pi=-i\frac{\partial}{\partial\xi}\, .
\end{equation}
In a way similar for to the Dirac equation, we propose that the square-root in the Hamiltonian \eqref{momentumoperator1} can be solved using the Dirac matrices $\alpha$ and $\beta$. This will give us the Hamiltonian
\begin{equation}\label{}
{\cal H}=\Omega^{1/2} \left(\alpha\cdot\hat p+m\beta\right) \Omega^{1/2}\, ,
\end{equation}
where $m$ is still a parameter associated to a "mass" of the FRWQ system. This Hamiltonian allows us to quantize a FRWQ Universe as a spin particle. The quantum equation is Eq.~\eqref{quantumeq}, where now $\Psi$ is a bi-spinor. Explicitly, using the operator \eqref{operDD} in the Hamiltonian, the quantum equation reads
\begin{equation}\label{eqCuanD}
i\frac{\partial\Psi}{\partial \theta}=-i\alpha^\xi\left[\frac{\partial}{\partial\xi}+\frac{1}{2}\frac{\partial\ln\Omega}{\partial\xi}\right]\Psi+m\beta\Omega\Psi\, ,
\end{equation}
where $\alpha^\xi$ stands for any of the matrices $\alpha^1$, $\alpha^2$ or $\alpha^3$.
Multiplying the equation by $\gamma^0=\beta$ ($\gamma^0\gamma^0=1$) and remembering that $\gamma^\xi=\gamma^0\alpha^\xi$, we have
\begin{equation}\label{diracQQQ}
i\gamma^0\frac{\partial\Psi}{\partial\theta}+i\gamma^\xi\left(\frac{\partial}{\partial\xi}+\frac{1}{2}\frac{\partial\ln\Omega}{\partial\xi}\right)\Psi=m\Omega\Psi\, .
\end{equation}
It is shown in the Appendix that this equation can be obtained directly from the known form of the Dirac equation in curved spacetimes. This gives validity to our quantization scheme.
Eq.~\eqref{diracQQQ} represents the quantized FRWQ system a spin particle. To study it, we can simplify it using the ansatz
\begin{equation}\label{}
\Psi=\left(
\begin{array}{c}
\psi \\
\chi \\
\end{array}
\right) e^{-\int (1/2)\partial_\xi \ln \Omega d\xi}\, ,
\end{equation}
where $\psi$ and $\chi$ are spinors. Then, from Eq.~\eqref{diracQQQ}, we get
\begin{eqnarray}
i\frac{\partial\psi}{\partial \theta}+i\sigma^\xi\frac{\partial\chi}{\partial\xi} &=& m\Omega\, \psi\, , \label{DiracFKG1} \\
i\frac{\partial\chi}{\partial\theta}+i\sigma^\xi\frac{\partial\psi}{\partial\xi} &=& -m\Omega\, \chi\, ,\label{DiracFKG2}
\end{eqnarray}
where $\sigma^\xi$ are the Pauli matrices.
One form to solve the previous set of equations is to assume that $\psi=\hat\psi\, e^{iE\theta}$ and $\chi=\hat\chi\, e^{iE\theta}$, with $\partial_\theta\hat\psi=0=\partial_\theta\hat\chi$. Then we find
\begin{eqnarray}
i \sigma^1\frac{\partial\hat\chi}{\partial\xi} &=& (E+m\Omega)\hat\psi\, , \\
i \sigma^1\frac{\partial\hat\psi}{\partial\xi} &=& (E-m\Omega)\hat\chi\, ,
\end{eqnarray}
which completely solved the problem if we can find the solution to the following equation
\begin{equation}\label{}
\frac{\partial}{\partial\xi}\left(\frac{1}{E-m\Omega}\, \frac{\partial\hat\psi}{\partial\xi}\right)+(E+m\Omega)\hat\psi=0\, .
\end{equation}
This equation has a general solution for the case $E=0$
\begin{equation}\label{}
\hat\psi(\xi)=C_1 e^{m\int^\xi\Omega(\xi') d\xi'}+C_2e^{-m\int^\xi\Omega(\xi') d\xi'}\, .
\end{equation}
However, a solution for $E\neq 0$ is not simple to be obtained.
Finally, we notice that if the previous assumptions are not taken, then we can find the set of coupled second order equations directly from Eqs.~\eqref{DiracFKG1} and \eqref{DiracFKG2}. These equations are
\begin{eqnarray}
\frac{\partial^2\psi}{\partial\theta^2}-\frac{\partial^2\psi}{\partial\xi^2}+im\frac{\partial\Omega}{\partial\xi}\, \sigma^1\chi+m^2\Omega^2\psi &=& 0\, , \\
\frac{\partial^2\chi}{\partial\theta^2}-\frac{\partial^2\chi}{\partial\xi^2}-im\frac{\partial\Omega}{\partial\xi}\, \sigma^1\psi+m^2\Omega^2\chi &=& 0\, ,
\end{eqnarray}
from where we can see that the spinors $\psi$ and $\chi$ do not satisfy simple Klein-Gordon equations. In the FRWQ model, the spinors are coupled to metric of the spacetime.
\section{Conclusions}
We study the unification between the FRW geometry and a scalar field that can be identified with the quintessence and the tlaplon field \cite{hrrs}. This scheme can be unified at the Lagrangian level and allows us to write a Fermat-like Lagrangian for the complete system, showing that the Universe behaves as a relativistic particle moving on a conformally flat spacetime.
We are now able to quantize the system as if it were a spinless particle obtaining a Klein-Gordon-like equation, and if it were a spin particle we get a Dirac equation. These quantization schemes can be considered as generalizations of the Wheeler-DeWitt Super-Hamiltonian formalism. We find particular solutions for both of these equations.
Our proposal establishes that the quintessence-tlaplon field could be necessary as a first step to construct a geometrically unified theory for the quantization of an expanding universe with quintessence dark energy.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,048 |
Honey Cafe has been on our list of places-to-go for quite some time now, so we were pleased to finally make it there for lunch today.
The cafe is light and bright inside, with vibrant yellow seats and cushions adding to the lively atmosphere. We liked the feature wall patterned with geometric shapes and the original, upturned cane baskets, which hang from the ceiling and brim with lush green foliage and leafy tendrils.
We took a seat outside in the sun, ordered a coffee and tea, and sifted through the menu. Honey offers all-day breakfast and lunch, with some unique meals such as mince on toast, and mushroom & savoury churros.
Rachel loved her breakfast platter – a tasty combination of an egg on toast, baby tomatoes, mushrooms, quinoa and spinach, alongside a pot of granola and poached fruit. I probably wouldn't order it again, but I thought I'd try something a little different so opted for the mediterranean terrine with charred capsicums, eggplant, feta and mozzarella.
To finish off we had a slice of the gluten-free carrot cake, which was sweet and fresh. Value for money was good and the service was attentive and efficient.
All in all, a trendy and pleasant cafe that is well designed and has lots on offer. A great spot on a sunny day!
Location: 12 Huron Street, Takapuna. Outdoor and indoor seating, dog friendly. Mon – Fri open from 7am. Saturday open from 8am.
My friend was telling me to go to this place, I was wondering what it was like. The food looks good! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,977 |
Q: Was Knick knack an actual game? In the nursery rhyme "This old man" played Knick knack on my knee, etc. I always assumed it was just nonsense rhyming. Was it a real thing?
This old man, he played one,
He played knick-knack on my thumb,
With a knick-knack paddywhack,
Give the dog a bone,
This old man came rolling home.
A: As suggested by the following source, knickknack may refer to tapping out a rhythm usin a spoon.
This traditional rhyme was first published in 1906 but almost certainly originates from earlier possibly from the time of the Irish potato famine.
The biggest clue to the meaning lies in the lyrics most particularly 'paddywhack' and to a lesser extent 'knick-knack'.
A knick-knack is a trinket or other trivial object. Knick-knack may also refer to the practice of tapping out a rhythm using spoons.
(www.learnarhyme.com)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,662 |
Established in 1991 Miller Whan & John Pty Ltd are experienced Agents in sales, auctions and property management with a proven track record born from a combined experience of over 118 years.
Servicing the South East of South Australia and the Western Districts of Victoria. We put our clients first, be they buyers, sellers or landlords, the honesty, integrity and satisfaction experienced by our clients is second to none.
Whilst not the biggest agency in town but we offer you a more personalised, satisfying real estate experience because of the attention we are able to give to every client. We ensure you are number one and that's the number you deserve. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,450 |
Asko Tapio Sahlberg, född 30 september 1964 i Heinola i Finland, är en sverigefinsk författare.
Sahlberg är bosatt i Kungälv utanför Göteborg. Han debuterade med romanen Pimeän ääni 2000. Eksyneet kom på svenska 2003 under titeln De vilsegångna.
Han har hyllats av både finländska och svenska kritiker samt är flerfaldigt prisbelönad och fick bland annat det sverigefinska litteraturpriset Kaisa Vilhuinen-priset.
Bibliografi (finska)
2000 Pimeän ääni
2001 Eksyneet
2002 Höyhen
2002 Hämärän jäljet
2004 Tammilehto
2005 Yhdyntä
2006 Paluu pimeään
2007 Siunaus
2010 He
2011 Häväistys
2013 Herodes
2015 Irinan kuolemat
2015 Yö nielee päivät
Bibliografi (svenska)
2003 De vilsegångna (Eksyneet)
2006 Eklunden (Tammilehto)
Finskspråkiga författare
Svenska författare
Födda 1964
Sverigefinländare
Sverigefinska författare
Män
Levande personer
Personer från Heinola | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8 |
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